This work is protected by copyright and other intellectual property rights and duplication or sale of all or part is not permitted, except that material may be duplicated by you for research, private study, criticism/review or educational purposes. Electronic or print copies are for your own personal, non-commercial use and shall not be passed to any other individual. No quotation may be published without proper acknowledgement. For any other use, or to quote extensively from the work, permission must be obtained from the copyright holder/s. VIBRATIONAL ENERGY TRANSFER PROCESSES IN GASEOUS SYSTEMS AT HIGH TEMPERATURES by G.E. MILLWARD A.R.I.C. A thesis submitted to the University of Keele in partial fulfilment of the requirements for the Degree of Doctor of Philosophy DEPARTMENT OF CHEMISTRY UNIVERSITY OF KEELE OCTOBER 1970 ACKNOWLEDGEMENTS In presenting this thesis I would like to acknowledge the contributions of the following people:-Dr. Peter Borrell for his valuable guidance and encouragement throughout this work. Professor H.D. Springall for providing the laboratory facilities for this project. Hiss Jacqueline Marsden for typing the thesis. The technical staff of the Chemistry Department, in particular Mr. Peter Holbrook and Mr. Frank Entwistle. The staff of the computer centre for programming assistance, and Miss Pat Highfield for instruction in use of the computer. The technical staff of the University Workshops, for the attention devoted to the construction of several parts of the apparatus. The University of Keele for financial support. All the work reported in this thesis was carried out by the author, under the supervision of Dr. Peter Borrell. A B S T R ACT A study has been made of the rate of vibrational excitation in shock heated CO for the temperature range l300K - 2500K, by observation of the infrared emission behind the incident shock wave. Care was taken to eliminate'impurity effects and the non uniformity of the flow was accounted for in the prescribed manner. The first measurements were made on the fundamental and overtone of CO in the pure gas and eO/Ne mixtures. The experimental data was reduced using kinetically derived equations, which incorporated the Landau-Teller ratios between the rate constants. For Teo-co the Landau-Teller constants were found to be A = -22.2; B = 160.2 and for teO-Ne A = -21.66; B = 148.9. The separate observations of the levels indicated that the population of the v = 2 level was by successive single quantum transitions. It was concluded that the Landau-Teller ratio of the rate constants k2l : klO = 2 : 1, was incorrect, in the case of the anharmonic oscillator. The coefficient was thought to lie between 2.2 and 2.4. It was demonstrated by computer calculations, using selected models, that the equations used in the reduction of experimental data were inadequate for real systems. Thus studies using the equations were thought to be subject to a small systematic error. The second series of measurements was performed on CO/C02; CO/N20 and COICOS mixtures, to measure the rate of vibration-vibration (V-V) energy exchange. Small concentrations of the faster relaxing gas (i.e. the polyatomic) were shown to dramatically reduce the relaxation time of the major component, CO. By using relaxation equations for a binary gas mixture, the probability of the v-v process was determined from the data. The probabilities were found to lie in the following sequence with respect to the energy discrepancy AE:-< < 64 -1 em These probabilities compared in magnitude and temperature dependence with the theories of Schwartz, Slawsky and Herzfeld and Rapp and Golden. CON TEN T S Page 1. INTRODUCTION 1.1 Preliminary Remarks 1 1.2 Experimental Techniques 5 1.3 Previous Shock Tube Studies of Vibrational Energy Transfer 17 1.4 Objects of the Present Study 22 References 24 2. EXPERIMENTAL 2.1 Introduction 30 2.2 Shock Tube 32 2.3 Vacuum and Pressure Technique 38 2.4 Diaphragms 40 2.5 Light Screens 42 2.6 Infrared Detector 47 2.7 Electronics Arrangement 49 2.8 Materials 51 2.9 Experimental Procedures 53 References 57 3. CALIBRATIONS AND RADIATION CHARACTERISTICS 3.1 Calibration Measurements 58 3.2 Radiation Characteristics 68 References 76 Page 4. VIBRATIONAL TRANSFER PROBABILITIES 4.1 Introduction 77 4.2 Parameter Relationships 79 4.3 Classical Method 81 4.4 Semi Classical Method 82 4.5 Quantum Mechanical Method 85 4.6 Empirical Method 91 4.7 Long Range Interaction Method 92 4.8 Discussion of Methods 94 Refer-ences 97 5. RELAXATION §QUATIONS 5.1 Introduction 101 5.2 Kinetic Approach 106 5.3 The Bauer Approach 116 5.lI- Comments on the Methods 122 References 127 6. TREATMENT OF RESULTS 6.1 Calculation of Relaxation Times 129 6.2 Calculation of Shock Parameters 133 6.3 Particle Time and Napier Time 138 References ll1-0 7. RESULTS 7.1 Energy of the Observed Emission lll-l 7.2 Effect of the Flow Correction ll1-6 7.3 Data of COlNe Mixtures for 1st Vibrational Level 150 7.lI- Data of CO/Ne Mixtures for 2nd Vibrational Level 158 7.5 Data of CO/polyatomic mixtures for 1st Vibrational Level References 8. DISCUSSION 8.1 Pure CO and CO/Ne Mixtures 8.2 CO/polyatomic Mixtures 8.3 S\Dllllary 8.4 Suggestions for Future Work References APPENDIX 1 FORMULAE A.l.i A.l.ii Tube Dimensions and Hot Flow Time Derivaticn APPENDIX 2 COMPUTER STUDIES A.2.i Flow Correction Programme A.2. ii S.S.H. Prograume A.2.iii Miscellaneous Programmes References Page 172 182 183 204 221 222 224 i iv i xiv xxxiii xxxiv 1. INTRODUCTION 1.1 PRELIMINARY REMARKS The molecules of a gas can possess energy in four different forms electronic, vibrational, rotational and translational. Each form of energy, excepting translational, has a discrete set of energy levels associated with it. The magnitude of the quanta associated with the levels decrease in the order:-Electronic » vibrational > rotational » translational The typical molecular energy spacings for the various forms of energy are shown below:-TABLE 1.1 Wavenumber Characteristic J mol- l cm-l Temp K ev Electronic 20,000 28,800 2.48 240,000 Vibrational 1000 1440 0.12 . 12,000 Rotational 10 14 0.0012 120 Transitions between the levels may be stimulated by electromagnetic radiation or molecular collisions. Neglecting radiative decay, both processes provide the disturbance which is necessary for the absorption - 2 -or emission of quanta of energy. The energy level spacings in the case of electronic energy are so large, that the excited states are not normally accessible by collision. The energy level spacings in the case of molecular vibrations and rotations do permit collisional excitation. In the case of translation the energy levels are so close that they are treated completely classically. The available thermal energy is equilibrated among the various degrees of freedom by the following processes:-1. 2. 3. ~. Rotation Translation Vibration (1) ~ Rotation * Translation Vibration (U ----:. Translation ~Vibration (1) ~ Vibration (2) * Translation The exchange of translational energy with the internal degrees of freedom, processes 1, 3, are called simple processes. If a vibrational quantum in one molecule is converted into a quantum of a different vibration, then it is called a complex process, i.e. ~. This last ease involves two vibrations and the difference between the two quanta is made up with translational energy. Process 2 which involves energy exchange between the internal degrees of freedom, vibrational and rotational, may alao be called a COIIIPlex process. The pl'Obabill ty per collision of energy exchange between vibration and translation is small and MIly collislona are often required to achieve energy conversion. The col!plex process is more - 3 -efficient, since less energy is required from translation. The efficiency of the complex process depends on the matching of the energy levels of the two vibrations. The present work is concerned with information that can be obtained about these types of processes, when the system is disturbed by an external forCe. After the disturbance the system requires a finite time to return to the new equilibrium. The return to equilibrium via the processes is known as a relaxation process. The rate at which the energy E of a relaxing system returns to its - 1 equilibrium value E has been shown to be, by Landau and Teller, proportional to its displacement from equilibrium:-= 1 (E - E) dE dt 'T or where T is the time for the energy displacement (E - E) to fall lIe of the initial value (EO - E) at t = 0, having the temperature held constant by a heat bath. The time scale for the molecular relaxation will be determined by the number of collisions within the region of observation. Since the collision rate is pressure dependent, the relaxation times will be similarly so. Convention dictates that the relaxation times should always be quoted at the standard pressure of 1 atmosphere. - 4 -In the present case the equilibrium was disturbed with a shock wave. The technique is well established. and has provided the majority of data on the temperature dependence of relaxation times. This method. together with several other important methods for studying energy transfer. will be reviewed below. - 5 -1.2 EXPERIMENTAL TECHNIQUES 1.2.a SHOCK TUBE TECHNIQUES The shock tube is a device in which a shock wave is produced by the sudden bursting of a breakable diaphragm, which divides the tube into high pressure and low pressure sections. The pressure of the driver gas at which the diaphragm bursts depends on the diaphragm material and thickness. The formation of a shock wave by the bursting is understood by regarding the driver gas as a piston, which moves in small incre.nts into the test gas. A pressure pulse is propagated. and this compression heats the gas slightly as it moves at the speed of sound. The second pulse also moves at the speed of sound but now in slightly hotter gas. The speed of sound increases with temperature. so that each pulse is moving faster than its predecessor. Eventually all the pulses coalesce to form a sharp front. i.e. the shock wave. This wave moves faster than the speed of sound. Thus the test gas 1II)1ecules, which can only move at the speed of sound, are bound at some stage to be ingested by the shock front. The result is that there is a build up of pressure and density in the shock wave. Originally the waves were treated 2 mathematically by Reimann, who ass .... d that the shock wave wa an adiabatic and reversible transition. RanlciDe 3 ex.ined the theory .. and showed that the process was not adiabatic. Finally, Rayleigh and Hugoniot5 both showed that the shock tranlitiOll aut iDvol.ve a change of entropy, and thus could not be adiabatic or reversible.· - 6 -Using a system of coordinates where the shock front is stationary, see Figure 1.1, The Rankine-Hugoniot conservation equations may be written:-FIGURE 1.1 STATE 2 SHOCKED GAS SHOCK FRONT STATE 1 u = W 1 s ( UNSHOCKED GAS where Ws = velocity of shock front v • velocity of gas molecules relative to the tube P,p,T • pressure, density and temperature The conservation of mass:-1.1 The conservation of IIOmentum:-= 1.2 The conservation of energy:-1.3 where H = enthalpy In 1.3, the term (iu12 - iu22) represents the amount of kinetic - 7 -energy which is changed into thermal energy, and thus the shock wave heats the gas. However, the heating produced by the shock wave is not the same as that produced in an adiabatic isentropic compression. This is due to the fact that AS > 0 for the former, while AS = 0 for the latter. The increase in entropy of the system which is compressed from state 1 to state 2 can be calculated:-q = Tds q = du + pdv (1st Law) • • ds = du + .2 dv T T But du = C dT v and RdT = pdv + vdp • • ds = dT [cv + R] .!Sa T dT v = R T P • • ds = C dT - Rdp p-T P T2 P2 Integrating AS = C I dT R I ~ p -T P T1 Pl AS = C In T2 Rln P2 P - -Tl P1 Since R II e - e and y II e Ie p v P v - 8 -liS C rVT1) ] = 1n 1 p (P2/P1/; X-l Thus:- T2 P2 X - = (-) exp (liSle ) 1.4 T1 Pl P The equation 1.4 shows that for the same value of (P2/Pl ), an isentropic process where liS = 0 (the exponential is one). the value of T2 must always be less than T2 across the shock front. where the exponential term is always greater than one. The change in entropy across the shock front may be calculated from:-AS = The ratios (P2/Pl > and (P2/Pl) have been derived. in terms of known quantities. from the conservation equations. The ratios are to be found in section 6.2 of Chapter 6. The shock wave is one of several waves produced after rupture of the diaphragm. The wave pattema are illustrated on the (x, t) diagram Figure 1.2. where x is the distance along the shock tube and t the time after the diaphragm bursts. The shock va.,. moves at a constant velocity in the te.t gas and is reflected on reaching the end plate. The test gas is h.ated by the initial sbock vave. and is heated again by the reflected shock. but to a l .... r d.gree. since the latter has a lower velocity. Following the .bock va". ia the ccatact .urface. the point at which the te.t gu •• t. the dri".r ,U. vhich 110"" vi th REFLECTED RAREFACTION ~ END PLATE An x-t diagram t DIAPHRAGM x TEST COLD TEST GAS END PLATE ~ cD C .., ~ -• I\J - 9 -a constant velocity, which is smaller than that of a shock wave. Thus at greater distances down the tube the separation between the shock front and the contact surface increases. The rarefaction fan, which is produced at the same time as the shock wave, moves into the dri ver gas at a sonic speed. The fan cannot coalesce to form a discontinuity and after reflection from the end plate accelerates towards the shock wave. The possibilities of interaction of the various wave forms depends upon the length of tube. This has been 6 dealt with elsewhere. The construction and operation of shock tubes is well defined and has been documented. 7,8,9,lO Particular emphasis has been placed on the measurement of shock speed, since this allows computation of the theoretical equilibri\DD state of the gas in the shock wave. Measurements of shock wave attenuation are also important, since in certain circumstances, boundary layer effects influence the 11 12 shock properties, ' see section 3.l.a of Chapter 3. The velocity is 118asured by devices which can respond rapidly, in the order of 1 l.IS. The methods available are pressure transducers, thin film temperature detectors and light detection devices. The techniques provide electric pulses which serve to trigger recording and timing devices. It is the region between the shock front and the contact surface, which is of particular interest to kineticiata, and it has facili tated an interesting range of chemical studies. There exists a large amount of data on the study of the uni.,leeular diasociations of - 10 -. 12 13 14 diatomic and polyatomic species, using shock techn1ques. ' , 10 The studies of chemical reactions in shock waves have been tabulated, 15 and have been updated by Strehlow. Recent ideas include cis-trans isomerization studies,16 and homogeneous exchange reactions using d•• 17 ra 1oact1ve tracers. Hooker18 has studied two phase systems, using a powder-gas aerosol for the test gas. However, the most exciting prospect seems to lie in the glow discharge shock tube,19 which has been effectively used by Gross20 and Cohen.2l ,22 The present work is concerned with the vibrational relaxation of gas molecules at high temperatures. The main measurement techniques involve those which can respond to changes of density; temperature and the emission of radiation. Shock front reflectance 23 methods have been used to measure rotational relaxation. Measurements • 24,25,26 of density changes have been made using a Mach-Zender 1nterferometer. The technique has been improved by the use of a laser light source. 27 28 29 30 Schlieren techniques have also been advanced by the use of lasers. ' , The vibrational temperature of the shocked gas may be followed by using the sodium line reversal method. 3l The infrared fluorescence methods allow the populations of individual vibrational levels to be followed during the relaxation period. The first work was done by Windsor, 32 33 Davidson and Taylor, ' who studied the infrared emission from the overtone of CO with a PbS detector. The technique is simple, the emission from the shock heated gas is pas.ed through a filter, to isolate the wavelength of interest, and on to a detector, which is - 11 -usually of the photoconductive or photovoltaic type. 34 The infrared 35 36 37 38 39 fluorescence technique has now been widely used ' , , , and its application in the present case is given in section 2.6 of Chapter 2. 40 41 42 43 Finally, the nozzle expansion flow technique, ' , , 44 45 which is an extension of the simple shock tube, has provided data ' which bear comparison to shock tube results. The shock wave raises the translational temperature, so that the vibrational degrees of freedom may be excited. The expansion flow utilises the stationary gas at the end of the shock tube after the shock baa reflected from the end plate. TIle gas is expanded through a nozzle, 80 that the gas is translationally cooled, and the deexcitation of the vibrationally excited gas may be followed. The vibrational temperature may be followed by the sodium line reversal technique. However the results may have been affected by impurities,46 drawn from the boundary layer. l.2.b ULTRASONIC METHODS At relatively low temperatures the ultruOllic technique has provided a large bulk of data an vibrational relaxation. TIlere are several good reviews on the subject, which &lao gift the types of systems that have been studied. 47 ,48,49 The bui. of the _thod i. as follows. The velocity of sound through the ._ i. given by:-- 12 -2 (1 + RIC )RT/H v = v where R = universal gas constant v = velocity of sound T = temperature Cv = heat capacity of gas H = molecular weight The rate of energy transfer through the gas has an inverse dependence on the effective heat capacity of the gas. If the period of the sound wave is long enough for equilibrium to be established between all modes, then the heat capacity is given by:-C = v C + trans where Ctrans ' Crot and Cvib are the translational, rotational and vibrational heat capacities. If the period between the waves is gradually decreased a point is reached at which energy cannot be transferred in and out of the vibrational modes rapidly enough to maintain equilibrium. This energy is then lost from the sound wave and dissipated in the gas. At some point no energy is transferred to the vibration, and the vibration makes no contribution to the specific heat of the gas. The sound wave now travels through the gas at higher velocity. From measurements of the velocity at various sound frequencies. the Napier time may be calculated. The method has had success in measurements of vibrational relaxation in a temperature - 13 -range up to lOOOK. However, it cannot be used for high frequency molecular vibrations which make a small contribution to the specific heat. 1.2.c MISCELLANEOUS METHODS Experimental techniques have been developed, for the stuqy of vibrational energy transfer, based on the radiative excitation of energy levels. Those vibrational modes of molecules which involve a change of dipole moment can take up quanta directly by absorption of radiation of the appropriate wavelength. A particular vibraticm will lose energy either by radiative decay of infrared fluorescence or by collisional deactivation, depending on the lifetimes of the two processes:-+ M + K.E. The radiative lifetime of an infrared active mode is independent of the gas pressure whereas the collisional lifetime is inversely proportional to it. For all but a few diatomic molecules with high vibration frequencies (i.e. CO) the radiative llfetime is alch longer than the collisional lifetime, and the latter dominates the vibrational relaxation. 50 An apparatus which combines radiative excitation with - 14 -sound wave detection is the spectrophone. The gas is pulsed with infrared radiation and the time lag for the modulated radiation to appear as a sound pulse (optic-acoustic effect) provides a measure • 51 52 53 of the rate of v1bration-translation energy transfer. ' , The infrared resonance fluorescence technique of Hillikan54 employs both spectroscopic excitation and deexcitation ( - -1) to study CO v = 2143 em ,one of the molecules whose radiative lifetime is shorter than its collisional lifetime, at room temperature. Millikan studied the half quenching of fluorescence from CO by small additions of other gases. From this he obtained the probabilities of energy transfer in heteromolecular collisions. Important features of the experiment were an optically thin gas and the elimination of wall collisions as an effective quencher of CO by containing the gas sample in a streaming jacket of Argon. The IIOst recent infrared excitation technique is the use of laser pulsing to modulate the population of a particular vibrational energy level. The development of the high power CO2-N2 gas laser55,56.57 has focussed considerable interest in the .. chanism and kinetic rates of vibrational energy exchange between CO2 and N2• This method of excitation, combined with the observation of fluorescence from another 58 vibrational mode has been used to study vibration-vibration exchange 59 60 61 in CH4 and CO2• There are good reviews on the subject. ' Another spectroscopic method, which gives information on vibration-vibration energy transfer, is flash spectroscopy. described - 15 -. h62 d 63 Th ch' d' ul . h by Norr~s an Callear. e te n1que, use part~c arly w~t NO, consists essentially of triggering a powerful light flash, filtered for suitable wavelengths, in a tube next to the reaction vessel. The NO molecules in their ground electronic state x2,.. (v=O) are raised to higher vibrational levels of the exicted electronic states, usually A2 r+. Relaxation of the vibrational energy can be studied in the excited state by observing the changing intensities of v=O and v=l progression in the fluorescent spectrum of the excited electronic state and the gro\md state by flash spectroscopy. In this last method the photoflash is followed, after a CClIltrolled delay, by a spectroscopic 2 flash, enabling the absorption band of the transition X ,.. (v=l) to the 2 + excited A r (v=0) to be photographed. The decay with ti_ of this band provides a measure of the vibrational relaxation of the v=l level in the gro\Dld electronic state of NO. Both fluorescent emission and flash spectroscopy have been used to study the relaxation of NO in homo and hetero molecular collisions. 1.2.d COMPARISON OF METHODS The spectroscopic methods are concemed with exciting molecules vibrationally. They provide a _ans of studying high frequency vibrations whose vibrational energy leWD are not appreciably populated at 3OOK. For these vibrations, as previously stated, there is li ttl. or no ultresonic dispersion or absorption. On the other hand it is possible to .aura relaxation processes - 1& -occuring in short times, by the ultrasonic technique. This is not always possible with the shock techniques. The spectroscopic and ultrasonic methods are complementary since in the former the direct perturbation of the vibrational energy facilitates the study of vibration-vibration energy transfer, while the latter is more suited to vibration-translation transfer. However, in the case of gas mixtures the two methods supplement each other. As with the acoustic method, the shock wave studies provide evidence primarily about vibration-translation transfer. Together the two techniques provide data on the temperature dependence of this process, which can be used to test the validity of theoretical calculations. - 17 -1.3 PREVIOUS SHOCK TUBE STUDIES OF VIBRATIONAL ENERGY TRANSFER The object of this section is to review shock tube results which are relevant to the present study. The perfonnance of the theoretical predictions is also given. There are several methods for the absolute calculations of Napier times, and these are given in Chapter 4. The most widely applied approach is the S.S.H. theory (section 4.5). It is a 3-Dimensional quantum mechanical treatment which predicts absolute values of the Napier times, to within an order of magnitude, for many systems over a wide temperature range. The details are tabulated.48 ,64,65 For simple non polar molecules 1 the dependence of temperature follows the Landau-Teller relationship, lnTaT-t. At lower temperatures and for non polar molecules the experimental times vary less steeply with temperature. The Millikan and White66 (M.W.) correlation, derived on an empirical basis, has had reasonable success in predicting vibration-translation Napier times (see section 4.6). However there are notable exceptions to the S.S.H. theory and H.W. correlation, particularly H2, D2, the hydrogen halides and NO. H 29 and D 28 show Napier times almost two orders of magnitude 2 2 longer than the S.S.H. theory predicts, while the M.W. correlation gives Napier times of 7 times too slow for D2 and 40 times tooslow for H2• The Napier times of HCl,67,68,69 HBr,70 HI70,71 and Del68 have been measured. At 2000K HCl haa a rough agreeMnt with the S.S.H. value but HBr and HI are much lower than predicted by the theory and the M.W. correlation. These discrepancies may be due to the efficiency of - 18 -vibration-rotation energy exchange, of which the theories take no account. 72 Moore has developed a vibration-rotation transfer theory, and it gi Yes a reasonable value for HC1, but not for HI. This may be due to the use of a single value of the repulsive potential parameter, t, to fit the experimental data of a wide range of molecules. In the NO case, it ia possible that an electronically excited state is formed, or that dimerization may have to be taken into account. A special potential function is needed for thia molecule. 73 74 7S 76 Linear triatomic molecules such as CO2 ' , , and N2082,83 have had much attention devoted to them. Opinion has been divided as to whether the moleculea poaaeaa more than one relaxation time. However, recent experimental evidence has hardened the attitude that all the modes relax together. 77 ,78,79 This was certainly the 80 case in Camac's study above 2000K. The theoretical S.S.H. results do not agree well with the experimental values, the former having a too steep te..,.rature dependence. H20 has b .. n studied in the shock tube and there is good agreement between various workers. These molecules are of interest in mixtures 81 since their vibrational modes will provide information on intermolecular vibration-vibration energy transfer, and also intramolecular vibration-vibration transfer. Carbon IIOnoxlcie, the bui. of the Pl'Uut work, baa been well atudied,32,36,84,85 The oriainal work in the reflected shock region gave a longer Napier ti_ tb.a the sub.eqUlnt atucU._ using different techniques. Thi. wa prabably dUl to the use of incident - 19 -shock wave parameters in the former work. The relaxation has been 86 87 studied in the presence of various collision partners ' and the Napier times agree well with the S.S.H. calculation, but are about a factor of 2 shorter than the H.W. predictions. The overtone of CO was studied and it was shown36 that the rate of populating the v=2 level by successive quantum jumps was ten times faster than the rate of direct excitation. The equilibrium population of the v=2 level of pure CO may be acheived in two ways:-Firstly COv=O then + co 0 v= k' 2 , , where or , k' -2 v = vibrational quantum number k' = bimolecular rate constants CO 2 + CO_O v= v-The mechanism was interpreted using a Landau-Teller model that. , , kOl : kl2 = 1 : 2 etc •• and it was concluded that the atepwise 36 process dominated. However no attempts were .ade to distinguish 1.S 1.6 1.7 between the population of the overtone by reactions 1.5 and 1.6 from 1.5 and 1.7. It was found that both _chan1_. reduced to the a .. - 20 -analytical equation, see Chapter 5. Higher vibrational levels of other molecules have been studied. In particular work on HCl67 and HI 70 showed that 'transitions where flv > 1 were possible, so that = = It was suggested that this result may be explained on the basis of efficient energy exchange from vibration to rotation. A large number of acoustical and spectroscopic measurements have been made in gas mixtures, which 'provide information about vibration-vibration energy transfer in heteromolecular collisioms. Considering a binary gas mixture, whose molecules A and B contain only one vibrational mode, then if the quantum of vibrational energy is denoted * the processes are:-1. A* + A ~ A + A ~ 2. A* + B ~ A + B 3. B* + B ~ B + B 4. B* + A ---::. B + A ~ s. A* + B ---::. A + B* ~ The self relaxation of B is more rapid than that of A. For IIIxtures of gases A and B, the relative Napier ti_s of the pure gases (1,3) and the relative matching of their vibrational energy levels (VA and VB) can be so chosen to be closely similar or widely different. If the matching of A and B is poor, 80 that the vibration-vibration exchange is slow, then at all compositions, the reciprocal Napier ti .. will be - 21 -linear, with slope depending on the relative rates of processes (1,2) for A, and (3,4) for B. Now consider the case in which the levels of A and B are nearly resonant so that for certain pairs of molecules the vibration-vibration reaction 5 will be more rapid than the simple deactivation of B. The mixture will show only one Napier time at any concentration, the time depending on the rates of processes 3,4. The most notable shock tube work in this field has been 38 by Taylor et al., who carried out experiments in th .. e gas mixtures -1 -1 -1 No-N2 (At • 456 em ); NO-CO (At • 267 em ); CO2-N2 (At = 18 cm ). The first two systems provided data OIl "near-resonant" vibration--vibration exchange. The CO2-N2 llixture provided data OIl resonant energy exchange. Since this work, other studies have been carried out on a variety of .yste ... 39 ,88,89,90,91,92 For diatoaic systems the general temperature dependence of the resonant rate. i. reasonable, but the magnitude of the experimental data i. considerably less than the S.S.H. theory predicts. Si.ple .olecules with additional vibrational modes have complex mechanisms for vibration-vibration energy tran.fer. 60 81 They are well studied at room temperature. and the theory bas been revised to take account of the possibilities of intra.olecular energy exchange.93,94 However, further work is required OIl the t811p8rature dependence of syet.. COfttaining such .olecuJ.u. - 22 -1.4 OBJECTS OF THE PRESENT STUDY i) The construction and development of a pressure driven shock tube. The tube was to be capable of producing conditions of high temperatures in gaseous systems, so that accurate infrared emission measurements could be made on the chosen systems. H) Infrared emission measurements on pure CO and COINe mixtures. CO is a molecule with a dipole moment, and thus lends itself to this type of experiment. As indicated in section 1.3 the vibrational relaxation of the fundamental and overtone have been studied under carefully controlled conditions. It was decided to remeasure the Napier time for these vibrational modes of CO. Then to study the effect of the addition of an inert gas on the mechanism of the population of the fundaHntal and the overtone. This was done with the intention of being able to distinguish between the two mechanisms of population of the overtone. The experimental results were to be compared with computed relaxation curves using various models for the population of upper levels. It was intended that the comparison of theory and experiaent would provide a stringent test of the models and assumptions used in the I'8duction of shock tube data. iii) Infrared emission .... urements on CO/polyatomic mixtures. 'lbe current inteNst in .hock work U.. in vibNtion--vibration exchange processes. Previous worker. haft used linear 38 90 triatOlll!c molecules • in a binary aixture ad c_ to the conclusion that energy IlUat be trmsfel'Nd fl'Olll th .. e molecule. - 23 -via the v3 antisymmetric stretching mode. Thus for a mixture with CO, the infrared emission fran the fundamental of CO, may give some measure of the rate of exchange between the fundamental of CO and the v3 mode of the triatomic. A series of mixtures which lend themselves to this study are (energy discrepancy in brackets) : -CO/C02 CO/N20 CO/COS (AE • 206 ca-l ) (At • 81 ca-l ) (AE. 6_ ca-1 ) The first sy.te. in the .erie., i. of special intere.t .ince the low di •• ociation energy of CO2, _an. that .0. CO is expected in electric di.charge. in CO2 -N2 or in CO2 lasera. ,",us it was hoped to obtain information on vibration-vibratiOll energy exchange in these .ystelllS. i v) ,",eoretical calculations. Extensi". calculations of the Napier times fOr the .yetems "ere carried out using the .. thod • • elected in Chapter ~. v) ConcluaiOlls. A Nuon.d .xplanation of the kinetics of the proce.... studi.d und.r ii ad iii "a waclvtuen. - 24 -REFERENCES 1. L.D. Landau and E. Teller, Z. PhyB. SorJjtltunion, 1936. lQ., 34. 2. B. Reimann, GeBamm8tttl W.~ke, 1876, 144. 3. W.J .M. Rankine, n-ans. Roy. SotJ. (Umdon). 1870, l&Q., 277. 4. J. Rayleigh, P1totJ. Roy. SotJ. (Imtdon), 1910, ~, 247. 5. H. Hugoniot, rJ. a. t"oottl PotllutJhn-lqw, 1889, i!, 1. 6. 1.1. G1us, UniVtl~Bity of ToJ'Onto AtlJ'mIaIltios Rtl11i.w, 1958, No .12. 7. J.N. Bradley, ShotJk WawB in ChtRrri.8tJry and PhyBitJ8, (Methuen, London) 1962. 8. A.G. Gaydon and I.R. Hur1e, r;,. Shook n.btl in High TtllPfJtlJ'(ZtuN ChtIml(J(lt PhYBitJB, (Chapman and Hall, London) 1963. 9. J.K. Wright, ShotJk TWbtIB, (Methuen, London) 1961. 10. t.F. Greene and J.P. Toennies, CMtrrloal, RtlaDtUms in ShotJk WatHlB. (Amo1d Publishers, Lcodon) 1964. 11. R.A. Strehlow and R.L. Belford, 7'etJh. Rllpt. A.A.E., !2=!, Univ. Illinois, Urbana, 1969. 12 • R. L. Belford and R.A. Strehlow, Ann. ~v. Phya. C1umr., 1969, 20, 247. -13. S.H. Bauer, Ann. RWv. Phys. ~., 1965, 16, 245. -14. J. Tree and H. Wagner, BtI~. 1J&tn8.ag.s Physi.k t:htIm., 1967, 1!t 937. 15. R.A. Strehlow, PI'o(JHs. in High TIIIIf'IIN'tI4N PhII.toe and ChtImi.Btzty, (C.H. Rouse Ed. PergllllOll Preaa, New York) 1969, 1. 16. P.M. JeffeN and W. Shaub, tI. AIr. ChImt. 800., 1969, 91, 7706. -- 25 -17. A.J. Kassman and D.S. Martini, J. Am. Chem. Soc., 1969, l!, 6237. lB. W.J. Hooker and R.D. Main, Physica, 1969, ~, 35. 19. R.A. Hartunian, W.P. Thompson and E. Hewitt, J. Chem. Phys., 1966, ~, 1765. 20. R.W.F. Gross, J. Chem. Phys., 1968,~, 1302. 21. R.W.F. Gross and N. Cohen, J. Ch~. Phys., 1968,~, 2582. 22. N. Cohen and R.W.F. Gross, J. Chem. Phys., 1969,2£, 3119. 23. E.F. Greene, G.R. Cowan and D.F. Hornig, J. Chem. Phys., 1951, 19, 427. 24. H.K. Zienkiewicz and N.H. Johannesen, J. Fluid M~oh., 1963, ]2, 499. 25. V.H. Blackman, J. Fluid ~ch., 1956,1,61. 26. D.R. White, J. ChIJl'fl. Phys., 1965, !.2., 447. 27. A.L. Besse and J.G. Kelley, Rev. Sci. Instr. , 1966, !I, 1497. 28. J.B. Kiefer and R.W. Lutz, J. ChIJI'fI. Phys. , 1966, ~, 658. 29. J.H. Kiefer and R.W. Lutz, J. c:h.". 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Legay and N. Legay-Sommaire, Co1Jt::>t. Rtmd., 1964, ill, 99. N. Legay-SoDlDaire, L. Henry and F. Legay, Compt. &Ind., 1967, l§,Q, 3339. J.T. Yardley and C.B. Moore, J. Chem. Phys., 1966,~, 1066. L.O. Hocker, M.A. Kovacs, C.K. Rhodes, G.W. Flynn and A. Javan, Phys. Rev. Lflttsrs, 1966, .ll, 233. 60. C.B. Moore, Ace. of ChIIm. Res., 1969, 1, 103. 61. R.L. Taylor and S. Bitterman, Rsv. Nod. Phys., 1969, ~, 26. 62. R.G.W. Norrish, TM StvtJ,j of ErulzttIIJ TJ'It:I1uIfer in Atoms and Mol-saul-es by Photochtmi.cal- Methods, (Interscience, Hew York) 1962. 63. A.B. Callear. Appt. Optics Sllppt •• 1965. 2. 145. 64. P. Borrell. TzvzIuJf8r arad Stol'Q(/8 of EMzttIIJ by No7.8ou14s, (Wiley - Inte1'8cienC8, Londco) 1969, 2., 180. 65. B. Stevens. eottisWncit Aoti~tion in Ga ••• (Pergaon Press. LeDdon) 1967. 201. 66. R.C. Millikan and D.R. White, J. Ch.. PhJJ •• , 1963, .!!. 3209. - 28 -67. P. Borrell and R. 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Caac, A1IOO RfIPOP't, 196", 195. 81. C.J.S.M. Simpson, T.R.D. Chandler, l7oc. IIDJI. Boo. (Lt1IIdtmJ. Bar. A. 1970, 317, 265. -82. W. Griffith, D. Brickl and V. B1aclcMn, P1rre. ~., 1956, lQl, 1209. - 29 -83. C.J.S.H. Simpson, K.B. Bridgman and T.R.D. Chandler, J. Chem. Phya., 1968, ~, 509. 84. A.G. Gaydon and I.R. Hurle, 8th Symp. Cbmbuation, (Williams and Wilkins, Baltimore) 1962, 309. 85. D.L. Matthews, J. Chem. Phya., 1961, ~, 639. 86. R.C. Millikan, J. Chsm. Phya., 1964, ~, 2594. 87. R.C. Millikan and L.A. Osberg, J. Cham. Phys., 1964,~, 2196. 88. D.R. White, J. ChBm. Phys., 1968, ~, 5472. 89. G. Kamimoto and H. Matsui, A.I.A.A. cToumaZ" 1969, I, 2358. 90. J.F. Roach and W.R. Smith, cT. Chem. Rhys., 1969, ~, 4114. 91. S.J. Colgan and B.P. Levitt, Private communication. 92. R.L. Taylor and S. Bitteman, .T. eMma Phys., 1969, 12. 1720. 93. K.F. Herzfeld, Di.saussi.ons F~ Soc., 1962, !!. 86. 94. K.F. Herzfeld, cT. CIuIm. Phys., 1967, .!I, 743. - 30 -2. EXPERIMENTAL 2.1 INTRODUCTION The shock tube has been used for energy transfer studies 1 for a number of years. The rise in temperature at the shock front provides rapid and uniform heating of the test gas, so that the various degrees of freedom exchange energy. By monitoring a suitable characteristic of the gas, information on the relaxation will be obtained. The tube consisted essentially of a metal or glass tube of circular cross section, divided into two sections by a diaphragm of cellophane or metal foil. The shorter of the sections contained a driver gas at high pressure which acted as a piston when the diaphragm was ruptured. A shock wave was driven into the experimental gas, at a considerably lower pressure, in the longer section of the tube. It was then possible to measure the velocity of the shock, and to observe the infrared emission from the shock heated gas, by utilising a detector and display unit. Since the shock wave travelled at about 1 to 2 millimetres per ~s, the relaxation process was completed within 500 ~s and electronic methods were necessary to measure the relaxation process. The following chapter describes the important parameters in the building of the shock tube, and in the choice and construction - 31 -of the ancillary equipment. Included also are details of the experimental procedure, together with some of the major difficulties which had to be overcome. - 32 -2.2 SHOCK TUBE 2.2.a CALCULATIONS OF TUBE DIMENSIONS Before any tube design was undertaken, it was necessary to calculate lengths of driver and test section required. The important quantity in these calculations was the hot flow time, (laboratory time) that is, the time between the passage of the shock front and the contact surface. to an observer stationed in the test section. In the ideal situation, the time is limited solely by the interaction of the wave systems produced when the diaphragm is burst, after they are reflected from the ends of the tube. Expressions relating the hot flow time to the combination of driver and test gas. 2 and shock speed have been derived. The pertinent equations are given in Appendix 1. The calculations are made assuming ideal gases. However. the growth of the boundary layer behind the shock front affected the flow, so that the hot flow time was considerably reduced. A detailed account of the boundary layer, and its effects on shock conditions will be given in Chapter 3. The calculations were made assuming that the driver gas was hydrogen, the test gas carbon monoxide. the shock wave travelled at Mach 5 and the hot flow time was 500 \.IS. The hot flow time was selected on the basis of previous results obtained co carbon 3 monoxide. Using these figures, the calculations showed that the ideal driver section was 1.5 m long, the minimum length of test - 33 -section was 3.61 m. However, the available laboratory space was restricted to a total length of 3.75 m. Thus the theoretical tube geometries were modified in accord with the ranges described by 4 Hall. This was an empirical approach, based on data of a number of tubes. The final lengths decided upon were:-Driver Section 1.15 m Test Section 2.55 m It was also important to calculate the distance the observation station was to be placed from the end plate. At a distance of 0.42 m from the end plate, calculations showed that the initial hot flow should not undergo any interaction from the reflected shock. 2.2.b TUBE MATERIALS AND DESIGN The shock tube will be considered in tems of its ind! vidual components, that is, the driver section, the connecting section and the test section. A tube of circular cross section was decided upon, since the absencecf corners made it easy to keep clean, and the circular cross section gave structural strength. There are two effects to be considered on change of the internal tube diameter. Firstly, a decreas8 in the , diameter leads to an increase in boundary layer effects, of the shock wave with the wall. Secondly. an increase in diameter leads to an increase in the pumping time. In this case it was decided to restrict - 34 -the pumping time, and to make the necessary corrections for the boundary layer. The driver section was made of brass, and was 1.15 m in length, with an internal diameter of 2.54 cm. See Figure 2.l.a. At either end of the section, a brass flange, diameter 12.5 em, was silver soldered to the tube. In one of the flanges were drilled three equally spaced holes, near the periphery of the flange. These were to house the securing bolts for the end plate. An '0' ring made the vacuum seal. The other flange had a seating for an '0' ring, 8 em diameter. Pins for locating the driver section with the connecting section were placed diametrically opposite at a distance of 11 em. Two grooves were cut into the edge of the flange, to accommodate the clamping screws of the connecting section. Located about midway on the tube was a hollow thread, which was silver soldered to a hole drilled into the section. This was the driver gas inlet. The connecting section, Figure 2.1.b, was the piece which allowed the driver section to be joined to the test section. It was made of brass, with brass flanges, and was of length 10 cm, with internal diameter 2.54 cm. At the end nearer the driver section the flange had an '0' ring 6.5 em in diameter. Also two holes were drilled to coincide with the locating pins OIl the foward flange of the driver. This ensured that both sections had their diameters in coincidence when the sections were clamped together. The clamping wu done by two butterfly screws, I 25 an Figure 2.1. a Driver S~ction __ ~~'NLET liS an Figure 2.I.b Connecting Section lOan 12·5cm - 35 -which fitted the grooves on the driver section's forward flange. Drilled at equidistant points near the periphery of the other flange were three holes, which coincided with the holes on a standard Q.V.F. backing flange, and which held the test section firmly against the connecting section. For spectroscopic work it would be advantageous to use a glass tube as the test section. However, in the present studies this presented two disadvantages. Firstly, the thick curved wall of the glass tube presented optical difficulties, and secondly pyrex b 1 3800 em-I glass was opaque e ow It was decided to use pyrex tubing . .-1 for the stud~es on the overtone of carbon mono~de, at 4260 em • -1 and for the studies on the fundamental, at 2143 cm , a pyrex tube-metal tube combination was used. The metal tube had calcium aluminate windows which transmitted in the region of interest. See Chapter 3. The pyrex tubing was standard Q.V.F. pipeline of 2.54 cm internal diameter. It had been tested to a maximum internal working -2 pressure of 689.5 kNm • The residual pressure in the tube after a -2 shock experiment was calculated to be never greater than 305 kNm • The pyrex tubing could be obtained in varying lengths. which were clamped together by two metal backing flanges. The joint between the glass tube was sealed by a P.T.F.E. crescent shaped ring. All the aforementioned parts are standard Q. V.F. pieces. The joint between the pyrex tube and the connecting section was made by clamping a metal - 36 -backing flange to the metal flange of the connecting section with a P.T.F.E. ring as the seal. For the measurement on the overtone of carbon monoxide the test section consisted of a 15 em length and a 2.4 m length of pyrex tubing. The 15 cm length had a 5 mm internal diameter gas exit and entry port glass blown on to it. At the end of the longer length, an end plate was clamped with a backing flange. For the studies on the fundamental of carbon monoxide, the following sections were used:- (i) 15 em section with gas port as before, (ii) 75 em section of pyrex tubing, (iii) 90 em section of metal tube, with one window mounting, (iv) 90 em section of metal tube with two window mountings. The couplings between glass-glass and glass-metal joints were as described before. The Figure 2.2.a shows the metal section with the two window mountings. The tube itself was constructed from stainless steel, with brass flanges, 8.75 em in diameter, and silver soldered to the tube. The joining of the two metal sections was arranged so that large steps at the joint were avoided. This was achieved by having a lip on the periphery of the flange of one tube, which fitted into a groove on the other. An '0' ring made the seal when the two flanges were bolted together. This system of mating the tubes gave not only a smooth tube, but also ensured a good vacuum. Mounted on the tubes were blocks of brass, which were machined to accommodate the window holders. The design of the holder is shown in Figure 2.2.b. The holders were clamped to the tube by means of screws, and an '0' ring made the seal. Before the I I ~ Figure 2.2.0 Metal Observation Section o o WIND~ 90cm Figure 2.2. b Window Holder SfCTION ACBQSS TUBE TUBE ~~~27i1J1:I ______ _ / /// ./ /'l //1 // /1 ~/ / I I ~ __ ===------i----ur.- - - - ---- '-------.;:::::.....-:~___r_--. \BASSIMNOOW MOUNTING. GLASS - 37 -windows were cemented in position, the holders were clamped in position and both tubes were honed to a diameter of 2.54 ± 0.0025 cm. The windows used were of two types (i) Calcium Aluminate glass type BS 39B obtained from Messrs. Barr and Stroud Limited. This glass had an -1 infrared transmission to about 1700 em ,with a transmittance of 50%. (Ii) Pyrex glass, which was obtained from Electronic Optics Limited. Both types of glass, were circular discs of 1.56 cm diameter and 0.63 cm maximum thickness, which had one surface ground to a radius of 1.27 em. The other face was flat. The windows, were cemented into the holder, using Araldite, so that they were flush with the inside wall of the tube. The first and second pairs of windows were pyrex, used to measure the velocity of the shock by means of light screens. The last window was used for observing the infrared emission. 2.2.c SUPPORTS Support for the shock tube and driver section was important. Considerable forces existed along the axis of the tube, in particular, when the diaphragm was broken and when the gas was brought to rest. Convenient rigid supports for the circular tube were standard Q.V.F. pipeline holders. These supports consisted of a split moulded rubber collar, kept in place by a U-bolt and back plate. The driver section was mounted on rollers to take up any recoil, and to facilitate changing of the diaphragm. - 38 -2.3 VACUUM AND PRESSURE TECHNIQUE The high vacuum apparatus is represented in Figure 2.3. The taps, wherever possible, were of the greaseless type. The ultimate vacuum was realised by means of an oil diffusion pump (N.G.N. type OPW 125, with entrainment head) which was backed by an Edwards Two Stage Rotary Pump (Type ED 35). The rotary pump was separated from the diffusion pump by a liquid nitrogen cooled trap, and similarly the diffusion pump was separated from the system by a trap. This combination of traps and pumps enabled the following vacua and leak rates to be attained:-Glass Tube Glass-Metal Tube Vacuum 106 rnNm- 2 133 mNm-2 Leak Rate 0.74 rnNm- 2s- l 1.11 rnNm -2s-l Test gases were stored in 5 1. and 20 1. bulbs, point A in the diagram. The gases were distilled by trap to trap distillation, along the distillation train, point B. It turned out, that the bulbs were lightly used, since high purity gases were required. These are described later. The pressure of test gases was measured on Edwards Barometrically Independent Capsule Dial Gauges, at point C, calibrated -2 -2 -2 in the ranges 0 - 2.66 kNm ,0 - 5.32 kNm ,0 - 101 kNm • The vacuum was measured by an Edwards Pirani Gauge Head, mounted in a B14 glass joint, and the vacuum was read from an Edwards Control Unit. u Figure Vacuum ~ 2.3 Sy~tem Figure 2.4 High Pressure System f ro YAel JUM SYS ~ VIA A GLASS COPPEB SEAI_ ~\P:....o.E.J,E..I.oID~IV~A~L.:.LV""""fS~ __ < H'fDB<)C'flJ G (NITROGEN H \----- (AUXILIARY GAS roBT 1 TO DRlVER SECTION YIA NYLON TuBING DRIvg:~ \PRESS\] GAlGE ~KING BoARD - 39 -The entry to the tube was at point D. The high pressure system is represented in Figure 2.4. All the taps on this system were Saunders-Edwards Speedivalves, which would stand the high pressures used in this section of the instrument. Discharge of any unwanted high pressure in the tube was via the tap at point E. The Bourdon gauge, at point F, was -2 obtained from Budenburg Company Limited, had a range of 0 - 1.38 MNm • The device also had an auxiliary pointer which enabled the driver pressure to be read accurately, after the shock. At point G, the driver gas hydrogen was taken from a cylinder. Nitrogen could be used as a driver gas, taken via point H. - 40 -2.4 DIAPHRAGMS In this study the diaphragms were allowed to burst naturally, by gradually increasing the driver pressure. The choice of diaphragm material was reduced to Aluminium foil, supplied by G.W. Aluminium Company Limited, and Melinex Polyester Film made by I.C.I. Tests were carried out on both types of material using carbon monoxide as the test gas. The aluminium foil, 0.01 em thick, -2 burst on average at about 380 kNm • In the first instance, the petals formed on bursting were torn off by the sharp edge of the connecting section. Thus the ~onnecting section was tailored to give an edge over which the petals could fold. In this condition, most of the subsequent runs showed that the diaphragm petalled from the centre, into four pieces, without fragmentation. For those experiments in which fragmentation did occur, there was no apparent tube damage and the fragments were removed after each experiment. In . -2 general 150 gauge Melinex (0.0038 em thickness) burst at 620 kNm • It was found, on inspection after an experiment, that this material split rather than folded or petalled back, to allow the greatest area 2 through which the driver gas could flow. It has been suggested that this type of diaphragm was unsatisfactory for spectroscopic work, since the material shredded and burned to gi va an undesirable illlpUl'i ty spectrum. It was thus decided to use aluminium diaphragms. The claphragm was clamped between staggered '0' rings on - 41 -the driver and connecting section. This gave good vacua in both sections. It was found that excessive tightening distorted the flanges, and that finger tightness, on the clamping screws was sufficient. - 4-2 -2.5 LIGHT SCREENS Initially the velocity of the shock wave was measured by two sets of light screens, after the method of Bleakney5 and his co-workers. 6 The arrangement is shown in Figure 2.S.a. In practice the light screens were independent of the frame. Their operation depended upon total reflection from the back of the shock wave of the incident light. The light source S, a 12 volt Tungsten projection lamp, was used in conjunction with a system of three knife edges kl , k2, k3 • The knife edges were arranged so that they lay in a line perpendicular to the axis of the flow. Then only rays lying in the plane so defined reach the photomultiplier P. The photomul tiplier was an E. M. I. type 9650B, of a 9-stage "squirre 1-cage" design, with a quartz envelope. The caesium antimony cathode 3 -1 3 -1 had a maximum spectral response between 33 x 10 cm and 25 x 10 em 3 -1 and a long wavelength cut-off at about 14- x 10 em • The rate of change of light flux on P was approximately proportional to the shock speed and the signal voltage rose rapidly with time. The output signal was fed to the circuit shown in Figure 2.S.h. The value of the load resistor, L, was large so that the output signal was large. The twin emitter follower, type 1$ 17, supplied by Venner Electronics Limited, was used as an impedance transformer. The emitter follower transmitted the signal from the high impedance source, to a low impedance at the output, in the co-axial cable of impedance about 50n. This combination FigurYl. 2. S. a Reflection Light Screen UGHI roUBCE S I --- == SHOCK Figure 2. s. b Photomultiplier Circuit TSI7 \wIN EMITTER fOllOWER PHOTOMl1.T1PLIER TUBE ~---a.-------_____ --lQ -8501 H.T. - 43 -of optics and electronics afforded a pulse rise time of the order of I \Js. The sensitivity of the method depended on the signal to noise ratio from the photomultiplier. The most important factor in noise generation was the amount of stray light. Attempts were made to reduce the stray light to a minimum by covering the screens with black cloths. In a series of test runs, using hydrogen as driver and nitrogen as test gas, the pulse depth was seen to falloff with increase in Mach number. The limit of operation was at a test gas -2 pressure of 0.9 kNm • This may be explained by considering the following equation, which was derived and used by Hornig and his , 'k 'I I t' 7,8 assoclates ln wor on rotatlona re axa lone where R = e = no = Po = t.p = R = 4 (I + tan e) 4 2 Po optical reflectivity = IntensitI of reflected lisht Intensity of incident light angle of incidence always large, that is grazing incidence =0 900 refractive index of the gas at S.T.P. density of the gas at S.T.P. total density change across the shock front - ~~ -All other things being equal, the quantity ~P was the factor which most influenced R. Since:-~P = (PA PA ~P = (-Pl where = density at the shock front = density of the downstream gas Although at higher Mach numbers the ratio PA/Pl increased the value of Pl , which is pressure dependent, decreased due to a lower downstream pressure. Thus at higher Mach numbers the light screens ceased to function. In 1965, J.H. Kiefer and R.W. Lutz,9 developed a quantitative laser schlieren technique to measure the vibrational relaxation of deuterium. The method used photomultiplier detection of the deflection of a narrow laser beam, the technique being sensitive to rapid gas density changes. It was decided to adopt several features of this technique to improve the performance of the light screens. The system decided upon is shown in Figure 2.6. At position A was the laser, a helium-neon type, supplied by Ferranti Limited, which emitted a . 3 -1 ( ') i . coherent beam of l1ght at 15.81 x 10 cm 632.8 nm. At pos t10n B was a beam splitter of 50\ transmittance and 50\ reflectance. The other mirrors Ml , M2 and M3 are of the front surfaced type. kl and Figure 2.6 Laser Light Screens A ( lUBE PHOTQMULDPUEB - 45 -k2 were knife edges onto which the laser light was focussed. The distance between the mirror H2 and the photomultiplier was 1 m. This length of optical lever was found to be satisfactory. The incident light beam crossed the tube almost normally to the direction of flow, e being 89.50 • It was assumed that the beam was not only reflected from the rear of the shock wave, but also it was refracted towards the higher density region. Test runs were carried out, using hydrogen as a driver gas and nitrogen as a test gas. Pulses of about 1 ps rise time were produced. Also pulse depths were increased by about 30\ over those obtained using corresponding conditions in the original technique. The system also operated to -2 a minimum test gas pressure of 0.2 kNm • Two problems emerged with the use of the laser system. Firstly, not having the beam exactly normal to the tube resulted in secondary "reflections from the walls of the tube, and secondly the spacial resolution of the beam was lost due to the curvature of the tube walls. The loss of spacial resolution was more acute in the case of the windows on the metal tube. To compensate for the di vergent effect of the windows. barrel shaped lenses of focal length 7.5 cm were used. Plate 2.1 shows the arrangement. To prevent any spurious reflections interfering with the incident beam, slits of adhesive tape were placed on the tube. Finally, in recent years advances in solid' state electJ'IGDi. '( PLATE 2 . 1 - 46 -have led to the production of photodiodes, which while retaining the essential characteristics, have a number of advantages over the photomultiplier tube. A P.I.N. photodiode was obtained from Plessey Microelectronics. At the present time a circuit which combined fast response with high gain had not been achieved. But, clearly, this is a field in which some further work is required. - 47 -2.6 INFRARED DETECTOR 2.6.a DETECTOR AND CIRCUIT The detector was a Mullard RPY 36 Indium-Antimonide photoconductive cell. The photoconductive element, with a time constant of <2.0 ~St was mounted in a glass dewar vessel and was cooled by liquid nitrogen. It had a sapphire window which trans-mitted to about 1800 cm-l • The element was mounted in the horizontal position with its longest edge parallel to the axis of the incident radiation. A surface aluminised mirror at 450 , reflected the incident radiation onto the element. During operating conditions, see Plate 2.1, a stream of dry air was played on the window to prevent condensation. Precautions were taken to exclude ambient radiation from the detector area, since visible radiation damaged the element. the electrical circuit of the detector is shown in Figure 2.7. The resistor-voltage combination was chosen to give a bias current of about 1.2 rnA, which registered on ammeter A. The two outputs from the detector were fed to the positive and negative inputs of a differential amplifier. This was done, in part, to smooth the noise generated in each resistor Rl and R2 • The variable resistance V.R. was used to prevent current surge in the detector and alllll8ter when tbe device was Sfitched on. The resistor, V.R., was wound from 250 kO to zero, and the current rose from about zero to the bias value. The leads on the circuit were made as short as possible, A 27kJ\ Figure 2.7 Detector Circuit 16jJF TO DIFFERENTIAL INPUT ON OSCILLOSCOPE ZlkJl E RPY 36 -9 RADIATION - 48 -but capacitor, C, was included in the circuit to smooth any stray capaci tance • A range of bias currents were tried from zero rnA to 1.6 rnA and it was found that the noise was minimal at 1.1 rnA. The recommended value was 1.1 mAo 2.6.b OPTICAL ARRANGEMENT The detector and its optical arrangement are shown in Figure 2.8. The optics were set up on a bench separated from the frame holding the shock tube. The slits were of 1 mm width. The filters, of the interference type, were supplied by S.T.C. Limited, and were cut-on or cut-off type. They were used to isolate the region of the fundamental of carbon monoxide. The pyrex glass tube, -1 which had a zero transmittance at 2143 em ,was used for the measurements on the overtone of carbon monoxide. The detector was aligned, without the filter in position, by placing a lamp and slit on the opposite side of the tube. The detector element was then aligned by eye so that the pencil of light produced by the slits, illuminated the detector uniformly. The detector could not be aligned electronically, in the biased condition, since there was a risk of damaging the element. Taking the limiting rays in the diagrams the optical rise time was calculated to be 2 ~s for a shock travelling at about 1.5 mm/~s. Figure 2.8 The Optical System-Not To Scale. -------- e __ -.fl--- --- ---e N --- -- --- --6 I ' I I I ---.:e~ IE. ('t) 6 o - 49 -2.7 ELECTRONICS ARRANGEMENT The electrical arrangement is shown in Figure 2.9. The outputs from the light screens were used to trigger a Venner microsecond time interval meter, type TSA 5536, and meaSUre the time for the shock to traverse the two screens. The minimum pulse peak to peak amplitude required to trigger the timer was 1 V. The trigger levels were adjustable, on both inputs, between 3 V and -3 V. The pulses were also displayed on a Hewlett-Packard 141A storage oscilloscope with a type 142LA Dual Trace Amplifier. This provided a check on the pulses entering the time interval meter. The output from the first light screen was also used to trigger the delay on the Hewlett-Packard 140A oscilloscope, so that the main sweep was triggered as the shock wave approached the central detection station. The output from the detector was fed via a 2 m length of double-shielded two conductor cable to a 1403A Differential Amplifier, which was plugged into the 140A oscilloscope. The amplifier had thirteen calibrated ranges from 100 mV/cm to 0.01 mV/cm. The bandwidth was 0.1 Hz to 400 kHz (0.9 ~s rise time) and the upper and lower limits of the bandwidth were independently selected. The bandwidth used was, upper limit >400 kHz, lower limit 100 Hz. This excluded interference from mains frequency. It was assumed that there was no distortion of the incoming pulse (frequency 2 1.6 kHz) due to the cut-off of the lower frequency components. The output from the Figure 2'.9, Electronics A rrangeme nt o H: o - 50 -detector was displayed on the oscilloscope and the transient was photographed using a Hewlett-Packard 196B, 1:1 ratio polaroid camera. The photographs were taken using 3000 ASA polaroid film. A u.v. light incorporated in the camera provided illumination of the graticules, which were exposed separately. From time to time performance checks were carried out on the oscilloscopes, camera and time interval meter. The high tension for the photomultipliers was supplied from a power supply. type Pm 25OO/R. obtained from Brandenburg Limited. This supplied an operating voltage of 850 V to the photomultipliers. The 10 V power supply required for the twin emitter follower was supplied from a Venner transistorized power supply. type TS 20/C. - 51 -2.8 MATERIALS I k 3,10 . l' d h ' ., d n past wor ~t was rea ~se t at ~mpur~t~es re uced the relaxation time of diatomic molecules. This was particularly so for polyatomic molecules with additional vibrational modes, and also for light molecules such as hydrogen. B.O.C. Limited Grade X gases were obtained, the gases being processed to the maximum feasible purity, under strict quality controlled conditions. The pure gases and gas mixtures were obtained in 1 1. pyrex flasks -2 filled to 101.3 kNm • The glass flasks had a 13.5 mm bore stem and the gas was contained by a 'swan-neck break seal'. The flasks were glass blown onto the system, and the seal could be broken by releasing a plunger. The following analyses on the test gases obtained from B.O.C.:-(1) Carbon Monoxide:-N2 <0.04\ O2 <0.001\ CO2 0.0016% H2 <0.0025\ Hydrocarbons 0.0013% (H) Neon:-He <0.05% N2 <0.0001% O2 <0.0001% CO2 <0.00005% H2 <0.00005% - 52 -(iii) Carbon Dioxide:-N2, O2 , CO, H2 all <0.0005% (iv) Nitrous Oxide:-N2 , O2, H2 all <0.0005% I Mixtures of gases were made up in the pyrex flasks, and were filled to to.I% tolerance. Carbonyl sulphide could not be obtained in Grade X quality, but a cylinder of 97.5% purity was obtained. The gas was purified and the mixtures were made up using the apparatus shown in Plate 2.2. This apparatus was connected to the vacuum system via a Q.V.F. P.T.F.E. bellows, and the pressure was reduced to 133 mNm- 2 • The carbonyl sulphide was then purified by a freeze-puMp-thaw cycle, and a measured amount allowed to expand into the 2 1. storage bulb. Then a measured amount of Grade X carbon monoxide was admitted to the bulb. The pressures were read on a mercury manometer. The flask was sealed off, and the mixture allowed to stand overnight so that diffusion could take place. The flask was then removed and glass blown onto the system. The driver gases were obtained from cOllDerCial cylinders •. In the case of hydrogen the purity was 99.9\, and the nitrogen was 99.5%. Before entering the driver system the gases were passed through a B.O.C. line filter, which removed pal"ticle. gNatar thaD 0.025 mm. PLATE 2 . 2 - 53 -2.9 EXPERIMENTAL PROCEDURES During this section reference is made to Figure 2.3 and Figure 2.4. The operational procedure for a shock experiment is enumerated below:-(i) The flask of test gas was glass blown onto the end of the distillation train at B. That part of the system was always well evacuated before the test gas was released. (ii) The shock tube and pressure gauge section, C, were evacuated by opening taps 5 and 6. It was found that a pumping time of not less than 75 minutes was required in order to obtain the vacua and leak rates mentioned in Section 2.3. (iii) After thorough evacuation tap 3 was closed and the test gas was admitted to the tube. the pressure being measured on the gauges at C. (iv) Taps, 4, 5, 6 were closed and the taps I and 2 were adjusted to cut off the diffusion pump. (v) Taps 7 and 3 were opened and the driver section was evacuated completely by the backing pump. (vi) While (v) was being done, the detector was filled with liquid nitrogen and the graticules of the recording scope were exposed and photographed. (vii) The taps 7 and 3 were closed and tap 1 opened. - 54 -(viii) The laser was switched on, the electronics set ready and the camera shutter opened. (ix) The shock was initiated by increasing the driver pressure. The camera shutter was closed, the laser and detector were switched off. The time taken, from the injection of the test gas to the shock going off, was typically ~ minutes, but at no time was it greater than 8 minutes. (x) The excess pressure in the tube was released and residual gases pumped away. The tube was then raised to atmospheric pressure with nitrogen. (xi) The diaphragm and end plate were removed, and the tube was cleaned by pulling a rag, soaked in diethyl ether, through the tube. This removed fragments of diaphragm material. (xii) A new diaphragm was clamped in position, and the end plate replaced. (xiii) The driver section was roughly pumped out, with the back-ing pump only. (xiv) The test section was then roughly pumped and the diffusion pump was turned into the system, to reduce the pressure further. (xv) The photograph was removed from the camera, developed and fixed. In a typical experiment, the test gas pressures were typically -2 -2 between 5.32 kNm - 0.53 kNm ,and the driver pressure was about 380 kNm-2• - 55 -Finally it is important to stress three major difficulties encountered while attempting to obtain a clean uniform flow. (i) In the present study, it was found that the diaphragm station was one of the most sensitive parts of the tube. It was pointed out in Section 2.4 that the diaphragm be made to burst evenly with little fragmentation, and with no tearing. This was partly achieved by bevelling the leading edge on which the diaphragm burst. Also it was found that a finger tightness was all that was required on the diaphragm clamp, since it was thought that overtightening may h ff d h • • d h b . h • i 11,12 ave a ecte t e openLng tLme an t e urstLng c aracterLst CSt (ii) It was made sure that the tubes were aligned as closely as possible at the joints~ Efforts were made to ensure that the windows andtheir holders fitted exactly as possible to the contour of the tube. This was done by honing them 'in situ'. At the ends of the pyrex tubing there was a slight increase in test section area, 0.35 cm2, which facilitated the clamping of the backing flanges. Thus to avoid any 'steps' in the tube, the metal sections, in glass-metal joints, were machined to meet this requirement. (iii) Before any of the Q.V.F. pipeline was used, it was cleaned with chromic acid, washed thoroughly with distilled water and dried. The metal sections were cleaned with organic solvents. The new tubes were always pumped thoroughly, approximately three days before being used. During operation, the cleanliness of the tube was maintained by pulling a rag moistened with an organic solvent - 56 -through the tube. This removed not only fragments of diaphragm, but also contaminants which came from the driver gas cylinder. Test runs were carried out using nitrogen as a test gas. The emission from the tube was studied using the RPY 36 detector. For the runs in which no cleaning took place, a steady increase in spurious emission was observed. In the cases where the tube was cleaned, and for short pumping times 30 minutes, a large emission was observed. This was ascribed to emission from the cleaning fluid, which was absorbed on the wall and windows of the tube. A longer pumping time reduced the emission to a level which was considered insignificant. The long pumping times necessitated that the backing pump and diffusion pump were maintained frequently. - 57 -REFERENCES 1. R.A. Strehlow, Tech. Rept. A.A.E., ~, Univ. Illinois, Urbana, 1968. 2. A.G. Gaydon and I.R. Hurle, ~e Shock TUbe in High Temperature ChemicaZ Physics, (Chapman and Hall Limited, London) 1963, Chapter IV. 3. W.J. Hooker and R.C. Millikan, J. Chem. Phys., 1963, ~, 214. 4. J.G. Hall, University of Toronto Aeronautics Review, 1958, No.12, 196. 5. W. Bleakney, High Speed Aerodynamics and Jet Pl'opuZsion~ (Oxford University Press, London) 1958, Vol. IX, 159. 6. W. Bleakney, D.K. Weimer and C.H. Fletcher, Rev. Sci. Instrum., 1949, 1Q, 807. 7. D.F. Hornig, High Speed Aerodynamics and Jet PropuZsion, (Oxford University Press, London) 1958, Vol. IX, 203. 8. D.F. Hornig, E.F. Greene and G.R. Cowan, J. Chem. Phys., 1951, 19, 427. 9. J.H. Kiefer and R.W. Lutz, J. c.hem. Phys., 1966, ~, 658. 10. M. Windsor, N. Davidson and R. Taylor, 7th SY"Posiunron Combustion, (Butterworths, London) 1959, 80. 11. C.J.S.M. Simpson, T.R.D. Chandler and K.B. Bridgman, Phys. FluidS. 1967, lQ. 1894. 12. D.R. White, GenemZ E'tectl'ic ReselD'Ch li2bol'atozoy Repol't. 58-RL-1999, 1958. - 58 -3. CALIBRATIONS AND RADIATION CHARACTERISTICS This chapter contains two sections. Firstly, a discussion of calibration experiments, with a particular emphasis on the estimation of the performance of the shock tube. The second section will contain some important features of the observed radiation. 3.1 CALIBRATION MEASUREMENTS 3.l.a SHOCK TUBE It is now well known that non idealities in shock tube flows affect the history of a gas sample. This is reflected particularly in terms of the hot flow time, which is always less than that predicted by ideal shock tube theory. In the ideal case, the shock wave and contact surface move with different, but constant velocities. Since the shock wave moves faster than the contact surface, the hot flow time increases as the shock progresses down the tube. In an actual shock tube flow the presence of a boundary layer causes the shock wave to decelerate and the contact surface to accelerate, thus reducing the hot flow time. The situation in the present case is ahown in Figure 3.1. It is seen that the hot flow ti. is reduced by about one half to one third of the theoretical value, over a range of Mach _.,. .... ,;:.[r HOT FLO/{ TIME I J,lS -tC-x-400 300 200 100 Figure 3.1 PURE CO IDEAL FLOW CALCULATIONS~ VALUES TAKEN FROM PURE CO Tfl..,tCES. o~------------~--------------~-------------. 4-5 5-0 5·5 MACH No. - 59 -numbers. This magnitude of hot flow time reduction has been observed generally. The flow conditions are shown on the (x, t) diagram, Figure 3.2.a, and Figure 3.2.b indicates the growth of the boundary layer in the shock tube. The presence of the boundary layer is due to the loss of heat to the walls by conduction and the loss of momentum by friction with the walls. Early measurementsl with a small diameter shock tube showed that non idealities could be acribed to the formation of a laminar boundary layer. Also, as the length-to-diameter ratio of the tube was increased and the initial downstream pressure was reduced, the boundary layer effects increased. The situation is further complicated by the fact that the shock wave may be decelerating over the period of measurement. These effects serve to influence the temperature and density of the hot flow. In recent years the effects of flow non idealities on shock parameters have been treated theoretically by Mirels.2,3,4 5 6 Strehlow and Belford' have arranged the equations into explicit form, which is of particular interest to kineticists. Since previous workers have calibrated their shock tube,7,8 the present study will also take account of the non idealities and the effect on shock parameters. The method used is ai ven below. The first stage of the calculatiCl1s is to find the real particle testing time, t , flO. the observed hot flow time. The p maximum hot flow time, t , at an observation station is given by:-m a b Figure 3.2 r / "/ / // / / / / // // // - -. -7 - - -~I f /. I // I "'\\ - DRIVER. GAS IN BOUNDARY LAYER. ///1 - TEST GAS IN BOUNDARY LAYER where - 60 -= -1 1 em torr-d = hydraulic diameter 4A A L = -= = = = = L 2 cross sectional rea of tube in em perimeter of the tube. em initial pressure. torr shock Mach number -1 velocity of sound in gas em s 3.1 The equation 3.1 was obtained by fitting Mirels theoretical curve (Figure 6. reference 2) of t /d2 Pl versus Mach number for a number m • of gases. The equation 3.1 fits the res ul ts of Mirels. over a range 2.5 < Hs < 8. for all cases to within 3\. The ideal particle heating time (tp)i is related to the hot flow time tt' by the equation; where (t )./t1 p 1 The real particle t = p = Equation 3.2 being valid for t 1/tm < 0.1. 3.2 - 61 -Errors in the temperature tend to move in the form of a temperature increase behind the shock, during the observation period. The rise in temperature has two causes:-a) the deceleration of the flow because of the boundary growth behind a steady shock. The relationship has been evaluated by Hirels to be:-= 3.3 where 6T = temperature increase 1< T2 = temperature behind the shock where the observation is made 1< H2s = value of Mach number behind the shock (subsonic) b = correction to boundary layer f 2 r (y-l) M_ + 2 M2s = 2YMs2- (:-1) M = shock Mach number. s b) the decleration of the shock wave produces the following temperature increase:-where t = p d = dlnM s dt Y = - 62 -calculated from equation refers to correction due YalMs = (Yal tm +1) -1 velocity decrement (\ m ) 3.2 dlnM s dt to deceleration 3.5 3.6 The derivation of 3.6 is shown in Appendix 1. The two effects may be treated independently and added to give the total effect of boundary layer growth and deceleration on the temperature. (!!.> T 2 TOTAL = Then the correction to the density is given by:-= (¥> / (y-l) 2 TOTAL P2 (P-)av 1 where P = av 3.7 3.8 To estimate the magnitude of the corrections the following factors must be known:-a) Diameter of the tube, a measurable quantity. b) The initial downstream pressure Pl , a measurable quantity. c) The hot flow time tl. measured from the infrared traces of - 63 -emission versus time. Since an average value of the density ratio and the shock temperature were taken, for the analysis of the results, an average value of the hot flow time was used. 'l1lus for the COINe runs, where in most cases equilibrium was reached just before the contact surface arrived, half the hot flow time was taken. In the CO/polyatomic runs, where in some cases equilibrium was reached near the shock front, t .. was taken as half the time fl'Olll onset of the shock to the onset of equilibrium. d) The shock Mach number, this is a value calculated frail the equations set out in Chapter 6, section 2. e) The ideal average shock te.aperature, is also calculated by the equations set out in Chapter 6, section 2. f) The ideal average shock density ratio, see Chapter 6, section 6. g) Velocity decreMnt. In the present study only two velocity measuring stations were possible. and the value of Y had to be estimated with the aid of the emission traces. It was found that for mst of the runs involving the CO/He and CO/polyatome systems, where the Mach nuaber ranged from 4.0 - 6.5 that the velocity decrement -1 vas on average 2\ m • Generally for Mach nUlllbera lower than 4 it was found that the velocity decrement was 1\ .-1. As an example of the COI'Nctiona involftd in a typical shock tube 1'\111, consider a shock wave of "s • 4.3, tN'ftlUDg ia CO in a 2.54 ell di ... ter tube at an initial pru ... of 1.OM k .. - 2 and an - 64 -initial temperature of 300 K with a velocity decrement of 1\ m-l • -1 For this case t m = 0.483 ms and d(lnHs)/dt = 19.1 s • The observations are made at tt = 60 ~s after the passage of the shock. The temperature shift becomes:-= 0.0285 + (0.00047) (19.1) = 0.037 In this case (T2 )av = 1903 K . •• AT = 71.2 K Ap = 0.51 If the velocity decrement is now set at 2\ m-1 , all other factors being the same then:-AT r; = 0.0285 + (0.00047) (38.2) = 0.046 AT = 88.2 K and Ap = 0.63 Thus it is shown that in the two cases above the te.perature is increased by 3.7\ in one case and 4.7\ in the other. The density increase is 9.3\ and 11.6\. Since the increases are 8aall. even in the case of the higher shock attenuation, it was felt that they could be added to the ideal tellperature (T2 ) and the density ratio av (P2/Pl)av' to give the final corrected flow properties in the shock - 65 -tube. Chapter 6 deals in detail with the calculation of the shock parameters of temperature and density. The ideal values of the shock parameters are compared with the corrected ones in Figure 6.6 of Chapter 6. In the present study the corrected shock parameters were always used to calculate the Napier time. How the Napier time is affected by using corrected and \Dlcorrected shock parameters, is shown in Chapter 7. A cOlllputer progra.ae was written to calculate the corrected shock parameters. The prograllllll8 is presented in Appendix 2. 3.1.b RISE TIME OF THE SYSTEM AND PREEMlSSION The limiting response ti .. of the whole system, optica, detector and electronics was deterained by using strong shocks in a C02/Ar mixture, with 5\ CO2• At a temperature of 2000 K the vibrational relaxation of the CO2 was less than 1 liS .10 The observed emission was that of the CO2 uYJIII8tric stretching IIOde -1 near 2350 CII • The rise of the infrared ellission sipal. wa displayed on an oscilloscope, a typical reCord is shown in FilUN 3.3. ~If the rise time is taken to be froa 10' to 90' of the equiliJ,)r1ua emission, then the record gives a rise tiae of about 8... Howe .. r, the record shows a significat (-12\ of total ems.ion) rise b ellission, prior to the ani val of the shock front. I~ 1. tboU&ht tbat preemission can arise fl"Oll two sources. FiNtly. resonance fluorescence, which arises froll the absorption and reemssion of Fig ure 3.3 - 66 -radiation, by the unshocked gas. As will be shown in Section 3.2.d the contribution to preemission from resonance fluorescence is very small. The second and more important pheomena, is multiple reflections from the tube walls which may cause radiation to enter the detection system prior to the shock arrival. 11 Brabbs, using 3% CO2 in Argon, has shown that it is possible to Observe emission from the shock front when the shock wave is about 0.1 m away from the observation point. The walls of the shock tube were coated with colloidal graphite, and Brabbsll then showed that a comparison of the equilibrium intensities for coated and uncoated gave the emission of the coated wall less by 30\. In view of the significant amount of preemission, it was felt that a realistic value of the rise time was =4 lAS. This value was consistent with that obtained, by considering the rise time of the combination:-In this case Tl T2 T3 = = = = optical rise time detector rise time amplifier rise time As mentioned in Chapter 2, section 2.6.b, the optical rise time was =2 lAS. The values of T 2 and T 3 were as stated in Chapter 2 • • •• = = 3 lAS - 67 -This value was not disadvantageous to the CO/Ne runs where the processes were completed within 50 ~s. However, in the case of the CO/polyatomic runs, it restricted the observations to a narrow range of concentrations of polyatomic species. - 68 -3.2 RADIATION CHARACTERISTICS 3.2.a SPECTRA The fundamental vibration-rotation band spectrum, Figure 3.4, was obtained with a gas cell, with sodium chloride windows. Figure 3.5 shows the spectrum of a mixture of 5\ N20 and 95\ CO. The P branch of the asymmetric stretching mode of the N20 is seen to overlap with some of higher rotational transi tions of the R branch of the CO. The overlap was also present with CO2 and COS. The transmission characteristics of the Calcium Aluminate glass is shown in Figure 3.6. The glass had a trans-mission of 45\ in the region of the fundamental of CO. A cut on filter, Figure 3.7, was used to isolate the region of the fundamental of CO, it had a transmission of about 80\ in the region of interest. The effect of the Calcium Aluminate glaas-filter combination was to remove several of the higher rotational transitions in the P branch of the CO i.e. j(19~18), j(20+l9) etc. One further disadvantage of the combination was that it allowed the emiaaion from the poly atomic species to enter the detection system. However, in the experiments up to a 2\ maximum concentration of polyatomic gas was used, and it was assumed that these specie. did not make a significant contribution to the radiation arriving at the detector. Observations CIl the oftrtone of CO were _de in a pyrex tube, ~ -===== ====: :;;; =--= ~ -=---C'i ==-==-.~~ ~~ o Q ------====-~-====--== -::=:::-8 N ,.. 'i: U &n ~ ~ ~ " '&.. ~ ~ ~:J .fil.' u. N z LIJ ~ ~ o 0 o ~o 3~Vlll~NV~! 8 "I"" ~ i Q~ -z NUJ ~ ~ o o Q o 0 ~ " 0 3:)NVf.1.I~SNV~ o 3 o TR.ANSMITTA NeE 0/0 ~ 0- CD 000 6 - 69 -the infrared spectra of this glass showed that there was a zero transmission in the region of the fundamental of CO. 3.2.b RADIATIVE LIFETIME The radiative lifetime, T2 , corresponds to the radiative process:-H , ... H X2 + hv v=O The radiative lifetime can be calculated using a formula derived originally by Ladenburg,12 which relates the radiative lifetime with the band strength:-1 ~ = = where = radiative lifetime s S = integrated absorption at S. T.P •. in -2 -1 em a1:ll v = frequency s -1 The values obtained13 for the band strengths of CO are:-0+1 S = 0+2 S = 1.7 - 70 -Thus the radiative lifetimes are found to be:-Fundamental Overtone = -2 3.288 x 10 s = 1.09 s However the second vibrational level may depopulate in a stepwise fashion:- i.e. CO .... v=2 21 12 COv=l + hv CO .... v=l 10 cOv=O + hv 12 If a harmonic oscillator is assumed then:-k21 = 2klO 10 21 12 -2 •• '[2 = - = 1.644 x 10 s 2 Thus the second level depopulates preferentially via a stepwise mechanism. The relaxation processes by collision are completed wi th 200 },IS or less. Thus the radiati v, process will seN8 to indicate the population of the observed state without affecting it significantly. - 71 -3.2.c CONCENTRATION OF EXCITED STATE AND DETECTOR OUTPUT The rate of population of a particular vibrational level was followed by observing the emission of the infrared radiation from this state, by a suitable detector. It is believed that the intensity of the emission, as recorded by the detector, is directly proportional to the concentration of molecules in the excited state. This is shown in the equations below:-(Ref. 14) D* = 3.9 where D* = A = Ilf = VS/VN = I = (Ref. 15) I = 1< = p = x = 10 = Substituting 3.10 in 3.9 D* = detectivity cm (HzP w-l area of sensitive element cm2 bandwidth of amplifier Hz signal to noise ratio incident light intensity on the sensitive element W spectral absorption coefficient 3.10 -1 -1 em atm ( CO 1) concentration of emitting species v= thickness of emitting layer em intensity of blackbody, VS/VN (Allf)' 10 1< x (COv=l) radiation W at. - 72 -At any wavelength ~d for a fixed path length:-* 10, D , A, 6f, K and x are constant .. 3.2.d RESONANCE FLUORESCENCE Resonance fluorescence arises from the absorption of radiation emanating from the shock front, and the reemission of the radiation, by gas downstream of the shock wave. Considering an infinitely long tube of emitting gas at pressure P2:-) . . ~ dx The increase in light along the incremental length dx is:-dIS = I' ISP2 Kdx 3.14 where Is = incident radiation intensity from shock -2 heated gas in watts em I' = radiation intensity in watts -2 em generated by the emitters in dx K = spectral absorption coefficient in -1 cm -1 atm - 73 -The spectral emissivity of the layer, £, is given by:-(Ref. 15) £ = KP2dx II = IOKP2dx 10 = intensity of radiation in watts em-2 • •• 3.14 becomes:-= of a black body 3.15 where the first term on the right represents the emitted light and the second tem measures the attenuation produced by the absorbers in dx. = Integrating and applying x = 0, IS = 0 = Thus a unit cube of shocked gas emits intensity IS into the unshocked gas and toward the detector. The UIO\Dlt of light absorbed, lAbs is given by:-lAbs = I - I II: S trans 3.16 - 74 -where I trans = -2 transmitted light watts em = downstream pressure atm The amount of light reemitted is determined by the radiative lifetime, T2, see section 3.2.b. Thus the fraction of molecules reemitting:-dN/N where At = = = time available before the shock wave reaches the observation point Assuming about 1/6 of the reemitted light reaches the detector. Thus the intensity of light reaching the detector prior to the shock is:-I = p.S. 6N 11 (l-exp(-Kpl»IS 6 Thus the ratio of light before the shock to the light after is:-I P.S. - = IS 6N N (l-exp(-Kpl)IS 6 IS In the present case 6t = 10-5 s - 75 -For the first vibrational level:-. •• I P.S. r;- = -s 5 x 10 (l-exp(-Kpl» In the case of the second vibrationallevel:-•• AN S - :; 10-N I P.S. r;- = -6 1.6 x 10 (l-exp(-Kpl» 3.17 3.18 Thus for the first vibrational level equation 3.17 shows that at a maximum the amount of resonance radiation reaching the detector prior to shock is 5 x 10-5. that reaching the detector after the shock. As is seen from equation 3.18 the amount of ligbt reaching the detector from the overtone is very much less. In tbe case of CO it was assumed that resonance fluoNscence made little or no contribution to any preemission seen on the infrared trace •• The above calculations aN only very approximate. Detailed calculations of sbock tube pNcursor radiation have been carried out 16 by Lapp, who concluded that in the case of co tbat the effect of resonance fluorescence was only of slight magnitude. - 76 -REFERENCES 1. R.E. Duff, Phys. Fluids, 1959, ~, 207. 2. H. Mire1s, Phys. Fluids, 1963, ~, 1201. 3. H. Mire1s, A.I.A.A. Journal, 1964, ~, 84. 4. H. Mire1s, Phys. Fluids, 1966, 9, 1907. 5. R.A. Strehlow and R.L. Belford, Teoh. Rept. A.A.E., Univ. Illinois, Urbana, 1969, ~. 6. R.L. Belford and R.A. Strehlow, Ann. Rev. Phys. ahem., 1969, 12, 247. 7. W.J. Hooker, Phys. Fluids, 1961, ~, 1451. 8. W.J. Hooker and R.C. Millikan, J. ahem. Phys., 1963, 1!, 214. 9. J.N. Fox, 1.1. McLaren and R.M. Hobson, Phys. Fluids, 1966, .2" 2345. 10. C.J.S.M. Simpson, K.B. Bridgman and T.R.D. Chandler, J. ahem. Phys., 1968, ~, 513. 11. T.A. Brabbs, J. c.hem. Phys., 1968, ~, 1433. 12. R. Ladenburg, Z. Physik, 1921, ~, 451. 13. W.S. Benedict and E.K. Plyler, Bneztgy 'htanBfel' In Hot Gases. National Buzteau of Standaztds. Washington. D.C., 1954, p. 57-13. 14. P.W. Kruse, L.D. McGlauchlin and R.B. McQuistan, EWmtmtB of Inft'azted Teohnology, (J. Wiley and Sons, New York) 1962, 265. 15. S.S. Penner, Quantitative Mo1.eau1.aI' Spectrosoopy and GaB Emissivi.ties, (Addison-Wesley Publishing Company Inc., Massachusetts) 1959, Chapter 1. - 77 -4. VIBRATIONAL TRANSFER PROBABILITIES 4.1 INTRODUCTION Exact solutions to rigorous calculations of the transition probability, P, for any process is not possible. This is due to the fact that realistic intermolecular potentials are not yet available, and some of the equations involved are too complex to be solved by computer. In the three quantitative approaches, i.e. classical, semiclassical and quantum mechanical, approximations are made to ease the problem. This chapter will briefly outline each of the methods, with special emphasis on those used to compare the present results with. There will be no deta~led derivations, since they are well 1 2 3 4 documented elsewhere. • " The relative merits of the assumptions and limitations of each method have been described. 2,3,4 Also mentioned are empirical methods, and those invol 'Ying long range forces. Energy transfer between translational and vibrational degrees of freedom, during a non reacti va collision can be described in te~ of the Ehrenfest Adiabatic Principle •. This principle states that the duration of the collision must be short ca.pared to the period of vibration, if energy exchange is to take place. Tbe classical situation is tmderstood by considering a vibrating _chanica! spring, whieb if - 78 -given a sudden blow will be left oscillating vigorously. The other extreme is that of a 'soft' force where the oscillator is able to adjust to the collision, during its period of vibration. This type of collision is adiabatic. The quantum mechanical case is that of a force field perturbing an oscillator. If the force field is only slowly varying then-the oscillator can readjust to the perturbation, whereas if the perturbation is changing rapidly, then there is a high probability of a quantum transition. The physical situation which applies to all three I-dimensional approaches, is represented in Figure 4.1. Atom A initially at infinite distance, approaches the simple harmonic oscillator Be, along their internuclear axis. The equilibrium internuclear distance of Be is y, and the displacement from equilibrium is X. The distance, r, between the centres of gravity of A and Be is:-r = d + X + y/2 The total energy of the two species will be cooserved throughout the collision and it is the relative kinetic energy which is converted into vibrational energy, the centre of gravity of Be being taken as S 6 stationary. Full details of the collision problem have been described. ' 5 7 Rapp' has compared the three approaches, and found that they give identical results at high velocities of approach when the amount of net energy transfer to the oscillator is 8mall. Figure 4.1 Two Particle Problem For A-Be Collisions - 79 -4.2 PARAMETER RELATIONSHIPS In the following sections the parameters involved in vibration excitation are as follows:-(a) P10' transition probability (b) ZlO' collision number (c) k lO ' rate constant (d) T. relaxation time The subscript 10 refers to the de-excitation process. Similar parameters, subscripted Ol,exist for the excitation process. In the case of a 2 state model, where the molecule has 2 vibrational levels then:-t = However in a multi state model:-T = At equilibrium = exp(-hv/kT) Then for T « hv/k T = 1 k lO Also P10 = t = Similarly P10 = where t = c n = P = n = MA, ~C = r = 0 0(2,2) = T = - 80 -l/ZlO ZlO t c klO t c n/1.271 x P viscosity liP 1 atmosphere pressure x r2 x 107 o masses of colliding species constant in L-J potential values viscosity data tabulated by Hirschfelder temperature K The values of 't were obtained froll a computer progra1llll8 see c Appendix 2. - 81 -4.3 CLASSICAL ME'lHOD The purely classical approach is where the external and internal coordinates are given by the classical equations of motion. The energy levels of the oscillator are considered to be a continuum, and only the total energy transfer is calculated. The most notable workers in the field, Benson and co-workers,8,9,lO,11,12 base their method on a 1 dimensional collision model, using a Morse interaction potential. The method has been extended to treat . ib . 13 d ib . ib ti 14 ch rotat10n-v rat10n an v rat1on-v ra on energy ex ange. Some of their earlier work gave erroneous results. lO and the more recent papers have been criticised by Rapp and Kassal. 2 Classical calculations have been carried out also by ParkerlS and Wilsonl6 ,17 and co-workers. The results of nUlllerical classical calculations on a large number of systems, \Ulder various conditions, are given by 18 Kelley and Wolfsberg. - 82 -4.4 SEMICLASSICAL METHOD In this method the molecule BC is treated as a quantum mechanical system with discre te energy levels. The motion of the incident particle A is treated classically. The first approaches 19 20 to the problem were made by Zener and Landau and Teller. Landau and Teller assumed that the interaction potential between the colliding species was exponential and of the form:-VCr) = A exp(-r/l) where VCr) = potential energy A = constant r = distance .between species nm 1 = range of repulsive forces nm They then showed that the probability of vibrational de-excitation, PIO :-PIO a exp(- Duration of Collision) Period of Vibration PIO = C exp(-l v/v) Thus the probability of energy transfer is favoured by a high 4.1 relative velocity, v, and by a low value of the vibration frequency, v. To obtain P 10 as a function of temperature equation ".1 was integrated over all possible velocities. The following expression was obtained:-- 83 -= 4.2 where Kl and K2 are constants which depend on collision across sections. Also NPIO = where N = number of 1 t collision (see section 4.2) per unit time klO = rate constant for deactivation t = relaxation time 1 NP IO (1 - exp(-h~/kT» = t , and N a pT-~ Thus equation 4.2 becomes t = K{ Tt expO<2/Tt) p(l - exp(-h~/kT» Therefore:-_. Int a T 3 approximately The original calculations have been extended. in particular by 21 22 23 24 Cottrell and Ream, Rapp and Sharp • and Talcayanagi. The ...l T 3 dependence was given by IIOst approaches. and in general the relaxation time can be expressed as:-lnT ...l. = A + BT 3 - 84 -In the process:-AB + CD ---=:::.. AB + CD (v=l) (v=Orc=-- (v=O) (v=l) there is a non zero, AE, interconversion of vibrational and translational energy. A semiclassical calculation, of the probabilities of near resonant vibrational energy transfer has been accomplished by Rapp and Golden. 25 ,26 The calculation was made with a two state approximation (i.e. 0, 1 and 1, 0), and using an exponential potential between B and C. The result is:-where = 3.7 T = temperature K. AE = difference in vibration frequency of the colliding species, measured in ca-1 4.4 Thus increase in AE tends to decrease the transitim probability for the exchange of vibrational energy. See Figure ".2. The above method 27 has been expanded to a Ilultistate procedure. Figure 4.2 102 Vibration Vibration Transfer Probabilities -3 10 PD01 10-4 -s 10 008 009 0-1 (TEMP.I K ) -1/3 - 85 -4.5 QUANTUM MECHANICAL MEniOD This section is divided into three parts. Firstly, a discussion of the method, secondly a discussion of how the molecular interaction potential may best be refined, and thirdly a comparison of tabulated and computed results. 4.5.a THE METHOD In this treatment the approaching and retreating atom is represented by a wave. If the wave corresponding to the retreating atom has the same wave number as the incident wave, then there has been no change in the internal energy of the oscillator. However, if the retreating atom has a different wave number, then there has been a change in internal energy. This method of treating the collision problem, in l-dimension, has been described in detail,l,28,29 and Jackson and Mott 30 have dealt with the solution of the integrals involved. The three dimensional molecular model is IIOre realistic. This problem has been treated and refined by Herzfeld and his 31 32 co-workers.' In this case the approaching atoll and the receding atom are represented by a system of spherical waves. The oscillator is treated as being spherically s,-tl'ical. that is the vibration is considered as a syaaetrical increase and decrease in the size of a • - 86 -sphere. In the original calculations an exponential potential was assumed between the colliding species, since it made the mathematics simpler. However, this type of collision problem could not be solved exactly, but only by an approximate method. At low collision velocities, where the total transition probability is small, the approximate solution is found by using the First Order Distorted Wave Approximation (F.O.D.W.A.). The final Schwartz-Slawsky-Herzfeld (S.S.H.) equation for estimating vibration-translation probabilities is:-where = x 1.017 x Z z Z x Zt x Y{2.2) o osc r r = distance of closest approach c = length constant in the Lennard-JonesEquation £ = well depth in Lennard-Jonespotential e = hv/k = characteristic temperature of the vibration Y(2.2) = a viscosity factor Z = orientation factor 0 Z = vibrational collision number osc Ztr = translational collision number The method has been refined further so that the probabilities of vibration-vibration energy transfer may be calculated. The details - 87 -of this equation are given in Appendix 2. For this type of calculation to be successful, a realistic estimation of the quantity, 1, the range of repulsive forces must be obtained. The value of 1 depends on the choice of intermolecular potential. The fitting of the purely repulsive potential, used in the derivation, to a reasonable potential is shown in the next section. ~. 5.b niE INTERMOLECULAR POTENTIAL At short intermolecular distances where repulsive forces dominate, the foregoing treatment is based on a purely repulsive potential. For real gases the field between two molecules is usually represented by the Lennard-Jones{L-J} potential. This potential contains an attractive, as well as a short range repulsive component. The attractive coaponent accelerates the particles during their approach. The equation is:-VL-J (r> [ r 6 (f> - r 12] (-;.> where I: = potential well depth VL_J (r) = potential energy with a minimum at r = 2* ro r = constant 0 The constant -£ is introduced into the exponential potential, to make the slopes of the two potentials compatible at saall distances, which are important in determining energy transfer probabilities. - 88 -The range of collision velocities has a minimum value of r at r c • There are two recognized methods of fitting the two potentials. Method A was developed originally by Schwartz, Slawsky and 29 Herzfeld. In this case the repulsive potential is made tangent to the L-J potential at r. Thus at this point the potential energy c and slopes of the potentials are set equal. It is found that:-r o = where Em = effective energy of collision 33 Method B is due to De Wette and Slawsky, who fitted the exponential potential by requiring that the curves have 2 points in common. The first point is as before, where both curves have some potential energy, and the second point is where VCr) = 0 and r = rAt From this condition it is found that:-.. The fitting of the exponential potential to the L-J potential is shown in Figure ~.3~ The value of ro/! as a function of 10glO EllIE for both methods is plotted in Figure ~.~. It is seen that .. thod B gi ves a reasonably constant value of r olt over a wide range of 10glO EllIE. Also, method B has been used by many workers since it Em r Vcr> F iCJ ur(Z 4.3 Fi tting Of Expon(Z ntia I To L-J Pot(Zntial I I \ l \ I , , \ \ \ \ \ --- METHOD A METHOD B \ o ~~-r--~~~----~----------------~ r -E 25 .!Q. I 20 J9 17 16 15 Figure 4.4 Comparison Of Methods A & 8 METHOD A i CO(300K) CO(772 r CO(2912K) 1 CO(l953K) 10 T------------.------------~----------~ o 1·0 2·0 - 89 -gives a smaller value of 1, which has usually led to better agreement with experiment. In this study method B was used. The important quantity obtained from the curve matching was the range of repulsive forces 1. 4.5.c COMPARISON OF COMPUTED AND TABULATED RESULTS A computer programme to deal with the S.S.H. calculations was devised. Details of the prograJlllle, together with the calculational method are set out in Appendix 2. The value of 1 was calculated by an iterative procedure, and for IIIOst systems lay between 1.5 and 2.5 nm. Figure 4.5 compares the computer results with those tabulated by Herzfeld and Litovitz l for method B, and for vibratian-translation energy transfer. The molecular constants used were obtained from 34 Hirschfelder et al, and are fOlUld in Table 4.1. The calculated values of 1 and Em/£ are compared to the tabulated valuesl in Table 4.2. It is seen that the agreement for the value of 1 is good, the discrepancy between the results never being greater than 2\. Tbe values of E1D/£ agree also to within 3\ of each other. Figure 4.5 shows a difference between the tabulated probabilities and the calculated ones. This is due to the selection 1 of a value for Z. Herzfeld chose a value of 3 for Z • whereas the o 0 present calculations find Zo from the following:-Molecule O2 N2 C12 CO2 TABLE 4.1 - -1 \I ern 1548.61 2330.22 562.64 667.0 Elk K 34.33 113.2 36.81 91.5 41.15 357.0 39.96 190.0 TABLE 4.2 Co 1.1.i ding Temperature Repulsion Parameter - lnm EmIt Species K Computed Herzfe1.d Computed Herzfeld Resu1.t & Litovit·z Result & Litovitz 2B8 1.7B7 1..787 25.21 25.35 O2-02 1372 1.B34 1.B33 70.66 71.33 1953 1.841 1.840 89.64 90.5 I 778 1.970 1.971 7B.9 79.39 H2-H2 4630 1.992 1.991 262.27 262.6 288 2.037 1.998 5.80 5.57 C~-C12 1000 2.119 2.090 13.29 13.15 1372 2.158 2.106 16.50 16.32 288 2.008 2.011 10.29 10.30 CO2-CO2 1.000 2.076 2.076 23.48 23.47 ----PI- O -3 10 -4 10 -S 10 -6 10 F=igur~ 4.5 " ...... " " " , - Herzfeld's Data. ----CompuleciData. " " ..... " , , " " " , " " " " °iO; , , , , -, ,N.:N2 IO~r---------~--------~----~·~ 007 0-08 009 ( TEMP./ K ;1/3 where - 90 -Z = 2.5 x (1/L)2 + L/l o L = interatomic distance of Be 1 = range of repulsive forces 4.6 From this equation it was1bund that for most systems Z ranged o between 5 and 10. Equation 4.6 is proceding toward a more realistic value for Z , but the recent work of Shin35 has refined it still o further. In conclusion, it is believed that the present computational method is sound, since the values of important molecular para_ters can be reproduced accurately. Although discrepancy exists between the probabilities, computed and tabulated, the present values may be more accurate due to the calculation of Z. Thus the procedure o contains no adj ustable parameters. - 91 -4.6 EMPIRICAL METHODS 36 Losev and Osipov suggested an empirical method for calculating vibrational relaxation tiMs. This was developed by 37 Millikan and White. It was found that linear plots of 10glOT ..J. (at unit pressure) against T 3 had a common origin for many diatomic gases of .imilar reduced ..... The relaxation tiMs of the gase. over a wide range of teaperatures are given by the fonnula:-where ~ = the reduced ... s of the colliding .ystem e = characteristic temperature of the'vibration. The equation is valid for the following lilli t.: -1.75 < p < '127 310 K < e < 3395 K 280 K < T < 8000 K Thi. relationship applies only to vlbNtion-traalation --raY exchanp. and take. DO account of vibration-vibration exob ..... - 92 -4.7 LONG RANGE INTERACTION MEnIODS De Wette and slawsky33 made a theoretical investigation of two cases in which long range forces are strong. They studied the dipole-dipole interaction of 2 HCl molecules and the quadrupole-quadrupole interaction in the case of 2 CO molecules. The inter-actions were divided into two possibilities a) long range (adiabatic) interaction, b) short range (nonadiabatic) interaction. The work confirmed that the effect of the long range forces was to accelerate the approaching particle, and this would affect the transition probability directly. The probability of vibration-vibration energy transfer between infrared active modes, due to dipole-dipole interaction, has been considered by Mahan. 38 The resonance interaction was d d··· h d· -3 Th cal ul i assume to 111U.n1S at great 1stances as r. e c at ons showed that; a) the resonant transfer of energy may be IDOre rapid than is indicated by the calculations using the repulsive potential, b) the probability decreases as the temperature increases. whereas the probability due to repulsive forces increases with increase in temperature. 39 These calculations have been extended by Yardley to the case of non-resonant vibration-vibration transfer, due to dipole-dipole interactions. Shama and Brau40 ,41 have shown that anoaaloua behaviour of the N2-C02 system below 1000 K, can be explained in terms of a - 93 -long range dipole quadrupole interaction. The probabilities calculated by this method show a negative temperature dependence, in contrast to the behaviour predicted by the S.S.H. theory. Recent experimental evidence has been provided by Roach and Srnith,42 who have worked on the N2-N20 system. However, it is assumed that above 1000 K short range repulsive forces will dominate the translation-vibration transfer process. The method will not be considered in the present study. • - 94 -4.8 DISCUSSION OF METHODS It is not easy to compare the theories with experiment, since vibrational energy transfer is extremely sensitive to the slope of the repulsive part of a potential. Thus the value of the range of the repUlsive force, 1, is not well known. Herzfeld43 found that a better evaluation of .. could be obtained by fitting the exponential potential to an L-J potential. A double exponential potential of the Morse type has been used,44 and Monchick and Mason45 have modified the L-J potential to a 12-6-3 potential which takes account of dipole-dipole interaction in polar molecules. Molecular beams enable measurements of the repulsive forces to be made more directly. Mueller and Marchi 46 have concluded that the L-J 12-6 potential does not provide a sound basis for the observed scattering pattems and a broader minimum is required. Further studies of this nature will lead to a better understanding of the interaction potentials between colliding species. The classical calculations of ParkerlS and Benson and 12 Berend have fitted their data to the Landau-TeUer theory, and have obtained reasonable agreement. The plots of Log P 10 are not siaple ...1. linear flmctions of T 3 over a large temperature range, but tend to show an 's' shaped behaviour. ODe of the reasons why this theory is not well applied is the need for careful COIIputing procedures to solve the integrals involved. - 95 -The S.S.H. theory has been widely applied since the equations are presented in an explicity soluble form. It is possible to calculate the relaxation times of pure gases and gas mixtures to within an order of magnitude, and log T is a linear ...l function of T 3. There are notable exceptions, where the relaxation times predicted by S.S.H. are different froll the experimental values. H247 relaxes abnortDally rapidly, as does NO. The case for NO has been explained48 by assUlling the forwation of an electronically excited state. Results for the hydrogen 49 halides, and in particular the HI/HI system show that Z is several powers of 10 smaller than the S.S.H. prediction. In the case of the hydrogen halides and some hydrides there is the possibility of vibration-rotation rather than vibration-translation energy exchange. Some 25 IDOlecul.es have been fitted to a seai-quantitative vibration-rotation mode150 with reasonable succe.s. However, this theory is unable to fit the hydrogen halides well. Vibration-vibration energy transfer probabilities were calculated froll equation 4.4, section 4.4. This method has had reasonable success in giving close agreeaent with rooa te.,.rature data5l and more recent high teaperature data,52 with the co-ao .cd N2 -NO systems. However the work of Vhi te 53 OIl 12-02 vibration 54 exchange had less agreement. Sato et al have studied v1br.tiClllal energy exchange, by the shoclc tube technique, and the experi..ntal - 96 -data agreed well with the equation derived by S.S.H. for vibration-vibration exchange. Recently, Riley and Kuppermann55 have made numerically exact quantum mechanical calculations of collisional energy transfer probabilities. They found that the F.O.D.W.A. used in derivation of the S.S.H. theory, was very accurate for the (0, 1) .. (1, 0) process. The one quantum translation-vibration processes were found to be an order of magnitude too large. Figure 4.6 compares the probabilities of vibration-vibration energy transfer calculated from equation 4.4, and those calculated by the S.S.H. method. Specific systems were taken for the S.S.H. calculations, as shown in the figure. The two approaches compare most favourably with increase in the value of 6E, the energy discrepancy. It is reasonable to assume that the quantum mechanical method should compare best with experimental data. The Millikan and White37 correlation has been crit~cised12 on the basis that the co~lation ignores some low temperature data not lying on the extrapolated curves. The equation contains no teZ'18 . relating to the intermolecular potential, and a coBlOn value for 1 is asslUDed. However as pointed out previously the transition probability is sensitive to L. The correlation does not work for H2, D2 the hydrogen halides and polyatolllc a.olecules in general. The correlation is successful in predicting vibration-translation probabilities for diatomic a.olecules. -3 10 -~ 10 Figure· 4.6 .... ~ ... ~·S95cm· \' :o~O) ,,&cP,oocm-' " ~ iN .... " .... "-....... ... ....... - SCmic.ialsical Theory. -s ---5.5, H. Theory. 10 " ... ..... .... .... ..... .... .... .. .... .... .... 007 0-08 009 0'1 (TEMP.I K)-I/3 .. - 97 -REFERENCES 1. K.F. Herzfeld and T .A. Litovitz, AbSOl"ption and IXsptll'sion of uttl'asonic Waves, (Academic Press. New York) 1959. 2. D. Rapp and T. Kassal, Gl'zDrrn:zn Reseaztch ~t:lrHmt RtlpOl't. 1968, ~-~. 3. J .L. Stretton, Tztansfeze and StoNgtl of EMl'W by NoU,Clil.tls, (Wiley, London) 1969, Vol 2, 58. 4. T. L. Cottrell and J.C. McCoubrey, No'l.tlcu'l.al' Entll'gJ/ Tztansftll' in Gases. (Butterworths. London) 1961. 5. D. Rapp. J. Chem. F*ys •• 1964. ~. 2813. 6. K. Takayanagi. Advczn. At. Mo'l.. F*ys •• 1965 • .!. 149. 7. D. RapP. J. Chsm. Phys., 1960. ,E, 735. 8. S.W. Benson. G.C. Berend and J.C. Wu, J. ahem. Rnys •• 1962. !Z.. 1386. 9. S.W. Benson, G.C. Berend and J.C. Wu. J. ~. ~s., 1963, !!!O' 25. 10. S.W. Benson and G.C. Berend, "T. CIumt. Plal/s •• 1963. 39, 2777. -11. S.W. Benson and G.C. Berend. "T. CIte",. PIt"s. , 19 sa. , !2, 1289. 12. S.W. Benson and G.C. Berend, "T. CIumt. Ph1Ja. , 1966, 1M. 470. -13. S.W. Benson and G:C. Berend. "T. cn.. P1qJs. , 1966, !!!.' 4241. S.W. Benson and G.C. Berend, J. Chat. Plal/s. , 1969, 51, 1480. -15. D.G. Parker, Phys. of nrAd., 1959, 2, 1M4. -16. E.B. AltemaD and D.J. Wilson. "T. CIaa-. Ph,,8., 1965,.!a, 1951. - 98 -17. R.J. Harter, E.B. Alterman and D.J. Wilson, J. Chem. PhY8., 1964, !Q" 2137. 18. J.D. Kelley and M. Wolfsberg, J •. Chem. PhY8., 1966, ~, 324. 19. C. Zener, Phys. Rev., 1931, lr, 556. 20. L.D. Landau and E. Teller, Z. Phys. SCMJBtunion., 1936, lQ, 34. 21. T.L. Cottrell and N. Ream, fitanB. FawzikqJ Soc., 1955, !!, 1453. 22. D. Rapp and T.E. Sharp, J. ahem. P.hys., 1963, ~, 2641. 23. T.E. Sharp and D. Rapp, J. Chem. PhY8., 1965, ~, 1233. 24. K. Takayanagi, Pl"ogl'. ThsoNt. PhII8. (lflIotoJ, 1963, Supp1. 25, 1. 25. D. RaPP and P.E. Golden, J. Chem. PhYB., 1964, ~, 573. -Ibid, 1964, ~, 5123. 26. D. Rapp, J. ahem. PhYB., 1965, !!!, 316. 27. A. Zelechow, D. Rapp and T.E. SharP, J. ahem. PhJI •• , 1968, ~, 286. 28. N.F. Mott and H.S.W. Massey, DI. 7'MDl'II of Atomic Co1.1.iBi0n8. (Oxford University Press, Oxford) 1949, 2nd EdD. 29. R.N. Schwartz, Z.I. Slawsky and K.F. Herzfeld, J. CIuIm. PhlJ •• , 1952, lQ, 1591. 30. N.F. Mott and J.M. Jackson. PJtoc. 1lDI/. Soc •• 1932. 137A, 703. -31. R.N. Schwarz and K.F. Herzfeld, ,J. ChaI. Phllfl •• 1954. ~, 767. 32. K.F. Herzfeld, J. CIunn. Ph1J... 1967, .!1. 743. F.W. de Wette and Z. I. 5 lawsky, PhYBiaa, 1954,' 20. 1169. -33. 34. J.O. Hirschfelder, C.F. Curtiss and R.B. Bird, M)t.owZaP - 99 -Theory of Gases and Liquids, (Wiley, New York) 1954. 35. H.K. Shin, J. Chem. Phys., 1967, !I, 3302. 36. S.A. Losev and A.I. Osipov, Soviet Phys. Usp. (EngZish Trans],.), 1962, ~, 525. 37. R.C. Millikan and D.R. White, J. Chem. Rnys., 1963,~, 3209. 38. B.H. Mahan, J. Chem. Phys., 1967,~, 98. 39. J.T. Yardley, J. Chem. Phys., 1968, lQ, 2464. 40. R.D. Sharma and C.A. Brau, Rnys. RSfJ. £StUN, 1967,.!!, 1273. 41. R.D. Sharma and C.A. Brau, tI. CJumr. Phye., 1969, 1Q., 924. 42. J.r. Roach and W.~. Smith, tI. Chsm. Rnys., 1969, !2, 4114. 43. K. r. Herzfeld, Thel'fTr)dynamics and Physics of M:ztUJI. (Princeton University Press. Princeton. N.J.) 1965. 44. S • L. Thomson, J. Chem. Rnys., 1968 • .!t!. 3400. 45. L. Monchick and E.A. Mason. tI. Cham. Phye •• 1961, ~, 1676. 46. C.R. Mueller and R.P. Marchi, tI. ChBm. Phys., 1963. l!, 745. 47. J.H. Keifer and R.W. Lutz, J. Cham. P*ys., 1966, ~, 668. 48. E.E. Nikitin, Opti1ca SpektJ'oBk., 1960, !. 8. 49. C.C. Chow and E.F. Greene, tI. CIwr!t. Phye., 1965,.!!!, 324. 50. C.B. Moore. J. ahem. E*ye., 1965,~, 2979. 51. N. Basco, A.B. Callear and R.G.W. Norrish, PJtoc. RON. Soo. (LondOn) 1961, A260, 459; 1962, A269. 180. - -52. R.L. Taylor, M. Camac, R.M. Feinberg, Sylp. Combut. 12th, 1967, .!!!' 1966. - 100 -53. D.R. White, J. Chem. Phys., 1968, ~, 5472. 54. Y. Sato, S. Tsuchiya and K. Kuratani, J. Chem. Phys., 1969, iQ, 1911. 55. H.E. Riley and A. Kuppermann, Chem. Phys. Letters, 1968, .!' 537. - 101 -5. RELAXATION EQUATIONS The following chapter will contain details of the mechanisms of the reactions used in this study, their respective kinetic equations and the solution of these equations. The introduction will deal with the mechanisms. Then methods will be presented which deal with a) a binary mixture of diatomic and inert gases, b) a binary mixture of gases with the possibility of a complex reaction. 5.1 INTRODUCTION In a pure diatomic gas, A, of two vibrational states, neglecting radiative loses, the reaction for excitation and de-excitation of the first vibrational level is:-AA kOl AO + A .. Al + A ---kAA 10 xAO xAl where A = diatomic molecule. subscripts 1. 0 refer to 1st excited state and ground state respectively. k~ ,k~ = bimolecular rate constants for deexcitation and excitation respectively - 102 -XAO ' xAl = concentrations of ground and excited state of A respectively The addition of an inert gas, B, to the system, introduces another reaction possibility:-AB kOl AO + B ---..:... Al + B ~AB klO AB AB bimolecular rate where kOl' klO = constants xB .- concentration of B Also:- xA = xAO + xAl ~ = ~ x = xA + xB (l-X) = xA -X = 1I01e fractico x • X •• = ~/x In this case the energy exchange can OCCUl' by collision of A vi th A or A with B. 111e overall rate constant for any aixtU1'e 1s 11 ... n by:-= 5.1 Assuming klO > kOl - 103 -where :: rate constant for a pure diatomic gas k~ :: rate constant for 1 molecule of A in pure B. Equation 5.1 has a linear dependence upon concentration, experimental results may be extrapolated to give k~. The excitation of the upper levels of a diatomic molecule is an example of a series mechanism, see Figure 5.l.a. The energy is fed into the molecule by translation, so that the lowest vibrational mode is vibrationally excited. This energy is then redistributed to other modes. kOl etc. In contrast to the series mechanism, the parallel _chani .. is the case where the vibrational modes can ezcbange energy with translatioo independently. Examples of this are 1) a pure polyatome molecule, which has several vibrational modes. ii) that of a binary mixture of diatomic or polyatoa!c gues where both species interchange energy with translation independently. In both cases the molecules relax towards equilibri\Dll without disturbing each other and two 2 ~O Figure 5.1 Relaxation Mechanisms. 2 O-.,..L--- o II o. SERIES ~CHANISM b. PARALLEL MECHANISM o 0 A B c. COMPLEX MECHANISM - 104 -relaxation processes are distinguishable. In the examples just considered the processes were those involving vibration-translation energy exchange. However, it is possible, in a binary mixture of diatomic or polyatomic gases, for a complex process to take place. The complex process involves vibration-vibration energy exchange. The reactions describing the process are given below. The colliding species are A and B (now diatomic), subscripted 1 if they are vibrationally excited. The rate constants are kl , k2 etc. for de-excitation and k_l , k_2 etc. for exe! tation • 1) vib.-trans. 2) vib.-trans. 3) vib.-vib. (complex process) vib .-trans. 5) vib.-trans. - 105 -The concentration of the various species are:-= = = = ~O xA = xAl + xAO xB = ~l + ~O x = xA + xB l-X xA X ~ = = -x x where X = mole fraction It is convenient to denote 1) as slower than 4) i.e. pure A has a longer relaxation time than pure B. The mechanism is shown schemat-ically in Figure S.l.c. ,Reactions 1, 2 and 4, 5 are si~le excitation and de-excitation of A and B. However reaction 3 couples the equilibria of 1, 2 and 4, 5. Thus the relaxation times are not only of 1, 2 and 4, 5 but independent linear combinations of the_, which gifts. sinale relaxation behaviour. The two methods used to analyse the _chanis. are &i Yen in the next section. - 106 -5.2 KINETIC APPROACH The basis of this method is to write down the mechanism. derive the rate equations and integrate them in terms of the concentration of a species which is measured experimentally. 5.2.a 1st VIBRATIONAL LEVEL The excitation to the first vibrational level can be considered in terms of two models. 5.2.a.i 2 STATE HODEL This model has two vibrational levels. the ground and the first. The reaction is written generally, so that the collision partner M is either another molecule A. or an inert atOll B. 'lbe concentration of M. x. is the sum of the concentrations of A and B. For clarity the concentration of AO and Al will be designated xo and Xl respectively. > < • • • = • • The billOlecular rate Calstant kOl and k 10 lIay be replaced by the concentration dependent quantities kOl and k10 equal. to k~l x and , klO x respectively. • •• dx l dt - 107 -= when t = 0, xl = 0 (a typical condition prior to a shock experiment) the equation integrates to:-5.2 Applying the equilibri\DII conditions (shocked gas fully relaxed) t -+ .. , Xl -+ x· 1 5 • 2 becomes:-Xl .. = kOl (kOl + klO) Xo • • • 5.3 The relaxation time T, T = 5.~ Euqation 5.3 gives an expoileJitial approach to equilibri\llo If equaticm 5 .3 is rearranged:-- 108 -= t T 5.3.a -1 Plotting the L.H.S. versus time. a straight line of slope t is obtained. 5.2.a.ii HARMONIC OSCILLATOR HODEL This model was first treated by Landau and Teller.l It is assumed that the molecule is a harmonic oscillator, and that transitions can only occur between adjacent vibrational levels, of which there are an infinite number. , , k. 1 . )- .) kj • j +l H + A. 1 >' A. + M --:.... Aj +l + M )- -.:;-- ) ~, k~+l,j k. . 1 ).)-A further condition is that the rate constants have a simple relation to each other:-k. 1 . J+ ,J k = (j+l) j,j-l j The equation for the change of popUlation of the j th level:-= k x - k x j-l,j j-l j,j-l j 5.5 and for the lowest state:-- 109 -= 5.6 The change in the total number of quanta is obtained by summing equations 5.5 and 5.6 over all levels. Then the relaxation time is found to be:-T = In both the two state model and the multistate model the following relationships may apply:-= exp(-hv/kT) so that providing:-T « hv/k the relaxation time in both cases reduces to:-t = 5.7 The rate constants are temperature dependent so tbat the equation 5.7 is only valid for oscillators relaxing in a canatant temperature be.t bath. In most shock tube work this assumption bas been macle, even for pu~ gases with vibrational modes. - 110 -S.2.b 2nd VIBRATIONAL LEVEL Three mechanisms explaining the population of the second vibrational level are possible. S.2.b.i DIRECT MECHANISM This reaction is analysed as a 2 state system, see section 5.2.a.i. The result 1s:-= 5.B 5.2.b.ii STEPWISE MECHANISM • kOl AO + M ~ Al + M ,. klO • k12 Al + M ". :::.. A2 + M k2l The rate equations &1'8:-dxo klO xl - kOl Xo - = dt dx l "(it dx2 cit - 111 -= = These equations were solved by the 'Operator Method'. full details of this technique are available. 2 The basis of the method is to let the derivative. d/dt. be designated by p. the operator. Treating P as a constant. many differential equations are reduced to ordinary algebraic equations. The equation can ~hen be solved for the unknown in terms of P. If the transform is Jcnown then the final solution can be derived. The operations of setting up the transfortaed functions and replacing the latter by the originals are based on the Laplace transform. There are a large number of tables (specially for Jcinetics)2 relating the transform to the original. The result of applying this method to the above rate equations. with the condition that x2 = 0 when t = 0:-= + Applying the Landau-Teller result:-l = = exp(-t(JcOl +klO» (JclO +JcOl> 5.10 - 112 -Thus equation 5.l0reduces to:-= 2 [ J k01 Xo 2 (k +k) 1-exp<-t(k'o+ko1» 01 10 • Applying the equilibrium conditions:-t • •• • •• Put 'T = -+ 00 Xo X2 -x 200 klO -1 x2 -+ x 200 = x200 (kOl +k lO ) kOl 2 [1-exp<-t ~ -' , , ... I ... P. ... ~ > - > N .J> ,II ,It --- 139 -Thus in the present studies the value of T was calculated from:-P2 P2 Pl T = Tl x (-) x (-) x - lAS atm p Pl av Pl av 760 where T p = particle relaxation time lAS atm Tl = laboratory relaxation time lAS Pl = downstream pressure _ P2 P2 average density and pressure ratios (p-) av· (P)av = at 1 1 temperature (T2) • av - 140 -REFERENCES 1. R. Gutteridge. Ph.D. Thesis, to be published. 2. A.G. Gaydon and loR. Hurle. The Shook Tube in High Tsmpsl'ature Chemical Physica, (Chapman and Hall Limited, London), 1963, Chapter II. 3. J .N. Bradley. Shook Waves in Chemi.s~ and Physics, (Methuen and Company Limited, London), 1962, Chapter II. 4. I.I. Glass, TheoPJJ and Pel'formanae of Siwple Shoak TubBs, Institute of Aerophysics, Rpt. No. 12, 1958. 5. R.C. Millikan, General EZsatl'ia RssBarah Laboratol'l/ Rpt •• (Schenectady, New York), 1964, No. 64-RL-3700C. 6. R.C. Millikan, General. E'Z.eatl'ia RB8Barch Laboratol'l/ Rpt •• (Schenectady, New York), 1965, No. 65-C-I06. 7 • J • A • N • A. F • , TMl'I'fr)chemiaal Tab l.e8, The Dow Chemical Coapany, Midland, Michigan, 1965. 8. G.S. Campbell and P.C. Klimas, A.I.A.A. J~l., 1967, !, 1713. - 141 -7. RESULTS This chapter contains the results obtained in the present study. Section 7.1 shows that the energy of the observed emissions corresponded to the energy of the 1 + 0 and 2 + 0 transitions. Section 7.2 shows the effect of the flow corrections on the relaxation times of the first vibrational level of pure carbon monoxide. 7.1 ENERGY OF THE OBSERVED EMISSION The equilibrium population of the ith vibrational level with energy E., 1 of a harmonic oscillator is given by:-n. = N exp(-Ei/kT)/p.f. 1 7.1 where N = total number of ~lecules n. = 1 number of molecules in level i T = absolute temperature K k = Boltzmann constant The partition function, p.f., is given by:-p.f. = exp(-t/2kT)/(1-exp(-t/kT» 7.2 £ = energy separation of the vibrational levels. - 142 -Substituting 7.2 in 7.1 n./N 1. = exp(-Ei/kT) (1-exp(-E/kT»/exp(-E/2kT) Rearranging:-In n./N = -(£./kT-£/2kT) + In(l-exp(-ElkT)) 1. 1. Differentiating:-d In n./N 1. N = -(E.-t/2)k-l + Elk (exp(E/kT) - 1) 1. Pl = - x 760 = density ratio = downstream pressure 7.3 n. corresponds to the concentration of oscillators in state i at 1. equilibrium, and is equated to I •• () -1 • of ln n. IN versus T ,should g1.ve 1. To a good approximation, a plot straight lines of slope -1 (E. - E/2)k • 1. The second term in 7.3, has a small value when the temperature is low. The term has more effect as the temperature increases, and the value of exp( E/kT) decreases. At 2000K the value of the second term is -1 -1 -800 K compared to (ti - £/2)k which i. "3100 K-l • Plots were carried out for the first and second vibrational levels of pure carbon monoxide. The Tables 7.1 and 7.2 contain the data, and Figure 7.1 shows the points plotted. For - 143 -the first vibrational level, the slope from Figure 7.1 was found to be 3333K-l and (E. - E/2) was 27.64 kJ mol- l • The energy of ~ the first vibrational level is known to be 25.64 kJ mol-I. Similarly for the seond vibrational level, the value of (E. - E/2) 1 -1 1 is 52.70 kJ mol and the known value is 50.95 kJ mol-. The figures indicate that the emission is due largely from the experimentally observed levels. Thus in the case of the first vibrational level contributions to the emission from transitions such as 2 + 1, 3 + 2 etc. are expected to be small. Similar applies to the second vibrational level where transitions such as 3 + 1, 4 + 2 are involved. - 14-4- -TABLE 7.1 FIRST VIBRATIONAL LEVEL Photo P1 (p/P1)av I ln n.lu (T2) (T )-1 x 104 lID 3. av 2 av No. mm K 1103 8.5 6.011 11.5 6.528 1890 5.291 1104- 12.5 5.755 12 6.245 1671 5.984 2101 11.9 5.799 11.5 6.3ll 1691 5.913 2102 9.9 5.949 12 6.429 1856 5.388 2103 9.1 6.058 12.5 6.536 1945 5.141 2104 8.0 6.098 12 6.617 1991 5.022 2105 7.0 6.224 12.5 6.771 2104 4.753 2106 5.9 6.231 12.5 6.941 2116 4.726 2107 8.7 6.054- 12 6.541 1925 5.194 - 145 -TABLE 7.2 SECOND VIBRATIONAL LEVEL Photo PI (p/P1 )av I In n. /U (T2) (T )-1 x 104 GO l. av 2 av No. nun K 1201 11.4 5.98 15.5 5.152 1773 5.64 1202 8.1 5.994 15.5 5.491 1818 5.501 1203 8.5 6.093 16 5.459 1930 5.18 1204 7.4 6.201 16 5.580 2022 4.945 2205 7.8 6.177 17 5.59 2014 4.965 2206 4.2 6.30 16 6.188 2184 4.58 . ..J ..J UJ UJ > > '=\ ~ -1 > 0 I- Z f./) 0 ex: u u:: w V) x 0 x ,.... Z ...... .5 C ..J £ Q I" Figure 7.1 x i J I x / 0 ~ ~ -D 9 -0 ~ 0 -)( '2 ~Ct Ln ~ b III .¢ 0 ~ - 146 -7.2 EFFECT OF FLOW CORRECTION The following section shows the effect, on the relaxation time, of applying the corrections, due to non uniform flow (Chapter 3). to the shock parameters (Chapter 6) of density and temperature. Table 7.3, Figure 7.2 show the uncorrected data with respect to the work of Hooker and Millikan. l The present work gives relaxation times which are lower by approximately 26\. If the corrections are applied to the data. using a 1\ m- l velocity decrement. the points are adjusted to give relaxation times approximately 4.2\ lower than the values of Hooker and Millikan. l This is shown in Table 7.4. Figure 7.3. The velocity decrement in the present ease was fo\Dld -1 to be 2\ m and the result of using this value in the corrections is shown in Table 7.5, Figure 7.4. The data gives good agreement wi th the previous work. The corrections for non \Dliform flow were applied in every case in the present work. A velocity decrement of 2\ m-l was observed for the majority of mixtures. and this value was used in the calculations. It is believed that the present calculational procedure is sound, and that the corrected values of the shock para_ters are lION realistic estimates of the conditions behind the shock wave. Belford and Strehlow2 have urged workers to always make estimates of the non uniform flow corrections. and to present them so that others .. , asse •• them. This has been dOlle by several workers. 3 .... 5.6. 7 TABLE 7'.3 PURE CO 1ST VIBRATIONAL LEVEL UNCORRECTED DATA Photo Pl Shock Shock Av.Shock .J. Lab Napier Spee~l Mach (Temp) 3 (P2/Pl)av (P2/Pl)av Time Time Ho. 1II1II DIll liS No. Temp K )JS )JS atm 1101 9.9 1.8164 5.154 1752 0.08295 5.358 31.21 43.9 95.6 1103 8.5 1.8528 5.257 1809 0.08206 5.401 32.50 38.4 75.4 I-' +: ...,J 1104- 12.3 1.7205 4.882 1606 0.08539 5.235 27.87 58.7 138.6 I 2101 11.9 1.7313 4.913 1622 0.08510 5.249 28.33 59.5 138.5 2102 9.9 1.8344 5.205 1780 0.08251 5.379 31.84 41.1 91.6 2103 9.1 1.8844 5.347 1860 0.08131 5.438 33.63 33.0 72.2 21oa. 8.0 1.9103 5.421 1903 0.08069 5.468 34.58 34.0 67.7 2105 7.0 1.9715 5.594 2004 0.07932 5.534 36.85 28.1 52.8 2106 5.9 1.9786 5.614 2016 0.07916 5.541 37.12 31.0 49.5 2107 8.7 1.8716 5.311 1840 0.08160 5.424 33.16 39.0 80.3 Figure 7.2 8 G> 0 0 o ,d~------r-------~--~----~---0·07 0·075 008 00 85 (TEMP/K)-1/3 TABLE 7." PURE CO 1ST VIBRATIONAL LEVEL - 1\ VELOCITY DECREMENT Photo P1 Shock Shock Av.Shock ....l Lab Napier Spe·~l Mach (Temp) 3 (P2'Pl)av (P2'Pl)av Time Time Ho. .. lID liS No. Temp K ~s ~s attn 1101 9.9 1.816 .. 5.1s.. 1832 0.8172 5.818 31.21 43.9 103.9 1103 8.5 1.8528 5.257 1875 0.08109 5.901 32.50 38.4 82.4 ~ ~ CD 110 .. 12.3 1.7205 4.882 1657 0.08450 5.655 27.87 58.7 149.7 2101 11.9 1.7313 4.913 1677 0.08416 5.689 28.33 59.5 150.2 2102 9.9 1.8aa.4 5.205 18 .. 2 0.08158 5.8"9 31.84 41.1 99.7 2103 9.1 1.8a..4 5.347 1929 0.08033 5.938 33.63 33.0 78.9 210.. 8.0 1.9103 5.421 1975 0.07970 5.988 34.58 34.0 74.0 2105 7.0 1.9715 5.594 2086 0.07826 6.104 36.85 28.1 58.2 U06 5.9 1.9786 5.614 2099 0.07809 6.111 37.12 31.0 54.6 2107 8.7 1.8716 5.311 1909 0.08061 5.934 33.16 39.0 83.9 Figure 7.3 TI JJs atm 2 10 Id~----~~----~---~----~---007 0·075 008 0085 (TEMP/K)-1/3 TABLE 7.5 PURE CO 1ST VIBRATIONAL LEVEL -2% VELOCITY DECREMENT Photo Pl Shoclc Shock Av.Shoclc ...L Lab Napier Spee~l Mach (Temp) 3 (P2/Pl)av (P/Pl )av Time Time No. DIll DIll liS No. Temp K l1S l1S atm 1101 9.9 1.8164 5.154 1852 0.08143 5.918 31.21 43.9 105.6 1103 8.5 1.8528 5.257 1890 0.08088 6.011 32.5 38.4 83.9 ...... ~ 1104 12.3 1.7205 4.882 1671 0.08427 5.755 27.87 58.7 152.2 \0 2101 11.9 1.7313 4.913 1691 0.08393 5.799 28.33 59.5 153.0 2102 9.9 1.8344 5.205 1856 0.08137 5.949 31.84 41.1 100.3 2103 9.1 1.8844 5.342 1945 0.08011 6.058 33.64 33.0 80.4 2104 8.0 1.9103 5.421 1991 0.07948 6.098 34.58 34.0 75.5 2105 7.0 1.9715 5.594 2104 0.07804 6.224 36.85 28.1 59.4 2106 5.9 1.9786 5.614 2116 0.07789 6.231 37.12 31.0 55.7 2107 8.7 1.8716 5.3ll 1925 0.08038 6.054 33.16 39.0 89.6 Figure 7.4 '!! ps atm I Id~----~------~--~----~---007 0075 0-08 0-085 (TEMP/K)-1/3 - 150 -7.3 DATA OF COINe MIXTURES FOR 1ST VIBRATIONAL LEVEL The data for each mixture is tabulated on a separate page and, for clarity, a Landau-Teller plot of the data follows each mixture. In certain cases where few data were available, e.g. section 7.3f and 7.3g, the gradients of the lines were drawn with reference to the slopes of previous data as well as the available points. A 1¥Pical experimental recOl'd of the infrared -1 emission due to the 1 + 0 transition (21 .. 3 em ) ia shown in Figure 7.5. The data corresponding to this trace is to be found in section 7.3b, photograph number 11 .. 4. The horizontal sweep speed is 20 lIS em-I (i.e. in the Figure 7.5 20 lls/largeaquare). and the emission of the CO rises vertically. and exponentially. The rise and the intensity shown on this figure bears co.parison to Figure 7.13. Figure 7.5 e ~ E<~-e N e u Ui-~ o N 7.3.a 90\ CARBON MONOXIDE -10% NEON Speed of Sound - 0.360 km S -1 Specific Heat Ratio - 1.417; Av. Mol. Wt. - 27.23 ; Photo Pl Shock Shock Av.Shock -l. Lab Napier Spee~l Mach (Temp) 3 (P2/Pl)av (P2/Pl)av Time Time No. IIIn mm liS No. Temp K lJS \.IS atm 1121 10.9 1.8164 5.04 1828 0.08178 5.718 30.1 30.4 75.1 1122 8.9 1.8653 5.18 1912 0.08058 5.804 31.67 40.1 86.3 112 .. 12.0 1.7699 4.91 1750 0.08298 5.633 28.47 48.0 121.5 ..... (JI ..... 1125 11.1 1.7814 4.94 1771 0.08264 5.667 28.85 47.4 113.0 1126 9.8 1.8405 5.11 1872 0.08113 5.776 30.82 44.0 101.0 1127 8.2 1.8972 5.27 1973 0.07973 5.889 32.78 36.3 75.6 2121 6.9 1.9576 5.43 2079 0.07835 5.962 34.92 32.0 60.4 2122 6.1 1.9786 5.49 2124 0.07779 6.043 35.70 31.0 53.7 1:1 J.lS atm 2 10 Figure 7.6 Id~----~------~------~---007 0-07S 008 0-085 (TEMP/KJI/3 7.3.b 80\ CARBON MONOXIDE - 20\ NEON Speed of Sound - 0.368 km s-l; Specific Heat Ratio - 1.435; Av. Mol. Wt. - 26.44 Photo PI Shock Shock Av.Shock J. Lab Napier Spee~1 Mach (Temp) 3 (P2/P1}av (P2/P1}av Time Time No. IIDl mm J,lS No. Temp. K J,lS ~s atm 1141 10.8 1.8105 4.92 1832 0.08172 5.545 28.72 40.8 92.3 1142 9.6 1.8349 4.99 1872 0.08114 5.571 29.50 44.2 91.8 I-' (J1 I\) 1143 8.9 1.8843 5.13 1957 0.07995 5.644 31.15 36.4 74.9 ll44 7.2 1.9370 ~.27 2057 0.07863 5.737 32.94 32.4 57.9 U45 6.6 1.9715 5.36 2111f. 0.07792 5.76 34.14 34.0 57.5 1146 11.5 1.8046 4.91 1913 0.08201 5.378 28.54 46.4 108.0 ll47 12.4 1.7587 4.78 1723 0.0834 5.406 27.08 56.0 ll3.8 Figure 7. 7 LI JJsa 10 I .. IO~----~------~------~---007 0075 008 0085 (TEMP/IQ -1/3 7.3.c 60\ CARBON MONOXIDE - 40\ NEON Speed of Sound - 0.385 km s-l; Specific Heat Ratio - 1.476; Av. Mol. Wt. - 24.88 Photo P1 Shock Shock Av.Shock ...1. Lab Napier Spee~l Mach (Temp) 3 (P2/Pl)av (P2/Pl)av Time Time • No. No. Temp K mm mm J,lS J,lS lJS atm 1.8779 1958 0.07993 ...... 1181 10.4 4.88 5.174 28.55 36.0 73.6 (J"I w I 1182 9.9 1.8844 4.90 1966 0.07983 5.15 28.76 37.2 71.7 1183 11.1 1.8105 4.71 1839 0.08162 5.071 26.52 48.3 94.8 1184 12.9 1.7971 4.67 1811 0.08204 5.027 26.12 43.4 96.7 1185 14.1 1.7699 4.60 1765 0.08275 4.99 25.3 48.0 112.4 Figure 7.8 I .. ,d~----~------~------~---007 0075 008 0085 (TEMP/K) -1/3 7.3.d 50\ CARBON MONOXIDE - 50\ NEON Speed of Sound - 0.394 km s-l; Specific Heat Ratio - 1.5; Av. Hol. Wt. - 24.09 Photo Pl Shock Shock Av.Shock -1. Lab Napier Spee~l Mach (Temp) 3 (P2/P 1) av (P2/P1)av Time Time No. mm DID }JS No. Temp K }JS }JS atm 1101 12.1 1.8844 4.78 1986 0.07956 4.928 27.54 29.18 63.1 ~ (]'I ~ 1102 15.0 1.7756 4.51 1779 0.08253 4.77 24.41 46.3 106.4 1103 16.7 1.7643 4.48 1746 0.08304 4.689 24.11 45.0 112.0 1104 17.8 1.7421 4.42 1708 0.08365 4.647 23.49 49.0 125.3 1105 19.4 1.7205 4.37 1672 0.08425 4.626 22.90 52.0 140.0 1106 11.1 1.9038 4.83 2006 0.07929 4.944 28.13 32.1 65.0 101 Tl JJSatm 10 Figure 7.9 .. Id~----~------~----~----()(j] 0'075 (T~~~IQ-1/3 0'085 7.3.e 40\ CARBON MONOXIDE - 60\ NEON Speed of Sound - 0.404 km s-l; Specific Heat Ratio - 1.526; Av. Mol. Wt. - 23.31 Photo Pl Shock Shock Av.Shock ...l. Lab Napier Spee~l Mach (Temp) 3 (P2/Pl)av (P2/P 1) av Time Time No. mm mm liS No. Temp K liS 1.18 atm I 1121 12.2 1.8528 4.58 1925 0.08038 4.67 25.44 36.8 70.2 I-' (11 (11 1122 14.5 1.8284 4.52 1878 0.08104 4.63 24.77 35.6 77 .9 ll23 19.9 1.7699 4.38 1691 . 0.08393 4.484 21.B9 50.1 128.8 1124 25.0 1.6152 4.00 1518 0.08700 4.328 19.25 70.0 191.8 1125 16.7 1.7205 4.38 1779 0.08253 4.569 23.18 41.1 95.7 Fi9ur~ 7. 10 'fl ps atm . I .. Id~----~------~------~---007 0-075 008 0085 (TEMP/K)-1/3 7.3.f 20\ CARBON MONOXIDE - 80\ NEON Speed of Sound - 0.426 km s-l; Specific Heat Ratio - 1.588; Av. Ho1. Wt. - 21.75 Photo P1 Shock Shock Av.Shock ...L Lab Napier Spee~l Mach (Temp) 3 (P2/P1)av (P2/P1)av Time Time No. mm mm },IS No. Temp K },Is },IS atm ..... 1161 20.0 1.7587 4.12 1768 0.0827 4.107 20.74 41.6 93.3 (J1 en 1162 23.0 1.7152 4.02 1689 0.08396 4.043 19.71 46.0 110.9 1163 25.0 1.6687 3.91 1612 0.08529 3.984 18.64 59.0 144.1 1164 20.6 1.7367 4.07 1729 0.08331 4.13 20.21 54.7 123.8 10 1'1 J,lS atm 2 10 Figure 7.11 .. IOIt---------r---____ ---....---007 7.3.S 10\ CARBON MONOXIDE - to\ NEON Speed of Sound - 0.439 km s-l; Specific Heat Ratio - 1.625; Av. Mol. Wt. - 20.963 Photo PI No. DID 1182 25.0 1183 22.9 1185 22.1 Shock Spee~l mm lJS 1.6737 1.8405 1.7532 Shock Av.Shock Mach No. Temp K 3.81 1578 4.19 1853 3.99 1706 -~ ~ (Temp) 3 (P2/P1)av (P2/P1)av Time lJS 0.08589 3.698 17.75 68.0 0.08142 3.835 21.53 37.2 0.08368 3.767 19.51 47.1 Napier Time \.IS atm 146.8 92.4 100.7 ...... U'I -...1 1d1 Figure 7. 12 TI _. JJsatm ,dr-----~------~-'----~---007 - 158 -7.4 DATA OF CO/Ne MIXTURES FOR 2ND VIBRATIONAL LEVEL The data for each mixture is tabulated on a separate page and, for clarity, a Landau-Teller plot of the data follows each mixture. A typical experimental record of the infrared emission due to the 2 + 0 transition (4260 cm-l ) is shown in Figure 7.13. The data corresponding to this trace is to be found in section 7.4.e photograph number 1251. The horizontal sweep is 20 ~s em-l (i.e. in the Figure 7.13 20 ~s/large square), and the emission of the CO rises vertically. The emission in this case does not rise exponentially, as does the emission in Figure 7.5, but it shows an initial 'toe'. The emission in this case is only 0.25 that shown in Figure 7.5. Fi 9 ure 7, 13 E .. u Cit ~ o N 7.4.a PURE CARBON MONOXIDE Speed of Sound - 0.353 km s-l; Specific Heat Ratio - 1.~0; Av. Mol. Wt. - 28.01 Photo PI Shock Shock Av.Shock _.1. Lab Napier Spee~l Mach (Temp) 3 (P2/Pl)av (P2/P1)av Time Time No. DIll mm ps No. Temp K lJS lJS atm 1201 11.4 1.7702 5.023 1773 0.08262 5.98 29.63 ~6.2~ 122.8 1202 8.1 1.8051 5.132 1818 0.08193 5.99~ 30.83 52.5 103.5 ..... U1 to 1203 8.5 1.8714 5.310 1930 0.08032 6.093 33.16 39.1 88.4 1204 7.4 1.9224 5.455 2022 0.07908 6.201 35.02 35.7 75.5 1205 7.8 1.9184 5.444 2014 0.07918 6.177 34.88 33.9 75.0 1206 5.9 1.9891 5.644 2136 0.07765 6.253 37.53 30.2 54.9 2201 3.6 2.1080 5.982 2369 0.07501 6.478 42.19 26.9 34.9 2203 2.3 2.2149 6.285 2583 0.07288 6.638 46.65 25.4 23.8 2206 4.2 2.0153 5.719 2184 0.07707 6.30 38.54 38.0 51.0 Figure 7. 14 TI J,lS otm Icf I Ol'-"-__ ---.. ___ --r-___ ~--007 0075 0<)8 0-085 (TEMPI K) -1/3 7.~.b 90\ CARBON MONOXIDE - 10% NEON Speed of Sound - 0.360 km s-l; Specific Heat Ratio - 1.417; Av. Mol. Wt. - 27.23 Photo PI Shock Shock Av.Shock J. Lab Napier Spee~l Mach (Temp) 3 (P2/P1)av {P2/P1 )av Time Time No. mm mm lAS No. Temp K liS liS atm 1221 6.3 1.9224 5.34 2022 0.07908 5.986 33.67 38.0 63.5 ...... 01 0 1224 4.5 2.0560 5.71 2267 0.07612 6.138 38.58 28.9 40.5 1225 7.5 1.9024 5.28 1984 0.07958 6.024 32.96 36.0 70.8 1226 8.3 1.8376 5.10 1878 0.08104 5.842 30.73 32.8 64.3 1227 9.8 1.7840 ~.95 1786 0.08242 5.75 28.95 51.3 110.1 1228 8.0 1.8868 5.24 1959 0.07991 5.897 32.43 37.3 75.0 2221 4.8 2.0514 5.70 2255 0.07626 6.114 38.4 30.0 44.5 Figure 7. 15 TI ).IS atm. .. 101 L------..----r----...,---007 0·075 0·08 0085 (TEMP/IQ-I/3 7.4.c 85\ CARBON MONOXIDE - 15\ NEON Speed of Sound - 0.364 km s-1; Specific Heat Ratio - 1.426; Av. Mol. Wt. - 26.84 Photo P1 Shock Shock Av.Shock _.1. Lab Napier Spee~l Mach (Temp) 3 (P2/01)av (P/P1)av Time Time No. DID mm J,lS No. Temp K lJS lJS atm I-' en 1232 9.7 1.8377 5.05 1878 0.08105 5.723 30.16 39.5 87.0 I-' 1233 9.7 1.8791 5.16 1947 0.08008 5.759 31.57 34.7 80.5 1234 7.6 1.9184 5.27 2021 0.07909 5.839 32.91 29.1 55.9 2231 13.3 1.7771 4.88 1776 0.08256 5.625 28.19 40.0 111.0 2232 14.4 1.7204 4.73 1681 0.0841 5.516 26.39 49.7 137.0 2233 5.9 2.0109 5.52 2188 0.07702 5.92 36.21 28.5 47.4 Figure 7.16 1:1 .. ps atm Icf ,d+-----~~----~-~------~---007 0-075 008 0085 (TEMP/K)-1/3 7.~.d 80% CARBON MONOXIDE - 20\ NEON Speed of Sound - 0.368 km s-l; Specific Heat Ratio - 1.435; Av. Mol. Wt. - 26.44 Photo P1 Shock Shock Av.Shock _.1. Lab Napier Spee~l Mach (Temp) 3 (P2/Pl)av (P2/Pl)av Time Time No. mm DIll 1.1S No. Temp K lJS lJS atm 1241 7.4 1.9427 5.28 2069 0.07848 5.762 33.13 28.0 52.0 12~2 5.9 1.993~ S.~2 2166 0.07728 5.86 34.9 27.0 43.6 ..... 0'1 "-> 1246 9.6 1.8430 4.99 1821 0.08188 5.66 29.48 43.1 90.9 1247 8.6 1.8638 5.07 1929 0.08033 5.652 30.45 38.5 74.9 2241 10.8 1.7875 4.86 1797 0.08225 5.529 27.99 44.1 96.8 2243 7.0 1.9635 5.34 2103 0.07805 5.772 33.85 28.7 51.6 2245 4.9 2.0423 5.55 2240 0.07643 5.805 36.61 27.5 37.7 2246 4.6 2.0483 5.57 2252 0.07621 5.81 36.88 25.0 32.5 2247 14.8 1.7204 4.68 1683 0.08406 5.40 25.9 49.8 135.6 2242 12.3 1.7466 4.75 1728 0.08333 5.462 26.7 53.4 126.0 Figure 7 . 17 Id~----~------~------~---0-07 0075 0·08 -1/3 0085 (TEMP/K) 7.4.e 75\ CARBON MONOXIDE - 25\ NEON Speed of Sound - 0.372 km S -1; Specific Heat Ratio - 1.444; Av. Mol. Wt. - 26.05 Photo No. 1251 1252 1253 1254 1255 1256 1258 P1 IIIIIl 11.0 9.9 7.8 5.8 4.6 10.9 15.4 Shock Spee~l mm llS . 1.8194 1.8562 1.9264 2.0242 2.0746 1.7635 1.7013 Shock Mach No. 4.89 4.99 5.18 5.44 5.58 4.74 4.58 Av.Shock Temp K 1856 1920 2044 2222 2328 1761 1651 (Temp)-t (P2/Pl)av (P2/P1)av 0.08136 5.486 28.46 0.08045 5.534 29.63 0.07879 5.624 31.95 0.07663 5.744 35.33 0.07545 5.B57 37.12 0.08280 5.395 26.72 0.08461 5.312 24.84 Lab Time llS 36.3 34.8 31.1 26.6 25.2 50.4 52.0 Napier Time llS atm 82.0 74.4 57.4 41.2 33.2 104.1 139.0 ...... m CIJ Figure 7. 18 TI jJS atm .. Id~----~------~------~---007 0·075 008 00 as (TEMP 1~-1/3 7.~.f 65\ CARBON MONOXIDE - 35\ NEON Speed of Sound - 0.380 km S -1 Specific Heat Ratio - 1.~65; Av. Mol. Wt. - 25.27 ; Photo PI Shock Shock Av.Shock ...l. Lab Napier Spee~l Mach (Temp) 3 (P2/Pl)av (P/Pl )av Time Time No. DID DID lJS No. Temp K lJS \.IS atm 1271 12.2 1. 7875 ~.70 1806 0.0821 5.205 26.38 ~0.7 89.8 1272 10.0 1.8488 4.86 ~ 1911 0.08058 5.287 28.2~ 36.2 71.2 01 ~ 1274 8.8 1.8868 4.96 1975 0.07969 5.313 29.43 33.4 60.4 1276 6.3 2.0065 5.28 2182 0.07710 5.40 33.3~ 28.5 42.6 1277 5.3 2.0699 5.44 2317 0.07556 5.44 34.50 23.5 31.7 1278 3.8 2.1227 5.58 2443 0.07425 5.702 37.36 22.8 24.3 2271 14.5 1.7335 4.56 1719 0.08348 5.088 2~.79 ~7.1 113.~ 2273 9.7 1.8714 4.92 1949 0.08005 5.299 28.94 3~.2 66.9 Figure 7. 19 TI J,JS atm " Id~----~------~------~---007 0075 . 0·08 0085 (TEMP/K)-1/3 7.~.g 60\ CARBON MONOXIDE - ~, NEON Speed of Sound - 0.385 km S -1 Specific Heat Ratio - 1.~76; Av. Hol. Wt. - 2~.8B ; Photo P1 Shock Shock Av.Shock -~ Lab Napier Spee~l Mach (Temp) 3 (P2/Pl)av (P/P1 )av Time Time No. DDD DDD }.IS No. Temp K }.IS J,lS atm 1282 11.9 1.8231 ~.7~ 1866 0.08132 5.122 26.B9 3~.9 75.3 1283 9.~ 1.8638 ~.8~ 1938 0.OB020 5.171 28.13 39.~ 70.B ...... 0'1 U1 1284 8.1 1.922~ ~.99 20~5 0.07879 5.26~ 29.9~ 31.4 52.B 1286 5.4 2.0378 5.30 2258 0.07620 5.38 33.69 26.3 33.9 1287 4.9 2.0653 5.37 2308 0.07566 5.391 34.63 26.8 32.3 2281 15.4 1.7269 ~.~9 1707 0.08366 5.005 2~.09 ~5.2 liO.5 2282 14.4 1.7533 ~.56 1748 0.08301 5.023 24.8~ 46.4 109.7 2283 8.5 1.9064 4.94 2014 0.07918 5.23 29.44 33.0 56.8 Tl )Js otm / Figure 7. 20 101~ __ ----.-__ --...--" __ .....-__ 0-07 0075 0-08 0085 C;TEM P I K) -I /3 7.4.h 50\ CARBON MONOXIDE - 50\ NEON Speed of Sound - 0.394 km 5-1 ; Specific Heat Ratio - 1.5; Av. Mol. Wt. - 24.09 Photo P1 Shoclc Shock Av.Shoclc _J. Lab Napier Spee~l Mach (Temp) 3 (P2/P1)av (P2/P1)av Time Time No. mm mm liS No. Temp K liS liS atm 1201 12.9 1.8303 4.64 1874 0.08111 4.85 25.95 34.3 73.3 1202 11.1 1.8563 4.71 1930 0.08031 4.923 26.71 33.6 64.6 ..... 0'1 0'1 1203 9.1 1.9144 4.86 2034 0.07892 4.993 28.45 30.9 51.2 1205 9.9 1.8868 4.79 1987 0.07954 4.98 27.62 32.5 58.2 1206 8.0 1.9952 4.96 2112 0.0779 5.046 29.68 28.6 45.1 1207 7.8 1.9184 4.87 2046 0.07877 5.006 28.56 36.6 47.2 1209 18.1 1.7108 4.34 1681 0.08411 4.736 22.64 47.2 120.6 2201 10.2 1.8946 4.81 1984. 0.07957 4.906 27.84 31.7 58.1 2202 12.2 1.8421 4.67 1893 0.08083 4.851 26.30 34.4 70.4 2203 14.9 1.7771 4.51 1786 0.08247 4.811 24.45 38.9 89.7 103 Figure 7.21 'tl }Is atm ,d~----~------~-~----~----007 o 075 008 0085 (TEMP/IQ-I/3 7.~.i ~5\ CARBON MONOXIDE - 55\ NEON Speed of Sound - 0.399 km s-l; Specific Heat Ratio - 1.513; Av. Mol. Wt. - 23.70 Photo P1 Shock Shock Av.ShocK _l. Lab Napier Spee~l Mach (Temp) 3 (P2/P1)av (PiP1) av Time Time No. mm DID l1S No. Temp K l1S liS atm ...... 01 1211 13.1 1.8051 4.52 1850 0.08145 4.759 24.69 35.5 71.9 -..J 1212 12.1 1.8266 4.58 1881 0.08101 4.777 25.28 37.4 71.9 1213 11.2 1.8600 4.66 1943 0.08005 4.826 26.23 32.9 61.3 2212 10.6 1.8829 4.72 1985 0.07957 4.855 26.90 31.6 57.6 2213 14.1 1.7980 4.51 1831 0.08173 4.742 24.5 38.5 83.0 2214 16.9 1.7237 4.32 1703 0.08373 4.643 22.48 46.8 108.5 Figure 7. 22 rl JJS atm 101-1--__ ---.. ___ --.--_" ----,..--007 0-075 0·08' 0·085 (TEMP/t9 -1/3 7.4.j ~\ CARBON MONOXIDE - 60\ NEON Speed of Sound - 0.404 km S -1 Specific Heat Ratio - 1.526; Av. Mol. Wt. - 23.31 ; Photo P1 Shock Shock Av.Shock _.1. Lab Napier Spee~l Mach (Temp) 3 (P2/P1)av (PiP1}av Time Time No. DIn mm lois No. Temp K lJS lJS atm 1221 13.4 1.8231 4.51 1880 0.08102 4.675 24.62 35.0 71.1 1222 11.9 1.8525 4.58 1934 0.08026 4.71 25.44 33.2 62.3 .... 0' en 1223 11.0 1.8896 4.68 2001 0.07936 4.749 26.47 31.2 56.8 1224 11.5 1.8868 4.67 1995 0.07943 4.737 26.39 31.8 60.1 1225 9.4 1.9184 4.15 2052 0.07869 4.771 27.29 29.2 47.0 1226 9.0 1.9346 4.79 2085 0.07827 4.803 27.77 29.7 46.9 1227 15.1 1.7601 4.36 1767 0.08271 4.58 22.92 43.7 91.2 1228 18.0 1.7302 4.28 1715 0.08353 4.533 22.15 40.0 95.1 1229 20.0 1.6880 4.18 1649 0.08464 4.473 21.08 49.8 122.8 Figure 7. 2 3 "(I JJS atm I ., IO~----~------~------~---007 0·075 008 0·085 (T EM PI K) -1/3 7.4.k 35\ CARBON MONOXIDE - 65\ NEON Speed of Sound - 0.409 km s-l; Specific Heat Ratio - 1.541; At. Mol. Wt. - 22.92 Photo Pl Shock Shock Av.Shock ...l. Lab Napier Spee~l Mach (Temp) 3 (P2/Pl)av (P2/P1)av Time Time No. mm mm lJS No. Temp K lJS lJS atm ..... 1232 12.2 1.8414 4.50 1919 0.08046 4.59 24.54 35.1 63.5 (1) 10 1233 11.3 1.8829 4.60 1993 0.07946 4.632 25.67 32.4 55.1 1234 10.0 1.9144 4.68 2058 0.07862 4.685 26.55 27.8 45.5 1235 9.0 1.9104 4.67 2035 0.07890 4.622 26.44 29.0 41.9 2231 20.0 1.6764 4.09 1626 0.08503 4.346 20.28 55.3 128.2 2232 17.8 1.7269 4.22 1711 0.08361 4.413 21.54 47.1 104.8 -rl . }Is atm 2 10 Figure 7. 24 Id~----~------~------~---0 .. 07 0.075 0·08 0·085 (TEMP/K)-1/3 7.~.1 20\ CARBON MONOXIDE - 80\ NEON Speed of Sound - 0.~26 km s-l; Specific Heat Ratio - 1.588; Av. Mol. Wt. - 21.75 Photo PI No. DID 1265 15.1 1266 16.7 1267 21.2 1268 14.7 2261 13.4 2262 19.4 2263 22.8 Shock Spee~l DID liS 1.8051 1.7737 1.6888 1.8087 1.8267 1.737 1.6764 Shock Mach No. 4.23 4.16 3.96 4.24 4.28 4.04 3.93 Av.Shock Temp K -~ (Temp) 3 (P2/Pl)av (P2/P1)av 1865 0.0812~ 4.212 21.86 1651 0.08460 4.041 21.09 1651 0.08460 4.041 19.09 1872 0.08113 4.213 21.95 1884 0.08097 4.208 22.39 1705 0.08371 4.05 19.91 1619 0.08515 3.981 18.81 Lab Time liS 40.3 58.9 57.5 ~4.6 42.1 55.2 61.3 Napier Time liS atm 73.7 126.3 123.7 80.1 69.8 113.6 137.7 ~ -.J 0 I~ Figure 7.25 II JJs atm ... 10'.&------.----.....------r--007 0075 008 0085 (TEMP/Krl/3 7.~.m 10\ CARBON MONOXIDE - 90\ NEON Speed of Sound - 0.~39; Specifc Heat Ratio - 1.625; Av. Ho1. Wt. - 20.96 Photo P1 No. lID 1281 20.0 1282 20.8 1283 25.0 1285 22.3 1286 18.7 Shock Spee~l 1IIIIl }.IS 1.7302 1.7367 1.6734 1.7045 1.7703 Shock Mach No. 3.93 3.95 3.81 3.88 ~.03 Temp K -~ (Temp) 3 (P2/P1)av (P2/P1)av Av.Shock 1734 0.08323 3.901 19.0 173~ 0.08323 3.866 19.11 1628 0.08499 3.808 17.75 1681 0.08410 3.822 18.43 1807 0.08209 3.9~9 19.90 Lab Time J,lS 48.6 48.7 56.4 52.6 41.3 Napier Time J,lS atm 94.8 98.6 125.4 111.3 79.7 ..... ...,J ..... TI pSCltm F i 9 u re 7. 26 . .. Id~----~------~------~---0-07 0-075 0·08 0-085 (TEMPI K) -1/3 - 172 -7.5 DATA OF CO/POLYATOMIC MIXTURES FOR 1ST VIBRATIONAL LEVEL The following section contains the data obtained on the CO/C02, CO/N20, and CO/COS systems. The data for each mixture of each system is tabulated on a separate page. After the data for each system a Landau-Teller plot, of all the mixtures, is presented. A typical experimental record of the infrared emission due to the 1 + 0 in CO is shown in Figure 7.27. The data corresponding to this trace is to be found in section 7.5.a.i -1 photograph number 1404. The horizontal sweep is 20 ~s aB (i.e. with Figure 7.27 20 ~s/large square) and the emission of the CO rises vertically. e u >--e N Figure7.27 < 7.5.a.i 99.9\ CARBON MONOXIDE - 0.1\ CARBON DIOXIDE Speed of Sound - 0.3529 km s-l; Specific Heat Ratio - 1.4; Av. Mol. Wt. - 28.03 Photo Pl Shock Shock Av.Shock .-.L Lab Napier Spee~l Mach (Temp) 3 (P2/Pl)av (PiP l)av Time Time Ho. .. _ ).IS No. Temp K ).IS lJS atm llfOl 12.1 1.7591 4.98 1705 0.08370 5.636 29.19 24.0 62.9 ~ .1402 9.9 1.8412 5.22 1836 0.08166 5.768 32.00 23.0 55.3 -..J w 1403 10.2 1.8347 5.20 1825 0.08182 5.749 31.76 23.4 57.3 1404 12.2 1.8228 5.18 1798 0.·08223 5.705 31.35 24.0 63.2 1405 13.3 1.7531 4.97 1695 0.08386 5.628 28.98 27.0 77.4 1406 14.0 1.7264 4.89 1649 0.08463 5.562 28.08 31.0 89.2 1..07 16.0 1.6791 4.76 1580 0.08586 5.508 26.56 35.0 107.7 1408 16.8 1.6592 4.70 1550 0.08640 5.47 25.93 36.0 112.9 7.5.a.ii 99.5\ CARBON MONOXIDE - 0.5\ CARBON DIOXIDE Speed of Sound - 0.3524 km s -1 . Specific Heat Ratio - 1.4; Av. HoI. Wt. - 28.04 • , Photo P1 Shock Shock Av.Shock _.L. Lab Napier Spee~l Mach (Temp) 3 (P2/Pl)av (P2/Pl)av Time . Time No. DIll 1IIID IJS No. Temp K IJS IJS atm 1412 12.8 1.7699 5.02 1718 0.08349 5.628 29.61 18.0 50.5 1413 13.9 1.7046 4.83 1617 0.08519 5.542 27.44 20.4 56.7 ~ ....:I ~ 1414 15.0 1.6439 4.66 1534 0.08671 5.496 25.50 24.0 66.2 1415 23.5 1.5136 4.29 1344 0.09061 5.284 21.56 30.0 105.5 1416 1l.0 1.7929 5.09 1758 0.08285 5.688 30.4 17.1 42.8 2413 15.9 1.6737 4.75 1564 0.08614 5.489 26.44 22.3 67.7 2414 10.3 1.7532 4.97 1698 0.08382 5.657 29.05 21.2 47.2 2415 9.3 1.8528 5.25 1863 0.08126 5.841 32.49 16.0 37.2 7.5.a.iii 99\ CARBON MONOXIDE - 1\ CARBON DIOXIDE Speed of Sound - 0.352 Jan s -1 Specific Heat Ratio - 1.4; Av. Mol. Wt. - 28.17 ; Photo P1 Shock Shock Av.Shock _.L. Lab Napier Spee~l Mach (Temp) 3 (P2/P1)av (P2/P1)av Time Time No. IIIID mm lJS No. Temp K lJs liS atm 1421 12.7 1.7643 5.01 1709 0.08364 5.613 29.52 12.6 34.9 1422 13.8 1. 7367 4.93 1662 0.08442 5.537 28.59 12.8 36.8 1423 15.1 1.6890 4.80 1590 0.08567 5.472 27.02 14.1 41.4 ...... -.J (J1 1424 16.4 1.6788 4.77 1572 0.08599 5.438 26.69 13.9 43.5 1425 18.3 1.6391 4.66 1514 0.08707 5.380 25.43 13.2 43.5 1426 19.5 1.5965 4.53 1451 0.08718 5.297 24.li 14.0 45.9 1427 23.0 1.5562 4.42 1393 0.08954 5.213 22.88 15.1 54.5 2421 24.5 1.5178 4.31 1341 0.09067 5.141 21.75 18.3 66.0 2422 26.7 1.4853 4.22 1296 0.09170 5.076 20.82 18.7 68.7 2424 9.9 1.8716 5.32 1879 0.08103 5.756 33.27 li.4 28.4 2425 8.8 1.8908 5.37 1882 0.08099 5.788 33.97 11.5 26.2 2426 12.5 1.7643 5.01 1705 0.08370 5.593 29.52 12.2 33.1 ~".." 11.'\, 1';< If 7.5.a.iv 98\ CARBON MONOXIDE - 2\ CARBON DIOXIDE Speed of Sound - 0.351 km S -1 Specific Heat Ratio - 1.4; ; Photo P1 Shock Shock Av.Shock J.. Spee~l Mach (Temp) 3 No. 1IIR 11m liS No. Temp K 1431 14.5 1.6839 4.80 1582 0.08582 1432 21.1 1.5432 4.40 1376 0.08989 1433 24.0 1.5178 4.32 1340 0.09071 1434 28.2 1.4579 4.15 1263 0.09249 1435 30.9 1.4352 4.09 1232 0.09327 1436 18.3 1.6294 4.64 1499 0.08736 2431 20.2 1.6011 4.56 1459 0.08816 2433 25.0 1.5095 4.30 1331 0.09091 2434 27.0 1.4973 4.26 1314 0.09129 Av. Mol. Wt. - 28.33 Lab Napier (P2/P1)av (P2/P1)av Time Time lJS lJS atm 5.457 27.03 9.0 25.3 5.194 22.64 9.6 31.2 ...... ...:I 01 5.132 21.90 9.6 34.1 , 5.038 20.17 10.4 39.2 4.987 19.54 10.9 43.2 5.349 25.28 8.8 28.6 5.315 24.4 8.1 27.9 5.128 21.65 10.0 36.5 5.117 21.29 10.0 38.7 T/ usat 2 10 I 10 0-01% CO2 x - O-S%C02 " A -I'O%C~ o - 2.0 (!) 0 T l-' ~ 0 -IE -• ... 0 tJ Ten ::l ;::-t: -0 -\-'8 0 0 0 • • .. Q 0 0 '","-~.f.> Figur~ 8.2 0 • -\ \ \ \ ()\ \ • i\ 0 ex) • ~ " 0 ('f) ..J ~Z - ~ CD 0 - UJ~ ~O' C +' ~ffi 0 '1-' ~ ° -• - 0 IE ... a -len fDO - ---..: ~ ('t) N - 0 0 -0 ~ - - • - 0 0 0 0 • 0 - • '0, • 0 0 () 0 Figure 8.3 0 \ -\ \ \ \ (1\ SX • 0 CD -l • ~ ~ 0 ex) ZUl C'\I ~~ ..0 r:-- W 0(( - ZUJ OZ ~~ +' C « I I ..0 .2 )(8 . ., C Ou 0 f .-., Ln LL u a • I- O~ LL 0 w ~~ -0 • 0 ~ en C!) > - 0 '1-' ('I 0 --• 'e 0 .... c -.en ::a. &1)0 _ ....... - CX) b '}-I 0 0 0 0 0 0 Q g Q 0 0 • 0 0 0 • 0 - 18S -the values of A and B in the Landau-Teller equation are:-A = -21.66 (-21.1) B = 148.9 (1~2.0) The values are higher by 2.7\ for A and 6.3\ fOr B than the values (in brackets) obtained by Millikan.S The eonstant B gives. steeper temperature dependenee. It is .ost pl'Obable that the present constants are more reliable since the effect of the diluent was studied over a number of concentrations. This is in contrast to the study of Millikan,S who must have obtained his values by extrapolation of the line between pure CO and 3\ CO in 97\ He. The temperature dependence of Teo-co Ed TeO-He is shown in Figure 8.~. This figure alao show. the value. given by the S.S.H. theory and the M.W. correlation. The constant. used in the S.S.H. calculations are shown below:-TABLE 8.1 r./k K _ -1 VB CII co 88.0 He 35.6 37.06 The S.S.H. calculation. &1.,. looci apee .. t with the experJaeDt&l value for pure CO, but tbe val •• aN about 3 tu.. aborter thaD the ••• ured value for TeO-He' Tbedi.CNpacy 1D the latter caH ~~--~--~-3 10 Irf TI JJ5 atm I 10 / / / PRE SENT WORK _. _. 5.S.H. TI-£ORY Figure 8.4 / / / .. - - - M W CORRELATION. CO-Ne /co-co / . co-co / / / /CO-Ne 100~ ____ ~ ____ ~ __ ~ ____________ _ o.07(TEMP/t ... ::J U c- CW) 0 .-~ . .'" 0 a-. )( .,.... 0 I -w a: .-... .. N w ~ ::J 0-e 0 u - 191 -obtained for the 5 level system. Thus for most of the computer runs, a 5 level system was used. b) For upper levels, the 2nd in particular, it wo observed that the L-T and L-T + resonance (L-T-R) models gave exactly the same relaxation curves. This effect wa independent of temperature and concentration of diluent. It wa also the case no matter hCIIW fast the resonance retes w.re _de with respect to the L-T V-T rates. (At about 2000K. for eact resonance the S.S.H. theory predicts a probability of 2.16 x 10-3, while for the -6 V-T process PIO is 4.8 x 10 at the s .. t811p8NtuN. Seve I'll 1 calculations were made where the v-v Nte coostanta W.N greater than the V-T constanta by factors of ten. The iDere .. iD the v-v constants did not affect the relaxatioo tiaa. In order to conserve COIIPuting time, the V-V Nte constants _" Mde a factor of 10 faster than the V-T constats. for lIOat of the oalcul.atiooa). The concordance of the relaxatiClll curves ~atNt.. a poiDt that 6 789 theoreticians' •• have argued. It 1s that the NaODaDC8 reactions serve only to redistribute tbe available vibNtional .. ray. thus enabling a quasi ateact, state distribut10a to be MintaiDecS. This diatribution slowly decays to equ1llbrl_ th~ V-T procea_. It is here that the distinctiClll between the 2 leftl + ruonmce and the L-T-R _cbaBis. beCOMa clearer. ID the fomar .,ata the upper vibrational le .. la can ODly abtaiD t1aeir eDel'I)' fI'OII translation throuah the 0 + 1 tra8iti_. Tbia .... that hiper - 192 -vibrational levels, in this model, would have to await significant population of the 1st vibrational level. In the L-T-R model energy is being fed from translation to every vibrational level. The computed relaxation curves showed that, for the 2nd level, the relaxation time obtained by the 2 level + resonance .,del wa longer than that for the L-T-R model. This is one piece of evidence which strongly suggests that the 2 level + resonance .,del is unrealistic. The kinetic analysis of this model is shown in section 5.2.b.iii of Chapter S. c) The relaxation time of the oscillator energy, which is a total property of the system, ia taken the time when the energy has reached 63.2\ of its equilibrilBll value. The tWu obtained were compared to the relaxation ti.s obtained, in the _ .. way, no. the populations. It was found that, for a 5 level L-T model at 15001< wi th temperature changing, the relaxation tiM for the oscillator energy was 17.9\ higher than that froa the populations. For a similar syatem at >3000K the discrepancy is >50\. In L-T plots of the experiMntal data, froa apectroscopic and interferometric .. uureMDts the acatter is such that theae discrepancies are not noticed. Further ~t will be ... in section e.l.c.iii. d) Relaxation ti.s obtaiDed _-.tn, a cout_t ba'th tnperature were different flOll those obtwe4 ill a cIa-sha bath. The relaxatiOll tiM in the case of a ccxaatat bath w. a.l.wqa areater than that in the temperature chuginl case. 'ftds la ct. to the fact - 193 -that exp(-hv/kT) is always greater in the former system, so that it has further to go to reach equilibrium. It is also usual in shock tube studies to take the average of the shocJc temperature and the final equilibrium temperature. The relaxation time ". quoted at the average temperature. It "a found that the difference between the average temperature and the t...,erature corresponding to the relaxation time ". about 10K in 2000lC. The discrepancy "as even less in the case of mixtures, "here the t.aperature changed l.ss as the concentration of diluent "as increased. Errors from this source were negligible. B.l.c.iii TEST OF IJIALYTlCAL EQUATI9!S FOR liT A!!P 2)0) LEViL This section "ill show the teats applied to the analysis of the relaxatiOll curves. The tests were ~p1ied equally to both levels, in pure CO and CO/He aixturea. Considering pure CO. For the fiNt vibrational le.,.l, the best po.sible model and conditiona, to suit the analytical equation 6.3, "a. an isother.al. two level syate.. The calculation. "ere clone at lSOOlC, with a constant bath. Takiq points off the relaxation curve and plotting -In(l-x(t)/x(-)) .. ol'dinate and t as abscissa, a straight line "a obtained, Fiaue 8.1. The raciprocal of the slope wa 't, this "as found to be 0.9 .. (the Yaluea of the gradations of the x axis of the Figura 8.5 vue arbitrari1, choa. - 194 -as 0.5 vs). The system was then revised to an L-T-R system with 5 levels. the bath temperature held constant as before. The plot of 6.3, showed an initial curvature. However, the best straight line was drawn through the points and t was found to be 1.0 lAS. This represented an 11\ increase over the previous value. A further approach to reality was then made by making the bath temperature change over the relaxation period. A plot of equation 6.3 in this case is shown in Figure 8.6 (N.B. Figure 8.6 now shows the two extremes, Le. the isothermal 2 level model and the more realistic multistate model, with changing temperature). There was a noticeable increase in curvature. over the isothermal 5 level cue. The best straight line gave a relaxation time of 1.18 liS an increase of 30\ over the ideal model. Experimentally the present results gaw 11 ttle or no curvature on plots of equation 6.3. It was thought that noise on 3 the traces may have obscured the effect. However, Matthews (interferometric study at >30001<) obtained a similar curvature by plotting -1n(l-(p/p » versus t. Where p,p are the densities • • at time t and equilibrium respectively. He obtained the relaxation time by taking the slope of the curve at its Ddd point. Analysis of the computed vibrational energy by this _thad, for a 5 level te~erature changing system. gives a value of 1.26 liS. This value is comparable to that obtained from the populations. It 1. realised that experimental inadequacies may account for about IS\ scatter, -L.n(I- (Ill.)) 2 ,I Figure 8.6 MULT'STATE a STATE CONSTANT TEMe o~----~----~----~------~----~----~---o 2 3 4 5 6 TIME IpS - 195 -about the average line on an L-T plot. However, systematic errors are introduced by taking values of the relaxation time from curves such as Figure 8.6 or from the curves given by Matthews. 3 There would also be a small random error due to say estimating the mid point of the curve. These three types of error prd>ably account for the scatter seen on L-T plots (see, for example, ref. 25, p.236, p.257) and why the spectroscopic results and interferometric results are the same. The same procedure was cal'l'ied out vi th the 2nd vibrational level. The ideal model vas that of an i.othermal 3 level system. The relaxation curves vere analysed by equation 6.4. The relaxation time obtained was exactly the .... as that for the 1st level. For the other two syste. which approeched reality, the .ame treatment was applied. The reaulting relaxation ti •• , vere the same as those obtained on the lst level, vith the .... discrepancies between the ideal and realistic 1M>dels. The relaxation ti.. of the lst and 2nd levels were coq>uted for mixture. of 50' CO - 50' .e and 10' CO - 90' lie. A 5 level L-T-R model, with chanlinl teaperature va _eel. Again the curvature was evident in plot. of equationa 6.3 aDd 6.... However, the curvature dilll1niahed slightly in the order:-Pure CO > 50' CO > 10' CO Thi. vas due to the fact that .. the CCIlceDtNtion o~ 411Udt VIS - 196 -was increased, the isothermal condition was approached. In the mixtures the relaxation times obtained for both 1st end 2nd levels were the same. Thus two facts emerge froll the above discussion:-a) that the relaxation times obtained by a realistic model are 30\ above those obtained by an ideal model, for both levels, using the same method of analysis. The results of the present study, and those which haft gone before,2 may be too high. The equations 6.3 and 6." are not adequate for analyaing the experimental curves. The inadequacy haa been de.mstrated by comparison with the interferometric work vhich gives rise to the same relaxation times. b) that in all conditicna of tupeNtuN and ccmcentration of diluent the second level gives the s_ relaxation tiM as the first, in the case of • huw:>nic oacillator. ". diSCNpllDC:Y between the results on the lat and 2nd leve18, iD tAe preHllt cae, is not resolved. One ass1.Iq)ticn remaina to be questioned. Are the Landau--Teller ratios of the rate coostants correct? This is dealt with in the next section. 8 .1. d 'l'HE WWAU-TELL£R RA 'I'E cotISTAliTS In the derivation of equatioos 6.3 _d 6 ....... sections 5.2.a and 5.2.b respectively. a baftlODic oacill.ator va .. s~d. The computer IDOdela used a hU'llODic oecillat01'. '1'bia.,.l h.acl the rate constantlrelated in the followiDI ,.aeral v.,:-k'+l ' ) ,) - 197 -k j , .)-1 = (j+l) j In the case of the 1st and 2nd levels:-= and = In the case of an anharmonic oscillator it might be expected that:-> 2 8.2 This arises because the rate constant for the 2 + 1 transition (2117 cm-l in CO) is greater than that fOr the 1 + 0 transition (2143 cm-l in CO). Using two different theoretical approach.. it is possible to put an approximate value to the coefficient in 8.2. in the anharmonic case:-i) Using the S.S.H. theory. with the cClDatanta given in Table 8.1. a value of klO can be obtained. Uaina the a_ constant •• _ -1 but now setting v = 2117 em it i. poe.ible to sat an apparent value d kapp. Th for k21• calle 10. e value .. -1 fOr klO 1a 1.07 x 10 a ad k~~P. is 1.23 x 104 .-1 at 1953K. Tbua the Ntio of Nte coutants in the anhal"lDOnic cue aay be writtenl-- 198 -k (..l!) = 2 x 1.23 k 10 anharmonic 'i":'07 = 2.23 ii) The vibrational relaxation of anharmonic oscillator molecules has been treated by Bray.10,ll In this treatment an exponential increase in V-T probabilities w .. given with increase in vibrational level. The equations allow direct calculation of the transition probabilities between the various levels of an anharmonic oscillator. The calculations are .. followa:-p = p( T) ,..!-. v.v-l ~-£V where v • vibrational quantu. number P( T) = constant, at constant teaperaturea, which can be chosen ellpirically. Thia factor ia elillinated in the present calculationa. £ = the anhar.onicity = F(y> • y • g = adiabaticity fllCtor which baa been evalued 12 empirically, to be 0.5(3-exp(-2y/3»exp(-2y/3) g(1-2£v) <0.5)+ if e' = 16 4 M y2,2c2 2N 1016 k 8.3 - 199 -The parameters in equation 8.3 have been described in Appendix 2, section A.2.ii. The value of e' was taken from the S.S.H. calculations. 6 Thus for 6' = 4.33 x 10 , t = 0.0061 and at 1953K:-Then for v = 1 From this:- -5 F • 2.63 x 10 •• P10 = P(T)F • • • For v = 2, all the other parameters being the .... as before:-From this:-• • • • • y :I 16.21 F :I -5 3.06 x 10 P2l • P(T)F 2.05 P2l • peT) 6.21 x 10-5 From the S.S.H. calculatiems 'f c (ti_ between colli.iems) at 19531< is 4.8 x 10-10 s. Thus the Nte ccnatantl are:-klO :I 5.4 x 10" peT) s -1 k2l • 1.3 x 105 peT) I -1 k 1.3 x 105 (~) . • 2._ •• It 10 anbamonic 5.4 x 10. - 200 -Thus the two different theoretical approaches both give approximately the same value for the Landau-Teller coefficient, in the anharmonic case. The above calculations apply to the rate constant, (klO)CO-CO' in the pure gas, but presumably the same reasoning may be extended to (klo)cO-Ne. There is very little experimental evidence to indicate whether the Landau-Teller ratios between the rate constants are 13 correct. The flash photolysis experiments of Callear and Smith indicate that k32 : k21 : klO are in the correct ratio. Similar 14 experiments on the high vibrational levels of O2 are consistent with a single step quantum j\lllp. The shock tube work of Hooker and Millikan2 also is consistent with the single quant_ juq>, with the ratios being correct for the lover levels. On the other hand work 15 16 on high vibrational levels of 12 ' haa indicated that treDsitions where Av = 2 playa significant part in energy transfer in 12• Also 17 . Chow and Greene have obserwd the 2nd, 3rd _d 4th levels of HI. They found that all the levels relaxed at tbe .... rate, with a simple exponential, not predicted by the LaDclau-Teller .odel. The present results indicate that the .tepwi.e .odel for the population of the upper atate. 1a correct. and Av > 1 traoaitiona play little or no part in energy trtmafer. Hove"r. it baa heeD ahown that ratio of k21 : klO ia not 2:1. It ia .oN likely that the Landau-Teller coefficient lie. betveeD 2.2 _d 2.... for tbeae Nte cOllstants. Unfortunately. no _thod for the extractiOll of • value for - 201 -the coefficient, from the present results was available. 8.l.e CONCLUSIONS The work on the 1st and 2nd vibrational levels of CO was reexamined, under carefully cont1'Olled conditions. The values of the L-T constants were in reasonable agreement with previous work. 2,5 A discrepancy between the results of work, on .u.xtures, of the 1st and 2nd vibrational levels was shown. The detailed conclusions are shown below:-a) One of the objects of these experi_nts wu to try to distinguish between the stepwise, section S.2.b.ii, and the resonance exchange, section 5.2.b.iii, mechanis. for the population of the second vibrational level. It was shown in .ection 8.l.c.ii that the 3 state model ass\lDed in 5.2.b.iii wa UDNalistic and this Mchan ism was discounted. It was concluded that the L-T-R _ch_iaa was the most realistic model. In this proceas, the N.onDce reactions maintain a quasi Boltzmann distribution, while the gen.ral fol'll of the relaxation is carried out through V-T proce..... Th. experi.ntal 18 19 works of Appleton and Millikan and White, on N2, .... to support this idea. It has been a.aon.trated theoNtica1lylO,ll that in the cue of high vibrational telipeNturu and low tnmslatianal tapeNtuN. v-v reactions play _ il!pOrtant role, in the relaatian procu.. Th. result is that vibratianal a.acti vation becOMa rapid, the Nt.. being - 202 -much faster than those predicted by the L-T model. In contrast . 41,42. h • i to some recent exper~ments wh~c show an enhanced vibrat onal 43 deexcitation rate, the work of Appleton et al shows that when T-V collisions control the relaxation process, the characteristic vibrational relaxation time of CO dilute in AI' is the same whether measured in a compression or expansion situation. Thus it seems that the L-T model may be applicable in cases of extreme vibrational excitation. b) The kinetic equations used to analyse the cCllllputed and experimental roelaxation curves, were found to be inadequate. They cannot cater for conditions of cbanging te~reture in multi-level harmonic oscillators. The relaxad.OIl tiMs obtained for the real conditions are often very different from tbose obtained from ideal conditions. The absolute value of tbe preaent reaulta may be too high because of tbis. Thus the work of Hooker _d Milllkan2 may have a small systematic error. Howev.r, it ia difficult to see bow the equations could be changed for the better. Johannesen and bis co-workers20,2l haft derived an exact procedure which detel'llinee the relaxatian ti_ at the local condi tiona of T and p,throughout the relaxatian region, behind the sbocJc. Blythe22 has compared this method with tbe appl'Oxiaate analysia using the average temperature. '111e differences between the two .thods were found to be of the order of 10\ - 20'. The exact _tbod h_ bad limited application, witb interferometric23 and 1 ... 1' scblieren2-- 203 -measurements. It would be worth while extending this technique to cover spectroscopic measurements, such as thosepresented here. e) It was thought that the difference between the 1st level and 2nd level results with the mixtures was genuine. The computer calculations showed that for all temperatures and concentrations of diluent. the lines in Figures 8.1. 2 and 3 should concur. It was assumed that the Av = *1 was still valid, and that the discrepancy was due to the fact that no account of anharmonicity was taken in the Landau-Teller model. It was shown that the Landau-Teller ratio between the rate constants k21 and klO was >2, the actual value being some wheN between 2.2 and 2.4. Thus further computer calculations using anharmonic oscillators are required. The general shape of the CUl'ftS for the 2nd level in Figures 8.1, 2 and 3 is more difficult to explain. The 'tailing off' at high concentrations of Ne may be due to the weakness of the infrared emission in these mixtures. As the temperature deCNa ... i.e. 1953K to 1628K. see Figuroes 8.1. 8.3, the rate constants k21 and klO are expected to decrease. However they do so in the __ relat! ve proportions. so that the discrepancy between the lst and 2nd levels remains constant at about 10\. - 20~ -8.2 CO/POLYATOMIC MIXTURES A detailed analysis will be presented for the CO/C02 system. Most of the arguments used in this analysis will be extended to the analysis of the CO/N20 and CO/COS syste.. Any minor amendments will be noted. 8.2.a CO/C02 SYSTEM A •• uming, initially that CO2 is • macula with a IlOund vibrational state and an excit.d atat. at 2S .. 9 eII-l , then the fift proce •••• which may govern the relaxatiCD of the CO/CO2 .Yllt •• are: kl 1) CO* + CO ~ CO + CO V-T , k_l k2 2) cO* + CO2 , , CO + CO2 V-T k_2 kS ::lI. CO2 * _ 206 eII-l 3) cot + CO2 < CO + V-V k_S k .. .. ) CO * + CO2 " CO2 + CO2 Y-T 2 , k_ .. 5) CO * + CO k5 ::lI. CO2 + CO '.T 2 'k -5 - 205 -where it indicates a vibrational energy quantum. It is also assumed that the rate of reaction (4) » (1). see Figure 7.28 at l372K -~ (T 3 = 0.09) for CO/C02 • Figure 7.28 shows an L-T plot of the results of the various mixtures of CO and CO2 • It is noted that small UK)WltS of CO2 have a dramatic effect on the relaxation time of CO. There are two explanations which arise depending on the rate of (3h-i) if (3) is much faster than all the other proces .... then it will maintain the vibratioo.al energy of the whole .ystem in local equilibrium. The vibrational energy will then relax via the fater of processes (4) or (5). There is a single relaxation tiM, with the rate controlling process being either ( .. ) or (5). If ( .. ) plays the dominant role. then a quadratic apendeDce of reciprocal. relaxation time on mole fraction of CO2 will be ,iven (thi. i •• hown in .ection 5.2.c and 5.3.a of Chapter 5). ii) if (3) is slower than (4) or (5) but f .. ter thE (1) or (2). then CO will relax by the routel- (3) followed by ( .. ) or (5). with (3) being the rate detendning step. This will gi .. a liDear dependence of the reciprocal relexatioa tiM OIl IIOle fNctioa of CO2 • There will be a double relaxatiOD procua. In general. except in the two ext~ .... abo .. , the relaxation proceaa cannot be .fiDed by a .bale Nlaatiaa tt.,nt can be regarded a a coupled cue of tteo Nluatlaa prooea.... It baa been observed, experi..ntally, by Taylor, C_c .. f'eJ.aMra (T.C.F.)26 - 206 -that mixtures of diatomic gases relax to some fraction of their equilibrium, then both gases relax with the .ame rate toward the equilibrium. The fraction to which they relax initially depends on the extent of v-v coupling, which is called, by T.C.F., the vibrational energy ratio (I. They show that for weak coupling (I ~ 1, and for strong coupling (I ~ 0. -1 Plots of t vs mole fraction of CO2 , Figure 8.7, at constant temperature l372K show a linear dependence. It i. noted that only a small concentration range haa been .tudi.d, and that the quadratic dependence may reveal i t.elf at higher conceDtrtltiooa of CO2 • How.ver -1 the linear dependence of t over this conceDtration range allow. the first qual! tati ve conclusion. It i. that CO-C02 aN weakly coupled (according to T.C.F. (I + 1) .uch that N.ctian (3) ia IIUch alower than the V-T proce •••• (q) and (5), but that it ia IIUCh fater th_ the V-T processes (1) and (2). Thus (3) dWnatea tbe NlaxatiOll of CO. The molecule CO2 haa q vibrational IIOdea, aDd not aa .ugeated earlier, just one. The a)'llbol CO2(XyZw) deDOt .. a CO2 IDOlecule with X qUllllta of the .)'IIDetric .tretching lOde, v1 ' excited. Y quanta of the doubly dagenerat. b.nding mde, v2' excited with angular ...ntws Z. and W quanta of the .. ~tric atretching .ode, v3, excited. The .,-tric .tNtch i. at 1388 CII -1; the beding .ode at 661 CII -1, and the .. ,...tric .tNtcb at 231t9 CII-l • Ihach work ... beeD carried out on the vibrational "laxation of the YU'iou. .,.. of CO2 • ". consenaus of pre.ent opinion ia that neither tbe ~tr1c atretching -I ,:;. I ~.1'tl1\1 }Iso ..... 3-0 1·0 Figure 8.7 -I Plot of l' vs Mole Fraction COl CO2 Mixtures At 1372 K 0-01 0'02 . Mole Fraction CO2 - 207 -mode nor the asymmetric stretching mode have significantly longer relaxation times than the bending mode, see section 1.3, Chapter 1. These measurements are in contrast to theoretical calculations27 ,28 which indicate that the asymmetric mode should have longer times than the other two. The calculations are sensi ti va to Lennard-Jones parameters which are used for calculating the potential energy function between colliding molecules. As indicated in Chapter ~ the parameters may not be known with sufficient accuracy. at the present 29 time. Rapp and Sharp have cri ticis.d the perturbation approximation used in the calculations. since it is invalid at high teaper8tures. The energy lewl diagraa for CO2 , Figure 8.8. .howa .ome of the lower levels for CO2 , md some of the poa.1bl. trm.i tiona for populating the upper state.. T-V proce •••• exci t. the lowe.t lying mode "2. There is a known Fend Re.onmce betw.en 2"2 md "1 80 that there is very rapid .nergy .xchmg.. Th... 1IOc1e. are always in .quilibrium. The vl and "2 then tND8fer en.rgy to IIIOde v3 via v-v processes. It is difficult to define th •• xact nature of the •• processes. since the picture regarding th_ i. coaplex. A few of the possible pathways are given b.lowl-1) Intr8lD01.cular v-v en.rgy tNn.fel'l-C02(~OO) + X ~ CO2(OOOl) + X + 199 ca-1 C02(1110) + X ~C02(QO01) + X - 272 aa-1 CO2(0330) + X ~ C02(OOOl) + X - "16 CIt-1 Figure 8.8 Energy Level Diagram For CO/C02 System ~------18 '065' '¢~ 0 Iff -IE ..!:!. >-~ '-w 0 0 c UJ ~ 0 .d 0 N .- N > .. 0 0 ~ 0 0 .--~ --S 0 0 &n N -0 ?u N ? 0 - 208 -ii) Intermolecular V-V energy transfer:-CO2(0110) + CO2(1000) ~C02(OOOl) + CO2 k2 ii) k ... • kS iii) k .... kS» k1, k2 iv) k_ 3 • k3 exp(6E/kT) where 6E • 206 aD-1 :. k_3 • 0.8 k3 :. kS > k_3, k3 v) X « 1 - 211 -Using these facts it is possible to make adjustments to the analytical equations:-Taking 5.31:-Using assumption iv) so that:-The two ass\.IDPtions used above may alao be applied to "1z-which !'educes to:-• • • - 212 -Rearranging: -Using assumptions ii), iv) and v) • • • This is the equation of a straight line, the gr.dient of vhich i. the rate constant le 3• Similar approxiaatiOll8 applied to the analysis of Tuesday and Boudart, s .. sectioa 5.2.c of Chapter 5, ,i'M the s_ equation. The gradient vas taJcen fl'01l FilUN 8.7 _d ... found to be 1.45 x 106 s-l, and thus the probability 5.2 x 10--. At 1372K, the -4 near resonance theory of S.S.H. gJ.".. • val_ of P10-01 of 1.96 x 10 , vhi1e the exact resonance value i. 1.097 x 10-3• The calculations of 32 "-3 lapp and Golden Ii va a value P 10-01 of 2.2 x 10 • The t...,.rature dependence of P 10-01 froll the expen-ta v_ obtained by plottiDI figures similar to 8.7 at diffeftDt tUlperatun. !be telipeNture dependence is shown in Filure 8.9, topther with tboae PWD by the theorie.. The agre...nt bet1MeD the obMrwcJ pl'OlbllbiU ti __ d calculated OIle. is lood CODaidariDl the UDcertmti. iD botb experi_nt and theory. The telllp8rature dapen_. __ to be .oN COI'NCt iD the case of bpp .. d Go1daD cue the the S.S.II. As _t1cDed pNvioualy ~OOI Figure 8,.9 ---- --. -- --. ---- . ---. -------------x ------x x )( EXPERIMEN TAL RESULTS- CO'CCl.2 SYSTEM. S.S.H. NEAR RESONAN:E. --- SSH EXACT RESONANCE. '-'-' RAPP & GOLDEN. ., '-- . -.. . ---- 213 -the discNpancy may be due to taking the geometric mean for Elk and the arithmetic mean of rO of the cOrNspcading two values for the pure gases. This may apply particularly in the cues of the CO/polyatom!c systems, since the potential well depth of the polyatomic species is about 2 to 4 times that for CO. Extrapolatica of the present resulta to 300K giftS a value -6 of p 10-01 of 2.3 x 10 • This value ia co.pared to a value of 8.0 x 10-4 auggested by Moore,3O which indicatea that the proceas (3) may be =-1+00 times more probable than the preHllt reaults suggest. However, at temperatures lClOOOl( 10llg rena- forces betvMft the dipole moment of CO2 (v3) and the quadrupole ...ant of CO becaM itIpol'tant. The calculaticas of Shal'lNl and BHU 33 indicate that the pl'Obabili ty was inversely proportiooal to teaperatUN .... HctiOll ".7, Chllpter 4. This has been demonstrated experi.antal1y by the sbock tube .tudie. 34 35 of Taylor and Bitterman and ao.ch and Saitb. 'ftu., it i. po.sible to ccaclude that there are two .eparate intu.ctiOll8 for near Nsonant exchange processes. At low te..,.ratUNS. 1CIDI rep forcea are important and the rate is proportional to T-1 , vldle at hiab temperature the interectiClll ia dcII!natec1 by .bort IWaP foroea. the rate being proportional to T. In thia cue the ... onaOl NMtiGll (I) is vrltt_ MI-- 214 --1 CO + N2~ - 80 an -1 The plot of T VB mole fraction of H20 is shown in Figure 8.10. The dashed line indicates the possibility of a quadratic dependence, and thus that the above reaction is fast coa.pued to )c4 for pure H20. However, it is pointed out that the Idxture containing 2\ H20, gave very fast experimental rise ti.a,· 5 .,., which were coq>arable with the rise ti.s of the s,.t_. It wu though that the point at 2\ H20 in Figure 8.10 wu \Ilreliable, Ed for the purposes of the present analysla It wUl be ipored. H20 has very .iailar propertie. to CO2 • HoIM_r, wbereas there is Ferad resonance between 2"'2 _d "'1 modea of CO2 , there 1. a wavenumber difference of 107 CII -1 between the .. CODd bu.aoic of "'2 and the "'1 IK)de of H20. Tb1e reaulta in a alower tr.afer of energy to the "'1 1100. frc8 the "'2 in H20 th_ In CO2 • 'ftle excitation of the "'2 mode is auch futer in H20. In thia caM it .., be expected that H20 would exhibit sepaNte NlaatiOll tt.a for the ftf'io_ IIOdes. 36 However Si~SOD et al, 1D a cueful ahock tube study up to 1700K, haw .hown that neither the "'1 .ode DOl' the "'3 mda haw aipificatly longer ralaxaticn ti_ t ___ the "'2 mM. The MgDitudea of the rete ccnatata, abtalDe4 by tbeol')' and experiaent are ahOND in Table 8.31-w •• ' ..... -'" ; .. r" 4-0 3-0 2-0 1-0 Figure 8. 10 Plot of f'~s Mole Fraction CO/N20 Mixtures At 1372 K / x / 'I' .. / x / /. 0-0·1 0·02 MoI~ Fraction N1..0 S.S.H. M.W. Correlation Expt. S.S.H. M.W. Correlation Expt. - 215 -TABLE 8.3 At 1372K -1 Reaction 1, kl/s 3 2.09 x 10 2 7.46 x 10 3 2.5 x 10 7 1.47 x 10 6 8.26 x 10 6.99 x 106 -1 Reaction 2, k2/s 2 1.94 x 10 2 2.96 x 10 7 2.12 x 10 7 1.03 x 10 In the S.S.H. calculationa, for H20. elk • 237.0 ~d .0 • 38.16 nm. In this case the aasumptiOOli), ii), iii) y) .. de in .. ctiOD 8.2.a still hold. However, since AE ia x.duced • Although in this case ka' k.3 are faster thaD in the CO/C02 case, reactions (4) and (5) in the CO/H20 haw 1DCNued by about a factor of 3 so that:-I ~':'l . The gradient of Figure 8.10 was found to be 2.75 • 10. ad thus -3 the probability is 1.05 x 10 • At 1372K. ~ ..... 1" ____ theory -It of S.S.H. gifts. value of '10-01 of 2.67 • lO 1IbU.e at ~ - 216 --3 resonance the value is 5.49 x 10 • The calculationa of Rapp and 32 -3 Golden give a value of P10- Ol of _.5 x 10 • The temperature dependence of PIO- Ol from the experi_nta was obtained by plotting figures similar to 8.7 at different temperaturea. The temperature dependence is shown in Figure 8.11, together with thoae gi.,.n by the theories. The experi_ntal data haa good qree.nt with the calculations, although the diacrepancie ... y be aseribed to choice of L-J parameters in mixture.. The experi-.ntal data .bov. an increase of a factor of 2 in P 10-01 for the COIH20 ayat_ 0 .. 1' the CO/C02 ayate •• The preaent reaulta were extr.polated to 300K, where a -5 value of 1.9 x 10 vas obtained for P 10-01. '!he oaly __ ure.ant at room temperature on thia .yat_ i. an old cl1apeNiOll __ Ul'eMnt 38 -_ of Eucken and Jaacka, who found a 'Val. of 2.3 x 10 for PCo-N O. 35 2 The moat recent cOilparable work i. that of Roacb ed s.1 th, who -1 .tudied the N2/N20 Byat_, AE • 107 CIIa ,ad fOWld tUt P 10-01 -4 was 7.7 x 10 at 3OOK. Thi. filUN .m. CGIIIpaNd to the pl'Uent value, reflecta again the illlpOrtance of the laDa rep toroea, for the near re.onance ccmcl1 tiOD at low t .... NtUIWI. 8.2.e CO/COS SYST.EI In this cue the NMD __ IUC'tJaa (3) 1a witteD .a-ka CO* + COS > co + COS. + .... -1 101 Figure 8.11 x EXPERIMENTAL RESULTS cal ~o SYS TEM S. S. H. NEAR. P-.ESONA t-.CE - - - s. S. H. EXACT RESONANCE -·_·RAPP & GOLDEN -------- --'- ---'-., --PIOO1 -'-- ---. ,----- '-- . --. '--'- . -. x x x )( x -:4 IOO~O-8------00~~-fiE--M-~-~---I~-O~~-9-------- 217 --1 The plot of T vs mole fraction of COS is shown in Figure 8.12. Only one ~ixture of CO/COS was studied, thus no conclusions about the quadratic behaviour of the CO/COS system can be made. COS has similar properties to CO2 and H20. As wi th H20 the rapid transfer of energy between the mode 2"'2 and mode "'1 is not present in COS, since the energy discrepancy is 197 em-I. The T-V excitation of the "'2 mode is much faster than CO2 and faster than H20. No high temperature data exist. for the relaxation of COS, presumably because of the pos.ibility of dis.ociation. However, as has been done in the previous case., it will be assumed that all the modes relax together and there is little dissociation. The magnitudes of the rate conatants obtained by theory and experiment are shown in Table 8.":--I 2 t x 10 }.lSi atm' 5-0 30 1·0 Figure 8 .12 Plot of T'vs Mole Fraction CO/COS Mixture At 1372K o~--------~----------~----0·01 0-02 Mole Fraction COS S.S.H. M.W. Correlation Expt. S.S.H. M.W. Correlation Expt. - 218 -TABLE 8.4 At l372K 2.09 x 103 7.46 x 102 2.5 x 103 -1 Reaction 4, k4/s 6 5.26 x 10 6 9.12 x 10 2 1.67 x 10 1.75 x 102 -1 Reaction 5, kS/s 1.0 x 107 1.29x 107 In the S.S.H. calculations for COS, E/k • 33SK and rO • 41.3 nm. The assumptions made in 8.2.a still apply. The comparison of the experimental data with the theoretical calculations is shown in Figure 8.13. As in the previous two cases, agreement between experi_nt aDd theory is good. The data showed an increase in the probability of a factor of 2 over the CO/H20 data. This incre .. e is IRON thaD expected, and IK)re than the theory predicts. It is possible that the COS was not perfrectly pure, and .. y have contained traces of water. The data was extrapolated to 1'00II tellperature and a value -,. of 1.3 x 10 wa obtained for Plo-Olo Again there is no recent 39 comparable work. tucken and Aybar, in 19"0, obtained a value of :) 1001 Filure 8.13 x EXR:.P.IMENTAL RESULTS- CO/COS Sf STEM - S.S.H. NEAR RESONANCE __ S. S.H. EXACT RESONANCE .-._. RAPP & GOLDEN -. - -'--'- '-. ------------ --. --. -- . --. X X - ----X X --'---X -I04~ ________ ~ __________ ~ ________ _ 0·08 0.085. ,,,\-I~O·09 \ TEMP/IV '1~ - 219 --4 6.8 x 10 for PCO-COS' 8.2.d CONCLUSIONS The conclusions for this section are enumerated below:-1) It has been shown experimentally that the V-V exchange process can control the vibrational relaxation of gas mixtures. This is the case when the minor component has a rapid T-V process available for excitation. By analysis of the vibrational relaxational data on the binary systems; CO/C02; CO/N20 and CO/COS, it has been shown that the smaller the energy discrepancy between the two species, the more efficient the V-V reaction. Thus the 3 systems studied were found to follow this sequence:-10 10 10 P(CO, CO2 ) < P(CO, H20) < P(CO, COS) V3 ~3 V3 AE = 206 80 -1 CIII This is shown in Figure 8.14, together with the probabilities of T-V processes calculated by the S.S.H. theory. This shows that V-V processes have a steeper temperature dependence than the T-V process. Thus as the gn te1llp8rature increnes the probabili tie. for V-v exchange and T-V exchange will approach each other. Therefore at lower temperatures (1000 - 3OOOK) where the T-V process of the major COlllpOnent is slow, V-V proce .... beCOM very ia.portant. - --~-- Figurerature dependence of the two proce.... are diff.rent. - 222 -8.4 SUGGESTIONS FOR FUTURE WORK 1) With regard to the work on COINe, the computer programme should be extended to aCCO\Ult for anharmonicity. It is possible then that a rate constant for k2l may be extracted from the present work. -1 2) This work has indicated the importance of the T temperature dependence of P10- Ol at temperatures lUIt. 11th, (Combustion Institute, Pittsburg. Pa.) 1967, 37. 9. E.R. Fisher and R.H. Kumaler, J. Chan. ~N." 1968, ~, 1075. 10.' K.N.C. Bray, P'tui.d MtJohanic LaboN'liDI1I PubU.oatij Fj = molecular weight of species j The computer programme used to calculate the corrected parameters is set out overleaf. The more important calculational symbols and their meanings are given in Table A.2.l where a symbol is part of the output, the output format will be included. The input symbols, their formats and meanings are detailed on the comment cards, Table A.2.2 lists the input information. COMPUTATION SYMBOLS FORTRAN XH ATlMIX U GAMIX XMWAV TC TO PRES DER TM TP TL iii TABLE A.2.l CALCULATIONS w s -y M.'i. 't C AT P4/Pl r MEANING Mach number (F6.3) Speed of sound in test gas (Fa.S) Speed of shock (Fll.S) Average specific heat ratio (F6.3) Average molecular weight (Fa.3) Flow duration of contact surface Hot flow duration (F15 .6) Diaphragm pres.una ratio (Fll.4) Radius of tube Maximum hot flow duration (£10.4) Particle time for hot flow (E10.4) Laboratory time FORTRAN XMS2 DELTB DLNMS DELTD DTOTAL DELRO TACT ROACT iv CALCULATIONS MEANING The Mach number of the hot flow behind the shock Temperature correction due to boundary layer Temperature correction due to deceleration (6T/T2)TOTAL Total temperature correction (F7.S) 6T Actual correction to temperature in K (FIO.2) Actual density correction (FlO .2) v TABLE A.2.2 INPUT INFORMATION CARD No. COL NOS. SYMBOL & UNITS FORMAT 1. Number of j ohs 1 - 5 NJOBS IS for one mixture 2. Experimental 1 - 10 XD JIll F10.4 parameters 11 - 16 GAMA4 F6.3 17 - 24 A4 mm s -1 F8.S 25 - 30 DER aD F6.3 31 - 37 YID , .. -1 F7.4 38 - 40 IS 13 3. Species 1 - 15 ZJl( 5A3 parameters 16 - 20 XHJ FS.2 21 - 27 XMWJ F7.3 28 - 34 GJ F7.3 4. Mole fraction. 1 - 10 FJ FlO.S Up to five species per mixture are allowed, and read across the card 5. Input Mach 1 - 6 XMS F6.3 nuabers for one job 7 - 12 DE F6.3 13 - 19 XMI F7.3 20 - 27 T1 K F8.3 vi CARD No. COL NOS. SYMBOL & UNITS FORMAT 6. Experimental 1 - 7 Pl torr. F7.3 input parameters 8 - 17 TL liS ElO.4 for one job 18 - 25 RHOS F8.4 26 - 33 T2 1< F8.2 7. Recycle card 1 - 2 ISSW 12 For any set of experimental parailetel'8 (card 2) and species parameters (cards 3, 4) thelle are a certain n1.llber of jobs (card 1). For any one job there will be a pair of experimental input cards (cards 5, 6). If the programme is lIecycled (card 7), then a ne" number of jobs (card 1) is lIead, and the specie. parametel'8 (cards 2, 3, 4) alle lIead. Then the experimental parametel'8 cards alle lIead in pairs lmtil the nlDber of job. is completed. C C G MILLWARO(KEEIF) CORRFCTlnNS FOR NON-UNIFORM FLOW IN SHrlCK TUR~S. C THEORY OF H.MIRELS DFVFLOPFD FOR EXPLICIT USF RY R.A.ST~FHLO~. c r: DIMENSION lJK(~.5).XNJ(~).XMWJ(5).CPJ(5).CVJ{5).CPAJ(5),rVAJ{5),FJ 1 (5) .GA.J(5) .GJ(5) COMMON I~.Tl.AY.RY,AT1~IX.AAMIX.GMIX.GAMIX.X~WAV.FJ,7JK,XMwJ.GJ,PR 1FS.TO.XM.U.JJ,XNJ 4 CONTINUF 1 FORMAT(30X,65H**********FLOW NONlJNIFORMITY-EFFE:CT ON SHOCK PARA~l-T 1ERS**********) ? FORMAT(30X.5~H lR) WRITE<:?,l) wRITF(?,;?) JJ=O ArJAPTFD BY r,.f:.MILLwAf~D 2'-:> f.·UGUST 19h C READ IN nATA C r C CARD 1 START COL 1. NJOPS=NUMBFR OF JORS(I5)FOP ON~ M!XTIIRE c r: READ(7.21)NJOBS 21 FORMAT= A N f) n ENS I T Y I "J C' R F A S E c CALL DFATH(DER,Pl,TL,RHOS,Yln,T2) XM=XM~ 1)3 CONTINUE U=XM*AT1MIX AM=U/AAMIX CY=AM**2 AZ=3.142*(OER**2) AP=2*3.142*DER TM=5.85*«4*AZ)**2)*Pl/«X~**1.7)*(AP**2)*ATlMIX'ln.n**5) TP=TL*RHOS*C1+(O.7*«TI /TM)iHO.5») XMS2=«(RY*CY)+2)/«?*GA~IX*CY)-8Y»**O.S ..... x r: [] F. L T ~ = ( ( T L ! T"'" ) * * n • " ) * ( ( ( X 1'1 S? it it ? ) * ~ Y ) ! ( ( )( "1 S 2 it * 2 ) -1 ) ) it ( -1 . 0 () ) nL~MS=(YTD*AT1MIX*XM)/«YJD*AT1~IX*TM)+1) nFL Tn=TP*( (2*( (CY-1 )**?) )/( (1+( (RY*CY)/?) )~( «2*GAMIX*CY)!>:n 6 PIIII-FCYCLING 3 CARDS ARE- Rr::(~lJI~Fr; II::: CARns 7,P.,9, (' 'rJHICH WII t CONTAI"! TH~ l-'h.fiAMF-T~RS nF THF- NFXT PAIR OF "10L!:CULES. r r; PEAOC7,500) IPW "inn FORMATCI?) (' PRINTING OF THF HFAOINGS. r c 1 FOR MAT C 1 n 0 H 1 CAL C U L A T I 0 f\J S 0 F V I R RAT ION A L ~ E L A X A T I 0 f\J 1 ~J F' U f' t: (3 A S t S A 1Nn GAS MIXTURES IJSINr, TH!= ~-n S.S.H. THFORY ) ? FORMAT(5?H AOAPTFO BY f~.r;UTTFRIDGF & G.f.MILLWAf./fl ?~ JULY 1~6e ) ~ FORMAT(3~HOGAS A DFACTIVATING GAS HC ) 4 FOPMAT(86HOCONSTANTS USE:f) F=PI 8~=ROI Tl. Ir\!IT.LO LO INC. A IN=AVAGAD C(CM/SEC) PLA~CKS ) WRITF(?,l) WRITEC?,2) WRITE(2,3) WRITE(2,4) WRTTFC2,41)F,gK,nL,RI I,A~,CF,H 41 FOPMAT(1~X,F7.4,lPF12.4,np2Fl0.5,lP3Fln.3) f, FORMAT(?9HOMOLFClJLAR C()N~T.ANTS EPS/K,7X,2HRO,r;,X,39Hr< >< ...... ...... .:?1'\-t?,(1) 1 f-. FOR MAT ( 1 H 0 • 1 9 X , 7 H T ( DE G\<) • ') Y , '5 H F M / F P , 7 X • 4 ~i R [1 / L , 6 X • 5 H R 0 / he, R X , ~ H L • t .• 1 ~ X , 6 H THE T AD • 7 X • tJ H Y ( 2 • ? ) , ) X • 221-1 F- X P ( - ~ P / K T ) 1 - t- '( P ) fjRITH?.16) XMW=XMA*XMRC/«XMA+XMRC)*AN) THJ:TA=YNlI*H*CE/AK TH~TAA=XNUA*H*CE/RK 1 1 ~ n 0 ,~ 0 I = 1 • 1'01 1LCI)=O.1500 ]~ AL=2.U*F*8L(I) ZEK(I)=XMW*(2.0*CE*F/lo.n**8*XNU*AL )**~/~K ZEMK=T~MP(I)/2.0*(ZFI«I)/TEMP(I»**O.333~3 Z E M K A = Z F ~1 K I C IJ g L C = n R * C 1 • - ( 0 • '5 * ( 1 • + S rJ P T ( Z F M I< A + 1 • ) ) ) * * C - • 1 f-, 6 6 7 ) ) / ( t:. L i) (; ( 1 r "1 K A + 1 • ) ) IF(8LC-BLCI»104.104.22 ?? RLC 1 )=RL( 1 )+RLT IF(O.?500-RL(T»27,27.1S 27 WRITFC?,24) ?4 FORMATC47HOBL LESS THAf\J n.150000R r,fH-ATFh' THAN 0.2'->000 GO Tn ?6 ~1 r,O Tn 15 104 nROC=(O.'5*Cl.+SQRTC1FMKA+l.»)**O.16h67 ROL=n~/RL C I) ZEnO=(?5*«~LCI)/nl)**?»+cnT/RLCI» RLN=RL (I) Y??=O.76*(1.+(1.1*CQ/TFMPeJ») IF(O.7-TFMP(Y)/ZFK(I»?k,?Q,?9 2A WRITF(?,10) 10 FORMAT(2?H KT/EPSILON Ton SMALL GO TO 26 ?q CONTINUE IF(1n.O-TFMPCI)/ZFK(I»11,13.13 11 WRITE(?.]?) 1? FORMAT(2?H KT/FPSILON TOU LARGF GO TO 26 1~ FXPEK(J)=EXPC-CQ/TFMP(J» ~ ...... ...... ...... r: r: CO"" P lJ T F COL LIS I n~, \I LJ M R ~ K 2J V I I-l RAT I 0 ~I T R A ~! S L AT! O~! I:: N ~ H r.; Y He A \J S r E- k • r r 70SC=(XMP*X~C*(XMA+xr1RC)*7FK(I»/«XM8**)+YMC**?)*X~A*F**?*T~~TA) ~ THE T A ( I ) = 1 • I ( 1 • - c: X P ( - T d>:' T A / T F tv! P ( I ) ) ) 7Th'AN( I )=( (F*THI::TA/ZFK( I) )**?)it(SQIH(1.S/F) )*( (TFMP( I )/ZI:K( I) )**11. 116A67)*CFXP(Cl.5*(lFK() )/T~MP(I»**O.3~~~~)-(THFTA/(?*TE PCI »») 1 F !l ( 1 ) = 1 • 0 1 7 * D ROC it it ? it Y ? !' ~ E T H f- T A ( T ) it 1 F n () * 70S C * Z T R 1\ f\l ( 1 ) ;h- X f'f::: t< ( I ) WRIT~(?,17)T~~P( 1),l~MKA,RnL,nRnC,PL~,IFK('),Y?2,f-XP~K( I),FT~ETA(I 1 ) 17 FORMAT(lHO,19X,F7.1,4X,F~.?,~X,FA.2,~X,r~.S,~X,F8.~,3X,1~El0.3,~X, 1 n PF6.3,4X,F7.4,8X,F7.4) RL(I)=RL~J 3fl CQNTINUF 04 FORMAT(lHO,23X,9?H****** VIRRATIONAL-TRA~SLATIONAI EN~~GY TRA~SF~H l--rOLLISION NUMBERS--RELAXATION TIM~S ******} WRITF(2,94) 03 FOR MAT ( 1 H 0 , 19 X , 7 H T ( Dr: G K ) , 4 X , 7 H T ( -1 I 3 ) , ~ X , ? H 7 n , 6 X , 4111 n S (: , (, X , ? H 1 T K A \j 1,1?X,3~710,lnX,4HTAlJC,l[lX,3HTAIJ) ~RITF(?,93) C COMPUTE RFLAXATION TIMFS VIA TA8ULAT~D VISCOSITY I)ATA. c DO 50 I=l,N T3( I )=1.0/TEMP( 1)**0.3.5333 7 ( J ) = 1 • 0 1 3 24 9 E -1 n * n R A * * 2 * SO R T ( 1 6 * A '" * F I X M A I ~ K / T F r-1 P ( I ) ) CTAU(I)=7Fn(I)/Z(I) XX=H-MP( I )/CQ CALL LOOt< A ( I ) = ( ? (:. 6 • 93 it ~ () '? T ( ? it X v, A it X M B C * T F "" P ( I ) / ( )( M A + X M g C ) ) ) / ( ( f)" it * 2 • ) * Y Y * 1(10.n**7.0» T A lie ( I ) = GAM A ( I ) 11 . ? 71/1 • (] 1 3249 F 6 TAU( I >=ZEOC I )*TAUe( I) W R I T E ( 2 , 51 ) T E M P ( I ) ,. T 3 ( I ) • Z F DO , Z 0 C; C , Z T R Ai\) ( I ) , 7 f n ( I ) , TAU r: ( I ) , T h U ( I ) ~t FOQMAT(lHO,19X,F7.1,~X,FP.5,~X,F~.3,?x,F?1.~X,1PF10.3,~V,El1.4,4X 1,lPE11.5,4X,1PF11.5) t:;n COI\IT I t\jIJE I F ( X f\J II A • F () • 0 • 0 ) G () T n 2 f' x X ...... < r. r. r. 0 '--1 P tJ T E r. () L LIS I 0" \J U M H l :~ c:. FOR N EAR R f=: S o,~ II I\) C F E: I\j F f,' G Y T R 1\ ,\J~, F f- q • r: ,.. \. 110 61 1=1,1\) 7 E L< A ( I ) = 1 6 • 0 * F If * 4 * X M A I A ~,' / 2 • 0 * ( R L ( I ) I 1 0 • 0 l: * .. * X I\) I I A it C ~ ) .~ If ! I >. K 7 E K A 8 C ( I ) = 1 6 • 0 * F * * 4 * X M :N l: ( f:< L ( I ) * * 2 I 1 q • n * ~ 1 f-, ) * C E:- * it ? It ( )( :Ii I i-x \J U A ) * If 2 1 IRK 7 E K RCA ( I ) = 1 6 • 0 * F * * 4 * X M Hit ( 8 L ( I ) * *? I 1 () • 0 If If 1 6 ) I: (: E If *? * ( ( Y M ,1 + Y ., C ) l: *? ) /4 1.0 7 F K 8 C ( I ) = 1 n • 0 it F * * 4 * X 1'1 H r I A 1\1 / 2 • 0 * ( R L ( I ) 11 n • 0 * ... >i If X N IJ * C t- ) * * 2 / H K 70SCI)=«4.0*XMA1* X~A2)/(XMA1**?+XMA2**))/4.n/F**4*«4.n*XM8*X~C 1)/(XMR**?+XMC**2»*7~K~(I )/THETAA*7EK8CC!)/THETA (,1 r:O~HII\J!lF ZONR=9.0 IF (7i==KARCCll.FO.O.O) GO Tf' 08 9~ FOQMATC1HO.24X,8RH****** VJ8~ATIOI\JAL-VTHRATInNAL ~~~~~Y lRA~SFrR-lNFAR R~SONANCE--COLLISln~ NUMRFRS ******) WRlTf(?,95) WRITF(?,9(,) 9 f-, F 0 Q MAT ( 1 ~ 0 • 32 X • 7 H T C n F. G K ) , 4 X , 7 H T ( -1/3 ) • ~ X • 2 ~ l (l • 10 X • 4 H 7 () S C • 1 (I X , ;::; H 7 T K lAN.9X.9HZ10CN.R.» no 73 I=l.N 7TRACJ)=(r**?)*CSQRT(3.0/2.0/F»*(CT~MPCT )/lFKA8CC!»**U.166A7) 1*«CTHFTA-THFTAA1/lEKARC(J»**2)*EXP(1.5*C(ZFKARCC!)1Tf-~~(!»** ?n.~3~~~)-(A8~(THFTA-THFTAA)/?n/TEMP(J»-(CO/TFMP(I)*~~» 71 ONRC T )=znNR*ZO')( I ).lTRA (T) WRTTF(?,97)TF~P( 1),T~(T),zn~R.10S(I).ZTRACJ).ZlnNH(!) 97 FORMAT(14n.3?X,F7.1,~x,r~.~.~X.F4.1,4X,E12.~.4X.Fl?~.4x.E12.5) 7 "!l CO!'llT I N 1.1 E c r. 0 ~ P I JT F: COLI 1ST 0 "J r<.HJ M R f f,! S fOP E X ACT L( E S 0 ~, A N C ~ F- N I:- .. U~ Y T "': 1\: S F F. f,> • C 9A rnRMAT(lHn.24X.89H.***** V!BRAT!O~AL-VJHPATTnNAL I:-~EkGY TRANSFFR--lFXACT RESONA.\lCF--COLLISJ()N NlJMHERS ***.**) wRTTF(?,9R) 99 rOPMAT(lHO.3?X.7HT(DFGK),4X,7HT(-1/3),5X,2HZ~.10X.4H!Osc.lnX,5HZT~ 1AN.9X.9H710(F.R.» x x < r: r: T 1-1 T S S U q q 0 I JT I N F T A K E" A V " L I! E: 0 F ')( X Fl..(O M T H f= "-1 A T N P HOI; fi /I W' F A \1 D C F I f\Jj) S A V A I U F n F Y Y • T HIS I S nON F 4 Y S 1FT I !\J r, T ~ R n t J r; H THE T A ~ 1I L /I T f- U r \!1~cnSITY PATA OF HIRSCHFFLDF-R,CIJRTISS A~![J tlIRII.T4t= \fAUH- OF YY r. T S T H p,1 11 S FOP! F U ~ T H F RCA Lei I L A T T 0 ~,I S I 'J T H [" M A JrJ P ~ () I~ R II IV! 'v' ~ • c SIjRROIJTI!\JF- 1,)()KLlP(XX,YY) COI'1MO~J M, X , Y n I 1'-1 ENS I O~! X ( 1 f) 0 ) , Y ( 1 no) no 17R J=1,M IF(XX.GT.X(J»GO TO 17R IF(XX.FQ.X(J»GO T0 17l TF(.NOT.(J.EQ.l»Gr) TO 1~1 WRITE(?179) YY=(XX-X(J»*«Y(J+l'-Y(J)'/(X(J+l)-X(J»)+yeJ' RETURN 177 YY=Y(J) RETURN 17'1 r;OI\ITINUE WRITF(2,lRO) tRr) FORMAT(/~9H P()INT AROVf- TArLE LIMITS-EXT'''APOLATIDN) 170 FOqMAT(/~9H pnINT RELOW TA8LF LIMITS-EXTRAPOLATION) 1~1 YY=(XX-X(J»*«Y(J)-Y(J-1 »/(X(J)-X(J-l»)+Y(J) RETtJRN FNn ~ < 1-" 1-" xxviii TABLE A.2.3 COMPUTATION SYMBOLS FORTRAN CALCULATIONS DR CQ £ XMW THETA e 'niETAA AL t' ZEK ZEMl< Em/k ZEMKA Em/E BLC ROL MEANING Average L-J constant Geometric mean of L-J potential well Reduced mass of collision partners Characteristic temperature of vibra.tion of Be Characteristic temperature of vibration of A (lPE10.3) Em effective energy of collision (FS.2) Range of repulai ve forces from Method B. (Fe.S) (FS.2) xxix FORTRAN CALCULATIONS MEANING Y22 Y(2,2) (OPF6.3) EXPEK exp(-E/kT) (F7.4) EllIETA (1_e- 6/ T)-1 (F7.4) ZOSC Z Vibrational collision number osc (F6.l) ZTRAN Z trans Translational collision number (lPE1O.3) ZED ZlO Collision number of deactivation (Ell.4) ...1. (F8.S) T3 T 3 XX T* Reduced temperature YY 0(2,2) GAMA viscosity TAUC t c Time between collisions (lPEll.S) TAU t Relaxation time (lPEll.S) ZEl

Study of vibrational excitation mechanisms of CO2 at high temperatures

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