T H E
J O U R N A L
OF
PHYSICAL CHEMISTRY Registered i n U. S. Patent Office
0 Copyright, 1977, by t h e American Chemical Society
VOLUME 81, NUMBER 1 JANUARY 13, 1977
Shock-’Tube Chemistry. 1. The Laminar-to-Turbulent Boundary Layer Transition John A. Bander and George Sanzone* Department of Chemistry, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 2406 1 (Received April 7, 1976) Publication costs assisted by the National Aeronautics and Space Administration
A model is proposed for the determination of the laminar-to-turbulent transition in the boundary layer of a chemical shock-tubeflow. The implications of this model for the accurate determination of reaction-rate constants are discussed.
I. Introduction In a kinetic study in the incident-shock region of a chemical shock tube, the arrival time of the contact surface is an important parameter since it determines the test-time available for the study. The prediction of contact-surface arrival times has been used as a test of a model for the determination of the laminar-to-turbulent transition in the incident-shock flow field. The implications of this model for the accurate determination of reaction rate constants in this flow field will be discussed.
consequence of this effect, the contact surface accelerates while the shock front decelerates until gas is being leaked through the boundary layer at the same rate it is being shock treated. When this steady-state condition is attained, the shock-front-to-contact-surfacedistance remains constant; this is the condition of “limiting flow”. The laboratory test time is much shorter than for ideal shock-tube flow and there are pressure and density gradients and therefore temperature gradients between the shock and contact fronts. Also, the laboratory-to-particle time ratio is no longer equal to a simple density ratioq3 Since two of the most important kinetic variables, observation time and temperature, are affected, the presence of nonidealities in shock-tubeflows has important implications for chemical kinetic studies. There appear to be two schools of thought as to the inclusion of boundary-layer corrections in the treatment of kinetic data. One school recognizes that the corrections have been difficult-to-apply averages over an assumed boundary layer flow state. They have opted to ignore the corrections and to treat their data on the assumption that shock-tubeflows are idea1.4~5These researchers point to what they feel is reasonable agreement with kinetic data obtained by other methods; they have tended to be interested in the determination of reaction mechanisms. The second school has been more interested in details of the theory of reaction rates or of reactive They have opted to apply boundary-layer corrections to their data. However the corrections employed have been approximations based on the assumption that the boundary
11. Incident Region Test Times The laboratory test time is the time interval between the passage of the incident shock front and the arrival of the contact surface. In the ideal shock tube, the shockfront-to-contact-surface distance increases uniformly with time. Consequently the laboratory test time also increases with time of flow (or with distance downstream of the shock-tube driver section). The ratio of laboratory test time to actual particle time for observed gas is equal to the ratio of gas densities across the shock fr0nt.l In the real shock tube, the shock-treated test gas in contact with the cold shock-tube walls will lose both heat and momentum. This creates a region of thermal, density, and velocity gradients at the walls called a boundary layer. The shock wave can be considered as generated by the pile-up of test gas in front of the expanding driver gas. In other words, the driver gas acts as a piston. However since shock-treated gas is channeled through the boundary layer, the driver acts as a leaky piston.* See Figure 1. As a 1
2
J. A. Bander and G. Sanzone
Figure 1. The “leaky-piston” effect. The test gas is channeled around the driver gas.
TABLE I: Reynolds Numbers and Transition Timesa Flow Reynolds No. Transition Reynolds No.
USDP 2
Re, = -
UZDP 2
Re, =-
ML2
0
PZ
’ 2.0
2.5
4.0
3.5
3.0
4.5
M A C H NUMBER
D
Figure 2. Laboratory test times vs. Mach number. Initial pressure p1 = 12 Torr. Dotted upper and lower envelopes represent purely laminar and turbulent cases, respectively. a D = hydraulic diameter; p a = shocked-gas density; M, = shocked-gas viscosity; us = shock-wave velocity; u, = shocked-gas velocity.
layer is either completely laminar or completely turbulent? Calculated effects of the boundary layer on properties such as the temperature can be significant. However the validity of these corrections might be questioned because of the purely-laminar or purely-turbulent assumption. Kinetic data have not previously been treated for the real case in which a boundary layer undergoes a laminar-toturbulent transition. ’ 111. The Laminar-to-Turbulent Transition Time The following model is suggested for the calculation of the laboratory time T at which the laminar-to-turbulent transition occurs. T is measured relative to the passage of the shock wave. Under the condition of limiting flow, total laboratory test times can be predicted for either purely laminar or purely turbulent boundary layersagTotal test time t is assumed to be the sum of a fraction of a fullylaminar test time L and a fraction of a fully-turbulent test time T. Thus
t = ( 7 / t ) L+ (1 - 7 / t ) T 1 1 t = - T + -{T2 2
2
+ 47(L-
(1)
T)}”2
(2)
Times L and T can be computed with recourse to the original papersg or to the very good approximations provided by Strehlow and Belford.lo,ll Note that if T = 0, the flow is completely turbulent and the test time is T. If the transition occurs just as the contact surface arrives, then T = t = L. The more general case, which describes most chemical shock-tube studies, involves transition times in the range 0 < T < L. The above model is viable only if an expression for T can be found. It is also to be noted that the transition is not instantaneous but occurs over a time interval which is assumed to be a negligible fraction of the test time. Although the transition time also depends on the roughness of the walls,12it is well established that the principal factors that affect it define a transition Reynolds n ~ r n b e r . ’ ~We J~ investigated the use of two transition Reynolds numbers based on two measures of boundary-layer thickness. See Table I. The first, ReA,is based on the distance a free stream particle travels in a transition time. The second, ReB, is based on the distance a particle travels from its initial position to the transition point. The shocked-gas parameters also define a flow Reynolds number. Using the shock-tube hydraulic diameter, we investigated two flow The Journal of Physical Chemisiw, Vol. 81, No. 1, 1977
INITIAL
PRESSURE
p,
ltorr)
Figure 3. Laboratory test times vs. initial pressure. Mach number for curves is 2.2. For experimental points, 2.0 IMa I2.4.
Reynolds numbers. The first, Rel, is based on the velocity of the shock wave; the second, Ren,is based on the velocity of the shocked gas relative t o the wall. Although the flow and transition Reynolds numbers use different characteristic properties, they are written for the same shock-tube flow state. By equating a flow and a transition Reynolds number, we obtained the four trial expressions for the transition phase given in Table I.
IV. Experimental Results All shock velocities and laboratory test times for shocks into argon over a range of initial pressures, pl, and a range of Mach numbers, Ma, were measured with a modified laser-Schlieren system.15 Experimental test times were compared with theoretical test times as predicted by eq 2 with the four expressions for the laminar-to-turbulent transition time given in Table I. Since the evaluations of the terms L and T i n the model assume the condition of limiting flow, it was necessary to reject a considerable number of shock experiments which showed severe attenuation of the velocity of the shock wave with distance along the shock tube. The 2-in. diameter shock tube used in this study has been described e l ~ e w h e r e . ’ ~ , ~ ~ From graphs of laboratory observation times as a function of Mach number, (see Figure 2) it was found that transition times rA2and rB1 fit the data. From graphs of laboratory observation times as a function of initial pressure, (see Figure 3), it was found that transition times T~~ and TA2 fit the data. On the basis of these observations, transition time T A was ~ considered to give the best overall agreement between experimental and predicted observation times. Note that the experimental points in Figure 3 correspond to a range of Mach number: 2.0 < Ma < 2.4. See Figure 4.
3
Absorption Spectrum of the CH302 Radical
transition time, it should now be possible to more easily obtain these density and temperature profiles and so to definitively determine the effect of shock-tube boundary layers on the measurement of chemical reaction rate constants. Acknowledgment. This work was supported in part by a grant from the Petroleum Research Fund, administered by the American Chemical Society, and in part by NASA Grant NSG 1020. We wish to acknowledge the helpful comments of R. Rover and D. L. Bernfeld. I
I
50
10 0
I N I T I A L PRESSURE
p1
References and Notes (1) Ideal shock tube flow is described in a number of textbooks. See, for example, E. F. Greene and J. P. Toennies, “Chemical Reactions in Shock Waves”, Academic Press, New York, N.Y., 1964. (2) A. Roshko, fhys. Fluids, 3, 635 (1960); W. J. Hooker, ibid, 4, 1451 (1961). (3) J. N. Fox, T. I.McLaren, and R. M. Hobson, fhys. Fluids, 9, 2345 11966). (4) k. D. Kern et al., J. fhys. Chem., 75, 171 (1971); 78, 2549 (1975); 79, 1483, 2579 (1974). (5) S. C. Baber and A. M. Dean, J. Chem. fhys., 60, 307 (1974). 161 , , M. Warshav. J. Chem fhvs., 57, 2223 (1972). Also see NASA Tech. Note TN-6-4795 (1970): (7) R. L. Belford and R. A. Stehlow, Annu. Rev. fhys. Chem., 20, 247 (1969). (8) W. E. Trafton, Ph.D. Thesis, University of Illinois, Urbana, Ill., 1973. (9) H. Mirels, fhys. Nuids, 6, 1201 (1963); 9, 1265, 1907 (1966). Also see AIAA J., 2, 84 (1964). 10) R. A. Strehlow and R. L. Belford, Technical Report No. AAE 69-01, University of Illinois, 1969. 11) R. J. D’Amato, Ph.D. Thesis, University of Illinois, Urbana, IiI., 1971. 12) W. P. Thompson and R. J. Ernrich, fhys. Fiuids, 10, 17 (1967). 13) A. J. Chabai and R. J. Emrich, Bull. Am. fhys. Soc., Ser. 2, 3, 291 (1958). 14) R. A. Hartunian, A. L. Russo, and P. V. Marrone, J. Aerospace Sci., 27, 587 (1960); Also see froc. Heat Trans. Fluid Mech. Inst., (1958). 15) J. A. Bander and G. Sanzone, Rev. Sci. Instrum., 45, 949 (1974). 16) J. A. Bander, MSc. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, Va., 1974.
(tort)
Figure 4. Laboratory test times vs. initial pressure: T = 7.42.
V. Conclusions This work is being extended to a shock tube of larger diameter. Although other expressions for the transition time will be sought, we suggest that the experimental results to date provide a reasonable test of the consistency of the model represented in eq 1. Incidentally, a choice of transition time TM implies that the laminar-to-turbulent boundary-layer transition time is simply the time for shocked gas to flow a distance of one shock-tube diameter. With this model, there now exists a method for the prediction of the laminar-to-turbulent transition time in any single shock-tube experiment provided the condition of limiting flow is met. The additional measurement of contact-front arrival time should be made as it provides a consistency check on the data treatment. The density and temperature of any point in the incident-shock flow regime are determined by the entire flow regime since the flow in this regime is subsonic. With a knowledge of the
Absorption Spectrum and Rates of Formation and Decay of the CH302 Radical C. J. Hochanadel,” J. A. Ghormley, J. W. Boyle, Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830
and Paul J. Ogren Central College, Pella, Iowa 50219 (Received May 6, 1976) Publication costs assisted by the U.S. Energy Research and Development Administration
Flash photolysis of azomethane was used to produce CH3radicals which reacted with O2 to form CH302. The CH302radical was found to have a broad absorption spectrum centered at 2350 A, with a maximum extinction coefficient of 870 M-l cm-’ based on loss of azomethane. The rate constant at 295 f 2 K for the formation reaction CH3 + 0 2 + M CH3O2 + M ( K 2 ) in the second-order limit was found to be 1.3 f 0.2 X lo9 M-’ s-l. The rate constant for the combination of peroxy radicals, CH302+ CH302 products (h3),was 2.3 f 0.3 X lo8 M-l , and for the combination of methyl radicals, CH3 + CH3 + M C2Hs+ M (kl), it was 3.1 rt 0.6 X 1O1O M-I s-I. The reactions were monitored by kinetic spectrophotometry of CH3absorption at 2160 A and CH302 absorption at 2480 A.
-
Introduction Peroxy radicals are important intermediates in many reaction systems in photochemistry, radiation chemistry, atmospheric chemistry, and combustion. Because of the current emphasis on the production and conversion of energy and the consequences of its usage, there is increased
--
urgency for establishing mechanisms and accurate rate constants for reactions of radicals of this type. Several years ago the absorption spectrum of the HOz radical was observed by us1 and also by Paukert and Johnston,2 and rates of its reaction with itself and with OH were measured. This study has now been extended to the CH302radical. The Journal of Physical Chemlstv, Vol. 81, No. 1, 1977
