GAS DYNAMICSICS
Impact on Chemistry
^oidiN reauune
Heating, reactions, _ and ablation in boundary layer on body r- Bow shock
Wake s h o c k ^
Jfiteke (ionized)
Catalysis at surface
Hot gas cap, reactions, ionization, and radiation
I he shock layer—a region of low-speed flow between detached bow shock and thin boundary layer on a blunt body—is a feature of hypersonic continuum flow field about a sphere. Extreme temperatures occur in the shock layer
76 C&EN JULY 18, 1966
D R . P E T E R P. W E G E N E R , Chairman, D e p a r t m e n t of E n g i n e e r i n g a n d A p p l i e d S c i e n c e , Y a l e U n i v e r s i t y , N e w H a v e n , C o n n .
The evolution within t h e p a s t 2 0 yeans on s o o f a b o d y o f information on highspeed gas flow—shock waves and isentnopic expansions t o supersonic speeds—has dramatically a d v a n c e d t h e s t u d y of fast gas-phase
reactions
and o t h e r rapid relaxation p r o c e s s e s in p h y s i c a l chemistry. W i t h this evolution has c o m e t h e d e v e l o p m e n t of a n e w b r a n c h of fluid m e c h a n i c s called g a s dynarr,!cs
S
trong interaction has always existed between fluid mechanics and chemistry. The motions of stirring, pouring and mixing, flows over obstacles, filtering, and the like have long been used by chemists and their predecessors. These techniques were based on common experience rather than on theoretical studies. • Fluid mechanical process technologies also became standard tools. Distillation processes and studies of steady and unsteady flows through pipes, valves, and vessels of all shapes appeared in the chemical laboratory. In these classical situations low flow speeds prevail, and the density of the fluid may be considered constant without indulging in unwarranted simplifications of the actual flow picture. Such flows are called incompressible. And gases, although they have a compressibility of 10 4 to 10 5 times greater than that of liquids, may, at least at low speeds, be treated similarly. Incompressibility implies that the change of density induced by the pressure variation in the flow field under consideration is negligible. If flow speed is increased to a significant portion of the speed of sound, this assumption becomes invalid because at high speeds density varies appreciably in flows, and compressibility must be taken into account. In such cases we ordinarily deal with gases, and if density is a function of pressure, the need of an equation of state arises. Thermodynamics appears on the scene of fluid mechanics. This new branch of fluid mechanics is called gas dynamics. Compressibility effects are also important if large gas masses such as our atmosphere are under consideration. The term gas dynamics, however., is reserved for
high-speed motion. With the extension of classical fluid mechanics to gas dynamics, useful tools for the chemist again become available. Some of these tools are discussed in this article. Traditionally, chemists have used examples of such high-speed flows in the study of detonations and explosions. However, I will omit combustion problems and discuss the simpler processes of shock waves, isentropic expansions to supersonic speeds, and the like. The mastery by research workers of these flows within the past 20 years or so has dramatically advanced the study of fast gasphase reactions and other rapid relaxation processes in physical chemistry. The object of this article is to discuss the fluid mechanics underlying these new techniques, rather than to present an historical accounting. This article probes the deeper reasons for achieving the remarkable time resolution and the unusual thermodynamic states inherent in the application of gas dynamics to physical chemistry. Incompressible
flows
Let us first return to further discussion of the more familiar incompressible flows. Many fundamental aspects of fluid motion, such as the classification of flows with and without shear, are changed at high speeds only in degree rather than in principle. Consider the familiar pipe flow often used by chemists in studies of reactions. The difference between laminar and turbulent flow can be recognized by visual observation of flow treated with dye. Or, they may be detected by finding radically different velocity profiles from wall to center. The dimensionless similarity param-
JULY 18, 1966 C&EN 77
Many chemical reactions occur In * high-ternperature air
; 0;
X2 + M + H , ^ X 2 + M
(1)
0 2 + M + 5.Xev*=±0 + 0 + M
(2)
N2 + M + 9-8 ev ^ N + N + M
(3)
NO + M + 6.5 ev^± N + 0 + M
(4)
NO + O + L 4 e v ^ 0 2 + N
(5)
N2 + 0 + 3.3 ev ^ NO + N
(6)
N2 + 0 2 + 1.9 e v ^ NO + NO
(7)
N +0
(8)
+2.8ev~>N0*+e
lVQOO 2,000 3,000 ^OQO 5,000;^ {\ - V ""TXdiegrees^Kelyin), -^:^^; Jj^X^fy%,
feSffli&i
£L&&4d apStates of 5 0 W 6 R. can readily be reached in gas dynamic plications. At these high temperatures, the effect of dissociation leading to PV/RT ^ 1 forces us to abandon the law of thermally perfect gases and to consider the role of caloric imperfections. The graph (after Wooley) shows the effect
eter controlling these types of flows is called the Reynolds number, which is defined as follows:
The picture of laminar flow at low Reynolds numbers and dominant viscous forces is one in which the momentum exchange between adjacent fluid layers occurs by the internal friction caused by viscosity. Conversely, at high Reynolds numbers, as in most pipe flows, inertial forces prevail except for an extremely narrow region near the wall. The turbulent main flow provides an effective momentum exchange by irregular bulk motion of large elements of fluid. The transition between these two modes of the flow usually occurs at pipe Reynolds numbers of about 2500; and, the process still defies full solution. If flows occur around obstacles in an extended space, or if the confining walls of an internal flow system are far from each other, the situation may be very different. In such open systems, as contrasted to pipe flow, it is often permissible to disregard the effects of viscous or turbulent momentum exchange in large parts of the flow field. Only in regions of highvelocity gradients at right angles to the main flow direction (high shear) is it unrealistic to treat the medium as an ideal fluid and to neglect viscosity. Shear regions always occur near solid 78 C&EN JULY 18, 1966
of dissociation at high temperatures on the enthalpy of nitrogen, computed for a dissociation energy of 9.756 e.v. In high-temperature air (3000 to 8000° K.), reactions such as these in the table occur. Initially workers thought that the system was a simple O-N mixture (after Wray)
surfaces at which the fluid speed drops to zero. The resulting division of a flow field—an inviscid free stream, the potential flow, and the narrow layers near surfaces to which all effects of friction are relegated—was first proposed by Prandtl early in this century. He introduced the concept of the boundary layer that is now used in describing the shear region near the wall. This concept marks the starting point of modern fluid dynamics. In turn, separation or breakaway of fluid from surfaces, such as seen in wakes, sharp turns, and the like is a direct consequence of the speed retardation in these boundary layers. These phenomena of real fluids were not accounted for in the classical mathematical theory of potential flows. Consequently, the classical approach failed to describe drag of bodies and other experimental observations. Qualitatively, these phenomena occur in flows at low speed and in those at high Mach numbers. Finally, we recall another fluid mechanical regime, also well known to the chemist, that retains its qualitative features in gas dynamics. So far, the described motions have been assumed to occur at pressures sufficiently high to ignore the molecular nature of the fluid. The fluid can then be treated as a continuum for which only certain bulk properties need be known. However, a differ-
ent regime of fluid mechanics occurs in vacuum systems where densities may be lowered to attain the state of free molecule flow ( F M F ) . In this state the mean free path of molecules is of the order of a typical linear dimension of the geometry of the flow. Such flows may be scaled by use of the Knudsen number ( K ) , which is another dimensionless similarity parameter defined by M^a*o(/uie,"paXlu ^ " t k e /
w^laAiA^
Kr The F M F regime is reached if the Knudsen number approaches or exceeds one. Such conditions, aside from laboratory vacuum systems, are encountered, for example, in studies of air chemistry at high altitudes in our atmosphere. Gas dynamics and
chemistry
The development of gas dynamics and the application of high-speed techniques to chemistry have shown a mutual interdependence. This recent history may be studied by determining why chemistry became important to fluid mechanics and in what manner the two fields converged in several areas. Again, we start from the viewpoint of fluid mechanics and define a demarcation between low-speed flow and flows where density is an important variable. Intuitively, we suspect this state occurs if the flow speed in-
creases to a point where the elastic properties of the fluid become important. Indeed, such effects enter if the flow speed approaches the speed of sound. We recall that small disturbances travel in a compressible gas with the acoustic velocity given by
where entropy is held constant in the differentiation. The speed of sound in normal air is close to 340 miles per second, a speed exceeding the usual speeds with which gases are passed through laboratory systems. At high speed, therefore, a similarity parameter in addition to the Reynolds number must be considered. This dimensionless parameter is the well-known Mach number (M) named after the Austrian physicist and philosopher, Ernst Mach (18381916), and it is defined by
We will have to deal with the Mach number because the gas dynamic processes discussed here depend on high speeds to produce a high time resolution for chemical processes. Quantitatively, the assumption of incompressibility ceases to be a good approximation of reality, once the local density difference, A/o, imposed by a flow field in terms of the undisturbed density (p) becomes significant. One can show that we have approximately P If we make a 5% density change in a flow field the limit of "small" (setting Ap/p — 0.05), the equation shows that for Mach numbers of 1/3 and less, the flow may be treated as incompressible. Therefore, at flow speeds up to about 250 miles per hour in normal air, compressibility of the air plays a small role and similitude based on Mach number need not be considered. The Reynolds number, therefore, reigns by itself, providing an equally sufficient key to the study of a fully submerged fish in the ocean, a propeller-driven airplane, or the flow processes in a refinery. At increasing Mach number compressibility can no longer be ignored. In particular, conditions change when the flow speed becomes supersonic (M > 1). Noise will then not precede a moving body, and the disturbances caused in the air are restricted to a roughly conical envelope trailing the object. Von Karman termed the resulting two major regions of a supersonic flow field the zones of silence and of action. This division into major regions demonstrates the sharp delineation of dis continuously changing
properties that are highly useful to the chemist. Flow regimes ordered according to the magnitude of the Mach number have become accepted as follows: • M < < 1—low-speed flow, density is constant. • M < 1—subsonic flow, compressibility must be considered. • M ~ 1—transonic flow, regions of sub- and supersonic flow simultaneously present. • M > 1—supersonic flow, no subsonic flows present except in narrow boundary layers and wakes. • M > > 1—hypersonic flow, say, M > 5, ideal gas laws break down. The term hypersonic, suggested by H. S. Tsien, does not simply represent a Greek equivalent for the Latin supersonic. Rather, the term characterizes an area of gas dynamics where action is restricted to a very narrow zone in which, owing to the extreme speed, the temperature increases strongly and the ideal gas laws are no longer applicable. This area of gas dynamics is, in fact, the regime where chemistry prominently and irrevocably enters the life of the aerodynamicist. High speeds of bodies in extended media, however, do not drastically alter the notions discussed in reference to boundary layers, separated flows, and the like. Indeed, because high Reynolds number flows in extended media are likely to occur because of the high speeds, it is once more often permissible to drop the viscous terms from the equations of motion in most of the flow field. The viscous terms cannot ever be neglected in the flow close to the wall irrespective of the free stream speed, because as always the flow speed becomes zero at the surface. As in low speed flow, this last condition is strictly valid only as long as the Knudsen number is small. The limits of continuum in a given application can readily be checked for all three similarity parameters discussed so far by appropriate subsitution of flow parameters in the following manner. From the relations between mean molecular speed and acoustic velocity, and mean free path and viscosity, as provided by kinetic theory, a simple relation between our similarity parameters given by
may be found for fixing the appropriate flow regime. The practical usefulness of these gas dynamic concepts has only lately entered chemistry. This late entry is at first surprising, because supersonic motion was known to artillerists in the last century. Ernst Mach per-
formed his classical experiments on supersonic flow before 1900, and Rankine, Hugoniot, Saint-Venant, Wantzel, Riemann, and others did much theoretical work on compressible flows in the past century. However, if we take international congresses as an indication of the interest evinced in a given field, we see that it took until 1935 for the first such gathering—the Volta Congress, sponsored by the Italian Academy in Rome and specifically devoted to compressible flows. The major reason for this late blooming of the field is probably associated with the development of applications to flight. The lack of sufficient power of internal combustion engines prevented aircraft from attaining sonic speeds. Similarly, the large expenditures of power and other technological advances required to operate wind tunnels supersonically were not available. It required the advent of jet power plants and large testing facilities in the period toward the end of and following World War II to mark the beginning of truly supersonic flight. This development also encompasses the activities in rocketry and in space, because the atmosphere must be traversed on the way out from our planet and on return. These aeronautical advances, in turn, led to the study and perfection of tools used by the chemist to resolve the time scale of gas reactions and other fast processes in physical chemistry hitherto considered to occur instantaneously. High
temperatures
Chemists have long attempted to study high-temperature gas reactions. Traditionally, flames, ovens, reactor bombs, rapid compression devices, flash photolysis methods, and the like have been used to produce temperatures up to a few thousand degrees. Electrical discharges were also employed, but the energy transferred to the reagent gases is often too high for the study of many reactions. The problems connected with these older methods stem from the lack of exact knowledge of the thermodynamic state to which the gases under study are exposed. Also, the difficulty arises how to resolve the sequence of states accurately. The better controlled but slower steadystate flow techniques such as those based on pipe flow are restricted to studies of reaction times longer than about one millisecond. More recently, times of rapid gas compression in the microsecond range have been achieved using the ingenious pitot-tube technique introduced by Kantrowitz(1942). Although vibrational relaxation phenomena can be JULY 18, 1966 C&EN 79
studied with this scheme, measurements of gas reactions have not been successful. Similarly, problems of interpretation have risen in understanding kinetic data obtained in ultrasonic interferometers. The rather ambiguous status of these high-temperature techniques and the resulting discrepancies in the quantitative findings on gas kinetics prevailed until the introduction of shock wave techniques into chemistry in the late 1940's. In parallel and initially quite separate from chemistry, the advent of research in hypersonic flows brought about a rapid broadening of the intellectual range of the aerodynamicist. Many a physical chemistry text found its way into his hands; and precedent and cross connections to earlier rocket research were rediscovered. This development has reached an impressive current extent. About 40% of all original contributions in two arbitrarily picked 1965 issues of the Journal of the American Institute of Aeronautics and Astronautics (AIAA) deal with chemical problems and ionized gases in flow systems. We can best see how this situation in aerodynamics came about by studying the conservation of energy in a small volume of gas mass, m, at two points on its path, denoted by 1 and 2, as expressed in the following equation:
H
\
/
o
E
nniBooo
-£.u I * + H l ~ £ a ! + H t - H 0 Distance
H represents the enthalpy, a natural variable for flow problems, since it is sum of the internal energy and the
The presence of the shock wave is transmitted backwards, as shown in this time-distance sequence
Laboratory fixed coordinates
Coordinates moving with shock front r 'p;/ 1; '~."f $, I Shock front
/ / / / / / / // // // // // /' /' ' "' /' /' / ' Shock frbnt
Particle path
X= 0 Time-distance graphs for shock wave fiow take on different configurations, depending on whether the coordinates are moving with the shock front or are fixed 80 C&EN JULY 18f 1966
V;
flow work. If the flow is brought to rest (u = 0 ) , say, at the stagnation point on the nose of some body, we find H 0 at this point. This stagnation enthalpy is larger than the enthalpy far from the body, by the amount of the kinetic energy stored in the sample air mass moving at the free stream speed. If the gas behavior is taken as ideal (an excellent assumption at lower Mach numbers), enthalpy can be expressed in terms of temperature. Then, for the stagnation temperature on the nose, the energy equation yields the following expression:
where T is the ambient air temperature, and y = C p / C v (where C p and C v are the molar heat capacities at constant pressure and volume, respectively). Assuming motion in the lower stratosphere (T = - 5 9 ° C.) we can compute the temperature increase, AT = T 0 — T on the heatinsulated nose of a flight article as given in the following table: M
u(m.p.h.)
327 653
T (° C.)
11 43 171 685
0.5 1.0 2.0 4.0 6.0
1,310 3,920
1,540
10.0
6,530
4,280
2,610
The temperature increase resulting from the motion is seen to be impressive at supersonic speeds. At
Mach numbers above 5 or so the temperature rises to values too high for the assumption of constant specific heats implicit in this calculation to remain valid. This effect causes a departure from the ideal gas behavior. In contrast to the departure from the calorically perfect behavior, the participating species individually remain thermally perfect gases. Intermolecular forces can be neglected in the region near the nose of the body at any altitude owing to the relatively low pressure at the stagnation point achieved by the inefficient compression in the shock wave. The increasing speeds and resulting increased temperatures lead in succession to a variety of processes in the gas mixture making up our air. These effects include, sequentially, the vibrational excitation and thermal dissociation of the diatomic molecular oxygen and nitrogen; reactions between the products, such as burning of nitrogen to form nitrous oxide; ionization of the species present; and, finally, electronic excitation. The latter processes are accompanied by radiation, and, in fact, the gas cap in front of a blunt body becomes a plasma. At typical entry conditions of a longrange missile, therefore, the electrical conductivity of the air near the nose is roughly equal to that of sea water. At even higher speeds than shown previously (such as that of bodies entering the atmosphere at the satellite speed of about 8 kilometers per second) the heating becomes extreme indeed. The kinetic energy of 1 gram mass moving at this speed, if converted into heat, is 7.5 kcal!
iliii Numbers refer to uniform llow regions of different
(The energy involved in the deceleration of unit mass from satellite speed to rest is equivalent to the energy released by roughly 10 times unit mass of TNT.) The approach to earth of a heavy object moving initially at satellite or higher speeds will, therefore, literally lead to astronomically high temperatures in the flow field. This heat buildup has long been observed in the fireballs produced by the entry of large meteoritic bodies in the atmosphere. Considering these facts, we expect technological solutions to the hypersonic entry problem to involve chemical processes in handling the required dissipation of the kinetic energy. This dissipation is, in fact, accomplished primarily by three means: • The transfer of energy to the ambient air (rather than to the body) via shock waves. This revolutionary fluid mechanical idea has led to the concept of high-drag blunt-body entry shapes (H. J. Allen and coworkers). • Radiation from the heated gas cap and body surface. • The energy consumption of ablating heat shields. A whole set of largely chemical problems had to be solved to predict the function and efficiency of these processes. Equilibrium properties of air at high temperatures had to be determined to obtain rational solutions for the bow-shock flow. Boundarylayer problems including reactions had to be solved. Catalysis on a variety of surfaces had to be understood, new material problems to find efficient ablators had to be solved, and other
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depend on i ^ (pj dfiddrh^ pfe$siiti0f
