W. W. Jones and J. P. Boris
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Flame and Reactive Jet Studies Using a Self-Consistent Two-Dimensional Hydrocode Walter W. Jones* and Jay P. Boris Naval Research Laboratory, Plasma Dynamics Branch, Washington, D.C. 20375 (Received July 6, 1977) Publication costs assisted by the Naval Research Laboratory
We are developing a self-consistent, two-dimensional (FCT) hydrocode to study ignition and propagation of flames in vapors and gas jets. Included are the important physical processes of particle diffusion, viscosity, and thermal conduction. The effects we are studying are those of (1)mixedness of initial fuel-oxygen, (2) initial fuel temperatures, and (3) streaming velocity of the jet. For initial tests we have used a one-stage reaction of the form A + B C + heat. where the heat release used was appropriate to methane combustion.
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Introduction We are developing a two-dimensional reactive flow hydrocode to study the interaction of chemical kinetics and hydrodynamics in reactive gas jets and the spread of flames in various types of fuels. The short term interest is to study the details of turbulent mixing and its effect on flame spread and burn efficiency. In the long run, we hope to look a t fire suppression techniques, such as nitrogen pressurization, and the dynamics of large fires. The model which we have developed is quite general. It has been designed as a flexible framework for our future development, such as studying open as well as closed boundries and their effect on the dynamics. The model consists of three distinct parts. The first is the basic compressible flow hydrodynamics,l the second consists of the basic transport, and the third is the chemical kinetics. Each of these sections of the model has been tested independently before being added to the overall computer code. This scheme allows us to test each section, and also ensures flexibility in the construction of the model. This forms the basic framework within which we can study the more interesting phenomena, such as turbulent mixing. Problem Description Initially we are looking a t a closed cylindrical combustion chamber. Figure 1 shows graphically the finitedifference grid. We have assumed cylindrical symmetry with hard impermeable walls. In all figures which follow, the left-hand side of the figure represents the axis of symmetry of the cylinder. When we inject fuel in the bottom center of the chamber, it will appear at the lower left corner of the grid. We report on three tests used to check the model. They are (1)a bouyant bubble of hot gas without reactions, but with the basic transport, such as diffusion and thermal conductivity, (2) a bubble of hot fuel in an oxygen atmosphere, and (3) a gas jet in an oxygen atmosphere. The preliminary calculations reported here lack turbulence modeling of any sort and assume an inviscid fluid. Theory The equations which we solve are the usual hydrodynamic and chemistry equations with appropriate source and sink terms added. The variables which we solve for are the mass density of each species, the total mass density, the velocity, and the total energy. In their original form the equations are -+ aPT/at= -V*pTv (11 -++ -+ a p , v p t = - v . p v v - VP + p g + vwv (2) -+ a e l a t = - V * € V - V-PV + V - K V T 7 [ae/at],,,, (3) where c = 1 / 2 p V + p g h + P / ( y - l),3 = total fluid veThe Journal of Physical Chemistry, Vol. 81, No. 25, 1977
locity, state
pT
= total density, and the ideal gas equation of
P = nTkT
(4)
Equations 1-4 are the typical conservation equations for mass, momentum, and energy. A problem which arises in the solution of these equations is that perturbations in pressure can travel at the speed of sound. This is, in general, much faster than any relevant fluid motion in the problems that we wish to consider. Thus we make the assumption that pressure can only vary asymptotically (either increase or decrease). Such an assumption makes it desirable to rewrite the conservation equations in a form where the variables are mass density, vorticity, and divergence. The divergence equation can then be solved algebraically (assuming no viscosity nor mass diffusion a t present).
3
+ V*(KTVT)
y P ( V ‘ v ) = -v*VP
where
(7)
-+
t;=vxv
Equation 7 is the asymptotic form of the time derivative of pressure, and is not allowed to oscillate (thus preventing sound waves). The assumption which is implicit here is that the system has a sufficient time to equilibrate and the variations due to propagating pressure waves do not significantly change the basic processes. This will limit us to relatively slow combustion, but most cases of interest fall into this category. If reactions are allowed, then one more set of equations must be solved. This is the set of mass conservation equations for the individual species. The form is similar to eq 7 but includes reactions and diffusion in the general case: -+
a p i / a t = - V * p i V - ZRijkPjPh kj
f
V.(DjVpi)
(8)
where Rl,k may be positive or negative, to account for sinks as well as sources of mass. In addition, the term [aP/at],hem must then be added to the right-hand side of eq 7 to account for the energy release. In the one stage process used, species 1 and 2 represent the fuel and oxygen respectively and 3 is the product.
Test of the Model As discussed earlier, we have used several tests on the model, and have made comparisons with the results obtained by “back of the envelope” calculations. The first of these test was a simple bouyant bubble of hot gas. In free space, a bubble of gas should rise at a rate which is
Flame and Reactive Jet Studies
2533
GRID POSITION
Flgure 1. b 1=195SEC
f=3.%5SEC
D=O
2m
I-
I-
zm
I..
2m
Flgure 2. h
R-
2m
-.....IS OXYGEN
Figure 3.
Rf ISFUEL
2m
R-.
---- IS PRODUCT
2m
dependent on gravity and the entrainment of cold air. In addition, depending upon the amount of diffusion present, the bubble should spread out somewhat and eventually equilibrate a t some level. Figure 2 shows the initial “bubble”, with two runs. The first (top) sequence is with no diffusion, and the second sequence is with normal (for air) diffusion. A third run was made with very large diffusion, and the expected happened, namely, the bubble dissipated. In both cases shown, the time is given at the top of the frame. The important parameters are radius of chamber = 2 m, radius of bubble = 0.5 m, height of chamber = 4 m, contents = one species only (no reactions), diffusion coefficient (ref 2), pressure = atmospheric pressure; grid = 20 X 20 (variable in the model). Contours are of constant density with the inner contour representing the lowest density which is chosen to show the density fluctuations. With our approximations, these contours also describe, approximately, the temperature. In these two runs, no chemical reactions took place, as noted above. A simple estimate shows that the bubble should have risen about 2.5 m in 2 s, and Figure 2 shows that this is approximately correct. The next check was to include the chemistry features, which implies mixing and reaction, with subsequent energy release. The initial test setup was the same as the previous run, except the hot bubble of gas consisted of fuel (methane) with oxygen surrounding it, at atmospheric pressure. A simple one stage reaction was assumed. To justify treating methane in such a cavalier fashion, we point out that we simply needed a chemistry scheme where the rates and reaction energies are known approximately, as a test of our model. (Now that the model is provisionally operational, much more carefully conducted and analyzed tests are being performed). The model performed as expected, that is, the fuel and oxygen reacted, releasing energy and producing waste. The primary difference between this test and the previous one was that the bubble (or actually a torus) rises somewhat faster and spreads out more. Such a test is difficult to quantify so one more test of the chemistry was necessary to ensure that it was self-consistent. We ran one case where there was no gravity and the bubble of hot fuel was placed in the center of the grid. The fuel and oxygen were assumed to be uniformly mixed. In this case, the bubble expanded until the internal pressures was equal to the external pressure, and then the gas began to dissipate the thermal energy diffusively and cool, with subsequent contraction. The fuel decrement in this case was 100%. Another test, in view of our interests, is to consider the problem of a gas jet. This requires inflow at a boundary. Such a test was run and the results are shown in Figure
Flgure 4.
The Journal 01 physical Cbamishy. Vol. 81. No. 25, 1977
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J. A. Miller and R. J. Kee
3. In this case, the initial conditions were essentially the same as in previous runs. However, as fuel was burned, more fuel was injected (hot but with zero velocity) into the bottom center of the test chamber. As can be seen, the fuel oxidizes as it rises and mixes with the oxygen. Figure 4 shows the actual computer printout from a test similar to that shown in Figure 3. It should be noted that the contour levels are not constant from frame to frame, but are chosen to show the position of each of the species. Conclusions We are developing a two-dimensional hydrocode with self-consistant transport and chemical kinetics. We use
a two-dimensional model, since we are interested in the problems of mixing and turbulence, which cannot be done in a satisfactory manner using one-point calculations, or one-dimensional models. While the results of our simple tests so far are extremely encouraging, considerable careful and quantitative analysis of the results is called for and detailed diagnostic routines have to be developed before more interesting and intricate problems can be attacked.
References and Notes (1) D. L. Book, J. P. Boris, and K. Hain, J . Comp. Phys., 18, 248 (1975). (2) F.W. Sears, "An Introduction to Thermodynamics, The Kinetic Theory of Gases and Statistical Mechanics", Addison-Wesley, Reading, Mass., 1964; T. R. Marrero and E. A. Mason, J . Phys. Chem. Ref. Data, 1, 3 (1972).
Chemical Nonequilibrium Effects in Hydrogen-Air Laminar Jet Diffusion Flames James A. Miller" and Robert J. Kee Combustion Research Division 835 1, Sandia Laboratories, Livermore, California 94550 (Received April 26, 1977) Publication costs assisted by Sandia Laboratories
A theoretical model for the structure of hydrogen-air laminar jet diffusion flames has been developed. The model includes the effects of nonequilibrium chemical kinetics and realistic transport properties. The reaction zone in such flames is found to resemble the recombination zone in premixed flames. Distributions of the free radicals H, 0, and OH are substantially in excess of local equilibrium values. The computed results show that the partial equilibrium approximation, which is often employed in premixed flames, is valid within the reaction zone but breaks down outside it. Hydroperoxyl is found to be an important intermediate species in the flame, being formed primarily by the recombination reaction H + O2 + M HOz + M and destroyed by H, 0, and OH radical attack. The absence of chemical equilibrium and the broadening of the energy release zone result in peak temperatures which are well below the adiabatic flame temperature. Broadening of the energy release zone is due significantly to the mobility of hydrogen atoms.
Introduction The purpose of the present investigation is to examine, by means of a theoretical model, the effects of nonequilibrium chemical kinetics on the structure of H2/air laminar jet diffusion flames. Such a detailed treatment has not appeared previously in the literature because of the inherent difficulties in handling the multiple, vastly differing, time and length scales characteristic of the problem. These disparate scales make the integration of the differential equations governing the behavior of the flame a troublesome task. Historically, it has been necessary to resort to simplifying assumptions about the chemical kinetics in order to obtain solutions. To illustrate the point and to define the problem more precisely, consider a fuel jet which is issuing vertically into ambient still air and that has been ignited for a sufficient time to allow a steady flow to develop (Figure 1). We can think of the flame length L as the characteristic convective length scale in the problem and the nozzle radius R as characteristic of the diffusive scales. The usual situation in diffusion flames is that the nozzle radius is small compared to the flame length. In terms of time scales, we define Td = R / V d and 7, = L / U , where U and v d are characteristic convective and radial diffusive velocities, respectively. The nature of the physical problem is such that Td and 7, are of the same order of magnitude (Le., diffusion of mass and energy must occur between r = R and r = 0 over the axial length L in order to burn all the The Journal of Physical Chemistry. Voi. 8 1, No. 25, 1977
fuel), and we may think of one time as being characteristic of the fluid and transport aspects of the problem. A third time scale of interest is TQ, the time associated with the chemical conversion of reactants into products and the accompanying evolution of chemical energy. At high temperatures, where chemical reactions occur very rapidly, TQ typically is very small compared to Td or 7,; that is, the Damkohler number is very large, Td/TQ >> 1,and intense chemical activity might be expected to confine itself to a relatively narrow region between the ambient air and the fuel stream. To be specific, Td/TQ = 103-104for problems of interest here. The problem is complicated further, and most importantly, by the fact that TQ is only one of the largest of a number of time scales associated with the complex chemical reaction. Times associated with the creation and destruction of reaction intermediates, particularly free radicals, typically are much smaller than r~ These disparate time scales lie at the heart of the computational problems in predicting the structure of laminar jet diffusion flames. Historically, diffusion flame problems have been solved and by taking one or both of the two limits T d / r Q Td/TQ 0. The first limit constitutes equilibrium flow and the second, frozen flow. Both eliminate the need for the explicit consideration of any of the chemical time scales. The Burke-Schumannl flame sheet approximation effectively results in infinitely fast reactions in the reaction zone, reducing this zone to a flame sheet. Frozen flow then
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