ࡱ>  t5@Objbj22 XX}4L/L/L/h/@1N@2 ;"<<<<<<KKKKKKK$ORQ*Ly?<<??L<<NJJJ?><<KJ?KJJKK<42 y_L/G\KK4N0NKSZJSKSK <<|JD=d=<<<LL)L/JL/Scale-Invariant Forms of Conservation Equations in Reactive Fields and a Modified Hydro-Thermo-Diffusive Theory of Laminar Flames SIAVASH H. SOHRAB Robert McCormick School of Engineering and Applied Science Department of Mechanical Engineering Northwestern University, Evanston, Illinois 60208 UNITED STATES OF AMERICA Abstract:- A scale-invariant model of statistical mechanics is applied to present invariant forms of mass, energy, linear, and angular momentum conservation equations in reactive fields. The resulting conservation equations at molecular-dynamic scale are solved by the method of large activation energy asymptotics to describe the hydro-thermo-diffusive structure of laminar premixed flames. The predicted temperature and velocity profiles are in agreement with the observations. Also, with realistic physico-chemical properties and chemical-kinetic parameters for a single-step overall combustion of stoichiometric methane-air premixed flame, the laminar flame propagation velocity of 42.1 cm/s is calculated in agreement with the experimental value. Key-Words:- Invariant forms of conservation equations in reactive fields. Theory of laminar flames. 1 Introduction The universality of turbulent phenomena from stochastic quantum fields to classical hydrodynamic fields resulted in recent introduction of a scale-invariant model of statistical mechanics and its application to the field of thermodynamics [4]. The implications of the model to the study of transport phenomena and invariant forms of conservation equations have also been addressed [5]. In the present study, the invariant forms of the conservation equations are described and the results are employed to introduce a modified hydro-thermo-diffusive theory of laminar premixed flames. 2 A Scale-Invariant Model of Statistical Mechanics Following the classical methods [1-3], the invariant definitions of the density rb, and the velocity of atom ub, element vb, and system wb at the scale b are given as [4]  EMBED Equation.DSMT4  , ub = vb-1 (1)  EMBED Equation.DSMT4  , wb = vb+1 (2) The scale-invariant model of statistical mechanics for equilibrium fields of . . . eddy-, cluster-, molecular-, atomic-dynamics . . . at the scale b = e, c, m, a, and the corresponding non-equilibrium laminar flow fields are schematically shown in Fig.1. Each statistical field, described by a distribution function fb(ub) = fb(rb, ub, tb) drbdub, defines a "system" that is composed of an ensemble of "elements", each element is composed of an ensemble of small particles viewed as point-mass "atoms". The element (system) of the smaller scale (b) becomes the atom (element) of the larger scale (b+1). The three characteristic length scales associated with the free paths of atoms, and elements, and the size of the system at any scale b are (lb = lb-1, lb, Lb = lb+1) where lb = <l2b>1/2 is mean-free-path of the atoms [5]. The invariant definitions of the peculiar and the diffusion velocities have been introduced as [4]  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4  (3) such that  EMBED Equation.DSMT4  (4) The above definitions are applied to introduce the invariant definitions of equilibrium and non-equilibrium thermodynamic translational temperature and pressure as [4]  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4  (5) and,  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4  (6) leading to the corresponding invariant forms of ideal "gas" laws [4]  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4  (7)  Fig.1 Hierarchy of statistical fields for equilibrium eddy-, cluster-, and molecular-dynamic scales and the associated laminar flow fields. 3 Scale-Invariant form of the Conservation Equations for Chemically-Reactive Fields Following the classical methods [1-3], the scale-invariant forms of mass, thermal energy, linear and angular momentum conservation equations [5] at scale b are given as  EMBED Equation.DSMT4  EMBED Equation.DSMT4  (8)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (9)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (10)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (11) where eb = rbhb, pb = rbvb , and pb = rbwb are the volumetric density of thermal energy, linear and angular momentum of the field, respectively and  EMBED Equation.DSMT4  is the vorticity. Also, Wb is the chemical reaction rate and hb is the absolute enthalpy [5]. The local velocity vb in (8)-(11) is expressed as the sum of convective wb = and diffusive velocities [5]  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4  (12a)  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4  (12b)  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4  (12c)  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4  (12d) where  EMBED Equation.DSMT4 are respectively the diffusive, the thermo-diffusive, the linear hydro-diffusive, and the angular hydro-diffusive velocities. For unity Schmidt and Prandtl numbers Scb = Prb = nb/Db = nb/ab = 1, one may express  EMBED Equation.DSMT4  (12e)  EMBED Equation.DSMT4  (12f)  EMBED Equation.DSMT4  (12g) that involve the thermal Vbt, the linear (translational) hydrodynamic Vbh, and the angular (rotational) hydrodynamic Vbrh diffusion velocities defined as [5]  EMBED Equation.DSMT4  (13a)  EMBED Equation.DSMT4  (13b)  EMBED Equation.DSMT4  (13c) Since for an ideal gas  EMBED Equation.DSMT4 , when  EMBED Equation.DSMT4  is constant and  EMBED Equation.DSMT4 , Eq.(13a) reduces to the Fourier law of heat conduction  EMBED Equation.DSMT4  (14) where kb and ab = kb/(rbcpb) are the thermal conductivity and diffusivity. Similarly, (13b) may be identified as the shear stress associated with diffusional flux of linear momentum and expressed by the generalized Newton law of viscosity [5]  EMBED Equation.DSMT4  (15) Finally, (13c) may be identified as the shear stress induced by diffusional flux of angular momentum (torsional stress) and expressed as  EMBED Equation.DSMT4  (16)) Substitutions from (12a)-(12d) into (8)-(11), neglecting cross-diffusion terms and assuming constant transport coefficients with Scb = Prb = 1, result in  EMBED Equation.DSMT4  (17)  EMBED Equation.DSMT4  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (18)  EMBED Equation.DSMT4  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (19)  EMBED Equation.DSMT4  EMBED Equation.DSMT4  (20) The above forms of the conservation equations perhaps help to better reveal the coupling between the gravitational versus the inertial contributions to total energy and momentum densities of the field. Except for possible externally imposed sources, eb, pb and pb have no internal sources as reflected in (9)-(11). However, in the presence of chemical reactions, the loss of gravitational mass could result in the production of inertial thermal energy or linear and angular momenta. For example, the first and the second parts of (18) respectively correspond to the gravitational and the thermal contributions to the total energy density of the field. For instance, the loss of gravitational mass induced by chemical reaction in the body of a person results in the generation of thermal energy (heat) in this persons body. Similarly, the first and the second parts of (19) respectively correspond to the gravitational and the inertial contributions to the total linear momentum density of the field. Now, one considers a stationary person with no initial linear momentum that suddenly starts to run, thus producing substantial linear momentum without the action of any external forces. In this case, there is no violation of the conservation of momentum, but rather because of chemical reactions in the body of such a person, the first part of (19) changes thus leading to a compensating change in the second part. Finally, the first and the second parts of (20) respectively correspond to the gravitational and the inertial contributions to the total angular momentum density of the field. For example, (20) may be used to describe the change of angular velocity of a ballet dancer. Here, the loss of mass by chemical reactions in the body of a spinning dancer that pulls the arms inwards, thus doing work against centrifugal forces, leads to an increase in the dancer's angular momentum. Because of the large value of the velocity of light c in the equation E = mc2, the actual loss of gravitational mass in the above examples will be exceedingly small. Substitutions from (17) into (18)-(20) result in the invariant forms of conservation equations [5]  EMBED Equation.DSMT4  (21)  EMBED Equation.DSMT4  (22)  EMBED Equation.DSMT4  (23)  EMBED Equation.DSMT4  (24) Equation (24) is the modified form of the Helmholtz vorticity equation for chemically reactive flow fields. The last two terms of (24) respectively correspond to vorticity generation by vortex-stretching and chemical reactions. Also, equation (23) is the scale-invariant equation of motion in reactive fields [5] that includes the reaction term (-vbWb/rb) representing generation Wb < 0 (annihilation Wb > 0) of linear momentum accompanied by release (absorption) of thermal energy associated with exothermic (endothermic) chemical reactions. It is known that as flames propagate, they convert stationary reactants to moving combustion products because of thermal expansion. Another important feature of the modified equation of motion (22) is that it involves a convective velocity wb that is different from the local fluid velocity vb. Consequently, when the convective velocity vanishes wb = 0, equation (23) reduces to the diffusion equation similar to mass and heat conservation equations (21)-(22). Because the convective velocity wb is not locally defined it cannot occur in differential form within the conservation equations [5]. This is because one cannot differentiate a function that is not locally, i.e. differentially, defined. To determine wb, one needs to go to the next higher scale (b+1) where wb = vb+1 becomes a local velocity. However, at this new scale one encounters yet another convective velocity wb+1 which is not known, requiring consideration of the higher scale (b+2). This unending chain constitutes the closure problem of the statistical theory of turbulence discussed earlier [5]. By summation of (8)-(11) over (b) one can arrive at the conservation equations at the next higher scale of (b+1). By such procedure, one can move from molecular-dynamic to cluster-dynamic scale or from cluster-dynamic to eddy-dynamic scale within the cascade of embedded statistical fields (Fig.1). The summation of Eq.(8) is simple since  EMBED Equation.DSMT4  (25) and  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (26) For Eq.(10), the summation of the first term is identical to that shown in (26). To treat the summation of the second term of (10), one starts with the relation based on (1)-(4)  EMBED Equation.DSMT4  (27) Multiplying (27) by (Yb+1 rbvb) and summing over (b) and (b+1) leads to  EMBED Equation.DSMT4   EMBED Equation.DSMT4  or  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (28) where Yb is mass fraction and use was made of the relation  EMBED Equation.DSMT4  from (4) in the last step. The summation of the energy (9) and vorticity (11) equations follow procedures similar to those used above in (27)-(28). 4 Connection Between the Modified form of Equation of Motion and the Navier-Stokes Equation The original form of the Navier-Stokes equation with constant coefficients is given as [1, 2]  EMBED Equation.DSMT4  (29) Since thermodynamic pressure Pt is an isotropic scalar, P in (29) is not Pt. Rather, the pressure P is generally identified as the mechanical pressure that is defined in terms of the total stress tensor  EMBED Equation.DSMT4  as [6]  EMBED Equation.DSMT4  (30) The normal viscous stress is given by (15) as  EMBED Equation.DSMT4  and since  EMBED Equation.DSMT4 because of isotropic nature of Pt, the gradient of (30) becomes  EMBED Equation.DSMT4  (31) Substituting from (31) in (29), the Navier-Stokes equation assumes the form  EMBED Equation.DSMT4  (32) that is almost identical to the modified equation of motion (23) with Wb = 0 except that in the latter the convective velocity wb is different from the local velocity vb. However, because (32) includes a diffusion term and the velocities wb and vb are related by EMBED Equation.DSMT4 , it is clear that (32) should in fact be written as (23). An example of exact solution of the modified equation of motion (23) was recently introduced [7] for the classical Blasius problem [2] of laminar flow over a flat plate. For this steady problem Eq.(23) in the boundary layer, with w'y = 0 and W = 0, reduces to  EMBED Equation.DSMT4  (33)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (33a)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (33b) where w'o is the constant free-stream velocity outside of the boundary layer and (x', y') are the coordinates along and normal to the wall, respectively. The local velocity v'x varies from v'x = = 0 at the wall y' = 0 to v'x EMBED Equation.DSMT4  w'o at the edge of the boundary layer at all axial positions. Therefore, the convective velocity w'x = , i.e. the mean value of the local velocity v'x inside the boundary layer, will have the constant value of w'x = w'o/2. Substituting for this convective velocity in (33) one obtains [7]  EMBED Equation.DSMT4  (34)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (35a)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (35b) in terms of the similarity variable EMBED Equation.DSMT4 , where vx = v'x /w'o, y = y'/lH, x = x'/lH, lH = n/w'o. The solution of (34)-(35) is the predicted velocity profile vx = erf h that is in excellent agreement with the experimental observations of Nikuradse [2]. The solution of (23) for the classical problem of Hagen-Poiseuille flow [2] in circular tubes has also been investigated [7]. It was found that the geometry of the predicted velocity profile involving Bessel function was quite similar to the classical parabolic profile and hence in agreement with the experimental observations. Finally, the exact solution of the modified Helmholtz vorticity equation (24) was recently reported [8] for the steady problem of non-reactive flow within a stationary liquid droplet that is located at the stagnation-point between two axisymmetric counter-flowing gaseous streams. It was found that such a spherical flow could be expressed by the dimensionless stream function [8]  EMBED Equation.DSMT4  (36) that describes two rings vortices that are located above and below the stagnation plane, rather than a single spherical Hill vortex for the classical problem of a droplet in a uniform stream [6]. Therefore, the preliminary investigations discussed above show that the modified equation of motion (23) does indeed lead to realistic solutions in agreement with experimental observations for these classical problems for which exact solutions of the Navier-Stokes equation are available. 5 Modified Hydro-Thermo-Diffusive Theory of Laminar Flames Theory of laminar flames is the most fundamental problem of combustion science and subject of many classical [9-18] as well as more recent [3, 19-24] studies. For one-dimensional propagation of a planar flame one introduces the dimensionless parameters  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (37) The adiabatic flame temperature Tb, the coefficient of thermal expansion c , and the Zeldovich number b, are  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (38) and one assumes that b >>1. Also, Q, E, and R are heat release per mole of fuel, activation energy, and universal gas constant. Prandtl, Schmidt, and Lewis numbers are assumed to be unity n = a = D, such that outside of reaction zone where L = 0, the q, y, and v fields will be similar under identical boundary conditions. Equations (21)-(23) for molecular-dynamic scale with the dimensionless coordinate, time, and velocity defined as x = x'/(a/v'o), t = t'/(a/v'2o), and v = v'/v'o become  EMBED Equation.DSMT4  (39)  EMBED Equation.DSMT4  (40)  EMBED Equation.DSMT4  (41) where v'o is the flame propagation speed. 5.1 Far-Field Convective Coordinate For laminar flames propagating in quiescent reactive fields, there is no forced convection w = 0, and (39)- (41) reduce to non-homogeneous diffusion equations with nonlinear sources. Because of thermal expansions in the flame, the stationary cold reactants are converted to moving hot combustion products resulting in a velocity jump across the flame sheet as schematically shown in Fig.2a.  EMBED Word.Picture.8  Fig.2a A propagating laminar flame viewed from far-field coordinate x'. When viewed from the perspective of the physical or far-field coordinate x', the flame appears as a mathematical surface of discontinuity without any spatio-temporal structures. Outside of the thin flame zone chemical reactions will be frozen L = 0, and the governing equations (39)-(41) become  EMBED Equation.DSMT4  f = y, q, v (42) To an observer in the coordinate x' only the mean velocity w'f = - v'o - v'b/2 will be detectable and this velocity will be without any spatial structure. If one introduces a coordinate system that moves with the flame z' = x' - w'f t', one obtains from (42)  EMBED Equation.DSMT4  f = y, q, v (43) and the flame becomes stationary with velocity jump (-v'o , -v'o -v'b) across the flame (z' = 0+ , z' = 0-) as shown in Fig.2b.  EMBED Word.Picture.8  Fig.2b A stationary laminar flame viewed from far-field coordinate z'. However, the hydro-thermo-diffusive structure of the flame will be hidden from this observer in the far-field coordinate z'. 5.2 Outer Convective-Diffusive Coordinate In terms of the convective-diffusive stretched coordinate x = x'/lT, where lT is the flame thermal thickness lT = a/v'o, the hydro-thermo-diffusive structure of the flame becomes visible as shown in Fig.3a. However, the convective-diffusive coordinate is being applied to the colder regions outside of the thin reaction zone such that chemical reactions will remain frozen L = 0 because b >>1. With linear approximation, the velocities (v, w, V) within the flame structure at the locations x = (-1/2, 0, 1/2) will be v = ( - vb, - vb/2, 0), w = (- 3vb/4, - vb/2, - vb/4), and V = (- vb/4, 0, vb/4) as schematically shown in Fig.3a, when vb = v'b/v'o. Hence, the flame hydrodynamic structure involves a variable convective velocity w, and the relation v = w + V is satisfied everywhere, while the sign of V as  EMBED Word.Picture.8  Fig.3a Hydrodynamic structure of a propagating laminar flame viewed from thermo-diffusive coordinate x.  EMBED Word.Picture.8  Fig.3b Hydrodynamic structure of a stationary laminar flame viewed from thermo-diffusive coordinate z. well as that of curvature of the velocity profile change across x = 0. 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It is important to emphasize however that mass is not strictly conserved across flame fronts and in fact an exceedingly small amount of gravitational mass (rest energy) of the reactants will be converted to other forms of energy (inertial energy) such as thermal energy of products of combustion and photons as discussed in sec.3. For the steady problem, the convective-diffusive coordinate that moves with the flame is z = x - w t where w = w'/v'o, z = z'/lT, and t = t'/(lT/v'o). One notes however that in terms of the stretched coordinate z, the velocity w = - 1 + vb (z - 1)/2 is no longer constant but has spatial structure as shown in Figs.3a-3b. One further notes that the average value of w at the flame center z = 0 is wf = - 1 - vb/2 that is exactly the mean flame speed as seen by the far-field coordinate z' discussed in Sec.5.1 above. For an observer that moves with the flame front, the velocity profile  EMBED Equation.DSMT4  remains invariant such that Eq.(43) becomes  EMBED Equation.DSMT4  f = y, q , v (44)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (45)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (46) Introducing the new variable  EMBED Equation.DSMT4  into (44)-(46) gives  EMBED Equation.DSMT4  f = y, q , v (47)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (48)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (49) with the solutions  EMBED Equation.DSMT4  (50) According to (50), the steady flame structure is given by a traveling error function schematically shown in Fig.4a. This result, while being in harmony, is fundamentally different from that discussed in an earlier study [25], and perhaps provides a clearer description of the steady flame structure. The predicted geometry of the temperature profile involving error function (50) is in close agreement with the experimental observations [26-28]. In addition, the predicted profiles are consistent with the measured temperature profiles of counterflow premixed flames [29] in the limit of vanishing rates of stretch. On the other hand, the temperature profile within the flame structure according to the classical theory of laminar flame is given by an exponential function as schematically shown in Fig.4b. More experimental measurements of temperature profiles in unsteady propagating laminar flames are needed in order to further test the validity of the modified versus the classical theory. The slope of the temperature profile of the outer convective-diffusive zone at the position of the reaction zone zi (Fig.4a) is obtained from (50) as  EMBED Equation.DSMT4  (51) that will be matched with the solution within the reaction zone to be described next.  EMBED Word.Picture.8  Fig.4a Flame structure according to the modified theory of laminar flame.  EMBED Word.Picture.8  Fig.4b Flame structure according to the classical theory of laminar flame. 5.3 Inner Reactive-Diffusive Coordinate The analysis of the thin reaction zone follows the classical methods [3, 19-24] and involves the stretched coordinate  EMBED Equation.DSMT4  (52) along with the temperature and concentration expansions  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  (53) that are introduced into Eqs.(39)-(40) to obtain, to the first order in e = 1/b << 1  EMBED Equation.DSMT4  (54)  EMBED Equation.DSMT4  (55) From the coupling of (54)-(55) and the boundary conditions at  EMBED Equation.DSMT4 , one obtains  EMBED Equation.DSMT4  such that Eq.(55) becomes  EMBED Equation.DSMT4  (56) that may also be expressed as  EMBED Equation.DSMT4  (57) The first integral of the above equation and matching of the slopes of the temperature profiles on either side of the reaction zone with the outer solution in Eq.(51) result in  EMBED Equation.DSMT4  (58) that relates the reaction zone position zi and hence the ignition temperature qi, to the flame propagation velocity v'o. The parameter B in (37) is related to the actual preexponential factor B' in the law of mass action [3] under Arrhenius kinetics by  EMBED Equation.DSMT4  (59) Also, the mass balance across the flame front ruv'o = rb(v'o + v'b) leads to  EMBED Equation.DSMT4  (60) By substitutions from (59)-(60) into (58), one obtains the analytic expression  EMBED Equation.DSMT4  (61) for the calculation of laminar flame propagation velocity. For single-step overall combustion of stoichiometric premixed methane-air flame at the flame temperature of 2100 K the relevant physico-chemical properties are nF = 1, rO = 1.38 kg/m3, WO = 32,  EMBED Equation.DSMT4 m2/s (thermal diffusivity of air at the flame temperature of 2100 K), E  EMBED Equation.DSMT4  46 kcal/mole,  EMBED Equation.DSMT4  m3/kmol-s [30], c  EMBED Equation.DSMT4  0.86, b  EMBED Equation.DSMT4  10, and the ignition temperature of qi = 0.99 that by (49) gives the reaction zone position  EMBED Equation.DSMT4 . Also, from the temperatures of reactants 300 K and combustion products 2100 K and the ideal gas law under constant pressure, one obtains the density ratio ru /rb = 7. With these realistic values of the physico-chemical properties, the value of flame propagation velocity calculated from (61) is about v'o = 42.1 cm/s in close agreement with the experimentally observed values [27, 28, 31-32]. Although this level of agreement between the theory and experiments is considered to be encouraging, it should be viewed with caution because of the well-known uncertainties in the overall chemical-kinetic parameters (E, B') [30]. The value of about v'o = 42 cm/s has also been obtained in a number of numerical investigations using complex multi-step kinetic models [33-36]. 6 Concluding Remarks A scale invariant model of statistical mechanics was applied to present invariant forms of mass, energy, linear, and angular momentum conservation equations in chemically reactive flow fields. The summation procedures for relating adjacent families within the hierarchy of statistical fields were described. Also, the coupling between the gravitational versus the inertial contributions to the energy-momentum density of the field was discussed. The connection between the modified equation of motion and the classical Navier-Stokes equation was established. The exact solution of the modified equation of motion for the classical problem of Blasius for laminar flow over a flat plate was presented. Also, a modified form of the Helmholtz vorticity equation was presented with a source of vorticity due to chemical reactions. The conservation equations at the molecular-dynamic scale were applied to present a modified hydro-thermo-diffusive theory of laminar flames. The predicted flame structure was found to be in agreement with experimental observations as well as numerical calculations. With realistic physico-chemical properties for one-step combustion of stoichiometric methane-air premixed flame, the flame propagation velocity of 42.1 cm/s was calculated in accordance with experimental observations. Acknowledgments: This research was supported by NASA under grant No. NAG3-1863. References: [1] de Groot, R. S., and Mazur, P., Nonequilibrium Thermodynamics, North-Holland, 1962. [2] Schlichting, H., Boundary Layer Theory, McGraw Hill, New York, 1968. [3] Williams, F. A., Combustion Theory, 2nd Ed.,Addison-Wesley, New York, 1985. [4] Sohrab, S. H., A scale-invariant model of statistical mechanics and modified forms of the first and the second laws of thermodynamics. Rev. Gn. Therm. 38, 845-854 (1999). [5] Sohrab, S. H., Transport phenomena and conservation equations for multicomponent chemically reactive ideal gas mixtures. Proceeding of the 31st ASME National Heat Transfer Conference, HTD-Vol. 328, 37-60 (1996). [6] Panton, R. L., Incompressible Flow, Wiley, New York, 1996. [7] Sohrab, S. H., Modified form of the equation of motion and its solution for laminar flow over a lat plate and through circular pipes and modified Helmholtz vorticity equation. Eastern State Section Meeting, The Combustion Institute, October 10-13, 1999, North Carolina State University, Raleigh, North Carolina. [8] Sohrab, S. H., Hydrodynamics of spherical flows and geometry of premixed flames near the stagnation point of viscous counterflows. Fifth International Microgravity Combustion Workshop, NASA, May 18-20, 1999, Cleveland, Ohio. [9] Mallard, E., and le Chatelier, H. L., Ann. de Mines 4:379 (1883). [10] Mikhel'son, V. A., Thesis, University of Moscow, 1889, see Collected Works, vol.1, Moscow: Novyi Agronom Press, 1930. [11] Taffanel, M. , Compt. Rend. Acad. Sci., Paris 157:714; 158:42 (1913). [12] Jouguet, E., Compt. Rend. Acad. Sci., Paris 156, 872 (1913); 179, 274 (1924). [13] Nusselt, W., Z. des Vereins Deutscher Ingenieure 59:872 (1915). [14] Daniell, P. J., Proc. Roy. Soc. London 126 A:393 (1930). [15] Jost, W., and Mffling, L. V., Z. Physik. Chem. A 181:208 (1937). [16] Lewis, B., and von Elbe, G., J. Chem. Phys. 2:537 (1934). [17] Kolmogoroff, A., Petrovsky, I., and Piscounoff, N., Bulletin de l'universit d' tat Moscou, Srie internationale, section A, Vol.1 (1937), English translation in Dynamics of Curved Fronts, P. Pelc (ed.), Academic Press, New York, 1988. [18] Zeldovich, J. B. and Frank-Kamenetski, D. A., Acta Physicochimica U.R.S.S., vol. IX, No.2 (1938). English translation in Dynamics of Curved Fronts, P. Pelc (ed.), Academic Press, New York, 1988. [19] Bush, W. B., and Fendell, F. E., Combust. Sci. Technol. 1:421 (1970). [20] Williams, F. A., Ann. Rev. Fluid Mech. 3:171 (1971). [21] Lin, A., Acta Astronautica 1:1007 (1974). [22] Joulin, G., and Clavin, P., Combust. Flame 35:139 (1979). [23] Buckmaster, J. D., and Ludford, G. S. S., Theory of Laminar Flames, Cambridge University Press, Cambridge, 1982. [24] Warnatz, J., Maas, U., and Dibble, R. W., Combustion, Springer, pp.21-30, New York, 1996. [25] Sohrab, S. H., Laminar flame theory revisited-Stationary coordinates for systems under rigid-body versus Brownian motions. Central States Section Meeting, The Combustion Institute, May 31-June 2, 1998, Lexington, Kentucky. [26] Fristrom, R. M., Grunfrelder, C., and Favin, J., J. Phys. Chem. 64:1386 (1960). [27] Gnther, R., and Janisch, G., Combust. Flame 19:49 (1972). [28] Andrews, G. E., and Bradley, D., Combust. Flame 19:275-288 (1972); 20:77-89 (1973). [29] Kurz, O., and Sohrab, S. H., Modified hydro-thermo-diffusive theory of laminar counterflow premixed flames. First Joint Western-Central-Eastern Section Meeting, The Combustion Institute, March 14-17, 1999, The George Washington University, Washington DC. [30] Sohrab, S. H., Ye, Z. Y., Law, C. K., Combust. Sci. Technol. 45:27 (1986). 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G_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  c==(T b "-T u )/T b <FDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APObjInfokmEquation Native X_1115460921frpF*y_*y_Ole G_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  b==E(T b "-T u )/RT b2ݸFDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APObjInfooqEquation Native _1115461003vtF*y_*y_Ole G_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   "y"t++w "y"x== " 2 y"x 2 "-Lye b(q"-1) d(x)ObjInfosuEquation Native _1115461002xF*y_*y_Ole     !"#$%&'()*+,-./012345678:;<=>?@ABCDEFGHIJOPQRSWXYZ[\^`abcdefghijklmnopqrstuvwxyz{|}~ݼFDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   "q"t++w "q"x== " 2 q"x 2 ++Lye b(q"-1) d(x)FDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   "v"t++w "v"x== " 2 v"x 2 ++ObjInfowy Equation Native  _1147844857| F*y_*y_PIC dvLye b(q"-1) d(x)`de ,e    q .  & Times ~@   ~ ~ ~0- !PRODUCTS=-SSiMETA {}2 PICT ~9wObjInfoK_1115461244nF*y_*y_- ! REACTANTS>-imiTimes ~@  ~ ~ ~0- !v' = 0$-j!x' ^PSymbol~@  ~ ~ ~0-! = 0 fTimes ~@  ~ ~ ~0- !x' = PSymbol~@  ~ ~ ~0-!- = Times ~@   ~ ~ ~0- !x ' = PSymbol~@  ~ ~ ~0-! & --$_zbz_z\_-_j_z &  & & & Times ~@   ~ ~ ~0-!v'b &  & Times ~@  ~ ~ ~0-!od &  & &  & Times ~@  ~ ~ ~0-! b & >kK{Times ~@  ~ ~ ~0- !FLAMER & Times ~@ !  ~ ~ ~0-!v'd & & PSymbol~@  ~ ~ ~0- ! = - d Times ~@ "  ~ ~ ~0-!v'd" &  & Times ~@  ~ ~ ~0-!bf* &  & &  & Times ~@ #  ~ ~ ~0-! d/ &  & 'wp  5?B,Times .+=PRODUCTS p"Rf ˡ  6@+ REACTANTS p" g^  ' ($v' = 0 p"  ] ~( ^x', Symbol) = 0  6( x' = )-  +x ' = ) pq\zb_bz_z\z_"_j  Xe(bv'  \f +o  Xe (b  p">k  J~T (RFLAME  Zg  (dv'   Z g#) = -  Z!g+)v'  ^)h0 +b  Z.g3 (d/ TFXDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APOle LObjInfoMEquation Native Np_1115461243F*y_*y_G_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   "f"t==n m  " 2 f"x 2!pFDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APOle TObjInfoUEquation Native V_1147844972 F*y_y_G_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  w x  dfdz=="-n m  d 2 fdz 2d T    q .PIC ]dMETA _ PICT XObjInfo  & Times ~@  ~ ~ ~0- !PRODUCTS@-iii- ! REACTANTSKCjkj---lTimes ~@$ "  ~ ~ ~0-!z' `PSymbol~@  ~ ~ ~0-! = 0 gTimes ~@$ #  ~ ~ ~0- !z' = PSymbol~@  ~ ~ ~0-!- = Times ~@$ $  ~ ~ ~0- !z' = PSymbol~@  ~ ~ ~0-! -kmk-FkT}Times ~@$ &  ~ ~ ~0- !FLAME Z & Times ~@  ~ ~ ~0-!v'b & & PSymbol~@$ '  ~ ~ ~0-! = -b Times ~@  ~ ~ ~0-! v'-b &  & Times ~@$ (  ~ ~ ~0-!od( &  & &  & Times ~@  ~ ~ ~0-! b- & & PSymbol~@$ )  ~ ~ ~0-!- b0Times ~@  ~ ~ ~0-!v'b: &  & Times ~@$ *  ~ ~ ~0-!bdB &  & &  & Times ~@  ~ ~ ~0-! bG &  & !v'& & & PSymbol~@$ +  ~ ~ ~0-! = -&Times ~@  ~ ~ ~0-! v'-& &  & Times ~@$ ,  ~ ~ ~0-!o( &  & &  & Times ~@  ~ ~ ~0-! & &  & 'Xp  8B?,Times .+@PRODUCTS p"hf ̡  ;F+ REACTANTS p"j\",  _  ( `z', Symbol) = 0  .( z' = )-  +z' = ) p" i^"Fk  R\ (ZFLAME  Xe  (bv'   Xe) = -  Xe)) v'  \'f. + o  X,e1 (b-  X/e;)-  X9eC) v'  \AfH +b  XFeK (bG  )(&v'   )) = -  )) v'  * + o  ) (& d0   y y 11Y$ 1Y< 0 0_1147844861 Fy_y_PIC dMETA (6"PICT  0  b   z .  & --$1Y1Y1b CAm1( ???Times ~@   ~ ~ ~0- !PRODUCTS9-WW7 ! REACTANTS@iwiTimes ~@ L  ~ ~ ~0- !v = 0A(E\E0!x\PSymbol~@   ~ ~ ~0-! = 0b&EFE$Times ~@ M  ~ ~ ~0-!V }!w<N!x = PSymbol~@   ~ ~ ~0-!-Times ~@ N  ~ ~ ~0-! PSymbol~@   ~ ~ ~0-!$Times ~@ O  ~ ~ ~0- !x = PSymbol~@   ~ ~ ~0-! & --$hukuhueh-hjhu &  & & & Times ~@ P  ~ ~ ~0-!v'b &  & Times ~@   ~ ~ ~0-!od &  & &  & Times ~@ Q  ~ ~ ~0-! b &  & !vf & & PSymbol~@   ~ ~ ~0- ! = - fTimes ~@ R  ~ ~ ~0-!vf &  & Times ~@   ~ ~ ~0-!bh% &  & &  & Times ~@ S  ~ ~ ~0-! f* &  & ' @ @ @@? @@?  @ @@  @@ @@  @ @@ @ @ `` p  1;?,Times .+9PRODUCTS y "W5 ˡ  8C+ REACTANTS y"id  + ((v = 0 y""EM"-  [|(\x, Symbol) = 0 y"&EK"FEKޡ  | ( }V  2M?Y(<Nw  4(x = )-) )  +x = ) yqeukhkuhueuh"hj  Xe(bv'  \f +o  Xe (b  \i (fv   \i ) = -  \i&)v  `$j+ +b  \)i. (f* dm   M{ { 1.@o$ ObjInfo_1147844863 Fy_y_PIC dMETA @ &m  b   | .  & --$@..o@o@.b CAo.@( ???Times ~@  ~ ~ ~0- ! REACTANTS(-v[v !PRODUCTS*//m mBMVyV>PSymbol~@ $  ~ ~ ~0- !z = 0C j!zTimes ~@  ~ ~ ~0-! = PSymbol~@ %  ~ ~ ~0-!- !zTimes ~@  ~ ~ ~0-! = PSymbol~@ &  ~ ~ ~0-!Times ~@  ~ ~ ~0-!v<PSymbol~@ '  ~ ~ ~0- ! = -1<V/VI`[n & & & !z R &  & Times ~@  ~ ~ ~0-!i X &  & &  &  &  & PSymbol~@ (  ~ ~ ~0-!- & & Times ~@  ~ ~ ~0-! PSymbol~@ )  ~ ~ ~0-!z &  & Times ~@  ~ ~ ~0-!i &  & &  &  &  & Times ~@ *  ~ ~ ~0-!vf & & PSymbol~@  ~ ~ ~0- ! = -1 - f Times ~@ +  ~ ~ ~0-!vf4 &  & Times ~@  ~ ~ ~0-!bh: &  & &  & Times ~@ ,  ~ ~ ~0-! f? &  & !vC & & PSymbol~@  ~ ~ ~0- ! = -1 - CTimes ~@ -  ~ ~ ~0-!vC. &  & Times ~@  ~ ~ ~0-!bE4 &  & &  & Times ~@ .  ~ ~ ~0- ! (1 + erfC9PSymbol~@  ~ ~ ~0-!zCaTimes ~@ /  ~ ~ ~0-!)/2 Cg &  & 'PICT M ObjInfo_1115462291Fy_y_Ole .@o< 0 0 @ @ @@? @@?  @ @@  @@ @@  @ @@ @ @ `` p  +,Times .+( REACTANTS { "vK  ",A(*PRODUCTS {"/"m 5"MV,"- Ρ  i, Symbol ( jz = 0  +Az) = ) -  *(z) = )  2?+-v) = -1 {"V"I`  Q Y( Rz  W\ +i   ( -  )  )z   +i  \i (fv  \ i5) = -1 -  \3i;)'v  `9j@ +b  \>iC (f?  9F(Cv  9F/) = -1 -  9-F5)'v  =3G: +b  98Fb (C9 (1 + erf  9`Fh)(z  9fFy))/2 ObjInfoEquation Native P_1115462033Fy_y_Ole       !%&'(,-./045678<=>BCDHIJKLPQRSTUWYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~4FDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  w==)#v*#=="-1++v b (z"-1)/2'ObjInfoEquation Native _1115462077Fy_y_Ole ݀FDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  [1"-v b (z"-1)/2] dfdz== d 2 fdz 2`ObjInfoEquation Native _1115462173Fy_y_Ole FDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  z!"!FDSMT5WinAllBasicCodePagesObjInfoEquation Native  _1115462099Fy_y_Ole Times New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  y"-1==q==v++1==0@&FDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APObjInfoEquation Native _1115462225Fy_y_Ole "G_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  z!"-"1 FDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  y==q"-1==ObjInfo#Equation Native $<_1115462408Fy_y_Ole )v++1++v b ==08FDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   v b V==1"-v b (z"-ObjInfo*Equation Native +T_1115462512^9Fy_y_Ole 11)/2HFDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  2V dfdV++ d 2 fdObjInfo2Equation Native 3d_1115462119Fy_y_Ole 9V 2 ==0FDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  V!"-"ObjInfo:Equation Native ;_1115462138Fy_y_Ole ?ObjInfo@Equation Native A_1115462678Fy_y_Ole EFDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  V!"\FDSMT5WinAllBasicCodePagesObjInfoFEquation Native Gx_1115635683Fy_y_Ole MTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  q==1"-y=="-(v++1)/v b ==(1++erfV)/2™eDSMT5WinAllBasicCodePagesObjInfoNEquation Native O_1147844868 Fy_y_PIC VdTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  `  "q"V()  V==V i  == exp("-V i2 ) p dLMETA u&PICT X ObjInfo_1147844869 Fy_y_$ >b    .  & --$6LL66Lb CDnL6( ???Times ~@  ~ ~ ~0- ! REACTANTSPSymbol~@ 0  ~ ~ ~0-!qeTimes ~@  ~ ~ ~0- ! = (1+erf ePSymbol~@ 1  ~ ~ ~0-!z)eTimes ~@  ~ ~ ~0-!/2e-ennTimes ~@ 2  ~ ~ ~0-!REACTION ZONE  ! STRUCTURE  ! PREHEAT t{ ! ZONEA~{ !PRODUCTSrPSymbol~@  ~ ~ ~0-!qE Geneva~@ 3  ~ ~ ~0-! ETimes ~@  ~ ~ ~0-!=E  Geneva~@ 4  ~ ~ ~0-! ETimes ~@  ~ ~ ~0-!1EPSymbol~@ 5  ~ ~ ~0-!q Geneva~@  ~ ~ ~0-! Times ~@ 6  ~ ~ ~0-!= 0DWR@:LHL7EE & --$EQQQE$-Q & +R@Jd|vsz!m!z- 6F's--Wdzi9Hr%r2I,%DA<62/+)'&%#! ~yslj & & & PSymbol~@  ~ ~ ~0-!x} &  & Times ~@ 7  ~ ~ ~0-!= 0 & PSymbol~@  ~ ~ ~0- !z = 0`+R&e !z = - !z =  & & & !z? &  & Times ~@ 8  ~ ~ ~0-!iE &  & &  &  & PPP@PSymbol~@  ~ ~ ~0- !x = - H| & !- & & Times ~@ 9  ~ ~ ~0-! 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