Ind. Eng. Chem. Res. 1992,31,669-681 Cikliiiski, P. J.; Bes, T. Analytical Heat Transfer Studies in a Spiral Plate Exchanger. R o c . XVZth Int. Congr. Refrig. (Paris) 1983, IZ, 449-454 (paper B.1-198). Jones, A. R.; Lloyd, S. A.; Weinberg, S. A. Combustion in Heat Exchangers. Proc. R. SOC.London, Ser. A 1978, 360, 97-115. Martin, H. Wdrrneiibertrager; Georg Thieme Verlag: Stuttgart, Federal Republic of Germany, 1988. Martin, H.; Chowdhury, K.; Linkmeyer, H.; Bassiouny, M. K. Straightforward Design Formulae for Efficiency and Mean Temperature Difference in Spiral Plate Heat Exchangers. Proceedings of the Eighth International Heat Transfer Conference (Sun Francisco);Hemisphere Publishing: Washington, DC, 1986; Vol.
669
VI, pp 2193-2797. Minton, P. Designing Spiral-Plate Exchangers. Chem. Eng. Progr. 1970, 77, 103-112. Strenger, M. R.; Churchill, S. W.; Retallick, W. B. Operational Characteristicsof a Double-Spiral Heat Exchanger for the Catalytic Incineration of Contaminated Air. Znd. Eng. Chem. Res. 1990,29, 1977-1984. Received for review January 22, 1991 Revised manuscript received August 5, 1991 Accepted August 13, 1991
Decay of Turbulence in a Tube following a Combustion-Generated Step in Temperature Lance R.Collins+and Stuart W. Churchill* Department of Chemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104
An essentially abrupt change in temperature owing to thermally stabilized combustion produces a correspondingly abrupt decrease in Reynolds number from the 3000-6000 range down to 650-1550 for an ethane-air flame in a 9.5-mm channel. Since the Reynolds number downstream from the flame front is below the transitional value for a tube (approximately 2100), the turbulence decays with distance from the flame front. The decay of turbulence and the consequent development of a parabolic velocity profile were investigated theoretically using a modified k-t model of Jones and Launder. The predictions of the velocity a t the centerline and axial gradient of pressure are in qualitative agreement with experimental results throughout the system, although the predicted approach to the asymptotic value for laminar flow was faster than that which was observed. This discrepancy identifies a nonphysical characteristic of the modified k-c model which was utilized in the current study.
Introduction Theoretical and experimental studies of the laminarization of turbulent flows go back to the early investigations of the decay of turbulence behind a screen in a wind tunnel (for a review of that behavior see Batchelor (1953) or Hinze (1975)). Because the kinetic energy of turbulence is continually being degraded to thermal energy as a result of viscous dissipation, there must be a mechanism to convert continually the energy of the mean flow. Since the mean flow behind screens in wind tunnels is generally uniform, and does not include a mechanism for producing additional turbulent kinetic energy, the preexisting energy decays. This decay is usually modeled in terms of a power law in space (or equivalently in time if the frame of reference is moving at the mean velocity). Wall-bounded flows have also been observed to laminarize under certain circumstances. Perhaps the most important example is boundary layer flow undergoing an acceleration due to a favorable pressure gradient (e.g., Pate1 and Head (1968), Launder and Jones (1969),Jones and Launder (1972a,b),Narasimha and Sreenivasan (1973,1979),and more recently Spalart (1986)). Examples of accelerated boundary layers include external flows over curved surfaces and internal flows in converging ducts. The dimensionless grouping that is usually assumed to control laminarization is the parameter of acceleration K. If K is greater than 3 X lo”, the boundary layer will generally revert to a laminar one, implying a decrease in thickness, thereby resulting in very *Author to whom inquiries should be addressed. Current address: Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802.
different rates of heat and mass transfer to the surface. The laminarization of internal flows has also been studied. For example, laminarization of fully developed turbulent flow of a gas in a tube heated at the wall was investigated by McEligot et al. (1970). In this case, the flow laminarizes because the viscosity of the gas increases with temperature and thereby cam the Reynolds number to decrease. They showed that the parameter which determines whether the flow will laminarize is closely related to the parameter of acceleration, K, thereby relating the laminarization in their system to that of the accelerated boundary layer. Laminarization in the thermally stabilized burner (TSB) is closely related to that in the heated pipe of McEligot et al. (1970). However, before discussing the process of interest further, it is worthwhile to provide a general description of thermally stabilized combustion. The TSB consists of one or more ceramic tubes 1cm or greater in diameter, through which premixed ethane and air are fed at one end and hot burned gases exit at the other. Separating the cold reactants from the hot products is a very thin (on the order of millimeters) zone of reaction whose axial location for a fixed fuel/air ratio and rate of flow is determined by an overall energy balance. All combustors recirculate some of the energy released by the reactions back to the cold reactant gases, heating them to the point of ignition. Typically this thermal feedback is accomplished by physical backmixing between burned products and unburned reactants as, for example, with a bluff-body stabilized flame. The TSB, by contrast, accomplishes this thermal feedback by two mechanisms that involve the wall of the channel. Energy from the hot products of combustion downstream from the flame front convects to the wall, which in turn radiates (-60%) and
QS88-5885/92/2631-QS69$03.QQ/Q0 1992 American Chemical Society
670 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
conducts (-40%) this energy upstream, whereupon it heats the incoming flow to the point of ignition by convection (Chen and Churchill, 1972a; Choi and Churchill, 1979). Backmixing in the TSB is quite negligible, which is the reason for the very thin zone of reaction. The thermal feedback is then determined wholly by the rate of recirculation of energy determined by the rates of conduction and radiation, which are functions of the conductivity and emissivity of the wall, respectively. As a consequence, the TSB has a limited range of flow for which a stationary flame can exist. However, the extreme simplicity of the field of flow in the TSB makes detailed fluid mechanical and chemical analysis of the system possible, which is unusual for practical combustors. The Reynolds number for flow in a tube is generally defined as Re = p U D / p or equivalently as Re = 4W/.lrDp. Stable combustion in a TSB with a diameter of 1 cm corresponds to a Reynolds number upstream from the flame front that is in the range of 3000-6000, resulting in turbulent flow. As the gas crosses the flame front, the temperature jumps from 300 K to slightly above the adiabatic temperature of a flame, 1700-2100 K (the temperature leaving the tube is of course necessarily below the adiabatic value). Conservation of mass requires that W remain constant throughout the TSB; however, the molecular viscosity increases substantially across the flame front. The result is that the downstream Reynolds number is reduced by the ratio of the molecular viscosities before and after the flame front ~ ( 3 0 0K)/p(Tf). This ratio for hydrocarbon and air flames is typically in the range of 4-5, corresponding to downstream Reynolds numbers of 60&1500, which are far below that for transition to laminar flow in a pipe (approximately 2100). Fluctuations inherited from the turbulent flow upstream from the flame front then decay to zero on the downstream side. Furthermore, the relatively flat velocity distribution characteristic of turbulent flow must rearrange into a parabolic profile. The mechanisms for laminarizationin the TSB is similar to that for flow in a heated pipe described by McEligot et al. (1970). In both circumstances the increase in the molecular viscosity lowers the overall Reynolds number and thereby causes the flow to laminarize. The unique aspect of laminarization in the TSB is that it occurs because of a nearly discontinuous change in Reynolds number at the flame front. Whether or not the flow will laminarize in a pipe depends on the rate of heating (i.e., the value of the parameter of acceleration), whereas in the TSB it depends solely on the final Reynolds number. The objective of this investigation was to describe the turbulence throughout the TSB as well as the evolution of the mean-field velocity using the low-Reynolds-number k-t model of Jones and Launder (1973). Experimental measurements of the time-mean velocity at the centerline and time-mean gradient of pressure for ethaneair flames are compared to the predictions.
Thermally Stabilized Combustion Modeling of the thermal behavior in the TSB was first done by Chen and Churchill (1972a) followed by Choi and Churchill (1979),who used a slightly different technique to integrate the equations. They both assumed plug flow with perfect radial mixing for the gas and a one-step global reaction for the kinetics. A one-dimensional energy balance for the wall of the tube which included radiative transfer and conduction was solved simultaneously with the energy and mass balances for the gas. Typical longitudinal profiles of the temperature of the gas and wall for two of the known seven stationary states are shown in Figure 1. The gas temperature is essentially uniform with
Figure 1. Longitudinal profiles of the temperature on the wall and in the gas of a TSB from the calculations of Chen and Churchill (1972a). Reprinted with permission from Chen and Churchill (1972). Copyright 1972 The Combustion Institute.
distance upstream from the flame front, suddenly jumps to near the adiabatic flame temperature, and then is reasonably uniform to the exit of the burner. This profile results from the large energy of activation associated with the various free-radical reactions of combustion. At ambient temperature premixed ethane and air react very slowly, but as the temperature increases the rates increase substantially, by as much as 15 orders of magnitude. This phenomenal change in the rates of reaction limits the zone of reaction to the thin layer in Figure 1. The thicker zone associated with conventional burners is a consequence of backmixing and nonuniformity. Chen and Churchill (1972a) were also able to determine the location of the flame front from their solution. For a given overall rate of flow and fuel-to-air ratio they predicted the existence of as many as seven steady-state solutions (Le., seven locations of the flame front). This multiplicity was later confirmed experimentally by Bernstein and Churchill (1977) for propane and by Goepp et al. (1980) for dropleb of hexane. More detailed chemical kinetic models were used by Tang and Churchill (1980a,b) and Pfefferle and Churchill (1984). These later models yielded more information on free-radical concentrations and on pollutant formation, but had little effect on the thermal behavior of the TSB, which is of primary concern in the current work. All previous models of the TSB assumed plug flow and perfect radial mixing and therefore neglected the laminarization entirely. The assumption of plug flow is usually considered adequate for describing chemical conversions in a tube because the turbulent fluctuations enhance mixing of momentum and species in the radial direction. However, downstream from the flame front where the turbulence is decaying, this assumption is on weaker ground. For example, Collins and Churchill (1990) (see also Collins (1987)) have shown that concentrations of species calculated from the developing velocity profile described herein differ from those for plug flow by as much as 40%.
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 671 Table I. Constants for k-c and k-c-1 Models of Turbulence c1
CZ
c,
b&
0,
1.44
1.92
0.09
1.0
1.3
oh 0.9
ow
(Y
+ ; ) V k ] + vtG - e - (2u(
Turbulence Model On the basis of the evidence of Figure 1, the flow in regions 1 and 3 is assumed to be isothermal and of low Mach number, which together imply that the gas is nearly incompressible in these regions; therefore the incompressible k-e model is used to describe the flow. The Mach number in region 2 is likewise small, but the large increase in temperature of the gas across the flame front causes an equivalent decrease in the density of the fluid. Consequently the incompressible k-e model is no longer valid and a variable-density k-e-1 model is required. The turbulence models for incompressible flow (regions 1and 3) and for variable-density flow (region 2) are summarized in the following two subsections, along with a discussion of the physical interpretation of some of the terms. k-c Model. The k-e model for turbulence has been perhaps the most widely used because it has been shown to be both computationally efficient and flexible in terms of geometries and regimes of flow. However, applying the k-e model to flows with boundaries requires assumptions about the effect of the wall on the turbulent structure (e.g., a law of the wall). Jones and Launder (1973) proposed a modified k-e model in which certain “constants” were allowed to be functions of the local turbulent Reynolds number defined as Rt = k2/ve. Additional empirical terms were also added to the k and e equations to correct for the effect of the boundary. The modified model can predict turbulent flows through the boundary layer and viscous sublayer right up to the wall, thereby avoiding the need for a separate wall function. This is particularly relevant for laminarizing flows, as previously demonstrated (Jones and Launder, 1972b) by calculations for a laminarizing boundary layer, because the near-wall behavior can deviate substantially from the “law of the wall” behavior observed in fully developed turbulence. This in turn can strongly affect the overall rate of decay of the energy of turbulence. Jones and Launder showed that the modified k-e model was capable of predicting the correct mean-field velocities and pressure drops for fully developed pipe flow, even for Reynolds numbers lower than those considered herein. For these reasons it was decided that the modified k-e model would be the best choice to describe the laminarizing flow in the TSB. The low-Reynolds-number k-e model of Jones and Launder is summarized below: V-t = 0 (1) = V.[(V + vt)(Vt + VvT)] (2)
gy}
at
1.0
From Figure 1, it appears that the TSB can be divided into three distinct sections: region 1,the nearly isothermal flow upstream from the chosen flame front; region 3, the nearly isothermal flow downstream from the flame front; and region 2, which joins them. Each region was modeled separately, with appropriate matching at the interfaces. Since regions 1and 3 are nearly isothermal and the effects of compressibility are minimal (Le., the Mach number is small), the k-e model for incompressible flow was used. Region 2, in contrast, has a rapidly varying density field and therefore required a variable-density k-e-1 model. The k-e and k-e-1 models are described and summarized in the next section.
at
ak + o.Vk = V.[
(3)
ae
- + t-ve = at
V*[
(+ Y
+ C lep G -
:)Vel
where ut =
Cj,k2/e
G = (Vt + VtT): Vt f2
= 1 - 0.3 exp(-R,2)
Rt = k 2 / m (5) The superscript T (eqs 2 and 5 ) means the transpose. The constants C1,C2,C,, ah,and a, were set to standard values from the literature (see Table I). f zand f,, as well as the final terms of the k and e equations, shown in the braces, are the modifications suggested by Jones and Launder to extend the range of the k-e model to smaller Reynolds numbers. f2 corrects for the effect of the Reynolds number on the rate of decay of homogeneous turbulence, and f, corrects for the region of the law of the wall in bounded flows. The k equation has been altered so that e can be set to zero at the wall (in general e is nonzero at boundaries), while the final empirical term in the e equation is to correct for errors in the calculated peak values of k . Variable-Density k-e-1 Model. Averaging in variable-density flows can potentially be done in two ways, the standard volume average as used above, and the densityweighted or Favre average which is defined as $ = G/p, where 6 is the density-weighted average of an arbitrary variable 4. The residual fluctuation for a Favre-averaged variable is then defined by 4” = 4 - $. The purpose of expressing the time-averaged governing equations in terms of Favre averages is that the number of higher order moments in the resulting equation is greatly reduced. Unfortunately the exact interpretation of the new correlations involving density-weightedfluctuations (i.e. 4”) is not well understood since these are often difficult to measure directly in practical flows. The moment modeling suggested in the review article by Jones and Whitelaw (1982) is based on a direct analogy with incompressible systems. Collins (1992) has shown that this analogy breaks down for vwiable-density flows with severe pressure gradients. Examples include the countergradient diffusion observed in turbulent flames (Libby and Bray, 1981) and demising observed in swirling flows (Wahiduzzaman and Ferguson, 1986). Collins points out that flows with relatively small pressure gradients may well be represented by simpler models like k-e-1, since these new correlations are not significant. Pressure gradients in the TSB are due solely to accelerations of the gas in the flame front and can be shown to be of order W ,where M is the Mach number. Therefore in the limit of M 0 the k-e-1 model should prove to be adequate. The Jones and Whitelaw model has been further modified here for low Reynolds numbers, in a manner analogous to that for the incompressible flow equations. The justification is that the resulting model is consistent with
-
672 Ind. Eng. Chem. Res., Vol. 31, No.3, 1992 Table 11. Physical Constants in k-1 4, cma/(g 8)
Model for an EthaneAir Flame' E&
3.83 x 10'2 Pm g / ( a 8) 1.84 x 1 v
C, [0.23(1 -+)]/(I
AH,ergs/(&! fuel)
K
1.06 x 1014 'I'M K , 298
+ 0.06Q)
C,
CP
3.73
7 R g / ( 7- 1)
4.65 X 10" n 0.77
7
1.3
- 0.08Q
Pr
SC
1.0
1.0
is the equivalence ratio.
the k-r model describing regions 1 and 3, and therefore permits matching between the three regions. As a result of the variations in density, an energy balance and a species balance for the concentration of the fuel are included. The resulting equations are summarized below: V.(P*) = 0
(6)
--
REGION1 T=3WK
Re=3WO-6wo
I
REGION 3 T ZlMK
-
RC=Mx)-15W
Figure 2. Schematic of regions 1, 2, and 3 in the TSB.
V.[ ( p
+ :)Vk]
- $kV.O
+ a G - pr - -VpVp H P2
where
PM, = pR$ p,
(ideal gas law)
= Cj,$k2/r
dT) = /dT/T,)"
R, = pk2/pr (12) The valuea for the two new constants, uhand u, are shown in Table I, and the physical constants for the model for an ethane-air flame are summarized in Table 11. Modeling the Turbulent Flow in the TSB The theoretical model for the combustor divides the system into three regions (see Figure 2). Region 1 is the gas upstream from the flame front, region 2 is the flame itself, and region 3 is the gas downstream from the flame front where the laminarization occurs (for the sake of nomenclature, subscripts 1,2, and 3 will be used to designate regions 1, 2, and 3, respectively). As discussed earlier, the temperature in regions 1 and 3 is assumed to be constant resulting in flow that is nearly incompressible.
Region 2 requires the variabledensity k-clmodel becam of the rapidly changing temperature and density. If the flame is treated as a hydrodynamic discontinuity separating regions l and 3, jump conditions can be derived. However as will be shown in a subsequent section, the jump conditions provide incomplete information making it necessary to seek a solution valid within region 2. Region 1. Region 1 includes the cold reactant gas up to the point that the temperature begins to rise. The Reynolds number of the flow in region 1 is typicnlly in the range 3o00-6W0, which is well above the transitional value for channels. Region 1 is similar to the entrance region in pipe flow where the radial velocity profile and intensities of turbulence evolve inta the fully developed characteristics of the particular Reynolds number of the gas (i.e., profiles which are functions of the radial coordinate, r, only). For the experimental conditions of this work, region 1 was never less than 50 diameters in length, and depending on the location of the ilame was often much longer. The flow entering the TSB was certain to include residual turbulence owing to the complex piping upstream; therefore the turbulence could be considered to be "tripped" at the entrance. Given the long distance ahead of the flame front and the ample tripping of turbulence at the entrance, the field of velocity was assumed to be fully developed before it reached the flame front (Le., region 2). Consequently, the relevant profiles for the calculation in region 2 are the fully developed ones for turbulent flow. The calculation of fully developed flow was done with the incompressible k-r model described in section 3. For fully developed turbulent flow in a tube that model reduces to
The boundary conditions are
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 673
r=O:
dii
dk
de
-0
Ti; =O, dr =O, dr -
r = R: ii=O, k = 0 , c = O (16) The equations were solved with a finite-difference algorithm using 100 radial grid points staggered so that a disproportionate number were placed near the wall to resolve the turbulent boundary layer. Calculations were done at a variety of Reynolds numbers to test the performance of the k-c model at low values. Region 2. The entire process of combustion is postulated to occur within region 2. Figure 1 shows that the increase in temperature from the ambient value to the value for the flame occurs in the longitudinal distance which is much smaller than the length or diameter of the burner. The flame is therefore postulated to act as a hydrodynamic discontinuity normal to the flow. Jump conditions for the discontinuity can be derived from the general governing equations by considering a control volume of zero length in the direction of flow. The unaveraged jump conditions for a flame are = (P43 pu2)1= (P + pu2)3
+ 1 C,T + ;u' + wAH l)
pii
I--
g dx
=
$[
(kh
= $[(Pi(&
$1
):
+
+ qAH
--) -4
+ Pt
d6 - q
(21)
(22)
dk = dr
(
= CpT + i u 2 )3
Averaging these conditions yields ( P i i l l = (Pa13 = (P + pa2 (P pa2 (C,T
piic,
aii
(PU),
(P
Flame fronts are often assumed to be sufficiently thin so that gradients in the direction of flow are much larger than in the tangential directions. The structure of a turbulent flame is then one-dimensional in the longitudinal coordinate. Dimensional analysis of the axisymmetric equations helps demonstrate this point (see Appendix A). The equations that describe region 2 in the limit of an infinite rate of reaction are
+ + 1 + ;a2 + ;1-u ' ~+ wAH
+p F ) 3
(CpT PM, = pR,T
+ p2+ p l-)t 2 (18)
The fluctuations in density and temperature have been ignored in eqs 17 and 18 because the bounding regions on either side of region 2 are nearly isothermal. The averaged jump conditions are not closed because second-order moments have been introduced into the equations. The problem is analogous to the classical one in turbulence. Without a relationship between the kinetic energy of turbulence and the mean-flow variables, the equations are indeterminate. Thus the problem encountered with all jump conditions (e.g., with flames, shocks, and detonation waves) extends to turbulent flow in the TSB. The momentum and energy balances simply give enough information to ensure that the total momentum and energy are conserved. For example, the heat of reaction liberated by the reactants at the flame front can become either thermal energy, mean-flow energy (i.e., (1/2)a2),or turbulent energy (i.e., ( 1 1 2 ) p ) . The problem is that these equations determine only the sum of the mean-flow and turbulent energies and not the partitioning between the latter two forms. Partitioning of energy and momentum between mean flow and turbulence in flows crossing discontinuities depends on the structure of the discontinuity (i.e., the microscopic thickness, shape, etc.). This implies that determining the effect of the flame on turbulence requires a detailed description through the flame. Therefore, a solution for the flame structure was sought using the variable-density k-e-1 model of turbulence.
The boundary conditions expressed in terms of the inner variable (Van Dyke, 1975) are = -co: = iil, P = Pi, p = p i , T = Ti, w = ~ 1 , k = k l , e = €1 z = +a: ii = iig, P = Pg, p = pa, T = Tp., w = 0, dk dc = 0, - = 0 (25)
dx
dx
The equations are completed by the relationships shown in eq 12. The values for a3,P3,p3, and T3are found from the jump conditions (eq 18), for which the effect of turbulent fluctuations on the jump conditions for the mean momentum and energy are ignored. This assumption will be evaluated by the solution. The fuel is assumed to react to completion and since only fuel-lean to stoichiometric mixtures are considered, the exit value for w is zero. The values of k and e are unknown on the downstream side. For consistency their respective gradients are assumed to approach zero outside the flame front. There appears to be an inconsistency at the interfaces between region 2 and regions 1 and 3 because the k-e model used in regions 1 and 3 is in terms of volume averages and the k-e-1 model used in region 2 is in terms of density-weighted averages. This is not the case, however, because near the edges of region 2 the fluctuations in temperature and density approach zero. As the fluctuations in density disappear, density-weighted averages of variables-approach the volume-average values (Le., lim ,+-, ii = pulp = p a l p = 0). $he numerical difficulty of integrating equations over an infinite domain can be circumvented by changing the independent variable from the coordinate variable, r , to the dependent variable ii, which has the finite domain, iil 5 ii 5 ii3. Using the chain rule, the following equations can be derived:
674 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
~r ddr[ r ( u + ~ ) ~ ] + C l ~k u t G - C j2 k2 - + 2 u u t
where G=
2[
(37)
The boundary conditions are
de
i[ dii
( E y + (E)'+ ($y + (!y]
(p
+ ;)ut
$1 +
")
Cliptu'( 2 - p 2 d/' dii dii
(29)
where from eqs 19 and 20 m = pii (constant) dfi = P + pii2 -(P + pa2), dx
4
,(P
49
(30)
+ Pt)
The boundary conditions become ii = a,: P = P1, p = p l , T = T1, w = w,, k = kl, t = t1 6, = ~ 3 P : = P3, p = p 3 , T = T3, w = 0, dk de _ 0, =, 0 (31) dii du The above is a typical analysis for a normal discontinuity; however, an additional consideration is required for a flame burning inside a tube. Near the wall of the tube there must be a thin boundary layer in which radial gradients are important. Because they are ignored in this analysis, the solution will not be valid within the boundary layer. This is not a serious limitation because the effect of the boundary layer on the kinetic energy of turbulence over most of the channel is small as compared to the effect of the rapidly expanding gas resulting from the reactions of combustion. The procedure for solving these equations was to integrate them numerically across the flame front for each radial grid point in region 1,generating the corresponding variables at the radial grid point entering region 3. This resulta in the initial profiles for all the relevant variables just downstream from the flame front. Region 3. Region 3 consists of the hot gaseous products downstream from the flame front, in which the residual turbulence inherited from region 2 decays to zero, and the velocity profile approaches the parabolic one characteristic of laminar flow. The laminarization is assumed to occur axisymmetrically with negligible diffusion in the longitudinal direction. The resulting equations for region 3 are
-1 r
ar
r = R: a = 0 , u=O, k = 0 , t = O x = xf: a(r) = U3(r), u = 0, k(r) = K3(r),
+ -aa =o ax
J 1 a ( r ) d(r/R)2 = constant (mean velocity) (34)
=
(38) The equations in region 3 are parabolic and therefore were integrated with a marching scheme in the longitudinal direction. Because of the rapid rearrangement of the profiles, the adaptive grid technique of Smooke (1982; see also Smooke and Koszykowski, 1984) was used to shift grid pointa radially from areas of less importance to areas where greater resolution was required. Experimental Setup The combustor was essentially the same as that of previous investigations (Chen, 1970; Choi, 1979). This system was enhanced to measure the pressure drop at the wall and the velocity at the centerline. The TSB consisted of a ceramic block 7.6 cm in diameter with seven cylindrical channels, each 9.5 mm in diameter, running ita length (see Figure 3). The channels were arranged so that the central channels was surrounded symmetrically by the other six. The outer tubes acted as guard heaters at the same temperature as the central channel, thereby reducing the driving force for radial heat loss from the center and rendering it nearly adiabatic. This configuration was chosen because of the difficulty of insulating a tube effectively at these extreme temperatures. The TSB had 22 Pt/90% Pt-10% Rd thermocouples that ran along the length of the burner at 2.5-cm intervals to measure the temperature of the wall as a function of axial location. These measurements were used to determine the location of the flame front as it drifted from one end of the burner to the other. Eight pressure taps were made along the central channel to measure the pressure drop over various segments. Each tap was 1.6 mm in diameter and was separated from ita nearest neighbor by 2.5 cm. A pure quartz Pitot tube was fabricated and inserted at the end of the burner to measure the velocity at the centerline. The Pitot tube was attached to an assembly which allowed it to be inserted and removed rapidly. This rapidity was necessary because the temperatures in the flame were sufficiently high (2100 K) so that the quartz degraded with time in the flame. The Pitot tube as well as the eight pressure taps were connected to a pressure transducer from MKS Instruments, which converted the registered pressure difference into an electrical signal. Switching amongst the various pressure taps and the Pitot tube was controlled by toggle valves. A typical experiment consisted of igniting the burner and stabilizing a flame near ita outlet. Then by choosing a rate of flow and equivalence ratio slightly outside of the stable range of operation, the flame was allowed to drift continuously from near the exit end toward the entrance. Owing to the very great thermal inertia of the large ceramic
1o.m
--
--
15-
n
4
56-m
111111111111111111111
,
thermocouples
pitot tube
pressure taps i
Fiure 3. Schematic of experimental setup.
block, this rate of drift was very slow (e.g., it took several h o w for the h e to reach the entrance of the burner). Pressure drops and velocities at the centerline were measured continuously, under the assumption that the rapid processes occurring in the gas were completely unaffected by the slow movement of the flame. Once the flame had reached nearly to the entrance, the rate of flow and/or equivalence ratio was adjusted again to force the flame to drift back toward the exit, thereby allowing a second set of readings to be taken at a different upstream Reynolds number and equivalence ratio. The location of the flame was determined from the thermocouple measurements of the temperature profie of the wall. On the basis of calculations and experimental determinations of stable flames in the TSB (Chen and Churchill, 1972a,b),the flame was assumed to be located at the point of inflection of the temperature profile of the wall. Naturally, there is some error associated with this determination owing to the finite spacing between the thermocouples, but the uncertainty is systematic in that all determinations would be expected to be shifted by the same amount (an amount presumably far less than the 2.5-cm spacing between the thermocouples). The Pitot tube recorded the impact pressure from the gas at the center of the channel just before it exited the burner. The impact pressure was converted to the centerline velocity by the formula ac= (ApR,T,/PMw)1/2.The temperature of the flame was assumed to be the adiabatic value since radial losses from the central channel were negligible. Errors associated with this measurement, related to the finite sue of the Pitot tube and its alignment, tended to reduce the measurement below the actual value. By considering the effecta of misalignment and the fmite size of the Pitot tube, the errors were estimated to be less than 20%. Every effort was made to minimizethese errors, but as will be shown in the following section, they are still evident. The pressure gradient was not directly measured by the pressure transducer; rather the pressure difference Ap
Table 111. Conditions for Calculations of Flow in the TSB Tr, K Re, (upstream) Re, (downatream) 0.70 1730 3000 114 4500 1162 6000 1549 0.85 1951 3000 104 4500 1056 M") 1409 1.00 2151 3000 653 4500 950 6000 1301
across a 2.5-cm length of the wall of the burner Az was measured. Ap/& represents an integrated value of the differential pressure drop, dp/dz, over the length, Az. An equal-area plot of the data was used to reconstruct the differential pressure gradient from the measured pressure differences (Churchill, 1979). The resulta are shown in the next section. Results and Discussion Integration of the governing equations for regions 1,2, and 3 (eqs 13-16,18, %31, and 32-38) resulted in the fully developed profiles upstream, the jump conditions for all variables across the h e , and the axisymmetric laminarization downstream from the flame. The field of flow in the TSB provides a severe test for the k-f model, since the model must describe the reacting and highly variable density flow in region 2 as well as the laminarizing flow in region 3. Calculations were made for three Reynolds numbers and three equivalence ratios (see Table III). The results for all three refions - are presented here followed by a discussion. Mean Flow. Reeion 1can be considered as eauivalent to isothermal, fullydeveloped turbulent flow in a pipe. The upstream Reynolds number for stable combustion in the TSB is in the range of 3OMHCC9, which is fairly close to the transitional value and is therefore difficult to model Jones and Launder (1973) have previously shown that k-e
676 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 10
20
1 6
u+
10
0 - t 0.00
.
1
0.01
I
0.02
I
0.03
0.04
1.2 1.1 1.o 0.9
0.8
In y+ Figure 4. Computed velocity profiles for Re = 3000 and 6000.
0.7
I
8wo 4
1 alQ=o2 x/D=o+ 3 fl=5 4 XJD-25
0.6
0.5 0.4
: 0.00
I
0.01
I
0.02
I
0.03
I 0.04
YQ Re
Figure 6. Dimensionlessplot of the pressure gradient (top) and the velocity at the centerline (bottom) vs dimensionlesa longitudinal distance. Points are experimental measurements with estimated errors.
4ooo-
m-
0.0
0.2
0.4
0.6
0.8
1.o
r/R Figure 6. Computed velocity profiles at three axial locations for Re = 3000 and = 1.0.
calculations of fully developed flow in a pipe result in excellent agreement with experimental measurements of the kinetic energy of turbulence, the turbulent viscosity, and the pressure drop even for low Reynolds numbers. Figure 4 shows a plot of computed values of u+ vs In yi for two Reynolds numbers. In the coordinates shown in Figure 4 the law of the wall would appear as a straight line. The k-c model predicts 3 distinct law-of-the-wall region, though the slope of both curves is larger than values for fully turbulent flow (approximately 2.5; Schlichting, 1979), due to the relatively small Reynolds numbers which are of interest in this study. The results for the mean flow in regions 1 , 2 , and 3 are summarized in Figures 5 and 6. The results for all Reynolds numbers and equivalence ratios can be related when shown as a function of the dimensionless variables r / R and x f DRes (see Appendix B for a derivation of this result). Radial profiles of the longitudinal velocity, ii, just behind the flame front, just ahead of the flame front, and at several locations downstream are shown in Figure 5 for an initial Reynolds number, Rel, of 3000 and an equivalence
ratio of 1.0. The large increase in temperature and consequent decrease in density of the gas at the flame front causes a nearly 7-fold increase in longitudinal velocity. Further downstream the turbulence is decaying, and consequently the velocity profile becomes more peaked as it approaches the parabolic form characteristic of laminar flow. For this example the laminarization is complete in approximately 25 diameters, which corresponds to x/DRe3 = 0.038. This distance is somewhat less than the distance required for development of laminar flow in the entrance region of a pipe because the starting profie for the latter is nearly flat, while the initial profile in these calculations is the appropriate turbulent profile, which is closer to the final profile. Longitudinal profiles of the experimental and theoretical velocity at the centerline and pressure gradient at the wall for an initial Reynolds number of 3477 and an equivalence ratio of 0.78, corresponding to a final Reynolds number of 876.5 and an adiabatic flame temperature of 1842 K, are shown in Figure 6. The predicted pressure gradient behaves qualitatively like the measured values, though the predicted rate at which the laminar asymptote is approached is much faster than was observed. Predicted velocities at the centerline are consistently larger (by as much as 20%) than the measured values over the entire length of the burner. The discrepancy is due in large part to errors associated with the Pitot tube measurement (see Experimental Setup). The discrepancies between theory and experiment are all well within the estimated error (as shown by the error bars in Figure 6b), though once again the predicted Velocities appear to approach the asymptotic value more quickly than the observed ones. An explanation for the slower approach to the laminar asymptote observed by both the pressure gradient and centerline
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 677
1
I
Table IV. Summary of the Effect of Combustion in Region 2 on the Turbulence ____ Re1 Ak,"% Al': % Aut: % 3000 1.0 34 1.7 77 4500 32 1.7 81 6000 30 1.6 82 3000 0.85 18 1.0 78 4500 17 1.0 81 6000 16 1.0 82 0.7 8 0.5 78 3000 4500 7 0.5 80 6000 7 0.4 81
" AB 100[(83- 81)/811%. 0.0 0.5
0.2
0.6
0.4
1.o
0.8
i
I
0.4
0.3 0.2
0.1 0
0.0
0.2
0.6
0.4
1.o
0.8
10 10 -3
-1
10 4 10 -5
2
/
10 -6
10 -7 10 -8 0.0
I
0.2
.
I
.
,
0.6
0.4
.
I
0.8
.
i
1.o
r/R Figure 7. Profdes of the characteristics of turbulence before (1)and after (2)the flame front. (top) Root-mean-square velocity; (middle) length scale of turbulence; (bottom) turbulent viscosity.
velocity will be postulated in the Discussion. Turbulence. Figure 7 summarizes the results for the effect of the flame front on the turbulence. Radial profiles of the root-mean-square velocity, which is defined as ri = k1/2, the turbulent length scale, which is defined as 1' = k3/2/e,and the turbulent viscosity! k , before and after the flame front are shown for an initial Reynolds number of 3000 and an equivalence ratio of 1.0. The effect of combustion on the kinetic energy of turbulence appears to be modest (less than 30% for all cases) as compared to the huge change in the mean-flow energy as the gas crosses the flame front (the mean-flow energy increases by the ratio of the densities squared, that is, by a factor of 35-50). It therefore appears that the fraction of energy released by the chemical reaction into mechanical energy is predominantly partitioned into mean-flow energy (as was expected), while the turbulence energy is enhanced only slightly. Likewise, the effect of the flame front on the length scale of turbulence also appears to be very small. However, the effect of the flame on the turbulent viscosity
is substantial, primarily because the empirical function f, depends exponentially on kinematic viscosity which increases across the flame front. Table IV gives a summary of the calculatedjump conditions for k,t, and /.c, for various combinations of initial Reynolds number and equivalence ratio. The question of the effect of combustion on the kinetic energy of turbulence and the length scale has remained an open question in the literature. The difficulty that arises experimentally for many systems of turbulent combustion is that the device which stabilizes the flame (e.g., a flame holder) often introduces turbulence into the flame, making difficult the resolution of the enhancement of turbulence due to combustion from that due to the stabilizing device. The TSB is unique in that it has no flame holder; accordingly changes in the kinetic energy of turbulence are due solely to reaction. Unfortunately the geometrical configuration of the TSB makes optical access for direct measurements of the turbulence very difficult. The calculations described previously indicate that the effect may be small, further explaining some of the conflicts in the literature. Figure 8 shows radial profiles of the root-mean-square velocity and the length scale further downstream in region 3. The nominal kinetic energy and length scale are seen to decrease with distance from the flame front as a result of the increased viscous dissipation in region 3. Simultaneously the maxima of both profiles approach the centerline, apparently because the energy decays more rapidly near the wall than at the center of the channel, Eventually the profiles reach a self-similar shape that decreases in magnitude with distance. Figure 9 is a semilog plot of the centerline value of root-mean-square velocity and length scale as a function of the longitudinal coordinate. Initially the behavior is complex, but far downstream the profiles approach straight lines indicating exponential decay. The dynamics of the self-similar regime that is approached far downstream from the flame front is the topic of a future paper (Collins and Churchill, 1992). Discussion. The discrepancy between the predicted and measured pressure drops and velocities at the centerline is significant. The k-t model predicts a much faster approach to the asymptotic value for laminar flow than was observed. The most likely explanation can be formulated by taking a more careful look at the modeling for region 2. The solution for region 2 indicates that the kinetic energy of turbulence and the length scale are changed only slightly by the presence of the flame. In contrast, the turbulent viscosity is drastically reduced, owing largely to the change in the empirical function, f,,. This sharp decrease in the turbulent viscosity profoundly affects the development of the velocity profile in region 3 because it effectively eliminates the dynamical contribution of turbulence downstream from the flame front. The unanswered question is whether this sharp decrease
678 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 0.02
1
P U
I
m=o
0.01
0.00
0.0
0.2
0.4
0.6
0.8
1.o
rlR 0.5
7
1
0.4
I' D
0.3
0.2
0.1
0.0 0.0
0.2
0.4
0.6
0.8
rlR
1.o
Figure 8. Profiles of the root-mean-square velocity and turbulence length scale at three axial locations in region 3.
PSI! U D
O
m
40
M
kY
xlD
Figure 9. Axial variation of the root-mean-square velocity and length scale for Re = 3000 and @ = 1.0.
is a physical phenomenon or an artifact of the low-Reynolds-number functions that have been added to the k--E model. The function f,, is used to account for the effects of the wall on the turbulent viscosity in bounded flows. Jones and Launder (1973) postulated this functional form by fitting the calculated turbulent viscosities to the law-ofthe-wall formula of Van Driest (1957). The consequence
of this empiricism is that the turbulent viscosity has an explicit dependence on the molecular viscosity. To gain insight into what occurs at an interface with a discontinuous viscosity, it is helpful to consider the simpler case of homogeneous turbulence, for example, turbulence that is generated by a grid and allowed to relax into the homogeneous form. The equilibrium spectrum for homogeneous turbulence at large Reynolds numbers is often divided into two ranges, the inertial range which is dominated by nonlinear transfer of energy in wavenumber space and the dissipation range where molecular viscosity is important. A sudden change in the molecular viscosity would result in an immediate increase in the rate of dissipation while the spectrum rearranged itself into a new equilibrium; however, the rate of decay would quickly relax to the value it held previously because that rate is independent of uiscosity at large Reynolds numbers. Furthermore, the , change in viscosity would not have an appreciable effect on the rate of transport due to turbulence since these rates are principally controlled by the large-scale eddies of the inertial range, which would be unaffected by the viscosity jump. It is possible to imagine that at smaller Reynolds numbers the viscosity may have some direct influence on the turbulent rates of transport, but the effect would be expected to be small. The above argument implies that the effect of the viscosity jump at the flame front on the empirical function f,, potentially exaggerates the direct effect of the change in molecular viscosity on the turbulent viscosity. The result of the precipitous drop in the turbulent viscosity is that turbulence is effectively eliminated from the dynamics in region 3. The process of laminarization (i-e., rearrangement of the velocity and approach of the pressure gradient to their respective asymptotes) was predicted to occur more quickly than was observed because the turbulent viscosity, which was effectively removed, would have tended to maintain the turbulent profiles, and thereby slow the rate of approach to the laminar asymptote. f,, is designed to estimate the effect of the presence of walls on turbulence. In the wall region the molecular viscosity can be important, and therefore f,, should be a function of the viscosity. Unfortunately this implies that step changes in the viscosity have an immediate effect on the turbulent transport parameters that is unphysical. This error in the k-t model is difficult to circumvent since the empirical functions are required to predict adequately flows at low Reynolds numbers. Further work is needed to determine the correct behavior of the empirical functions at discontinuities in the density, viscosity, or both. Conclusions Turbulent flow in a tube that is suddenly heated by exothermic reactions at a flame front can laminarize owing to property changes that accompany the sudden increase in temperature. Laminarization occurs because the increase in temperature that accompanies the exothermic reaction causes a corresponding decrease in the Reynolds number to below the transitional value. Turbulent energy inherited from the flow upstream from the flame front decays to zero while the time-mean velocity rearranges into the parabolic distribution characteristic of laminar flow. This mechanism for the laminarization is closely related to that of previous studies of laminarizing flow in tubes heated at the wall. However, in this case the heating occurs in a thin flame zone and is therefore nearly instantaneous. The question of whether or not the downstream flow laminarizes does not arise consequently for this system since the downstream Reynolds number was in all cases far below the transitional value.
Ind. Eng. Chem. Res., Vol. 31, No. 3,1992 679 Previous studies of the TSB indicated that for modeling purposes the device could be divided into three regions: the cold gaseous reactants upstream from the flame front, the hot gaseous produds downstream from the flame front, and the flame region itself, where the chemical reactions occurred. The isothermal regions (regions 1and 3) were modeled with the modified k-e turbulence model of Jones and Launder. The region of the flame (region 2) was modeled with the variable-density k-e-1 model proposed by Jones and Whitelaw and modified by analogy to the k-e model. Measurements of the longitudinal pressure drop and velocity a t the centerline were made for an ethane-air flame and were compared with the predictions. The flow upstream from the flame front was assumed to be fully developed. The k-e model was solved for the radial profdes of the mean velocity as well as the variables of turbulence, k, e, and pv The flame front was treated as a normal hydrodynamic discontinuity. Because of the presence of turbulence, however, the standard jump conditions did not contain sufficient information to determine all variables, and therefore a numerical solution for the flame itself was sought. A modified k-e-1 model was solved using a stretched coordinate in the longitudinal direction to account for the rapid change. The results implied that the kinetic energy of turbulence and length scale were nearly unchanged across the flame, whereas the turbulent viscosity was reduced by nearly an order of magnitude. This was principally a consequence of the change in the empirical function f,,,which is an exponential function of the molecular viscosity. This function, which is designed to represent the “law of the wall” in the 12-6 model, causes the turbulent viscosity to be a strong function of the molecular viscosity, thereby exaggerating the effect of the flame. The large drop in the turbulent viscosity across the flame front causes the turbulence to be removed effectively from the dynamics of the laminarization downstream from the flame front. This is perhaps responsible for the difference in the rate at which the predicted and measured time-mean velocity at the centerline and pressure gradient approached their respective asymptotes for laminar flow. Further research on the proper behavior of turbulence transport coefficients (e.g., the turbulent viscosity) at hydrodynamic discontinuities is required for a better understanding of the effect of combustion on turbulence.
Acknowledgment L.R.C. was supported in this research by AT&T Bell Laboratories, through their Cooperative Research Fellowship Program. The experimental work was supported by the National Science Foundation, Grant CPE-80203967.
Nomenclature C = heat capacity of gas, J/(kg K) $ = constant in k-e model C2 = constant in k-e model C, = constant in k-e model D = diameter of channel, m fl = dimensionless function in k-e model f 2 = dimensionless function in k-e model f = dimensionless function in k-e model d = square of the mean strain tensor, l/s2 AH = heat of reaction, J/kg k = kinetic energy of turbulence, m2/s2 K = (u/u2)(du/dx)= acceleration parameter =;adid profile of k just downstream from flame front, 1’ k$2 / e = length scale of turbulence, m M , = molecular weight of gaseous mixture, kg/mol
:
p = dynamic pressure, kg/(m s) Ap = difference between impact pressure and ambient
pressure, kg/ (m s)
P = thermodynamic pressure, kg (m s) q = net
/
rate of reaction, mol/(m s)
r = radial coordinate, m R = radius of channel, m Re = p U D / p = Reynolds number R , = ideal gas constant, J/(mol K) Rt = k2/ve = Reynolds number of turbulence
T = temperature of gas, K Tf= adiabatic flame temperature, K Tier = reference temperature, K u = velocity in the x direction, m/s u, = centerline velocity in the x direction, m/s u+ = o / ( T , / P ) ’ / ~ = dimensionless velocity, m/s U = characteristic velocity in the x direction, m/s U3 = radial profile of u just downstream from flame front, m/s u = velocity in the r direction, m/s v = velocity vector, m/s V = characteristic velocity in the r direction, m/s w = mass fraction of fuel x = axial coordinate, m x , = characteristic axial distance, m xf = flame location Greek Symbols e = rate of dissipation of turbulent energy, m2/s3 E3(r) = radial profile of e just downstream from flame front,
m2/s3 p = molecular viscosity, kg/(m s) po = molecular viscosity at reference temperature, kg/ (m e) pt = turbulent viscosity, kg/ (m s) Qab = molecular diffusivity, m2/s
v = kinematic viscosity, m2/s ut = turbulent kinematic viscosity, m2/s p = density of gas, kg/m3 p o = characteristic density in region 2, kg/m3 Uk = constant in k-e model u, = constant in k-e model
= turbulent Prandtl number
Uh
u, = turbulent Schmidt number 7, = shear stress at the wall Subscripts 1 = region 1
2 = region 2 3 = region 3
Superscripts
-
- = time-average value = density-weighted average value
= root-mean-square value ’ = fluctuation from time-mean value
A
”
= fluctuation from density-weighted average value
Appendix A. Dimensional Analysis for Region 2 The axisymmetric equations representing region 2 can be generically represented by the energy equation, which has the following form:
All variables in the above expression are dimensionless (overbars and tildes are left out for simplicity). U and V are the characteristic velocities while xa and R are the characteristic length scales in the x and r directions, re-
680 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
spectively. x , is defined as ( p o U ) / Q , where Q is the characteristic rate of reaction. The Reynolds number is defined for the purpose of this analysis as Re* = Ux,/v, and the molecular Prandtl number is given by Pr = v/a. From the definition of xa, as the rate of reaction increases the length scale in the longitudinal direction decreases. In the limit as Q approaches infinity, the energy balance reduces to
Equation A2 implies that, to order x,/R (presumably a small parameter), the radial derivatives can be ignored in region 2. On the basis of these arguments, the governing equations for region 2 were assumed to be one-dimensional.
Appendix B. Dimensional Analysis for Region 3 The governing equations for region 3 (eqs 32-38) can be made dimensionless by defining the following dimensionless variables:
where the overbars signify dimensionless variables (not time averaging). Substituting these expressions into the governing equations and rearranging gives the following dimensionleas equations (note, overbars have been dropped below for convenience):
ak
ak ar
ax:
”[ ):
r dr r ( 1 +
ut
f2
=
$1 +
CJPRe2k2 e
= 1 - 0.3 exp(-Rt)
($)2+&[($$2+(E)2+(f)2]
The boundary conditions become au r=O - -- 0 , u = o , -ak= o , ar ar
u=O,
k=0, t = O
In general, the dependent variables can be written functionally as
where f represents a functional relationship. The effect of the Reynolds number of the flow, Rea, on the final solution appears explicitly in the dimensionless x variable as well as in the expressions for the dimensionless turbulent viscosity and R,. It is also implicitly involved in the boundary conditions as can be seen by considering the velocity profile at the entrance to region 3 for two Reynolds numbers. The larger Reynolds number will correspond to a larger upstream Reynolds number (Rel)and will therefore have a flatter velocity profile initially. Similar arguments are true for k and e. The influence of the Reynolds number on the initial profiles might be expected to diminish with distance downstream. Furthermore, the turbulent viscosity quickly becomes small as compared to the molecular viscosity in region 3, removing its influence over the solution. Functionally this corresponds to the following simplification of eq A l :
Registry No. Ethane, 74-84-0.
v,G- e-2(?>’
R, = Re32k2/e G=
u=O,
This expression becomes more accurate with increasing distance from the flame front. Equation A2 implies that the curves from all combinations of Reynolds numbers and equivalence ratios can be collapsed onto a single curve if plotted in the above coordinates. This postulate was confined throughout region 3, except very near the flame front where deviations between the various solutions exist.
i l u ( r ) d(r)2 = 1 u-+u-=
r = R:
-at -- 0 dr
Literature Cited Batchelor, G. K. The theory of homogeneous turbulence; Cambridge University Press: New York, NY, 1953. Bernstein, M. H.; Churchill, S. W. Multiple Stationary States and NO, Production for Turbulent Flames in Refractory Tubes. Sixteenth Symposium (International) on Combustion;The Combustion Institute: Pittsburgh, PA, 1977; pp 1737-1745. Chen, J. L.-P.An Experimental and Theoretical Investigationof the Stabilization of Propane-Air Flames in Insulated Refractory Tubes. Ph.D. Thesis, University of Pennsylvania, Philadelphia, PA, 1970. Chen, J. L.-P.; Churchill, S. W. A Theoretical Model for Stable Combustion inside a Refractory Tube. Combust. Flame 1972a, 18, 27-36. Chen, J. L.-P.;Churchill, S. W. Stabilization of Flames in Rsfractory Tubes. Combust. Flame 1972b,18,37. Choi, B. Combustion of Uniformly Sized Droplet Sprays in a Refractory Tube. Ph.D. Thesis, University of Pennsylvania, Philadelphia, PA, 1979. Choi, B.; Churchill, S. W. A Model for Combustion of Gaseous and Liquid Fuels in a Refractory Tube. Seventeenth Symposium (International) on Combustion; The Combustion Institute: Pittsburgh, PA, 1979; p 917. Churchill, S. W. The Interpretation and Use of Rate Data. The Rate Concept, revised ed.; Hemisphere: Washington, DC, 1979. Collins, L. R. Prediction and Observation of Decay of Turbulence and Reaction Behind the Flame Front in a Thermally Stabilized
Znd. Eng. Chem. Res. 1992,31,681-687 Burner. Ph.D. Thesis, University of Pennsylvania, Philadelphia, PA, 1987. Collins, L. R. A Reynolds Stress Model for Low Mach Number Variable-Density Flow. In review, 1991. Collins, L. R.; Churchill, S. W. Effect of Laminarizing Flow on Poetflame Reactions in a Thermally Stabilized Burner. Znd. Eng. Chem. Res. 1990,29,456-463. Collins, L. R.;Churchill, S. W. A Numerical Study of the Asymptotic Rate of Decay of Turbulence in Tubes. In review, 1991. Goepp, J. W.; Tang, H.; Lior, N.; Churchill, S. W. Multiplicity and Pollutant Formation for the Combustion of Hexane in a Refractory Tube. AIChE J. 1980,26,855-858. Hinze, J. 0.Turbulence, 2nd ed.; McGraw Hill: New York, NY, 1975. Jones, W. P.; Launder, B. E. Some Properties of Sink-Flow Turbulent Boundary Layers. J. Fluid Mech. 1972a,56,337-351. Jones, W. P.; Launder, B. E. The Prediction of Laminarization with a Two-Equation Model of Turbulence. Znt. J. Heat Mass Transfer 1972b,15,301. Jones, W. P.; Launder, B. E. The Calculation of Low-ReynoldsNumber Phenomena with a Two-Equation Model of Turbulence. Znt. J. Heat Mass Transfer 1973,16,1119. Jones, W. P.; Whitelaw, J. H. Calculation Methods for Reacting Turbulent Flows: A Review. Combust. Flame 1982,48,1. Launder, B. E.; Jones, W. P. Sink Flow Turbulent Boundary Layers. J. Fluid Mech. 1969,38,817-831. Libby, P. A.; Bray, K. N. C. Countergradient Diffusion in Premixed Turbulent Flames. AIAA J. 1981,19,205. McEligot, D. M.; Coon, C. W.; Perkins, H. C. Relaminarization in Tubes. Znt. J. Heat Mass Transfer 1970,13,431. Narasimha, R.; Sreenivasan, K. R. Relaminarization in Highly Accelerated Turbulent Boundary Layers. J. Fluid Mech. 1973,61, 417.
681
Narasimha, R.; Sreenivasan, K. R. Relaminarization of Fluid Flows. Adv. Appl. Mech. 1979,19,221-309. Patel, V. C.; Head, M. R. Reversion of Turbulent to Laminar Flow. J. Fluid Mech. 1968,34,371. Pfefferle, L. D.; Churchill, S. W. The Stability of Flames Inside a Refractory Tube. Combust. Flame 1984,56,165-174. Schlichting, H. Boundary Layer Theory, 7th ed.; McGraw Hill: New York, NY, 1979. Smooke, M. D. Solution of Burner-Stabilized Premixed Laminar Flames by Boundary Value Methods. J. Comput. Phys. 1982,48, 72-105. Smooke, M.D.; Koszykowski, M. L. Two-Dimensional Fully Adaptive Solutions of Solid-Solid Alloying Reactions; Sandia Report NO. SAND83-8909,1964. Spalart, P. R. NASA Report No. TM-88220, 1986. Tang, S. K.; Churchill, S. W. A Theoretical Model for Combustion Reactions Inside a Refractory Tube. Chem. Eng. Commun. 1980a,9,137-150. Tang, S. K.; Churchill, S. W. The Prediction of NOx Formation for the Combustion of Nitrogen-Doped Droplets of Hexane Inside a Refractory Tube. Chem. Eng. Commun. 1980b,9,151-157. Van Driest, E. R. J. Aerosol Sci. 1957,23,1007. Van Dyke, Perturbation Methods in Fluid Mechanics; The Parabolic Press: Stanford, CA, 1975. Wahiduzzaman, S.; Fergwon, C. R. Convective Heat Transfer from a Decaying Swirling Flow within a Cylinder. Proceedings of the Eighth International Heat Transfer Conference,San Francisco; Hemisphere: Washington, DC, 1986;Vol. 3,pp 987-992.
Received for review April 25, 1991 Revised manuscript received August 7, 1991 Accepted August 22,1991
Thermally Stabilized Combustion as a Means of Studying the Devolatilization of Coal Christina Chan,?Norio Arai,t Noam Lior,s and Stuart W. Churchill* Department of Chemical Engineering, The University of Pennsylvania, 220 South 33rd Street, Philadelphia, Pennsylvania 19104-6393
Thermally stabilized combustion of premixed ethane and air in a ceramic tube results in a virtual step function in temperature, composition, and velocity at the flame front. If the mixture is fuel-rich, the burned gas consists almost wholly of CO, COz, HzO, and N2. For a tube greater than 30 mm in diameter, the stable range of flow is in the turbulent regime both upstream and downstream from the flame front. An experimental technique exploiting this environment for study of the devolatilization of coal is described, and illustrative results are presented. As compared with other techniques using heated grids or entrainment, the conditions outside a particle are more uniform and the rate of heating is greater, but the velocity and temperature of the particle must be determined from momentum and energy balances rather than by direct measurement.
Introduction When coal is heated in a vacuum or an inert atmosphere as much as 75 wt% , depending primarily on ita rank, can be decomposed and volatilized. Hence, devolatilization is an important facet of the processing of coal by combustion, gasification, hydropyrolysis, and coking. The rate and extant of devolatilization depend on the rate of heating, the ultimate temperature, and the pressure and
*Towhom correspondence should be addressed. 'Current address: E. 1. d u Pont Marshall Laboratory, 3500 Grays Ferry Road, Philadelphia, PA 19146. Current address: Department of Chemical Engineering, Nagoya University, Nagoya 464,Japan. Department of Mechanical Engineering and Applied Mechanics, The University of Pennsylvania.
*
composition of the gaseous environment, as well as on the type of coal and ita granulation. Because of ita importance, the devolatilization of coal has been the subject of many investigations, both experimental and theoretical. However, because of the variety of coals, the number of variables, and the complexity of the process itself, this behavior is not yet completely understood or effectively generalized [see, for example, Solomon and Hamblen (1986)and Niksa (1988)l.A major diffiiulty arises in designing experiments from which the rate of devolatilization can be determined accurately, and for which the environmental conditions are not only defined unambiguously but also represent practical applications. A new experimental technique for this purpose is proposed herein and compared with existing methods. Attention is focused on the principles involved in the new methodology, but representative facilities and
0888-588519212631-0681$03.00/00 1992 American Chemical Society
