756
Ind. Eng. Chem. Res. 1992,31, 756-159
Can One- and Two-Equation Turbulence Models Be Modified To Calculate Turbulent Heat Transfer with Variable Properties? Shang-Woo Chyou and Charles A. Sleicher* Department of Chemical Engineering, University of Washington, Seattle, Washington 98105
When fluid properties, particularly viscosity, have large spatial variations caused by temperature variations, heat-transfer rates in turbulent flow cannot always be accurately predicted. A good model of this process would be useful in developing a correlation for a wide range of conditions. Here we have studied a variety of empirical modifications to zero- and one-equation turbulence models in an attempt t o apply them to variable property cases. None were as successful in predicting heat transfer in a pipe as the simple empirical correlations of Sieder and Tate, Sleicher and Rouse, or Petukhov and Popov.
Introduction In the literature there is a large mass of data on heat transfer in pipes. The data encompass a wide range of parameters, including laminar and turbulent flow, physical properties, flow rate, length to diameter ratio, pipe curvature, wall roughness, variation of physical properties with temperature, and heabtransfer rate. Almost all of the data, however, are heating data; good data on the rate of transfer from a hot fluid to a cold pipe are few. Consequently, correlations for cooling have been developed from heating data and have not been thoroughly tested. Because data on cooling of fluids in pipes are limited, it would be very useful in developing a correlation for cooling to have a good physical model of the heat-transfer process. The idea would be to use both heating and cooling data to evaluate constants or functions in the model and then use the model to extrapolate to conditions of cooling for which there are no useful data. Such calculations could then form the basis for an empirical correlation for cooling conditions. This paper reports a study of the application of zero- and one-equation turbulence models to this problem. (No calculations for laminar flow have been made.) Turbulence Models. It is well-known that analytical solutions to problems of turbulent flow require a “closure approximation”. That is, one must use some kind of ad hoc or physical assumption or model of the turbulence in order to reduce the number of equations. The problem arises because of the nonlinear term in the transport equation for the velocity, which is here taken to be the Navier-Stokes equation. If the velocity component Viis written Vi = Vi + ui, where Vi is the steady (time-averaged) part and ui is the turbulent fluctuation, and substituted into the term Uj(dUi/dxj) in the equation of motion, after time-averaging, a new unknown term arises The transport equation for the double correlation, uiuj, is very complicated and includes yet another unknown, a triple correlation. The transport equation for the triple correlation includes a quadruple correlation and so on. Thus a %1osuremapproximation is required to close or end the otherwise infinite set of equations that describe the flow. Eddy viscosity and mixing length models are simple closure approximations that have been proven to be very useful for a wide range of practical problems. They are called “zero-equation” models because they contain no differential equation for any turbulence quantity. The key to their success is to calibrate the model to the flow, that is, to evaluate the constants and functions in the model under conditions not very different from the conditions to which the model is intended to apply. These simple
G.
models cannot be used, however, in geometries much different from those used to calibrate the model. For example, mixing lengths determined in a pipe are not useful in calculating drag and heat transfer in a cavity. Many attempts have been made in the past two decades to develop turbulence models that are more generally applicable than mixing length and eddy viscosity models. One-equation models contain a transport equation for one turbulence quantity, the turbulent kinetic -energy - per unit uZ2+ u ~ ~in) addi, mass, K = 0.5q2 = 0 . 5 z = 0.5(? tion to the continuity equation and transport equation for Ui.The eddy viscosity is then modeled by ut = Clql where 1 is a prescribed length scale. This equation yields poor results near a wall, and Norris and Reynolds (1975)have proposed an empirical correction to account for suppression of turbulence by the wall:
+
ut =
Clql[l- exp(-C2qy/v)l
(1)
One term of the transport equation for kinetic energy is the local mean rate of energy dissipation, e. I t is known at high Reynolds that e can be modeled by c = c3q3/1 numbers and by e = C4vq2/12a t low Reynolds numbers. This led Norris and Reynolds (1975)to use
E = C q 1 + - q;;u] which is presumably a good model both near and far from a wall. Note that the one-equation models of turbulence have three adjustable constants in addition to an adjustable function, the length scale (0, which must be specified as a function of position. This flexible empiricism makes the models very adaptable but requires either an ad hoc assumption about 1 or enough knowledge of the flow field to be able to describe 1. To circumvent this problem, the two-equation models of turbulence were developed. These models contain differential equations for two turbulence quantities, the effect of which is to give 1. The most well-known of these models in the K-e model, which contains a transport equation for e as well as K. The model does not avoid empiricism; a number of constants and functions must be specified, but presumably the effect of the empiricism is diminished because one more turbulence equation is satisfied. Note that with this model one in general must solve six coupled partial differential equations for the flow field, even in the constant property case: continuity, the three scalar equations for the velocity, and the equations for K and e. For variable property heat transfer, one must also solve the energy equation and property equations for any variable property.
0888-5885/92/263~-0156$03.00/0 0 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 757
Analysis Constant Property Cases. The work reported here used both zero- and one-equation models, together with a number of empirical damping functions on the turbulent diffusivity. The objective was to find one set of constants, one 1 function, and one damping function that would give calculated results that agree with selected heat-transfer data covering the wide range of Reynolds number, Prandtl number, and physical property variation. For the zeroequation case we used continuity equation
These equations have been used with success by Hassid and Poreh (1975) for constant property, non-Newtonian flow and by El Hadidy et al. (1982) for constant property heat transfer with liquid metals. Each group used different values of the constants and different 1 functions. Variable Property Cases. In order to incorporate variable properties into these models, we experimented with two types of modifications to the models of Hassid and Poreh (1975) and El Hadidy et al. (1982). The first type is to redefine y+. The modifications we tried were
(3) mean velocity equation
-(VU, ax a -
(ii)
+ -(rVQ r adr
=r adr [ r ( u F - i i i j ) ] - ~ $ (4) mean enthalpy equation
(iii)
together with
Q = 2 n S0R O r dr = nR2Ub
(6)
In this case closure was obtained by assuming
a0 aR uv =-ut - and vh = -at (7) dr ar Here ut and at were based upon the works of Reichardt (1951) and Notter and Sleicher (1971), which for the constant property case can be summarized as follows: ut
- = 0.0083(y+)2 Y
;)[
2v = L(l+ 15 ""= ct
1
y+ < 45
+ 2(;)']
(8)
y+ > 45
0.00093(y+)3 y+ < 45 [ l + 0.0067(y+)211/2
(9)
(VI
(ii)
a
-(OK) dr
a + -(rVK) = r dr -vu -au + c + dr
"Ir[
r dr
v
$! - .( k +
;)]I
(iii)
(12)
together with a closure approximation (vii) The eddy viscosity is then calculated from eq 1,at from eqs 10 and 11, and E from a generalization of eq 2:
Y$Y E(y)R
E(y) =
:[
1 - (1 -
i) ] 312
(15)
All definitions but 15(iii) reduce to the usual definition of y+ when physical properties are constant. The length scales were modified by using y+R/R+for y and y+/R+for y / R . Equation 15(iv) is the model of Goldmann (1954), which has sometimes been applied to turbulent flow problems, for example by Hanna and Sandal1 (1978). The second type of modification is to multiply ut by a factor that will damp or amplify depending on whether viscosity decreases or increases near the wall. In general, such factors would be complex functions of flow field and property field, but in this study we explored only a few explicit functions of properties and position:
(10)
where the turbulent Prandtl number, fit,was set to 0.755 (1/1.3). These equations give excellent results, for they were empirically calibrated to the constant property cases of flow in a pipe. For the one-equation model we used eqs 3-5 plus the turbulent kinetic energy equation
Yt =
(viii)
758 Ind. Eng. Chem, Res., Vol. 31, No. 3, 1992 Table I. Comparison of Data, Simulations, and Correlations (MBS = Malina and Sparrow (1964), A&E = Allen and Eckert (1964), Sirn = Simulation) ref; fluid Thr O C R h Reh Ph/k NUh error, % 60 3 101 OOO 1.25 401 M&S; water -1 396 Sim, Goldmann 4 416 Sim, local damping -16 345 Seider-Tate 3 411 Sleicher-Rouse 85 75 M&S; oil 12OOO 2.2 249 229 Sim, Goldmann -8 208 -16 Sim, local damping 266 7 Seider-Tate 6 265 Sleicher-Rouse 15.6 13 OOO 8 1.50 114 A&E; water 94.6 -17 Sim, Goldman 100 Sim, local damping -13 109 Seider-Tate -4 110 -4 Sleicher-Rouse 15.6 8 A&E; water 13 OOO 2.8 116 69.8 Sim, Goldmann -40 99.2 -14 Sim, local damping 119 3 Seider-Tate 115 -1 Sleicher-Rouse 2.7 66.8 47 700 2.1 189 Dickinson; water 109 -42 Sim, Goldmann 173 -8 Sim, local damping 202 Seider-Tate 7 204 Sleicher-Rouse 8 59.7 3 Dickinson; water 3.0 1230 428OOO 880 -28 Sim, Goldmann 1597 Sim, local damping 30 1340 Seider-Tate 9 1340 Sleicher-Rouse 9 49.7 3.6 Chyou; water 11 700 1.9 77 79.8 Sim, Goldmann 4 Sim, local damping 8 80.4 69 Seider-Tate -10 75 Sleicher-Rouse -3 42.5 16.9 Chyou; 1-propanol 11 400 2.7 131 123 Sim, Goldmann -6 131 Sim, local damping 0 119 Seider-Tate -9 Sleicher-Rouse -6 123
Two-Equation Models. We did not investigate K-t models because we believe that the one-equation models are likely to give better results for our problem for the following reason. In the two-equation models the length scale, 1, is determined by solving a differential equation containing several assumptions. On the other hand, 1 has been empirically determined for constant property flows in pipes, which are geometrically similar to our cases, Le., the eddy viscosity and length scales are "calibrated" for conditions very close to ours. Therefore, our concern was to try to find corrections for variable properties to the length scale and to eddy diffusivity. Moreover, El Hadidy et al. (1982)and Hassid and Poreh (1975)report good results with one-equation models. Calculation Procedure. The calculation procedure was straightforward. The initial velocity profiles (at x = 0, the thermal entrance) were calculated from the equations already given. The equations were nondimensionalized and then numerically integrated with a CrankNicholson scheme. The independent variables were { = x / R and 71 = ( r / R ) 2 . In the {direction the step size was proportional to 5; and steps were nonuniform in the 71 direction. The pressure at each step was determined by linear extrapolation from the previous step and was then adjusted by iteration until the continuity and momentum equations were satisfied. Calculations were done on a VAX computer. The program was checked by calculating the classical constant property Graetz problem for both constant wall temperature and constant wall heat flux. Results agreed
within 0.1%. The turbulent flow calculations of Notter and Sleicher (1975)were also checked. Agreement was within 2% for a wide range of Reynolds and Prandtl numbers, but the present results are 4 4 % low in the range loo00 < Re < 15000 with 1< Pr < 3. This difference is unsettling, but the cause is not known.
Results For comparison of the turbulence models to data we chose data from four studies that were selected for the applicability of the data to our problem and the quality and range of data. The four were the study data of Allen and Eckert (1964)on the heating of water, Malina and Sparrow (1964)on the heating of water and oil, Dickinson (1968)on the heating of water, and Chyou (1986)on the cooling of water and 1-propanol. Both zero- and oneequation models with both types of variable property corrections were compared to the data. In these models both viscosity and density were their tabulated functions of temperature. In all cases the effects of density variations were very small compared to those of viscosity, so dilatation and body forces were unimportant. (In the case of gases, both density and viscosity variation with temperature might be significant.) The results are easily summarized. The one-equation models gave unsatisfactory results even at low heat flux unless the turbulent Prandtl, vt/at,number was taken to be quite small, 0.4. This is contrary to empirical evidence, so the one-equation model was not investigated in detail against the data at high heat flux.
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 759 The zero-equation models all gave satisfactory results (generally within 10% of the data) when the heat flux is small, which is of course expected since the diffusivity profiles were calibrated with those data. At high heat fluxes it is necessary to determine the constants in the various damping/amplification factors applied to ut. A value for these constants was determined for each factor by fitting the simulations to data. Some of the results are summarized in Table I, where “Sim, Goldmann” refers to the zero-order model with y+ defined by eq 15(iv) and “Sim, local damping” refers to the zereorder model with ut modified by eq 16(vi). In most casea those models each gave the best results for their class of modification. Cooling data were simulated better than heating data. That may in part be due to the more restricted range of parameters for the cooling data, but it could also be caused by the likelihood that the destabilizing effect of heating on turbulence is more complex and more pronounced than the stabilizing effect of cooling. The primary promise of turbulence models is that one set of empirical constants in the models might be found that would fit a wide range of parameters and a variety of geometries. For the case of variable property liquids this goal has proved elusive. None of the models investigated were satisfactory. No model gave results that were within 20% of all of the data. It is enlightening, if a little discouraging, to note that the simple empirical correlation of Sleicher and Rouse (1975) and of Petuhkov and Popov (1970) and even the old standby, the Sieder-Tate equation, give more satisfactory results than any of the much more complicated turbulence models that we have tried. This study is by no means conclusive. Perhaps we are wrong about the K-e model; perhaps there is some way to modify it to give satisfactory results for variable property flows. In any case, it would be useful to have both data and modeling ideas on what happens to turbulence in a fluid that is being heated or cooled in a pipe and to retain a healthy skepticism about the usefulness of K-e theory to problems of heat transfer in liquids.
Nomenclature h = enthalpy fluctuation H = enthalpy k = instantaneous value of turbulent kinetic energy, 0.5uiui K = local turbulent kinetic energy I = local value of turbulent length scale p = pressure fluctuation Prt = turbulent Prandtl number, ut/at q = local turbulent velocity scale Q = volumetric flow rate r = radial coordinate R = pipe radius u = velocity fluctuation in axial direction
ui = velocity fluctuation in i direction u* = friction velocity, ( ~ ~ / p ) l / ~ U, = bulk average velocity Ui = instantaneous velocity in i direction u = velocity fluctuation in radial direction z = axial coordinate y = coordinate normal to wall, R - r y+ = yu*/v a = thermal diffusivity at = turbulent thermal diffusivity e = local mean dissipation of turbulent kinetic energy Y = kinematic viscosity ut = turbulent kinematic viscosity p = density rW = shear stress at wall overbars = local time averages
Literature Cited Allen, R. W.; Eckert, E. R. G. Friction and Heat-Transfer Measurements to Turbulent Pipe Flow of Water (Pr = 7 and 8) at Uniform Wall Heat Flux. J. Heat Transfer 1964, 86, 301-310. Chyou, S.-W. Heat Transfer of a Variable Property Turbulent Fluid to a Colder Pipe Wall. Ph.D. Dissertation, University of Washington, Seattle, WA, 1986. Dickinson, D. R. AEC R & D Report BNWL-631;Atomic Energy Commission: Richland, WA, March 1968. El Hadidy, M. A.; Gori, F.; Spalding, D. B.Further Results on the Heat Transfer to Low-Prandtl-Number Fluids in Pipes. Numer. Heat Transfer 1982,5,107-117. Goldmann, Kurt. Heat Transfer to Supercritical Water and Other Fluids with Temperature-Dependent Properties. CEP Symp. Ser. 1954,50 (No. ll),105-113. Hanna, 0. T.; Sandall, 0. C. Heat Transfer in Turbulent Pipe Flow for Liquids Having a Temperature-Dependent Viscosity. Trans. ASME, J. Heat Transfer 1978,100, 224-229. Hassid, S.; Poreh, M. A. Turbulent Energy Model for Flows with Drag Reduction. Trans. ASME, J. Fluids Eng. 1975,97 (June), 234-241.
Malina, J. A.; Sparrow, E. M. Variable-Property, Constant-Property, and Entrance-Region Heat Transfer Results for Turbulent Flow of Water and Oil in a Circular Tube. Chem. Eng. Sci. 1964,19, 953-961.
Norris, L. H.; Reynolds, W. C. Turbulent Channel Flow with a Moving Wavy Boundary. Stanford University Department of Mechanical Engineering Report FM-10; Stanford University: Stanford, CA, 1975. Notter, R. H.; Sleicher, C. A. The Eddy Diffusivity in the Turbulent Boundary Layer Near a Wall. Chem. Eng. Sci. 1971,26,161-171. Petukhov, B. S.; Popov, N. V. In Petukhov, B.S. Heat Transfer and Friction in Turbulent Pipe Flow with Variable Physical Properties. In Aduances in Heat Transfer; Hartnett, J. P., Thomas, F. I., Eds.; Academic: New York, 1970; Vol. 6, pp 503-564. Reichardt, H. Representation of Velocity Distribution of Turbulent Flow in Pipes (in German). 2.Angew. Math. Mech. 1951, 31, 208-219.
Sleicher, C. A.; Rouse, M. W. A Convenient Correlation for Heat Transfer to Constant and Variable Property Fluids in Turbulent Pipe Flow. Int. J. Heat Mass Transfer 1975, 18,677-683. Received for review April 29, 1991 Accepted June 28,1991
