Nahrain University, College of Engineering Journal (NUCEJ) Vol.10, No.1, 2007 pp.53-64
Basim O. Hasan Chemistry. Engineering Dept.- Nahrain University
Turbulent Prandtl Number and its Use in Prediction of Heat Transfer Coefficient for Liquids
Basim O. Hasan, Ph.D
Abstract: eddy conductivity is unspecified in the case of heat transfer. The classical approach for obtaining the A theoretical study is performed to determine the transport mechanism for the heat transfer problem turbulent Prandtl number (Prt ) for liquids of wide follows the laminar approach, namely, the momentum range of molecular Prandtl number (Pr=1 to 600) and thermal transport mechanisms are related by a under turbulent flow conditions of Reynolds number factor, the Prandtl number, hence combining the range 10000- 100000 by analysis of experimental molecular and eddy viscosities one obtain the momentum and heat transfer data of other authors. A Boussinesq relation for shear stress: semi empirical correlation for Prt is obtained and employed to predict the heat transfer coefficient for du the investigated range of Re and molecular Prandtl ( ) 1 number (Pr). Also an expression for momentum eddy m dy diffusivity is developed. The results revealed that the Prt is less than 0.7 and is function of both Re and Pr according to the following relation: and the analogous relation for heat flux: Prt=6.374Re-0.238 Pr-0.161 q dT The capability of many previously proposed models of ( h ) 2 Prt in predicting the heat transfer coefficient is c p dy examined. Cebeci [1973] model is found to give good The turbulent Prandtl number is the ratio between the accuracy when used with the momentum eddy momentum and thermal eddy diffusivities, i.e., Prt= m/ diffusivity developed in the present analysis. The h. Thus Eq.(2) can be written as: thickness of thermal sublayer decreases with Reynolds number and molecular Prandtl number. q dT ( m /Prt ) 3 Keywords: Turbulent Flow, Heat Transfer c p dy Coefficient.
1. Introduction Thus if one knows the eddy diffusivity and turbulent Prandtl number, Prt, the heat transfer problem can be Despite many years of intensive research into solved [Simpson et. al. 1970]. A number of turbulent diffusion, it is still poorly understood and can experimental and theoretical investigations have been only be rather crudely predicted in many cases [Philip devoted to obtain the eddy diffusivity but these for and Webester 2003]. Because of highly complex turbulent Prandtl number are less turbulent flow mechanism, the prediction of the As the complex nature of the turbulent transport transport rates necessarily involves the formulation of process is not yet well understood, the development of conceptual models which embody many simplifying models for the prediction of Prt requires the assumptions [Gutfinger 1975]. In the momentum introduction of many tenuous assumptions regarding problem the eddy viscosity remains unknown while the the behavior of turbulence. The validity of these
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assumptions may be verified only indirectly by have concluded, on the basis of their own results, that comparing the predicated Prt values with the measured Prt is varyingly affected by the molecular Prandtl ones. There are basically two routes by which Prt may number, the flow Reynolds number, and the distance be evaluated experimentally [Gutfinger 1975]: from the wall. The suggested trends of Prt with these 1. Utilizing experimental time-average velocity and parameters for the case of liquid metals (Pr<0.1) differ temperature (or concentration) profiles together with from those for ordinary fluids (Pr>0.7). For liquid the integrated Reynolds equations. From equations (1) metals Prt is greater than unity and decreases with and (2) increasing Reynolds number and the distance away from the wall (Carr and Balzhiser, 1967). This effect is du / 1 opposite to that for air (Sleicher, 1958), where Prt is dy less than one, and increases towards unity with Pr increase in Re and distance from the wall. Although the t q dT 4 / 1 above trends have provided a basis for the formulation C p dy of models for the prediction of Prt to be considered in predicting heat transfer coefficient, the picture is by no means conclusive, and many studies have produced Thus, if the momentum and heat flux variation with the results which exhibit a different behavior. The value of distance from the wall are known, m, h , and hence Pr=0.7 to 1 is of practical importance because it Prt can be evaluated. represents most gases. Also the value of Pr=7 2. Direct measurement of the Reynolds transport terms represents water and light liquids near room temperature [Kays 1993]. Quarmby and Quirk (1974) ( u' ' , T' ' ) and use of the definition of the eddy have concluded, on the basis of an extensive study of diffusivity: turbulent flow in a plain circular tube with Prandtl du numbers varying from 0.7 to 1200 and Reynolds t u' ' m numbers ranging from 5230 to 23550, that Prt is a dy simple function of the nondimensional distance from q dT the wall, independent of both the Prandtl number and t T' ' h dy Reynolds number. They found that Prt varies smoothly from 0.5 at the wall to unity at the tube center. Hence, Brienkworth and Smith (1969) (Pr=6) and Eckelman u' ' T' ' and Hanratty (1972) (Pr=0.7) suggest on the other m and h hand, that Prt is constant and equal to approximately du / dy dT/dy unity over the whole of the boundary layer. Yet a different trend was reported by Simpson et al. (1970) who found that for boundary layer flows of air dy or Pr u' ' dT (Pr=0.77), the local value of Prt near the wall is t T' ' dy du greater than unity, displaying a maximum value of Thus, Prt can be evaluated from the experimentally approximately 1.4 at the wall and decreasing to determined velocity and temperature fields, and the approximately unity in the outer region. Shlanchyauskas et al. (1974) for Pr>1 found that Prt is turbulent fluctuating terms u' ' and T' ' . constant and equal to 0.75. In view of the contradictory The characteristic feature of all the reported experimental data it is significant to note the analysis experimentally determined Prt is the great scatter of the of Deissler (1963) based on the statistical nature of the data. The results of different investigations lack in turbulent process. Deissler suggests that Prt approaches agreement and are often conflicting. This situation unity as the velocity gradient increases, regardless of arises directly from the necessity to differentiate the the molecular Prandtl number of the fluid. Deissler also two measured profiles and determine the shear stress suggests that the effect of Pr is much greater at low and heat flux profiles for the calculation of Prt. The values of Pr (i.e., liquid metals) than at higher ones. differentiation procedure greatly amplifies the This is consistent with the results of both Sliecher uncertainties associated with the experimentally (1958) and Quarmby and Quirk (1974). Miyak [1992] measured velocity and temperature profile. These and Gurniki et. al [2000] stated that the Prt is between difficulties arise from the inaccessibility of probes to 0.33 and 0.5. Aravinth [2000], developed a resistance- the thin region adjacent to the wall in which the in-series model to quantitatively predict the heat and velocity and temperature gradients concentrate, mass transfer processes for turbulent fluid flow through particularly at high molecular Pr [Gutfinger 1975]. tubes and circular conduits under uniform wall Experimentally determined turbulent prandtl numbers temperature condition. Kays [1994] examined the have been surveyed by Kestin and Richardson (1963) available experimental data on Prt for the two and then by Blom and de Veres (1968). The results of dimension boundary layer in circular tube and flat surveys together with the more recent data of Bolm plate. Churchill [2002] analyzed the viscosity and eddy (1970), Simpson et al. (1970), Sleicher et al. (1973), conductivity in fully developed turbulent pipe flow and and Quarmby and Quirk (1969,1972,1974) and redefined the Prt directly in terms of the time-averaged Gutfinger (1975) are summarized in Table 1. Although fluctuations and stated that it remains an essential the results are greatly different, various investigators parametric variable. Toorman (2003) showed that Prt
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vary from 1 to 10 in the near wall region and fluctuates satisfactory empirical theory. This in turn has around 1 in the fully turbulent region. Crimaldi et al. necessitated the formulation of semi-empirical theories [2006] measured the distribution of Prt in the in an attempt to both rationalize the existing data and to laboratory boundary layer and developed an analytical provide a starting point for heat and mass transfer model for Prt and found that it is significantly larger calculations. In the present work it is aimed to obtain a than unity even at large distance from the wall. model for average Prt from available experimental data The above comments are sufficient to indicate the for wide range of Reynolds and molecular Prandtl general inconsistencies and lack of agreement which numbers. Also it is aimed to examine the capability of are characteristic of much of the published work. These many previously proposed models for Prt to predict inconsistencies, and low accuracy of the available heat transfer coefficient by comparing them with the measurement, have hindered the development of a experimentally determined heat transfer coefficient.
Table (2) Models for Turbulent PRANDTL Number for Various Authors.
Equation for Prt Author 1 Diessler (1952) n Re Pr 1 exp n Re Pr Prt n=experimental constant=0.000153 1 1 135 Re 0.45 exp 0.25 Azer and Chao (1961) 0.46 0.58 0.25 0.6 Prt= Pr high turbulence intensity 2 6Pr Tyldesley and Silver Pr low turbulence intensity (1968) t 9 Pr 2 9Pr Pr high turbulence intensity t 3 9Pr Graber (1970) 1 =0.91+0.13Pr0.545 Prt 0.7 2. Theoretical Aspects The shear stress due to molecular and turbulent transport of momentum is given by Eq.(1) NUCEJ Vol.10, NO.1 Hasan 55 + * + * w * y since u =u/u , y =yu / , and =u y (1 ) 2 1 R dy 10 d(u u*) f Eq.(1) becomes, (1 m ) 0 1 m * * u yu d( ) Since; dT qw= Cp( ) h dy 11 du or (1 m ) ; hence: u*2 dy h dT du qw= Cp( ) 12 (1 m ) d(y / ) dy 5 w Hence: r Since where R=radius and * h dT qw= Cpu (1/Pr+ ) 13 w R dy r=distance from the center=R-y; hence R y R y y (1 ) 6 y dy u* Cp T w R R R dT 14 1/Pr /v q Eq.(5) becomes: 0 h w Tw y du hence: (1 ) (1 m ) 7 R dy y * dy (T-TW )=(qw/u Cp) 15 0 1/Pr h /v f where R+=(Ru*/ )= Re Eq.(15) gives the temperature at any point (y+). If the 8 temperature profile is written in dimensionless form, thus: i.e.: y (1 ) T+ =(T-Tw)/(Tb-Tw) du R dy 8 m If Eq.(14) is integrated from the wall(y+=0 and T=Tw) 1 to the center of turbulent core (y+ =R+ and T=Tb), one obtain Equation (8) can be integrated to the limits that at R y+=0, u+=0 and at y+=y1+, u+=ub. Where u+b is the * dy dimensionless average bulk fluid velocity(ub/u*) and (Tb-Tw )=(qw/u Cp) 16 y1+ is the distance from the wall beyond which the 0 1/Pr h /v velocity becomes equal to ub. Therefore Eq. (8) becomes dividing Eq. (15) by Eq. (16) gives: y y (1 ) 1 y u R dy 9 dy b 0 m 1/ Pr / v 1 T TW 0 h T+= 17 T T R u u b W dy since b b 2 , u 1/ Pr h / v b u* f f 0 u b 2 Since Q=hA(Tb TW ) substitution in Eq. (9) yields h=q/(Tb-Tw) Sub. in Eq.(16) R dy u* Cp/h= 1/Pr /v 0 h Since Nu=hd/k NUCEJ Vol.10, NO.1 Turbulent Prandtl Number 56 c- Near turbulent core region: R dy * The large eddies in the core region and the small u Cpd/Nu k= 18 variation in turbulent intensity in the central region 0 1/Pr h /v makes the eddy viscosity constant. Therefore in the present analysis the momentum eddy diffusivity for u*=ub f / 2 =(Re / d) f / 2 central region will be considered constant as did by hence from Eq.(18): Mizushina et. al. [1971] and Hinze [1975 ]: Nu= f/2RePr R dy 19 m / =0.07 R+ y+>y2+ 21c 1/Pr / 0 h Since Prt= m/ h, thus Eq.(19) becomes: Now, for predicting momentum eddy diffusivity expression near the wall, Eq. (21a) is inserted in Eq.(10) and the experimental value of friction factor is f/2RePr substituted in Eq.(10). The well known Blasius Nu correlation for friction factor will be adopted, i.e.: R dy 20 0 1/Pr ( m ) / Pr f=0.079Re-0.25 22 t Hence Eq.(10) becomes Accordingly, if the variation of m/ with y+ and Prt are known, the value of Nu can be estimated from y y (1 ) Eq.(20). 1 5.032Re0.5 R dy 23 3 3. New Momentum Eddy Diffusivity Model 0 1 Ay The eddy diffusivity behavior in the viscous sublayer, The upper limit of integration in Eq.(23) is obtained by damped turbulence layer, and turbulent core affect equating equations (21a) and (21b), i.e.: greatly the rate of heat (or mass) transport between the A y1+3= 0.45 y1+ wall and bulk. Previous studies [Von Karman 1939, 0.45 or y Lin et. al. 1953, Deissler 1952, Wasan and Wilk1964, 1 A Rosen and Tragradh 1995, Meignen and Berthoud 1998, Lam 1988, Gurniki et. al 2000, and Wang and Thus Eq. (23) becomes Nesic 2003] showed that the eddy diffusivity is y 0.5 (1 ) function of many variables such as the distance from (0.45/A) the wall (y+), Re, and Pr. Most studies [Wasan and 5.032Re0.5 R dy 24 3 Wilke 1964, Townsend 1961, Hughmark 1969, Shaw 0 1 Ay and Hanraty 1964, Escobedo et. al. 1995, Papavassiliou Substitution of Re in Eq.(24) and performing the 1997] showed that in the near wall region (the region integration numerically the value of A can be obtained of major importance in the property transport) m / by trial and error for each value of Re. Fig. (1) shows proportional to y+3, i.e, m / =A y+3. a plot of A vs Re as obtained from Eq.(24). From best In the present analysis the region of interest is divided fit method: into three zones for momentum eddy diffusivity variation: a- Near wall region: A=0.0064Re-0.322 25a m / =A y+3 0 Best fitting of experimental results for m / obtained by Sleicher et. al. [1958] indicated that outside the y2+=0.156R+ 25c viscous sublayer B=0.45. Levich [1962] showed that B=0.4. NUCEJ Vol.10, NO.1 Hasan 57 Now to investigate whether Prt is unity over the whole substituted in Eq.(19), i.e., h= m (or Prt=1) to range of Re and Pr the expressions of momentum eddy estimate Nu and compare with experimental Nu. Thus diffusivity developed in Eqs. (21a, 21b, 21c) is Eq. (19) becomes: f/2RePr Nu y y 1 dy 2 dy R dy 26 0 0.322 3 1/Pr 0.0064Re y y 11/Pr 0.45y y2 1/Pr 0.07R Insertion of Re and Pr with f from Eq.(22) and , i.e., h and m are not equal. They also indicate that performing the integration numerically, Nu can be at a particular Pr, the higher the Re is the higher the obtained. The results are plotted in Fig. 2 as compared difference between Nu from Eqs.(26) and (27). Also at with the experimental Nu of Friend and Mitzner a particular Re the higher the Pr is the higher the [1958]. Friend and Mitzner performed an experimental difference. heat transfer study for wide range of Pr and obtain the following relation for 0.5 > Pr> 800: 4. Examination of the capability of proposed Prt models to predict nu value f Re Pr Figure 2 indicates that Prt is not unity. To Nu 2 27 1/ 3 investigate which models of Prt presented in Table 1 1.2 11.8 f / 2(Pr 1) Pr gives accurate results, these models are substituted into equation (20) with m/ taken from Eqs. (21). Thus Fig. 2 reveals that the Nu obtained from Eq.(26) is Eq.(20) becomes: different from that obtained from experimental results indicating that the turbulent Prandtl number is not unity f/2RePr Nu y y 1 dy 2 dy R dy 28 0.322 3 0 1/Pr (0.0064Re y / Pr ) y 11/Pr (0.45y / Pr ) y 1/Pr (0.07R / Pr ) t t 2 t Now inserting of various models from Table 1 in Eq. prevent confusion) with experimental Nu of Friend and (28) and performing the integration numerically, with Mitzner (Eq. (27)). The figure shows that most models y1+ and y2+ from Eqs.(25b) and (25c) and f from Eq. exhibit considerable deviation. The model of Cebeci (22), the Nu value can be obtained for each value of Re [1973] exhibits good agreement for high Pr values. and Pr. Fig. 3 shows comparison between Nu predicted from various models (some models are avoided to NUCEJ Vol.10, NO.1 Turbulent Prandtl Number 58 3 Pr=7 2 Azer and Chao Deissler Cebeci 1000 9 Friend and Mitzner (Exp) 8 7 Tyldesley and Silver 6 5 4 u 3 N 2 100 9 8 7 6 5 4 3 8 9 2 3 4 5 6 7 8 9 10000 100000 Re Figure (3) Comparison of Nu Obtained from Various Prt Models and Experimental Nu for Pr=7. 5. More accurate model for turbulent where y+ varies from 0 to y1+. By performing the PRANDTL number integral u+ is obtained for each y+. Fig. 4 shows the velocity profile at various Re as compared with other To obtain more accurate expression for Prt from authors. The figure shows that the velocity profile experimental results of Friend and Mitzner, Nu is obtained from present analysis is in good agreement estimated from Eq. (27) of Friend and Mitzner and with previous experimental and theoretical studies. substituted in Eq. (28). Performing the integration of Also the figure reveals that very close to the wall the the denominator numerically, Prt is obtained by trial velocity profile is linear and Re has no effect on u+. and error for each value of Re and Pr. In present analysis the variation of Prt with the radial distance is 30 ignored, i.e., the average turbulent Prandtl number is determined. Using statistical method the following Present Analysis [Re=10000] relation is obtained: Wasan and Wilke Sleicher et al (exp) Von Karman Prt=6.374Re-0.238 Pr-0.161 20 29 C.C=0.98 + The results shows that for liquids of Pr > 1 the Prt is u less than one indicating that the thermal eddy diffusion 10 is larger than momentum eddy diffusion. Also Prt decreases with increasing Re and Pr indicating that the increase in thermal eddy diffusion is larger than that in momentum eddy diffusion. Many studies [Deissler 1952, Jenkis 1951, Azer and Chao 1961] showed that 0 for liquid metals (Pr< 0.1) the Prt is higher than one. Also it is to be mentioned that Miyak [1992] and Gurniki et. al [2000] found that the Prt is between 0.33 0 10 20 30 40 50 and 0.5. y+ Figure (4) Velocity Profile as Compared with other 6. Velocity Profile in the Near Wall Region Workers The velocity profile in the near wall region is obtained by substituting Eq. (21a) into Eq.(8) and integrating 7. Temperature Profile both sides, i.e.: To obtain temperature profile Eq.(17) is used with h= m/Prt and Prt from Eq. (29) and the expression of y m/ from Eqs.(21) for each region of y+. Thus Eq. y (1 ) u R dy 30 (17) becomes: 0.322 3 0 1 0.0064Re y NUCEJ Vol.10, NO.1 Hasan 59 y dy 1 / Pr ( / v) / 6.374 Re 0.238 Pr 0.161 T 0 m y y 1 dy 2 dy R dy 31 0 0.0064 Re 0.322 y 3 y 1 0.45 y y 0.07 R 1/Pr 1/Pr 2 1/Pr 6.374 Re 0.238 Pr 0.161 6.374 Re 0.238 Pr 0.161 6.374 Re 0.238 Pr 0.161 The denominator of Eq.(31) is constant because the limits of integration are constants while the integral in 1.0 the numerator varies with y+, i.e., for each value of y+ there is a value of numerator and consequently a value of T+. The expression of m/ in the numerator is taken 0.8 from Eqs.(21a) according to y+, the limit of integral. Re=40000 The values of T+ that are obtained from Eq. (31) are Pr=600 0.6 Pr=100 shown graphically in Fig. 5 for Re=40000. The same + Pr=40 T trend in Fig. 5 is for other values of Re (10000 and Pr=20 10000). The figure shows the effect of Pr on Pr=7 0.4 temperature profile. In a turbulent boundary layer, the gradient is very steep near the wall and weaker farther from the wall where the eddies are larger and turbulent 0.2 mixing is more efficient [Lienhard 2001]. As the Pr increases the temperature profile close to the wall becomes more flat (the slope increases) indicating that 0.0 the dimensionless thickness of the of thermal sublayer 0 5 10 15 20 ( +T) decreases. Fig. 6 reveals that increasing Re y+ leads to slight increase in +T. Using statistical Figure (5) Temperature Profile Obtained from the methods the following relation is obtained for +T: Models of Eddy Diffusivity and Prt of Present Analysis at Various Pr and Re=40000. + T=8.635Re-0.067Pr-0.245; C.C=0.99 32 1.0 or 0.8 2 T/d=8.635 Re-1.067Pr-0.245 33 f 0.6 Pr=7 Re=10000 + Re=40000 Hence the thickness of thermal sublayer decreases with T Re and Pr. Levich [1962] proposed the following Re=100000 relation for the ratio of viscous sublayer to thermal 0.4 sublayer: 0.2 b 1/3 Pr 34 T 0.0 0 5 10 15 20 Hence the thickness of viscous sublayer is y+ Figure (6) Temperature Profile Obtained from the Models of Eddy Diffusivity and Prt of Present 2 b/d =8.635 Re-1.067Pr0.088 35 Analysis at Various Re and Pr=7. f Predicting Heat Transfer Coefficient The expressions of momentum eddy diffusivity and turbulent Prandtl number developed in, Eqs. (21) and Eq. (29) respectively are substituted in Eq.(20) to obtain Nu. Figs. 7 to 9 show a comparison between Nu predicted from present analysis with other experimental works. It is evident that Nu predicted NUCEJ Vol.10, NO.1 Turbulent Prandtl Number 60 from present analysis is in good agreement with other 5 4 experimental works for the whole range of Re and Pr. Pr=600 3 Present Analysis 7 Colburn 6 Petkhov Pr=7 2 Notter and Sleicher 5 Present Analysis 4 Allen and Eckert Slaiman and Hasan 3 Petkhov u 1000 N 9 8 7 u 2 N 6 5 4 3 100.00 9 8 2 7 8 9 2 3 4 5 6 7 8 9 10000 100000 6 Re 8 9 2 3 4 5 6 7 8 9 10000 100000 Re Figure (9) Comparison between Nu Obtained from Present Analysis with Other Works at Pr=600.. Figure (7) Comparison between Nu Obtained from Present Analysis with Other Works at Pr=7. CONCLUSIONS 3 The following points are concluded for the investigated range of Re and Pr: 1- Turbulent Prandtl number plays an important role in 2 Pr=100 present analysis determining the value of heat transfer coefficient and Colburn the assumption of unity turbulent Prandtl number is Petkhov very poor for fluids of Pr>1, i.e., the momentum eddy 1000 Notter and Sleicher 9 diffusivity is not equal to thermal eddy diffusivity. 8 7 2- The Turbulent Prandtl number depends on both 6 Reynolds and molecular Prandtl numbers and its value u N 5 is generally less than 0.7. 4 3- The models of turbulent Prandtl number and 3 momentum eddy diffusivity developed in present analysis give more accurate results in predicting the 2 heat transfer coefficient than previously proposed models. 4- The thickness of thermal sublayer is function of Re and Pr while the thickness of viscous sublayer is 100 8 9 2 3 4 5 6 7 8 9 strongly affected by Re and slightly by Pr. 10000 100000 Re Figure (8) Comparison between Nu Obtained from Present Analysis with Other Works at Pr=100. 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