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A new method to predict convective heat transfer in a tube with twisted tape inserts for turbulent flow
International Journal of Thermal Sciences 41 (2002) 955–960 www.elsevier.com/locate/ijts
A new method to predict convective heat transfer in a tube with twisted tape inserts for turbulent flow P.K. Sarma a,∗ , T. Subramanyam a , P.S. Kishore a,1 , V. Dharma Rao a , Sadik Kakac b a Department of Mechanical Engineering, College of Engineering, Andhra University, Visakhapatnam 530003, India b University of Miami, Coral Gables, FL 33124, USA
Received 13 June 2001; accepted 7 November 2001
Abstract The article presents a new approach in predicting the convective heat transfer coefficient in a tube with twisted tape inserts of different pitch to diameter ratios. A modification is proposed to the classical van Driest eddy diffusivity expression to respond to the case of swirl flow generated by the tape inserts. The universal constant K in the relationship is found to be a function of pitch to diameter ratio of the tape and Reynolds number of the flow in the tube. The proposed analysis is compared with various correlations available in the literature showing reasonable agreement between them. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: Convective turbulent swirl flow; Twisted tape insertions; Universal constant
1. Introduction Convective heat transfer in a tube with tape inserts is dynamically different from the one without tapes. It is well established [1–12] that the swirl created in the medium facilitates efficient transportation of heat from the tube wall both for laminar and turbulent regimes. Several correlations have been suggested to include the effects of increased turbulence due to spiraling flow, increased circulation created with heating due to large centrifugal force and the tape fin effects. Studies of Smithberg and Landis [1] include some of the turbulence boundary layer characteristics in the estimation of the influence of spiraling effects and associated tape fin effects. Lopina and Bergles [2] have developed a correlation by considering the contributions of spiral convection and centrifugal convection. The principle of superposition is applied in the estimation of mean heat transfer coefficient. However, their equation includes the fin effects through a factor of multiplication to the equation. It is assigned a value depending on whether the tape is loosely or tightly fitted inside the tube. Analysis of Thorsen and Landis [3] is an extension of the investigation of Smithberg and Landis [1]. The * Correspondence and reprints.
E-mail addresses: [email protected] (P.K. Sarma), [email protected] (T. Subramanyam), [email protected] (V.D. Rao), [email protected] (S. Kakac). 1 Research scholar.
effect of radial temperature gradients is considered including the property variations and buoyancy effects for heating and cooling of the gases. Manglik and Bergles [4,5] presented a comprehensive treatment of swirl flow in two parts for laminar and turbulent regimes. Mostly the treatment of the data is accomplished by regression analysis with the aid of dimensionless parameters. The effect of the tape thickness is also considered in the correlations. The proposed equations relate to the momentum and heat transfer for a wide range of Reynolds numbers. The existing equations for prediction of friction coefficient vary over a wide range. Bergles [6] gave a comprehensive review and survey related to techniques to augment convective heat and mass transfer. Various techniques to augment heat transfer are listed in the handbook of heat transfer [6]. In contrast to the existing approaches in the literature the present article is an attempt to formulate and investigate the problem with tape inserts as an extension of fully developed turbulent convective heat transfer in tubes without tape and is treated as a special case of it. The concepts are verified with some data available in the literature.
2. Formulation Turbulent convective heat transfer in tubes is a wellstudied phenomenon and the interrelationship between momentum and heat transfer phenomena is well established.
1290-0729/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 1 2 9 0 - 0 7 2 9 ( 0 2 ) 0 1 3 8 8 - 1
956
P.K. Sarma et al. / International Journal of Thermal Sciences 41 (2002) 955–960
Nomenclature A A+ Cp D DH f H h K k m Nu Pr R R+ Re T T+ t u
cross sectional area of the tube . . . . . . . . . . . . m2 constant in van Driest expression specific heat at constant pressure . . J·kg−1 ·K−1 inside diameter of the tube . . . . . . . . . . . . . . . . m hydraulic diameter of the tube . . . . . . . . . . . . . m friction coefficient helical pitch of the twisted tape for 360◦ twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . m heat transfer coefficient . . . . . . . . . . W·m−2 ·K−1 universal constant in van Driest expression thermal conductivity . . . . . . . . . . . . W·m−1 ·K−1 flow rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . kg·s−1 Nusselt number = hD/k Prandtl number = µCp /k inside radius of the tube . . . . . . . . . . . . . . . . . . . m dimensionless radius Reynolds number = 4m/πDµl temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ◦ C dimensionless temperature thickness of the tape . . . . . . . . . . . . . . . . . . . . . . m velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m·s−1
Some of the momentum transfer relationships proved to be of immense utility in the study of turbulent convective heat transfer. Thus, it is proposed to treat swirl flow generated by tape inserts as a special case of fully developed turbulent flow. The following assumptions are made in the formulation of the problem. (1) The swirl flow is fully developed both from hydrodynamic and thermal point of view. The property variations of the medium due to temperature gradients are ignored. The fin effects in the analysis are ignored. (2) The wall is under isothermal conditions and the eddy diffusivity expression of van Driest [7] holds good for swirl flow as well. However, the universal constant K would be dependent on the degree of swirl generated by tape. The constants in the relationship can be functions of Reynolds number, Re and tape twist ratio, H /D. Extension of momentum transfer studies with tape inserts in the tube would pave the way to establish convective heat transfer. The various effects of centrifugal forces and other secondary flows on heat transfer can be indirectly assessed and reflected in the magnitude of K of van Driest expression [7]. + + 2 ∂u+ εm 2 +2 = K y 1 − Exp −y /A ∂y + ν
(1)
where K = 0.4 and A+ = 26 for fully developed turbulent flow in tubes.
u∗ um u+ y y+
shear velocity . . . . . . . . . . . . . . . . . . . . . . . . m·s−1 mean velocity . . . . . . . . . . . . . . . . . . . . . . . . m·s−1 dimensionless velocity = u/u∗ distance measured normal to the tube wall . . m dimensionless distance = yu∗ /ν
Greek symbols ρ µ τ ν εm εH η
density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kg·m−3 absolute viscosity . . . . . . . . . . . . . . . kg·m−1 ·s−1 shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . N·m−2 kinematic viscosity . . . . . . . . . . . . . . . . . . m2 ·s−1 turbulent eddy viscosity . . . . . . . . . . . . . . m2 ·s−1 thermal eddy diffusivity . . . . . . . . . . . . . . m2 ·s−1 dimensionless variable
Subscripts B C H l tape W
mean bulk centerline hydraulic liquid tape wall
(3) Neglecting the physical presence of the tape in the tube for further analysis, the shear distribution in the tube is assumed to vary linearly across the tube y+ τ (2) = 1− + τW R where y+ =
yu∗ , ν
R+ =
Ru∗ ν
and τW = 0.5ftape ρu2m ,
um =
m Aρ
The friction coefficient ftape , relationship of Smithberg and Landis [1] is used in the present study. −1.2 H − 0.5 (3) (Re φ)−n ftape = 0.046 + 2.1 D where
−0.5 H n = 0.2 1 + 1.7 and D 1 φ= 1 + 2/π
Re =
4m , πDµl
φ is the tape correction factor due to its presence in the tube. Many equations for ftape are available in the literature and it is observed that these equations are different from one another. Hence, the choice is purely based on its accuracy to predict value close to the experimental heat transfer observations.
P.K. Sarma et al. / International Journal of Thermal Sciences 41 (2002) 955–960
Consistent with the assumptions for the fully developed flow the velocity distribution can be obtained from 1 − y + /R + du+ = (4) dy + 1 + εm /ν Combining Eq. (4) with Eq. (1) it can be shown that du+ dy + =
from the following simplified form of the energy conservation equation: +
dT εm d Pr 1 + =0 (10) + dy ν dy + where T+=
−1 +
+ 1 + 4K 2 y +2 [1 − y + /R + ][1 − exp(− Ay + )]2 +
2K 2 y +2 [1 − exp(− Ay + )]2
The following equation for R + can be deduced making use of the expression for τW appearing in Eq. (2)
Dum ftape + (6) R = 2ν 2 where R m um = (7) and m = ρu2π(R − y) dy Aρ 0
The expression of Smithberg and Landis [1], i.e., Eq. (3) for ftape is used to calculate R + . Eq. (7) in dimensionless form can be written as R + u+ 1 − y + /R + dy +
(8)
and y + =
yu∗ ν
y + = 0, y + = R+ ,
T+ =0 T+ =1
(11)
Eq. (10) assumes that turbulent conduction across the cross section of the tube for swirl flow is more dominant than the convective contributions. By definition of the heat transfer coefficient dT k h=− (12) (TW − TB ) dy y=0 Eq. (12) in dimensionless form TW − TC dT + Nu = 2R + + + dy y =0 TW − TB
(13)
The temperature ratio term can be evaluated once the profiles are determined. It can be shown that R+ + + u T [1 − y + /R + ] dy + TW − TB = 0 R + (14) TW − TC u+ [1 − y + /R + ] dy + 0
0
Thus, Eqs. (1)–(8) would enable us to fix the value of K by straight forward iteration procedure for the input values of Re and H /D. K = K[H /D, Re]
TW − T TW − TC
The boundary conditions are
(5)
4m =4 Re = πDµl
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Thus, Eqs. (5), (8), and (10) are solved together to obtain Nusselt number from Eq. (13). The results are further discussed.
(9)
The calculation steps are as follows: • Prescribe Re and H /D. • Calculate ftape using Eq. (3). • Calculate R + . Obtain u+ = u+ (y + ) for 0 < y + < R + by numerically integrating Eq. (5). • Iterate on K till R.H.S = L.H.S of Eq. (8). Hence, fix the value of εm /ν to be subsequently used in evaluating heat transfer coefficients. Thus, Eq. (9) would serve as a priori in the estimation of convective heat transfer with tapes. Thus, a modification of the eddy diffusivity expression, Eq. (1) would serve as the essential tool in the estimation of convective heat transfer.
3. Heat transfer coefficients In convective heat transfer studies εH is taken as εm for Pr > 1. Hence based on this assumption, neglecting convective effects for thermally developed conditions the temperature profile across the cross section can be determined
4. Validation and discussion As per the model the dependence of the universal constant K in Eq. (1) is determined following the iteration procedure already indicated for different values of Re and H /D. However, the value of A+ is taken as 26. It can be seen from the results of Fig. 1 that for small values H /D, as Re increases the value of K monotonically decreases. On the other hand, for H /D > 15, the value of K is more or less constant and tends to an asymptotic value 0.4. Employing regression, the following expression is proposed in the evaluation of K for the range of parameters 10 000 < Re < 100 000 and 4 < H /D < 15. K = 3.3891 Re−0.0914(H /D)−0.3737
(15)
The regression equation could be obtained with a standard deviation of 3%. Hence, in the next step of the analysis it is imperative to check the usefulness of Eq. (15) in predicting convective heat transfer coefficients with tape inserts. Though the tape thickness effects are not to be found explicitly in Eq. (15) it is to be understood that the estimates of K reflect the experimental conditions prevailing
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P.K. Sarma et al. / International Journal of Thermal Sciences 41 (2002) 955–960
Table 1 List of various empirical correlations from literature for Nusselt number Reference Smithberg and Landis [1]
Equation
Remarks, nomenclature Re = DνH u¯
Nu = Re Pr A1 /B1 700 D DH Pr0.731 B1 = 1 + Re f H D √ ) + 0.023 D Re−0.2 Pr−2/3 φ1 A1 = 50.9(D/H D Re f 1/2 H φ1 = 1 + 0.02192 (H /D) f
Lopina and Bergles [2]
NuH
= 0.023[α Re]0.8 Pr0.4 +F
α = [1 + (π D/H )2 ]0.5
1
excluded in Figs. 2–4 Manglik and Bergles [5]
Nu = 0.023 Re0.8 Pr0.4 1 + 0.769 2D H φ2
Sarma et al. [8]
D 2.065 Re0.67 Pr1/3 Nu = 0.1012 1 + H
in the investigations of Smithberg and Landis [1] in the acquisition of the data. To include the tape thickness effects, it is essential that more experimental data must be available in the literature. Hence, tentatively to test the correctness of the universal constant K obtained from the friction data, the relationship is utilized further in predicting turbulent convective heat transfer dissipation from the tube wall. With the value of the universal constant K taken from Eq. (15) and making use of the eddy diffusivity expression, i.e., Eq. (1) both velocity and temperature profiles are obtained in the evaluation of asymptotic value of the mean heat transfer coefficient from Eq. (13). The present
u¯ = m/(ρA) A = free flow cross section
Note: The centrifugal convection F1 is
Fig. 1. Variation of K in the eddy diffusivity expression of van Driest with Reynolds number for 4 < H /D < 22.
2 /4)−tD) DH = 4((ππDD+2D Nu = hD k
Re = DνH u¯ NuH = hDk H π 0.8 π +2−2t/D 0.2 φ2 = π −4t/D π −4t/D t = thickness of the tape insert D = diameter of the tube H = pitch of the helical tape Nu = hD/k Re = 4m/π Dµl
Fig. 2. Comparison of present theory with various authors for H /D = 6.
theoretical analysis will be compared with some of the correlations available in the literature. Generally referred correlations are listed (in Table 1) for comparison purposes. Manglik and Bergles [5] conducted investigations for the range H /D = 6, 9, 12 and their correlation is observed to agree with the data within ±10%. In Figs. 2, 3 and 4 the predictions from the present equations are shown plotted with correlations of various investigators. Evidently, the predictions from the present theory are in satisfactory agreement with correlations of various investigators. Thus, the subsequent effort is to go into the dynamics of the flow as per the model.
P.K. Sarma et al. / International Journal of Thermal Sciences 41 (2002) 955–960
Fig. 3. Comparison of present theory with various authors for H /D = 9.
Fig. 5. Velocity profiles for different H /D ratios.
Fig. 4. Comparison of present theory with various authors for H /D = 15.
Fig. 6. Temperature profiles for different H /D ratios.
5. Velocity and temperature profiles The velocity variation is plotted in Fig. 5. It shows a profound influence of H /D on both the magnitude and nature of the profiles. Decrease in H /D leads to higher shear resistance at the wall and maximum dissipation of energy leading to decrease in local velocities in the flow. Further, it can be seen that the profile becomes more or less uniform in the core. The relative decrease in the magnitude of velocity with H /D decreasing is due to increase in the flow path of the particle and increase in turbulence over the cross section. The temperature profiles in Fig. 6 indicate the effect
959
of H /D on wall temperature gradients. Evidently, as H /D decreases the temperature gradients become steeper in the wall region leading to increase in heat dissipation from the wall. Thus as H /D decreases the heat transfer coefficient increases. The variation of turbulent eddy viscosity can be seen from Fig. 7. As can be seen from this figure, the magnitude of eddy viscosity in the core of the tube increases with decrease in the value of H /D. These trends support enhancement in shear resistance and wall heat flux. In view of satisfactory agreement of the present theory with the studies of other investigators, the theoretical results are subjected to regression to obtain a correlation equation to
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P.K. Sarma et al. / International Journal of Thermal Sciences 41 (2002) 955–960
heat transfer coefficient since it reasonably agrees with the generally referred equations in the literature and also with our data as shown in the figures. (4) The analysis in a way is similar to Colburn’s analogy since it makes use of momentum transfer characteristics in the estimation of heat transfer from the tube wall under diabatic conditions with swirl induced by tape inserts. In the present study εm is modified to include swirl effects through the universal constant K as a function of Reynolds number and H /D.
Acknowledgements The authors thank the University Grants Commission, New Delhi for the assistance. References Fig. 7. Variation of eddy diffusivity with H /D.
predict heat transfer coefficient with twisted tape inserts for water. 0.87 0.4 Nu = 0.023 Re0.8 1 + 2π/(H /D) Pr (16) Eq. (16) predicts the values with an accuracy of 3% of average deviation and a standard deviation of 4%.
6. Conclusions The following conclusions can be drawn from the analysis. (1) The new model proposes that the enhancement in heat transfer is due to increase in the turbulent eddy diffusivity in the flow due to twisted tape insert. Accordingly modification in the eddy diffusivity expression of van Driest [7] has yielded a satisfactory approach in the estimation of asymptotic values for the fully developed thermal and hydrodynamic conditions of swirl flow induced by tape. (2) Eq. (1) with the value of the modified universal constant K computed from Eq. (15) will enable us to estimate theoretically the heat transfer coefficient for the ranges of 4 < H /D < 15 and Pr = 5. It may be construed that the model is an extension of the convective heat transfer for turbulent flows with twisted tape inserts in a tube. (3) Correlation Eq. (16) can be employed to estimate the
[1] E. Smithberg, F. Landis, Friction and forced convection heat transfer characteristics in tubes with twisted-tape swirl generators, Trans. ASME J. Heat Transfer 86 (1964) 39–49. [2] R.F. Lopina, A.E. Bergles, Heat transfer and pressure drop in tapegenerated swirl flow of single phase water, Trans. ASME J. Heat Transfer 91 (1969) 434–442. [3] R. Thorsen, F. Landis, Friction and heat transfer characteristics in turbulent swirl flow subjected to large transverse temperature gradients, Trans. ASME J. Heat Transfer 90 (1968) 87–98. [4] R.M. Manglik, A.E. Bergles, Heat transfer and pressure drop correlations for twisted tape inserts in isothermal tubes: Part I—Laminar flows, Trans. ASME J. Heat Transfer 115 (1993) 881–889. [5] R.M. Manglik, A.E. Bergles, Heat transfer and pressure drop correlations for twisted-tape inserts in isothermal tubes: Part II—Transition and turbulent flows, Trans. ASME J. Heat Transfer 115 (1993) 890– 896. [6] A.E. Bergles, Techniques to augment heat transfer, in: W.M. Rohsnow, et al. (Eds.), Hand Book of Heat Transfer Applications, 2nd Edition, McGraw-Hill, New York, 1985. [7] E.R.G. Eckert, R.M. Drake Jr., Analysis of Heat and Mass Transfer, McGraw-Hill, Kogakusha, Japan, 1972. [8] P.K. Sarma, V.D. Rao, T. Subramanyam, Convective heat transfer and pressure drop for high Prandtl number fluids with twistedtape turbulence promoters, Final Report [Ref: BHEL/ PSVK: 96: 0210,19,1996], Hyderabad, India, 1998. [9] A.W. Date, Prediction of fully developed flow in a tube containing a twisted tape, Internat. J. Heat Mass Transfer 17 (1974) 845–859. [10] B. Donevski, M. Plocek, J. Kulesza, M. Sasic, Analysis of tube side laminar and turbulent flow heat transfer with twisted tape inserts, in: S.J. Deng, et al. (Eds.), Heat Transfer Enhancement and Energy Conservation, Hemisphere, New York, 1990, pp. 175–185. [11] S.W. Hong, A.E. Bergles, Augmentation of laminar heat transfer in tubes by means of twisted tape inserts, Trans. ASME J. Heat Transfer 98 (1976) 251–256. [12] W.J. Marner, A.E. Bergles, Augmentation of highly viscous laminar heat transfer inside tubes with constant wall temperature, J. Experimental Thermal Fluid Sci. 2 (1989) 252–267.
A new method to predict convective heat transfer in a tube with twisted tape inserts for turbulent flow P.K. Sarma a,∗ , T. Subramanyam a , P.S. Kishore a,1 , V. Dharma Rao a , Sadik Kakac b a Department of Mechanical Engineering, College of Engineering, Andhra University, Visakhapatnam 530003, India b University of Miami, Coral Gables, FL 33124, USA
Received 13 June 2001; accepted 7 November 2001
Abstract The article presents a new approach in predicting the convective heat transfer coefficient in a tube with twisted tape inserts of different pitch to diameter ratios. A modification is proposed to the classical van Driest eddy diffusivity expression to respond to the case of swirl flow generated by the tape inserts. The universal constant K in the relationship is found to be a function of pitch to diameter ratio of the tape and Reynolds number of the flow in the tube. The proposed analysis is compared with various correlations available in the literature showing reasonable agreement between them. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: Convective turbulent swirl flow; Twisted tape insertions; Universal constant
1. Introduction Convective heat transfer in a tube with tape inserts is dynamically different from the one without tapes. It is well established [1–12] that the swirl created in the medium facilitates efficient transportation of heat from the tube wall both for laminar and turbulent regimes. Several correlations have been suggested to include the effects of increased turbulence due to spiraling flow, increased circulation created with heating due to large centrifugal force and the tape fin effects. Studies of Smithberg and Landis [1] include some of the turbulence boundary layer characteristics in the estimation of the influence of spiraling effects and associated tape fin effects. Lopina and Bergles [2] have developed a correlation by considering the contributions of spiral convection and centrifugal convection. The principle of superposition is applied in the estimation of mean heat transfer coefficient. However, their equation includes the fin effects through a factor of multiplication to the equation. It is assigned a value depending on whether the tape is loosely or tightly fitted inside the tube. Analysis of Thorsen and Landis [3] is an extension of the investigation of Smithberg and Landis [1]. The * Correspondence and reprints.
E-mail addresses: [email protected] (P.K. Sarma), [email protected] (T. Subramanyam), [email protected] (V.D. Rao), [email protected] (S. Kakac). 1 Research scholar.
effect of radial temperature gradients is considered including the property variations and buoyancy effects for heating and cooling of the gases. Manglik and Bergles [4,5] presented a comprehensive treatment of swirl flow in two parts for laminar and turbulent regimes. Mostly the treatment of the data is accomplished by regression analysis with the aid of dimensionless parameters. The effect of the tape thickness is also considered in the correlations. The proposed equations relate to the momentum and heat transfer for a wide range of Reynolds numbers. The existing equations for prediction of friction coefficient vary over a wide range. Bergles [6] gave a comprehensive review and survey related to techniques to augment convective heat and mass transfer. Various techniques to augment heat transfer are listed in the handbook of heat transfer [6]. In contrast to the existing approaches in the literature the present article is an attempt to formulate and investigate the problem with tape inserts as an extension of fully developed turbulent convective heat transfer in tubes without tape and is treated as a special case of it. The concepts are verified with some data available in the literature.
2. Formulation Turbulent convective heat transfer in tubes is a wellstudied phenomenon and the interrelationship between momentum and heat transfer phenomena is well established.
1290-0729/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 1 2 9 0 - 0 7 2 9 ( 0 2 ) 0 1 3 8 8 - 1
956
P.K. Sarma et al. / International Journal of Thermal Sciences 41 (2002) 955–960
Nomenclature A A+ Cp D DH f H h K k m Nu Pr R R+ Re T T+ t u
cross sectional area of the tube . . . . . . . . . . . . m2 constant in van Driest expression specific heat at constant pressure . . J·kg−1 ·K−1 inside diameter of the tube . . . . . . . . . . . . . . . . m hydraulic diameter of the tube . . . . . . . . . . . . . m friction coefficient helical pitch of the twisted tape for 360◦ twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . m heat transfer coefficient . . . . . . . . . . W·m−2 ·K−1 universal constant in van Driest expression thermal conductivity . . . . . . . . . . . . W·m−1 ·K−1 flow rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . kg·s−1 Nusselt number = hD/k Prandtl number = µCp /k inside radius of the tube . . . . . . . . . . . . . . . . . . . m dimensionless radius Reynolds number = 4m/πDµl temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ◦ C dimensionless temperature thickness of the tape . . . . . . . . . . . . . . . . . . . . . . m velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m·s−1
Some of the momentum transfer relationships proved to be of immense utility in the study of turbulent convective heat transfer. Thus, it is proposed to treat swirl flow generated by tape inserts as a special case of fully developed turbulent flow. The following assumptions are made in the formulation of the problem. (1) The swirl flow is fully developed both from hydrodynamic and thermal point of view. The property variations of the medium due to temperature gradients are ignored. The fin effects in the analysis are ignored. (2) The wall is under isothermal conditions and the eddy diffusivity expression of van Driest [7] holds good for swirl flow as well. However, the universal constant K would be dependent on the degree of swirl generated by tape. The constants in the relationship can be functions of Reynolds number, Re and tape twist ratio, H /D. Extension of momentum transfer studies with tape inserts in the tube would pave the way to establish convective heat transfer. The various effects of centrifugal forces and other secondary flows on heat transfer can be indirectly assessed and reflected in the magnitude of K of van Driest expression [7]. + + 2 ∂u+ εm 2 +2 = K y 1 − Exp −y /A ∂y + ν
(1)
where K = 0.4 and A+ = 26 for fully developed turbulent flow in tubes.
u∗ um u+ y y+
shear velocity . . . . . . . . . . . . . . . . . . . . . . . . m·s−1 mean velocity . . . . . . . . . . . . . . . . . . . . . . . . m·s−1 dimensionless velocity = u/u∗ distance measured normal to the tube wall . . m dimensionless distance = yu∗ /ν
Greek symbols ρ µ τ ν εm εH η
density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kg·m−3 absolute viscosity . . . . . . . . . . . . . . . kg·m−1 ·s−1 shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . N·m−2 kinematic viscosity . . . . . . . . . . . . . . . . . . m2 ·s−1 turbulent eddy viscosity . . . . . . . . . . . . . . m2 ·s−1 thermal eddy diffusivity . . . . . . . . . . . . . . m2 ·s−1 dimensionless variable
Subscripts B C H l tape W
mean bulk centerline hydraulic liquid tape wall
(3) Neglecting the physical presence of the tape in the tube for further analysis, the shear distribution in the tube is assumed to vary linearly across the tube y+ τ (2) = 1− + τW R where y+ =
yu∗ , ν
R+ =
Ru∗ ν
and τW = 0.5ftape ρu2m ,
um =
m Aρ
The friction coefficient ftape , relationship of Smithberg and Landis [1] is used in the present study. −1.2 H − 0.5 (3) (Re φ)−n ftape = 0.046 + 2.1 D where
−0.5 H n = 0.2 1 + 1.7 and D 1 φ= 1 + 2/π
Re =
4m , πDµl
φ is the tape correction factor due to its presence in the tube. Many equations for ftape are available in the literature and it is observed that these equations are different from one another. Hence, the choice is purely based on its accuracy to predict value close to the experimental heat transfer observations.
P.K. Sarma et al. / International Journal of Thermal Sciences 41 (2002) 955–960
Consistent with the assumptions for the fully developed flow the velocity distribution can be obtained from 1 − y + /R + du+ = (4) dy + 1 + εm /ν Combining Eq. (4) with Eq. (1) it can be shown that du+ dy + =
from the following simplified form of the energy conservation equation: +
dT εm d Pr 1 + =0 (10) + dy ν dy + where T+=
−1 +
+ 1 + 4K 2 y +2 [1 − y + /R + ][1 − exp(− Ay + )]2 +
2K 2 y +2 [1 − exp(− Ay + )]2
The following equation for R + can be deduced making use of the expression for τW appearing in Eq. (2)
Dum ftape + (6) R = 2ν 2 where R m um = (7) and m = ρu2π(R − y) dy Aρ 0
The expression of Smithberg and Landis [1], i.e., Eq. (3) for ftape is used to calculate R + . Eq. (7) in dimensionless form can be written as R + u+ 1 − y + /R + dy +
(8)
and y + =
yu∗ ν
y + = 0, y + = R+ ,
T+ =0 T+ =1
(11)
Eq. (10) assumes that turbulent conduction across the cross section of the tube for swirl flow is more dominant than the convective contributions. By definition of the heat transfer coefficient dT k h=− (12) (TW − TB ) dy y=0 Eq. (12) in dimensionless form TW − TC dT + Nu = 2R + + + dy y =0 TW − TB
(13)
The temperature ratio term can be evaluated once the profiles are determined. It can be shown that R+ + + u T [1 − y + /R + ] dy + TW − TB = 0 R + (14) TW − TC u+ [1 − y + /R + ] dy + 0
0
Thus, Eqs. (1)–(8) would enable us to fix the value of K by straight forward iteration procedure for the input values of Re and H /D. K = K[H /D, Re]
TW − T TW − TC
The boundary conditions are
(5)
4m =4 Re = πDµl
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Thus, Eqs. (5), (8), and (10) are solved together to obtain Nusselt number from Eq. (13). The results are further discussed.
(9)
The calculation steps are as follows: • Prescribe Re and H /D. • Calculate ftape using Eq. (3). • Calculate R + . Obtain u+ = u+ (y + ) for 0 < y + < R + by numerically integrating Eq. (5). • Iterate on K till R.H.S = L.H.S of Eq. (8). Hence, fix the value of εm /ν to be subsequently used in evaluating heat transfer coefficients. Thus, Eq. (9) would serve as a priori in the estimation of convective heat transfer with tapes. Thus, a modification of the eddy diffusivity expression, Eq. (1) would serve as the essential tool in the estimation of convective heat transfer.
3. Heat transfer coefficients In convective heat transfer studies εH is taken as εm for Pr > 1. Hence based on this assumption, neglecting convective effects for thermally developed conditions the temperature profile across the cross section can be determined
4. Validation and discussion As per the model the dependence of the universal constant K in Eq. (1) is determined following the iteration procedure already indicated for different values of Re and H /D. However, the value of A+ is taken as 26. It can be seen from the results of Fig. 1 that for small values H /D, as Re increases the value of K monotonically decreases. On the other hand, for H /D > 15, the value of K is more or less constant and tends to an asymptotic value 0.4. Employing regression, the following expression is proposed in the evaluation of K for the range of parameters 10 000 < Re < 100 000 and 4 < H /D < 15. K = 3.3891 Re−0.0914(H /D)−0.3737
(15)
The regression equation could be obtained with a standard deviation of 3%. Hence, in the next step of the analysis it is imperative to check the usefulness of Eq. (15) in predicting convective heat transfer coefficients with tape inserts. Though the tape thickness effects are not to be found explicitly in Eq. (15) it is to be understood that the estimates of K reflect the experimental conditions prevailing
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Table 1 List of various empirical correlations from literature for Nusselt number Reference Smithberg and Landis [1]
Equation
Remarks, nomenclature Re = DνH u¯
Nu = Re Pr A1 /B1 700 D DH Pr0.731 B1 = 1 + Re f H D √ ) + 0.023 D Re−0.2 Pr−2/3 φ1 A1 = 50.9(D/H D Re f 1/2 H φ1 = 1 + 0.02192 (H /D) f
Lopina and Bergles [2]
NuH
= 0.023[α Re]0.8 Pr0.4 +F
α = [1 + (π D/H )2 ]0.5
1
excluded in Figs. 2–4 Manglik and Bergles [5]
Nu = 0.023 Re0.8 Pr0.4 1 + 0.769 2D H φ2
Sarma et al. [8]
D 2.065 Re0.67 Pr1/3 Nu = 0.1012 1 + H
in the investigations of Smithberg and Landis [1] in the acquisition of the data. To include the tape thickness effects, it is essential that more experimental data must be available in the literature. Hence, tentatively to test the correctness of the universal constant K obtained from the friction data, the relationship is utilized further in predicting turbulent convective heat transfer dissipation from the tube wall. With the value of the universal constant K taken from Eq. (15) and making use of the eddy diffusivity expression, i.e., Eq. (1) both velocity and temperature profiles are obtained in the evaluation of asymptotic value of the mean heat transfer coefficient from Eq. (13). The present
u¯ = m/(ρA) A = free flow cross section
Note: The centrifugal convection F1 is
Fig. 1. Variation of K in the eddy diffusivity expression of van Driest with Reynolds number for 4 < H /D < 22.
2 /4)−tD) DH = 4((ππDD+2D Nu = hD k
Re = DνH u¯ NuH = hDk H π 0.8 π +2−2t/D 0.2 φ2 = π −4t/D π −4t/D t = thickness of the tape insert D = diameter of the tube H = pitch of the helical tape Nu = hD/k Re = 4m/π Dµl
Fig. 2. Comparison of present theory with various authors for H /D = 6.
theoretical analysis will be compared with some of the correlations available in the literature. Generally referred correlations are listed (in Table 1) for comparison purposes. Manglik and Bergles [5] conducted investigations for the range H /D = 6, 9, 12 and their correlation is observed to agree with the data within ±10%. In Figs. 2, 3 and 4 the predictions from the present equations are shown plotted with correlations of various investigators. Evidently, the predictions from the present theory are in satisfactory agreement with correlations of various investigators. Thus, the subsequent effort is to go into the dynamics of the flow as per the model.
P.K. Sarma et al. / International Journal of Thermal Sciences 41 (2002) 955–960
Fig. 3. Comparison of present theory with various authors for H /D = 9.
Fig. 5. Velocity profiles for different H /D ratios.
Fig. 4. Comparison of present theory with various authors for H /D = 15.
Fig. 6. Temperature profiles for different H /D ratios.
5. Velocity and temperature profiles The velocity variation is plotted in Fig. 5. It shows a profound influence of H /D on both the magnitude and nature of the profiles. Decrease in H /D leads to higher shear resistance at the wall and maximum dissipation of energy leading to decrease in local velocities in the flow. Further, it can be seen that the profile becomes more or less uniform in the core. The relative decrease in the magnitude of velocity with H /D decreasing is due to increase in the flow path of the particle and increase in turbulence over the cross section. The temperature profiles in Fig. 6 indicate the effect
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of H /D on wall temperature gradients. Evidently, as H /D decreases the temperature gradients become steeper in the wall region leading to increase in heat dissipation from the wall. Thus as H /D decreases the heat transfer coefficient increases. The variation of turbulent eddy viscosity can be seen from Fig. 7. As can be seen from this figure, the magnitude of eddy viscosity in the core of the tube increases with decrease in the value of H /D. These trends support enhancement in shear resistance and wall heat flux. In view of satisfactory agreement of the present theory with the studies of other investigators, the theoretical results are subjected to regression to obtain a correlation equation to
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heat transfer coefficient since it reasonably agrees with the generally referred equations in the literature and also with our data as shown in the figures. (4) The analysis in a way is similar to Colburn’s analogy since it makes use of momentum transfer characteristics in the estimation of heat transfer from the tube wall under diabatic conditions with swirl induced by tape inserts. In the present study εm is modified to include swirl effects through the universal constant K as a function of Reynolds number and H /D.
Acknowledgements The authors thank the University Grants Commission, New Delhi for the assistance. References Fig. 7. Variation of eddy diffusivity with H /D.
predict heat transfer coefficient with twisted tape inserts for water. 0.87 0.4 Nu = 0.023 Re0.8 1 + 2π/(H /D) Pr (16) Eq. (16) predicts the values with an accuracy of 3% of average deviation and a standard deviation of 4%.
6. Conclusions The following conclusions can be drawn from the analysis. (1) The new model proposes that the enhancement in heat transfer is due to increase in the turbulent eddy diffusivity in the flow due to twisted tape insert. Accordingly modification in the eddy diffusivity expression of van Driest [7] has yielded a satisfactory approach in the estimation of asymptotic values for the fully developed thermal and hydrodynamic conditions of swirl flow induced by tape. (2) Eq. (1) with the value of the modified universal constant K computed from Eq. (15) will enable us to estimate theoretically the heat transfer coefficient for the ranges of 4 < H /D < 15 and Pr = 5. It may be construed that the model is an extension of the convective heat transfer for turbulent flows with twisted tape inserts in a tube. (3) Correlation Eq. (16) can be employed to estimate the
[1] E. Smithberg, F. Landis, Friction and forced convection heat transfer characteristics in tubes with twisted-tape swirl generators, Trans. ASME J. Heat Transfer 86 (1964) 39–49. [2] R.F. Lopina, A.E. Bergles, Heat transfer and pressure drop in tapegenerated swirl flow of single phase water, Trans. ASME J. Heat Transfer 91 (1969) 434–442. [3] R. Thorsen, F. Landis, Friction and heat transfer characteristics in turbulent swirl flow subjected to large transverse temperature gradients, Trans. ASME J. Heat Transfer 90 (1968) 87–98. [4] R.M. Manglik, A.E. Bergles, Heat transfer and pressure drop correlations for twisted tape inserts in isothermal tubes: Part I—Laminar flows, Trans. ASME J. Heat Transfer 115 (1993) 881–889. [5] R.M. Manglik, A.E. Bergles, Heat transfer and pressure drop correlations for twisted-tape inserts in isothermal tubes: Part II—Transition and turbulent flows, Trans. ASME J. Heat Transfer 115 (1993) 890– 896. [6] A.E. Bergles, Techniques to augment heat transfer, in: W.M. Rohsnow, et al. (Eds.), Hand Book of Heat Transfer Applications, 2nd Edition, McGraw-Hill, New York, 1985. [7] E.R.G. Eckert, R.M. Drake Jr., Analysis of Heat and Mass Transfer, McGraw-Hill, Kogakusha, Japan, 1972. [8] P.K. Sarma, V.D. Rao, T. Subramanyam, Convective heat transfer and pressure drop for high Prandtl number fluids with twistedtape turbulence promoters, Final Report [Ref: BHEL/ PSVK: 96: 0210,19,1996], Hyderabad, India, 1998. [9] A.W. Date, Prediction of fully developed flow in a tube containing a twisted tape, Internat. J. Heat Mass Transfer 17 (1974) 845–859. [10] B. Donevski, M. Plocek, J. Kulesza, M. Sasic, Analysis of tube side laminar and turbulent flow heat transfer with twisted tape inserts, in: S.J. Deng, et al. (Eds.), Heat Transfer Enhancement and Energy Conservation, Hemisphere, New York, 1990, pp. 175–185. [11] S.W. Hong, A.E. Bergles, Augmentation of laminar heat transfer in tubes by means of twisted tape inserts, Trans. ASME J. Heat Transfer 98 (1976) 251–256. [12] W.J. Marner, A.E. Bergles, Augmentation of highly viscous laminar heat transfer inside tubes with constant wall temperature, J. Experimental Thermal Fluid Sci. 2 (1989) 252–267.