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Thermodynamic optimization of a coiled tube heat exchanger under constant wall heat flux condition
Energy 34 (2009) 1122–1126
Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Thermodynamic optimization of a coiled tube heat exchanger under constant wall heat flux condition Ashok K. Satapathy* Department of Mechanical Engineering, National Institute of Technology, Rourkela 769 008, Orissa, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 February 2009 Received in revised form 23 April 2009 Accepted 25 April 2009 Available online 24 May 2009
In this paper the second law analysis of thermodynamic irreversibilities in a coiled tube heat exchanger has been carried out for both laminar and turbulent flow conditions. The expression for the scaled nondimensional entropy generation rate for such a system is derived in terms of four dimensionless parameters: Prandtl number, heat exchanger duty parameter, Dean number and coil to tube diameter ratio. It has been observed that for a particular value of Prandtl number, Dean number and duty parameter, there exists an optimum diameter ratio where the entropy generation rate is minimum. It is also found that with increase in Dean number or Reynolds number, the optimum value of the diameter ratio decreases for a particular value of Prandtl number and heat exchanger duty parameter. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Coiled tube heat exchanger Entropy generation rate Optimization
1. Introduction In a thermodynamic process the loss of exergy is primarily due to the associated irreversibilities which generate entropy. Most convective heat transfer processes are characterized by two types of exergy losses, e.g. losses due to fluid friction and those due to heat transfer across a finite temperature difference. The above two interrelated phenomena are manifestations of thermodynamic irreversibility and investigation of a process from this standpoint is known as second law analysis. However, there exists a direct proportionality between the wasted power (the rate of available work lost) and the entropy generation rate. If engineering systems and their components are to operate such that their lost work is minimized, then the conceptual design of such systems and components must comply with the minimization of entropy generation [1,2]. EGM (entropy generation minimization) is the method of modeling and optimization of the devices accounting for both heat transfer and fluid flow irreversibilities. Coiled tube heat exchangers are often used to obtain a large heat transfer area per unit volume and to enhance the heat transfer coefficient on the inside surface of the tube. Because of the increased turbulence (due to constantly varying flow direction), the heat transfer coefficient for a coiled tube is much greater than that of a straight tube. The centrifugal forces arising due to the swirl
* Tel.: þ91 661 2463522; fax: þ91 661 2472926. E-mail addresses: [email protected], [email protected], [email protected] 0360-5442/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2009.04.028
flow in a coiled tube give rise to secondary flow pattern, which consists of two vertices perpendicular to the axial flow direction. As a result, the heat transport occurs not only by diffusion in the radial direction but also by convection. The contribution of such secondary convective transport dominates the overall process and significantly enhances the rate of heat transfer per unit length of the tube, as compared to that of a straight tube of equal length. Consequently, the heat exchangers with coiled tube can provide a larger heat transfer area per unit volume and thereby makes the unit more compact. The minimization of entropy generation in ducts attributed to fluid flow and heat transfer is of considerable practical interest with regard to heat transfer augmentation techniques. The main objective is to increase the wall-fluid heat transfer coefficient as compared to that of an unaugmented surface. A parallel objective, however, is to achieve this improvement without an increase in the pumping power as demanded by the forced convection arrangement. These two conflicting objectives lead to the concept of optimal design of an engineering system: a modification that enhances the heat transfer is likely to also augment the mechanical pumping power requirement. The effect of a proposed augmentation technique on thermodynamic performance can therefore be evaluated by comparing the entropy generation rate before and after the implementation. The method of EGM was originally applied to a straight tube with smooth surface in a pioneering work by Bejan [3]. The method of optimizing augmentation techniques applied to the design of ducts with surface roughness was proposed by Bejan and Pfister [4]. Also spiral tubes with internal ribs were optimized by Oullette and
A.K. Satapathy / Energy 34 (2009) 1122–1126
Nomenclature A B0 cp d D Dn f G h k _ m Nu Pr
2 cross-sectional area of the tube, A ¼ (pd2)/4, mp ffiffiffiffiffiffiffiffiffiffiffi _ heat exchanger duty parameter, B0 ¼ ðq0 rmÞ= kT m5 specific heat of tube-fluid, J/kg K internal diameter of tube, m mean coil diameter, m Dean number, Dn ¼ (Re d)/D Darcy friction factor _ mass velocity, G ¼ m=A, kg/m2 s heat transfer coefficient, W/m2 K thermal conductivity of the tube-fluid, W/m K mass flow rate, kg/s Nusselt number, Nu ¼ (hd)/k Prandtl number, Pr ¼ (mcp)/k
Bejan [5]. Additional references on thermodynamic optimization of smooth and roughened tubes are cited by Bejan [2]. The irreversibility analysis was studied by Sahin [6,7] in various duct geometries with constant wall heat flux and laminar flow. Sahin [8] further investigated the entropy generation and pumping power in a turbulent fluid flow through a smooth pipe subjected to constant heat flux. Irreversibility analysis was successfully carried out for the helical coils or double-sine duct by Ko and Ting [9,10] and Ko [11]. The optimal design work for rectangular curved ducts based on the minimal entropy generation principle was first reported by Ko and Ting [12]. In the study, the entropy generation due to laminar forced convection in a curved rectangular duct with constant wall heat flux was investigated. Ko [13–15] subsequently extended to laminar forced convection in a curved rectangular duct with internal ribs. Recently Ko and Wu [16] reported analysis of entropy generation induced by turbulent forced convection in a curved rectangular duct with external heating. The work by Shokouhmand and Salimpour [17,18] also deals with entropy generation analysis of fully developed laminar forced convection in a helical tube with uniform wall temperature. The purpose of the present study is to optimize the geometry of a coiled heat exchanger tube for both laminar and turbulent flow conditions using the EGM method. In particular, attention has been focused on a helical coiled tube with a single helix, subjected to a constant boundary heat flux. The flow is assumed to be fullydeveloped both hydrodynamically and thermally. The relevant optimizing parameters are the Dean or Reynolds number and the ratio of coil to tube diameter, which adequately represent the flow geometry. The non-dimensional entropy generation rate, when plotted against Dean or Reynolds number, shows a distinct minimum and, therefore, it is possible to maximize the ratio of heat transfer to pumping power, in consistence with second law analysis. 2. Analysis _ in a passage of cross-sectional area A The flow of fluid at a rate m is considered where the fluid is under a favorable pressure gradient (Fig. 1). If the heat is transferred to the fluid at a rate q0 , the entropy generation rate per unit length based on average heat transfer and fluid friction is given [1,2] by: 0 S_ gen
_3 f q02 d 2m ¼ þ _ p St r2 T dA2 4T 2 mc
q0 Q Re 0 S_
1123
heat transfer rate per unit tube length, W/m dimensionless heat flux, Q ¼ q0 /(kT) Reynolds number, Re ¼ (Gd)/m entropy generation rate per unit tube length, W/m K Stanton number, St ¼ h/(cpG) temperature, K distance coordinate, m
gen
St T x
Greek alphabets coil to tube diameter ratio, d ¼ D/d density of tube-fluid, kg/m3 viscosity of tube-fluid, kg/m s scaled dimensionless entropy generation rate, 0 J ¼ S_ gen =ðkQ 2 Þ
d r m J
0 safely applied to a helical coiled tube provided that S_ gen is calculated on the basis of unit center-line length of the tube. The Stanton number St is based on hydraulic diameter of the tube (equal to d) and f is Darcy friction factor. The first part of the quantity on right hand side of Eq. (1) arises due to heat transfer, while the second part is due to friction. It is evident from Eq. (1) that a high Stanton 0 number contributes to reducing the heat transfer share of S_ gen , while a higher friction factor has the effect of increasing the entropy generation rate due to viscous effects. In a round tube of internal diameter d, Eq. (1) assumes the form
0 S_ gen ¼
_ 3f q02 32m þ 2 2 5 2 pkT Nu p r Td
The following non-dimensional quantities are then defined:
Q ¼
q0 ; kT
Re ¼
_ 4m ; pmd
B20 ¼
where T, cp and r are fluid bulk properties. The above equation is derived for a straight tube of any arbitrary cross-section and can be
_2 q02 r2 m 5 kT m
(3)
where Q is non-dimensional heat flux, Re is Reynolds number and B0 is the heat exchanger duty parameter that represents the heat load. Using above dimensionless quantities Eq. (2) thereby reduces to
J¼
1
pNu
þ
p3 fRe5 32 B20
(4)
where J is scaled non-dimensional entropy generation rate given 0 by: J ¼ S_ gen =ðkQ 2 Þ. 2.1. Laminar flow The friction loss in flow through helical coiled tubes has been studied by Ito [19] and the following correlation has been recommended to predict the friction factor:
f ¼ 0:37ð64=ReÞDn0:36
(5)
Janssen and Hoogendoorn [20] have experimentally studied the heat transfer in a single helical coiled tube subjected to a constant heat flux and presented the data as
Nu ¼ 0:7Re0:43 Pr 1=6 ðd=DÞ0:07 (1)
(2)
(6)
Substituting the values of friction factor f and Nusselt number Nu from Eqs. (5) and (6) respectively into Eq. (4) and after recasting the constants, the non-dimensional entropy generation rate can be derived as
1124
A.K. Satapathy / Energy 34 (2009) 1122–1126
a
b
Fig. 1. Geometry of the coiled tube heat exchanger.
J ¼
0:455 0:145 Pr 1=6 Dn0:43 d
2
Dn4:36 d þ 4:47 B20
(7)
Thus for every Dean number, a thermodynamically best design can be achieved by adopting an optimal value of d, for which J would be a minimum. This minimum can be obtained by differentiating J with respect to d and equating the derivative to zero. Under such operation, solving for d gives:
dopt
B0:932 0 ¼ 9:86Pr0:078 Dn2:233
(8)
Similarly following the aforementioned procedure and solving for Dn yields
Dnopt ¼
B0:418 0
(9)
0:448 2:613Pr 0:035 d
Thus, once the heat exchanger duty parameter B0 and Prandtl number Pr are decided upon, the thermodynamically best values of d or Dn can be readily calculated.
2.2. Turbulent flow
dopt þ 3:455 10:707B0 ¼ 0:632 2:229 Pr Re d0:55 opt The value of optimum Reynolds number is finally derived as
Reopt ¼
d 0:945B0:4024 0
0:221
Pr 0:081 ð3:455 þ dÞ
0:2012
64 Re 1=20 Re d2
3. Results and discussion The scaled non-dimensional entropy generation rate J has been computed using Eqs. (7) and (12) for laminar and turbulent flow respectively, after assigning practical values to B0, Pr and d. The function J has been plotted against Dn for various values of radius ratio in Fig. 2. It is seen that for each diameter ratio, there exists an optimum value for the Dean number, for which the scaled nondimensional entropy generation rate is minimum. For turbulent
(10)
Xin and Ebadin [22] measured the average Nusselt number using constant heat flux boundary condition and reported the following correlation for Nusselt number in the turbulent regime:
3:455 Nu ¼ 0:00619Re0:92 Pr 0:4 1 þ
(11)
d
Substituting the values of f and Nu from Eqs. (10) and (11) respectively in Eq. (4), and after simplification gives
J ¼
51:423 Re4:05 þ 15:503 0:1 þ 3:455=dÞ B2 d
Re0:92 Pr 0:4 ð1
(12)
0
Differentiating J with respect to d and Re separately, and equating to zero yields
(14)
For practical applications it is possible to determine the optimum tube diameter which leads to minimum irreversibility. This is due to the fact that, as the tube diameter increases, although the exergy loss due to fluid friction increases but the exergy loss due to heat transfer term decreases.
In turbulent flow the Dean number does not correlate the flow measurements data well, whereas Reynolds number can be used as a model parameter to predict the friction factor accurately. The friction factor in the turbulent regime is given by Mori and Nakayama [21]:
f ¼
(13)
Fig. 2. J vs. Dn curve for laminar flow.
A.K. Satapathy / Energy 34 (2009) 1122–1126
1125
Fig. 3. J vs. Re curve for turbulent flow.
Fig. 5. J/Jmin vs. Re/Reopt curve for turbulent flow.
flow, similar characteristic curves are also plotted for J vs. Re in Fig. 3 and an optimum Re is observed. It is interesting to note that J varies steeply towards right than towards left at the optimum Dean number (Fig. 2) or at the optimum Reynolds number (Fig. 3). Hence, while selecting the optimum Dn or Re for practical applications, it should always be ensured that these operating parameters should lie towards left in order that the entropy generation J will be relatively minimum. The thermodynamically optimum value of Dean and Reynolds numbers has been derived in Eqs. (9) and (14) for laminar and turbulent flow respectively. The values of Jmin are calculated using Eqs. (7) and (12) after substituting the optimal values of Dn and Re. Then the ratio J/Jmin is calculated for different combinations of B0, Pr and d. These results are presented graphically in Figs. 4 and 5 for laminar and turbulent flow respectively. These graphs indicate that J/Jmin becomes unity as Dn/Dnopt or Re/Reopt approaches unity and that the rate of entropy generation increases sharply on either
side of the optimum. These curves also predict the performance of an optimal design relative to any other design. The ratio J/Jmin can be used to monitor any design relative to the best design that can be achieved, under the same constraints. 4. Conclusion The present study shows that in coiled tube heat exchanger applications with constant heat flux boundary condition, the diameter ratio of the coil to tube is an important criterion and should be set at the optimum value. The significance of this optimum is that it will minimize the degradation of thermal energy and viscous dissipation of mechanical energy, if this optimum is incorporated in the design. The results also show that with increase in Dean or Reynolds number, the optimum value of diameter ratio decreases for a particular value of Prandtl number and duty parameter. It is felt that the present analysis, in principle, may be extended to heat exchangers of various other geometries for different boundary conditions. Acknowledgment The partial financial support provided by Grant No. F.26-14/ 2003, TSV, Dt. 14.01.04, MHRD, Government of India, is gratefully acknowledged. References
Fig. 4. J/Jmin vs. Dn/Dnopt curve for laminar flow.
[1] Bejan A. Entropy generation through heat and fluid flow. NY: John Wiley; 1994. p. 105–9. [2] Bejan A. Entropy generation minimization: the new thermodynamics of finitesize devices and finite-time processes. J Appl Phys 1996;79:1191–218. [3] Bejan A. A study of entropy generation in fundamental convective heat transfer. ASME J Heat Transf 1979;101:718–25. [4] Bejan A, Pfister PA. Evaluation of heat exchanger augmentation techniques based on their impact on entropy generation. Lett Heat Mass Transf 1980;7:97–106. [5] Oullette WR, Bejan A. Conservation of available work (exergy) by using promoters of swirl flow in forced convection heat transfer. Energy 1980;5:587–96. [6] Sahin AZ. Thermodynamics of laminar viscous flow through a duct subjected to constant heat flux. Energy 1996;21:1179–87. [7] Sahin AZ. Irreversibilities in various duct geometries with constant wall heat flux and laminar flow. Energy 1998;23:465–73.
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[8] Sahin AZ. Entropy generation and pumping power in a turbulent fluid flow through a smooth pipe subjected to a constant heat flux. Exergy 2002;2: 314–21. [9] Ko TH, Ting K. Entropy generation and thermodynamic optimization of fully developed laminar convection in a helical coil. Int Comm Heat Mass Transf 2005;32:214–23. [10] Ko TH, Ting K. Optimal Reynolds number for the fully developed laminar forced convection in a helical coiled tube. Energy 2006;31:2142–52. [11] Ko TH. Numerical analysis of entropy generation and optimal Reynolds number for developing laminar forced convection in double-sine ducts with various aspect ratios. Int J Heat Mass Transf 2006;49:718–26. [12] Ko TH, Ting K. Entropy generation and optimal analysis for laminar forced convection in curved rectangular ducts: a numerical study. Int J Therm Sci 2006;45:138–50. [13] Ko TH. Numerical investigation of laminar forced convection and entropy generation in a helical coil with constant wall heat flux. Num Heat Transf A 2006;49:257–78. [14] Ko TH. Numerical investigation on laminar forced convection and entropy generation in a curved rectangular duct with longitudinal ribs mounted on heated wall. Int J Therm Sci 2006;45:390–404.
[15] Ko TH. A numerical study on entropy generation and optimization for laminar forced convection in a rectangular curved duct with longitudinal ribs. Int J Therm Sci 2006;45:1113–25. [16] Ko TH, Wu CP. A numerical study on entropy generation induced by turbulent forced convection in curved rectangular ducts with various aspect ratios. Int Comm Heat Mass Transf 2009;36:25–31. [17] Shokouhmand H, Salimpour MR. Optimal Reynolds number of laminar forced convection in a helical tube subjected to uniform wall temperature. Int Comm Heat Mass Transf 2007;34:753–61. [18] Shokouhmand H, Salimpour MR. Entropy generation analysis of fully developed laminar forced convection in a helical tube with uniform wall temperature. Heat Mass Transf 2007;44:213–20. [19] Ito H. Friction factors for turbulent flow in curved pipes. J Basic Eng 1959:123–34. [20] Janssen LAM, Hoogendoorn CJ. Laminar convective heat transfer in helical coiled tubes. Int J Heat Mass Transf 1978;21:1197–206. [21] Mori Y, Nakayama W. Study on forced convective heat transfer in curved pipes. Int J Heat Mass Transf 1967;10:681–95. [22] Xin RC, Ebadian MA. The effects of Prandtl numbers on local and average convective heat transfer characteristics in helical pipes. ASME J Heat Transf 1997;119:467–73.
Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Thermodynamic optimization of a coiled tube heat exchanger under constant wall heat flux condition Ashok K. Satapathy* Department of Mechanical Engineering, National Institute of Technology, Rourkela 769 008, Orissa, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 February 2009 Received in revised form 23 April 2009 Accepted 25 April 2009 Available online 24 May 2009
In this paper the second law analysis of thermodynamic irreversibilities in a coiled tube heat exchanger has been carried out for both laminar and turbulent flow conditions. The expression for the scaled nondimensional entropy generation rate for such a system is derived in terms of four dimensionless parameters: Prandtl number, heat exchanger duty parameter, Dean number and coil to tube diameter ratio. It has been observed that for a particular value of Prandtl number, Dean number and duty parameter, there exists an optimum diameter ratio where the entropy generation rate is minimum. It is also found that with increase in Dean number or Reynolds number, the optimum value of the diameter ratio decreases for a particular value of Prandtl number and heat exchanger duty parameter. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Coiled tube heat exchanger Entropy generation rate Optimization
1. Introduction In a thermodynamic process the loss of exergy is primarily due to the associated irreversibilities which generate entropy. Most convective heat transfer processes are characterized by two types of exergy losses, e.g. losses due to fluid friction and those due to heat transfer across a finite temperature difference. The above two interrelated phenomena are manifestations of thermodynamic irreversibility and investigation of a process from this standpoint is known as second law analysis. However, there exists a direct proportionality between the wasted power (the rate of available work lost) and the entropy generation rate. If engineering systems and their components are to operate such that their lost work is minimized, then the conceptual design of such systems and components must comply with the minimization of entropy generation [1,2]. EGM (entropy generation minimization) is the method of modeling and optimization of the devices accounting for both heat transfer and fluid flow irreversibilities. Coiled tube heat exchangers are often used to obtain a large heat transfer area per unit volume and to enhance the heat transfer coefficient on the inside surface of the tube. Because of the increased turbulence (due to constantly varying flow direction), the heat transfer coefficient for a coiled tube is much greater than that of a straight tube. The centrifugal forces arising due to the swirl
* Tel.: þ91 661 2463522; fax: þ91 661 2472926. E-mail addresses: [email protected], [email protected], [email protected] 0360-5442/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2009.04.028
flow in a coiled tube give rise to secondary flow pattern, which consists of two vertices perpendicular to the axial flow direction. As a result, the heat transport occurs not only by diffusion in the radial direction but also by convection. The contribution of such secondary convective transport dominates the overall process and significantly enhances the rate of heat transfer per unit length of the tube, as compared to that of a straight tube of equal length. Consequently, the heat exchangers with coiled tube can provide a larger heat transfer area per unit volume and thereby makes the unit more compact. The minimization of entropy generation in ducts attributed to fluid flow and heat transfer is of considerable practical interest with regard to heat transfer augmentation techniques. The main objective is to increase the wall-fluid heat transfer coefficient as compared to that of an unaugmented surface. A parallel objective, however, is to achieve this improvement without an increase in the pumping power as demanded by the forced convection arrangement. These two conflicting objectives lead to the concept of optimal design of an engineering system: a modification that enhances the heat transfer is likely to also augment the mechanical pumping power requirement. The effect of a proposed augmentation technique on thermodynamic performance can therefore be evaluated by comparing the entropy generation rate before and after the implementation. The method of EGM was originally applied to a straight tube with smooth surface in a pioneering work by Bejan [3]. The method of optimizing augmentation techniques applied to the design of ducts with surface roughness was proposed by Bejan and Pfister [4]. Also spiral tubes with internal ribs were optimized by Oullette and
A.K. Satapathy / Energy 34 (2009) 1122–1126
Nomenclature A B0 cp d D Dn f G h k _ m Nu Pr
2 cross-sectional area of the tube, A ¼ (pd2)/4, mp ffiffiffiffiffiffiffiffiffiffiffi _ heat exchanger duty parameter, B0 ¼ ðq0 rmÞ= kT m5 specific heat of tube-fluid, J/kg K internal diameter of tube, m mean coil diameter, m Dean number, Dn ¼ (Re d)/D Darcy friction factor _ mass velocity, G ¼ m=A, kg/m2 s heat transfer coefficient, W/m2 K thermal conductivity of the tube-fluid, W/m K mass flow rate, kg/s Nusselt number, Nu ¼ (hd)/k Prandtl number, Pr ¼ (mcp)/k
Bejan [5]. Additional references on thermodynamic optimization of smooth and roughened tubes are cited by Bejan [2]. The irreversibility analysis was studied by Sahin [6,7] in various duct geometries with constant wall heat flux and laminar flow. Sahin [8] further investigated the entropy generation and pumping power in a turbulent fluid flow through a smooth pipe subjected to constant heat flux. Irreversibility analysis was successfully carried out for the helical coils or double-sine duct by Ko and Ting [9,10] and Ko [11]. The optimal design work for rectangular curved ducts based on the minimal entropy generation principle was first reported by Ko and Ting [12]. In the study, the entropy generation due to laminar forced convection in a curved rectangular duct with constant wall heat flux was investigated. Ko [13–15] subsequently extended to laminar forced convection in a curved rectangular duct with internal ribs. Recently Ko and Wu [16] reported analysis of entropy generation induced by turbulent forced convection in a curved rectangular duct with external heating. The work by Shokouhmand and Salimpour [17,18] also deals with entropy generation analysis of fully developed laminar forced convection in a helical tube with uniform wall temperature. The purpose of the present study is to optimize the geometry of a coiled heat exchanger tube for both laminar and turbulent flow conditions using the EGM method. In particular, attention has been focused on a helical coiled tube with a single helix, subjected to a constant boundary heat flux. The flow is assumed to be fullydeveloped both hydrodynamically and thermally. The relevant optimizing parameters are the Dean or Reynolds number and the ratio of coil to tube diameter, which adequately represent the flow geometry. The non-dimensional entropy generation rate, when plotted against Dean or Reynolds number, shows a distinct minimum and, therefore, it is possible to maximize the ratio of heat transfer to pumping power, in consistence with second law analysis. 2. Analysis _ in a passage of cross-sectional area A The flow of fluid at a rate m is considered where the fluid is under a favorable pressure gradient (Fig. 1). If the heat is transferred to the fluid at a rate q0 , the entropy generation rate per unit length based on average heat transfer and fluid friction is given [1,2] by: 0 S_ gen
_3 f q02 d 2m ¼ þ _ p St r2 T dA2 4T 2 mc
q0 Q Re 0 S_
1123
heat transfer rate per unit tube length, W/m dimensionless heat flux, Q ¼ q0 /(kT) Reynolds number, Re ¼ (Gd)/m entropy generation rate per unit tube length, W/m K Stanton number, St ¼ h/(cpG) temperature, K distance coordinate, m
gen
St T x
Greek alphabets coil to tube diameter ratio, d ¼ D/d density of tube-fluid, kg/m3 viscosity of tube-fluid, kg/m s scaled dimensionless entropy generation rate, 0 J ¼ S_ gen =ðkQ 2 Þ
d r m J
0 safely applied to a helical coiled tube provided that S_ gen is calculated on the basis of unit center-line length of the tube. The Stanton number St is based on hydraulic diameter of the tube (equal to d) and f is Darcy friction factor. The first part of the quantity on right hand side of Eq. (1) arises due to heat transfer, while the second part is due to friction. It is evident from Eq. (1) that a high Stanton 0 number contributes to reducing the heat transfer share of S_ gen , while a higher friction factor has the effect of increasing the entropy generation rate due to viscous effects. In a round tube of internal diameter d, Eq. (1) assumes the form
0 S_ gen ¼
_ 3f q02 32m þ 2 2 5 2 pkT Nu p r Td
The following non-dimensional quantities are then defined:
Q ¼
q0 ; kT
Re ¼
_ 4m ; pmd
B20 ¼
where T, cp and r are fluid bulk properties. The above equation is derived for a straight tube of any arbitrary cross-section and can be
_2 q02 r2 m 5 kT m
(3)
where Q is non-dimensional heat flux, Re is Reynolds number and B0 is the heat exchanger duty parameter that represents the heat load. Using above dimensionless quantities Eq. (2) thereby reduces to
J¼
1
pNu
þ
p3 fRe5 32 B20
(4)
where J is scaled non-dimensional entropy generation rate given 0 by: J ¼ S_ gen =ðkQ 2 Þ. 2.1. Laminar flow The friction loss in flow through helical coiled tubes has been studied by Ito [19] and the following correlation has been recommended to predict the friction factor:
f ¼ 0:37ð64=ReÞDn0:36
(5)
Janssen and Hoogendoorn [20] have experimentally studied the heat transfer in a single helical coiled tube subjected to a constant heat flux and presented the data as
Nu ¼ 0:7Re0:43 Pr 1=6 ðd=DÞ0:07 (1)
(2)
(6)
Substituting the values of friction factor f and Nusselt number Nu from Eqs. (5) and (6) respectively into Eq. (4) and after recasting the constants, the non-dimensional entropy generation rate can be derived as
1124
A.K. Satapathy / Energy 34 (2009) 1122–1126
a
b
Fig. 1. Geometry of the coiled tube heat exchanger.
J ¼
0:455 0:145 Pr 1=6 Dn0:43 d
2
Dn4:36 d þ 4:47 B20
(7)
Thus for every Dean number, a thermodynamically best design can be achieved by adopting an optimal value of d, for which J would be a minimum. This minimum can be obtained by differentiating J with respect to d and equating the derivative to zero. Under such operation, solving for d gives:
dopt
B0:932 0 ¼ 9:86Pr0:078 Dn2:233
(8)
Similarly following the aforementioned procedure and solving for Dn yields
Dnopt ¼
B0:418 0
(9)
0:448 2:613Pr 0:035 d
Thus, once the heat exchanger duty parameter B0 and Prandtl number Pr are decided upon, the thermodynamically best values of d or Dn can be readily calculated.
2.2. Turbulent flow
dopt þ 3:455 10:707B0 ¼ 0:632 2:229 Pr Re d0:55 opt The value of optimum Reynolds number is finally derived as
Reopt ¼
d 0:945B0:4024 0
0:221
Pr 0:081 ð3:455 þ dÞ
0:2012
64 Re 1=20 Re d2
3. Results and discussion The scaled non-dimensional entropy generation rate J has been computed using Eqs. (7) and (12) for laminar and turbulent flow respectively, after assigning practical values to B0, Pr and d. The function J has been plotted against Dn for various values of radius ratio in Fig. 2. It is seen that for each diameter ratio, there exists an optimum value for the Dean number, for which the scaled nondimensional entropy generation rate is minimum. For turbulent
(10)
Xin and Ebadin [22] measured the average Nusselt number using constant heat flux boundary condition and reported the following correlation for Nusselt number in the turbulent regime:
3:455 Nu ¼ 0:00619Re0:92 Pr 0:4 1 þ
(11)
d
Substituting the values of f and Nu from Eqs. (10) and (11) respectively in Eq. (4), and after simplification gives
J ¼
51:423 Re4:05 þ 15:503 0:1 þ 3:455=dÞ B2 d
Re0:92 Pr 0:4 ð1
(12)
0
Differentiating J with respect to d and Re separately, and equating to zero yields
(14)
For practical applications it is possible to determine the optimum tube diameter which leads to minimum irreversibility. This is due to the fact that, as the tube diameter increases, although the exergy loss due to fluid friction increases but the exergy loss due to heat transfer term decreases.
In turbulent flow the Dean number does not correlate the flow measurements data well, whereas Reynolds number can be used as a model parameter to predict the friction factor accurately. The friction factor in the turbulent regime is given by Mori and Nakayama [21]:
f ¼
(13)
Fig. 2. J vs. Dn curve for laminar flow.
A.K. Satapathy / Energy 34 (2009) 1122–1126
1125
Fig. 3. J vs. Re curve for turbulent flow.
Fig. 5. J/Jmin vs. Re/Reopt curve for turbulent flow.
flow, similar characteristic curves are also plotted for J vs. Re in Fig. 3 and an optimum Re is observed. It is interesting to note that J varies steeply towards right than towards left at the optimum Dean number (Fig. 2) or at the optimum Reynolds number (Fig. 3). Hence, while selecting the optimum Dn or Re for practical applications, it should always be ensured that these operating parameters should lie towards left in order that the entropy generation J will be relatively minimum. The thermodynamically optimum value of Dean and Reynolds numbers has been derived in Eqs. (9) and (14) for laminar and turbulent flow respectively. The values of Jmin are calculated using Eqs. (7) and (12) after substituting the optimal values of Dn and Re. Then the ratio J/Jmin is calculated for different combinations of B0, Pr and d. These results are presented graphically in Figs. 4 and 5 for laminar and turbulent flow respectively. These graphs indicate that J/Jmin becomes unity as Dn/Dnopt or Re/Reopt approaches unity and that the rate of entropy generation increases sharply on either
side of the optimum. These curves also predict the performance of an optimal design relative to any other design. The ratio J/Jmin can be used to monitor any design relative to the best design that can be achieved, under the same constraints. 4. Conclusion The present study shows that in coiled tube heat exchanger applications with constant heat flux boundary condition, the diameter ratio of the coil to tube is an important criterion and should be set at the optimum value. The significance of this optimum is that it will minimize the degradation of thermal energy and viscous dissipation of mechanical energy, if this optimum is incorporated in the design. The results also show that with increase in Dean or Reynolds number, the optimum value of diameter ratio decreases for a particular value of Prandtl number and duty parameter. It is felt that the present analysis, in principle, may be extended to heat exchangers of various other geometries for different boundary conditions. Acknowledgment The partial financial support provided by Grant No. F.26-14/ 2003, TSV, Dt. 14.01.04, MHRD, Government of India, is gratefully acknowledged. References
Fig. 4. J/Jmin vs. Dn/Dnopt curve for laminar flow.
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