Effective parameters on second law analysis for semicircular ducts in laminar flow and constant wall heat flux B

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1 International Communications in Heat and Mass Transfer 32 (2005) Effective parameters on second law analysis for semicircular ducts in laminar flow and constant wall heat flux B Hakan F. Oztop* Department of Mechanical Engineering, Firat University, Elazig, Turkey Abstract Entropy generation for semi-cylindrical ducts is obtained analytically for laminar flow and subjected to constant wall heat flux boundary conditions. Affecting parameters such as heat flux rate, Reynolds number and cross sectional are studied for entropy generation. It is concluded that cross-sectional area and wall heat flux have considerable effect on entropy generation. For the increasing value of these parameters, both entropy generation and pumping power ratio are increased at fixed Reynolds number. D 2004 Elsevier Ltd. All rights reserved. Keywords: Second law analysis; Laminar flow; Semi-circular duct 1. Introduction Heat transfer devices are accompanied by irreversibilities and therefore entropy generation is due to temperature gradients and pressure. For efficient optimal thermodynamic design entropy generation must be reduced. In this context, geometry of duct (cross-sectional area) is an important parameter on entropy generation. Various cross-sectional ducts are used in heat transfer devices due to the size and volume constraints to enhance heat transfer with passive method. Pressure drop and heat transfer analysis in various shaped ducts were summarized by Shah and London [1]. Also, B Communicated by J.W. Rose and A. Briggs. * Corresponding author. Tel.: x6331; fax: address: /$ - see front matter D 2004 Elsevier Ltd. All rights reserved. doi: /j.icheatmasstransfer

2 H.F. Oztop / International Communications in Heat and Mass Transfer 32 (2005) Etemad and Bakhtiari [2] obtained general solutions for fully developed fluid flow and heat transfer characteristics in complex geometries. Semicircular ducts are used in heat exchanger design in various engineering application with different segment angles. These pipes can carry Newtonian or non-newtonian fluid under the constant wall temperature or constant heat flux boundary conditions. Hong and Bergles [3] obtained the thermal entry length solutions for semicircular duct for both constant heat flux and constant temperature boundary conditions for Newtonian fluid. Etemad et al. [4] made an experimental study for viscous non-newtonian forced convection heat transfer in semicircular and equilateral triangular ducts. They discussed the Rayleigh number effects on heat transfer. They found that Rayleigh number influences the Nusselt number. Nag and Mukherjee [5] analyzed the thermodynamic optimization of convective heat transfer through a duct under the constant wall temperature boundary conditions. However, second law analysis in convective heat transfer was outlined in detail by Bejan [6,7]. He obtained entropy generation equations for convection heat transfer and different heat transfer devices. Two different studies were made to show geometrical effects on entropy generation by Xahin [8,9]. In the first, he obtained the optimum shape of duct subjected to constant wall temperature. In the second, he investigated the irreversibilities in various duct geometries with constant wall heat flux. Both of them were made for laminar flow regime and circular, rectangular, triangular, square and sinusoidal cross-sectional pipes. He found that circular geometries are the best choices for both temperature boundary conditions in entropy minimization. Narusawa [10] investigated the mixed convection numerically in three dimensional rectangular cavities with heating at the bottom. Also, he examined entropy generation for this geometry. In laminar flow condition, the effect of variable viscosity on the entropy generation for circular duct was investigated by Xahin [11]. He showed that an optimum length may be obtained which minimizes total energy loses due to both entropy generation and pumping power. A similar study was made for liquid flow with variable viscosity effect for constant wall temperature in turbulent flow regime by Xahin [12]. Oztop et al. [13] investigated second law analyses for hexagonal duct and different fluids. He found that constant viscosity assumption may yield a considerable amount of deviation in entropy generation and pumping power results from those obtained in the temperature dependent viscosity case. To the best of the author s knowledge the entropy generation in semicircular ducts with constant wall heat flux has not yet been investigated. The present paper reports an analytical study of entropy d Fig. 1. Physical model for semi-circular duct problem.

3 268 H.F. Oztop / International Communications in Heat and Mass Transfer 32 (2005) q h Control volume m T T+dT d h x L x+dx q Fig. 2. Physical model of the duct. generation in laminar flow. The effects of Reynolds number, heat flux and geometrical dimensions on entropy generation are analyzed. 2. Physical model of problem The physical model of semicircular duct is depicted in Fig. 1. The hydraulic diameter of any duct is given by D h ¼ 4A P ð1þ where A is the cross-sectional area and P is perimeter. The hydraulic diameter for semicircular crosssectional area can be written as D h ¼ 2 p ffiffiffiffiffi 2p p ffiffiffi A : ð2þ p þ 2 3. Second law analysis The control volume for the analysis of entropy generation with constant heat flux boundary condition and dx length is given in Fig. 2. In this figure ṁ is mass flow rate, T 0 is the inlet temperature and L is the length of duct. The total entropy generation within a control volume, shown in Fig. 2, can be written as follows dṡs gen ¼ ṁmds d Q : T w For an incompressible fluid, ð3þ ds ¼ C p dt T dp ðqtþ : ð4þ

4 H.F. Oztop / International Communications in Heat and Mass Transfer 32 (2005) The mass flow rate is given by ṁm ¼ qua: ð5þ The pressure drop can be written as dp ¼ f qu 2 2D dx ð6þ f is the Darcy friction factor. A dimensionless total entropy generation based on the flow stream heat capacity rate (ṁc p ) is defined as w ¼ ṠS gen ¼ ṠS gen ð7þ ṁmc p Q=DT where DT is the increase of the fluid temperature in the duct, T L T 0. The heat transfer is d Q ¼ ṁmc p dt ¼ hpðt w TÞdx ð8þ where h is the average heat transfer coefficient as depicted in Fig. 2 which is taken to be constant along the surface of the duct for constant thermo-physical properties. Eq. (8) can also be written as d Q ¼ ṁmc p dt ¼ qpdx: Integrating this equation, over the control volume the bulk temperature variation of the fluid and the total heat transfer rate along the duct is [8] T ¼ T 0 þ 4q=qUDC p x: ð10þ And then, for the constant wall heat flux boundary conditions, the total entropy generation is obtained by integration of Eq. (2) using Eqs. (2), (9) and (10). The total entropy generation can be written as w ¼ ln½ðre þ sp 1 Þð1 þ sþ= ðre þ sre þ sp 1 ÞŠ: ð11þ ð9þ In this equation P 1 and P 2 are, P 1 ¼ 4Nu k=pr ð12þ P 2 ¼ l 3 ðfreþ=8q 2 D 3 hq: ð13þ Values of Nu and ( fre) for fully developed laminar flow are given by Shah and London [3] for a variety of duct geometries. In these equations some parameters can be made dimensionless as follows k ¼ L D s ¼ T w T T 0 St ¼ h ¼ Nu quc p RePr : ð14þ ð15þ ð16þ

5 270 H.F. Oztop / International Communications in Heat and Mass Transfer 32 (2005) Required pumping power The power required to overcome the fluid friction in the duct in dimensionless form is PPR ¼ ADPU Q : ð17þ This equation is valid for both constant wall temperature boundary conditions and constant wall heat flux boundary conditions. For constant wall heat flux boundary conditions, the pumping power to heat transfer ratio for fully developed laminar flow becomes PPR ¼ P 2 Re 2 : ð18þ 5. Results and discussion An analytical study was carried out for the range of Reynolds number of 0bReb3000. Constant wall heat flux was the boundary condition considered for the analysis. Water has been used for working fluid whose properties are listed in Table 1. Fig. 3 shows dimensionless entropy generation for different cross-sectional area of semicircular cylinder at different Reynolds number values. As can be shown that as the cross-sectional area is increased dimensionless entropy generation decreases. Lower value of entropy generation is obtained for larger A. For very low Reynolds number values there is no considerable difference of total entropy generation for different cross-sectional areas. Entropy generation for different wall heat flux can be seen from Fig. 4. In this figure, as the heat flux values are increased dimensionless entropy generation increases. This is because of the fact that the dimensionless total entropy generation is affected by both heat transfer and by viscous friction. The smaller heat flux value gives less entropy generation. For all heat flux values, the entropy generation values tend to decrease initially and then increases while the Reynolds number is increased. For higher Reynolds numbers and for the q=500 W/m 2 and q=250 W/m 2 values dimensionless entropy generation is almost the same but for lower Reynolds number as wall heat flux increases b values increased. Fig. 5 shows pumping power ratio Table 1 Thermo physical propeties of water Water C p (J/kgK) 4182 Pr 7 T w (K) 293 A(Ns/m 2 ) U(kg/m 3 ) 998.2

6 H.F. Oztop / International Communications in Heat and Mass Transfer 32 (2005) A= ψ A= A= Re Fig. 3. Variation of dimensionless entropy generation for various cross-sectional areas and Reynolds number, q=500 W/m 2. for different wall heat flux at different Reynolds numbers. As can be seen from this figure, as wall heat flux is decreased pumping power ratio increases. Also, this result is valid for circular duct [9]. The cross-sectional area of the semicircular duct is another effective parameter on pumping power ψ q=500 q=1000 q= Re Fig. 4. Variation of dimensionless entropy generation for wall heat flux and Reynolds number, A= m 2.

7 272 H.F. Oztop / International Communications in Heat and Mass Transfer 32 (2005) PPR q=250 q=500 q= Re Fig. 5. Pumping power ratio vs. Reynolds number for different wall heat flux in semi-circular ducts, A= m 2. ratio. When Figs. 5 and 6 are compared with each other it is seen that both wall heat flux and crosssectional areas show similar trend on pumping power to heat transfer ratio. As Reynolds number is increased, pumping power increases depending on cross-sectional area and wall heat flux values. For A= PPR 1 A= A= Re Fig. 6. Pumping power ratio vs. Reynolds number for different cross-sectional areas of semi-circular duct, q=500 W/m 2.

8 H.F. Oztop / International Communications in Heat and Mass Transfer 32 (2005) lower cross-sectional areas, higher pumping power is obtained. But, there is no minimum value of entropy generation for any value of cross-sectional area. 6. Conclusion Second law analysis of laminar flow subjected to constant wall heat flux has been obtained for semicircular ducts. From this study some conclusions can be drawn as follows:! As Reynolds number is increased total entropy generation is decreasing.! With the increase of temperature difference, s, total entropy generation increased depending on Reynolds number.! When cross-sectional area is increased total entropy generation is increasing for the same Reynolds number. Further, an increase in cross-sectional area causes to increase required pumping power. Nomenclature A Cross-sectional area of duct, m 2 C p Specific heat capacity, J/kg K D h Hydraulic diameter, m f Friction factor h Average heat transfer coefficient, W/m 2 K k Thermal conductivity, W/mK L Length of duct, m m mass flowrate, kg/s Nu Average Nusselt number, hd H /k p Perimeter of duct, m P Pressure, N/m 2 PPR pumping power to heat transfer ratio, ADPU/Q Pr Prandtl number, lc p /k Q Total heat flux, W Re Reynolds number, qud H /l s Entropy, J/kg K Ṡ gen Entropy generation, W/K St Stanton number, h /(quc g p ) T Temperature, K T 0 Inlet fluid temperature, K T w Wall temperature of the duct, K U g Fluid bulk velocity, m/s x Axial distance, m DP Total pressure drop, N/m 2 DT Increase of fluid bulk temperature, K l Viscosity, Ns/m 2 k Non-dimensional axial distance, L/D H Non-dimensional group, 4Nuk/Pr P 1

9 274 H.F. Oztop / International Communications in Heat and Mass Transfer 32 (2005) P 2 Non-dimensional group, l 3 ( fre)/(8q 2 D 3 h q) w Non-dimensional entropy generation q Density, kg/m 3 s Non-dimensional inlet wall-to-fluid temperature difference (T 0 T w )/T w Acknowledgements The Author thanks to Prof. Dr. A.Z. Xahin from KFUPM in Dhahran because of his valuable contribution to this study. References [1] R.K. Shah, Laminar flow forced convection in ducts, Academic Press, [2] S.Gh. Etemad, F. Bakhtiari, Int. Commun. Heat Mass Transf. 26 (1999) 229. [3] S.W. Hong, A.E. Bergles, Int. J. Heat Mass Transfer 19 (1976) 123. [4] S.Gh. Etemad, A.S. Mujumdar, R. Nassef, Int. Commun. Heat Mass Transf. 24 (1997) 609. [5] P.K. Nag, P. Mukherjee, Int. J. Heat Mass Transfer 30 (1987) 401. [6] A. Bejan, Entropy generation through heat and fluid flow, John Wiley & Sons, [7] A. Bejan, J. Heat Transfer 101 (1979) 718. [8] A.Z. Xahin, Heat Mass Transf. 33 (1998) 425. [9] A.Z. Xahin, Energy 23 (1998) 465. [10] U. Narusawa, Heat Mass Transf. 37 (2001) 197. [11] A.Z. Xahin, Exergy, Int. J. 2 (2002) 314. [12] A.Z. Xahin, Heat Mass Transf. 35 (1999) 99. [13] H.F. Oztop, A.Z. Sahin, I. Dagtekin, Int. J. Energy Res. (article in press).

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