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Heat transfer and sensitivity analysis in a double pipe heat exchanger fi lled with porous medium Kamel Milani Shirvan a, Soroush Mirzakhanlarib, Soteris A. Kalogirouc, Hakan F. Oztop d, Mojtaba Mamourian a,* aDepartment of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran bYoung Researchers and Elite Club, Qazvin Branch, Islamic Azad University, Qazvin, Iran cCyprus University of Technology, Department of Mechanical Engineering and Materials Sciences and Engineering, Limassol, Cyprus dDepartment of Mechanical Engineering, Technology Faculty, Frat University, Elazig, Turkey a r t i c l ei n f o Article history: Received 18 June 2016 Received in revised form 1 February 2017 Accepted 13 July 2017 Available online 21 July 2017 Keywords: Response surface methodology Sensitivity analysis Heat exchanger effectiveness double pipe heat exchanger Heat transfer Porous medium a b s t r a c t In this paper, 2-D numerical investigation and sensitivity analysis are performed on heat transfer rate and heat exchanger effectiveness of a double pipe heat exchanger fi lled with porous medium. The Darcy eBrinkmaneForchheimer model is applied to model the fl ow fi eld in the porous zone. The sensitivity analysis is performed utilizing the Response Surface Methodology. The studied parameters are: Reynolds number (50 ? Re ? 250), Darcy number (10?5? Da ? 10?3), temperature difference between hot and cold fl uids (30 ?DT ? 70) and the porous substrate thickness (1/3 ?d? 1). The obtained results showed that enhancement of the Nusselt number due to the increase in Reynolds and Darcy numbers is in the vicinity of the 77.84% for the case withd 2/3 and Da 10?5to 10?3, and 203.25% for the case withd 1 and Re 50 to 250. Furthermore, increasing porous substrate thickness reduces the mean Nusselt number untild 2/3 and then increases it. In addition, it is found that the heat exchanger effectiveness increases with the Re number and reduces with enhancement of the Da number. The sensitivity analysis showed that the sensitivity of the mean Nusselt number to the Re and Da numbers and the porous substrate thickness is positive, while the sensitivity of the heat exchanger effectiveness to the Re number is positive but to the Da number is negative. 2017 Elsevier Masson SAS. All rights reserved. 1. Introduction Due to the important role of convective heat transfer in porous media and its several technological applications in related in- dustries such as food processing, geothermal heat extractions, solar collector technologies, spread of pollutants underground, storage of grains, heat removal from nuclear reactors, exothermic reactions in packs, bed reactors, electronic boxes, etc., it has been a subject of interest in several fundamental researches in last decades. Among recent studies in this fi eld, some researchers have studied heat transfer performance in heat exchanger without considering a porous medium 1e5. For example, an experimental investigation on the effectof perforated discontinuous helical turbulators on heat transfer characteristics of double pipe water to air heat exchanger has been carried out by Sheikholeslami et al. 6. The results showed that increasing the open area ratio and pitch ratio reduces the friction factor and Nusselt number. Also, the thermal perfor- mance increases with area ratio; but increasing the pitch ratio decreases the thermal performance. Chen and Dung 7 have per- formed a numerical investigation on heat transfer of parallel and counter fl ow double tube heat exchangers with alternating hori- zontal or vertical oval cross section pipes. They presented the temperature and pressure contours and velocity vectors at several selected cross sections. Also, axial averaged Nusselt number and overall heat transfer coeffi cient distributions and heat transfer enhancement factor based on three different parameters are ob- tained in this paper. Also, Bayer et al. 8 have performed a strategic optimization for a borehole heat exchanger. The results revealed that by supplying a given cooling and heating demand, the thermal impact on the long-term conditions in the ground can be minimized. However, some researchers have studied heat transfer perfor- mance in channels, tubes and heat exchangers fi lled with the * Corresponding author. P.O.B. 91775e1111, Mashhad, Iran. E-mail address: mamourianum.ac.ir (M. Mamourian). Contents lists available at ScienceDirect International Journal of Thermal Sciences journal homepage: /10.1016/j.ijthermalsci.2017.07.008 1290-0729/ 2017 Elsevier Masson SAS. All rights reserved. International Journal of Thermal Sciences 121 (2017) 124e137 porous materials. Targui and Kahalerras 9 have performed a nu- merical study to investigate the effect of porous baffl es and fl ow pulsation on a double pipe heat exchanger performance. Their re- sults revealed that the addition of an oscillating component to the mean fl ow increases the heat transfer. Additionally, the maximum performance of the heat exchanger is observed for the case in which only the fl ow of the hot fl uid was pulsating. A numerical analysis of fl ow and heat transfer characteristics in a double pipe heat exchanger with porous structures inserted in the annular gap has also been done by Targui and Kahalerras 10. Two different confi gurations have been considered in this study: on the inner cylinder; and on both the cylinders in a staggered fashion. The maximum heat transfer rates are observed when the porous structures are attached in second confi guration at small spacing and high thicknesses. Alkam and Al-Nimr 11,12 have studied the transient developing forced convection fl ow in concentric tubes and circular channels, which were partially fi lled with porous materials. They found that the external heating penetration is more effective in the porous substrate than that of the clear fl uid region. A numerical investigation on combined convectioneradiation heat transfer rate and the pressure drop in a porous solar heat exchanger has been performed by Rashidi et al. 13. They found that increasing the Darcy number reduces the pressure drop ratio in the vicinity of 58% and 23% ford 1/3 and 1, respectivelyand Da 10?6 to 10?2. Dehghan et al. 14 have done an investigation on com- bined heat transfer in heat exchangers fi lled with a fl uid saturated cellular porous medium. They found that to simulate and predict of thermal performance of solar energy harvesting systems, the semi- analytical methods (like HPM, VIM, DTM, and HAM) can be used. Pavel and Mohamad 15 have carried out an experimental and numerical study on the heat transfer enhancement for gas heat exchangers fi lled with metallic porous materials. The numerical code developed did not consider the radiative heat transfer. Jung and Boo 16 have analyzed the radiation heat transfer in a high- temperature heat pipe heat exchanger. The results showed that the consideration of the radiant heat transfer enhances the heat transfer rate and makes the temperature distribution more uni- form. Another investigation on combined heat transfer in heat exchangers fi lled with a fl uid saturated cellular porous medium has been performed by Soltani et al. 17. They analyzed exactly the effects of porous medium shape and radiation parameters on the thermal performance. Aguilar-Madera et al. 18 have investigated the convective heat transfer inside a channel, which was partially fi lled with a porous material. The results revealed that utilizing a channel fully fi lled with the porous material enhances the heat transfer rate. Moreover, in some studies, researchers have focused on Response Surface Methodology (RSM) to optimize the heat transfer rate, such as Mamourian et al. 19 and Milani-Shirvan et al. 20. In these studies, the effective parameters on thermal perfor- mance in solar heat exchangers are investigated by Response Sur- face Methodology. According to the literature review and to the best knowledge of the authors, despite the important applications of convective heat transfer in porous medium a numerical investigation and sensi- tivity analysis of effective parameters on heat transfer rate between two pipes of a double pipe heat exchanger fi lled with porous me- dium has not yet been considered. Furthermore, a sensitivity analysis is needed to evaluate the effects of Reynolds and Darcy numbers and porous substrate thicknesses on heat transfer rate and heat exchanger effectiveness inside a double pipe heat exchanger fi lled with porous medium. The sensitivity analysis of these parameters is done using the RSM method. Current investi- gation aims to obtain the optimal conditions to enhance the heat transfer rate and heat exchanger effectiveness in a double pipe heat exchanger in order to provide a useful guideline for researchers in the energy related fi elds. In general, the motivation in this article is based on the investigation of optimal conditions and sensitivity of the mean Nusselt number and heat exchanger effectiveness, of double pipe heat exchanger fi lled with porous media using the fi nite volume method (FVM) and RSM models. Nomenclature ANOVAAnalysis of Variance Cp specifi c heat at constant pressure (J/kg K) CCFFace Centered Central Composite Design CF inertia coeffi cient Dhhydraulic diameter, (m) DaDarcy number Eheat exchanger effectiveness FVMFinite Volume Method h convective heat transfer coeffi cient (W/m2 K) kthermal conductivity (W/m K) Kpermeability of the porous layer (m2) Llength of the heat exchanger m$ mass fl ow rate (kg/s) NuNusselt number ppressure (Pa) Pdimensionless pressure PrPrandtl number rradial coordinate (m) RMSERoot Mean Square Error Rdimensionless radial coordinate ReReynolds number Rkthermal conductivity ratio, ke/kc RSMResponse Surface Methodology Ttemperature (K) uaxial velocity (m/s) Rdimensionless radial coordinate Udimensionless axial velocity nradial velocity (m/s) Vdimensionless radial velocity xaxial coordinate (m) Xdimensionless axial coordinate Greek symbols porosity rdensity (kg/m3) gbinary parameter qdimensionless temperature mdynamic viscosity (kg/m s) Subscripts ccold eeffective hhot iinner mmean oouter pporous wwall K. Milani Shirvan et al. / International Journal of Thermal Sciences 121 (2017) 124e137125 2. Problem statement The schematic geometry of studied double pipe heat exchanger fi lled with porous medium is indicated in Fig. 1. According to Fig. 1, ri, roand rpare inner, outer and porous ra- diuses, respectively and L is the length of the heat exchanger. The hot and cold fl ows are assumed to be laminar, steady and incom- pressible with uniform velocity and temperature at the inlet of the heat exchanger; and the top and bottom walls are adiabatic. In addition, the hot and cold fl uids are considered to enter the inner cylinder, porous layer and annular gap, with uniform velocities of uh, upand ucand constant temperatures of Th, Tpand Tc, respec- tively. To formulate the model some assumptions are made as follows: - The 2-D, fl ow is incompressible, laminar and steady, with no internal heat generation and neglecting viscous dissipation. - The porous medium is assumed to be homogeneous, isotropic and saturated with a single phase fl uid. - The thermo-physical properties of the solid matrix and the fl uid are assumed to be constant. - The fl uid and solid phases in porous region are assumed to have the same temperatures (i.e., local thermal equilibrium, LTE ex- ists). However this assumption is valid until there is no signifi - cant temperature difference between the fl uid and solid phases 21. - The thermal resistance of the inner cylinder is neglected since, it is assumed that its thickness is very small 22. - To combine the viscous and inertia effects, the fl ow is modeled using the Brinkman-Forchheimer extended Darcy model in the porous layer; utilizing the Navier-stokes equations in the fl uid domain, and the thermal fi eld by the energy equation. 3. Governing equations According to above assumptions and the schematic shown in Fig. 1, the governing equations must be solved for two zones: the clear fl uid; and the porous medium. The following dimensionless variables are assumed in order to obtain the dimensionless form of the governing equations as follows: X x Dh;R r Dh;U u uc;V v uc;P p pcu2 c ;q T ? Tc Th? Tc;Dh 2?r0? ?r i rp ? (1) here Dhis the hydraulic diameter, ucis the inlet axial velocity of the cold fl uid and Th is hot fl uid temperature at the heat exchanger input. Continuity equation: vU vX 1 R vRV vR 0(2) Momentum equations: ? g ? 1 2 ? 1 ? 1 ? U vU vX V vU vR ? ?vP vX 1 Re v2U vX2 1 R v R ? R vU vR ?! ? g DaRe U ? gCF ffiffi ffiffi ffiffi Da p ? ?V !? ?U (3) ? g ? 1 2 ? 1 ? 1 ? U vV vX V vV vR ? ?vP vR 1 Re v2V vX2 1 R v R ? R vV vR ?! ? g DaRe V ? gCF ffiffi ffiffi ffiffi Da p ? ?V !? ?V (4) Energy equation: U vq vX V vq vR 1 RePr gRk? 1 1 v2q vX2 1 R v R ? R vq vR ?! (5) wheregis a binary parameter, which can take the value of 0 in the clear zone (hot and cold fl uid), and 1 in the porous zone; and ? ?V !? ? ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi U2 V2 p . In addition, Forchheimer coeffi cient CF is calcu- lated using the following equation: CF 1:75 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1502 p(6) The dimensionless parameters of the above mentioned gov- erning equations are: Re ucrcDh mc ;Da K D2 h ;Rk ke kc (7) The local Nusselt number along the outer wall of the inner cylinder is calculated to investigate the effect of the porous layer on heat transfer as follows: Nux hDh kc kevq vR ? ? ? ?RR i qw?qm (8) hereqmis the dimensionless mean temperature, which is calcu- lated by Ref. 22: qm Z Ro Ri jUjqRdR Z Ro Ri jUjRdR (9) Fig. 1. The schematic geometry of double pipe heat exchanger fi lled with porous medium. K. Milani Shirvan et al. / International Journal of Thermal Sciences 121 (2017) 124e137126 In addition, the dimensionless mean velocity is obtained using the following Eq. (22): Um Z Ro Ri jUjRdR Z Ro Ri RdR (10) The mean Nusselt number is obtained from: Num 1 L ZL 0 NuXdX(11) More details about the governing equations can be found in Ref. 38. 3.1. Boundary conditions As shown in Fig. 1, a uniform fl ow is assumed at the heat exchanger entrance as follows: 8 : 0RRi: U ?r 0 ri ?2 ? ?r p ri ?2 ? 1;V 0;q 1 RiRRp: U r2 0 r2 p? r 2 i ? 1;V 0;q 0 Rp : 0RRi: vU vX 0;V 0; vq vX 0 RiRRp: vU vX 0;V 0; vq vX 0 RpRR0: vU vX 0;V 0; vq vX 0 (13) Axisymmetric conditions are used at the axis of the heat exchanger, given from: vU vR 0; V 0; vq vR 0(14) Moreover, no-slip condition is applied at the heat exchanger walls as the velocity boundary conditions, and is given from the following relations: At the inner cylinder wall: R Ri: U 0; V 0; vq vR ? ? ? ?h kevq vR ? ? ? ?p (15) At the outer cylinder wall: R Ro: U 0;V 0; vq vR 0(16) The boundary conditions at the interface between the porous and the clear fl uid zones is considered by several earlier published papers, such as Gobin and Goyeau 23, Chandesrisand Jamet 24,Valdes-Parada et al. 25,26, Goyeau 27, Ochoa-Tapia and Whitaker 28,29 and Rashidi et al. 30. At the interface between the porous and the clear fl uid zones, the coupling conditions are assumed, which follow the continuity of the pressure, velocity components, stresses, temperatures and heat fl uxes. The continuity of velocity components at the interface between the porous and the clear fl uid zones is given as following: uf up; vf vp(17) Here f and p are indicators of fl uid and porous zones, respectively. The continuity of the shear stress at the interface is also given by: mfvUf vR mevUp vR (18) In Eq. (18), at the interface region, it is assumed that the velocity and the shear stress in the porous medium and the fl uid are equal 31. me is an artifi cial quantity, which indicates the porous medium effective viscosity; and is related to the momentum equation Brinkman term. A minor effect has been reported by Alazmi and Vafai 31 on the velocity distribution due to signifi cant variations in the effective viscosity (from 1mfto 7.5mf). They also observed no changes in the Nusselt number and temperature due to the changing of effective viscosity, even in this wide range 31. As a result, in present study,meis set equal tomf. In fact, this equality is a good approximation in the range of 0.7 1; and has beenwidely used in other studies 32e37. In addition, it must be noted that, to model the water as the working fl uid, the Prandtl number (Pr) is assumed to be fi xed at 7 in all computations. The heat exchanger effectiveness is determined as the ratio of the actual rate of heat transfer (Qa) to the maximum possible heat transfer rate (Qmax) in the heat exchanger from Ref. 22: E Qa Qmax ?m$C p ? cqoc ?qic Cminqih?qoh (19) hereqocandqoh are the dimensionless cold and hot fl uids tem- peratures at the heat exchanger outlet. Also, it must be noted that m$ h m$ cand Cmin minm$Cpc;m$Cph?. It must also be noted that the commercial software ANSYS FLUENT is employed to perform the simulations. 4. Numerical solution and validation The governing equations are discretized numerically utilizing the fi nite volume method by considering of above mentioned boundary conditions and an axisymmetric problem. In addition, using the second order upwind scheme, the convection term in the governing equations is discretized and the velocity and pressure fi elds are coupled utilizing the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm. To discretize the diffusion terms the GreeneGauss is used. The convergence criterion of the summation residual has been assumed to be less than10?7. More- over, discretization of the c