Numerical investigation on self-coupling heat transfer in a counter-flow double-pipe heat exchanger filled with metallic foams

Applied Thermal Engineering 66 (2014) 43e54

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Numerical investigation on self-coupling heat transfer in a counter-flow double-pipe heat exchanger filled with metallic foams H.J. Xu, Z.G. Qu*, W.Q. Tao MOE Key Laboratory of Thermo-Fluid Science and Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China

h i g h l i g h t s
Model with self-coupling heat transfer is established.
Fully-developed region in counter-flow heat exchanger with metallic foams is identified.
Applicable condition for local thermal non-equilibrium model is presented.
Optimal porosity and pore density ranges are obtained for higher effectiveness.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 23 July 2013 Accepted 24 January 2014 Available online 4 February 2014

The self-coupling heat transfer in a counter-flow double-pipe heat exchanger filled with metallic foams is numerically investigated. The Forchheimer extended Darcy equation with a quadratic term is adopted for the momentum equation, whereas the local thermal non-equilibrium model is applied for establishing energy equations with thermal dispersion. The domain-extension method, pressure correction near the porous-solid interface, and the large coefficient method are specially employed for the porous/fluid-wallporous/fluid coupling problem. The velocity and temperature fields of solid and fluid are obtained. The effects of various parameters on pressure drop per unit length, heat transfer coefficient, and heatexchanger effectiveness are also presented. The thermally fully developed region is located in the middle section of the heat exchanger, where the local convective heat coefficient is unalterable. Effectiveness can be improved by decreasing porosity, increasing pore density, or increasing the foam solid thermal conductivity. The applicable range of the thermal conductivity ratio for the local thermal equilibrium model is kf1/ks1 > 102. The local thermal non-equilibrium model should be adopted when kf1/ks1 < 103. Ranges for porosity (less than 0.9) and pore density (greater than 10 PPI) are recommended to ensure higher effectiveness (greater than 0.8). Ó 2014 Elsevier Ltd. All rights reserved.

Keywords: Self-coupling Double-pipe heat exchanger Metallic foam Counter-flow

1. Introduction Metallic foams are excellent candidate materials for exchanging a large amount of heat within a small volume for their advantages such as low density, high specific surface area, high solid thermal conductivity, and strong flow-mixing capability. Potential applications of metallic foams in thermo-fluid engineering include solar collectors, fuel cells, heat sinks, heat exchangers, and so on. Over the last two decades, thermal transport in metallic foams has been theoretically and practically developed. In particular, the development of the cosintering technique and new manufacturing methods for metallic foams facilitated the use of the new heat transfer enhancement material in compact heat exchangers, thus motivating the research on convective heat transfer in metallic foams. * Corresponding author. Tel./fax: þ86 029 82668036. E-mail address: [email protected] (Z.G. Qu). 1359-4311/$ e see front matter Ó 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2014.01.053

The convective heat transfer in metallic foams has been studied to some extent [1e8]. Lu et al. [1] theoretically investigated the flow and heat transfer in metallic foams based on the fin analysis theory, in which the structure of the metallic foam is simplified as a cubic with inter-connected cylinders. Calmidi and Mahajan [2] performed experimental and numerical studies on forced convection in parallel-plates filled with metallic foams and obtained good agreement between the findings by adjusting some empirical parameters. Zhao et al. [3] conducted similar experimental and numerical studies on forced convection in a rectangular duct filled with metallic foams as well as a parameter study on flow and heat transfer. Calmidi [4] performed an experimental study on heat conduction in metallic foams filled with air and water and presented a large amount of experimental data on the effective thermal conductivity of metallic foams. Based on the experimental data, Boomsma and Poulikakos [5] considered the metallic foam cell as a three-dimensional tetrakaidecahedron and developed an analytical

44

H.J. Xu et al. / Applied Thermal Engineering 66 (2014) 43e54

Nomenclature a asf c CI Da df dp h hsf k kd kf ks K L Nu Num p P Pr q r r1 r2 r3 R Re Red T Tfin1 Tfin2 u,v

thermal diffusivity, m2 s1 specific surface area, m1 specific heat capacity, J kg1 K1 inertial coefficient Darcy number fiber diameter, m pore diameter, m convective heat transfer coefficient, W m2 K1 local convective heat transfer coefficient, W m2 K1 thermal conductivity, W m1 K1 thermal dispersion conductivity, W m1 K1 thermal conductivity of fluid, W m1 K1 thermal conductivity of solid, W m1 K1 permeability, m2 length of double-pipe heat exchanger, m Nusselt number mean Nusselt number pressure, N m2 dimensionless pressure Prandtl number heat flux, W m2 radial position, m inner radius of inner tube, m outer radius of inner tube, m inner radius of outer tube, m dimensionless r coordinate Reynolds number pore Reynolds number temperature, K inlet temperature for fluid in inner pipe, K inlet temperature for fluid in annular duct, K x,r velocity components, m s1

correlation of effective thermal conductivity of metallic foams. Zhao et al. [6] performed an experimental study on natural convection in a vertical cylindrical enclosure filled with metallic foams and then numerically investigated the local thermal nonequilibrium effect under experimental conditions. Xu et al. [7] theoretically investigated the forced convective heat transfer in a channel filled with metallic foams using analytical, numerical, and fin-analysis methods. Phanikumar and Mahajan [8] conducted experimental and numerical work on natural convection in the metallic foam layer placed in an open enclosure. They highlighted that the local thermal non-equilibrium effect is significant [6e8]. Overall, the use of metallic foam as heat transfer enhancement materials is feasible, but at the expense of significant pressure loss. For the low pressure-drop flow in an enhanced metallic-foam duct, the configuration of a partially filled metallic-foam duct is recommended [9e12]. Based on basic duct structures, metallic foams can be employed in compact heat exchangers by sintering on solid plates or tubes. The conjugation of the heat transfer on both sides of the solid wall has to be well handled for fluid dynamics computation, which is a special porous/solid conjugate problem. Early works on convective heat transfer in porous media employed the local thermal equilibrium model. Alkam and Nimr [13] and Targui and Kahalerras [14] respectively performed numerical simulations on the thermal performance of a double-pipe heat exchanger filled with porous substrates and of a double-pipe heat exchanger filled with porous baffles employing the local thermal equilibrium model, which neglected the thermal resistance of the tube wall. Lee and Vafai [15]

um1 um2 umr U,V x X

mean velocity of fluid in inner pipe, m s1 mean velocity of fluid in annular duct, m s1 mean velocity ratio dimensionless velocity components in X,R direction axial position, m dimensionless x coordinate

Greek symbols binary parameter 3 porosity 3 effectiveness q dimensionless temperature m dynamic viscosity, kg m1 s1 r density, kg m3 rr density ratio f heat, W fm1 heat transfer at the inner surface of the inner tube, W fm2 heat transfer at the outer surface of the inner tube, W u pore density (0.0254/dp), PPI (pores per inch)

g

Subscripts e effective f fluid I inertial int interface m mean nb neighbor p porous P point r relative s solid w wall 1 physical quality for the inner pipe 2 physical quality for the annular space

indicated that the local thermal equilibrium assumption is inaccurate when the thermal conductivity difference between solid and fluid is large, and the thermal performance for convective heat transfer should be addressed using the local thermal nonequilibrium model. Lee et al. [16] and Betchen and Straatman [17] theoretically proposed a discretizing method with the local thermal non-equilibrium model for the flow and heat transfer at the porous-solid interface, but the implementation of this method was not presented. Lu et al. [18,19] obtained analytical solutions for the fully developed forced convective heat transfer in a double-pipe heat exchanger filled with metallic foams, but the dividing-wall heat flux is artificially set as uniform without the quadratic and thermal dispersion terms. Du et al. [20] numerically studied the heat transfer performance of a double-pipe heat exchanger with parallel flow with local thermal non-equilibrium model, but the heat flux along the tube was derived from the fluid temperature without considering the internal temperature distribution. Based on above review, the effect of the solid wall is usually neglected, and the practical thermal coupling process of the two sides of the fluid separated by the interface wall under the local thermal non-equilibrium model can rarely be observed for the numerical modeling of the heat exchanger filled with porous medium. To this end, by considering the local thermal nonequilibrium, thermal dispersion, and tube-wall heat conduction effects, the conjugate flow and heat transfer in double-pipe heat exchangers filled with metallic foams with counter flow is numerically simulated. Flow characteristics, thermal performance, and effectiveness are presented.

H.J. Xu et al. / Applied Thermal Engineering 66 (2014) 43e54

rf1

2. Numerical method

3

2.1. Problem description A two-dimensional numerical simulation of the cylindrical coordinates on conjugate flow and heat transfer in counter-flow double-pipe heat exchangers filled with metallic foams is conducted. The schematic configuration of the counter-flow doublepipe heat exchangers filled with metallic foams is shown in Fig. 1(a), whereas the computational domain is shown in Fig. 1(b). The inner radius of the inner tube is r1, whereas the outer radius of the inner tube is r2. The inner radius of the outer tube is r3, whereas the thickness of the outer tube is excluded in the numerical simulation. The outer surface of the double-pipe heat exchanger is adiabatic. Across the tube-wall, heat is transferred from the hot fluid flowing through the annular duct to the cold fluid flowing through the inner pipe in the opposite direction.


u

2 1

45

"

  vu 1 vðruÞ vp m v2 u 1 v vu þv ¼  þ f1 r þ vx r vr vx vr 31 vx2 r vr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p rf 1 CI1 ffiffiffiffiffiffi u2 þ v2 u:  p K1

# 

mf1 K1

u

(1b)

rf1 3


u

2 1

vv 1 vðrvÞ þv vx r vr



"  # m vp mf1 v2 v 1 v vv þ r  f1 v þ vr vr 31 K1 vx2 r vr ffi rf1 CI1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffi u2 þ v2 v: K1

¼ 

(1c) " kse1

 #   v2 Ts 1 v vTs r T ¼ 0:  h þ a  T s sf1 sf1 f r vr vr vx2

(1d)

 3 "  #   v2 T v rTf vT 1 1v vTf f f 5 4 rf1 cf1 u r ¼ kfe1 þ kd1 þv þ r vr r vr vx vr vx2   þ hsf1 asf1 Ts  Tf : 2

2.2. Governing equations and boundary conditions Momentum transport in porous media can be described with Darcy model, Brinkman model or Forchheimer model while thermal transport in porous media can be described with local thermal equilibrium model (one-equation model) or local thermal nonequilibrium model (two-equation model). By considering the quadratic term for flow, thermal dispersion effect, and local thermal non-equilibrium effect for temperature, the Brinkmane Frochheimereextended Darcy equation is employed for momentum transfer, and a two-equation model is adopted for energy transport. The dimensional governing equations can be obtained as follows:

(1e) (2) In the tube wall region (r1 < r < r2),

" ks

 # v2 T 1 v vT r ¼ 0: þ vr vx2 r vr

(1f)

(1) In the inner tube region (0 < r < r1), (3) In the annular region (r2 < r < r3),

vu 1 vðrvÞ þ ¼ 0: vx r vr

(1a)

annulus (hot) r2

r1

vu 1 vðrvÞ þ ¼ 0: (1g) vx r vr # "   
rf2 vu m 1 vðruÞ vp mf2 v2 u 1 v vu þ v ¼  þ r  f2 u u þ 2 2 vx r vr vx r vr vr 3 K2 vx 3 2 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p rf 2 CI2 ffiffiffiffiffiffi u2 þ v2 u:  p K2

inner tube (cold)

(1h)

r3

rf2 3

L




2 2

u

vv 1 vðrvÞ þv vx r vr

(a)Schematic diagram



"  # m vp mf2 v2 v 1 v vv þ r  f2 v þ vr vr 32 K2 vx2 r vr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p rf2 CI2 ffiffiffiffiffiffi u2 þ v2 v:  p K2

¼ 

(1i)

annulus (hot)

r2 inner tube (cold)

r3

r1

" kse2

 #   v2 Ts 1 v vTs r  hsf2 asf2 Ts  Tf ¼ 0: þ 2 r vr vr vx

(1j)

 3 "  #   v2 T v rTf vT 1 1 v vT f f f 5 ¼ kfe2 þ kd2 rf2 cf2 4u r þv þ r vr r vr vx vr vx2   þ hsf2 asf2 Ts  Tf : 2

L (b) Computational domain Fig. 1. Double-pipe heat exchanger filled with metallic foams for counter-flow arrangement.

(1k)

46

H.J. Xu et al. / Applied Thermal Engineering 66 (2014) 43e54

The digital subscript “1” represents the physical quality for the inner tube, whereas “2” represents that in the annular space. Porosity and pore density are two basic morphological parameters for metallic foams. Porosity 3 is defined as the ratio of the void volume to the total volume, whereas pore density u is defined as the number of pores per linear inch (PPI). According to these two parameters, the correlations of the permeability K, inertial coefficient CI, specific surface area asf, local heat transfer coefficient hsf, effective thermal conductivities of fluid and solid (kfe, kse), and thermal dispersion conductivity kd can be obtained from Table 1. The values of constant parameters in Eqs. (1a)e(1k) can be found in Table 2. The corresponding default values for porosity, pore density, Reynolds number, and solid thermal conductivity are 0.9, 10 PPI, 1000, and 16.3 W/(m K) (taking stainless steel as the typical material), respectively. In the parameter study, any value of the above variables can be changed. The boundary conditions for the computational domain shown in Fig. 1(b) are listed below.

Table 2 Constant parameters. Value

Unit

cf1 kf1

1005 0.0259 1.205 1.81  105 1005 0.0283 1.093 1.96  105 0.01 0.012 0.022 20 50 0.4

J/(kg K) W/(m K) kg/m3 kg/(m s) J/(kg K) W/(m K) kg/m3 kg/(m s) m m m  C  C m

rf1 mf1

cf2 kf2

rf2 mf2

r1 r2 r3 Tfin1 Tfin2 L

x ¼ L;

(1) At the boundary for inlet of inner-pipe and outlet of annulus,

8 s > 0 < r < r1 : u ¼ um1 ; v ¼ 0; vT > vx ¼ 0; Tf ¼ Tfin1 > < vTf s : x ¼ 0; r1 < r < r2 : u ¼ v ¼ 0; vT vx ¼ vx ¼ 0 > > > : vTf vTs vu vv r2 < r < r3 : vx ¼ vx ¼ vx ¼ vx ¼ 0

Parameter

8 > 0 < r < r1 : vu > vx ¼ > < > > > :

vTs vx

r1 < r < r2 : u ¼ v ¼ 0;

vTf vx

¼

vTs vx

¼

r2 < r < r3 : u ¼ um2 ; v ¼ 0;

¼ 0 vTf vx vTs vx

:

¼ 0 ¼ 0; Tf ¼ Tfin2

(5) (2) To identify the dimensionless parameters that characterize the thermal transport process in metallic foams, the above governing equations are normalized using the following qualities. The dimensionless variables are:

x r u v ;R ¼ ;U ¼ ;V ¼ ; r1 r1 um1 um1 p ¼ : rf 1 u2m1

X ¼ vu vTs vTf ¼ 0; v ¼ 0; ¼ ¼ 0: vr vr vr

¼

2.3. Normalization

(2) At the center line,

r ¼ 0;

vv vx

(3)

qfðsÞ ¼

TfðsÞ  Tfin1 Tfin2  Tfin1

;P

(6) (3) At the boundary of outer-tube with the adiabatic condition,

Some other important dimensionless parameters are:

R2 ¼

vTs vT þ kfe2 f ¼ 0; Ts ¼ Tf : r ¼ r3 ; u ¼ v ¼ 0; kse2 vr vr

(4) ¼

r r2 r u k ; R ¼ 3 ; umr ¼ m2 ; rr ¼ f 2 ; afe1 ¼ fe1 ; ad1 rf 1 r1 3 r1 um1 kf1 kd1 k k ks ;a ¼ fe2 ; ad2 ¼ d2 ; as ¼ kf 1 fe2 kf2 kf2 kse1 (7a)

(4) At the boundary for outlet of inner-pipe and inlet of annulus, Table 1 Correlations of foam parameters. Parameter

Correlation ðdf =dp Þ1:11 d2p 0:00073ð1  3 Þ pffiffiffiffi 0:132 0:00212ð1  3 Þ ðdf =dp Þ1:63 = K

Permeability, K

K ¼

Inertial coefficient, CI

CI ¼

Specific surface area, asf

asf ¼

Local heat transfer coefficient, hsf hsf

Reference 0:224

3pdf ½1eðð13 Þ=0:04Þ  ð0:59dp Þ2

Effective thermal conductivity, ke

[18]

8 > 0:76Re0:4 Pr 0:37 kf =df ; > d > < ¼ 0:52Re0:5 Pr 0:37 kf =df ; d > > > : 0:26Re0:6 Pr 0:37 k =d ; d

[4] [4]

f

f

[18] ð1  Red  40Þ

40  Red  103

3 10  Red  2  105

ke ¼

1 pffiffiffi 2ðRA þRB þRC þRD Þ

RA ¼

4l ð2e2 þplð1eÞÞks þð42e2 plð1eÞÞkf

ðe2lÞ2 ðe2lÞe2 ks þð2e4lðe2lÞe2 Þkf pffiffiffi 22e pffiffiffi 2 RC ¼ 2 2pl ks þð2 2pl Þkf 2e RD ¼ e2 k þð4e 2 Þk s f rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffi pffiffiffi 3 2ð2ð5=8Þe pffiffiffi 223 Þ; l ¼ e ¼ pð34 2eeÞ

[5]

RB ¼

Thermal dispersion conductivity, kd

kse ¼ ke jkf ¼0 ; kfe ¼ ke jks ¼0 pffiffiffiffi kd ¼ CD rf cf K u; CD z0:1

0:16

[4]

H.J. Xu et al. / Applied Thermal Engineering 66 (2014) 43e54

ase2 ¼ ¼

23 2 umr ðR3 R2 Þ 1 vðRVÞ vP U vV Re2 vX þ V R vR ¼ rr vR þ

hsf1 asf 1 r12 hsf2 asf2 r12 kse2 ; Bioe1 ¼ ; Bioe2 ¼ ; Nusf1 kse1 kse1 kse2



hsf1 asf1 r12 h a r2 ; Nusf2 ¼ sf2 sf2 1 kf1 kf2 (7b)

m c m c K K Da1 ¼ 21 ; Da2 ¼ 22 ; Pr1 ¼ f1 f 1 ; Pr2 ¼ f2 f 2 : Re1 kf 1 kf 2 r1 r1 r $u $2r1 r $u $2ðr3  r2 Þ ¼ f 1 m1 ; Re2 ¼ f 2 m2 mf1 mf2

(7c)

In this paper, the values of R1, R2, and R3 are respectively 1, 1.2, and 2.2 in the numerical calculation. By substituting these dimensionless qualities into the dimensional formulas of Eqs. (1)e(5), the dimensionless governing equations can be obtained as follows: (1) In the inner tube region,

vU 1 vðRVÞ þ ¼ 0: vX R vR

(8a)

"  # vU 1 vðRUÞ 23 1 v2 U 1 v vU 2 vP þV ¼ 3 1 þ R U þ vX R vR vX Re1 vX 2 R vR vR 

2 23 21 3 CI1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U  p1ffiffiffiffiffiffiffiffiffi U 2 þ V 2 U Re1 Da1 Da1

2 23 21 3 CI1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V  p1ffiffiffiffiffiffiffiffiffi U 2 þ V 2 V Re1 Da1 Da1

    v2 qs 1 v vqs R  Bioe1 qs  qf ¼ 0 þ 2 R vR vR vX

C

I2 ffi  p2ffiffiffiffiffiffi Da

3

2

(8i)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U2 þ V 2V

qf þ V R1 U vvX

vðRqf Þ vR

¼ 2

ðafe2 þad2 Þumr ðR3 R2 Þ Re2 Pr2

h

v2 qf vX 2

 umr ðR3 R2 Þ qs  qf þ2 Nusf2 Re 2 Pr2

 i v R vq f þ R1 vR vR 

(8j)

(8k)

The dimensionless boundary conditions are as follows: (1) At the boundary for inlet of inner-pipe and outlet of annulus,

8 qs ¼ 0; q ¼ 0 > > 0 < R < R1 : U ¼ 1; V ¼ 0; vvX f > > < q v q v X ¼ 0; R1 < R < R2 : U ¼ V ¼ 0; vXs ¼ vXf ¼ 0 : > > > > : R < R < R : vU ¼ vV ¼ vqs ¼ vqf ¼ 0 2 3 vX vX vX vX

(9)

(8c)

vU vqs vq ¼ 0; V ¼ 0; ¼ f ¼ 0: vR vR vR

(10)

(3) At the boundary of outer-tube with the adiabatic condition,

R ¼ R3 ; U ¼ V ¼ 0;

vqs kfe2 vqf þ ¼ 0; qs ¼ qf : vR kse2 vR

(11)

(4) At the boundary for outlet of inner-pipe and inlet of annulus,

(8e)

8 vq f vq s vV > > 0 < R < R1 : vU > vX ¼ vX ¼ vX ¼ vX ¼ 0 > L < qf qs X ¼ ; R1 < R < R2 : U ¼ V ¼ 0; vvX : ¼ vvX ¼ 0 r1 > > > > q : R < R < R : U ¼ u ; V ¼ 0; v s ¼ 0; q ¼ 1 mr 2 3 f vx (12)

"  # v2 q s 1 v vqs as R ¼ 0 þ vR vX 2 R vR

(8f)

(3) In the annular region,

vU 1 vðRVÞ þ ¼ 0 vX R vR

(8g)

2

23 2 umr ðR3 R2 Þ 1 vðRUÞ 2 vP U vU Re2 vX þ V R vR ¼ rr vX þ 23 2 umr ðR3 R2 Þ  2 Re U 2 Da2

v V vX 2



i v R vV þ 1R vR vR

"  #   v2 q s 1 v vqs R  Bioe1 qs  qf ¼ 0 þ vR vX 2 R vR

R ¼ 0;

(2) In the tube wall region,

3

2

2

(8b)

(8d)

  "  # afe1 þ ad1 v2 qf 1 v vqf 1 v Rqf vqf R ¼ 2 þV þ U R vR vX Re1 Pr1 vR vX 2 R vR   Nusf1 qs  qf þ2 Re1 Pr1

23 22 umr ðR3 R2 Þ V Re2 Da2

h

(2) At the center line,

"  # vV 1 vðRV Þ 23 1 v2 V 1 v vV 2 vP þV ¼ 3 1 þ R þ U vX R vR vR Re1 vX 2 R vR vR 

47 2 3 2

h

v2 U vX 2

2

2.4. Parameter definitions The Reynolds numbers for fluid flow in the inner and outer pipes are:



i v R vU þ R1 vR vR

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi U2 þ V 2U  ffiffiffiffiffiffi Da 2 C p2 I2

3

The thermal conductivities of the solid phases in the two foam regions and the thermal conductivity of the tube wall may differ from one another. Moreover, the parameter difference (porosity and pore density) between the inner tube and the annular space is considered in the present numerical model and may facilitate flexible thermal design for tube-in-tube heat exchangers with different foam parameters in both sides of the tube wall.

(8h) Re1 ¼

rf1 $um1 $2r1 mf 1

(13a)

48

H.J. Xu et al. / Applied Thermal Engineering 66 (2014) 43e54

Re2 ¼

rf2 $um2 $2ðr3  r2 Þ mf 2

(13b)

The pressure drop per unit length for the inner-pipe and annulus spaces can be calculated as

dp=dx ¼

pin  pout L

(14)

The convective heat transfer coefficients on the inner and outer surfaces of the inner pipe are given by

hm1 ¼

hm2 ¼



fm1

A1 Twm1  Tfb1 

fm2

A2 Tfb2  Twm2



(15a)



(15b)

Taking the outer surface of the inner pipe as a benchmark, the overall heat transfer coefficient across the tube wall can be calculated by Ref. [21]

  1 1 1 r 1 ¼ þ ln 2 þ k2 A2 hm1 A1 2pLks hm2 A2 r1

(16)

In the practical design and verification of a conventional lowtemperature heat exchanger, the main task is to determine the effectiveness using the heat exchange unit method. The effectiveness of heat exchangers can be defined as

3

¼

  i h max Tfin2  Tfout2 ; Tfout1  Tfin1 Tfin2  Tfin1

(17)

For counter-flow double-pipe heat exchangers, the effectiveness can be expressed as 3

¼

1  eNTUð1Cr Þ 1  Cr eNTUð1Cr Þ

(18)

where Cr and NTU can be obtained with

h

i min prf1 um1 r12 cf 1 ;prf2 um2 r32  r22 cf2 h Cr ¼

i max prf1 um1 r12 cf 1 ;prf2 um2 r32  r22 cf2 NTU ¼

k2 A2 h

i 2 min prf 1 um1 r1 cf1 ;prf2 um2 r32  r22 cf 2

(19a)

(19b)

2.5. Special numerical treatment The steady SIMPLE algorithm [22] with a self-developed FORTRAN code is used for solving the governing equations with the full-field method. The SGSD scheme [23] is adopted to discretize the advection terms of the momentum and fluid energy equations. The control volume is divided using the staggered grid. The grid independence check is implemented as shown in Table 3 and 252  52 is selected as the final grid system for numerical simulation. To model the flow and non-equilibrium heat transfer in the porous-solid system numerically, some special numerical treatments are developed. For the solid temperature equation in Eqs. (8d), (8f), and (8j), the corresponding velocity components are set as zero before iteration because of the absence of advective terms. The harmonic mean method is used to obtain the diffusive

coefficient for the momentum and energy equations along with the porous/solid interface. Velocity and fluid temperature cannot be observed in the solidwall region and are extended to this region using the domain extension method. To ensure that the velocity of the solid-wall region is zero and the fluid temperature at the porous fluid interface is equal to the solid temperature, the velocity is firstly forced to set as zero, whereas the fluid temperature is compulsorily set as the value of the solid temperature before iteration. Secondly, the large coefficient method [22] is adopted to ensure zero velocity and equality between the fluid and solid temperatures in the solid-wall region during the iteration process. In the solid-wall region, pressure is meaningless. To mitigate the effect of solid-wall pressure on the fluid pressure, the pressure coefficient for the marginal control volume in the solid wall is set to zero in the pressure correction equation. For the two porous-solid interfaces as shown in Fig. 2, the pressure correction for representative control volumes of P1 and P2 can be written as 0 0 0 0 aP1 PP1 ¼ aE1 PE1 þ aW1 PW1 þ aS1 PS1 þb

ðaN1 ¼ 0Þ: (20a)

0 0 0 0 ¼ aE2 PE2 þ aW2 PW2 þ aN2 PN2 þb aP2 PP2

ðaS2 ¼ 0Þ: (20b)

where a is the coefficient for control volume in the discretized pressure correction equation, and P’ is the pressure correction. E1, W1, N1, and S1 are the four neighboring control volumes of P1, whereas E2, W2, N2, S2 are the four neighboring control volumes of P2. Thus, the pressure correction in representative control volumes of P1 and P2 can be obtained through the iteration without the effect of pressure in solid wall. Owing to the employment of the local thermal non-equilibrium model, two temperatures respectively for solid and fluid exist in the porous domain, while only one solid temperature exists in the tube wall region. To conveniently handle these variables in the whole computational domain and match the non-equilibrium model for the tube wall, the fluid temperature is extended to the tube-wall region, which is called the pseudo fluid temperature. The coefficients in advective term, the diffusive term and the source term were all set as zero for the pseudo fluid temperature equation in the tube-wall region. The pseudo fluid temperature in the tube-wall region was artificially made equal to the solid temperature, which is used to calculate the discretized coefficient of fluid temperature equation near the porous-solid interface. To make the pseudo fluid temperature in the tube-wall region equal to the solid temperature, the large coefficient method can be used, as shown below.

aP qf;P ¼

X

anb qf ;nb þ b:

(21a)

aP ¼ A; b ¼ A$qs;P ; A/ þ N

(21b)

In Eq. (21), the coefficient is aP ¼ A and the source term is b ¼ A  qs,P, where A is a very positively large number. Thus, the Table 3 Grid independence check. Grid density 102 152 252 352

   

22 32 52 72

k2 62.357 66.822 68.967 69.918

Relative error of k2

3

Relative error of

6.68% 2.50% 1.42%

0.766 0.788 0.797 0.801

2.73% 0.96% 0.53%

3

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porous region W2

upper interface

49

N2 P2

E2

S2

tube wall region

radial direction lower interface

N1 W1

porous region

P1

E1

S1 Fig. 2. Schematic diagram of pressure correction near the porous-solid interface. Fig. 3. Comparison between the present numerical model and an analytical solution [18].

pseudo fluid temperature in the tube-wall region is equal to the corresponding solid temperature. 3. Results and discussion 3.1. Validation Due to the lack of experimental data for forced convection in double-pipe heat exchangers, the computational results on condition without inertial term are compared with the analytical solution developed by Lu et al. [18] to verify the numerical model used in this study. Fig. 3 shows the comparison of velocity profiles obtained with the present numerical model and the analytical solution of Lu et al. [18], in which the quadratic term of the numerical work is dropped to approach the BrinkmaneDarcy model used in the analytical solution. The numerical result without inertial term agrees well with the analytical solution, from which the numerical method is analytically verified. 3.2. Velocity distribution and pressure drop characteristics Fig. 4 shows the velocity profile of the metallic foam filled double-pipe heat exchangers. The foam parameters of the inner pipe are equal to those of the annulus space. The porosity is set as 0.9, and two different pore densities, 5PPI and 30 PPI, are investigated. The velocity distributions of the metallic foam filled innerpipe and annulus are both uniform because of the presence of foam ligaments which lead to very high flow resistance and very uniform permeability. Simultaneously, the velocity distribution for 30 PPI is more homogeneous than that for 5 PPI, which is attributed to the fact that with the increase in pore density, the permeability of porous foam is reduced and the proportion of tube-wall viscous force in the total flow resistance is reduced. The velocity distribution in the inner pipe exhibits characteristics similar to that in the annulus. Fig. 5(a) and (b) respectively shows the pressure drop per unit length for the representative inner-pipe as functions of porosity and pore density. With a decrease in porosity or an increase in pore density, the pressure drop per unit length increases. This is attributed to the fact that a decrease in porosity can increase the volume fraction of the obstructing solid, whereas an increase in pore density can result in denser foam ligaments, therefore, the permeability of metallic foam is decreased and the flow resistance of foam duct is increased.

3.3. Temperature distribution Fig. 6(a) and (b) respectively shows the temperature contours of solid and fluid. In the tube-wall region (R1 < R < R2), the solid temperature exhibits a linear distribution. From the outer tube radius R3 to the center line, the solid temperature profile is monotonic, whereas the fluid temperature exhibits a peak value near the outer tube wall (R ¼ R3). Although the fluid and solid temperature profiles are not perpendicular to the outer tube wall individually, the total heat flux expressed with fluid temperature gradient and solid temperature gradient at the outer tube wall is zero, as indicated in the adiabatic boundary condition at R ¼ R3 in Eq. (11). Fig. 7 shows the axial distributions of the cross-sectional mean temperatures of the inner fluid, outer fluid, and tube wall. The three temperature profiles are almost linear. The tube-wall mean temperature is between the outer fluid mean temperature and the inner fluid mean temperature. The temperature increment from the inlet and outlet for cold fluid is higher than the temperature decrement for hot fluid because the product of mass flow rate and fluid specific heat for the cold fluid (0.01 W/K) is much lower than that for the hot fluid (0.06 W/K).

Fig. 4. Dimensionless velocity profile in metallic foam filled double-pipe heat exchanger.

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relative difference between solid and fluid temperatures becomes more evident as kf1/ks1 decreases. The difference between solid temperature and fluid temperature is negligible for kf1/ks1 ¼ 102, whereas this difference cannot be neglected for kf1/ks1 ¼ 103. Thus, the local thermal equilibrium model can be applied in the condition of kf1/ks1 > 102 to estimate the heat transfer, whereas, the local thermal non-equilibrium model should be adopted on the condition of kf1/ks1 < 103. 3.4. Local heat transfer coefficient Fig. 10(a) and (b) respectively shows the distributions of the local convective heat transfer coefficient and local wall heat flux along the axial direction. The local heat transfer coefficients firstly decrease sharply near the entrance and are nearly unchanged in the middle of the heat exchanger in the range 3 < X < 7, thus showing that the heat transfer is in the fully developed region. It was noted that higher local heat transfer coefficient was gained at 0 < X < 3 for the inner pipe and at 7 < X < 10 for the annulus because of the entrance effects. This finding implies that the fully developed regions for the two fluids are located in the middle section of the heat exchanger.

Fig. 5. Effects of key parameters on pressure drop per unit length.

Fig. 8 presents the temperature profiles for solid and fluid at four different cross-sections, x/L ¼ 0.2, 0.4, 0.6, and 0.8. From the position x/L ¼ 0.2 to 0.8, the fluid and solid temperatures increase. The solid temperature varies more gently compared with the fluid temperature at each cross-section. In the inner pipe, the fluid temperature is lower than the solid temperature, whereas in the annular space, the fluid temperature is higher than the solid temperature. This finding implies that heat is firstly transferred from the hot water to the foam ligaments in the annular space and then to the inner-pipe foam ligaments through the tube-wall. Finally, heat is transferred from the inner-pipe foam ligaments to the fluid nearby. Fig. 9 shows the effect of thermal conductivity ratio (kf1/ks1) on the temperature profiles of solid and fluid for three thermal conductivity ratios kf1/ks1¼102, 103 and 104. In the numerical calculation, the inner fluid thermal conductivity is fixed as 0.0259 W/(m K), whereas the thermal conductivity ratio (kf1/ks1) can be adjusted by changing the solid thermal conductivity. The temperature range from the center-line (R ¼ 0) to the outer tube wall (R ¼ R3) is significantly reduced when kf1/ks1 changes from 102 to 104. Taking the solid phase as an example, the ranges of dimensionless solid temperature for kf1/ks1 ¼ 102, 103 and 104, are respectively about 0.04e1.00, 0.59e0.95 and 0.66e0.69. The

Fig. 6. Solid and fluid temperature contour line (3 1 ¼ Re1 ¼ Re2 ¼ 2000).

32

¼ 0.9, u1 ¼ u2 ¼ 10 PPI,

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51

hm1  kf1/ks1 curve can be divided into three ranges: kf1/ks1 < 104, 104 < kf1/ks1 < 101 and kf1/ks1 > 101. In the range of kf1/ ks1 < 104, the heat transfer coefficient exhibits a flat trend as kf1/ks1 increases where the conductive thermal resistance is negligible. This finding implies that the convective thermal resistance occupies most of the total thermal resistance and dominates the heat

Fig. 7. Axial distributions for cross-sectional mean temperatures of inner fluid, outer fluid, and the tube wall.

The heat flux shown in Fig. 10(b) can be calculated with the following equation:

qw ¼ ks

Tjr¼r2  Tjr¼r1 r2  r1

(22)

Correspondingly, the local heat flux is relatively high at the two ends of the heat exchanger because of the entrance effect shown in Fig. 10(b). The effects of porosity, pore density, and thermal conductivity ratio on the mean heat transfer coefficient of the representative inner-pipe are respectively shown in Fig. 11(a)e(c). As shown in Fig. 11(a), the heat transfer coefficient gradually decreases as porosity increases, which is attributed to the fact that an increase in porosity can decrease the volume fraction of the solid phase and decrease effective thermal conductivity. As shown in Fig. 11(b), the heat transfer coefficient increases with increasing pore density. This observation can be attributed to the fact that an increase in pore density can improve foam surface area and densify foam ligaments, which facilitates an increase in the velocity gradient near the solid wall, as shown in Fig. 4. As shown in Fig. 11(c), the

Fig. 8. Solid and fluid temperature profiles for at different cross-sections (3 1 ¼ 3 2 ¼ 0.9, u1 ¼ u2 ¼ 10 PPI, Re1 ¼ Re2 ¼ 1000, ks1 ¼ ks2 ¼ 259 W/(m K), ks ¼ 16.3 W/(m K)).

Fig. 9. Effect of solid thermal conductivity on solid and fluid temperature (3 1 ¼ 3 2 ¼ 0.9, u1 ¼ u2 ¼ 10 PPI, Re1 ¼ Re2 ¼ 1000, ks ¼ ks2 ¼ ks1)

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with 3 > 0:8: As shown in Fig. 12(b), the effectiveness monotonically increases from about 0.76 to about 0.84 when pore density varies from 5 PPI to 30 PPI. However, the increasing amplitude of effectiveness gradually decreases. The recommended applicable range for pore density is u > 10 PPI with 3 > 0:8: As shown in Fig. 12(c), the effectiveness curve can be divided into three ranges of

Fig. 10. Local distribution along the axial direction for heat transfer.

transfer process. In the range of 104 < kf1/ks1 < 101, the heat transfer coefficient decreases as the thermal conductivity ratio kf1/ ks1 increases (ks1 decreases), this results from the fact that an increase in the thermal conductivity ratio can increase the conductive thermal resistance and enlarge the temperature difference between solid and fluid. In the range of kf1/ks1 > 101, the heat transfer coefficient becomes flat with an increase in kf1/ks1, in which the temperature difference between solid and fluid is approaching zero, and the contribution of convection is negligible This implies that in the range kf1/ks1 > 101, thermal resistance of conduction dominates the heat transfer process. 3.5. Comprehensive heat transfer performance The effectiveness reflects the heat transfer potential for heat exchangers and the effectiveness of counter-flow double-pipe heat exchanger can be obtained from Eq. (18). The effects of porosity, pore density, and foam solid thermal conductivity on the effectiveness of metallic foam filled double-pipe heat exchangers are presented in Fig. 12(a)e(c). As shown in Fig. 12(a), the effectiveness gradually decreases approximately from 0.95 to 0.4 as porosity increases from 0.8 to 0.975, which is attributed to the fact that an increase in porosity can evidently decrease effective thermal conductivity. The recommended applicable range for porosity is 3 < 0.9

Fig. 11. Effects of key parameters on heat transfer.

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kf1/ks1 < 5104, 5104 < kf1/ks1 < 101, and kf1/ks1 > 101. In the range kf1/ks1 < 5  104, the effectiveness approaches 1 as kf1/ks1 increases. The effectiveness decreases as the thermal conductivity ratio increases (ks1 decreases). When kf1/ks1 > 101 holds, the effectiveness decreases slowly with increasing of kf1/ks1. 4. Conclusions In this work, the numerical model for the porous-solid coupling problem of forced convective heat transfer in a metallic foam fullyfilled double-pipe heat exchanger is established with the local thermal non-equilibrium model. The velocity distribution for metallic foam fully-filled double-pipe heat exchanger is quite uniform and more flattened velocity profiles can be obtained by increasing pore density. The pressure drop can be decreased by increasing porosity or decreasing pore density. The fully developed region for the two fluids locates in the middle section of the heat exchanger, whereas the two locations with higher local heat fluxes are at the two ends of the heat exchanger. The heat transfer coefficient can be improved by decreasing porosity or increasing pore density. The local thermal equilibrium model can be applied on the condition of kf1/ks1 > 102 to estimate the heat transfer, whereas, the local thermal non-equilibrium model should be adopted on the condition of kf1/ks1 < 103. The effectiveness of the heat exchanger can be increased by decreasing porosity or increasing in pore density. The recommended applicable range for porosity and pore density are 3 < 0.9 and u > 10 PPI to guarantee higher effectiveness ð3 > 0:8Þ: Acknowledgements This work was financially sponsored by the National Natural Science Foundation of China (No. 51322604) and the National Key Projects of Fundamental R/D of China (973 Project: 2011CB610306). References

Fig. 12. Effects of key parameters on the effectiveness of the heat exchanger.

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