Correlation studies of hydrodynamics and heat transfer in metal foam heat exchangers

Accepted Manuscript Correlation Studies of Hydrodynamics and Heat Transfer in Metal Foam Heat Exchangers Shaolin Mao, Norman Love, Alma Leanos, Gerardo Rodriguez-Melo PII:

S1359-4311(14)00515-8

DOI:

10.1016/j.applthermaleng.2014.06.035

Reference:

ATE 5742

To appear in:

Applied Thermal Engineering

Received Date: 21 February 2014 Revised Date:

9 June 2014

Accepted Date: 15 June 2014

Please cite this article as: S. Mao, N. Love, A. Leanos, G. Rodriguez-Melo, Correlation Studies of Hydrodynamics and Heat Transfer in Metal Foam Heat Exchangers, Applied Thermal Engineering (2014), doi: 10.1016/j.applthermaleng.2014.06.035. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Correlation Studies of Hydrodynamics and Heat Transfer in Metal Foam Heat Exchangers

Abstract

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Shaolin Mao1, Norman Love, Alma Leanos, Gerardo Rodriguez-Melo Department of Mechanical Engineering, The University of Texas at El Paso, TX 79968, USA

This study presents the correlations of both hydrodynamics and heat transfer in a metal foam heat exchanger. The present work is focused on the application to dry cooling such as air-cooled

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condensers (ACCs). In particular empirical correlations for the permeability, form drag coefficient, friction factor, and the overall heat transfer coefficient for different samples of

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metallic foam have been validated and verified with available experimental data and numerical simulations. The modified correlations used in this study are established through the validation & verification studies of metal foam heat exchangers. In order to address the difference, finned tube heat exchangers are used to compare to the metal foam heat exchangers with the same geometry

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size and layout. For fully wrapped metal foam heat exchangers, the prediction using empirical correlation is consistent with computational fluid dynamics (CFD) simulations. However, the scenarios become complicated for partially wrapped metal foam heat exchangers. The numerical

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results show that there is an optimal choice of the porosity of metal foam in which the wall heat transfer coefficient and pressure drop reach the design goal. Overall, the heat transfer capability

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of metal foam heat exchangers can supersede conventional compact heat exchangers given optimal scenario.

Key words: metal foam, metallic foam, dry cooling, air-cooled condenser, porous medium, Darcy-Forchheimer model

1

Corresponding author. Voice: +1 915 747 5830. E-mail: [email protected], [email protected]

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I.

Introduction

The stochastic or periodic metal and carbon foam materials are considered as one of the

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most promising types of methods that can enhance heat transfer through increases in surface area, structural strength, and contains many multifunctional capabilities. For example, the high ratio of contact area to volume, embedded tortuosity, and outstanding heat transfer capability make the

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bulk glass metallic foam one of the best candidates to replace traditional finned tube heat exchangers with great potential in dry cooling for industrial power plants (Conradie and Kroger

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1996). In some cases this type of technology may partially or completely supersede the current wet cooling tower technologies used in thermal power plant (Shi and Bushart 2012; Rich Aull 2012; Hooman and Gurgenci 2010). There is a great interest in research and industrial applications of bulk cellular foams in the areas of nuclear waste storage, impact reduction,

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medical devices, environmental protection, and aerospace exploration (Lu 2002; Boomsma et al. 2003). For more details about thermal-hydraulic transport in high porosity cellular ceramic and metallic foam materials, a recent state-of-the-art knowledge review article by Zhao (2012)

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presents a good overview. There are also huge resources about foam material properties which can be obtained directly from manufacturers.

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In the last decade, heat transfer and pressure drop of ceramic and metallic foam heat exchangers have been extensively investigated in both microscopic structure and macroscopic thermo-hydraulic behavior (Mahjoob and Vafai 2008; and references therein). Calmidi and Mahajan (2000) characterized the heat transfer behaviors of aluminum foams under air forced convection in a wind tunnel. Bhattacharya et al. (2002) experimentally and analytically studied

the effective thermal conductivity
 , permeability , and the inertial coefficient ℱ of high

porosity aluminum and carbon foams, in which they showed that ℱ only depends on the porosity 2

ACCEPTED MANUSCRIPT , not on the pore diameter  and shapes. Boomsma et al. (2003) studied the thermal resistance of water forced convection for open-cell aluminum foams compact heat exchangers on the heater-foam interface and obtained two to three times lower thermal resistance than those

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comparable commercial products. Modified correlations for the overall heat transfer were suggested in their work. Carbon foam heat exchangers have been investigated by Wu et al. (2004) and Gallego and Klett (2003) for the uniqueness of its heat transfer capability. Odabaee and

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Hooman (2007) performed numerical investigation of metal foam heat exchangers in both overall heat transfer and the pressure drop for geothermal power plant air-cooled condensers.

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The metal foam design in their simulation has shown 2 to 6 times higher of heat transfer capability than conventional design with the same operating conditions. Dai et al. (2012) compared metal foam design vs. conventional louver-fins design and shows a lighter and smaller size with the same thermal-hydraulic performance of the heat exchangers. More recently, Mancin

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et al. (2010; 2013) experimentally measured heat transfer coefficient and pressure drop for a series of aluminum and copper foam samples under air forced convection conditions. From the optimization point of view, the pressure loss reduction and the heat transfer increase are

exchangers.

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identified as two critical parameters to meet the goal of geometry and weight of metal foam heat

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The motivation of this study is two-fold. First the determination of the criteria to compare metal foam heat exchangers to conventional finned tube heat exchangers is presented. Historically both metal foams and finned tubes can be treated as porous media, which is conversely defined by its porosity , i.e., the fraction of the total volume occupied by void and

free to the fluid flow. In addition to the porosity , the pore size  and  shape and relative density, i.e. number of pores per unit length (PPI, inch) are important factors to influence the

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fluid flow in metal foams. Most transport experiments conducted on porous medium have been based on beds of packed particles with more or less uniform spherical shape for individual particle (Ergun 1952; Kaviany 1995). Because the geometric features of metal foams show a

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large difference from the structure of fins, the strict quantitative comparison is not a trivial task. A literature survey shows that different research groups have led to different, and even contradictory, conclusions for the overall heat transfer capability of metal foam heat exchangers

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(Mahjoob and Vafai 2008). In order to establish the underlying correlations of flow and heat transfer, as the first step, this study will introduce equivalent parameters for the uniqueness of

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metal foams. The discussion of strict criteria for the comparison will help to clarify the confusion in the design and analysis of metal foam heat exchangers. The second objective is to use the evaluation criteria established here to validate and verify most available correlation models for the permeability, form drag coefficient, overall heat transfer and the pressure drop. To explore

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the second objective, linear and non-linear regression analysis techniques will be applied for fitting widely spread experimental data and simulation results. As a supplementary design tool, the 2D/3D CFD simulation of typical metal foam heat exchangers are undertaken to further

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examine the modified correlation models (Mao, Cheng et al. 2000; Mao, Feng et al. 2000; Yang et al. 2011; Gerbaux et al. 2010; Hooman and Gurgenci 2010). The overall goal is to determine

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the right correlation of pressure drop and heat transfer factors which can be directly applied to conceptual design and prototyping of metal foam heat exchangers for ACCs. II.

Governing Equations for Porous Medium

Flow through a porous medium, which has been studied in a variety of applications since Darcy’s work in the nineteenth century, is a long-standing challenge facing thermal-hydraulics and multiphase flow simulations due to the complicated structure of the materials, widely

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disparate length scales and turbulent flow time scales involved. In order to overcome the difficulties associated with the multiple scales in a porous medium, the concept of representative elementary volume (r.e.v.) has been used to simplify the governing equations of momentum and

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energy transport (Bear 1972). The equations of motion are written with respect to the averaged fluid velocity  for simplicity, the average symbol is dropped out hereafter. Basically the average

process in the r.e.v. can be conducted through two distinct ways, the averaged fluid velocity over

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the volume of fluid of the r.e.v., and the seepage velocity



  over the total volume of the r.e.v. the averaged fluid velocity is given by

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including both the fluid phase and the solid phase. The relation between the seepage velocity and





 (1)  =  The governing equations for mass, momentum and energy conservation are written as (Nield

and Bejan 2013)





+ ∇ ∙ 



  =0

∇ = −



  −  



#



√









  

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 !"  +  !"



∇$ = ∇ ∙ %
∇$& + '((() 

(2) (3) (4)

where the so-called Darcy-Forchheimer model is used to simplify the momentum equation as

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shown in Eqn. (3), which has been widely used for engineering design and analysis. The

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subscript  denotes fluid quantities. The inertial effects of fluid flow are neglected in Eqn. (3) while the friction force and pressure drop balance each other. This methodology found a great success in modeling fluid flow in granular particulate systems such as a bed of packed beads (Kaviany 1995; Bottura and Marinucci 2008). However, most of the results are for low porosity  ≤ 0.6 while metal foams of interest have a high porosity (≥ 0.80) and high tortuosity. It is

important to note that the Darcy-Forchheimer model here is for homogeneous and isotropic porous medium to calculate the pressure drop which does not count for anisotropic and transient 5

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scenarios in which a general form has been used by Yakinthos et al. (2012) to model recuperative heat exchangers for aero engines.

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In order to compare empirical results and CFD simulations the heat exchanger equation for thermal balance is also introduced below. In general, the interfacial heat transfer coefficient around a tube depends on the flow dynamics in the metal foam as well as the boundary

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conditions on the tube such as the wall, geometry cylinder or square shape. Bearing this mind, the analytical solution can be obtained from the heat exchanger thermal balance equation, which

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is a zero dimensional energy balance equation with heat exchange between hot solid wall and cold fluid only. The heat exchanger governing equations read (Kakac et al. 2012), %1 − &0 0

1$2 13

+ ℎ5 $2 − $  = 0

%$6 3 − $78 &2)   = '′′′)

(5) (6)

where subscript  denotes quantities of fluid while 0 stands for quantities of solid materials.

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Assuming the tube surface temperature and metal foam temperature are the same, $2 , while the

fluid temperature, $, is the average of inlet and outlet air temperature, i.e., $ = %$78 + $6 3 &/2 .

The heat flux is calculated by

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'′′′) = ℎ5 =%$2 − $ &

(7)

The analytical solution of the volume heat transfer coefficient is obtained from Eqns. (5)-(7),

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ℎ5 = >%1 − &0 0 %1 − 22)

= &  !

(8)

In Eqn. (8), the constant > is directly related to the transition solution of heat exchanger thermal balance equation. The correlation relation and CFD results are then compared to the analytic solutions for typical ACCs scenarios in Section 3 of this paper. A typical size and layout of metal foams for air-cooled condensers (ACCs) are shown in Figure 1. It is a single row open-cell metal foam used for compact heat exchanger with 6

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aluminum or copper for both foams and tubes. The current study considers heat transfer through the metal foam-tube interface for single phase only; however, the fluid flow in the tube is considered as two-phase with condensation from vapor to liquid water in the thermal cycle of

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power generation. Most empirical correlations of the permeability, inertial coefficient and effective thermal conductivity available for regular pore size and distribution (e.g. packed beads) cannot be directly applied to open-cell metallic/ceramic foams with the porosity  ≥ 0.8 for

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applications of heat exchangers (Ozmat et al. 2004). For the finned tube bundle air-cooled condensers, Hooman and Gurgenci(2010) suggested the following correlation for the form drag

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coefficient obtained from the CFD modeling of porous medium, ? = 0.55 × [9.887%1 − & × % − 0.323& − 0.8443]

(9)

However, it is unclear whether this correlation can be adapted to metal foam air-cooled condensers in general. Experimental and numerical studies have shown that the form drag

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coefficient depends on porosity not directly on the pore size and shape (e.g., Calmidi and Mahajan 2000). Numerical modeling combined with experimentation is used to verify empirical and/or semi-empirical correlations in recent years (e.g. Hugo et al. 2011). Even though there are

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lots of data for other applications, in metal foam heat exchangers for ACC applications, the case

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of open-cell metal foams has yet limited success when using numerical approaches due to the scarcity of data from accurate experimental measurements. A sequence of test cases about metal foam samples used for heat exchangers has been selected from available data to compare to above correlations for permeability and inertial coefficients for heat exchangers. The best practice of validation will be conducted in the regime of turbulent flow for most scenarios of the air side in ACCs, and inlet and outlet boundary conditions will be tuned here to address the optimal values of interfacial overall heat transfer and 7

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acceptable pressure drop through the porous matrix. Without diluting our attention, the modeling and simulation are assuming that the system is under locally thermal equilibrium (LTE) state and the air flow through the metal foam is uniform and without boundary effects so that the mass

III.

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flow rate is obtained by the production of the front area and the inlet air velocity.

Correlation Results for Metal Foam Heat Exchangers

3.1 Characteristic parameters of metal foam heat exchangers

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In order to address metal foam geometric characteristics 30 metal foam samples were taken

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from available literature and have been chosen to compare to the relation between the permeability and porosity. Two types of correlation formulations are studied here. The first type is based on Carman-Kozeny model as shown by Eqn. (10) which is widely used to compute the permeability for tube bundle porous medium for conventional heat exchangers (Table 1 & 2), one focuses on overall performance (Table 1), the other highlights individual pore topology and

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size parameters (Table 2). A length scale 3 , the diameter of the tubes, is needed in the model. In

addition, the constant H is a fitting value to be consistent with the geometry and the structure of

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the porous medium.

Table 1: Metal foam sample material characteristic parameters (Mancin et al. 2013; Nawaz et al. 2010) Porosity

Fiber Diameter (m)

PPI

K

I

Sample 1: Al

0.956

0.000445

10

1.82

0.102

Sample 2: Al

0.93

0.000324

40

0.634

0.086

Sample 3: Al

0.896

0.000484

10

2.65

0.106

Sample 4: Cu

0.933

0.0005

5

0.97

0.051

Sample 5: Cu

0.905

0.000403

10

1.21

0.056

Sample 6: Cu

0.936

0.000244

40

4.5

0.221

Sample 7: Al

0.9272

0.00025

10

1.2

0.097

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Table 2 Mechanical characteristics of metal foam samples used for Eqns. (10) and (11)



60mm

"

30mm

0.71

N/A

N/A

0.9486

0.25mm

3.13mm

0.9272

0.50mm

4.02mm

0.9726

0.35mm

2.58mm

0.9005

0.24mm

1.98mm

0.952



N P Q

3.61 × 10JK 2L 1.2 × 10 2

L

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JM

1.2 × 10JM 2L 2.7 × 10 2 JM

L

0.9 × 10JM 2L

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O  = R%SJ& P



0.562 × 10 2 JM

L

ℱ

0.148 0.097 0.097 0.097 0.088

0.0976

(10)

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In tube bank heat exchangers the value for H is typically 100 (e.g., Bejan and Morega 1993;

Hooman and Gurgenci 2010; Mao, Cheng et al. 2010). Recently, Tadrist et al. (2004) applied Carman-Kozeny model to study the permeability of metal foam heat exchangers and found a

widely disparate value of H = 100~865 in Eqn. (10). Instead of the diameter of tube of finned NP Q

R%SJ&P

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=

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tube structure, the fiber thickness of metal foam,  is used in Eqn. (11)

(11)

Figure 2 shows the distribution of two constants used in the Carman-Kozeny model for

metal foam heat exchangers. The factor H is related to the permeability while factor > is related

to form drag coefficient. The term > is a constant empirical value depending on material and geometry of metal foams, the empirical formula by Tadrist et al. (2004) is applied here (Eqn. 13).

We see that the constant spreads in a variety of margin from 100 to 400 for H and four times of 9

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the model should be careful for different materials.

The second type of correlation directly relies on the pore shape, size and the relative density rather than that based on empirical constant as shown in Eqn. (10) and (11). From the

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engineering application point of view, it is straightforward to study the microscopic structure of

metal foams, which can shed light on the relation between the porosity  and the permeability . 

2



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The following correlations have been used to study metal foams by Du Plessis et al. (1994): = 36U%U−1& 2

(12)

Where the diameter  or the element width  of the open-cell metal foam pores, based on the

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cubic representative unit cells or CRUCs concept, and the tortuosity U are two unknowns to be determined before using Eqn. (12) or (15) to calculate the permeability or the pressure drop. Overall, Eqn. (12) is a better prediction of permeability than the empirical assumption in Eqn.

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(10) given the pore geometry of metal foams. Equations (10)-(11) do not consider underlying microscopic structure, which might not be reasonable for metal foams.

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3.2 Correlation of Momentum exchange of metal foam heat exchangers

The structure of the metal foam heat exchanger has two rows of tubes with parallel layout, slightly different from the layout as shown in Figure 1, the diameter of the tube is 30 mm, and the metal foam wrapping thickness is 30 mm, no gaps between metal foam wrapping are allowed between two neighboring tubes. In other words, the metal foam fills all interstices of the tube bundle where the distance between two neighboring tubes is 60 mm. The total number of tubes

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ACCEPTED MANUSCRIPT used in this compact heat exchanger is 12 with the geometry size of 12 × 0.92 × 0.252. Despite having the same layout of geometry and material (Cu), the pressure drop illustrates a big disparate distribution between Sample 4 and Sample 6 (Table 1).Two samples have close

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porosities, however, the permeability and form drag coefficients show big difference. The major difference between Sample 2 (Al) and Sample 4 (Cu) is the relative density, even though the porosities are close each other. It is not clear whether the material or the structure play the

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critical role to lead a big disparate of pressure drop considering the fact that both permeability and form drag coefficient cannot explain the phenomenon. Given the same porosity, or slight

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change in porosity, the pressure drop seems constant for different samples at the same fluid flow inlet conditions. Data show significant change for distinct relative density and pore geometry of metal foams while the porosity is very close each other for Sample 2 and Sample 7, both are Al. Using these samples we compare four models for the prediction of pressure drop, ∆, in

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metal foams, which is quantified using the form drag coefficient ? or the inertial coefficient ℱ =  /√ in the Darcy-Forchheimer model. Alternatively the friction factor has been used to

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compute the pressure drop in metal foams. The first three are from archived publications and the fourth model is proposed here based on CFD simulation and regression analysis of legacy data.

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The first model is used by Tadrist et al. (2004), which are an empirical formula for the inertial coefficient, ℱ=

WX

√

=

Y%SJ&  Q N

, > = 0.65~2.6

(13)

The subscript  denotes pore fiber (Figure 1) instead of the fluid quantities. The inertia

coefficient ℱ is distinguished from the friction factor  discussed in the following. The correlation presented here provides a concise expression with clear physical meaning, but needs 11

ACCEPTED MANUSCRIPT extensive validation to determine the value of fitting constant > for metal foam structures numerically and/or experimentally. After combining Eqns. (11) and (13), the form drag coefficient is recast, ? =

√H 3/2

? =

2.05U%U−1& √ 2 %3−U& 

1

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>

(14)

The second type of pressure drop correlation is from Du Plessis (1994),

(15)

Where the diameter or the width of the open-cell metal foam pores, based on the cubic

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representative unit cell or CRUC concept, and the definition of tortuosity U are the same as

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before. Because both the length scale and the tortuosity are not empirical parameters, it can be directly scaling up to large-scale engineering applications. For a given metal foam product, the width of open-cell pores is not difficult to obtain. After using Eqn. (12) to replace the tortuosity U, the above formula becomes, 1 2.05[U%U−1& 3−U

=

2.05

2 2

  6√\15−]9+  ^

(16)

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? = 

Eqn. (16) provides alternative approach to calculate the form drag coefficient without knowing

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the tortuosity directly.

The third type is from Paek et al. (2000) for the friction factor correlation, _

`a c√ b dP

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−

= e = fg + 0.105 = d√ + 0.105 S

h



(17)

where 
is the familiar friction factor  used in Fluid Mechanics textbooks except the

characteristic length √. However, the empirical constant 0.105 in Eqn. (17) has its limitation and cannot be extended to a variety of scenarios of metal foam heat exchangers directly.

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Figure 3 shows the comparison among the predictions from four correlations of the form drag coefficient, which cover a variety of porosity. It is seen that the correction from Paek et al. (2000) is insensitive to porosity due to an empirical constant. The form drag coefficient

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monotonically decreases with the porosity of metal foam for two models by Tadrist et al. (2004) and Du Plessis et al. (1994). Because the correlation in Hooman and Gurgenci (2010) was established from CFD simulation based on finned tube heat exchangers, it exhibits a maximum at

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tube heat exchangers for metal foam topological design.

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a value of  ≈ 0.65. This shows that it is probably true to strict validate correlation from finned

Figure 4 shows the change of pressure drop of metal foam heat exchangers at different Reynolds number (velocities of air flow) through an ACC unit without considering phase change of the air side for these cases. Figure 4 also shows two distinct group data included in this

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comparison. The first group, the very left three cases for the same metal foam sample. The only change is the air flow rate. It is seen as a non-linear relation between the pressure drop and the air velocity, which is close to a quadratic curve. The second group, four cases shown in the

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middle of the figure, compares how different materials and metal foam geometries influence the

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pressure drop. For the same air flow rate, the change in magnitude of the pressure drop can be as much as twice as large among the four cases, again showing the importance of metal foam geometry for system optimization design. Following the notation in finned tube heat exchangers, the hydraulic diameter is defined as four times the flow passage volume divided by the total heat transfer area: jℎ =

4klmin l

(18)

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the air flow direction, klpqr is the minimum free flow passage volume and A is the total heat

transfer area. Without considering the metal foam, the tube diameter of heat exchangers can be

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chosen as the characteristic length ℓ = jt ; while the pore diameter can be used as the

characteristic length for metal foam heat exchangers ℓ = ju . The Reynolds number is defined

based on the characteristic length v = wqx ℓ wqx /ywqx . From the definition of friction factor , Δ Δ{

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we have = − 2ℓ |7} | |7} | |7} 

(19)

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Where |7} is the averaged velocity of fluid instead of the seepage velocity of fluid  . The

generic correlation for the friction factor can be obtained using Darcy-Forchheimer model, Eqn.

 =  L 

Lℓ

 P

+

LℓP S

 fg

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(3),

(20)

Where the above friction factor is referred to as Darcy friction factor, which has a relation with

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Fanning friction factor  = 4wrrqr€ . Compared to the empirical correlation in Eqn. (17), a general correlation considering the contribution of linear and nonlinear viscous forces is obtained

here. Eqn. (20) implies that both permeability  and the inertia coefficient ℱ or form drag

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coefficient ? evolve only with the solid matrix, here the pore diameter, independent of the flow

field. Furthermore, if the permeability and form drag coefficient are constant, the friction factor

 is only a function of velocity or Reynolds number. The friction factor introduced by Paek et al. (2000),  , has the following relation with , 
=  √ 2ℓ

(21)

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Now if we substitute Carman-Kozeny model into Eqn. (20) to replace the permeability, we have the friction factor below,  =|+

S

fg

(22)

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where parameters | |8  are constants, independent of the flow field. The fitting constants are

| = 1.9,  = 125.0 (Fig. 6). This would recover the relation in Eqn. (17), considering the fact =

w

fg

+

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that √ is a length scale with unit of 2. In general, the friction factor can be recast as ‚

fg ƒ

(23)

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Therefore the correlation has three fitting parameters, |,  and !, constant at a given flow regime. For low Reynolds number flow, we can use the following coefficients through regression analysis | = 25.6,  = 0.50, and ! = 0.04

(24)

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Figs. 5 and 6 illustrate the comparison between the general correlation Eqn. (20) and

empirical formulas in Eqns. (21), (22) and (23). The classic results by Ergun for packed beads

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was included in Figure 5 to illustrate the difference between the general correlation for metal foam and that obtained from granular materials. The fitting parameters in Eqn. (22) overall show

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a good agreement with available experimental data for both low and high Reynolds number for metal foam heat exchangers. It also confirms that the flow friction factor is only a function of the porosity of porous media, not directly related to the microscopic shape such as filament diameter, size, and the relative density PPI. 3.3 Correlation for the interfacial heat transfer coefficient of metal foam heat exchangers

Heat transfer in forced flow in metal foam heat exchangers is even more complicated than finned tube heat exchangers due to the fact that a significant thermal dispersion, the so-called 15

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hydrodynamic mixing of interstitial fluid, exists at the pore scale. The correlation of heat transfer for bared tubes may not be used directly in metal foam heat exchangers due to the thermal dispersion effect. Another challenge for heat transfer correlation is the experimental difficulties

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of accurate measurement in porous media environment. From an application point of view, simple correlations could be obtained through analogue to bared tubes. For an internal flow, considering a channel packed with a porous medium for fully developed flow, the relative heat

ℎ0 } %„73ℎ 6}6 0 27|&

ℎ0 } %„73ℎ6 3 6}6 0 27|&

≈


2


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transfer augmentation effect is written as follows (Nield and Bejan 2013),

(25a)

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where
2 = %1 − &
0 + 
 is the effective thermal conductivity, and
 is fluid thermal conductivity. However, it is not clear whether the heat transfer augmentation for an external flow in porous media observes a similar relation or not.

The macroscopic correlation of the volumetric heat transfer coefficient ℎ5 %…/2† & or the

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local interfacial heat transfer coefficient ℎ0 %…/2L & is presented as Nusselt number versus Reynolds number and Prandtl number as follows ‡ ˆ = !‰ + !S v r Š}S/†

(25b)

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Where !0 and !1 are two fitting constants for a range of Reynolds number, which covers the

limit value !0 for a laminar flow in porous media, while the power 8 = 0.3~0.8 counts for

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different situations. For intermediate or high Reynolds number flow, the constant !0 can be

negligible, Eqn. (25b) becomes the classic formula (Bird et al. 2002), ‡ ˆ = 0.023v ‰.‹ Š}S/†

(25c)

Several specific correlations have been established in past decade for typical metal foam samples with given porosities so far. For example, the following metal foam-tube interfacial heat

16

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transfer relation was correlated by Calmidi and Mahajan based on wind-tunnel experiments (2000), ‡ ˆ =

tΠN e

= # vN‰.K Š} ‰.†M , vN = 

d N Ž

, Š} =

W e

, # = 0.52

(26)

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where the subscript 0 denotes the quantity on the interface between the metal foam and the tubes. The parameter ℎ0 is evaluated at the interface between a tube and the surrounding wrapped metal

foam. The constant ! is the specific heat of air, $ is a correlation constant, the velocity  again

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denotes the Darcy velocity or the seepage velocity, and  / is the intrinsic velocity of fluid

flow in porous medium. The value of  is determined by the diameter of pore fiber (Fig. 1),

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y and  are the kinetic and kinematic viscosity of air, respectively. Recently, Mancin et al. (2013) introduced the following heat transfer estimate from experimental and modeling results, ‡ ˆ = 0.418v ‰.K† Š}S/†

(27)

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where Eqn. (27) is suitable for flow 30 ≤ v ≤ 200. The definition of Reynolds number and Prandtl number are same as before in Eqn. (26).

In order to validate the general correlation in Eqn. (25) and address the differences among

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different heat transfer coefficient predictions, a total of 23 cases of metal foam samples (out of the 30 samples) have been used in heat exchangers.

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Figure 7 shows the change of the interfacial heat transfer coefficient ℎ0 with the air flow

velocity while other flow conditions remain the same for all metal foam samples. It is seen that, when Reynolds number vN increases, the interfacial heat transfer coefficient increases for the

same metal foam sample monotonically. However, it illustrates a distinct distribution of the interfacial heat transfer coefficient for the same Reynolds number, e.g. sample #2 and #3 are less

than half value of the ℎ0 obtained in sample #5 and #6. On the other hand, the interfacial heat transfer coefficient increases drastically with high air flow rate at a nearly linear relation for 17

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sample #2 and #3. From the optimal design point of view, there is a trade-off between the improvement of overall heat transfer effect and the acceptance of total pressure drop through the metal foam matrix. In order to address this relation, the metal foam heat exchanger, conventional

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type finned tube heat exchanger, and simple tube heat exchanger without fins are compared for two major thermal-hydraulic features. Finally, we see that the interfacial heat transfer coefficient

quantification of metal foam heat transfer capability.

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reported in the literature show a wide range of variability which results in a large certainty of

Figure 8 includes 21 from the total 30 samples by removing redundant tests for pressure

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drop and heat transfer considering the optimal design of heat exchanger performance curve. By changing the air side flow rate, better heat transfer is obtained with the price of increase of pressure drop. One thing that should be avoided is set-up of the operating point above the threshold of performance curve in Fig. 8 (the performance curve was not shown here). The rate

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of change in heat transfer is far smaller than the rate of change in pressure drop if the operating point above the threshold. It is seen that data for metal foam can be separated into three groups: the four points on the top right corner; five to six points in the middle and the left clustered

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points. Samples #1 & #2 in Figures 4 and 7 represent the first group, Samples #3 & #4 (Figures 4&7) for the middle and Samples #5-#7 (Figures 4&7) for the last group. It is important to note

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that the correlation between pressure drop and the heat transfer not only depends on the flow rate and distribution of pressure, temperature, but also on the metal foam materials and geometric parameters.

The correlation of interfacial heat transfer coefficient was adapted for metal foams as well as classic results of Eqn. (25c) (Bird et al. 2002) in Figure 9 for comparison. In order to focus on fluid physics, the Prandtl number is chosen as a constant, i.e. Š} = 0.72 for all cases here. The 18

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correlations between Calmidi et al.(2000) and Mancin et al. (2013) show a close result even though different fitting parameters have been used by two research groups. When Reynolds number increases, the three models are close to each other. At the low limit, the prediction shows

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large differences using the classic model. As a final comment, the 0D model or heat exchanger thermal balance equation leads to over-prediction of results compared to the other three

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correlations, in particular, at the high Reynolds number flow. For this reason, two dimensional and three-dimensional CFD simulations are needed to validate empirical or semi-empirical models. IV.

Numerical results for Metal Foam Heat Exchangers

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In this section commercial CFD tool ANSYS FLUENT has been applied to simulate two scenarios at the stage of conceptual design of metal foam heat exchanger for ACCs. The first scenario is partially wrapped metal foam around each tube while the second scenario will be

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fully wrapped metal foam around each tube in the core area of the heat exchangers. Both metal foam and tubes are aluminum. The geometry size of the metal foam heat exchanger unit is

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1.0m× 0.2m× 0.55m. There are total 112 tubes in the unit with 30mm for the inner diameter j

and 1.6mm thickness for each tube. Figure 10 illustrates the first scenario as well as the computational set-up. The staggered layout of tubes is considered here to enhance flow mixing. The pitch k0 between two neighboring tubes is 60 mm, the wrapper depth of aluminum metal

foam is ℎ = 10 22 , The tube center distance from the edge of the core area k1 is 50 22. The

geometry, layout and size for the second scenario are the same as the first scenario except fully 19

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wrapped metal foam in the core size of heat exchangers. Three samples of aluminum metal foam (Table 3), total of six cases, are simulated. In each case, the air flow is from the left side to the right side of the computational domain, while a mass flow rate boundary condition is imposed at

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the inlet and an outflow boundary condition at the outlet, respectively. The top and bottom are periodic boundary conditions to save computing time. All six cases are focusing on the steadystate fields and the distribution of pressure drop and heat transfer coefficient will be obtained for

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steady flows only.

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The following assumptions are considered for all cases. First, the airflow is assumed uniform along the front area, the use of isotropic porous medium in principle will neglect the

nonuniform air flow. The inlet air velocity for the baseline is 5.5 2/0 which is set as the reference value. Second, the Darcy-Forchheimer model is applied to calculate pressure drop

through trial and error approach to determine the form drag coefficient ? in Eqn. (3). Because

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Table 3 has already provided the inertial coefficient for each metal foam sample, the trial and error iteration that is needed in most heat exchanger simulation is skipped by direct using the quantity provided here. Third, a methodology based on non-dimensional Nusselt number is used

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to predict the local heat transfer coefficient (Mao, Cheng et al. 2010). Finally, the CFD

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simulation focuses on the air side flow and heat transfer without considering phase change in the vapor side of ACCs, for simplicity and without loss of generality, a constant temperature on the

„ tube wall is assumed for all tubes, $„ 78 = $6 3 = 348. In other words, we only consider the

phase change that vapor condenses to liquid water without considering any supercritical heat in

the tube. The ambient air temperature is assuming a constant, 298 while the ambient pressure

is 1 |32. For turbulent flow simulation in porous medium, the standard
−  model is used with recommended parameters in metal foam matrix, even though other turbulent models can be used 20

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v = 4.0 × 10‘ ~5.0 × 10K to compare with the baseline velocity 5 = 5.52/0.

In addition to the momentum equation for CFD simulation, two additional terms are used to

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represent the effects of porous medium on air flows, the effective thermal conductivity
 =

%1 − &
0 + 
 is applied in the energy equation with
0 and
 represents thermal

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conductivity of metal foam and the air, respectively. Different porosity will lead to slightly different effective thermal conductivity for the same materials. After using an appropriate

air flow are computed locally as ') = ℎ’6!|’ %$ − $„ &l0

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temperature for the heat exchanger tubes, the heat flux between the heat exchanger and the dry

(28)

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where l0 is the heat exchanger surface area per volume (22 /23 ).

Table 3: Permeability and inertia coefficient of three samples used in the simulations Symbol  %2L & >%2JS &

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Property Porosity Permeability Inertia Coefficient

Sample 1 0.90 6.6 × 10J‹ 389

Sample 2 0.91 4.79 × 10J“ 1088

Sample 3 0.95 7.2 × 10J‹ 1107

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Finally, the pore size of the metal foam samples varies from 5 to 20 PPI (number of pores per inch) so the mesh size in both 2D and 3D CFD simulation will be consistent with this value. The minimum mesh size varies from 0.05 mm to 0.2 mm for the 6 test cases. Non-uniform bodyfitted stretching meshes are generated using commercial tools. Smaller mesh sizes are applied in the region of metal foam-tube interfaces and in the region of metal foam-space interfaces. In order to test mesh independence, three meshes A (coarse), B (intermediate), and C (fine) have

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been generated. The numerical results are almost unchanged between mesh B and mesh C. For this reason, the intermediate mesh B will be used as the baseline in all six test cases. Figure 11

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shows two mesh profiles (B and C). Figure 12 illustrates the velocity vectors for the first scenario (Sample 1 and 3 in Table 3), the only difference among three metal foam samples are the air flow in the gap between wrapped

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tubes. The lower porosity in solid matrix, the more air will be squeezed to the gap. This is understandable considering that low porosity lead to less flow passage inside the solid matrix.

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However, the effect on heat transfer is more complicated than the distribution of flow volume due to the fact that thermal dispersion is anisotropic in the domain. This partial wrapping will allow for the thermal design to take into account more options while more difficult to predict using conventional empirical correlations. A quantification comparison among velocity change is shown in Figure 13, in which velocity magnitude was computed along the line that cuts through

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the centers of two bottom cylinders. Before and after the tube bank the profile is similar, however, a big change exhibits between the two tubes among three cases. Figure 14 shows the steady-state static temperature and pressure contours for the first scenario (Sample 1 in Table 3).

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Because of the staggered layout used in the design, the distribution of temperature and pressure

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are not symmetric in the y direction (perpendicular to the air flow direction). The advantage is a possible enhancement of flow mixing, so-called thermal dispersion, and heat transfer between solid tube/matrix and fluid. In order to estimate the overall heat transfer and the pressure drop, the area-weighted average surface heat transfer coefficient and total pressure drop are output for three samples. As a comparison, Fig. 15 shows the static pressure contours for fully wrapped metal foam heat exchanger scenario (Sample 1 and 3 in Table 3). It is seen that larger temperature gradient for partially wrapped metal foam (Fig. 14) across the tube/metal foam 22

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matrix, which may cause higher flow and heat transfer mixing. On the other hand, the total surface area per unit volume (22 /23 & is much higher in Fig. 16 than that in Fig. 14. From the

optimization point of view, there may be critical value of the wrapped depth for metal foam

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matrix.

Figure 17 shows the change of heat transfer coefficient ℎ versus porosity  (right panel)

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and for pressure drop (left panel) versus porosity  . It is seen that the porosity reduced from 0.95 to 0.91, the pressure drop increased from 236.05 Pa to 358.97 Pa (52.07% gain), while the

surface heat transfer coefficient increased 550.42 …/2L ∙  to 825.32 …/2L ∙  (49.9% gain).

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However, change of porosity from 0.91 to 0.90, the gain of heat transfer is 7.7% and 2.4% of pressure drop, respectively. The cause of the different trend is due to the fact that metal foam is partially wrapped around each tube in the domain, a challenge facing the thermal-hydraulic

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design of metal foam heat exchangers.

In order to quantify the impact of porosity on the flow and heat transfer in metal foam heat exchangers, Fig. 18 shows the profile of static pressure and static temperature along the air flow

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direction (horizontal direction) for fully wrapped metal foam. In all three samples (Table 3), the pressure profiles are widely spread even though the porosity slightly changes from 0.90 to 0.91.

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However, the temperature changes are less sensitive, only a large variant of static temperature occurred when the porosity increased from 0.91 to 0.95. An interesting conclusion here is that the results are consistent with the 0D model discussed before. It is fair to say that the sensitivity of pressure drop to the porosity is higher than the interfacial heat transfer coefficient for metal foam heat exchangers, either partially or fully wrapped structure.

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V.

Conclusion

Extensive parameters observed in this study have been tested for metal foam heat exchangers in order to optimize the overall heat transfer capability and the pressure drop in the

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air-side of air-cooled condensers (ACCs). A variety of metal foam samples have been selected from archived literature in recent years to compare to 2D/3D CFD simulations using the porous

medium model. The form drag coefficient ? was found to be insensitive to the metal foam pore

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size and its shape but only a function of the porosity  of the metal foam sample. The Carman-

Kozeny model that has been widely applied in finned tube heat exchangers design and analysis

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needs further validation before being directly applied for metal foam heat exchanger design and analysis. The interfacial heat transfer coefficient obtained from CFD simulation is in good agreement with the empirical correlation as given by Calmidi and Mahajan (2000) for low Reynolds numbers, however, it shows discrepancy for turbulent flow with high Reynolds

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numbers. It is seen a similar trend between the analytic solution of surface heat transfer coefficient to the area-weighted average heat transfer coefficient from CFD simulations. The overall objective of this paper was to determine a correlation between pressure drop and

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heat transfer which can be used to assist in the design on metal foam heat exchangers for ACCs.

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For this purpose a methodology to compare metal foams to finned heat exchangers was developed and these evaluation criteria have been made available to those interested in the design and prototyping of metal foam heat exchangers. New empirical correlations have been suggested in this study from the parametric studies to better fit in the available experimental data from the published literatures and numerical results from CFD simulations based on the porous medium model. These include the Darcy friction factor Eqn. (22) and Eqn. (23) obtained by least squares regression analysis, and the surface heat transfer coefficient through the consistence

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comparison between Eqn. (25), Eqn. (26) and an analytical solution. It is seen that classical packed beads and other granular systems may not be accurate for metal foam structures. There is a trade-off between the increase of overall heat transfer and the acceptance of pressure drop for a

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given flow field. From the optimal design point of view, the metal foam thickness, wrapped depth around tubes, and other parameters could cause significant impact on both heat transfer

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and pressure drop.

Finally, the numerical modeling of the related metal foam heat exchangers suggests that a

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higher overall heat transfer capacity of 2-3 times compared to conventional finned tube heat exchanger with the same number of tubes and geometry size and layout can be obtained. The pressure drop from metal foam is in the acceptable range of the optimizing design to use in the thermal power plant for dry-cooling applications. Furthermore simulation studies are needed to

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accurately address the thermal optimization of the system.

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1.

VI. References Aull, R., Cooling tower technology advances, NSF-EPRI Joint Workshop on Advancing

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Power Plant Water Conserving Cooling Technologies, ASME 2012 IMECE, Houston, Texas, Nov 9-15, 2012

Bear, J., Dynamics of Fluids in Porous Media, Elsevier, New York, 1972

3.

Bejan, A. and A.M. Morega, Optimal arrys of pin fins and plate fins in laminar forced-

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2.

convection, ASME J. Heat Transfer, 115(1), 75-81, 1993

Bhattacharya, A., Calmidi, V.V. and R.L. Mahajan, Thermophysical properties of high

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4.

porosity metal foams, Int. J. Heat Mass Transfer, 45, 1017-1031, 2002 5.

Bird, R.B., Stewart, W.E. and E.N. Lightfoot, Transport Phenomena, 2nd Ed., John Wiley & Sons, 2002.

6.

Boomsma, K., Poulikakos, D. and F. Zwick, Metal foams as compact high performance

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heat exchangers, Mechanics of Materials, 35, 1161-1176, 2003 Bottura, L. and C. Marinucci, A porous medium analogy for the helium flow in CICCs, Int.

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J. Heat Mass Transfer, 51, 2494-2505, 2008 Calmidi, V.V. and R.L. Mahajan, Forced convection in high porosity metal foams, ASME J.

9.

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Heat Transfer, 122, 557-565, 2000 Conradie, A.E. and D.G. Kroger, Performance evaluation of dry-cooling systems for power plant applications, Applied Thermal Engineering, 16(3), 219-232, 1996 10. Dai, A., Nawaz, K., Park, Y., Chen, Q., and A.M. Jacobi, A comparison of metal-foam heat exchangers to compact multilouver designs for air-side heat transfer applications, Heat Transfer Eng. 33(1), 21-30, 2012

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11. de Lemos, M.J.S., Turbulence in Porous Media: Modeling and Applications, 2nd Ed., Elsevier, 2012 12. Du Plessis, P., Montillet, A., Comiti, J. and J. Legrand, Pressure drop prediction for flow

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through high porosity metallic foams, Chem. Eng. Sci. 49(21), 3545-3553, 1994

13. Ergun, S., Fluid flow through packed columns, Chem. Eng. Prog. 48, 89-94, 1952

14. Gallego, N.C. and J.W. Klett, Carbon foams for thermal management, Carbon 41, 1461-

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1466, 2003

15. Gerbaux, O., Buyens, F., Mourzenko, V.V., Memponteil, A., Vabre, A., Thovert, J.F.,

155-165, 2010

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Adler, P.M., Transport properties of real metallic foams, J Colloid Interface Science, 342,

16. Hooman, K. and H. Gurgenci, Porous medium modeling of air-cooled condensers, Transp. Porous Med. 84, 257-273, 2010

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17. Hugo, J.M., Brun, E. and F. Topin, Metal foam effective transport properties, in Evaporation, Condensation and Heat Transfer, A. Ahsan (Ed.), ISBN 978-953-307-583-9, pp.279-302, InTech, 2004

York, 1995

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18. Kaviany, M., Principles of Heat Transfer in Porous Media, 2nd Ed., Springer-Verlag, New

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19. Kakaç, S., Liu, H. and A. Pramuanjaroenkij, Heat Exchangers: Selection, Rating, and Thermal Design, CRC Press, 3rd Ed. 2012 20. Lu, T., Ultralight porous metals: from fundaments to applications, ACTA Mechanica Sinica, 18(5), 457-479, 2002

21. Mahjoob, S. and K. Vafai, A synthesis of fluid and thermal transport models for metal foam heat exchangers, Int. J. Heat Mass Transfer, 51, 3701-3711, 2008

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22. Mancin, S., Zilio, C., Cavallini, A. and L. Rossetto, Pressure drop during air flow in aluminum foams, Int. J. Heat Mass Transfer 53, 3121-3130, 2010 23. Mancin, S., Zilio, C., Diani, A. and L. Rossetto, Air forced convection through metal foams:

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experimental results and modeling, Int. J. Heat Mass Transfer, 62, 112-123, 2013

24. Mao, S., Cheng, C., Li, X.C. and E.E. Michaelides, Thermal/structural analysis of radiator for heavy-duty truck, Applied Thermal Engineering, 30, 1438-1446, 2010

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25. Mao, S., Feng, Z.G. and E.E. Michaelides, Off-highway heavy-duty truck underhood thermal analysis, Applied Thermal Engineering, 30, 1726-1733, 2010

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26. Nawaz, K., Bock, J., Dai, Z. and Jacobi, A., Experimental studies to evaluate the use of metal foams in highly compact air-cooling heat exchangers, International Refrigeration and Air Conditional Conference, 2010, Paper 1150, Purdue University 27. Nield, D.A. and A. Bejan, Convection in Porous Media, Fourth Ed., Springer, New York,

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2013

28. Odabaee, M. and Hooman, K., Application of metal foams in air-cooled condensers for geothermal power plants: an optimization study, Int. Commun. Heat Mass Transfer 38,

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838-843, 2011

29. Ozmat, B, Leyda, B. and B. Benson, Thermal application of open cell metal foams,

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Materials and Manufacturing Processes, 19(5), 839-862, 2004 30. Paek, J.W., Kang, B.H., Kim, S.Y. and J.M. Hyun, Effective thermal conductivity and permeability of aluminum foam materials, Int. J. Thermophys. 21(2), 453-464, 2000 31. Shi, J. and S. Bushart, EPRI technology innovation water conservation program overview, NSF-EPRI Joint Workshop on Advancing Power Plant Water Conserving Cooling Technologies, ASME 2012 IMECE, Houston, Texas, Nov 9-15, 2012

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32. Tadrist, L, Miscevic, M., Rahli, O. and F. Topin, About the use of fibrous materials in compact heat exchangers, Exp. Thermal Fluid Sci. 28(2-3), 193-199, 2004 33. Wu, Z., Caliot, C., Flamant, G. and Z. Wang, Numerical simulation of convective heat

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transfer between air flow and ceramic foams to optimize volumetric solar air receiver performances, Int. J. Heat Mass Transfer 54, 1527-1537, 2011

34. Yakinthos, K, Missirlis, D., Sideridis, A., Vlahostergios, Z., Seite, O. and A. Goulas,

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Modeling operation of system of recuperative heat exchanger for aero engine with combined use of porosity model and thermo-mechanical model, Engineering Applications

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of Computational Fluid Mechanics, 6(4), 608-621, 2012

35. Yang, B.J., Mao, S., Altin, O., Feng, Z.G. and E.E. Michaelides, Condensation analysis of exhaust gas recirculation (EGR) system for heavy-duty trucks, ASME J Thermal Sci Eng Appl, 3(4), 0410007, 2011

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36. Zhao, C.Y., Review on thermal transport in high porosity cellular metal foams with open

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cells, Int. J. Heat Mass transfer 55, 3618-3632, 2012

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Figure 1: A single row metal foam wrapped tube (left) as well as two bulk metallic foam samples (middle and right); A typical pore topology is also shown on the bottom of the central panel, in which the edge length of an open cell is  , and the pore fiber thickness is  .

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Figure 2: Correlation factors H and > used for permeability (e.g. Carman-Kozeny model) and the inertial coefficient (Eqn. 13) in metal foam heat exchangers for aluminum and copper.

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Figure 3: Comparison of the form drag coefficient ? from four models. The Hooman and Gurgenci (2010) correlation results were based on finned tube heat exchangers. The other three were based on metal foam porous medium.

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Figure 4: Pressure drop versus fluid flow velocity for seven metal foam samples (out of the 30 samples) used in the correlation study.

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Figure5: Darcy friction factor used for finned tube heat exchanger and comparison to packed bed results (Ergun 1952). The involved characteristic length is the diameter of the tube ℎ .

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Figure 6: Comparison of Darcy friction factor among experimental data and correlation results for metal foam heat exchangers. The characteristic length used here is the pore thickness 

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Figure 7: The heat transfer coefficient at the metal foam-tube interface versus fluid flow velocity for seven metal foam samples in the correlation study

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1.60E+04

Foam

1.40E+04

Finned Simple

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Pressure Drop (Pa)

1.20E+04 1.00E+04 8.00E+03 6.00E+03

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4.00E+03

0.00E+00 0.00

100.00

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2.00E+03

200.00

300.00

Heat Transfer Coefficient

400.00

500.00

(W/m2K)

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Figure 8: Comparison of heat transfer and pressure drop change for different type of heat exchangers (metal foam, finned tubes and bared tubes) with the equivalent geometric size and the layout of tube bundles.

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Figure 9: Nusselt number versus Reynolds number for different correlation models as well as the analytical solution

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Figure 10: CFD simulation setup for the first scenario, partially wrapped metal foam is considered (ℎ is the depth of the wrapper around each tube)

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Figure 11: The baseline mesh (top, 170421 cells in 2D) and finer mesh (bottom, 362572 cells in 2D) for the first scenario simulation as shown in Fig. 10.

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Figure 12: Comparison of x direction velocity vectors for the first scenario (partially wrapped metal foam): (a) Sample 1 and (b) Sample 3.

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Figure 13: The Velocity magnitude profile along the line which cuts the centers of the two cylinders, (a)  = 0.90, (b)  0.91 and (c)  0.95 .

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Figure 14: Steady state temperature distribution (top) and static pressure contours (bottom) for the first scenario (Sample 1 in Table 3)

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Figure 15: Steady state pressure distribution for the second scenario (fully wrapped metal foam heat exchanger): (a) Sample 1 and (b) Sample 3.

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Figure 16: Static temperature distribution for the second scenario (fully wrapped metal foam heat exchanger): (a) Sample 1 and (b) Sample 3.

Figure 17: Area-weighted average pressure drop (left panel) and surface heat transfer coefficient (right panel) versus porosity for the first scenario. 45

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Figure 18: Pressure drop at different position along air flow direction (top panel); total temperature change along air flow direction (bottom panel) for the fully wrapped metal foam scenario. Case 1:  = 0.90 , Case 2:  = 0.91 and Case 3:  = 0.95 .

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Highlight for ATE-2014-5772

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• • •

Systematically examined correlations for flow and heat transfer in bulk metallic foam heat exchangers Established novel correlations for friction factor in metallic foam with applications in ACCs Compared analytical and numerical studies of heat transfer and air flow in metallic foam matrix 2D and 3D computational fluid dynamics (CFD) simulations have been conducted for optimization design and analysis in this R&D work

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•