Heat transfer and fluid flow characteristics of tubular channel partially filled with grooved metal foams

International Communications in Heat and Mass Transfer 108 (2019) 104336

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Heat transfer and fluid flow characteristics of tubular channel partially filled with grooved metal foams

T

⁎

Hamdi E. Ahmed , Obaid T. Fadhil, Wisam A. Salih Department of Mechanical Engineering, College of Engineering, University Of Anbar, 31001 Ramadi, Iraq

A R T I C LE I N FO

A B S T R A C T

Keywords: Metal foam Helical grooves Pipe heat exchanger

The metal foam is a type of porous media that can be characterized by high porosity, tortuous flow paths, high strength to weight ratio and high thermal conductivity. Due to these features the metal foam is used in many engineering applications such as heat exchanges. Therefore, a new design of a circular pipe partially filled with a grooved metallic foam is proposed in order to improve the hydraulic and thermal performance, and consequently reduce the pumping power losses and the metal foam volume. In the current work, the non-Darcy laminar forced convection flow is considered under the thermal boundary condition of constant wall heat flux. The governing equations are solved using the finite volume method (FVM) with temperature-dependent water properties. The variable parameters are; the pitch of the helical grooves, the number of helical grooves and the aspect ratio (Ri: diameter ratio of the metal foam to the pipe). The results show that for Ri = 0.55, four helical grooves having two pitches provides the optimal increase in the Nu number (7%) and the PEC is around 1.21 with a reduction in the pumping power and the amount of the metal foam about 25% and 16.74%, respectively, at Re = 1000.

1. Introduction

features in the heat transfer and fluid flow characteristics when compared with solid materials. They are porous materials which have a cellular structure and have many interesting combinations of mechanical, thermal and physical properties such as; ultra-light, uniform pore size, high porosity, high permeability, high thermal conductivity, and high stiffness with a low specific weight. Metallic foams can be made from aluminum, copper, titanium, steel and nickel [2]. The fluid flow in such applications is usually forced convection because the high pumping power required due to high flow resistance [3]. The goal of researchers is reducing the thermal boundary layer and consequently reduce the thermal resistance. Porous media could destroy the growth of the thermal boundary layer, and reduce the thermal resistance due to the conduction through the porous matrix and the convection between the fluid and the heated wall and between the fluid and the porous media [4]. The major problem encountered when the classic pipe heat exchanger is; the high cost of energy and materials which motivated toward economic and efficient hydrothermal systems, low heat transfer rate which might cause a failure to the device, the low thermal conductivity of most of the basic fluids such as water which requires additional pumping power for forcing through the pipe when the metal foam is inserted. Mahdavi et al., [2] obtained numerically that when the porous

The pipe heat exchangers require innovative designs for improving the hydrothermal performance under the current industrial revolutions. The porous media is one of the attractive methods which used to improve the heat transfer as an insertion. Simultaneously, the remarkable improvement in the heat transfer is associated with an additional pressure drop because of the frequent collision of the liquid with the edges of the porous matrix and the flow impedance which takes place. The heat transfer enhancement has significant attention by researchers to develop the heat exchangers in order to obtain high thermal efficiency, low manufacturing cost, lightweight, and small size. Therefore, energy cost and environmental considerations are motivating efforts to invent more effective thermal systems over the existing designs. This can be achieved when the heat removal of a given device is increased associated with a decrease in the pressure losses. The porous media are implemented in a wide range of engineering applications such as; automobile industries, air conditioning, refrigeration, chemical reactors, electronic chips cooling, solar energy systems, nuclear reactor cooling, heat sinks, heat exchangers and etc., [1]. The metal foam is a form of the porous media which is used to fill the pipe heat exchanger partially or fully. The metal foams are innovative materials that exhibit several ⁎

Corresponding author. E-mail addresses: [email protected] (H.E. Ahmed), [email protected] (O.T. Fadhil).

https://doi.org/10.1016/j.icheatmasstransfer.2019.104336

0735-1933/ © 2019 Elsevier Ltd. All rights reserved.

International Communications in Heat and Mass Transfer 108 (2019) 104336

H.E. Ahmed, et al.

Nomenclatures A Ac C CF cp D d de dp ƒ H h K k ke L Nu Ǭ P PEC PP PPI

PPs Q q″ Re Ri T u V Vr W ΔP

convection heat transfer area, m2 cross-sectional area, m2 Forchheimer's factor, m−1 Inertia Coefficient specific heat, J/kg·K pipe diameter, m metal foam diameter, m edge thickness, m pore size, m friction factor Height, m convective heat transfer coefficient, W/m2·K Permeability, m2 thermal conductivity, W/m·K effective thermal conductivity, W/m·K Length, m Nusselt number volumetric flow rate, m3/s Pressure, Pa Performance evaluation criteria pumping power, W pores per inch

saving pumping power, % heat transfer rate, W heat flux, W/m2 Reynolds number Aspect ratio, Ri = d / D Temperature, K axial velocity, m/s volume, m3 volume reduction, % Width, m pressure drop, Pa

Greek symbols ε μ ρ α

Porosity fluid dynamic viscosity, Ns/m2 Density, kg/m3 Thermal diffusivity, m2/s

Subscripts e ƒ

Effective Fluid

H.J. Xu et al., [4] studied the effect of the porosity, PPI, aspect ratio and metal foam on the hydrothermal performance of partially filled pipe with foam placed on the inner wall. It was stated that the Nu number increased and the friction factor decreased when the porosity increased and the optimum Nu number was at the porosity ranged from 0.92 to 0.94. When the PPI increased, the friction factor increased and the Nu number decreased. The optimum hydrothermal performance was at Ri = 0.3, and it decreased over and less this value. Higher Nu number was with using copper, aluminum, and then followed by the stainless steel foam. Qu et al., [12] studied the effect of the porosity, PPI and Ri of the pipe partially filled with metal foam placed on the wall of a pipe on the Nu number and pressure drop. When the porosity increased, the Nu number increased with decreasing the pressure drop. The higher Nu number was when the porosity around 0.93. The pressure drop increased and Nu number decreased with increasing the PPI, however, the pressure drop and Nu number increased with increasing the thickness of metal foam. Yang et al., [13] studied the effect of the partially filling of the porous medium positioned on the inner wall and at the pipe center. When the pipe was almost filled with the porous medium at the center, the thermal performance of the pipe was higher than the foam attached on the wall. The Nu number increased with

medium thickness and Darcy number increased, a higher heat removal was obtained for the pipe partially filled with the porous matrix positioned on the inner wall under the laminar forced convection heat transfer. Poulikakos and Kazmierczak [5] enhanced the heat transfer of a circular pipe partially filled with porous media for Darcy number range of 10−5 to 10−1. It was found that the Nusselt number increased with increasing the aspect ratio and the Darcy number. Hadim [6] carried out a numerical study for laminar forced convection heat transfer of pipe filled partially and periodically with metal foam. Darcy numbers were varied from 10−6 to 0 with a porosity of 0.97. It was shown that the Nusselt number increased when the Darcy number decreased. The heat transfer rate was almost the same increase especially at low Darcy number, while the pressure drop was much lower in the partially filled channel. Kim et al., [7] studied the forced convection in pipes partially filled with metal foam positioned either at the core or attached to the inner wall. By assuming the Brinkman-Forchheimer with Darcy model, it was found that the Nusselt number increased as the Forchheimer number decreased when porous media placed in the inner wall pipe, and opposite behavior was monitored when the porous was at the core. Pavel and Mohamad [8] found experimentally and numerically that a smaller porosity could positively affect the heat transfer and negatively the pressure drop. Huang et al., [9] enhanced the heat transfer experimentally and numerically in a pipe by inserting metallic porous metals at the pipe core. It was monitored that the heat transfer and the friction factor increased with decreasing the porosity. Taylor et al., [10] studied numerically the effect of the partial filling of the porous media inserted at both pipe center and at the inner wall of the pipe. By considering the Darcy–Brinkman–Forchheimer model and the LocalThermal-Equilibrium assumption, it was displayed that for the porous attached to the inner wall, the Nusselt number increased with increasing the thickness of porous with a high value of thermal conductivity. Shokouhmand et al., [11] studied numerically the effect of porous media position for both; centers and attached to the wall under laminar forced convection. They found that the position of the porous medium insert had a significant effect on the thermal performance of the channel. They also found that if a porous medium located in the center of the channel, the pressure drop was higher than that when adjacent to the walls.

Fig. 1. Schematic diagram of the pipe with metal foam having helical grooves. 2

International Communications in Heat and Mass Transfer 108 (2019) 104336

H.E. Ahmed, et al.

Fig. 2. Cross-sectional diagram of the pipe partially filled with grooved metal foam (G: groove).

[16]. A tetrakaidecahedron unit cell was used in this study. It was found that metal foam can speed up the melting rate of a phase change material. The time of the melting was 37.5% lower than the case without metal foam. This showed that the efficiency of the thermal energy storage had been increased and the reaction time had been reduced. A keen review of the past studies shows that investigating the effect of the helical grooves in a metal foam attached on the internal surface of a pipe seems not to have been explored yet. Hence, the present study attempts to fill this research gap numerically under the laminar flow regime. The grooves pitch, number of grooves, aspect ratio Ri are the variable parameters investigated here.

Table 1 Values of the constants a, b, and c of Eq. (12).

k ρ μ cp

a

b

C

−0.8185 754.93 0.02466 5491.90

7.86 × 10−3 1.88983 −0.0001397 −8.39408

−1.03 × 10−5 −3.62 × 10−3 2.01 × 10−7 1.34 × 10−2

increasing the porous matrix volume and the optimal value was at Ri < 0.3. Yang et al. [14] studied the thermal energy storage enhancing by using metal foam, they simulated a shell and tube unite and employed a metal foam with a porosity of 0.94, and PPI of 15, for phase change materials domains. The effects of the metal foam location and porosity on the heat storage were analyzed. Yang et al. [15] also showed the melting behaviors of pure phase change materials and phase change materials embedded in the metal foam. The tests are achieved at 0°, 30°, 60° and 90° inclination angles, respectively. The melting time is reduced by comparing with the case at 90° by 12.28%, 22.81% and 34.21% at 0°, 30° and 60°, respectively. The influence of melting rate with metal foam was studied numerically by Yang et al.

2. Mathematical model description 2.1. Physical model and assumption A horizontal pipe partially filled with grooved metallic foam attached on the inner surface of the pipe having a diameter of (D = 50 mm). The pipe length is L/D = 10 as shown in Fig. 1. The ratio of the metal foam diameter to the pipe diameter (aspect ratio, Ri = d / D) is taken as 0.25, 0.4 and 0.55.

Fig. 3. Grid generation of (a) fluid and porous media domain for straight grooves, and (b) fluid domain for helical grooves. 3

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3

2.5

2.5 Nu f

2.4

f

120 2.35

115

110

Parameter deviation %

2.45

125

2.3

1

2

3

4

5

6

No. of nodes (× 10e05)

7

8

Error of Nu Error of f

2

1.5

1

0.5

0

2.25

2

4

5

No. of nodes (× 10e05)

7

8

Fig. 4. Grid independent test; (a) Nu and ƒ, (b) relative error of Nu% and ƒ %.

∂ 1 ∂ 1 ∂ (ρ u) + (ρ rv ) + (ρ w ) = 0 ∂z f r ∂r f r ∂θ f

Rectangular grooves with several grooves heights (H) and grooves widths (W) are explored. The heights are 6 and 10 mm whereas the width is 5 mm. Helical grooves (HG) are investigated with several grooves number, namely, from one to seven grooves are considered as shown in Fig. 2. One to seven grooves are undertaken and one to four pitches are assumed for the helical grooves. For the smooth case, the volume of the metal foam is calculated by

V=

π 2 (D − d 2) × L 4

(2)

For the hollow region, the momentum equation is: z-component:

∂u ∂u ∂ 2u w ∂u ⎞ 1 ∂ ⎛ ∂u ⎞ 1 ∂2u ⎤ ∂P + +v + r ρf ⎛u =− + μf ⎡ 2 + ⎢ ∂r ∂ r ∂θ ⎠ z r ∂r ⎝ ∂r ⎠ r 2 ∂θ 2 ⎥ ∂z ⎝ ∂z ⎦ ⎣ (3)

(1) r-component:

In order to estimate the total amount of the volume of the metal foam, the volume of the grooves removed from the porous matrix is subtracted, the volume of the grooved metal foam is calculated by using Ansys-Fluent after modelling for each sample by separated the domain to hollow region and metal foam region. The wall of the pipe is subject to a constant heat flux (4000 W/m2) along the surface of the pipe. At this heat flux, no water boiling might take place and avoiding the overheating which may cause damage to the system when the case tested experimentally. Eight Reynolds numbers are covered (Re = 250, 500, 750, 1000, 1250, 1500, 1750 and 2000). In order to simplify the governing equations, the following assumptions are adopted;

∂v ∂v w2 w ∂v ⎞ +v − + ρf ⎛u ∂r r r ∂θ ⎠ ⎝ ∂z 2v 2 ∂ ∂ ∂P ⎛ 1 ∂rv ⎞+ 1 ∂ v − 2 ∂w ⎤ =− + μf ⎡ 2 + 2 ⎢ ∂z ∂r ⎝ r ∂r ⎠ r ∂θ 2 r 2 ∂θ ⎥ ∂r ⎦ ⎣ ⎜

⎟

(4)

θ -component:

∂w ∂w w ∂w vw ⎞ ρf ⎛u +v + − ∂r r ∂θ r ⎠ ⎝ ∂z 2w 1 ∂P 1 1 ∂ 2w 2 ∂v ∂ ∂ ∂rw ⎞ =− + μf ⎡ 2 + ⎛ + 2 2 + 2 ⎤ ⎢ ∂z r ∂θ r ∂θ r ∂θ ⎥ ∂r ⎝ r ∂r ⎠ ⎣ ⎦

i. Steady, incompressible and laminar fluid flow. ii. The Aluminum foam is rigid, isotropic, homogeneous and fully saturated with liquid. iii. Local-Thermal-Equilibrium (LTE) is considered between the solid and liquid (Ts = Tf). iv. Non- Darcy's law is valid (i.e. inertial influences are significant). v. Thermal diffusion, radiation, and viscous dissipation are neglected. vi. The thermo-physical properties of the water are assumed to be variable with temperature (temperature dependent properties) in a polynomial form.

(5)

Energy equation:

∂T ∂T ∂ 2T w ∂T ⎤ 1 ∂ ⎛ ∂T ⎞ 1 ∂2T ⎤ +v + + (ρcp )f ⎡u r = k⎡ 2 + ⎢ ⎥ ∂r r ∂θ ⎦ r ∂r ⎝ ∂r ⎠ r 2 ∂θ 2 ⎦ ⎣ ∂z ⎣ ∂z

(6)

For the porous region, the momentum equation [5]: z-component:

ρf ∂u ∂u w ∂u ⎞ ⎛u +v + ∂r ε 2 ⎝ ∂z r ∂θ ⎠ 2 μ μf 1 ∂ ⎛ ∂u ⎞ 1 ∂2u ⎤ ∂P CF f ∂ u + 2 2 − ρf 1/2 =− + ⎡ 2 + r u |U | − u ⎥ ∂ ∂ ∂ ∂ ∂z ε ⎢ z r r r r θ K K ⎝ ⎠ ⎦ ⎣

2.2. Governing equations

(7) In forced convection heat transfer, the convection process is governed by the basic conservation principles of mass, momentum, and energy. The continuity equation for hollow and porous media region is;

r-component:

4

International Communications in Heat and Mass Transfer 108 (2019) 104336

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20

16

Yang et al. Present

Xu et al. Present 12

Nu

Nu

15

10

8

4

5

0 0.2

0.3

0.4

0.5

0.6

Ri

0.7

0.8

0.9

0 0.2

1

0.3

0.4

0.5

(a)

0.6

Ri

0.7

0.8

1

(b)

240

3

10

220

Qu et al.

200 180 160

Re= 250 Re= 500 Re= 1000 Re= 1500 Re= 2000 Present

102

140

dP/dx

Local Nusselt number

0.9

120 100 80

101

60 40

Mahdi et al. Present

20 0

0

79

158

237

0

10

316

Circumference distance (mm) around rectangular cylinder

0.2

0.3

Ri

0.4

0.5

Fig. 5. Validation of the simulation with (a) Nu number of Yang et al. [13], (b) Nu number of Xu et al. [4], (c) local Nu number of Raed et al. [22], and (d) pressure gradient of Qu et al. [12].

ρf ε2

2 ⎛u ∂v + v ∂v − w + w ∂v ⎞ ∂r r r ∂θ ⎠ ⎝ ∂z 2 μf ∂2v ∂ 1 ∂rv ∂P ⎞ + 1 ∂ v − 2 ∂w ⎤ − ρf CF v =− + ⎡ 2+ ⎛ 2 ∂θ 2 ε ⎢ z r r r r r 2 ∂θ ⎥ K1/2 ∂r ∂ ∂ ∂ ⎝ ⎠ ⎣ ⎦ μf |U | − v K ⎜

∂T ∂T w ∂T ⎤ 1 ∂ ⎛ ∂T ⎞ 1 ∂2T ⎤ ∂ 2T +v + (ρcp )f ⎡u r = ke ⎡ 2 + + ⎢ ⎥ ∂r r ∂θ ⎦ z r ∂r ⎝ ∂r ⎠ r 2 ∂θ 2 ⎦ ∂ ⎣ ∂z ⎣

⎟

(10)

where |U | = u2 + v 2 + w 2 , and μf = μe [10]. ke is the effective thermal conductivity [17,18]. (8)

ke = εkf + (1 − ε ) kaluminum foam

(11)

θ-component: The permeability is estimated by using the following formula [20,21,24].

ρf

⎛u ∂w + v ∂w − wv + w ∂w ⎞ ε 2 ⎝ ∂z ∂r r r ∂θ ⎠ 2 μ 1 ∂P 1 ∂ 2w 2 ∂v CF ∂ 1 ∂rw ⎞ f ∂ w w =− + ⎡ 2+ ⎛ + 2 2 − 2 ⎤ − ρf 1/2 r ∂θ ε ⎢ z r r r r r ∂θ ⎥ K ∂ ∂ ∂ ∂θ ⎝ ⎠ ⎣ ⎦ μf |U | − w (9) K

K = dp2 × 0.00073(1 − ε )−0.224 (de / dp)−1.11

(12)

where u, v and w are the velocity components in z, r and θ directions, respectively, and T is the temperature, P is the pressure, μf is the viscosity of the fluid and ρf is the fluid density.

Energy equation: 5

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130

5

4HG Ri= 0.55

120

4HG Ri= 0.55

Smooth 1P H6 2P H6 3P H6 4P H6 1P H10 2P H10 3P H10 4P H10

4.5

110 100

4

90 3.5

f

80 Smooth 1P H6 2P H6 3P H6 4P H6 1P H10 2P H10 3P H10 4P H10

70 60 50 40 30 20

0

500

1000

1500

Re

3 2.5 2 1.5

2000

0

500

(a) 1.3

1000

Re

1500

2000

(b)

4HG Ri= 0.55

4HG Ri= 0.55

1P H6 2P H6 3P H6 4P H6 1P H10 2P H10 3P H10 4P H10

35

PPs %

1.2

1P H6 2P H6 3P H6 4P H6 1P H10 2P H10 3P H10 4P H10

45

25

1.1 15

1.0

0

500

1000

1500

Re

5

2000

0

500

1000

Re

1500

2000

Fig. 6. Effect of pitch number of four helical grooves on the (a) Nusselt number, (b) friction factor, (c) PEC and (d) PPs% at Ri = 0.55.

373.15 K.

2.3. Boundary conditions At z = 0 (inlet); v = w = 0, u = ui, T = Ti. ∂u ∂v ∂w At z = L (outlet); ∂Z = ∂Z = ∂Z = 0 , Pgage = 0 (Pa), ∂u ∂z

∂T ∂Z

3. Numerical procedure

=0

∂T ∂z

= v = w = 0, =0 At axisymmetric axis, r = 0; at r = R; u = v = w = 0, q′′ = constant (4000 W/m2) ∂u ∂v ∂w ∂T At θ = 0; ∂θ = ∂θ = ∂θ = 0 , r ∂θ = 0 ∂u

∂v

∂w

3.1. Solution method The finite volume method (FVM) is used for solving the governing equations and Navier- Stokes equations along with the corresponding boundary conditions by using ANSYS-FLUENT v.18. The 2nd upwind differencing scheme is adopted for the convective terms, while the SIMPLE algorithm used to solve the flow field. The term of diffusion in the energy and momentum equations is approximated by the 2nd upwind difference. At convergence, the discrete conservation equations (continuity, momentum and energy) are obeyed for all cells. Solution no longer changes with more iteration that can show in monitoring convergence with residuals. In the present work, the residual of the continuity, momentum and energy equations is about 1 × 10−6.

∂T

At θ = π; ∂θ = ∂θ = ∂θ = 0 , r ∂θ = 0 The water temperature at the inlet is taken as 300 K while the inlet water velocity depends on the Reynolds number (250 ≤ Re ≥ 2000). The velocity at the wall of the pipe is assumed to be zero. At a temperature of 300 K, the thermo-physical properties of water at the inlet are considered as published by [19]. The porosity and PPI of thee porous matrix are assumed as ε = 0.93, PPI = 10, respectively.

M = a + bT + cT 2

(13)

where M represents any property of water, a, b, and c are constants and tabulated in Table 1, and T is the water temperature in K. The range of the temperature of the above equation is taken from 273.15 K to 6

International Communications in Heat and Mass Transfer 108 (2019) 104336

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130

2P H10 Ri= 0.55

120

100

4

90

3.5

80

f

Nu

Smooth 1HG 2HG 3HG 4HG 5HG 6HG 7HG

4.5

110

70

3

Smooth 1HG 2HG 3HG 4HG 5HG 6HG 7HG

60 50 40 30 20

2P H10 Ri= 0.55

5

0

500

1000

Re

1500

2.5 2 1.5

2000

0

500

(a) 1.3

1000

Re

1500

2000

(b)

2P H10 Ri= 0.55

1HG 2HG 3HG 4HG 5HG 6HG 7HG

50

40

PEC

PPs %

1.2

2P H10 Ri= 0.55

60

1HG 2HG 3HG 4HG 5HG 6HG 7HG

1.1

30

20

10

1

0

500

1000

Re

1500

0

2000

0

500

(c)

1000

Re

1500

2000

(d)

Fig. 7. Effect of a number of the helical grooves on the (a) Nusselt number, (b) friction factor, (c) PEC and (d) PPs% at Ri = 0.55.

number and friction factor. So, the mesh of 632,000 nodes is adopted in the current simulations. Fig. 4(a) shows the values of Nu and f with the grid density, and Fig. 4(b) shows the relative error of Nu % and f %.

3.2. Grid independent test After generating the mesh and controlling it in the 3D model, it is necessary to estimate the quality and dependency of the mesh for a prior run to ensure the result accuracy. Therefore, enough number of cells is required for accurate results. As illustrated in Fig. 3(a) and (b), the fluid and porous media domain are meshed hexagonally making an interaction at the interface between the fluid and porous media and connecting the nodes at the interface surfaces. The grid independent test is examined in order to find a fine enough grid system. Six different sizes of the grid are tested from 100,000 up to 825,000 approximately. The mesh generated for the present analysis involves 632,000 nodes roughly, where the relative error between the fifth and sixth grid is 0.03% for the Nusselt number and 0.14% for friction factor. The increase in mesh elements over than 632,000 has < 0.03% variation in Nusselt number and friction factor. The latter mesh provides a slight deviation, shorter time for iteration and low memory size required on the computer compared with other finer grids. The finest grid (sixth grids) is tested to make sure that there is no change in Nu

3.3. Numerical calculation The modified inertial coefficient can be calculated from the relation of the Forchheimer factor for the aluminum foam [20–23]:

(1 − ε )1.5226 ⎞ CF = 29.613 × ⎜⎛ ⎟ dp ⎝ ⎠

(14)

where ε is the porosity and dp is the pore size of the metal foam [24]. The heat transferred to the pipe by supplying heat flux on the wall can be written as [19]:

q′ ′ = −k

∂T ∂r

r = r (wall)

(15)

The heat flux on the outer wall surface of the pipe equals the heat 7

International Communications in Heat and Mass Transfer 108 (2019) 104336

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Fig. 8. Vectors of velocity and isotherm contours for (a) smooth case, (b) 4HG with (H10, W5) at Ri = 0.55 and Re = 1000.

transferred by convection from the inner surface to the fluid and the metal foam, which is represented by the following equation:

q′ ′ = h (Tw − Tb)

the hydrothermal performance of the thermal system and defined as ∗

Nu ⎞ PEC = ⎛ × ⎝ Nu ⎠

(16)

∗

where Tw and Tb are the heated wall temperature and the water bulk temperature, respectively. The local Nusselt number is estimated by

Nuθ, Z =

h θ, z D q′ ′ D = ke ke (Twθ, z − Tbθ, z )

1 A

(17)

PP = Ǭ × ∆P = u × Ac × ∆P

Z = L θ = 2π

∫ ∫

(18)

where A is the total heat transfer circumferential surface area of the pipe, which is estimated as =πDL, where L is the pipe length and D is the pipe diameter. The Reynolds number is defined according to the pipe diameter and the velocity of the fluid at the pipe inlet as:

Re =

PPs =

Vr =

(23)

Vold − Vnew × 100 (%) Vold

(24)

(19)

The Darcy friction factor in the pipe is calculated by

2D∆P f= L ρf ui 2

PPold − PPnew × 100 (%) PPold

The volume reduction of metal foam is estimated as

ρf . ui . D μf

(22)

where Ac is the cross-sectional area of the flow channel. The saving of pumping power is estimated as

Nuθ, Z rdθdz

Z=0 θ=0

(21)

∗

where Nu and f represents, respectively, the Nusselt number and friction factor of the grooved metallic foam cases, whereas Nu and f represent the Nusselt number and friction factor of the pipe with smooth metallic foam, respectively. Pumping power (PP) is the volumetric flow rate at the corresponding pressure drop Δp which is estimated as

Where the average Nusselt number can be calculated by integrating the above equation over the total heated surface

Nu =

f f∗

3

3.4. Validation of simulation For the purpose of verifying the accuracy of the numerical solution, several comparisons are carried out with the open literature. First, validation is achieved with the laminar average Nusselt number of the cylindrical pipe partially filled with metal foam for all dimensionless

(20)

where ΔP is the pressure difference between the inlet and the outlet. The Performance Evaluation Criteria (PEC) is used for representing 8

International Communications in Heat and Mass Transfer 108 (2019) 104336

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110

Re= 1000 Ri= 0.4

9

Re= 1000 Ri= 0.4

8 105

f

Nu

7

6

100

5

Smooth 1HG 2HG 3HG 4HG

95

1

2

No. of pitch

3

Smooth 1HG 2HG 3HG 4HG

4

4

1

2

(a)

No. of pitch

3

4

(b)

Re= 1000 Ri= 0.4

1HG 2HG 3HG 4HG

1.12

50

Re= 1000 Ri= 0.4

1HG 2HG 3HG 4HG

40

PEC

PPs %

1.08 30

20

1.04

10

1 1

2

No. of pitch

3

0

4

(c)

1

2

No. of pitch

3

4

(d)

Fig. 9. Effect of pitch numbers on (a) Nusselt number, (b) friction factor, (c) PEC and (d) PPs% at Ri = 0.4 and (H10 W5) for different values of groove numbers.

save some material of the porous matrix which leads to reducing the pumping power required. The saving the porous matrix material could cause lighter weight and lower cost thermal system. A pipe partially and circumferentially filled with metal foam without grooves is considered the standard case in the current study. The Nusselt number, friction factor, pumping power and foam volume is estimated. The objective of this research is removing some amount of material from the metal foam in the form of grooves. The groove dimensions, the number of the grooves pitches, the number of grooves, the effect of the aspect ratio are the varied parameters examined here. The effect of these parameters on heat transfer rate, in terms of Nusselt number, and the fluid flow, in terms of friction factor and pumping power, is estimated and analyzed. In addition, the PEC is evaluated in order to show the current design's superiority and priority. The saving in the pumping power and foam volume are also calculated.

radiuses (Ri) reported by Yang et al., [13] and Xu et al., [4] is shown in Fig. 5(a) and (b). A good agreement is explicitly observed where the maximum deviation is < 4.6% and 7.5%, with the above two references, respectively. In addition, the local Nusselt number of laminar forced convection of a rectangular channel having a transverse cylindrical heater at Re = 200 as studied by Mahdi et al., [25]. The Nu number was estimated on the circumferences of the heater. Fig. 5(c) shows an excellent agreement where the maximum deviation is found to be < 1%. Second, the verification of the pressure drop of the cylindrical pipe partially filled with metal foam reported by Qu et al., [12] is validated with different dimensionless radii (Ri) and a wide range of Reynolds numbers as shown in Fig. 5(d). The comparison is found to be in acceptable as the maximum deviation is lower than 6%. 4. Results and discussion

4.1. Effect of the pitch of four helical grooves

This paper studies the improvement of the hydrothermal design of pipe heat exchanger by making grooves in the metallic foam while keeping the thermal performance constant or higher. This study aims to

The effects of the number of grooves, from one to seven grooves, 9

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Fig. 10. Vectors of velocity and isotherm contours at Ri = 0.4, (H10, W5) and Re = 1000 for (a) smooth case, (b) 2HG with 4P.

increases the Nu increases. This interment in Nu persists to 4HG and then going down as shown in Fig. 7(a). It is worth to be mentioned that 6HG has approximately the same Nu number compared with the smooth case, while 7HG slows lesser Nu and it is designed is unless. All other cases transfer the heat greater than the smooth one. This thermal trend can be obviously seen at Re > 1000, as the helical grooves disturb the thermal boundary layer greatly. As it is explained early, ƒ decreases as the amount of foam removal increases. This amount increases with increasing the number of grooves. Therefore, the lowest ƒ obtained is for 7HG compared with the smooth case as shown in Fig. 7(b). Besides, a great reduction in the frictional losses observed when Re increases at it is lower values, while ƒ decreases slightly at the highest values of Re. By merging the thermal and hydraulic performance to gather in order to show the optimal thermal hydraulic performance (PEC), four to six helical grooves outperformance is monitored particularly at (500 < Re < 1250). The maximum value of PEC recorded is about 1.21 as shown in Fig. 7(c). Where Fig. 7(d) shows the PPs% increases with increasing the volume of metal foam removed. The minimum and maximum PPs% registered is around 2% to 45% at the lowest flow rate. Higher flow rate, higher pumping power required. The max volume reduction is 29%. These cases exhibit an increase in Nu while a decrease of ƒ, PP and volume reduction. Thus, several advantages can be extended as; improvement of heat transfer rate, less pressure drop for given flow rates, high thermal and hydraulic performance, lower pumping power, cost saving on the total life-cycle basis, improving plant run length, remove the part of the metal and lighter in weight. Fig. 8(a) and (b) show the effect of 4HG and 2P on the velocity vectors and temperature contours, compared with a smooth metal foam, where Re = 1000, Ri = 0.55 and the dimensions of the groove (H10 W5). It can be seen the fluid is penetrated deeply in the foam

and the numbers of the pitch, from one to four pitches, are studied. It is certain that the helical grooves cause a secondary flow due to the swirl eddies normal to the flow direction. It could cause a mixing between the cold and hot fluid layers and consequently increases the Nusselt number since deeper penetration for the fluid through the porous matrix. In addition, a decrease in the friction factor is recorded with the grooved foam compared to the un-grooved matrix. Two helical grooves dimensions are studied here; 10 × 5 mm2 and 6 × 5 mm2. They are compared with the smooth porous media results. The Nusselt number increases with increasing Re as shown in Fig. 6(a). It can be seen that the eight grooved metal foam designs show higher Nu with respect to the smooth case. Among those cases, 4HG, (2P H10) shows the superiority of Nu particularly of higher Re where recorded about 7% compared with the smooth case. Further pitches (3 and 4) begin to decline in Nu. The friction factor decreases significantly in the 4HG, especially in the cases of (1P H10) and (2P H10) were recorded about 32% in (2P H10) compared with the smooth case. Fig. 6(b) shows ƒ decreases with Re. Fig. 6(c) shows the cases of H10 provide greater PEC than that of the cases of H6. More confirmation, (1P H10) and (2P H10) show the superiority in hydro-thermal design compared to others. In addition, the range of Re (500 < Re < 1250) is recorded the best which it recorded at 1.21. It can show that at Re > 500, there is no significant change in the PPs% which can be neglected. The (1P H10) and (2P H10) show the best PPs% for the whole range of Re. A saving in PPs% (1P H10), around 32% is liable to be remarked as shown in Fig. 6(d) where the max volume reduction is 19.41% In 4HG (H10, W5) and four pith. 4.2. Effect of the helix of two pitches helical grooves The current results confirm that as the number of helical grooves 10

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Fig. 11. Effect of a number of grooves and pitches on (a) Nusselt number, (b) friction factor, (c) PEC and (d) PPs% with Ri = 0.25 for different groove numbers.

original smooth case, which is; 1HG (2, 3 and 4P). No further increase in Nu when P increases, due to more effect for turbulence intensity on Nu. All other designs have lower Nu as the smooth case. The designs of three and four helical grooves are useless thermally due to a large amount of foam removal and then lesser conduction heat transfer. The presence of helical grooves in the metal foam inside the pipe leads to a reduction in the friction factor, where an amount of the metal foam is removed. Fig. 9(b) shows that ƒ decreases as the number of grooves increases, and as the number of pitch decreases. In contrast, all proposed designs show smaller ƒ compared to a smooth case. However, PEC displays that the best hydrothermal design is 2HG (1 or 2P) as shown in Fig. 9(c). It can be attributed to the fact that even 4HG provides the modified Nu lesser than the original one Nu, but it also shows quite smaller f. The turbulence intensity increases explicitly with an increasing number of the pitch but till 4P as there is no more change in Nu and also a slight change in f. Therefore, the optimal design, in general, is (1P and 2P). The max PEC is roughly 1.1 for the case of (2HG 1P). As the number of grooves increases, the amount of removing metallic foam increase and the pumping power of fluid decreases. More pumping

metal in the case of 4HG more than the smooth one due to secondary water flow. More fluid penetrations could cause a thinner boundary layer and then higher heat transfer. Thus, 4HG could reduce the pumping power required in simultaneously increase the heat removal. 4.3. Effect of the aspect ratio From the previous sections, when Ri = 0.55, it is found that the best hydrothermal performance case of helical grooves in 4HG and the dimensions are (H10, W5) with 2P has the higher hydrothermal design. Therefore, the dimensions (H10, W5) are used in the next sections. The Re number is kept constant at 1000 to show the effect of other variable parameters. The effect of the number of pitches per number of helical grooves is studied here and the result of smooth metal foam in the pipe is also graphed for comparison with the result of using the grooves. For reducing the number of the set of simulations, the numerical simulation is carried out at Re = 1000. The average Nu for smooth foam at Ri = 0.4 is obtained to be 107.53 as shown in Fig. 9(a). The designs of 1HG to 4HG show that three cases only exhibit higher Nu than the 11

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Fig. 12. Vectors of velocity and isotherm contours at Ri = 0.25, (H10, W5) and Re = 1000 for (a) smooth case, (b) 1HG with 4P.

temperature is recorded compared to the smooth case, although 4HG are used.

power is required when the number of pitches increased. Therefore, the greatest PPs% recorded is for 4HG (1P) which is 47% as shown in Fig. 9(d). The max volume reduction is 15.12% for 4P and 4HG. Fig. 10(a) and (b) shows the effect of 2HG and 4P on the velocity vectors and temperature contours, compared with a smooth metal foam, where Re = 1000, Ri = 0.4 and (H10 W5). A great reduction in the velocity vector at the beginning of the grooved pipe due to the lower amount of foams in case (b) used compared to the case of (a). It shows the 2HG and 4P leads to secondary flow through the grooves, which reduces the wall temperature, so it leads to higher temperature gradients. When Ri is considered 0.25 as illustrated in Fig. 11(a) shows that Nu increases with decreasing the number of helical grooves. At the same time, Nu increases with increasing number of pitches. However, all the suggested designs here exhibit lower Nu number compared to the smooth case. Nu behavior is very close when the number of pitch increases more than 3P. By the way, the friction factor at the proposed designs is less than the smoothed case. Moreover, f decreases with increasing the number of grooves and/or decreasing number of the pitch. As shown in Fig. 11(b), the lowest f in the 4HG and 1P. The best PEC observed is for the case of one and two pitches, while one and two grooves are found to be the best as knowing in Fig. 11(c). In general, the maximum PEC is recorded around 1.03 in 1HG and 2P. The maximum PPs% observed is for 4HG at the single pitch and this reduces with increasing the number of the pitch for all cases, see Fig. 11(d). The reduction in PPs% is apparently in 1HG design. Max volume reduction is 13.08% for 4P and 4HG. The velocity vectors and thermal contours of the water flow through a pipe with and without 1HG and 4P along the pipe length is shown in Fig. 12(a) and (b), where Ri = 0.25, the groove dimensions (H10, W5) and Re = 1000. Due to the reduction in the convection heat transfer for the case of (1HG 4P) because of the low porous media region, higher

4.4. The overall PEC The PEC is calculated for the pipe partially filled with grooved metal foam. Fig. 13(a)–(d) displays the effect of the Ri for different numbers of helical grooves as well as the variable numbers of the pitch respectively. The value of Reynolds number and the dimensions of grooves are fixed through the current section. It can be observed that PEC increases significantly by increasing aspect ratio. Moreover, this increment becomes remarkable when the number of grooves increases to 4HG. Higher aspect ratio means a low amount of foam, and then low effect for the conduction heat transfer. In addition, the fluid flowing through the grooves becomes closer to the heated pipe wall and eventually more heat absorbed by the fluid. In addition, increasing the number of helixes, higher PEC is registered up to two pitches and then PEC is going down. As a result, maximum PEC obtain is at 4HG 2P (H10, W5) at Ri = 0.55, where PEC = 1.21 at Re = 1000 as seen in Fig. 13(b). 5. Conclusions In this work, laminar forced convection in a pipe partially filled with grooved metal foam is numerically explored under steady state condition and compared with smooth metallic foam partially filled pipe. The investigation aims to optimize the hydrothermal performance of the system by using different designs of grooves in the metallic foam, and showing the influence of water flow through the pipe under constant heat flux on the heat transfer rate and the friction factor. The following conclusions can be summarized briefly as: 1. The superiority of the number of helical grooves and number of 12

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Fig. 13. Effect of Ri on PEC for (a) 1P, (b) 2P, (c) 3P and (d) 4P, for different numbers of grooves.

pitches when Ri = 0.55 is observed in (4HG 2P) (H10, W5), which shows the highest reduction in the pumping power around (25%), and a decline in the weight of the system of 16.74%, with an increase in the Nusselt number around 7%, and the PEC is around 1.21 at the Re = 1000. 2. For Ri = 0.4, it shows the best number of helical grooves and number of pitches is (1HG 2P) decrease the pumping power 14% and the amount of material decreased 3.5%, Nusselt number increased 1.3%, and the PEC is around 1.1 at the Re = 1000. 3. In the Ri = 0.25, it shows the best number of helical grooves and number of pitches is (1HG 2P) decrease the pumping power 16% and the amount of material decreased 4%, Nusselt number decreased around 2%, and the PEC is around 1.03 at the Re = 1000. The current findings are recommended when the thermal performance is not carried. 4. It is observed that the PEC increases significantly by increasing Ri from 0.25 to 0.55. The maximum PEC obtain in (4HG) (H10 2P) at Ri = 0.55, where the PEC reached 1.21.

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