The gradient porous materials (GPMs)-filled pipe structure has been proved to be effective in improving the heat transfer ability and reducing pressure drop of fluid. A GPMs-filled pipe structure in which radial pore-size gradient increased nonlinearly has been proposed. The field synergy theory and tradeoff analysis on the efficiency of integrated heat transfer has been accomplished based on performance evaluation criteria (PEC). It was found that the ability of heat transfer was enhanced considerably, based on the pipe structure, in which the pore-size of porous materials increased as a parabolic opening up. The flow resistance was the lowest and the integrated heat transfer performance was the highest when radial pore-size gradient increasing as a parabolic opening down.

## Introduction

Porous materials have been widely used in industrial production. Because porous material has become one of the most effective materials to enhance the heat transfer in many applications [1–6], such as heat exchangers, heat pipes, electronic components heat sinks, solar thermal collectors, and printed circuit heat exchanger. In order to study the heat transfer and flow resistance characteristics of the pipe, setting equal wall temperature or equal wall heat flux was the most commonly way in the numerical simulations [1–2]. The hydraulic and thermodynamic properties of the three-dimensional porous microchannel radiator with various structures were investigated by computational fluid dynamics (CFD) simulation, and it presented that the porous media could improve the heat transfer ability of the radiator, but the pressure drop increased obviously [3]. The numerical simulation and sensitivity analysis of the heat transfer efficiency of heat exchanger were carried out. After that, the optimal combination for different Reynolds number, Darcy number, and porous substrate thickness has been obtained, which maximized the heat transfer coefficient [4]. The filling of porous materials in solar collector was an effective method to improve its heat collecting efficiency [5–6], and the field synergy theory and the principle of volumetric dissipation were used [6].

Different from homogeneous porous material (HPM), gradient porous material (GPM) was one functionally graded material with a gradual change of porosity and pore-size. In generally, plenty of techniques were available to produce GPMs, such as centrifugal separation of the lactescency [7], powder metallurgy [8], melting processing [9], and graded polymer processing [10–13]. GPMs have been used extensively, such as medical transplant, scaffolds for tissue engineering, and thermal barrier coatings [7,14–16]. Wang et al. [17] studied the heat transfer enhancement and fluid flow resistance in the pipes filled with GPMs by CFD simulation, on pore-size gradient and porosity gradient along both the *Z*-direction and *R*-direction, respectively. However, GPMs have been considered to enhance the heat transfer in few of previous works.

In this paper, the pore-size gradient increasing nonlinearly in fully filled pipes, the effects of GPMs on the enhancement of heat transfe,r and fluid flow performances have been simulated by CFD. The simulation results obtained from the proposed structures has been compared with linear gradient increase of pore-size, HPMs, and nonporous situations, which was analyzed based on the field synergy theory and performance evaluation criteria (PEC).

## Model Descriptions

### Physical Model.

This paper adopted a two-dimensional model with an axisymmetric configuration, which is the same as the work performed by Wang et al. [17], as shown in Fig. 1(a). The length *L* and the diameter *D* of the pipe is 1 m and 0.2 m, respectively. The fluid entered a GPM-filled pipe with a constant temperature *T*_{in} and a constant velocity (*u*_{in}). The wall is set to have a constant temperature (*T _{w}* = 1000 K).

AISI304 (Delft, The Netherlands) served as the porous material in this model, whose thermal-physical properties are illustrated in Table 1. Air is used as the model fluid in this work. The temperature of the porous material will increase sharply because of the heat exchange occurs between air and the porous material while *T _{w}* = 1000 K and

*T*

_{in}kept as constant. The variation of air physical properties was neglected [17]. However, the physical properties of air change greatly, which makes it necessary to take the changes into consideration shown in Table 2 [18].

In this work, six different porous material-filled pipe configurations are considered when the porosity *ε* = 0.98 and *R _{p}* = 1.0. The function graphs of different configurations are shown in Fig. 1(b). The six kinds of configurations are marked as (A), (B), (C), (D), (E), and (F) consecutively.

- (A)
The HPMs-filled pipe with

*d*= 0.001 m._{p} - (B)
The HPMs-filled pipe with

*d*= 0.008 m._{p} - (C)
The HPMs-filled pipe with

*d*= 0.016 m._{p} - (D)
The GPM-filled pipe with the linear gradient increase on pore-size along the radial direction (

*d*= 0.15_{p}*r*+ 0.001). - (E)
The GPM-filled pipe with the gradient increase as a parabolic opening up along the

*R*direction (*d*= 1.5_{p}*r*^{2}+ 0.001). - (F)
The GPM-filled pipe with the gradient increase as a parabolic opening down along the

*R*direction (*d*= P_{p}_{2}(*r*) = −1.49947*r*^{2}+ 0.299*r*+ 0.001).

### Governing Equations.

Local thermal equilibrium is usually assumed to exist between the fluid phase and solid phase, especially when the volumetric heat transfer coefficient is very high and no heat is released in either the solid phase or the fluid phase [19]. Mohammad assumed the local thermal equilibrium condition to investigate the effect of porous materials on heat transfer in their study. Moreover, the Forchheimer–Brinkman Darcy model is adopted assuming laminar, boundary layer flow with no internal heat generation, and neglecting viscous dissipation [1,20].

### Boundary Conditions.

## Numerical Simulations

### Numerical Method.

The numerical simulation was conducted by commercial CFD software. The porosity kept as a constant and the pore-size gradient were calculated by user defined functions. In order to perform the grid study, the physical model is meshed in different sizes of $723\xd773$, $1000\xd7100$, $1277\xd7127$ in the axial and radial directions, respectively. The structure of *d _{p}* = 0.016 m is chosen to check the grid of independence, and the results are shown in Fig. 2(a). The mash size of $1000\xd7100$ is chosen and has been used in the computations, because the value of Nusselt number almost keeps constant with the same Reynolds number when the mesh size keeps increasing.

### Validation of Computational Fluid Dynamics Model.

To check the validation of the model developed in part two, the numerical results are compared with the experimental results [1]. The validation simulation was performed following the same geometric and flow parameters. As shown in Fig. 2(b), the average Nusselt number against Reynolds number exhibited positive agreement between the numerical simulation results and the experimental results with maximum deviation less than 10%. The existed deviations resulted from the heat losses, the inevitable difference between the actual porous structure in the experiment and the simplified structure in the simulation, and instrumental inaccuracy.

## Results and Discussion

### The Velocity Contours and Fully Developed Velocity Profiles.

The velocity distribution under different conditions is presented in Fig. 3(a). The value of fully developed velocity increases when *d _{p}* changed from 0.001 to 0.016 m in HPMs-filled pipes. However, the velocity contours in GPM-filled pipes by

*R*-direction are considerably different from those in HPM-filled pipes. First, a high velocity area of the fully developed fluid is found near the wall of the pipe in configuration (D). Second, the high-velocity area also existed in configuration (E) and velocity magnitude is higher than that in configuration (D). But the low velocity area always existed near the whole axis of the pipe. Third, the high-velocity area of the fully developed fluid in configuration (F) existed in the middle of the pipe.

Figure 3(b) presents the fully developed velocity profiles. Configuration (B) has been chosen as an example of HPM situations. It could be easily observed that the velocity distribution magnitude in the GPM-filled pipe along the radial direction is considerably different from nonporous and HPM situations. The velocity maximum value in configuration (E), when dimensionless radial coordinate (*R*) is 0.79, is higher than that in configuration (D) (*R* = 0.71). But, the velocity maximum in configuration (E) is much lower than that of configuration (D) when *R* < 0.6. The velocity maximum appears while *R* = 0.5 in configuration (F), and it is almost equal to that of configuration (D). The average value of velocity magnitude in configuration (F) is the highest. The maximum velocity of fully developed flow in all of the GPM-filled pipes is higher than that of HPM-filled pipes, while the value of maximum velocity of HPM-filled pipes is higher than the one of the nonporous pipes.

### Effect of the Change of Re on Average Nu_{m}.

_{m}

As shown in Fig. 4(a), the heat transfer coefficient (Nu* _{m}*) is the highest in configuration (E), and Nu

*in configuration (E) is much larger than that of nonporous pipes. Moreover, Nu*

_{m}*is lower in configuration (D) than that of configuration (E). However, Nu*

_{m}*in configuration (F) is very close to that of configuration (C), and a little higher than that of configuration (B). In summary, Nu*

_{m}*in configuration (E) is improved considerably.*

_{m}### Friction Factor Variation With Different Reynolds Numbers.

As illustrated in Fig. 4(b), the friction factor is much higher in configuration (A) than that of the other configurations. Presenting Reynolds numbers against friction factor, the friction factor is relatively high in configuration (E), when Reynolds numbers range from 850 to 1400. The friction factor is the lowest in configuration (F) than that of the other configurations besides nonporous situation. For the configuration (F), the value of Nu* _{m}* increases slightly than the other configurations, but the friction factor decreased considerably, as shown in Figs. 4(a) and 4(b). So, a better integrated heat transfer performance can be obtained from configuration (F).

### The Performance Evaluation Criteria Analysis.

*R*-direction and configuration (C) by the performance evaluation criteria [21]

Nonporous situation is used as control.

As shown in Fig. 4(c), the integrated heat transfer performance in configuration (F) is the best. The integrated heat transfer performance in configuration (E) is higher than that of configuration (C) when Reynolds numbers is from 600 to 1000, and lower when Reynolds numbers is from 1400 to 1900. The value of PEC for these two configurations is very close when Reynolds number varies from 1000 to 1400. Compared with the other three configurations, the integrated heat transfer performance of configuration (D) is inefficient, as described in Fig. 4(c). It also shows that the integrated heat transfer performance of configuration (E) and configuration (F) improves positively.

### Field Synergy Analysis.

Figure 4(d) shows that configuration (E) has the smallest $\phi $ among all the configurations and kept $\phi =5\u2009deg$ when *R* = 0. It also indicates that velocity field and temperature gradient are synergized well considerably. Furthermore, a large $\phi $ at *R-*direction appears for the configuration (D) comparing with that of configuration (E). Similar values of $\phi $ have been obtained for the configuration (C) and (F). And the largest $\phi $ exists in nonporous pipe. The changing trends of $\phi $ in Fig. 4(d) and Nu* _{m}* in Fig. 4(a) agree well with each other, and Nu

*increases when the field synergy angle decreases. Above all, it demonstrates that the heat transfer performance is enhanced, since the synergic influence is improved in configuration (E).*

_{m}Three kinds of homogeneous structures, *d _{p}* = 0.001 m,

*d*= 0.008 m, and

_{p}*d*= 0.016 m; a linear structure,

_{p}*d*= 0.15

_{p}*r +*0.001 m; two kinds of nonlinear structures,

*d*= 1.5

_{p}*r*

^{2}+ 0.001 m and

*d*=

_{p}*P*

_{2}(

*r*). Compared with nonporous, homogeneous structures and linear structure, nonlinear structures obtain the more effective ability to balance the flow resistance and heat transfer enhancement. The heat transfer is enhanced by reducing the influence of boundary layer on the flow resistance and heat transfer. On the other hand, the heat transfer is enhanced by the synergic effects between the velocity field and temperature gradient with the reducing flow resistance.

## Conclusions

In this work, the pore-size gradient increasing nonlinearly in fully filled pipes was proposed. The effects of GPMs on the enhancement of heat transfer and fluid flow performances have been simulated by CFD simulation.

- (1)
The evolution of the air physical properties is taken into account when the heat transfer and fluid flow characteristics are investigated under different pipe structures.

- (2)
The heat transfer performance (the value of Nu

) is the highest for the parabolic opening up (_{m}*d*= 1.5_{p}*r*^{2}+ 0.001). However, the friction factor is the lowest for the parabolic opening down (*d*= −1.49947_{p}*r*^{2}+ 0.299*r*+ 0.001) except nonporous pipe. The highest value of PEC have been obtained when*d*= −1.49947_{p}*r*^{2}+ 0.299*r*+ 0.001. But, the field synergy angle is the smallest when*d*= 1.5_{p}*r*^{2}+ 0.001. - (3)
The nonlinear GPMs could more effectively balance the heat transfer enhancement and the flow resistance reduction than the linear GPMs, while the linear GPMs has a better performance than HPMs.

## Funding Data

Anhui Province key research and development project (1704a07020087).

Fundamental Research Funds for the Central Universities (JZ2017HGTB0208).

National Natural Science Foundation of China (Grant No. 51408181).