# A Volume Averaging Theory for Convective Flow in a Nanofluid Saturated Metal Foam

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modified Buongiorno Equations for Nanofluids

_{BO}is the Boltzmann constant and d

_{p}is the nanoparticle diameter. d

_{p}can be anywhere of the order of 1 nm to 100 nm. Li and Nakayama [25] investigated the effect of temperature-dependent thermophysical properties on convective heat transfer rates, and found that variations of base fluid properties due to temperature variation are small enough to be neglected as compared with those effects of nanoparticle volume fraction and temperature. Aladag et al. [26] pointed out that nanofluids with high particle volume fraction or within the range of low temperature may deviate from Newtonian characteristics. Therefore, we shall not consider the nanofluids of very high volume fraction or of very low temperature. Detailed discussions on these effects on the thermophysical properties may be found in Corcione [27].

## 3. Clear Nanofluid Convective Flow in a Circular Tube

## 4. Volume Averaging theory

_{int}represents the interfaces between fluid and solid matrix within a local averaging volume. Note N

_{i}is the unit vector pointing outward from nanofluid side to solid side. All dependent variables for nanofluid and metal phases are decomposed according to Equation (12). Upon exploiting the foregoing spatial average relationships, the microscopic Equations (1) to (4) are integrated over a local averaging volume. The set of macroscopic governing equations thus obtained for nanofluid and metal phases in a nanofluid saturated metal foam with uniform porosity ε may be given as follows:

## 5. Mathematical Modeling

_{v}is the volumetric interstitial heat transfer coefficient. Upon implementing the foregoing mathematical models, the equations reduce to

## 6. Thermal Dispersion

## 7. Nanoparticle Mechanical Dispersion

## 8. Mathematical Model for Hydro-Dynamically and Thermally Fully Developed Flows

## 9. Results and Discussion

^{−4}, ${N}_{BT}$ = 0.5 and ${\varphi}_{B}$ = 0.02. Figure 12 shows the effects of the interstitial Nusselt number $N{u}_{v}$ on fluid and solid temperature profiles.

^{−4}. $N{u}_{v}$ = 1 and ${\varphi}_{B}$ = 0.02, so as to investigate the effect of the ratio of macroscopic Brownian and thermophoretic diffusivities, ${N}_{BT}$, on nanoparticle volume fraction profile, velocity profile and both fluid and solid temperature profiles.

#### 9.1. Asymptotic Solutions for the Case of Nearly Uniform Nanoparticle Distribution (${N}_{BT}>>1$)

#### 9.1.1. Channel Flows

#### 9.1.2. Tube Flows

#### 9.2. Asymptotic Solutions for the Case of Nearly Local Thermal Equilibrium ($N{u}_{v}>>1$)

#### 9.2.1. Channel Flows

#### 9.2.2. Tube Flows

#### 9.3. Heat Transfer Performance Evaluation

^{9}− 1.3 × 10

^{15}(corresponding to $\langle u\rangle $= 0.01 to 1 m/s, 2R = 0.02 m, ${d}_{m}$ = 0.001 m) for the tube. The figure clearly indicates that heat transfer coefficient for the tube filled with a nanofluid saturated metal foam is much higher than that for a tube filled with a base fluid. The ratio increases towards 80 as increasing the pumping power $P.P.$, in which thermal dispersion becomes significant. Naturally, higher volume fraction of nanofluid results in higher heat transfer coefficient, especially when $P.P.$ is large such that thermal dispersion is quite significant.

## 10. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

List of Symbols | |

A | surface area (m^{2}) |

${a}_{f}$ | specific surface (1/m) |

A_{int} | interfacial area between the fluid and solid (m^{2}) |

c | specific heat of nanofluid (J/kgK) |

c_{p} | specific heat of nanoparticle (J/kgK) |

c_{s} | specific heat of solid phase (J/kgK) |

$Da$ | Darcy number (-) |

${D}_{B}$ | Bronwian diffusion coefficient (m^{2}/s) |

${D}_{T}$ | thermophoretic diffusion coefficient (m^{2}/s) |

${d}_{m}$ | mean pore diameter (m) |

${d}_{p}$ | nanoparticle diameter (m) |

$f,F,g,G$ | profile functions (-) |

Hg | Hagen number (-) |

$h$ | wall heat transfer coefficient (W/m^{2}K) |

${h}_{v}$ | volumetric heat transfer coefficient (W/m^{3}K) |

H | channel height (m) |

$k$ | thermal conductivity of nanofluid (W/mK) |

${k}_{BO}$ | Boltzmann constant (J/K) |

${k}_{dis}$ | dispersion thermal conductivity (W/mK) |

${k}_{stag}$ | stagnant thermal conductivity (W/mK) |

$K$ | permeability (m^{2}) |

$N{u}_{v}$ | Interstitial Nusselt number (-) |

Le | Lewis number (-) |

${n}_{j}$ | unit vector pointing outward from fluid side to solid side (-) |

${n}_{BT}$ | microscopic Brownian and thermophoretic diffusivity ratio (-) |

${N}_{BT}$ | macroscopic Brownian and thermophoretic diffusivity ratio (-) |

$N{u}_{H,R}$ | Nusselt number (-) |

$N{u}_{v}$ | Interstitial Nusselt number (-) |

$p$ | pressure (Pa) |

$Pe$ | Peclet number (-) |

Pr | Prandtl number of nanofluid (-) |

P.P. | dimensionless pumping power (-) |

${q}_{0}$ | wall heat flux (W/m^{2}) |

$r$ | radial coordinate (m) |

R | tube radius (m) |

$t$ | time (s) |

T | absolute temperature (K) |

${u}_{i}$ | velocity vector (m/s) |

$V$ | representative elementary volume (m^{3}) |

${x}_{i}$ | Cartesian coordinates (m) |

x, y, z | Cartesian coordinates (m) |

$\gamma $ | parameter associated with temperature ratio (-) |

$\epsilon $ | porosity (-) |

${\epsilon}^{*}$ | effective porosity (-) |

${\varsigma}_{k}$ | transverse thermal dispersion coefficient (-) |

$\eta $$\left(\zeta =1-\eta \right)$ | dimensionless radial coordinate (-) |

$\mu $ | viscosity (Pa·s) |

$\nu $ | kinematic viscosity of nanofluid (m^{2}/s) |

$\rho $ | density of nanofluid (kg/m^{2}) |

$\varphi $ | nanoparticle volume fraction (-) |

Special Symbols | |

$\tilde{\phi}$ | deviation from intrinsic average |

$\overline{\phi}$ | average over the cross-section |

$\phi *$ | dimensionless variable |

$\langle \phi \rangle $ | Darician average |

${\langle \phi \rangle}^{f,s}$ | intrinsic average |

Subscripts and Superscripts | |

B | bulk mean |

bf | base fluid |

dis | dispersion |

f | fluid phase |

p | nanoparticle |

s | solid phase |

w | wall |

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**Figure 1.**Clear nanofluid convective flow in a tube: hydrodynamically and thermally fully developed flow in a tube subject to constant heat flux.

**Figure 2.**Effects of nanoparticle volume fraction on velocity, temperature and volume fraction profiles in a tube with ${N}_{BT}$ = 0.2 and $\gamma =0$: (

**a**) velocity profiles; (

**b**) temperature profiles; and (

**c**) volume fraction profiles.

**Figure 6.**Innovative heat transfer enhancement: (

**a**) synergy effects of nanofluid saturated metal foam on heat transfer; and (

**b**) possible mechanism of heat transfer enhancement.

**Figure 7.**Averaging volume in nanofluid saturated metal foam: averaging volume is smaller than macroscopic characteristic scale but smaller than pore scale.

**Figure 8.**Pore scale passage consideration: pore scale distributions of velocity, temperature and nanoparticles.

**Figure 9.**Transverse thermal dispersion in nanofluid saturated metal foam: comparison of empirical correlation and present formulas.

**Figure 10.**Fully developed flow subject to constant heat flux: (

**a**) channel; and (

**b**) tube. Hydrodynamically and thermally fully developed flow is established in a channel /tube subject to constant heat flux, filled with a nanofluid saturated metal foam, where Darcian velocity is higher close to the wall since the viscosity there is lower.

**Figure 11.**Velocity overshooting in a tube flow: numerical results based on the full momentum equation.

**Figure 12.**Effects of $N{u}_{v}$ on the temperature profiles in a channel filled with a nanofluid saturated metal foam.

**Figure 13.**Effects of $N{u}_{v}$ on the nanoparticle distribution in a channel filled with a nanofluid saturated metal foam.

**Figure 14.**Effects of $N{u}_{v}$ on the velocity distribution in a channel filled with a nanofluid saturated metal foam.

**Figure 15.**Effects of ${N}_{BT}$ on the nanoparticle volume fraction distribution in a channel filled with a nanofluid saturated metal foam.

**Figure 16.**Effects of ${N}_{BT}$ on the velocity distribution in a channel filled with a nanofluid saturated metal foam.

**Figure 17.**Effects of ${N}_{BT}$ on the temperature distribution in a channel filled with a nanofluid saturated metal foam.

**Figure 18.**Effects of ${\varphi}_{B}$ on the heat transfer coefficient ratio in a channel filled with a nanofluid saturated metal foam.

**Figure 19.**Effects of ${N}_{BT}$ on the heat transfer coefficient ratio in a channel filled with a nanofluid saturated metal foam.

**Figure 20.**Effects of $N{u}_{v}$ on the heat transfer coefficient ratio in a channel filled with a nanofluid saturated metal foam.

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**MDPI and ACS Style**

Zhang, W.; Li, W.; Yang, C.; Nakayama, A.
A Volume Averaging Theory for Convective Flow in a Nanofluid Saturated Metal Foam. *Fluids* **2016**, *1*, 8.
https://doi.org/10.3390/fluids1010008

**AMA Style**

Zhang W, Li W, Yang C, Nakayama A.
A Volume Averaging Theory for Convective Flow in a Nanofluid Saturated Metal Foam. *Fluids*. 2016; 1(1):8.
https://doi.org/10.3390/fluids1010008

**Chicago/Turabian Style**

Zhang, Wenhao, Wenhao Li, Chen Yang, and Akira Nakayama.
2016. "A Volume Averaging Theory for Convective Flow in a Nanofluid Saturated Metal Foam" *Fluids* 1, no. 1: 8.
https://doi.org/10.3390/fluids1010008