# Sensitivity Analysis of Entropy Generation in Nanofluid Flow inside a Channel by Response Surface Methodology

^{*}

## Abstract

**:**

_{2}O

_{3}-water nanofluid flow inside a channel is considered. The total entropy generation rates consist of the entropy generation rates due to heat transfer and friction loss are calculated by using velocity and temperature gradients. The continuity, momentum and energy equations have been solved numerically using a finite volume method. The sensitivity of the entropy generation rate to different parameters such as the solid volume fraction, the particle diameter, and the Reynolds number is studied in detail. Series of simulations were performed for a range of solid volume fraction $0\le \varphi \le 0.05$, particle diameter $30\text{\hspace{0.17em}nm}\le dp\le 90\text{\hspace{0.17em}nm}$, and the Reynolds number 200 ≤ Re ≤ 800. The results showed that the total entropy generation is more sensitive to the Reynolds number rather than the nanoparticles diameter or solid volume fraction. Also, the magnitude of total entropy generation, which increases with increase in the Reynolds number, is much higher for the pure fluid rather than the nanofluid.

## 1. Introduction

_{2}O

_{3}nanoparticles with water as the base fluid. Their results revealed that the increase in heat transfer coefficient by adding the nanoparticles to the base fluid is about 32% at ϕ = 2% (ϕ is solid volume fraction of nanoparticles). Beside this advantage, they observed an increase in pressure loss for the nanofluid in comparison to the pure water.

_{2}O

_{3}and TiO

_{2}. Their results indicated that adding the Cu nanoparticles to the base fluid (water) generates more entropy in comparison to the other nanoparticles. This was due to the high density of Cu particles. Khaleduzzaman et al. [7] performed an exergy analysis on water-alumina nanofluid for an electronic liquid cooling system. They found that the friction factor increased with the rise of the solid volume fractions of nanoparticles. In another research, Khairul et al. [8] performed an exergy analysis on the metal oxide nanofluid flow in a corrugated plate heat exchanger. They observed an increase in the friction factor, pressure drop and pumping power with increase in particle volume fraction and volume flow rate of nanofluids.

_{2}O

_{3}-water nanofluid in circular microchannels. They observed that the viscous dissipation has a negligible effect on the entropy generation due to the fluid friction irreversibility. Hajialigol et al. [12] investigated the exergy characteristics of nanofluid flow in a 3-D microchannel under a magnetic field. They found that the contribution of thermal entropy generation in the total is prominent in comparison to the frictional and magnetic one. OzgunKorukcu [13] performed an exergy analysis for the laminar and steady flow across a square obstacle placed in the channel with strong blockage.

## 2. Problem Statement and Computational Model

_{2}O

_{3}-water nanofluid flow between two parallel plates with half width D and length L with inlet uniform velocity (U

_{∞}) and temperature (T

_{∞}) is considered. The plates are maintained at a constant temperature (T

_{w}).

- (1)
- The flow is considered to be two-dimensional, laminar, steady and incompressible.

- ✓
- This range of Reynolds numbers is significant for designing many devices such as micro devices and compact heat exchangers, which are two important applications of this geometrical (channel).
- ✓
- Higher Reynolds numbers are beyond the limit where two dimensional simulations can be performed [17].
- ✓
- It is safe to drop the viscous dissipation effects in the energy equation at this range of Reynolds number.

- (2)
- Bottom half of the channel is considered in simulation due to the symmetrical shape.

- Conservation of mass equation:$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$
- Momentum equation in x and y directions [18]:$${\rho}_{eff}\left[u\frac{\partial u}{\partial \text{\hspace{0.17em}}x}+v\frac{\partial u}{\partial \text{\hspace{0.17em}}y}\right]=-\frac{\partial p}{\partial x\text{\hspace{0.17em}}}+{\mu}_{eff}\left[\frac{{\partial}^{2}u}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial}^{2}u}{\partial \text{\hspace{0.17em}}{y}^{2}}\right]$$$${\rho}_{eff}\left[u\frac{\partial v}{\partial \text{\hspace{0.17em}}x}+v\frac{\partial v}{\partial \text{\hspace{0.17em}}y}\right]=-\frac{\partial p}{\partial y\text{\hspace{0.17em}}}+{\mu}_{eff}\left[\frac{{\partial}^{2}v}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial}^{2}v}{\partial \text{\hspace{0.17em}}{y}^{2}}\right]$$
- Energy equation [18]:$${\rho}_{eff}{C}_{eff}\left[u\frac{\partial T}{\partial \text{\hspace{0.17em}}x}+v\frac{\partial T}{\partial \text{\hspace{0.17em}}y}\right]={k}_{eff}\left[\frac{{\partial}^{2}T}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial}^{2}T}{\partial \text{\hspace{0.17em}}{y}^{2}}\right]$$
_{eff}, k_{eff}, ρ_{eff}and μ_{eff}are effective specific heat, conductivity, density and viscosity, respectively. - The effective density is given by [19]:$${\rho}_{eff}=(1-\varphi ){\rho}_{f}+\varphi {\rho}_{p}$$
- The effective specific heat is measured by using the following equation [20]:$${C}_{eff}=\frac{(1-\varphi ){\rho}_{f}{C}_{f}+\varphi {\rho}_{p}{C}_{p}}{{\rho}_{eff}}$$
- The effective dynamic viscosity is defined in following form [21]:$${\mu}_{eff}={\mu}_{f}+\frac{{\rho}_{p}{V}_{B}{{d}_{p}}^{2}}{72N\delta}$$$$\begin{array}{l}N={{\mu}_{f}}^{-1}\left[\left({n}_{1}{d}_{p}+{n}_{2}\right)\varphi +\left({n}_{3}{d}_{p}+{n}_{4}\right)\right]\\ {n}_{1}=-0.000001113\frac{\text{kg}}{{\mathrm{m}}^{2}\mathrm{s}},{n}_{2}=-0.000002771\frac{\text{kg}}{\text{ms}}\\ {n}_{3}=0.00000009\frac{\text{kg}}{{\mathrm{m}}^{2}\mathrm{s}},{n}_{4}=-0.000000393\frac{\text{kg}}{\text{ms}}\end{array}$$
_{B}are the distance between nanoparticles and Brownian velocity of the nanoparticles, respectively. These parameters are defined by [21]:$${V}_{B}=\frac{1}{{d}_{p}}\sqrt{\frac{18{K}_{B}T}{\pi {\rho}_{p}{d}_{p}}}$$$$\delta =\sqrt[3]{\frac{\pi}{6\varphi}}{d}_{p}$$_{p}and K_{B}are nanoparticle diameter (= 30 nm) and Boltzmann constant (= 1.38 × 10^{−23}J·K^{−1}), respectively. - Finally, the effective thermal conductivity is calculated by [23]:$$\frac{{k}_{eff}}{{k}_{f}}=1+64.7\times {\varphi}^{0.7460}{\left(\frac{{d}_{f}}{{d}_{p}}\right)}^{0.3690}{\left(\frac{{k}_{p}}{{k}_{f}}\right)}^{0.7476}\times {\text{Pr}}^{0.9955}\times {\text{Re}}^{1.2321}$$
_{f}is molecular diameter of the water (= 0.3 nm). The Prandtl and Reynolds numbers in this equation are calculated by:$$\text{Pr}=\frac{\mu}{{\rho}_{f}{\alpha}_{f}}$$$$\text{Re}=\frac{{\rho}_{f}{K}_{B}T}{3\pi {\mu}^{2}{l}_{BF}}$$_{BF}is the mean free path of water (= 0.17 nm) and µ is calculated by [18]:$$\mu =2.414\times {10}^{-5}\times {10}^{\frac{247.8}{T-140}}$$

- At the inlet of the channel, a uniform flow is assumed. This boundary is defined by:$$u={U}_{\infty},v=0,T={T}_{\infty}$$
- At the channel wall, no slip and constant temperature boundary conditions are imposed. These boundaries are:$$u=0,v=0,T={T}_{w}$$
- Zero gradient boundary conditions are used at the outlet of the channel [25]. These boundaries are given by:$$\frac{\partial u}{\partial x}=0,\frac{\partial v}{\partial x}=0,\frac{\partial T}{\partial x}=0$$
- Symmetry conditions are assumed at the centerline. These boundaries are given by:$$v=0,\frac{\partial u}{\partial y}=0,\frac{\partial T}{\partial y}=0$$

^{−7}.

No. | Grid Size | Nusselt Number | Percentage Difference |
---|---|---|---|

1 | 150 × 20 | 6.057 | 1.6 |

2 | 300 × 40 | 6.154 | 1.1 |

3 | 600 × 80 | 6.222 | 0.3 |

4 | 1200 × 160 | 6.241 | – |

_{2}O

_{3}-water nanofluid flow inside a horizontal circular duct with a constant surface temperature. Numerical and experimental results are presented for Nusselt number ratio at the solid volume fraction ϕ = 0.01 and nanoparticle diameter dp = 40 nm. The Nusselt number ratio is defined as the ratio of the Nusselt number for the nanofluid to that of pure water. As shown in Figure 3, the numerical results are in agreement with the experimental data. The relative error is about 7%, which is within the error of the experimental data that was reported, as ±6% [4].

**Figure 3.**Comparison between the numerical results and experimental data for the variation of the Nusselt number ratio with Reynolds number.

Parameters Symbol | Level | |||
---|---|---|---|---|

−1 | 0 | 1 | ||

Re | A | 200 | 500 | 800 |

dp (nm) | B | 30 | 60 | 90 |

ϕ | C | 0.01 | 0.03 | 0.05 |

^{2}, B

^{2}and C

^{2}), 3 two-factor interaction terms (AB, AC and BC) and 1 intercept term. Also, Res is the dimensionless total entropy generation rate and is calculated by coded values [29].

^{a}+ 2a + b experiments is employed to specify the response results. a = 3 and b = 6 are the number of factors and the number of center points, respectively. As a result, 20 experiments with the Reynolds number (A), the particle diameter (B) and the solid volume fraction (C) as the independent coded variables are needed for this problem. A series of these experiments for nanofluid flow inside the channel are presented at Table 3.

Standard Order | Coded Value | Real Value | Responses | ||||
---|---|---|---|---|---|---|---|

A | B | C | Re | dp | ϕ | N_{t} | |

1 | −1 | −1 | −1 | 200 | 30 | 0.01 | 0.014441 |

2 | 1 | −1 | −1 | 800 | 30 | 0.01 | 0.043037 |

3 | −1 | 1 | −1 | 200 | 90 | 0.01 | 0.013952 |

4 | 1 | 1 | −1 | 800 | 90 | 0.01 | 0.031240 |

5 | −1 | −1 | 1 | 200 | 30 | 0.05 | 0.015356 |

6 | 1 | −1 | 1 | 800 | 30 | 0.05 | 0.059118 |

7 | −1 | 1 | 1 | 200 | 90 | 0.05 | 0.014793 |

8 | 1 | 1 | 1 | 800 | 90 | 0.05 | 0.038779 |

9 | −1 | 0 | 0 | 200 | 60 | 0.03 | 0.014956 |

10 | 1 | 0 | 0 | 800 | 60 | 0.03 | 0.041098 |

11 | 0 | −1 | 0 | 500 | 30 | 0.03 | 0.037220 |

12 | 0 | 1 | 0 | 500 | 90 | 0.03 | 0.027376 |

13 | 0 | 0 | −1 | 500 | 60 | 0.01 | 0.026047 |

14 | 0 | 0 | 1 | 500 | 60 | 0.05 | 0.033898 |

15 | 0 | 0 | 0 | 500 | 60 | 0.03 | 0.031349 |

16 | 0 | 0 | 0 | 500 | 60 | 0.03 | 0.031349 |

17 | 0 | 0 | 0 | 500 | 60 | 0.03 | 0.031349 |

18 | 0 | 0 | 0 | 500 | 60 | 0.03 | 0.031349 |

19 | 0 | 0 | 0 | 500 | 60 | 0.03 | 0.031349 |

20 | 0 | 0 | 0 | 500 | 60 | 0.03 | 0.031349 |

Source | DOF | Sum of Squares | Contribution | Adj Mean Squares | F-Value | p-Value | – |
---|---|---|---|---|---|---|---|

Model | 9 | 0.002478 | 99.46% | 0.000275 | 203.17 | <0.0001 | Significant |

Linear | 3 | 0.002249 | 90.26% | 0.000750 | 553.16 | <0.0001 | – |

A | 1 | 0.001954 | 78.40% | 0.001954 | 1441.41 | <0.0001 | – |

B | 1 | 0.000185 | 7.43% | 0.000185 | 136.62 | <0.0001 | – |

C | 1 | 0.000110 | 4.43% | 0.000110 | 81.45 | <0.0001 | – |

Square | 3 | 0.000039 | 1.58% | 0.000013 | 9.68 | 0.0030 | – |

AA | 1 | 0.000033 | 1.34% | 0.000023 | 16.72 | 0.0020 | – |

BB | 1 | 0.000004 | 0.14% | 0.000005 | 3.98 | 0.0074 | – |

CC | 1 | 0.000002 | 0.09% | 0.000002 | 1.74 | 0.0217 | – |

Interaction | 3 | 0.000190 | 7.62% | 0.000063 | 46.68 | <0.0001 | – |

AB | 1 | 0.000121 | 4.85% | 0.000121 | 89.11 | <0.0001 | – |

AC | 1 | 0.000060 | 2.40% | 0.000060 | 44.09 | <0.0001 | – |

BC | 1 | 0.000009 | 0.37% | 0.000009 | 6.85 | 0.0026 | – |

Residual Error | 10 | 0.000014 | 0.54% | 0.000001 | – | – | – |

Lack-of-Fit | 5 | 0.000014 | 0.54% | 0.000003 | – | – | – |

Pure Error | 5 | 0.000000 | 0.00% | 0.000000 | – | – | – |

Total | 19 | 0.002492 | 100% | – | – | – | – |

^{2}, AB, AC and BC are significant terms for the response (N

_{t}). As a result, the relationship between response and independent variables can be summarized in following mathematical relationship:

^{2}) and adjusted R-squared (R

^{2}-adj) are presented in Table 5 to perform more examination of the model accuracy. It can be seen that there is an excellent mathematical relationship between the independent variables and response as R-squared and adjusted R-squared values are in the vicinity of unity [28].

N_{t} | ||
---|---|---|

Term | Coefficient | p-Value |

Constant | 0.03117 | <0.0001 |

A | 0.01398 | <0.0001 |

B | −0.00430 | <0.0001 |

C | 0.00332 | <0.0001 |

A^{2} | −0.00287 | 0.0020 |

B^{2} | 0.00140 | 0.0740 |

C^{2} | −0.00093 | 0.2170 |

AB | −0.00389 | <0.0001 |

AC | 0.00273 | <0.0001 |

BC | −0.00108 | 0.0026 |

– | R^{2} = 99.46% | R^{2}-adj = 98.97% |

## 3. Results and Discussion

^{−12}) at the centerline to a maximum value (i.e., 7.5 × 10

^{−3}) at the channel wall due to the zero temperature and velocity gradients at the center of the channel, and the comparatively high values of these gradients around the channel wall. It is worth mentioning that the entropy generation has a peak at the entrance of the channel (see red lump) and it decreases along the channel. In addition, the total entropy generation increases with increase in the Reynolds number. Note that the frictional entropy generation becomes more significant for the higher Reynolds numbers. This augmentation in the total entropy generation is in the vicinity of 113% for 200 < Re < 800. It is worth mentioning that the entropy generation is dominated by the heat transfer irreversibility for low values of the Reynolds numbers.

**Figure 5.**Entropy generation contours for (

**a**) Pure fluid flow at different values of Reynolds number; (

**b**) Nanofluid flow at Re = 500, dp = 60 nm and different values of solid volume fractions of nanoparticles; (

**c**) Nanofluid flow at Re = 500, ϕ = 0.03 and different values of nanoparticles diameters; (

**d**) Nanofluid flow at ϕ = 0.03, dp = 60 nm and different values of Reynolds numbers.

**Figure 7.**Predicted responses as a function of different factors. (

**a**) Effects of A and B; (

**b**) Effects of A and C; (

**c**) Effects of B and C.

B | C | Sensitivity | ||
---|---|---|---|---|

$\frac{\partial Nt}{\partial A}$ | $\frac{\partial Nt}{\partial B}$ | $\frac{\partial Nt}{\partial C}$ | ||

−1 | −1 | 0.0151 | −0.0032 | 0.0044 |

0 | 0.0179 | −0.0043 | 0.0044 | |

1 | 0.0206 | −0.0054 | 0.0044 | |

0 | −1 | 0.0113 | −0.0032 | 0.0033 |

0 | 0.0140 | −0.0043 | 0.0033 | |

1 | 0.0167 | −0.0054 | 0.0033 | |

1 | −1 | 0.0074 | −0.0032 | 0.0022 |

0 | 0.0101 | −0.0043 | 0.0022 | |

1 | 0.0128 | −0.0054 | 0.0022 |

**Figure 8.**Sensitivity analysis results (

**a**) A = 0 and B = −1; (

**b**) A = 0 and B = 0; (

**c**) A = 0 and B = 1.

## 4. Conclusions

- The total entropy generation for nanofluid increases with increase in the Reynolds number and solid volume fraction. These augmentations are in the vicinity of 175% and 30% for 200 < Re < 800 and 0.01 < ϕ < 0.05, respectively.
- The total entropy generation decreases with increase in the nanoparticles diameter. This reduction is in the vicinity of 32% for 30 < dp < 90.
- The magnitude of total entropy generation, which increases with increase in the Reynolds number, is much higher for pure fluid rather than the nanofluid.
- The change in nanoparticles diameter has negligible effect on the entropy generation rate for low values of the Reynolds number.
- The total entropy generation is more sensitive to the Reynolds number rather than the nanoparticles diameter or solid volume fraction.
- The sensitivities of the total entropy generation to the Reynolds number and nanoparticles diameter increase with increase in the solid volume fraction.
- The sensitivities of the total entropy generation to the Reynolds number and the solid volume fraction decrease with increase in nanoparticles diameter.

## Author Contributions

## Conflicts of Interest

## Nomenclature

a | number of factors (-) |

ANOVA | analysis of variance (-) |

K_{b} | Boltzmann constant (-) |

Be | Bejan number (-) |

b | number of center points (-) |

C | specific heat at constant pressure (J·kg ^{−1}·K^{−1}) |

CCD | central composite design (-) |

CCF | central composite face centered (-) |

D | half of the channel gap (m) |

d_{f} | molecular diameter of base fluid (nm) |

dp | nanoparticle diameter (nm) |

DOE | design of experiments (-) |

$h$ | heat transfer coefficient (W·m ^{−2}·K^{−1}) |

$k$ | thermal conductivity (W·m ^{−1}·K^{−1}) |

L | length of the channel (m) |

l_{BF} | mean free path of water (-) |

N_{g} | dimensionless local volumetric entropy generation rate (-) |

N_{t} | dimensionless total entropy generation rate (-) |

$p$ | pressure (Pa) |

Pe | Peclet number (Re×Pr) |

Pr | Prandtl number ($\nu /\alpha $) |

Re | Reynolds number (ρU _{∞}Dµ^{−1}) |

Res | response (-) |

RSM | response surface methodology (-) |

${S}_{g}^{\u2034}$ | entropy generation rate (W·m ^{−3}·K^{−1}) |

$T$ | temperature (K) |

$u,\text{\hspace{0.17em}}v$ | velocity component in x and y directions, respectively (m·s ^{−1}) |

x, y | rectangular coordinates components (m) |

## Greek Symbols

$\alpha $ | thermal diffusivity of fluid (m ^{2}·s^{−1}) |

$\mu $ | dynamic viscosity (kg·m ^{−1}·s^{−1}) |

$\nu $ | kinematic viscosity (m ^{2}·s^{−1}) |

$\rho $ | density of the fluid (kg·m ^{−3}) |

ϕ | solid volume fraction (-) |

δ | distance between particles (nm) |

∞ | free stream (-) |

## Subscripts/Superscripts

B | Brownian (-) |

eff | effective |

f | fluid |

P | particle-pressure (-) |

s | solid |

w | wall |

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**MDPI and ACS Style**

Darbari, B.; Rashidi, S.; Abolfazli Esfahani, J.
Sensitivity Analysis of Entropy Generation in Nanofluid Flow inside a Channel by Response Surface Methodology. *Entropy* **2016**, *18*, 52.
https://doi.org/10.3390/e18020052

**AMA Style**

Darbari B, Rashidi S, Abolfazli Esfahani J.
Sensitivity Analysis of Entropy Generation in Nanofluid Flow inside a Channel by Response Surface Methodology. *Entropy*. 2016; 18(2):52.
https://doi.org/10.3390/e18020052

**Chicago/Turabian Style**

Darbari, Bijan, Saman Rashidi, and Javad Abolfazli Esfahani.
2016. "Sensitivity Analysis of Entropy Generation in Nanofluid Flow inside a Channel by Response Surface Methodology" *Entropy* 18, no. 2: 52.
https://doi.org/10.3390/e18020052