Entropy generation minimization: Darcy-Forchheimer nanofluid flow due to curved stretching sheet with partial slip

doi:10.1016/j.icheatmasstransfer.2019.104445

International Communications in Heat and Mass Transfer

Volume 111, February 2020, 104445

https://doi.org/10.1016/j.icheatmasstransfer.2019.104445 Get rights and content

Abstract

This paper explores the Darcy-Forchheimer two-dimensional flow of nanofluid due to curved stretching sheet. Brownian motion and thermophoresis effects are taken in to account. Bejan number and entropy generation are analyzed in presence of MHD, convective boundary conditions, partial slip and viscous dissipation. Nonlinear ordinary differential systems are developed through transformations. Convergent series solutions are constructed by using NDSolve of MATHEMATICA. Behavior of involved variables on flow characteristics is shown through graphs. Velocity reduces for higher slip parameter and Forchheimer number. Temperature and concentration have direct relation with thermal and solutal Biot numbers. An increase in entropy generation is seen for higher curvature parameter, porosity parameter and Brinkman number. Decrease in Bejan number is observed for higher estimations of Brinkman number and slip parameter. Comparative study of present results with previous information in a limiting sense is made.

Introduction

Entropy generation now is well organized concept in thermodynamics. It helps us the ways to reduce or control the irreversibility factor in system. It is specially used in channels to study the heat transfer at both nano and micro levels. Entropy generation minimization (EGM) is utilized in curved pipes, chillers, gas turbines, fuel cells etc. Bejan number is able to predict the losses due to friction and thermal irreversibility. Bejan [1,2] initially gave idea of entropy generation with the dissipation of energy in convective flow. Entropy generation optimization in electroosmotic flow is analyzed in the studies [3,5]. Khan et al. [4] studied the irreversibility in hybrid nanofluid flow. Flow of Prandtl-Eyring fluid with entropy generation is elucidated by Khan et al. [6]. Analysis of entropy generation in nano fluids through heated plate of heat exchanger in wavy channel flow is studied by Esfahani et al. [7]. They concluded that the amplitude and Reynolds number have more effect on thermal irreversibility. Rashidi et al. [8] analyzed the thermal irreversibility in the solar still with nanoparticles. Computation of irreversibility in Darcy's hybrid nanofluid flow is done by khan et al. [9]. Ellahi et al. [10] studied entropy generation in boundary layer slip flow due to moving plate. Khan et al. [11] analyzed the entropy generation in peristaltic transport of nanofluid with single and multi wall carbon nanotubes. Xie and Jian [12] carried out analysis for micro parallel channels in view of entropy of two-layer magnetohydrodynamic electroosmotic flow. Ranjit and Shit [13] discussed the heat transfer irreversibility in electro-osmotic flow pumping by uniform peristaltic wave. The study shows that under different physical conditions the generation of entropy is maximum near the wall. Khan et al. [[14], [15], [16]] studied the irreversibility in case of non-Newtonian fluid model and nanofluid.

Fluid flows filling porous medium have wide applications in mechanical system like in enhanced oil recovery, underground spreading of substance misuse, nuclear waste document, grain amassing, warm insurance outlining, redesigned recovery of oil stores and warmth pipe development. Semi-empirical equation was developed by Darcy for small velocity and under weak porosity conditions. For higher Reynolds number such empirical equation is inadequate. Forchheimer [17] developed relation named as Darcy–Forchheimer equation which is the modified form of Darcy expression. Here nonlinear factor through velocity is introduced. Muskat [18] called it a Forchheimer factor. Some refs. Regarding this study can be seen through [[19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35]].

Our motivation in this article is to find entropy generation in flow by curved stretching sheet with slip effects. To our knowledge, entropy generation in nanofluid flow by curved stretching sheet is not yet discussed. Darcy-Forchheimer effect is further examined here. Additional effects of viscous dissipation, slip condition, MHD and convective boundary conditions are also taken. The entropy generation, velocity, Bejan number, concentration and temperature are analyzed. Solutions for velocity, concentration and temperature are tackled by NDSolve of MATHEMATICA. Physical interpretation is arranged.

Section snippets

Mathematical modeling

Here our main object is to study irreversibility in Darcy-Forchheimer flow of nanofluid by curved stretching sheet of radius R with coiled shape. Partial slip effects, convective condition and viscous dissipation are also studied. Here B₀ is the strength of constant applied magnetic field. Due to curved stretching sheet the governing equations are modeled by curvilinear coordinates (r, s). Due to stretching and slip conditions the velocity in radial direction is set as $U_{w} = as + L (\frac{\partial \overset{⌣}{u}}{\partial r} - \frac{\overset{⌣}{u}}{r + \overset{⌣}{R}})$

Entropy generation

This section deals with the entropy generation in flow by curved sheet with heat transfer irreversibility, viscous dissipation irreversibility, Joule heating irreversibility, porous medium irreversibility and mass transfer irreversibility. Entropy generation in dimensional variables is $\begin{array}{c} S_{gen}^{'''} = \underset{heat transfer irreversibility}{\underset{⏟}{\frac{k}{{\overset{⌣}{T}}_{\infty}^{2}} {(\frac{\partial \overset{⌣}{T}}{\partial r})}^{2}}} + \underset{viscous dissipation irreversibility}{\underset{⏟}{\frac{μ}{{\overset{⌣}{T}}_{\infty}} Φ}} + \underset{mass transfer irreversibility}{\underset{⏟}{\frac{RD}{C_{\infty}} {(\frac{\partial \overset{⌣}{C}}{\partial r})}^{2} + \frac{RD}{{\overset{⌣}{T}}_{\infty}} (\frac{\partial \overset{⌣}{C}}{\partial r} \frac{\partial \overset{⌣}{T}}{\partial r})}} \\ \underset{Joule heating irreversibility}{+ \underset{⏟}{\frac{σ B^{2} {\overset{⌣}{u}}^{2}}{{\overset{⌣}{T}}_{\infty}}}} + \underset{porous medium}{\underset{⏟}{\frac{μ {\overset{⌣}{u}}^{2}}{{\overset{⌣}{T}}_{\infty} k^{*}}}} \end{array}\}$

Discussion

Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12, Fig. 13, Fig. 14, Fig. 15, Fig. 16, Fig. 17, Fig. 18, Fig. 19, Fig. 20, Fig. 21, Fig. 22 are sketched for analysis of velocity, Bejan number, concentration, entropy generation and temperature against involved parameters.

Fig. 2 analyzes the flow characteristics of nanofluid for higher values of magnetic parameter (M = 0, 0.5, 1.0, 1.5, 2.0). One can easily see the reduction in velocity $({\tilde{f}}^{'} (ξ))$ for higher

Conclusions

Main findings of the problem are as follows:

•
Increase in velocity $(\tilde{f} (ξ))$ of the fluid is noticed against curvature parameter (A) while opposite trend is seen for Forchheimer number (F_r).
•
Temperature enhances for larger curvature parameter (A) and thermal Biot number (γ₁).
•
Entropy generation (N_G(ξ)) is maximum for higher curvature parameter (A), porous medium (β) and Brinkman number (Br).
•
Bejan number (Be) enhances for curvature parameter (A) and slip parameter (L₁) while it decays for higher Brinkman

Declaration of Competing Interest

There is no conflict of interest amond authors.

Acknowledgments

This work was supported by the Deanship of Scientific Research (DSR), Saudi Arabia, King Abdulaziz University, Jeddah, Saudi Arabia, under Grant No. (D-146-130-1441). The authors, therefore, gratefully acknowledge the DSR technical and financial support.

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Entropy generation minimization: Darcy-Forchheimer nanofluid flow due to curved stretching sheet with partial slip

Abstract

Introduction

Section snippets

Mathematical modeling

Entropy generation

Discussion

Conclusions

Declaration of Competing Interest

Acknowledgments

Energy

Int. J. Heat Mass Transf.

Comp. Methods Programs Biomedicine

Comp. Methods Programs Biomedicine

Int. J. Heat Mass Transf.

Renew. Energy

Energy

Energy

J. Mol. Liq.

J. Colloid Interface Sci.

J. Mol. Liq.

Int. J. Therm. Sci.

Results Phys.

J. Mol. Liq.

Phys. Letters A

Phys. Letters A

Results Phys.