A dissipative and entropy-optimized MHD nanomaterial mixed convective flow for engineering applications

Abstract

Background and objective: Nanomaterials play significant roles in numerous industrial and engineering applications, like nuclear plants, paper production, thermal power plants, glass fibres, manufacturing of medicines, medical instruments, micro-electronics and polymer sheet extrusion. In view of such important applications, in this study, we discuss the magnetohydrodynamic flow of a nanofluid over an inclined surface by employing the Darcy–Forchheimer model. The Buongiorno model is applied to understand the various important aspects of the nanofluid. Radiation, magnetic field, dissipation and entropy generation in a chemically reactive flow are also discussed. Methodology: The governing nonlinear expressions were transformed into a dimensionless system through adequate transformations. The obtained non-dimensional systems were computed by the NDSolve approach. Results: Physical illustrations for the flow, temperature, concentration and entropy rate via emerging variables were examined. Here an enhancement in velocity was seen for the mixed convection variable, while opposite impacts on flow and temperature were noticed through the Hartman number. A higher Eckert number was obtained with a rise in temperature, while a decrease in concentration was noticed for the thermophoresis variable. An augmentation in the entropy rate was detected for radiation, while the thermal transport rate was boosted by thermophoresis.

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1 Introduction

Nanofluids are a mixture of nano-sized (1–100 nm) solid particles (metals, carbides, oxides and carbon nanotubes) and conventional fluids (water, lubricant oil, ethylene and propylene glycol). Nanofluids play a vital role in the heat-transfer processes of conventional liquids. Nanomaterials play significant roles in numerous industrial and engineering applications, like nuclear plants, paper production, thermal power plants, glass fibre, manufacturing of medicines, medical instruments, micro-electronics and polymer sheet extrusion. The thermal conduction processes of conventional materials can be improved by the addition of nano-sized particles. Various industrial processes face a low-heat transportation problem caused by the utilization of conventional fluids. This issue is often tackled by using nanofluids and hybrid nanomaterials. The enhancement of the thermal conductivity of conventional materials through the addition of nano-sized metallic particles was first reported by Choi.¹ Buongiorno² later gave a mathematical model and discussed seven slip mechanisms for the enhancement of the thermal characteristics of fluids, with Brownian motion and thermophoresis as the most important factors. Waini et al.³ reported on the magnetohydrodynamic radiative flow of a Reiner–Philippoff nanoliquid subjected to a random motion and thermophoresis. Heat transfer in the electrically conductive flow of a nanoliquid with thermophoresis and random motion was discussed by Kalpana et al.⁴ The hydromagnetic convective flow of an Eyring–Powell nanomaterial considering multiple diffusions was analysed by Patil and Kulkarni.⁵ Random and thermophoresis diffusions for a Casson nanoliquid flow considering a gyrotactic microorganism was studied by Upreti et al.⁶ Ohmic heating in the magnetized convective flow of a Reiner–Rivlin nanoliquid considering the entropy rate was explored by Khan et al.⁷ Cattaneo–Christov fluxes for the bioconvective flow of a Maxwell nanomaterial subjected to the Arrhenius activation energy were examined by Bagh et al.⁸ Some further related studies on nanomaterials are listed in ref. 9–20.

Measurement of the wastage energy in any thermodynamical system refers to the entropy. Entropy generation is an important thermodynamical approach that is used to analyse the thermal performance of systems in industrial and engineering fields. Entropy is produced due to heat transfer, molecule collision, thermal radiation, Joule heating, diffusion, fluid friction, and spinning motion, etc. In any thermodynamical system, a significant part of the thermal energy is wasted, meaning it is not fully utilized for useful work. Therefore, several researchers have paid attention to this important issue. Bejan^21,22 was the first to introduce entropy minimization in convective flow. Entropy in the squeezing flow of a CNT nanoliquid was examined by Dawar et al.²³ Hayat et al.²⁴ investigated an unsteady nanoliquid flow considering Sort and Dufour characteristics. A few important studies concerning the entropy rate are mentioned in ref. 25–32.

Our prime objective here was to analyse the magnetohydrodynamic convective flow of a nanoliquid by an inclined surface. The Darcy–Forchheimer relation was considered and is discussed in this context. Radiation, magnetic field, and dissipation were considered in relation to energy. Brownian motion and thermophoresis diffusion were accounted for. The entropy rate for a chemically reactive flow was studied. Nonlinear partial differential equations were reduced to non-dimensional systems through suitable transformation. The ND-solve technique was implemented for the computations. Graphical discussions were used to consider the flow, Nusselt number, entropy rate, drag force, concentration, and temperature via sundry variables.

2 Statement

The hydromagnetic mixed convective flow of a nanoliquid over an inclined surface is studied. The Darcy–Forchheimer relation was considered here. Random and thermophoresis diffusions were accounted. Dissipation, magnetic field Joule heating, first-order reaction, and radiation were also taken into account. Entropy optimization for the chemically reactive flow was elaborated. Flow tests in the presence of a constant magnetic field were conducted. A sketch of the problem is presented in Fig. 1.³³

Fig. 1 Flow sketch.

Here, the governing expressions satisfy:^24–28

with

Letting:^34,35

we get

subject to conditions

Here the non-dimensional parameters are , , , , , , , , , , , , , and .

2.1 Solution

First-order truncation:

In 1st order truncation we assume the derivatives w.r.t. ζ are zero. i.e. then eqn (7)–(10) become,

subject to the conditions

Second-order truncation:

Here we consider , , , , , and denote by primes. Expressions (7)–(10) become

Differentiating eqn (15)–(18) w.r.t. ζ and neglecting the terms , , , , and we get

3 Physical quantities

3.1 Coefficient of skin fraction

One may write

with shear stress τ_w given by

We now have

3.2 Nusselt number

One may define

in which heat flux q_w obeys

Finally, we may have

Re_x^−1/2Nu = −(1 + Rd)θ′(0). (28)

3.3 Sherwood number

Mathematically

in which mass flux J_w obeys

From eqn (21) and (22) we have (31)

Re_x^−1/2Sh = −ϕ′(0). (31)

In which depicts the local Reynolds number.

4 Entropy rate

Mathematically one can write

In the dimensionless version, we have

In the above expression, the dimensionless variables are , , , and .

5 Analysis

The ND-solve technique was implemented for the solution. The velocity, surface drag force, concentration, Nusselt number, temperature, Bejan number, and entropy rate against the emerging variables were explored. A comparative analysis of the recent outcomes with those by Wang³⁶ is highlighted in Table 1. Here an outstanding consensus could be noticed.

Table 1 Comparative analysis of Re_x^−1/2Nu with the outcomes reported by Wang³⁶

Pr	Wang^36	Recent outcomes
0.07	0.0656	0.065621
0.20	0.1691	0.169109
0.70	0.4539	0.453901
2.00	0.9114	0.911409
7.00	1.8954	1.895412
20.00	3.3539	3.353925

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5.1 Velocity

A plot of the velocity against Hartman number (Ha) is depicted in Fig. 2. A decreasing behavior was noticed for higher Hartman numbers due to the increment in resistive force. The effect of (Fr) on (f′(ζ,η)) is portrayed in Fig. 3. A reduction in velocity was noticed for higher Forchheimer (Fr) numbers due to the stronger resistance produced in the fluid flow. The influence of velocity (f′(ζ,η)) via the porosity parameter (λ) is illustrated in Fig. 4. An increase in the porosity variable corresponded to a more intensive viscous force Therefore the velocity (f′(ζ,η)) decayed. A plot of the velocity against the mixed convection variable is illustrated in Fig. 5. An increasing impact on the velocity was noted through utilizing the mixed convection variable.

Fig. 2 f′(ζ,η) via Ha.

Fig. 3 f′(ζ,η) via Fr.

Fig. 4 f′(ζ,η) via λ.

Fig. 5 f′(ζ,η) via Ri.

5.2 Temperature

A plot of temperature versus the Hartman number (Ha) in shown in Fig. 6. A larger approximation of the Hartman number increases the Lorentz force, which creates extra heat in the system. Thus the temperature increased. Fig. 7 exhibits the temperature (θ(ζ,η)) performance versus the (Ec) Eckert number. The Eckert number increases the kinematic energy of the thermal system, which increases the temperature. A reduction in thermal diffusivity was observed against the Prandtl number, which decreased the temperature (see Fig. 8). The temperature (θ(ζ,η)) behavior with the thermophoresis parameter (Nt) is displayed in Fig. 9. The thermophoresis variable increased the temperature distribution. Fig. 10 shows the impact of radiation (Rd) on the temperature. An enhancement in the thermal field was observed with the radiation variable due to the production of additional energy in the system.

Fig. 6 θ(ζ,η) via Ha.

Fig. 7 θ(ζ,η) via Ec.

Fig. 8 θ(ζ,η) via Pr.

Fig. 9 θ(ζ,η) via Nt.

Fig. 10 θ(ζ,η) via Rd.

5.3 Concentration

The influence of concentration (ϕ(ζ,η)) with the reaction parameter (k₁) is shown in Fig. 11. An enhancement in concentration was observed with the reaction parameter. The random motion (Nb) impact on the concentration is explored in Fig. 12. Here, the concentration of the nanoliquid had an enhancing effect through the random motion variable. Fig. 13 depicts the concentration performance for the Schmidt number. An augmentation in concentration was seen through the Schmidt number. Fig. 14 depicts (ϕ(ζ,η)) against (Nt). A decreasing trend for concentration was seen through the thermophoresis variable variation.

Fig. 11 ϕ(ζ,η) via k₁.

Fig. 12 ϕ(ζ,η) via Nb.

Fig. 13 ϕ(ζ,η) via Sc.

Fig. 14 ϕ(ζ,η) via Nt.

6 Entropy rate

Fig. 15 illustrates the entropy performance considering the Brinkman (Br) number. Clearly, the entropy rate increased with the Brinkman number. The effect of the variation of the Hartman (Ha) number on (S_g) is portrayed in Fig. 16. A higher estimation of Hartman number decreased the entropy rate. Fig. 17 shows the impact of (S_g) via radiation. A decrease in entropy was detected against the radiation variable. From Fig. 18, it could be noticed that the entropy reduced with higher porosity.

Fig. 15 S _g via Br.

Fig. 16 S _g via Ha.

Fig. 17 S _g via Rd.

Fig. 18 S _g via λ.

7 Quantities under interest

Graphical illustrations for the skin friction coefficient and Nusselt number are addressed.

7.1 Skin friction coefficient

Fig. 19–21 display the impacts of the velocity gradient with the Hartman number, mixed convection variable, and Forchheimer number. An increasing behavior was observed against rising values of (Ha) and (Ri), but the opposite scenario held for (Fr).

Fig. 19 via Ha.

Fig. 20 via Ri.

Fig. 21 via Fr.

7.2 Heat transfer rate

The impact of emerging variables ((Nt), (Nb) and (Ec)) on (Re^1/2Nu) are displayed in Fig. 22–24. Clearly, the thermal transport rate increased versus higher random motion (Nb) and thermophoresis (Nb) variables. From Fig. 7b, it could be detected that the heat transport rate declined for higher Eckert numbers.

Fig. 22 Re^1/2Nu via Ha.

Fig. 23 Re^1/2Nu via Nb.

Fig. 24 Re^1/2Nu via Nt.

8 Conclusions

Key observation are listed below.

➢ Mixed convection and porosity had opposite impacts on the flow.

➢ Hartman number yielded an opposite outcome for the flow and entropy rate.

➢ Flow was decreased by Forchheimer number.

➢ An enhancement in Hartman number led to enhanced drag force and higher temperature.

➢ Higher Eckert numbers led to an enhancement in temperature.

➢ Prandtl number led to a reduction in temperature.

➢ An opposite trend for entropy and thermal field was witnessed through considering radiation.

➢ An enhancement in temperature and the heat transport rate were noticed for the thermophoresis variable.

➢ Concentration was reduced versus a higher thermophoresis variable.

➢ Increasing trend for concentration was noted through the Schmidt number.

➢ Brinkman number corresponded an enhancement in entropy.

➢ Thermal transport rate surged upwards against the random motion variable.

➢ Decrease in entropy rate was witnessed versus the Brinkman number.

➢ Higher Eckert numbers decreased the temperature gradient.

Further investigations in the future should consider bioconvection and motile microorganisms, of which almost nothing is known yet. Such attempts for mixed, bioconvection and Marangoni convection, activation energy and slip and melting conditions should be examined.

Abbreviations

u,v	Velocity components
k r	Reaction rate
x,y	Cartesian coordinates
L	Reference length
Ω	Inclination angle
R	Molar gas constant
ν f	Kinematic viscosity
ψ	Stream function
g	Gravity
Rex	Local Reynolds number
β	Thermal expansion coefficient
Ha	Magnetic variable
ρ f	Density
Gr	Grashoff number
σ f	Electrical conductivity
Ri	Mixed convection variable
B 0	Magnetic field strength
Re	Reynolds number
ρ p	Nanoparticle density
λ	Porosity variable
ρ f ∞	Nanofluid density
Fr	Forchheimer number
	Inertia coefficient
Nt	Thermophoresis variable
C b	Drag force coefficient
Pr	Prandtl number
k p	Porous medium permeability
Rd	Radiation variable
T	Temperature
Nb	Brownian motion variable
T w	Wall temperature
Ec	Eckert number
T ∞	Ambient temperature
k 1	Reaction variable
α f	Thermal diffusivity
Br	Brinkman number
	Ratio of heat capacitance
C f	Skin friction coefficient
σ*	Stefan–Boltzmann constant
τ w	Wall shear stress
D T	Thermophoresis coefficient
Nu	Nusselt number
k*	Mean absorption coefficient
Sh	Sherwood number
D B	Brownian diffusion coefficient
q w	Heat flux
c p	Specific heat
J w	Mass flux
C	Concentration
S g	Entropy rate
C w	Wall concentration
α 1	Temperature difference variable
C ∞	Ambient concentration
L 1	Diffusion variable
u ∞	Reference velocity
α 2	Concentration difference variable

Conflicts of interest

There are no conflicts to declare.

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