ARTICLE

J. Cent. South Univ. (2019) 26: 1161-1171

DOI: https://doi.org/10.1007/s11771-019-4078-7

Effects of homogeneous-heterogeneous reactions and thermal radiation on magneto-hydrodynamic Cu-water nanofluid flow over an expanding flat plate with non-uniform heat source

DOGONCHI A S¹, CHAMKHA Ali J^{2, 3}, HASHEMI-TILEHNOEE M¹,SEYYEDI S M¹, RIZWAN-UL-HAQ⁴, GANJI D D⁵

1. Department of Mechanical Engineering, Aliabad Katoul Branch, Islamic Azad University,Aliabad Katoul, Iran;

2. Mechanical Engineering Department, Prince Sultan Endowment for Energy and Environment, Prince Mohammad Bin Fahd University, Al-Khobar 31952, Saudi Arabia;

3. RAK Research and Innovation Center, American University of Ras Al Khaimah, P.O. Box 10021,Ras Al Khaimah, United Arab Emirates;

4. Department of Electrical Engineering, Bahria University, Islamabad, Pakistan;

5. Mechanical Engineering Department, Babol Noshirvani University of Technology, Babol, Iran

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: This study presents the effect of non-uniform heat source on the magneto-hydrodynamic flow of nanofluid across an expanding plate with consideration of the homogeneous-heterogeneous reactions and thermal radiation effects. A nanofluid’s dynamic viscosity and effective thermal conductivity are specified with Corcione correlation.According to this correlation, the thermal conductivity is carried out by the Brownian motion. Similarity transformations reduce the governing equations concerned with energy, momentum, and concentration of nanofluid and then numerically solved. The influences of the effective parameters, e.g., the internal heat source parameters, the volume fraction of nanofluid, the radiation parameter, the homogeneous reaction parameter, the magnetic parameter, the heterogeneous parameter and the Schmidt number are studied on the heat and flow transfer features. Further, regarding the effective parameters of the present work, the correlation for the Nusselt number has been developed. The outcomes illustrate that with the raising of the heterogeneous parameter and the homogeneous reaction parameter, the concentration profile diminishes. In addition, the outcomes point to a reverse relationship between the Nusselt number and the internal heat source parameters.

Key words: nanofluid; non-uniform heat source; homogeneous-heterogeneous reactions; thermal radiation; Brownian motion

Cite this article as: DOGONCHI A S, CHAMKHA Ali J, HASHEMI-TILEHNOEE M, SEYYEDI S M, RIZWAN- UL-HAQ, GANJI D D. Effects of homogeneous-heterogeneous reactions and thermal radiation on magneto- hydrodynamic Cu-water nanofluid flow over an expanding flat plate with non-uniform heat source [J]. Journal of Central South University, 2019, 26(5): 1161–1171. DOI: https://doi.org/10.1007/s11771-019-4078-7.

1 Introduction

It is well known that many engineering systems have low thermal performance. There are several methods to over this problem. One of these methods is the use of nanofluids. The nanofluid is the combination of base fluids and metallic nano-scale particles. Because this combination has the thermal conductivity higher than regular fluids,it has gained the notice of many researchers [1–37]. CHAMKHA et al [1] looked at the effect of magnetic field on mixed convection in lid-driven trapezoidal cavities filled with Cu-water nanofluid. SHEREMET et al [2] considered the free convection of a nanofluid in a porous wavy cavity using the mathematical model devised by Buongiorno. They also considered the thermal dispersion impact in their study. SHEREMET et al [3] analyzed nanofluid free convection in a porous cavity with the partially heated wavy wall under the influence of thermophoresis. TAYEBI et al [4] examined the buoyancy-driven heat transfer improvement in a sinusoidal heated cavity using hybrid nanofluid. GHASEMI et al [5] have studied entropy generation of nanofluid natural convection within a porous enclosure. They [6] also investigated magneto-hydrodynamic free convection of nanofluid within porous enclosures under magnetic fields with different conductivity ratios. DOGONCHI et al [7] studied the impact of Joule heating on two-phase nanofluid flow among non-parallel plates. They also investigated how buoyancy nanofluid flow over a stretching sheet was impacted by the Cattaneo–Christov heat flux model [8]. ALSABERY et al [9] utilized Buongiorno’s two-phase model and explored mixed convection of Al₂O₃-water nanofluid in a double lid-driven square cavity. ELLAHI et al [10] studied the aggregation effect on water-based Al₂O₃- nanofluid over porous wedge in mixed convection. The effects of buoyancy impact and thermal radiation on magneto-hydrodynamic nanofluid flow above an extended area considered by RASHIDI et al [11]. BHATTI et al [12] explored impacts of thermal radiation and thermo-diffusion on Williamson nanofluid above a permeable shrinking/ stretching surface. DOGONCHI et al [13] explored the influences of thermal radiation and Hartman numbers on Go-water nanofluid in a porous medium. GHALAMBAZ et al [14] investigated the phase-change heat transfer in a cavity that was heated from below in the presence of nanoparticles. The flow of magneto-hydrodynamic nanofluid between parallel plates is taken into consideration thermal radiation and Cattaneo-Christov heat flux impacts examined by DOGONCHI et al [15]. The entropy generation of nanofluid flow past a stretching permeable surface was explored by ABOLBASHARI et al [16]. Cu-water nanofluid flow and heat transfer in a converging-diverging channel have been studied by SARI et al [17]. ZAIB et al [18] have perused transient boundary layer nanofluid flow and heat transfer over a contracting wall. YOUSOFVAND et al [19] have explored magneto-hydrodynamic mixed convection and entropy generation of electromagnetic pump filled with nanofluid. Thermal radiation impact on magneto Carreau nanofluid has been examined by WAQAS et al [20]. Radiative flow of nanofluid considering partial slip by rotating disk has been studied by HAYAT et al [21]. In another study, HAYAT et al [22] explored stagnation point third- grade nanofluid flow under magnetic field.

Recently, the study of flow in the presence of chemical reaction due to its extensive applications, for instance, the dispersion and formation of fog, food and chemical processing, the hydrometallurgical industry, water and air pollution, etc., has become one of the favourite topics of researchers. In such procedure, chemical reaction leads to the molecular dispersal of species which cannot be ignored. In many systems such as biochemical systems, catalysis, and combustion, chemical reactions include homogeneous heterog- eneous reactions. HAYAT et al [38] investigated the nanofluid flow in the presence of heterogeneous- homogeneous reactions. KAMESWARAN et al [39] examined the effect of heterogeneous-homogeneous reactions in a nanofluid flow over a porous expanding sheet. MERKIN [40] explored heterogeneous homogeneous reactions in boundary layer flow. SHEIKH et al [41] analyzed homogeneous heterogeneous reactions under the impact of stagnation point flow of Casson fluid flow due to a stretching/shrinking surface with uniform suction and slip. SAJID et al [42] explored the effects of heterogeneous-homogeneous reactions on magneto-hydrodynamics nanofluid flow taking into account thermal radiation. HAYAT et al [43] inspected the nanofluid flow through a permeable area under conditions of convection and heterogeneous- homogeneous reactions.

In recent years, boundary layer flow and heat transfer over an expanding wall is considered by many researchers [44–49] due to extensive applications in engineering and industrial processes such as glass blowing and crystal growing, materials fabricated by extrusion, paper production. BHATTACHARYYA et al [44] investigated magneto-hydrodynamic boundary layer flow and heat transfer over a wall under slip condition. ISHAK [45] has studied the influence of thermal radiation on boundary layer micropolar fluid flow and heat transfer over an expanding wall. Micropolar fluid flow and heat transfer over an expanding surface under internal heat generation and viscous dissipation impacts were examined by MOHAMMADEIN et al [46]. BHARGAVA et al [47] have explored micropolar fluid flow over a nonlinear expanding wall. DOGONCHI et al [48] have studied the influence of thermal radiation on buoyancy flow and heat transfer of nanofluid over an expanding wall. In their study the impact of Brownian motion is also considered. Nanofluid film over a transient stretching wall was studied by NARAYANA et al [49].

This reported review portrays that the consequences of non-uniform heat source and homogeneous-heterogeneous reactions on the magneto-hydrodynamic Cu-water nanofluid flow across a stretching plate considering the thermal radiation effect have not investigated yet. So, this paper studies the consequences of a non-uniform heat source on the magneto-hydrodynamic Cu-water nanofluid flow across a stretching plate considering the thermal radiation effect and homogeneous-heterogeneous reactions. The CORCIONE model [50] is also used to simulate the nanofluid. The impacts of effective parameters including the radiation parameter, the nanofluid volume fraction, the internal heat source parameters, the magnetic parameter, the heterogeneous parameter, the homogeneous reaction parameter, and the Schmidt number on heat transfer and flow characteristics are perused and presented graphically.

2 Problem description

This work considers a steady, Newtonian, incompressible viscous and two-dimensional laminar flow of a Cu-water nanofluid over a stretching flat plate (Figure 1).

The plate’s surface and the free stream are held at a stable temperature T_w and T_∞, respectively such that T_w>T_∞. Also, the considered impacts are the homogeneous-heterogeneous reactions, non- uniform internal heat generation and/or absorption as well as thermal radiation. Further, we assume the impact of a consistent magnetic field with strength of B₀. The homogeneous-heterogeneous reactions [38] are stated as:

rate=ab²k_c                  (1)

rate=ak_s                                        (2)

where b and a are the concentrations of the species B and A, respectively and k_i (i=c, s) explains constant rate. Employing the regular boundary layer approximations, the governing equations for conservative momentum, energy and concentration are stated as: [38, 39, 44]

                              (3)

          (4)

     (5)

                 (6)

                 (7)

In these equations, u and v denote x- and y-direction velocities correspondingly; T, σ_f, B₀, U_∞ and q_rad are the temperature, the fluid’s electric conductivity, the magnetic field, the free stream velocity and the radiative heat flux, respectively; the diffusion coefficients are D_A and D_B. Further, q′″ is the inconstant heat absorbed or generated per unit volume. The value of q′″ is defined as:

       (8)

Here B* and A* denote the temperature- and space- dependent heat absorption and/or generation parameters, respectively. It is worth mentioning that A* and B* positive values define the internal heat source and the negative values of them define the internal heat sink.

Figure 1 Schematic of problem

In accordance with the Rosseland approximation, q_rad is determined as:

                    (9)

where σ* andare the Stefan-Boltzmann constant and the mean absorption coefficient of the nanofluid, respectively. Using and Eqs. (8) and (9), Eq. (5) are turned to:

 (10)

In Eq. (10), ρ_nf and (ρC_p)_nf are the nanofluid effective density and heat capacitance, respectively, which are defined as:

                      (11)

            (12)

The nanofluid’s effective viscosity and thermal conductivity are specified by the CORCIONE correlation [50], in which the following is considered:the impacts of the particle volume fraction, particle size, temperature dependence and types of particle and combinations of base fluid on thermal conductivity.

           (13)

     (14)

                           (15)

                            (16)

here Pr, T_fr and k_b are the Prandtl number of the base liquid, the freezing point of the base liquid, and the Boltzmann’s constant, respectively. Table 1 shows the base fluid and nanoparticles’ characteristics [51].

The pertinent boundary conditions are:

Table 1 Thermo-physical properties of water and nanoparticle at T=310 K [51]

,

     (17)

We define these parameters:

        (18)

According to Eq. (18), the governing equations are turned to:

             (19)

    (20)

                   (21)

                  (22)

Here A₁ and A₂ denote constant parameters which are defined by:

                (23)

subject to the boundary conditions:

          (24)

here M_n, N, Sc, K, K₁ and δ comprise the magnetic parameter, the radiation parameter, the Schmidt number, the strength of homogeneous reaction, the strength of heterogeneous reaction and the diffusion coefficient ratio, respectively.

In many cases, if A and B are the same size, so D_A and D_B should be the same. Therefore, g(η)+h(η)=1. Finally, by applying boundary conditions Eqs. (21) and (22) are simplified to:

                 (25)

                   (26)

The Nusselt number as the one of the main interest of the researchers is defined as follows:

       (27)

Applying Eq. (18), Eq. (27) is turned to:

                   (28)

3 Results and discussion

This section’s aim is to provide the influences of the dynamic parameters of this study, the radiation parameter (N), the nanofluid volume fraction (f), the internal heat source parameters (A^* and B^*), the magnetic parameter (M_n), the heterogeneous parameter (K₁), the homogeneous reaction parameter (K), and the Schmidt number (Sc) on the heat and flow transfer features of the magneto-hydrodynamic Cu-water nanofluid above a stretching flat plate considering Brownian motion. In this study, we also consider the impacts of the homogeneous-heterogeneous reactions, non- uniform heat source, and thermal radiation. The Corcione model [50] is used to simulate nanofluid. We solve the governing equations with the fourth-order Runge-Kutta method. In addition, the present outcomes are compared with the outcomes of BHATTACHARYYA et al [44] for pure fluid, N=0, A^*=0, B^*=0, Sc=0 and K=0. The collation of the outcomes is portrayed in Table 2 where a great conciliation is viewed. Furthermore, the base fluid and nanoparticles’ thermophysical properties are presented in Table 1.

Figure 2 illuminates the impact of the volume fraction of nanofluid on the concentration and velocity profiles. In this work, we assumed that there is no slip between nanoparticles and the base fluid and also that they are in the thermal equilibrium. The density of the base fluid rises when nanoparticles are added to it, making the base fluid denser. On the other hand, we know that the denser fluids have lower velocity than regular fluids. So, as seen in Figure 2, as the nanofluid’s volume fraction rises, the velocity profile diminishes. Further, with the increase of the nanofluid’s volume fraction, the concentration profile decreases, and its alternation is more gradual than when the volume fraction of nanofluid is at a higher value.

Table 2 Comparison between present and BHATTACHARYYA et al [44] results for f ′(η) (f=0, N=0, A^*=0, B^*=0, Sc=0, K=0)

Figure 2 Effect of volume fraction of nanofluid on velocity and concentration profiles

Figure 3 illuminates the volume fraction of nanofluid and radiation parameter’s impacts on the temperature profile and Nusselt number. First of all, we should point out the fact that the Nusselt number is an abating function of thermal boundary layer thickness. As the radiation parameter and volume fraction of nanofluid rise, the temperature profile rises and subsequently, the thickness of the thermal boundary layer also rises. So, it is expected that the Nusselt number will diminish with an increase of the volume fraction of nanofluid and radiation parameter. But this did not happen. In accordance with Eq. (28), the Nusselt number is delineated as the multiplication of three terms: (k_nf/k_f), (1+(4/3)N) and θ′(0). It is clear that as the radiation parameter and volume fraction of nanofluid increase, the (k_nf/k_f) and (1+(4/3)N) increase, respectively. On the other hand, the augmentation of these terms will prevail the reduction of the θ′(0). So, the Nusselt number will grow and the volume fraction of nanofluid and radiation parameter will increase.

Figure 4 illuminates the effects of A^* and B^* on the Nusselt number and the temperature profile. It should be mentioned that in this work we consider the internal heat source case (A^* and B^*>0). Because there is a heat source in the system, it is anticipated that with an increase of heat source parameters, the temperature profile increases. This is shown in Figure 3. Also, it can be surmised that the impact of A^* on the profile of the temperature is more prominent than B^*. On the other hand, since the Nusselt number and the thickness of the thermal boundary layer have an inverse relationship, the heat transfer decreases with the ascending of the A^* and B^*.

Figure 3 Effects of volume fraction of nanofluid and radiation parameters on temperature profile and Nusselt number

Figure 4 Effects of internal heat source parameters on temperature profile and Nusselt number

Figures 5 and 6 illustrate the impact of the magnetic parameter on the temperature and concentration profiles, velocity, and Nusselt number. It is found that the concentration and velocity profiles diminish with an increase in the magnetic parameter while the temperature profile has a reverse trend and the variation is more progressive than when the magnetic parameter is set at a lower value. Moreover, as the magnetic parameter increases, the Nusselt number goes up because the thickness of the thermal boundary layer diminishes with the ascendancy of the magnetic parameter. Figure 6 shows the impacts of Schmidt number, homogeneous reaction, and heterogeneous parameters on concentration profile. As shown the minimum value of the concentration happens close to the surface and it diminishes with increasing of the homogeneous reaction and heterogeneous parameters while it ascends with the ascending of the Schmidt number.

Figure 5 Effect of magnetic parameter on velocity and concentration profiles

Figure 6 Effect of magnetic parameter on temperature profile and Nusselt number

Finally, we obtain the Nusselt number correlation in terms of active parameters of the present study which is defined as follows:

(29)

Figure 7 Effects of homogeneous reaction and heterogeneous parameters and Schmidt number on concentration profile

The R-squared for this correlation is equal to 0.9955. The above correlation is valid for M_n=1–3, A^*=0.05–0.15, B^*=0.05–0.15, f=0.01–0.03 and N=1–3.

4 Conclusions

The problem of the magneto-hydrodynamic Cu-water nanofluid across a stretching flat plate in consideration of Brownian motion is studied numerically. The impacts of the homogeneous- heterogeneous reactions, non-uniform heat source, and thermal radiation are also taken into consideration. Further, the correlation for the Nusselt number with regards to active parameters is obtained. The primary results of this study follows:

1) The velocity profile augments by increasing of the M_n and it decreases with f.

2) The concentration profile augments with the increasing of the M_n and Sc and it reduces with the raising the values of K, K₁ and f .

3) The temperature profile is an ascending function of the A^*, B^*, N and f while it is a diminishing function of the M_n.

4) Nusselt number is proportional with M_n, N and f while it is against the parameters A^* and B^*.

Nomenclature

A, B

Chemical species

A₁, A₂

Constant parameters

A^*, B^*

Internal heat source parameters

B₀

Magnetic field

C_p

Specific heat

D_A, D_B

Diffusion coefficients of chemical species

d_s

particle size

f

Dimensionless velocity

g

Dimensionless concentration

K₁

Heterogeneous parameter

K

Homogeneous reaction parameter

k

Thermal conductivity

k^*

Mean absorption coefficient

k_b

Boltzmann’s constant

M_n

Magnetic parameter

N

Radiation parameter

Nu

Nusselt number

Pr

Prandtl number

q_rad

Radiative heat flux

Sc

Schmidt number

T

Temperature

T_fr

Freezing point of the base liquid

u, v

Velocity components in x and y directions, respectively

θ

Dimensionless temperature

f

Nanofluid volume fraction

ρ

Density

μ

Dynamic viscosity

ν

Kinematic viscosity

σ

Electrical conductivity

σ^*

Stefan–Boltzmann constant

η

Dimensionless variable

f

Base fluid

nf

Nanofluid

s

Solid nanoparticles

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(Edited by YANG Hua)

中文导读

均相和非均相反应和热辐射对Cu-水纳米流体在非均匀热源平板上磁流体力学的影响

摘要：考虑均相和非均相反应和热辐射效应，研究了非均匀热源对纳米流体在膨胀板上流动的影响。用Corcione关系式描述了纳米流体的动力黏度和有效热导率的关系。根据这个关系式得出，导热是由布朗运动进行的。通过相似变换减少了与纳米流体的能量、动量和浓度有关的控制方程，并进行数值求解。研究了内热源参数、纳米流体的体积分数、辐射参数、均相反应参数、磁场参数、非均相参数和Schmidt数等对热和流体流动的影响。最后，考虑研究中的有效参数，改进了Nusselt数的关系式。结果表明，随着非均相参数和均相反应参数的变大，浓度曲线变稀。结果表明，Nusselt数与内部热源参数之间存在反向关系。

关键词：纳米流体；非均匀热源；均相反应；热辐射；布朗运动

Received date: 2018-08-29; Accepted date: 2018-10-01

Corresponding author: DOGONCHI A S, MSc; Tel: +98-935-4841866; Email: sattar.dogonchi@yahoo.com; ORCID: 0000-0003-3153- 1851