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Abstract

Recently, a great amount of research was done on wavy conduits due to the importance of heat and flow transfer in wavy channels and conduits, and their special use in various industries. The improvement of heat transfer in these ducts attracted the attention of many researchers, and experimental and numerical studies are devoted. The effect of the presence of a porous medium and magnetic field on forced convection heat transfer and a nanofluid flow in a complex wave-shaped channel with a special boundary condition (Non-uniform with respect to location) was studied for the first time. For this purpose, the heat transfer of a nanofluid in a two-dimensional sinusoidal channel containing a porous medium in the attendance of a magnetic field was analyzed. First, the nanoparticles were optimized using the design of the experiment method, and then modeling was performed using selected optimal particles. To improve heat transfer in a sinusoidal channel, the factors like the injection of optimized nanoparticles, porous medium, and magnetic field usage were used. The equations were discretized using the Fluent software with a finite volume method. The validation of problem outcomes was done using experimental and numerical studies. Porous medium in four different Darcys (10⁻⁵, 10⁻⁴, 10⁻³, and 10⁻²), and applying magnetic fields in four different Hartman (0, 4, 7, and 10) were examined. The results show that the channel wave, and magnetic field intensity improved heat transfer. Furthermore, using a porous medium with high Darcy can increase the Nusselt number. Hence, in the same conditions (Re = 500, Ha = 10) with Darcy number increasing from 10⁻⁵ to 10⁻², Nu becomes equal to 5.011. This proposed system is an industrial-utility system that can increase heat transfer efficiency.

Keywords

Forced heat transfer numerical study finite volume method nanofluid porous medium magnetic field

Introduction

Heat efficiency in heat exchangers is one of the important issues in engineering sciences. In recent years, efforts were made to design and manufacture cooling transducers. The flow and heat transfer (HT) in the sinusoidal channel in the presence of a magnetic field (MF) play an important role in thermal industries. Moreover, porous medium (PM) is widely used to increase thermal efficiency. Adding nanoparticles (NPs) to basic fluids, such as water, changed thermophysical properties, and enhanced thermal efficiency. Simultaneous use of PM and MF to increase HT is a new technique. The thermal conductivity (k) of NPs is always higher than base fluid, and all theories believe that adding some NPs increases k. All experiments and analytical studies observed increased k regardless of type, shape, and volume fraction (VF).¹

One important factor in designing heat exchangers is raising the HT surface ratio to the system volume. Changing the channel geometry to sinusoidal channels can fulfill this important design goal. Sinusoidal ducts increase thermal efficiency which has a relatively small effect on pressure drop. In recent years, wavy (sinusoidal) channels were used to increase HT. Sakanova et al.² studied the thermal performance of nanofluid (NF) in a wavy microchannel to compare it with a rectangular channel. Based on their observations, using wavy walls in microchannels improved HT significantly compared to flat walls. Bitam et al.³ examined turbulent current, and HT inside a wavy tube embedded in a cylinder. The results of their study showed that by enhancing the Reynolds number (Re), HT improved, and wall temperature decreased. Young et al.⁴ numerically examined a sinusoidal cylindrical tube's natural convection HT properties in a rectangular chamber. Based on their observations, the number of sinusoidal tube waves caused thermal efficiency. Sumit et al.⁵ studied the effect of channel wall amplitude on the HT and laminar flow characteristics in a sinusoidal channel and concluded that the Re and increasing the channel amplitude increased the local Nusselt number (Nu) number and the average Nu.

Applying an external MF was one of the active procedures to increase the HT rate.⁶ Previous studies showed that applying an MF could affect the HT of flow. Recently, MHD received much attention and many applications in various industries, including polymer and metallurgy industries, heat exchangers, boilers, chemical catalytic reactors, and solar collectors.⁷ The interaction between an MF and fluid motion was one of the design models to improve HT and flow control. MF caused regular orientation of NF particles, and it caused more communication among NPs. MF affected forced convection HT and natural convection HT. Using an MF in natural convection reduced the velocity near the wall and reduced HT.⁸ But, in forced convection, MF flattened velocity profile, it increased the velocity near the walls and, in turn, increased the HT coefficient. In fact, by increasing the intensity of MF, the velocity near the walls increased, and at the same time, the maximum velocity in the middle of the channel decreased. This tendency of flow to the wall improved the HT rate. In recent years, several studies surveyed the HT of NFs using an MF in channels and heat exchangers. Elsaid and Abdel-Wahed⁹ explored the effect of MHD on mixed convective HT in a channel, they found that MHD increased RN and increment flow instability which incremented the HT rate. Sheikholeslami and Ganji¹⁰ studied the current and HT numerical simulation of two NFs among parallel plates. They found that increasing Ha's number increased HT. Sheikholeslami and Shehzad¹¹ studied the effect of a non-uniform MF caused by an electric current-carrying wire. According to their observations, by the enhancement intensity of MF, HT was reduced. Ma et al.¹² studied the effect of MF on forced convection HT of hybrid NF (Ag-MgO) in a channel. The flow pattern and HT characteristics were analyzed by Ha number, Reynolds number, and VF of NPs. The results showed that the HT rate increased by increasing VF or decreasing the Ha value. Benos and Sarris¹³ studied the natural HT of NFs in a cavity with an MF by generating internal energy. In their study, using an external MF reduced HT. Chegini and Amanifard¹⁴ studied the effect of nonuniform MF caused by using electric current on a semi-insulated horizontal tube to increase HT and improve thermal efficiency, the effect of changing different parameters on the HT coefficient. Bezaatpour and Goharkhah¹⁵ examined the effect of MF on HT in heat wells. The results of their work show that by the enhancement of the intensity of applied MF, the convective HT coefficient increased. Many researchers studied HT in a PM. Murat et al.¹⁶ numerically studied the parameters of HT, and NF flow in the presence of an MF for forced convection in a tube. In all studied flows, it was shown that in terms of the presence of an MF and NP_S, the velocity of fluid flow decreased, but, on the other hand, Nu had a direct relationship with MF. Han et al.¹⁷ numerically studied forced convection HT and entropy generation in a microchannel in the presence of an MF and concluded that Ha number improved the HT rate. Hussain et al.¹⁸ studied the convective HT in the presence of MF with hybrid NF (Al₂O₃-Cu-water) in a wavy channel. In their work, the effect of basic elements, such as Reynolds number, VF of NPs, and wave amplitude on HT was studied. Soltanipour and Pourfattah¹⁹ studied the effect of an MF on forced convection HT of two-phase ferrofluid in a semi-porous tube. The results show that the applied MF and the PM simultaneously increase the HT rate. Kalpana et al.²⁰ numerically studied the flow characteristics of a hybrid NF (Ag-TiO₂-Water) in a channel in the presence of an MF. The results of their work showed that regarding the MF, the boundary layer became thinner and increased the HT rate. Besides, the change in the values of the magnetic parameter, VF of NPs, and the amplitude of the wavy wall increases the HT rate. Mohammadi et al.²¹ studied the effect of an MF on forced HT inside the tube. The results of their numerical research showed that the presence of NPs in the base fluid (water) increased the Nu compared to pure water. Moreover, applying an MF on the tube has caused a much greater increase in the HT in terms of creating transverse currents in the flow.

One of the ways to improve HT noticed in recent years is using porous media. PM can increase the convection HT coefficient in terms of the increase in the fluid's contact surface, which has a higher thermal conductivity coefficient than working fluid. In recent years, several studies surveyed the convection HT inside a pipe and the channel in which the porous material is placed. Sheikhnejad et al.²² simulated the forced laminar convective current into a channel half-filled by porous material with a circular cross-section under an MF. They studied the effect of Darcy number on the rate of HT. Ashorynejad and Zarghami²³ studied a channel saturated with NF that had a PM in the presence of an MF. Their results showed that the MF caused the trapping of the working fluid in the porous layers, and for the flow with high Darcy numbers in a constant pressure gradient, the HT rate increased. Izadi et al.²⁴ modeled the natural convective HT in a porous chamber by attending two variable MFs. Nu decreased by increasing the hydrodynamic conductivity. Arasteh et al.²⁵ studied the coefficient of hydraulic-thermal performance in a sinusoidal heat exchanger with a PM and obtained the optimal condition of channel wavelength, diameter, and thickness of porous medium. Wang et al.²⁶ studied the numerical simulation of current and HT features of a screw tape geometry from a PM in two triangular tubes and a circular tube. The triangular tube has better thermal efficiency than the circular tube. Yerramalle et al.²⁷ studied the combined convective HT in a porous channel. The study showed that with the enhancement of the height of the porous space inside the channel, the HT increased. Ibrahim et al.²⁸ studied forced convection HT of NF flow in a channel with porous blocks in the presence of an MF. According to obtained results, in the presence of an MF, the pressure drop and convection HT coefficient increase by increasing the Reynolds number. Besides, the local friction coefficient and the Nu number increased by increasing the Ha number.

Based on the comprehensive studies carried out so far, no research was done in studying the HT of a porous sinusoidal channel with an external MF and alternating heat flux (non-uniformity with respect to the location). Therefore, considering the importance of HT in converters and channels in the industry, this research aims to simulate this geometry to investigate the effect of different parameters (channel wave, PM, and MF) on HT and flow. Some of the most important innovations of this study are:

Applying a non-uniform heat flux relative to the location (sinusoidal) on the wall of the wavy channel and preparing the UDF code to model this boundary condition.

The optimization of NPs (for achieving the optimal proportion of hybrid NPs) using the design of experiment (DOE) method by Design Expert software. This method obtained the optimal ratio between two particles of the same composition. This NP was used in the results section.

Simultaneous use of MF and PM in a co-phase sinusoidal channel to study HT, flow, and pressure drop.

Problem description

As shown in Figure 1, NF flow is studied via a porous sinusoidal channel under alternating heat flux and a uniform transverse MF $\vec{B}$ . The desired geometry with specified dimensions is shown in Figure 1. The channel wave equation is in the form of equation below²⁹

y (x) = α . [\cos ((2 π / λ) . L) - 1]

(1)

Figure 1.

Schematic representation of geometry studied in the present study.

Governing relations and boundary conditions

The NF flow is assumed to be laminar, continuous, and incompressible, and the fluid causes Newtonian behavior, while the viscosity losses are neglected. It is assumed that NPs and base fluid particles are in thermal equilibrium. The governing equations include the mass conservation equation, the momentum equation, and the energy equation. Since an external MF B is applied to the entire domain of study, a source term called Lorentz force is added to the momentum equation. Furthermore, since a monotonous MF is exerted on the domain of the solution, a source term was appended to the equation of momentum to account for the Lorentz force owned by the MF. Besides, a source term for the poriferous medium is appended to the equation of momentum as a Darcy term. Equation (2) is the continuity equation.³⁰

\nabla . \vec{V} = 0

(2)

The equation of momentum using MF and PM is as follows.³¹

[\frac{1}{φ} \nabla (\frac{\vec{V} . \vec{V}}{φ})] = - \frac{1}{ρ} \nabla P + \frac{ϑ}{φ} Δ \vec{V} + \frac{1}{ρ} + \vec{j} \times \vec{B} - \frac{ϑ}{k} \vec{V}

(3)

where ρ, ϑ, K,

V

, and T are density, kinematic viscosity, PM permeability, flow velocity, and temperature. Thus, in this equation,

\frac{1}{ρ}

\vec{j} \times \vec{B}

the term is related to applying the external MF to the whole geometry. –

\frac{ϑ}{k} \vec{V}

is called Darcy term related to the porous medium. In fact, the pressure drop caused by the PM appears as the source term in the momentum equation. MF and the flow density are obtained from the following equations.²³

\vec{F} = \vec{J} \times \vec{B}

(4)

In equation (4),

\vec{j}

is the electric current density, which is calculated from Ohm's law that is defined as follows:

j = σ (\vec{E} + \vec{V} \times \vec{B})

(5)

where

\vec{E}

and

\vec{B}

are applied electric and MFs, respectively. Furthermore,

\vec{V}

and σ represent the velocity field and the electrical conductivity of the fluid, respectively. No external electric field was applied to the solution domain. This

\vec{E}

is zero in equation (5). Equation (6) is the energy equation. Equation (5) is rewritten to assume the thermal balance among the solid and fluid phases in the PM.³¹

φ (ρ C_{p})_{f} V . \nabla T_{f} = (1 - φ) \nabla . (K_{s} . \nabla T_{s}) + φ \nabla . (K_{f} . \nabla T_{f})

(6)

where f and s are related to liquid and solid phases, respectively.

C_{p}

is the specific heat of fluid phase at constant pressure, k is the thermal conductivity,

- k_{s}

\nabla T_{s}

is the conductive heat flux via the solid, so the term

\nabla

. (

k_{s}

\nabla T_{s}

) is the net heat conduction rate per unit volume of the solid. Also, (1−

φ

) represents the ratio of the cross-sectional area occupied by the solid phase to the total cross-sectional area of the environment. The term V.

\nabla T_{f}

is the rate of temperature change in the volume element through the convection of fluid to it. The product of this expression in

ρ C_{p}

is the rate of change of thermal energy of the fluid per unit volume of the element, through convection. Darcy term is added to equation (6). To calculate the pressure drop due to PM.²³

\nabla P = - \frac{μ}{K} \vec{V}

(7)

where μ, K, P, and V are dynamic viscosity, PM permeability, pressure, and flow velocity, respectively. To calculate the channel input velocity, Re is used based on the flow regime, which is defined as follows³²:

Re = \frac{ρ_{nf} u_{in} D_{h}}{μ_{nf}}

(8)

Consequently, the inlet velocity of NF flow is calculated based on Re number from the following equation.

u_{in} = \frac{Re μ_{nf}}{ρ_{nf} D_{h}}

(9)

where

ρ_{nf}

u_{in}

D_{h}

, and

μ_{nf}

are density, flow inlet velocity, hydraulic diameter, and dynamic viscosity of NF, respectively. The thermal entrance (

x_{fd, t}

) length can be expressed as follows³³:

{(\frac{x_{fd, t}}{D})}_{lam} \approx 0.05 Re . \Pr

(10)

where Pr is the Prantel number. The hydrodynamic entrance (

x_{fd, h}

) length is calculated via the following relation³³:

{(\frac{x_{fd, h}}{D})}_{lam} \approx 0.05 Re

(11)

ΔP is the pressure drop between the inlet and outlet of the channel as follows:

Δ P = P_{in} - P_{out}

(12)

The local HT coefficient and the Nu are computed as below²³:

h_{local} = \frac{Q^{″}}{(T_{w} – T_{bulk})}

(13)

N u_{local} = \frac{h_{local} D_{h}}{k_{nf}}

(14)

N u_{avg} = \frac{1}{L} \int_{0}^{L} N u_{local} d x

(15)

where

T_{w}

is the temperature of the channel wall and

T_{bulk}

is the NF bulk temperature. The effective dimensionless parameter is related to the porous environment of Darcy dimensionless number, which is expressed as the ratio of permeability to the cross-sectional region³¹:

Da = \frac{K}{D_{h}^{2}}

(16)

where K is PM permeability. The hydraulic diameter (D_h) is twice the height of the channel (D). The porosity coefficient of a PM is defined as equation (17) and its value varies between zero to one.²³

ε = \frac{V_{V}}{V_{T}}

(17)

where V_T and V_V are The total volume of the PM and The volume of the cavity in the porous medium, respectively. Moreover, Ha number is used to express the intensity of an MF, which is defined as equation (18).⁸

Ha = B_{o} L \sqrt{\frac{σ_{nf}}{μ_{nf}}}

(18)

where

B_{o}

and L are the MF strength and the length of channel, respectively. Moreover,

σ_{nf}

is the electrical conductivity of the NF. It is assumed that the NPs added to the base fluid (water) are in thermal balance with the base fluid and do not slip relative to it. By this assumption, they can be considered a single-phase fluid. Therefore, all governing relations comprising the continuity, momentum, and energy equation are used for NFs.³⁴ Therefore, the equivalent thermophysical attributes for the NF are calculated. The effective density and specific heat capacity are computed using equations (19 and 21).⁸

ρ_{h n f} = (1 - φ) ρ_{f} + φ ρ_{h p}

(19)

(ρ C_{p})_{h n f} = (1 - φ) (ρ C_{p})_{f} + φ (ρ C_{p})_{h p}

(20)

where

φ

is the VF and

(ρ C_{p})_{h n f}

is the specific heat of the NF. Moreover, the NF effective viscosity is determined using the Brinkman³⁵ equation:

μ_{h n f} = \frac{μ_{f}}{{(1 - φ)}^{2.5}}

(21)

Considering the same spherical shape and size for NPs, the k is obtained from the Maxwell-Grant approximation of equation (22).³⁶

\frac{k_{h n f}}{k_{f}} = \frac{k_{h p} + (n - 1) k_{f} + (n - 1) (k_{h p} - k_{f}) φ}{k_{h p} + (n - 1) k_{f} – (k_{h p} - k_{f}) φ}

(22)

where n is the particle shape factor. It is assumed that the NPs are spherical, so n = 3.

\frac{k_{h n f}}{k_{f}}

is the ratio of the HT coefficient of the NF to the thermal conductivity.

k_{h n f}

is the effective thermal conductivity of the NF,

k_{f}

is the thermal conductivity of the base fluid, φ is the VF of the particles. To obtain the effective electrical conductivity of the NF, the Maxwell relation³⁷ is used, which is as follows.

\frac{σ_{h n f}}{σ_{f}} = 1 + \frac{3 (\frac{σ_{h p}}{σ_{f}} - 1) φ}{(\frac{σ_{h p}}{σ_{f}} + 2) – (\frac{σ_{h p}}{σ_{f}} - 1) φ}

(23)

Equations (24–28) show the relationships of VF, density, specific heat capacity, and the conductivity of thermal and electrical for hybrid particles, respectively:

φ_{h p} = φ_{1} + φ_{2}

(24)

ρ_{h p} = \frac{φ_{1} ρ_{1} + φ_{2} ρ_{2}}{φ}

(25)

(C_{p})_{h p} = \frac{φ_{1} {(C_{p})}_{1} + φ_{2} {(C_{p})}_{2}}{φ}

(26)

k_{h p} = \frac{φ_{1} k_{1} + φ_{2} k_{2}}{φ}

(27)

σ_{h p} = \frac{φ_{1} σ_{1} + φ_{2} σ_{1}}{φ}

(28)

In the above equations, φ is the VF of NPs; The hnf, f, and p are related to hybrid NF, base fluid (water), and NPs, respectively. The attributes of the base fluid and NPs used in the present study are given in Supplemental Tables 1 and 2, respectively.^38–40 In the geometry of the present study, the channel walls under alternating (sinusoidal) heat flux equations are as follows:

q^{″} = 50000 + 1000 s i n (x / 0.1)

(29)

The condition of non-slip is established in the upper and lower channel walls. The boundary condition of the velocity inlet and pressure outlet is used at the input and output of the channel, respectively. The velocity profile at the inlet is uniform, and the fluid inlet temperature is supposed to be 300 K. The thermal equilibrium is supposed to be between the solid phase and the fluid phase in the PM and at the interface between the fluid region and the PM region, the interface boundary condition is used. In other words, the governing equations in the joint chapter between two regions are satisfied using this boundary condition.

Numerical solution method, validation, and grid study

In the present study, NPs optimization was performed first by DOE method and the mixture model. With this method, the optimal ratio between two compound particles was obtained. Then, modeling was performed using the selected optimal particle. The geometry grid is structural, with quadrilateral elements. A two-dimensional and reliable model for conservation equations with pressure-based solvers code was used to numerically solve the governing equations and discretize the solution region. A pressure-based solver was used for incompressible flow. The finite volume technique solves the governing relations and the corresponding boundary condition. The standard method and the second-order approximation were used to discretize the equations’ pressure and other terms. The energy equation is solved to study temperature changes and HT. The SIMPLE (Semi Implicit Method for Pressure Linked Equations) algorithm is used to solve the pressure and velocity components simultaneously. When the residual values of all parameters are less than 10⁻⁶, the problem solver converges.

To check the independence of the mesh, the Average Nu is calculated in four different meshes and the appropriate mesh size is determined by comparing them. The results are given in Supplemental Table 3. The error difference in the calculation of network 3 and mesh 4 is about 0.3%. Finally, mesh number 3 with 30,000 cells was selected as the appropriate mesh for channel analysis. Because by shrinking the mesh more than this value, a significant change in Nu was not detected. The structural grid for the domain used in this study is shown in Figure 2.

Figure 2.

Structural grid for domain.

The validation of the result was done with the experimental result of Kim et al.⁴¹ To validate the result of the simulation, for this purpose, the fluid inside a two-dimensional channel was examined with a constant HF, without internal heat source production. Re number value of 1620 was used for the validation study. Figure 3 shows the result of the numerical experimental data of Kim et al.⁴¹ Silva and Delemos⁴² were used to validate the fluid current in the attendance of porous material. Silva and de Lemos presented their results in velocity profiles developed for a channel half the width filled with PM. Figure 4 shows the validation performed. This figure shows the velocity profiles acquired in the current solution and the velocity profiles reported by Silva and Delemos. The maximum error in this comparison was around 2%. The porosity and Darcy number in the Silva study were equal to 0.6 and 0.0004, respectively.

Figure 3.

Collation among the numerical and experimental data⁴¹ for Nu.

Figure 4.

Comparison of the fully advanced velocity profile with the previous study⁴².

Results and discussions

The forced HT process of NF in a sinusoidal channel with alternating heat flux in the presence of a PM and a uniform MF was studied. In the results, in the “Optimization of NPs (choosing the best combination)” section , suitable NPs are selected. The “The effect of waves” section is about the channel ripple effect. “The effect of MF” section examines the effect of MF. “Effect of the PM” section describes the effect of PM.

Optimization of NPs (choosing the best combination)

In this part, NPs in the desired geometry are examined and selected without the presence of an MF. Then, the NPs are optimized in the presence of MF, and the suitable composition of NPs is selected. NPs were optimized to select a combination of 6 available NPs. Design Expert statistical software was used for this purpose. To choose the best combination of six NPs, the NPs were optimized. This software works based on statistical methods. To perform this test in this software, the DOE method and the mixture model are used to select the optimal particle. This method obtains the optimal ratio between two particles of the same composition. The data (NP_S) were transferred to Design Expert software as input, and the output ( $\frac{Nu}{N u_{0}}$ ) was extracted by statistical methods in this software. In fact, by doing this method, the best combination with better HT is selected. The present study carried out the experiment in two VFs of 2 and 4%. In this method, five experiments were performed for each combination of NPs in each VF (concentration).

Furthermore, five types of hybrid NP compositions were used. In short, it combines six present NPs two by two. Combines both particles with five different combinations. The optimal ratio of NPs in each composition is presented for two different VFs. The data presented are shown in the Tables 1 and 2. Considering the main parameter is the average Nu, for this reason, we choose $\frac{Nu}{N u_{0}}$ the output parameter (target) in this experiment. By looking at the table, without an MF, the hybrid composition is optimal for a VF of 2%, the combination of aluminum oxide + magnesium oxide ( $A l_{2} O_{3} + MgO$ ). Thus, the optimal composition for VF is 4% of magnesium oxide (MgO). By looking at Table 2, in the presence of an MF at a VF of 4%, the combination (Fe₃O₄+Cu) is the optimal combination that has the highest $\frac{Nu}{N u_{0}}$ value. According to the thermophysical properties of NPs, basic particles (water), the geometry of matter, boundary conditions, and experiments in the presence of MF, the combination of particles (Fe₃O₄+Cu) is the best combination. In fact, the hybrid composition (Fe₃O₄+Cu) improved the forced convection HT by about 15%. The results section continues with this optimal combination.

Supplemental Figure 11 shows the $\frac{Nu}{N u_{0}}$ ratio at different Re for the base fluid and the optimal hybrid NF (Fe₃O₄+Cu) (selected in the presence of an MF). As can be seen, Nu is directly related to the increase of Re and VF of NPs. The reason for this is that with the increase in the VF of the particles, the density of the NF increases, which increases the momentum and strengthens the convection HT mechanisms. The reason is to improve the thermal conductivity properties of the fluid. The increasing trend of this coefficient is higher at higher Reynolds numbers with higher NP VFs. By increasing the VF of hybrid NPs (Fe₃O₄+Cu), $\frac{Nu}{N u_{0}}$ is improved by about 15%. Physically, the convection HT coefficient is proportional to $\frac{k}{δ_{t}}$ . where k is the thermal conductivity and $δ_{t}$ is the thickness of the thermal boundary layer. Simultaneously by adding NPs to the base fluid (water) and the formation of NF, while the HT coefficient increases, the convection of fluid also increases.

The effect of waves

This section studies the effect of channel wall waves on the flow and HT. According to Figure 5 (a), it can be seen that changing the geometry is one of the important techniques to improve the HT rate. Because increasing the number of waves, increasing the mixing of the mass flow, or decreasing the thickness of the thermal boundary layer, increases the HT. In this part, the wavy channel with different waves was studied, the average Nu for the channel was calculated in different cases, and the outcomes are presented in Figure 5(a). With increasing Re, the Nu enhances. The enhancement of the Re number decreased the thickness of the thermal boundary layer. At a constant Re rate, the Nu is enhanced by increasing the number of waves. In fact, by increasing the number of channel waves, the resulting vortex flow caused a change in the flow regime, disrupted the thermal boundary layer, and increased the convective HT. At six waves and Re 600, the increase in Nu was about 10% compared to 4 waves. In the explanation of Figure 5(a), for a specific axial position, the value of the Nu number always increased for higher Re numbers. In forced convection HT, when the flow velocity rate increased, the HT rate increased because the thickness of the thermal and dynamic boundary layer (velocity) decreases. In fact, based on $h \propto \frac{1}{δ_{t}}$ equation, the convection HT coefficient increased with the decrease in the thickness of thermal boundary layer. One of the ways to reduce the thickness of the boundary layer was to increase the velocity (Reynolds number).

Figure 5.

The influence of channel waves on flow characteristics. (a) impact on the average Nu (b) impact on the Ratio of heat transfer (HT) coefficient and reduce pressure.

Figure 6.

Velocity profile for different Ha in (Re = 500, Da = 0.0001).

The ratio of the HT coefficient and reduced pressure for different waves is presented in Figure 5(b). By an increased number of waves, the ratio of HT coefficient and reduced pressure increases. In short, the increase in the number of waves affects HT and pressure drop, and the optimal state should be achieved so that both HT improves and pressure drop does not increase. In fact, by increasing the wave number of the channel, the fluid flow deviates from its smooth path which has a greater tendency to create a vortex flow, and this factor causes a drop in pressure (pressure difference between the beginning and end of the channel.)

The effect of MF

The velocity profile is shown in the presence of an MF and PM to study the effect of the MHD parameter on the flow. Using an MF, a drag-like force called Lorentz force was created. This force is the factor that resists the NF flow and causes a decrease in velocity in the middle of the channel. Regarding the constant flow rate and mass conservation, the decrease in velocity in the center line of the channel increases the velocity in the vicinity of the surfaces. By increasing the Ha number, the maximum velocity value decreases in the center of the channel and simultaneously increases near the channel wall shown in Figure 6.

The effect of Ha number on average Nu in various Re is given in Supplemental Table 4. This table shows that the Nu considerably increases as the Ha number enhances. This increase is because as the MF enhances, the velocity in the center of the channel decreases and increases near the channel wall. It causes a strong velocity gradient along the walls and increases the heat exchanged between the NF and the channel wall. The highest average Nu occurs at Ha 10 with 65.51, and the lowest mean Nu at Ha 0 with 33.7. (at Re number= 500). The reason for these changes is that as the intensity of MF increases, the velocity and flow near the walls increases, while the maximum velocity in the middle of the channel decreases. This tendency of the flow to the wall increases the convective HT. This factor creates a strong velocity gradient along with the walls and increases the heat exchanged between the NF and the channel wall. It can be said that by applying an MF (increasing the Ha number), the velocity near the walls increases, and this factor causes the thickness of the velocity boundary layer, which is an unfavorable factor in HT, to decrease. Nu and HT rate increase with decreasing thickness of the boundary layer. When the flow moves near the walls (slope), it increases the temperature gradient at this location, which is an enhancing factor for HT. The velocity contour is given in Figure 7. As the Ha increments, the maximum velocity decreases, indicating that the velocity profile flattens. In fact, as the Ha increments, the maximum velocity in the center decreases and increases near the channel wall at the same time. As mentioned, as the MF increases, the velocity decreases in the center of the channel and increases near the channel wall. It is known that by applying an MF, a tensile force called the Lorentz force is created. This force is the factor that resists the NF flow and reduces the flow rate. Due to the constant flow rate and conservation of mass, the decrease in velocity in the center line of the channel increases the velocity in the vicinity of the surfaces. In fact, as the Ha number increases, the maximum velocity decreases in the center of the channel and simultaneously increases near the channel wall. As the strength of the MF increases, the possibility of creating a vortex decreases, and this is due to the increase in velocity near the walls. Consequently, increasing the velocity near the walls and decreasing the center strengthens the exchanged heat.

Figure 7.

Velocity magnitude contours in a poriferous channel in the axial direction with Re = 500 and Da = 0.0001 for (a) Ha = 4, (b) Ha = 7, and (c) Ha = 10.

Figure 8(a) shows the variation of the local Nu number at each Ha number. During the channel, the Nu number changes in a waveform. The reason is the shape of the channel and the boundary conditions (sinusoidal flux). In all Ha numbers, the maximum Nu number is at the beginning of the channel, and then, the first wave of channel, and the lowest is at the last wave of the channel. As can be seen, the local Nu number is at its highest value at the beginning of the channel and decreases as it progresses along with the channel. Since the boundary layer is resistant to HT, the highest local HT coefficient (Nu) is observed at the beginning of the channel, i.e., where the boundary layer thickness is zero. The value of this coefficient gradually decreases by increasing the thickness of the boundary layer, and consequently, the increase of HT resistance.

Figure 8.

The influence of MF on flow characteristics. (a) impact on the local Nu. (b) impact on the ratio of HT coefficient and reduce pressure. MF: magnetic field; HT: heat transfer.

The ratio of HT coefficient and reduced pressure in the wavy channel for various Hartmans numbers is presented in Figure 8(b). As the Ha number increases, the intensity of the applied MF enhances. Increasing the Ha number will increase Nu; therefore, the HT coefficient will be enhanced. The ratio of this coefficient will be greater than one, and by increasing Ha number, this ratio will be larger. In examining the effect of the magnetic force on pressure drop, it can be seen that with the increase of Ha number, the electromagnetic force increases, and as a result, more force is applied in the opposite direction. This force acts as a resisting force to the flow, and as a result, creates a greater pressure drop (pressure difference between the beginning and the end of the channel). Hence, $\frac{Δ P}{Δ P_{0}}$ in Ha 10 is about 2.24 times compared to Ha 4.

Effect of the PM

This section studies the effect of the porous layer installed near the channel wall on the flow and HT. Using porous media can improve HT due to its unique capabilities. Figure 9 shows the velocity contour for different Darcys. By observing this contour, it is clear that at higher darcies, the temperature gradient near the walls is higher. According to the relation of Darcy number which is defined as Da = $\frac{K}{D_{h}^{2}}$ , there is a direct relationship between Darcy number and permeability. Therefore, by increasing the Darcy number (permeability of the porous layer), the velocity gradient near the channel wall increases, which increases the temperature gradient, convection HT coefficient, and Nu.

Figure 9.

Velocity magnitude contours in a poriferous channel in the axial direction with Re = 500 and Ha = 10 for (a) Da = 0.00001, (b) Da = 0.0001, and (c) Da = 0.001.

The effect of average Nu on changes in the Darcy number and Re is given in Supplemental Table 5. As the Darcy number enhances, the permeability of PM will increase, and the flow convection will be improved, which will increase the convective HT coefficient and consequently increase the Nu. The enhancement in convective HT is related to the permeability of the PM. As Darcy number enhances, the flow convection in the PM is improved, and the heat exchanged increases. In a constant Re (500) in Darcy 0.00001 to Darcy 0.01, the average Nu is about 5 times more. At a constant Re, as the Darcy number enhances, the average Nu in the channel also enhances. In fact, by increasing the Darcy number, the permeability of the PM will increase, and the flow convection will be enhanced which will increase the convective HT coefficient, and consequently increase the Nu. Besides, in a fixed Darcy, the Nu in the channel increases with Re enhancement. In summary, it can be said that in all cases, the average Nu increases with the permeability of PM (Darcy number). Figure 10(a) shows the variation of the local Nu at each Darcy number. In all cases, the highest local Nu is at the beginning of the channel. This coefficient gradually decreases along with the way. Darcy number is one of the characteristics of fluid flow in porous media. Since a PM is used in the walls of the channel, by increasing the Darcy number, the permeability of porous space increases, consequently, the velocity of fluid flow via this space increases and the heat exchange capability of this space increases. In porous media with high permeability, NF has high velocity and momentum, and this increase in velocity leads to improved heat exchange. At very low Darcy numbers, the low permeability of the PM causes the medium to act as a solid impermeable barrier, reducing the flow in this region and thus greatly reducing the heat exchange. This is well illustrated in Figure 10(a). It is clear. By increasing the Darcy number, from 0.00001 to 0.01, the maximum HT coefficient, which is in the first wave of the channel, improves 3.4 times.

Figure 10.

The influence of PM on flow characteristics. (a) impact on the local Nu. (b) impact on the Ratio of HT coefficient and reduce pressure. PM: porous medium; HT: heat transfer.

The ratio of HT coefficient and reduced pressure in the wavy channel is presented for different Darsi's in Figure 10(b). Figure 10(b) shows that as Darcy increases, the HT coefficient ratio enhances. In fact, by the enhancement of Darcy number, the permeability enhances and strengthens the HT coefficient. Furthermore, increasing the permeability means reducing the viscous resistance and reducing the pressure drop. At weak (low) Darcy numbers, the porous layer reduces the amount of flow, and the pressure drop increases. As a result, the porous layer in the channels should be properly designed. By looking at Figure 10(b), the Darcy number has an inverse relationship with $\frac{Δ P}{Δ P_{0}}$ .

Conclusion

In this study, the effect of different parameters on HT was studied. The forced flow of single-phase NF was assumed to be laminar and incompressible. The geometry of the channel with sinusoidal walls and in the presence of MF, and PM was studied. First, an experiment was conducted to select the appropriate (optimal) NP among the available NPs. By this method, the optimal ratio between two mixed particles was obtained. It was observed that in the presence of MF, the hybrid composition (Fe₃O₄+Cu) is the optimal composition because it had the highest $\frac{Nu}{N u_{0}}$ . For this reason, this combination was used for the results section. After selecting suitable NPs, the effect of channel wave, MF and PM on forced convection HT and flow was investigated. In general, it was observed that channel wave, MF and PM lead to different thermal behavior in this geometry. In summary, the results of this study are presented as follows:

For the studied sinusoidal channel, Nu number of the Hybrid NF (Fe₃O₄+Cu+H₂O) was higher than that of the base (Water) fluid and enhanced by increasing NPs concentration.

According to the results, the Nu increased by increasing channel waves. In fact, by increasing the number of channel waves and creating vortex flow, the flow regime, and the boundary layer change and increase the convective HT.

The value of Nu increased by increasing Re number. Because the thickness of the thermal boundary layer decreased by increasing flow velocity. Consequently, by reducing the thickness of the thermal boundary layer, the HT was improved.

The heat exchanged using the MF always had an upward trend. As the Hartmann number enhanced, the intensity of the MF enhanced. As the MF intensity perpendicular to the direction of the flow increased, the Nu increased so that under the same conditions (Re = 500, Da = 0.0001), the intensity of the MF increased from Hartmann 4 to 10, the Nu increase was 59%.

Increasing Darcy number strengthened the convective flow, reduced the viscous resistance, and improved HT. The highest average Nu was 154.4 in the Darcy number 0.01, and the lowest average Nu was 26 in the Darcy number 0.00001.

In studying the effect of the magnetic force on pressure drop, it was observed that the pressure drop increases by increasing Hartmann number. It was because, in the presence of an MF, more force was used to the flow against the direction of fluid movement and as a result, the pressure drop (pressure difference between the beginning and the end of the fluid) increases.

Despite the PM and the increase in Darcy number (increasing the permeability of the porous layer and reducing the viscous resistance), the HT had always improved and the pressure drop had a decreasing trend. Hence, in a constant condition, by increasing Darcy number from 0.001 to 0.01, the convection HT coefficient increased by about 20% and at the same time, the pressure drop decreased by 110%. So, using the PM had better results than the use of the MF.

Simultaneous use of PM and MF in a sinusoidal channel improved HT. The results of this study can provide useful guidelines for designers and manufacturers of heat exchange devices, especially channels and sinusoidal converters in the industry.

Table 1.

The optimal ratio of hybrid NPs in the channel without the presence of MF.

VF (%) $φ_{hnf}$	Al₂O₃	Cu	Ag	MgO	TiO₂	Fe₃O₄	( $\frac{Nu}{N u_{0}}$ )
4	-	-	-	4	-	-	1.09559
	4	-	-	-	-	-	1.09339
	-	-	0.914687	-	3.08531	-	1.08914
	-	1.073	-	-	-	2.927	1.08583
	-	-	1.07686	-	-	2.92314	1.08512
2	0.104344	-	-	1.89566	-	-	1.0469
	2	-	-	-	-	-	1.04551
	-	1.15293	-	-	-	0.847073	1.04463
	-	-	0.450416	-	1.54958	-	1.04359
	-	-	0.53472	-		1.46528	1.04158

MF: magnetic field; NPs: nanoparticles; VF: volume fraction

Table 2.

The optimal ratio of hybrid NPs in the wavy channel with the presence of MF.

VF (%) $φ_{hnf}$	Al₂O₃	Cu	Ag	MgO	Fe₃O₄	( $\frac{Nu}{N u_{0}}$ )
4		0.83006			3.16994	1.15993
			1.27686		2.72314	1.14831
					4	1.13673
	3.18532			0.81468		1.13251
			2.084	1.916		1.12614
2			2			1.10184
		0.85672			1.14328	1.09192
					2	1.08023
	0.75433			1.24567		1.07429
		1.2469		0.7531		1.68321

MF: magnetic field; NPs: nanoparticles; VF: volume fraction

Supplemental Material

sj-docx-1-pie-10.1177_09544089221149000 - Supplemental material for Numerical analysis of heat transfer of hybrid nanofluid in a porous sinusoidal channel with magnetic field and an alternating heat flux

Supplemental material, sj-docx-1-pie-10.1177_09544089221149000 for Numerical analysis of heat transfer of hybrid nanofluid in a porous sinusoidal channel with magnetic field and an alternating heat flux by Nejat Sheikhpour, Arash Mirabdolah Lavasani and Gholamreza Salehi in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Nejat Sheikhpour

Arash Mirabdolah Lavasani

Gholamreza Salehi

Supplemental material

Supplemental material for this article is available online.

Appendix

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