The Numerical Simulation of Nanofluid Flow in Complex Channels with Flexible Wall

Nomenclature

1  Introduction

The topic of improving heat transport in intricate geometries is crucial. This advantage results from its use in a wide variety of heat exchangers, chemical treatment, electrical stations, and electronic device applications in industry and engineering [1]. Nanoparticles, cavities, and fluid structure interaction can be used to improve heat transmission in heat exchangers (FSI) [2].

Fluid structure interaction (FSI) is used in a wide range of cutting-edge systems, including power plants, chemical therapies, the food industry, the paint industry, mixing devices, micro-scale biological investigations, cooling of electronic components and heat exchangers [3], clinical evaluation and medical devices [4]. Electronic component cooling is one of the most critical barriers to system development in terms of being faster, smaller, and more reliable [5].

The focus of this research is FSI installed in flexible walls. Multiple researchers looked into this kind, such as [6]: studied numerically the behavior of convection inside a cavity containing a nanofluid, where the left wall of the cavity moves vertically towards the top and has a cold temperature, while the right wall has a high temperature, and the rest of the cavity wall is isolated. The following parameters were used in a study to study their effect on heat transfer: internal Ra (between 103 and 106), (Ri) between 0.01 and 100, (Ha) between 0 and 50, magnetic field inclination angle (between 0° and 90°), (E of a flexible wall between 5 × 102 and 106) and the effects of nanoparticle volume fractions ranging from 0 to 0.05. The average heat transfer falls when the Richardson number lowers and the Hartmann number and internal Rayleigh number increase. and also conducted another study [7], a numerical representation of the mixed convection that occurs in a square hollow that is filled with SiO2 nanofluid and volumetric heat production with a flexible wall an inner spinning cylinder and. The hollow walls and the surface of the cylinder are considered to be adiabatic, and the top wall is kept cold and the bottom wall hot. For different solid nanoparticle shapes, the effects of external Rayleigh number (103–5 × 105), internal Ra 104–106, E = 5 × 102–106, nanoparticle volume fraction (0–0.03) in fluid flow, the angular rotational speed of the cylinder (from −2000 to 2000) and heat transfer are numerically studied (spherical, cylindrical, brick, and blade). The elastic modulus of the flexible wall and internal Rayleigh number decrease while the outward Rayleigh number enhances local and averaged heat transmission. Cylinder rotation improves heat transmission for all nanoparticle kinds. Cylindrical nanoparticles promote heat transmission better than spherical ones, and also [8] studied numerically explored nanofluid mixed convection, elastic-wall, 3D, inner cylinder and trapezoidal chamber. The impacts of the Ri from (0.05–50), side surface elastic modulus (103–105), side wall inclination angle (0°–20°), and volume friction percentage (0–0.04) on heat transfer of fluid flow in a 3D lid-driven trapezoidal hollow were studied numerically. The relevant factors affect these traits. Flexible side surfaces influence on the rate of transmission of heat. The angles of wall inclination between 0° and 10°, flexible side walls increase and decrease space, respectively, affecting heat transmission. For side wall inclination angles of θ = 0°, increasing the elastic modulus from 1000 to 105 raises Nuavg about 9.80%. At the maximum volume friction percent increases heat transmission linearly by 25.30%, there is another study for [9] about flexible wall, where MHD with nanofluid in a flow square chamber with a lid-driven flexible side wall was quantitatively examined. The cavity’s top wall travels at a steady speed and is cooler than the bottom wall. Insulation covers other cavity walls. For the following parameters: the effect of the flexible wall’s Young’s modulus (104 N/m2–2.5 × 105 N/m2), with Ri = 0.01–5, a Ha = 0–5, and φ = 0–0.04, on flow and heat transmission is quantitatively studied. The average of transmission of heat decreases with Ha and Ri values. As the flexible wall’s Young’s modulus drops for E = 104 N/m2, the mean heat transmission increases by 66.5% compared to E = 2.5 × 105 N/m2. Reference [10] employed a square cavity to probe the effect of natural convection. The cavity was divided into halves by the flexible wall into two triangles. For the purpose of eliciting the movement of the flexibale wall, an arbitrary Lagrangian-Eulerian (ALE) is employed. A number of factors, including fluid characteristics and wall stiffness, were investigated for their effects and also [1] FSI was utilized in order to investigate the process of heat transmission in a compartment with an elastic bottom wall, a lid-driven by mixed convection, and laminar flow. The findings revealed that the flexible bottom wall improves heat transmission. Reference [11] utilized a square elastic-walled chamber with variable Hartmann from (0–200) and Rayleigh numbers from (105–108) and an adjustable magnetic field direction (0–180 degrees). Reference [12] investigated the transport of heat by mixed convection in a cavity that has a wall that is flexible. The impact of heat transfer in a heat exchanger was investigated by using many parameters, such as the ratio of the height of the channel to the height of the cavity, H/D = 0.5–1.1, the radius of the cavity, Ri = 0.1–100, and the length of the heat source = 0.5–1.5. As can be seen from the data, H/D influences the Nusselt number by around 5%. Reference [13] investigated the numerical of the wall FSI for a backward-facing step in nanofluid forced convection. The bottom wall of the step was flexible. The ranges of these parameters were employed in this investigation: Re = 25–250, E = 104–106, and the volume percentage of solid particles = 0–0.035. Heat transmission rates were found to be enhanced when the elastic bottom wall was utilized. At Re = 250 and E = 104, heat transmission is at its maximum. For all Reynolds numbers, the flexible bottom wall has an average increase in Nusselt number of roughly 6.1% compared to a rigid bottom wall at E = 104. Reference [14] used the FSI technique to study its effect on heat transfer, and a cavity in the shape of an “L” with a flexible wall and an inclined wall was used. The study was completed with a magnetic field, whose angle of inclination ranged between 0 and 90 degrees. In addition to that, the following parameters were used: Ha = 0–50, Ra = 104–106, φ = 0–0.04, Ri = 0.03–30, and E = 104–108. The results obtained showed that the flexible wall affected the heat transfer significantly. Nu increases when the size of the nanoparticles used increases. Reference [15] studied wall FSI, utilizing elastic wall cavities. Conventional mixing of a nanofluid consisting of CuO and water was induced in a cavity by the use of an angled magnetic field. Simulations numerically are obtained by utilizing the Arbitrary-Lagrangian-Eulerian method. The numerical effects of changing parameters such as the Reynolds number (from 100 to 500), the magnetic inclination angle (from 0 degrees to 90 degrees), the Hartmann number (from 0 to 40), the elastic modulus of the flexible wall (from 104 to 108), and the nanoparticle volume fraction (from 0 percent to 3 percent), were studied. It was found that the average Nusselt number went up by between 9 and 9.5 percent when the maximum amount of nanoparticles was added, both with and without a magnetic field. Ismael [16] created vortex using a flexible wall with upstream and downstream baffles. According to the findings, heat transport is improved by 94% in channels that are equipped with baffles as opposed to channels that do not have baffles at Re = 250 and [17] analyzing the performance of a right-wall FSI in a square cavity with natural convection and a solid cylinder within. Since the solid cylinder affected the passing solution and altered the interaction between the structure and the flow, the results show that the flexible wall of the hollow maintained its S form. Increased wall flexibility can lead to a 2 percent rise in the average Nusselt number. Reference [18] employed a cavity to study fluid–structure interaction (FSI) by vertically dividing it with a thin flexible wall. The left vertical wall was heated in a sinusoidal way that changed over time, whereas the isothermal cooling method was used to chill the right vertical wall. To tackle this problem, a Lagrangian–Eulerian technique was used. The following characteristics were used to describe impact heat transfer: elasticity modulus (5 × 1012–1016), Prandtl number (0.7–200), and Rayleigh number (104–107). The outcomes reveal frequency of temperature changes has no significant effected on (Nu) and flexible membrane distortion. When the Prandtl number of a fluid goes up, convective heat transfer and membrane stretching go up and Al-Amir et al. [19] employed a hexagonal cavity to investigate the effects of magneto hydrodynamic forced convection on a CNT-water nanofluid. Parameters like the Hartmann number (Ha) = 0–60, the Re = 100–1000, φ = 0–0.1 were used, along with the modulus of elasticity of wall that used in the left and bottom walls between 104–107. Recently, it has come to light that the bottom enclosure wall is very sensitive to the elastic wall.

Also, there are many previous studies that focus on their studies on corrugated channel, such as [20]: studied corrugated channel entry convective heat transfer. Water was tested with two-channel spacing at 20° corrugation. I50 ≤ Re ≤ 4000 flows were tested. Low Reynolds-number flow visualization revealed longitudinal vortices, whereas higher Reynolds numbers exhibited spanwise vortices. Corrugated channels showed 130% and 280% greater friction factors than parallel-plate channels for Re > 1500. Corrugated channels outperformed parallel-plate channels in heat transfer at similar mass flow rate, pumping power, and pressure drop per unit length. Reference [21] studied what happens to heat transfer and pressure drop in a corrugated channel experimentally and explained the effect of each change in phase shaft and channel spacing on heat transfer and pressure drop. parameters were used in his study: Re = 3220–9420, a constant corrugation rate, and a constant wall temperature. According to the findings, the average heat transfer coefficient increased by 2.6 to 3.2 and the pressure decreased by 1.9 to 2.6. Reference [22] studied numerically and experimentally the heat transfer in three channels of different shapes (straight, sinusoidal, and trapezoidal). In channels, SiO2-water was used as the working fluid with volumetric friction of 0%–1%. Whereas the results of his study revealed that the average Nu increases and pressure drop decreases as the volumetric friction of the nanoparticles increases, The corrugated trapezoidal channel promotes more heat transfer than the straight or sinusoidal channel. Reference [23] studied to evaluate the thermal and flow implications that a magnetic source has on mixed convection conditions within an unique arc-shaped lid-driven cavity issue. Ferrofluid made of Fe3O4, and water is what fills it up. Several parameters were applied, including Richardson numbers (about 0.04 to 40), magnetic numbers (about 0 to 100), and solid volume fractions (about 0 to 0.05). According to the findings, an increase in the rate of heat transmission of up to 338.35% is possible when Ri is equal to 0.04, thanks to the activities of the magnetic field, arc-shaped walls, and magnetite nanoparticles suspended in suspension. Reference [24] studied Numerically, the thermal properties of nanofluids inside a semicircular zigzag channel. Water was used as a basic liquid with four types of nanoparticles (CUO, SiO2, ZnO, and Al2O3) with volume frictions between 2 and 8 percent and a diameter between 20 and 80 nanometers. The results obtained in his study explained that the thermal performance effects of a semicircular zigzag channel are greater than those of a flat channel. The results also revealed that increasing the diameter of the nanoparticles resulted in a decrease in Nu, but it increased as volume friction and Re increased. Reference [25] employed silicon dioxide (SiO2)-water nanofluid as a working fluid in two different forms of corrugated channels, including the straight channel (SC) and the new form of a trapezoidal-corrugated channel (TCC). It was done using nanofluids with SiO2 volume fractions from 0.0% to 2.0% with Reynolds numbers (10,000–30,000). When compared to straight channel, the utilization of corrugated channel (TCC) resulted in an increase of heat transfer rates of up to 63.59%, a drop in pressure of up to 1.37 times, and an increase in thermal performance of up to 2.22 times. There are many previous studies that used particles of Al2O3 in working fluid, such as: Naphon [26] investigated heat transport in a wavy channel with variable phase angles (20–60) with a range of Re =400–1600. Where it was found that the wavy surface affects pressure drop and heat transmission. Reference [27] looked at the convection currents in a V-shaped channel at phase angles between 0 and 180 degrees. These findings demonstrated the significant impact the channel’s distinctive V shape had on thermal conduction. Reference [28] examined nanofluid thermal characteristics in a trapezoidal channel using water as the base fluid and four nanotypes (SiO2, ZnO, Al2O3, and CuO) with concentrations of 2%, 4%, 6%, and 8%. SiO2 is the best heat-transfer liquid. Reference [29] investigated Al2O3 performance flow with concentrations (0%–4%) in triangular, trapezoidal, and sinusoidal channels with Re = 6000–22000. Trapezoid shapes had high nusselt numbers also [5] focus in his study at Al2O3 where investigated the behaviour of a 90° elbow-shaped nanofluid of Al2O3 and water. A multiphase mixture model was used to numerically apply it to the forced convection and 3D model. Several Reynolds numbers (between 10,000 and 100,000), nanoparticle volume fractions (between 0.02 and 0.06), and nanoparticle sizes make up the simulation settings (10 to 40). The findings demonstrated that the Reynolds number with a reduced DP and an increase in the volume of the nanoparticle fraction prevented the formation of vortices in the flow. The volume percentage and diameter of the nanoparticles both decrease as the pressure drop rises. Towards the exterior wall, significant heat transfer rates are seen. Reference [30] looked into the relationship between the Reynolds and Nusselt numbers by employing a wavy channel with a phase angle of =0–180 and a Reynolds number (Re) of 1000 to 10000. Nu increased when Re increased, as predicted by the data.

In recent years, there has been an uptick in interest in the study of heat transport in a broad variety of engineering geometries. The majority of studies only considered one-sided, regular-shaped flexible walls (square, rectangle, triangle, etc.). Here, we model fluid flow in a complex channel comprised of six elastic walls. The effort aims to improve heat transmission in cases when cooling is impossible due to the formation of vortices as a result of the structure’s complexity. The vortices were released by the elastic wall's vibration and replaced by additional vortices. This motion is continuous with the movement of the wall, resulting in heat exchange between the wall and the fluid. In this research, heat transfer and fluid flow fields in a heat exchanger with a complex channel were analyzed to examine how the Reynolds number, Cauchy number (modulus of elasticity), and elastic wall location in a heat exchanger with a complex channel shape impact the flow field and heat transfer.

2  Theoretical Formulations

2.1 Model Description

A description of the geometry that is applied throughout this investigation is found in Fig. 1. The geometry is represented by a channel that extends over two dimensions, with the length marked by the letter L1 and the height signified by the letter H1 as shown in Table 1. The intricate channel has a flexible wall that is attached to the wall of the channel in six different locations along the channel wall. The temperature of the wall of the complex channel is kept at a relatively high level TH at all times. The temperature of the Al2O3–water nanofluid that is moving through the complex channel has a uniform temperature Tc and velocity u0. The findings of this research are applied to a number of different geometric shapes that have ratios of L2/H2 that range from 0 to 6, as well as flows that are unstable, incompressible, laminar, and two-dimensional that are forced convective fluxes. The subjects of the parameter assessments were the Reynolds number, which varied from 60 to 160; the Cauchy number of the baffle, which ranged from 10−4 to 10−8; and lastly, the volume percentage of the nanoparticles, which went from 0 to 0.06.

Figure 1: A diagrammatic depiction of the situation of a complicated channel that includes Flexible wall

2.2 Governing Equations

In dimensional vector form, the governing equations of the unstable fluid and the elastic-dynamic structure are written as:

Continuity Equation ∇⋅u∗=0(1)

Momentum Equation∂u∗∂t∗+(u∗−w∗)⋅∇u∗=−1ρf∇p∗+υf∇2u(2)

Thermal energy ∂T∗∂t∗+(u∗−w∗)⋅∇T∗=αf∇2T∗(3)

Thermal energy in solid∂T∗∂t∗=αf∇2T∗(4)

For elastic structure domain, it is possible to write the equations that describe the nonlinear elastic displacement, and the energy of the wall as follows [18]:

ρsd2ds∗dt∗2+∇σ∗=Fv∗(5)

dT∗dt∗=αs∇2T∗(6)

Fv∗ in Eq. (5) is representing the gravitational forces per unit volume and written as [12]:

Fv∗=ρsgy(7)

where (u*) is the velocity vector, (w*) identify the speed of the moving coordinates, (p*) identify the fluid pressure, (T*) identify the fluid/solid temperature, (Fv∗) identify the gravitational force exerted on the heated wall measured in terms of unit volume. However, (Fv∗) is ignored in the current work as the wall elasticity is fastened at one end and unrestrained at the other and (σ*) identify the Cauchy stress tensor. ρf and ρs represent the densities of fluid and solid respectively. αf and αs represent thermal diffusivities of the fluid and the solid, respectively, gy is the gravitational forces, vf identify the kinematic viscosity of the fluid and β is volumetric thermal expansion coefficient.

This stress may be described using the following form of equations. The elastic wall is subjected to a stress tensor as a result of the fluid flow pressure taking the nonlinear geometry variation [12]:

Where F=(I+∇ds∗), (I: unity matrix), J = det. (F) and S is the second Piola-Kirchhoff stress tensor that relevant to the strains in the hot wall [12] as:

S=C,ε=12(∇ds∗+∇ds∗T+∇ds∗T∇ds∗)(8)

where C is a function of dimensional modulus of elasticity and possin’s ratio as shown below and the colon is the double-dot tensor product.

C=C(C⋅υ)(9)

where E is the modulus of elasticity of the wall or and v is the Poisson's ratio.

2.3 Dimensional Boundary Conditions

a)   On the elastic wall, the boundary conditions are continuity of dynamic movement and kinematic forces [18]

∂ds∗∂t∗=u∗(10)

σ∗⋅n=−p∗+Mf∇u∗(11)

where (u*) is the velocity vector, (p*) identify the fluid pressure, el (σ*) identify the Cauchy stress tensor, (n) identify is the normal vector and (μf) identify is the viscosity of fluid.

b)   The energy balance may be defined at the boundary between the fluid and the solid as [18]:

Kf∂T∗∂n|f=Ks∂T∗∂n|s(12)

where Ks and Kf are the thermal conductivities of solid and fluid, respectively, and n is a normal vector.

c)   The elastic wall in the present work with different positions and free displacement fixed with ∂ds∗∂t∗=0 at all corner this mean end and starting of piece of elastic line. The condition of the pressure constraint is applied at the exit hole at the right wall, i.e., p* = 0 [18].

d)   The fluid inlet velocity is u∗=uin

e)   T∗=Th at y = lower and upper wall. 0<y≤L.

f)   T∗=Tc at x=0.0<y≤H.

The following non-dimensional parameter definitions were utilized in order to non-dimensionalize the governing equations [2]:

θ∗=T∗−TcTpn−Tc,X=xL,Y=yL,U∗=u∗uo,Ug∗=uguo,V∗=v∗uo,Vg∗=vgvo,P=pρfuo2,vT=vnfvf,ρT=ρnfρf,Reρf.u0.Lμf,δ=σnfσfKr=knfkf,(ρCp)T=(ρCp)nf(ρCp)f,αT=αnfαf,σ=σ∗E∗,E=E∗.L2ρnf.αnf,Fv=(ρnf−ρs)LEv,ds=ds∗L,τ=u0.tL,Ca=ρf.uo2E

By substituting the non-dimensional parameters from before, the dimensional governing Eqs. (5)–(8) may be ex