Nanofluid in Enclosure 1 1Affiliation December 23, 2021 Abstract Introduction Numerous researches and experiments focus on the impact of particles with higher thermal conductivity in fluids compared to conventional fluids. These experimental works illustrate that nanofluid has a higher thermal conductivity and a higher heat transfer rate than the conventional fluid ([1], [2] and [3] ). There have been many investigations to model mechanisms of thermal conductivity of the nanofluid. [4] presented a model of convective transport in nanofluid. He found that the most important slip mechanisms of the interaction of velocity between nanoparticles and the base fluid are Brownian diffusion and thermophoresis. Brownian diffusion is the random movement of nanoparticles immersed in a base fluid and it produces by its continuos collisions with the surrounding molecules of base fluid. Thermophoresis nanoparticles disperse under the presence of a temperature gradient. However, [5] and [6] observed that the impact of Brownian diffusion on thermal conductivity of the nanofluid is very small and negligible. Natural convection in enclosures are considered as an important part for many heat transfer applications. An effective heat transfer system is required in many engineering applications such as electronic cooling systems ([7]). The conventional fluid inside an enclosure has a limitation of heat transfer enhancement because the fluid has a low thermal conductivity. In order to achieve a higher heat transfer in the enclosures, nanoparticles are dispersed in the base fluid. The beneficial attributes of the nanofluid consist of a higher stability associated with low clogging rates and better heat transfer potential. The study of the microstructure of nanofluid assists to enhance the thermal performance of the system [8]. Consequently, the changes in nanofluid composition affect the energy transport process in the enclosure. Hence, the artificial interaction of particles and the liquid molecules increases in energy transport. 1 Model Description and Mathematical Formulation The natural convection in a two-dimensional rectangular enclosure was investigated by [9]. This work considers the flow in a rectangular enclosure with a width of W and a height of H, as shown in Figure 1. The left wall of enclosure operates at elevated temperature (Th) and the right wall of enclosure remains at lower constant temperature (Tc) and the top and botten of enclosure wall kept constant with adiabatic condition. The nanofluid in the enclosed space is displayed as a dilute solid-liquid fluid with uniform nanoparticles, such as (Al2O3), scattered inside a base fluid, such as water, ([9].). 1 Figure 1: Schematic of the physical configuration. The shaded area is base fluid and the dark dots present the nanoparticles. W is width and H is height. Th and Tc represent temperature with different value. The model assumes that the compression work and viscous dissipation remain infinitesimally small. Hence, it enhances the conservation of the mass, momentum, and energy in the two-dimensional steady-state natural convection flow in the enclosure system. The model incorporates the following governing equations in dimensionless form. Vorticity equation: ∂(uω) ∂x + ∂(vω) ∂y = P rf cp,mf kmf µmf ∂ 2ω ∂x2 + ∂ 2ω ∂y2 + Ra2 f ρmfβmf cp,mf kmf ∂θ ∂x . (1) Stream function equation: ∂ 2ψ ∂x2 + ∂ 2ψ ∂x2 = −ω. (2) Energy equation: ∂(uθ) ∂x + ∂(vθ) ∂y = ∂ ∂x k ∗ ef ∂θ ∂x + ∂ ∂y k ∗ ef ∂θ ∂y . (3) Here u and v are the dimensionless horizontal and vertical velocities and Prrf is the Prandtl number. The physical properties in the equations (1) and (3) has selected formulas. The thermophysical properties are: Thermal connectivity: k ∗ mf = km kf where km is the thermal conductivity of nanofluid and kf is the thermal conductivity of base fluid. Also, the subscripts m present nanofluid and f present base fluid. Next, k ∗ ef = ke kf where ke = km(1 + 3P φem p ) is the effective thermal conductivity which relates to heat transfer enhancement mechanisms of the nanofluid and km = kf ” 2 + k ∗ pf + 2φ(k ∗ pf − 1) 2 + k ∗ pf − φ(k ∗ pf − 1) # , 2 Figure 2: The comparison between thermal conductivity formula (5) and (4). which is known as the Maxwell formula and φ is the volume fraction, P ep is the particle Peclet number and kpf = kp kf . In this work, there are two formulas for effective thermal conductivity ratio, k ∗ ef = k ∗ mf = ” 2 + k ∗ pf + 2φ(k ∗ pf − 1) 2 + k ∗ pf − φ(k ∗ pf − 1) # , (4) and k ∗ ef = k ∗ mf (1 + 3P φem p ). (5) The second formula (5) has more effective thermal conductivity then the first formula (4), see figure 2. Dynamic viscosity : µ ∗ mf = µm µf , where the dynamic viscosity of nanofluid Brinkman’s (?) formula is µm = µf (1 − φ) −2.5 , (6) which is frequently used in many studies. The other formula found by using experimental data µm = µf (1 + 7.3φ) + 123φ 2 , (7) as shown This work examines the effect of different dynamic viscosity values. The equation (6) has less effective dynamic viscosity then the equation (7). Density: ρ ∗ mf = ρm ρf , where the density of nanofluid ρm and the density of base fluid ρf . 3 Specific heat: c ∗ p,mf = cp,m cp,f with cp,m = 1 ρm [(1 − φ)ρf cp,f + φρpcp,p]. Thermal expansion coefficient: β ∗ mf = βm βf . Here thermal expansion coefficient of nanofluid is βm and thermal expansion coefficient of base fluid is βf . Four different types of nanofluid were investigated useing different formulas of dynamic viscosity and thermal conductivity. Studing these models help understand the effects of different formulas for dynamic viscosity and thermal conductivity on heat transfer. The boundary condition on the left wall x = 0 is ψ = 0 and θ = 0.5 and the on the right wall x = 1 is ψ = 0 and θ = −0.5. Also, we have boundary condition on the top wall y = 1 and bottom wall y = 1 walls which are ψ = 0 and ∂θ ∂y = 0. 2 Numerical Simulation of Natural Convection of Nanofluid in a Square Enclosure Chebychev collocation method combined with Newton linearization are used to discretize the equations (1), (2), and (3) in both x-direction and y-direction. We have to find the boundary conditions of function ω via integral constraints ([10]). As we did in the second chapter, we integrate equation (2) with respect to y Z 1 0 ∂ 2ψ ∂x2 + ∂ 2ψ ∂y2 + ω dy = 0, (8) which gives ∂ψ ∂y y 0 + Z y 0 ∂ 2ψ ∂x2 + ω dy = 0. (9) The boundary conditions on y = 0, 1 give Z 1 0 ∂ 2ψ ∂x2 + ω dy = 0. (10) Substitute that ψ(x, y) = ψ¯(x, y) + δψ(x, y) and ω(x, y) = ¯ω(x, y) + δω(x, y) into (10) to obtain Z 1 0 ∂ 2 δψ ∂x2 + δω dy = − Z 1 0 ∂ 2ψ¯ ∂x2 + ¯ω dy. (11) For the other condition, we can integrate equation (8) with respect to y which gives [ψ] y 0 + Z 1 0 Z y 0 ∂ 2ψ ∂x2 + ω dy1 dy = 0. (12) Using the boundary conditions of function ψ on y = 0, 1 we obtain Z 1 0 Z y 0 ∂ 2ψ ∂x2 + ω dy1 dy = 0. (13) 4 Assume that ψ(x, y) = ψ¯(x, y) + δψ(x, y) and ω(x, y) = ¯ω(x, y) + δω(x, y) and substitution into (10) gives Z 1 0 Z y 0 ∂ 2 δψ ∂x2 + δω dy1 dy = − Z 1 0 Z y 0 ∂ 2ψ¯ ∂x2 + ¯ω dy1 dy. (14) We can approximate the integrals in equation 14 of the form R h(y)dy where h(y) = Z y 0 g(y1)dy1, by applying the recurrence relation for Chebyshev polynomial. This technique was described in more details by [11].