Modelling the effects of temperature on the flow of fluid through loamy soil with spatial dependent thermo-physical property

Introduction

In power industry, which involves the method of generating electricity through the heat of the fluid that flows in the ground, the study of the fluid dynamics is of a great importance. Moreover, because of the flow of these fluids through the ground, the temperature of both the ground and the fluid are affected by a number of factors. This of no doubt has drawn the attentions of researchers toward this subject. For example, Akinpelu et al. [1] examined how solar radiation affects both the clay-loam soil and sandy-loam soil using convectional boundary condition. They reported in their findings that the solar radiation generally affects the temperature of both soil samples. In other word, the temperatures of these soils rise as the intensity of the radiation also increases. In addition, they found out that as the moisture content of these soils increased, their thermal conductivities also increased, this in turn increased their temperature when the solar radiation is involved; though this is at different levels and rates. Mohammed [2] in his own work includes chemical reaction and heat generation while studying the effects of radiation on mass flow passing via a very high porous medium. Having examined the physical parameters that are involved in the work, he showed in his results that the thermal and solutal buoyancy forces, heat generation and the permeability of the medium all increased the rate at which the fluid flows. However, since the fluid considered was emitting radiation, the temperature and the speed of the fluid flow reduced as the radiation increased. Olaleye et al. [3] modelled the temperature of loamy soil as being affected by solar radiation. Having examined some appropriate heat related physical quantities that surfaced in the study, they revealed that the solar radiation alongside heat generation are capable of shooting up the warmth of loamy soil when they are enlarged. On the other hand, when all the material parameters are kept constant, after some time, the temperature of the soil will begin to drop, especially as the depth of the soil increases.

In their own study, Mahender and Srikanth [4] includes viscous dissipation and magnetic field while researching into the free convection flow of the fluid pass a vertical plate that is porous in nature. They concluded that as a result of the temperature produced by the source of heat and viscous dissipation, the temperature and the velocity of the fluid also increased. Meanwhile, the Schmidt number reduced the thickness of the concentration boundary layer. Alabison et al. [5] studied the effects of warmth generation and radiation on convective fluid movement through a permeable medium using an intermittent temperature boundary state. It was evident in their result that the increasing buoyancy forces alongside the warm generation boost the temperature and increase the speed of the flow. But the increasing Schmidt number, Chemical parameter and Prandtl number lowers the temperature and the fluid's velocity. Some others include Hud et al. [6] who modelled heat transport in greenhouse amid a surface soil warming structure. Sharma and Bismeeta [7] considered the way uneven thermal conductivity and viscosity together with Dufour and Soret influence the movement of fluid through an upright cone in free convective way. Considering the above works among host of others, this research focuses on modelling the effects of temperature on the flow of fluid through loamy soil using spatial dependent thermo-physical property. Since the soil is a porous medium which allow the passage of heat and fluid, the choice of spatial dependent thermo-physical property makes the work more relevant in application.

Mathematical analysis

The Navier-Stoke heat equations were modified under the following conditions. The flow is considered infinite along the horizontal axis leaving the equation to be a function of the vertical axis (z) and time, t (being unsteady). The soil is optically thin environment. That is, it allows the passage of fluid and heat. At the boundary layer, the temperature is considered to be variable due to different factors. Also, both the suction velocity and the thermal conductivity are taken to be variable. With the Boussinesq's approximation, the governing equations in their dimensional forms are:

Momentum equation

Where 𝑧̅ , is the vertical axis, 𝑡 ̅ is the time, 𝑣̅ is the component velocity along 𝑧̅ direction, 𝑤 ̅ is the suction velocity. 𝑇 ̅ is temperature, 𝑇 ̅ ∞ is free stream temperature, 𝑇 ̅ 𝑤 is wall temperature. 𝜗 is kinematic viscosity, 𝑉 ̅ 𝑝 and 𝑉 ̅ ∞ are wall and free stream velocities respectively. 𝑔 𝑎𝑛𝑑 𝜌 are the acceleration due to gravity and density respectively. 𝛽 is volumetric coefficient of thermal expansion, 𝐶 𝑝 is specific heat capacity, 𝐾 ̅ is soil's permeability, 𝑘 𝑎𝑛𝑑 𝜑 are thermal conductivity and porosity respectively. 𝜏̅ 𝑎𝑛𝑑 𝑞 ̅ 𝑟 are the amplitude of variation and radiative heat flux respectively. The bars signify that the parameters are in dimensional form.

In accordance to study made by Krishna and Reddy [8] , the radiative heat flux is defined to be:

α is absorption coefficient.

In line with Nwaigwe [9] , variable suction velocity is known to be:

Where 𝑤 0 is initial suction velocity, A is suction parameter and 𝜔 ̅ is frequency of oscillation.

In agreement with Akinpelu et al. [1] , Mohammed [2] and Sharma [10] , non-dimensional quantities used are:

Moreover, in agreement with the work of Olaleye et al. [3] and Akinpelu et al. [11] , the thermal conductivity is considered to be spatial dependent, varying linearly with position, and given as:

𝑘 0 and 𝛾 are constant and variable thermal conductivity (parameter) respectively. Using equations ( 6)-( 9), equations ( 2)-( 5) were reduced to:

Where 𝑃 𝑟 is Prandtl number, 𝐺𝑟 is thermal Grashof number, γ is thermal conductivity parameter, R is radiation parameter and Q is heat generation parameter.

Method of solution

Equations ( 10)-( 11) are partial differential equations of second order. In order to reduce them into ordinary differential equations using perturbation method, the assumed solution is given as:

Differentiating equations ( 14)-( 15) and substituting them into equations ( 10)- (11), while taking the order of epsilon and ignoring higher order 𝑜(𝜀 2 ), we have

In order to obtain the solution to equations ( 18)-( 19), as employed by Umavathi [12] , perturbation method of second phase is required. The variable thermal conductivity parameter is used as the perturbation factor:

Therefore, the velocity and temperature distributions respectively are obtained to be:

𝜃 = 𝜃 00 + 𝛾𝜃 01 + 𝜀𝑒 𝑖𝜔𝑡 (𝜃 10 + 𝛾𝜃 11 )

Results and Discussion

In the previous section, the model for the work has been developed which are the velocity and the temperature profiles (equations 34 and 35). The following data/information about the loamy soil were retrieved so as to be specific about the soil working on and to be able to examine the physical parameters that emerged in the model which has to do with temperature/heat. Namely: Solar radiation parameter (R), buoyancy force parameter (Gr), heat generation parameter (Q) and the prandtl number. [13] , Soil Permeability [14] Moreover, making use of other parameters involved in the study as used by some existing works like Mohammed [2] among host of others unless otherwise stated, With the aid of Matlab R2009b software, the effects of these temperature based physical parameters were examined on the velocity of the fluid as it passed through the loamy soil. The results are presented on graphs below. As the radiation increases it is seen that the speed of flow of the fluid also increases. This is because both the soil and the fluid absolve the electromagnetic energy which afterward enhances their temperature. As the temperature of the fluid is raised, its viscosity is reduced, and hence increases the speed of flow of the fluid. In figure 2, which is the effect of rising heat generation on the velocity of fluid's flow through loamy soil, the fluid's velocity also increased as the internal heat mounted. This follows from the above; the internal heat increased the fluid's temperature which lowers its viscosity and in turn increased the flow rate. Figure 4 refers to the effect of increasing Prandtl number (Pr) on the velocity of fluid's flow through loamy soil. As the Prandtl number increases, the thermal conductivity of the fluid reduces and its temperature also reduces. As a result, the viscosity increases which reduces the velocity of the flow. When all other parameters are kept constant, as time goes on, at increasing depth of the soil, the flow of the fluid decreased. This is shown in figure 5.

In figure 6, it is clearly seen that the increasing permeability of the soil increases the velocity of the fluid through the soil.

Conclusion

Unsteady one dimensional heat equation with spatial dependent thermal conductivity with some relevant conditions was used to model the effects of temperature on the flow of fluid through loamy soil with spatial dependent thermo-physical property. The model was formed after solving the equations governing the study which are the transient temperature and velocity. According to the results, it is evident that the solar radiation, thermal buoyancy and internal heat boost will increase the flow of a fluid through the loamy soil. But when these factors remain unchanged, the flow speed will gradually decline with time.

References

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