Variations in ground temperature in the presence of radiative heat flux and spatial-dependent soil thermophysical property

Introduction

Ground temperature plays a dominant role and dictates the type or species of crops that can be planted in a particular region. It has been established that significant relationship exists between soil temperature, agricultural yield and time requirement for germination. Fahim and Ghulam [2] examined that the quality and quantity of agricultural products depends upon factors that include ground temperature which significantly affects the budding and growth rates of plants. With increase in ground temperature, chemical reactions speed up and cause seeds to germinate and regulate rate of plant development and growth. Information on ground temperature is also necessary for construction projects. These include design of airports, road pavements, excavation of foundations and the design and construction of underground space for buildings. Engineers and Architects need the knowledge of the factors that determine the ground temperature and how the quality varies with time and depth from the surface as reported by Williams and Gold [11] . Williams and Gold [11] and Daniel [1] investigated that the ground temperature varies depending on several factors which can be grouped into three general categories: meteorological, terrain and subsurface variables. The meteorological elements include solar radiation, air temperature and rain. Examples of some thermo-physical properties of the soil that affect the temperature of the ground are specific heat capacity, thermal conductivity and thermal diffusivity. In an investigation conducted on the problem of ground temperature with suction velocity and radiation by Nwaigwe [7] , the results revealed that increase in Prandtl number or radiation decreases the ground temperature. However, the author imposed among other conditions that the thermal conductivity is constant and also assumed that the temperature of the surface is maintained at a constant temperature, w T T    . Daniel [1] in his work revealed that the thermal conductivity of the ground is seldom uniform. It varies with the depth of the ground, time and temperature of soil.

More so, the surface temperature is periodic and can be expressed as a function of time. Similar results were reported by Nofziger and Wu [6] . Ishikawa [4] studied the thermal heat transfer in the soil and confirmed the significant of surface temperature for understanding the temperature phenomena in the soil. By their findings, the surface temperature corresponds to the temperature at the top active layer of the ground. It is measured at the uppermost centimeters of the ground in order to avoid the radiation influences and focuses on the seasonal changes of the ground temperature. Gao, et al. [3] reported that surface temperature varies canopy layer and soil water content. Changes in soil porosity and thermal conductivity have little or no effect on it. In this research work, effort is therefore made to present a mathematical model on ground temperature variation that combines some soil parameters such as spatial dependent thermal conductivity, prandtl number and Radiation with the boundary conditions involving a periodic surface temperature.

Formulation of the problem

Consider the heat flow into the ground through the surface in the vertical direction (z-axis). It is assumed that the surface temperature T  of the ground oscillates as a sinusoidal or harmonic function of time around an average temperature w T  . It is therefore expressed as a function of time. It is also assumed that the ground temperature is high so that the radiative heat transfer is significant. The thermal conductivity of the soil depends on the depth z. However, the density  is constant and does not depend on z.

is the radiative heat flux and w is the velocity of the fluid flow. Equation ( 2) is subject to the boundary conditions

Where, w is the initial suction velocity. Following Israel-Cookey et al [5] and for the purpose of this work, w is taken to be ) 1 (

A and  are very small parameters such that 1  A  . It is assumed that the ground is an optically thin environment. As reported by Tarkher et al [10], radiative flux takes the form

Where  is the absorption coefficient.

Since the thermal conductivity k of the ground is assumed to be spatial dependent, and then set to be

Where, 0 k is a constant thermal conductivity and  is a very small parameter such that 1   . Substituting equations (3a) -(3b) into equation ( 2), it gave

Introducing the following non-dimensional parameters as used in Nwaigwe [7] and Okedayo [8] v

Where, R = radiative parameter, Pr = Prandtl number, A = suction parameter, and A0 = amplitude of variation.

Method of Solution

Since equation ( 6) is a partial differential equation (p.d.e.) and  is small, a time-dependent perturbation method is adopted for the flow variable. A solution is assumed in the form of

Substituting equation (7) into equations ( 5) and ( 6) and simplifying the results, the following coefficients of the orders of  are collected:

Putting equations (8a) and (8b) into equations ( 9) -( 12) and compiling the orders of  with their corresponding boundary conditions, it yielded:

Results and Discussion

In the preceding section, the problem of variation in ground temperature under the influence of spatial dependent thermal conductivity and radiation was formulated and solved by means of perturbation method. The expression for the ground temperature was obtained. To illustrate the variations in this quantity, its numerical value was computed with respect to the variations in the governing parameters namely, the Prandtl number (Pr), radiation term (R), frequency of oscillation (ω), depth (z) and thermal conductivity (k). The graphical results are also presented. respectively. These results correspond to that of Israel-Cookey et al. [5] and Nwaigwe [7] . Figure 4 revealed that increase in the frequency of oscillation decreased the period or cycle of oscillation. This corresponds to the study of Stroud [9] . Figure 5 showed that as the depth of the ground increased the amplitude or the ground temperature decreased. Table 1 revealed that increase in the thermal conductivity decreased the ground temperature.

Conclusion

Variation in ground temperature under the influence of spatial dependent soil thermal conductivity and Radiation has been studied. The non-dimensional governing equation was solved using perturbation method. It was observed that the ground temperature decreased with increase in Prandtl number, Radiation term and soil thermal conductivity. These quantities speed up the rate of heat diffusion and thereby decreased the temperature. Ground temperature undergoes cycle with increase in the depth and frequency of oscillation. These changes are confined to near the surface of the ground in the case of the change in depth of the soil.

Appendices z

References

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