scholarly journals Analytical solution for two-dimensional groundwater flow in presence of two isopotential lines

2005 ◽  
Vol 41 (12) ◽  
Author(s):  
Ekkehard Holzbecher
Keyword(s):  
Analytical Solution ◽  
Groundwater Flow ◽  
Two Dimensional

An analytical solution of a two-dimensional temperature field in a hollow cylinder under a time periodic boundary condition using Fourier series

2009 ◽  
Vol 223 (8) ◽  
pp. 1889-1901 ◽  
Author(s):  
G Atefi ◽  
M A Abdous ◽  
A Ganjehkaviri ◽  
N Moalemi
Keyword(s):  
Boundary Condition ◽  
Temperature Field ◽  
Fourier Series ◽  
Temperature Distribution ◽  
Analytical Solution ◽  
Hollow Cylinder ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Separation Of Variables ◽  
Two Dimensional

The objective of this article is to derive an analytical solution for a two-dimensional temperature field in a hollow cylinder, which is subjected to a periodic boundary condition at the outer surface, while the inner surface is insulated. The material is assumed to be homogeneous and isotropic with time-independent thermal properties. Because of the time-dependent term in the boundary condition, Duhamel's theorem is used to solve the problem for a periodic boundary condition. The periodic boundary condition is decomposed using the Fourier series. This condition is simulated with harmonic oscillation; however, there are some differences with the real situation. To solve this problem, first of all the boundary condition is assumed to be steady. By applying the method of separation of variables, the temperature distribution in a hollow cylinder can be obtained. Then, the boundary condition is assumed to be transient. In both these cases, the solutions are separately calculated. By using Duhamel's theorem, the temperature distribution field in a hollow cylinder is obtained. The final result is plotted with respect to the Biot and Fourier numbers. There is good agreement between the results of the proposed method and those reported by others for this geometry under a simple harmonic boundary condition.

A Model for a Geothermal Well in a Hydraulic Groundwater Flow for Ground Source Heat Pump Systems

2011 ◽  
Author(s):  
Kevin D. Woods ◽  
Alfonso Ortega
Keyword(s):  
Thermal Conductivity ◽  
Analytical Solution ◽  
Groundwater Flow ◽  
Heat Pump ◽  
Temperature Response ◽  
Ambient Air ◽  
Heat Pumps ◽  
Thermal Reservoir ◽  
Geothermal Well ◽  
Ground Source

Heat pumps are mechanical systems that provide heating to a space in the winter, and cooling in the summer. They are increasingly popular because the same system provides both cooling modes, depending on the direction of the cycle upon which they operate. For proper operation, the heat pump must be connected to a constant temperature thermal reservoir which in traditional systems is the ambient air. In ground source heat pumps however, subterranean ground water is used as the thermal reservoir. To access the subterranean groundwater, “geothermal” wells are drilled into the formation. Water from the building heating or cooling system is circulated through the wells thereby promoting heat exchange between the coolant water and the subterranean formation. The potential for higher efficiency heating and cooling has increased the utilization of ground source heating ventilating and air conditioning systems. In addition, their compatibility with a naturally occurring and stable thermal reservoir has increased their use in the design of sustainable or green buildings and man-made environments. Groundwater flow affects the temperature response of thermal wells due to advection of heat by physical movement of groundwater through the aquifer. Research on this subject is scarce in the geothermal literature. This paper presents the derivation of an analytical solution for thermal dispersion by conduction and advection from hydraulic groundwater flow for a “geothermal” well. This analytical solution is validated against asymptotic analytical solutions. The traditional constant linear heat source solution is dependent on the ground formation thermal properties; the most dominant of which is the thermal conductivity. The results show that as hydraulic groundwater flow increases, the influence of the ground formation thermal conductivity on the temperature response of the well diminishes. The diminishing influence is evident in the Peclet number parameter; a comparison of thermal advection from hydraulic groundwater flow to thermal conduction by molecular diffusion.

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