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|Unsteady Convective Heat Transfer in Liquid Saturated and Unsaturated Porous Media with Reference to an Energy Storage System
|Tathgir, R. G.
|1-Equation Model;2-Equation Model;Glass-Water Bed;Steel-Water Bed;Porous Medium;Forced Convection
|Flow and heat transfer in porous media has received much attention due to its importance in various fields of engineering: mechanical, chemical, ground water hydrology, petroleum engineering, soil mechanics and environmental science. The examination of transport in porous media relies on the knowledge gained in studying these phenomena in plain media. The mathematical modeling of convective heat transfer in a porous medium is carried out by volume averaging of governing energy equations for the fluid and the solid phases. The interphase convective heat transfer coefficient hsf couples these two energy equations at the interface. In addition, this model also contains effective thermal conductivity and dispersion tensors and is expected to give reasonably good predictions. This model is called the thermal non-equilibrium model or the 2-equation model. It is advantageous to use this model, as against the thermal equilibrium or 1-equation model, when there is significant heat generation in one of the phases, during sudden heating/cooling of the medium and when the thermal properties of the two phases are distinct. The volume-averaged energy equations based on thermal equilibrium and thermal non-equilibrium between the two phases are developed. The finite difference discretization of the model is carried out by using implicit format for the time derivatives, central difference scheme for second order partial-spatial derivatives and the upwind scheme and QUICK approaches for modeling of convective terms. The resulting algebraic equations are solved by Gauss-Seidel iterations using sufficiently fine grid in space and large number of time-steps. The numerical simulation is first compared with analytical results obtained by neglecting various terms of the mathematical model such as interphase heat transfer and convection. These include 2-D unsteady state conduction, 1-D & 2-D unsteady advection-diffusion in plain flow i.e. no solid phase present in the medium. Further 1-D form of thermal equilibrium model is compared with the analytical solution. It ensures the accuracy of the numerical simulation and proper selection of grid and time steps. A laboratory scale experimental set up is developed to validate the results of numerical simulation and provide vital information about the transport phenomenon in porous medium. It ensures that the numerical simulation can model real life problems such as an energy storage system. The validation is performed for two distinct boundary conditions related to step and frequency response of the porous bed. The step response is the response of an initially cold domain to the hot incoming fluid whereas the frequency response relates to alternate flow of hot and cold fluids through the porous bed. The main components of experimental set up are porous bed, constant head tanks, storage tanks, pumping system, electric immersion heater, temperature controller, thermocouples, data acquisition card, rotameter, electronically controlled solenoid values and electronic timer. The glass beads and steel spheres are closely packed in a PVC tube to make two different porous beds. It ensures that the validation is under widely varying thermal conditions of the bed. An immersion heater of 1 kW capacity is inserted inside the hot water constant head tank. It is controlled by the PID temperature controller and maintains water to a designated constant value within 0.3 K. Electronically controlled solenoid valves are used to supply the cold and the hot water to the bed. The electronic timer actuates the valves for a specific period so that step and frequency response experiments of the porous bed can be conducted. The temperature is measured by K-type single sleeve thermocouples inserted in the bed at the required positions. A 16 channel NI-4351 type data acquisition card collects the temperature history of the bed through the thermocouples and stores it in a PC. The flow rate of water is measured by a calibrated rotameter of capacity 170 lph. The step response refers to the thermal response of an initially cold domain to hot fluid, which flows through one of its ends. For an energy storage system, it relates to charging of the bed. The results of experiments are compared with numerical simulation using the 1 and 2-equation models of heat transfer. The comparison is obtained qualitatively by examining the unsteady temperature profiles at various locations and quantitatively by comparing the front speed and the spread of the thermal front traveling through the bed. The validation is carried out for both the glass-water and the steel-water beds. A sensitivity analysis is performed to find the effect of various parameters on the comparison obtained. The response for glass-water bed is distinct from that of the steel-water bed. The thermal non-equilibrium between the fluid and the solid phases during the step response is discussed. The response of the bed subjected to a time varying inlet temperature is of waveform having peaks and valleys. During the hot phase, the temperature level of porous medium rises and reaches a maximum value at the end of the phase. During the cold phase; the level falls and reaches a minimum value at the end of the phase. Hence, the frequency response signifies the unsteady behavior of the porous medium subjected to pulsating boundary conditions. The experimental results of frequency response are compared with numerical simulation using the 1 and 2-equation models of heat transfer. In addition, the global properties such as attenuation of temperature, phase lag, pulse speed and spread are discussed to provide a qualitative validation. The analysis is again carried out for both the glass-water and the steel-water beds. The response is obtained from both the porous beds using distinct Peclet numbers and frequencies of pulsation. A reference frequency is defined as the reciprocal of the time period required for the fluid to travel to the other end of the bed during each phase. The amplitude of pulsations in glass-water bed is higher than that in the steel-water bed but the phase lag is lower. The extent of thermal non-equilibrium between the two phases also varies cyclically in a fashion similar to the temperature fluctuations. The amplitude of this pulse decreases with distance from the inflow plane but increases with Peclet number. Its value for steel-water bed is higher as compared to the glass-water bed. Energy storage using a porous medium is an attractive method of storing momentarily available excess energy and reusing it at a later point of time. It is obtained by using fixed porous solid mass such as closely packed mesh screens or spherical beads through which hot and cold fluid alternately flow. The storage and the retrieval of energy from the fixed porous solid mass are obtained by back and forth flow of hot and cold fluids through the mass. This type of flow is referred to as oscillatory flow. During the hot phase, heat flows from the fluid to the solid phase and temperature of the fluid phase is higher as compared to that of the solid. During the cold phase, the solid phase temperature is higher. Hence, the estimation of temperature differential between the two phases is important in the working of the storage system. The thermal performance of a porous medium for energy storage is discussed in the present study. It depends on the thermal properties of the solid and the fluid phases. The other parameters of interest are the frequency of oscillation, storage space, Reynolds number, particle size, porosity, number of cycles elapsed, thermal non-equilibrium and the effect of heat loss from the bed. Various thermal properties of the solid phase are considered. Instantaneous temperature distributions inside the porous bed are plotted for judging the thermal performance of the bed. Amplitude of temperature fluctuations in the bed is considered. The extent of thermal non-equilibrium between the two phases is discussed and the resulting difference between 1 and 2-equation models is also considered. Optimum parameters that yield the maximum effectiveness are obtained. When the pore space is not fully filled with water and air is present in the pores, the medium is called unsaturated. The flow channels in the unsaturated medium become narrower, the flow path becomes more tortuous as compared to that in the saturated medium, surface tension adds resistance to flow, and the overall flow rate reduces. The main property of the unsaturated medium is the water content present in the pores, c. The relative hydraulic conductivity Kr measures how easily can water flow through this medium as compared to that of the saturated medium. Its value depends on the water content present in the pores and the negative pressure head in the pores. The velocity profiles are obtained in the medium using the continuity equation and flux rate using the modified Darcy’s law. The temperature profiles are subsequently obtained using the 2-equation model. The governing equations are solved by finite difference discretization. Thus, the present study concentrates on energy storage in porous medium. The finite difference based solution is compared with lab scale experiments. The knowledge gained is useful in the design of an energy storage system capable of handling summer air conditioning loads. The problem studied bring out interesting features of convective heat transfer in saturated porous medium and unsaturated soils, which are important from application point of view.
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