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incorporate low- and no-flow conditions. Additionally, a conductive heat-transfer term was added to account for heat conduction through the tank walls. Self-degradation of the thermocline in the axial direction is neglected due to the low relative values during standard operation; this is a known limit of the model during times of no flow. 4.1.1.1 Energy Balance for the Fluid The equation set for the fluid section of the thermocline is shown below in equations (20-27) and is directly coupled with the filler material. The initial term to define is the geometrical definitions of the fluid section, . Equation 20 defines the fluid cross sectional area where, is the cross-sectional area of the fluid, is porosity of the tank, and R is radius of the tank. (21) Equation 21 defines the fluid energy balance where, is the enthalpy at node i, is heat conduction through the walls, is the temperature of the fluid at node i, is the filler temperature at node i, and is the heat capacity of the fluid. (22) Equation 22 defines the average fluid velocity through the thermocline, where is the fluid mass flow is fluid moving from the hot side to the cold side, is fluid moving from the cold side to the hot side. is the fluid density at node i, and is the fluid velocity. (23) Equation 23 defines the effective heat transfer area between the beads and the surrounding fluid where, is the heat-transfer area of filler per unit length of the tank, is the surface shape factor, and is the radius of filler beads. (24) (25) (26) Equations 24-26 define the convective heat transfer coefficient interaction between the fluid and beads where, is the convective heat transfer coefficient, is the characteristic radius, is the diameter of the filler material, and is the Reynolds number. 4.1.1.2 Energy Balance for the Filler Material This section describes the equation set for the filler material inside the thermocline. Equation 27 is the filler material energy balance, where is the heat capacity of the filler and is the filler density. 4.2 Nodalization The thermocline system is split into axial nodes that each incorporate a fluid and a solid component. During charging, the flow runs from the hot side to the cold side; it runs in the opposite direction during discharge. Boundary conditions for each node depend on the direction of the flow and the advection of the previous node’s values. For example, if the thermocline is operating in discharge mode, node N-1 will receive boundary values from node N-2. Whereas, if in charging mode, node N-1 will use boundary conditions from node N. Figure 28 provides a representation of this nodal split. (27) 22PDF Image | Thermal Energy Storage Model Development
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