A thermal network model (Shabgard et al., 2010) is used to determine energy storage and discharge rates. The solution domains are identified by the dashed lines in Figs. 2.2 and 2.3. The top (bottom) HTF channel is not a part of the domain during the charging (discharging) process because there is no HTF flow in that channel.
Heat transfer between the HTF and PCM is determined by application of a thermal network model based upon the following assumptions. All PCM properties are assumed to be constant in both the liquid
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and solid phases, and PCM phase change occurs at a fixed temperature. Conduction in the direction of the HTF flow, within both the HTF and PCM, is neglected. The advective effects of liquid and vapor flow within the heat pipes are neglected, as are thermal resistances due to evaporation and condensation inside the heat pipes (actual evaporation and condensation heat transfer coefficients are of the order of 107 W/m2⋅K). The effect of close-contact melting that may occur due to the falling of suspended solid PCM onto the bottom channel during melting is neglected.
In order to deal with spatial variations of the HTF temperature in the flow direction, the length of each PCM unit is sub-divided into discrete numerical modules through which the HTF temperature distribution is assumed to be linear in its flow direction. Ultimately, a numerical experiment showed that with a module length of 2 m the results of thermal network model are independent of the module length.
Figure 2.5 depicts a representative module and pertinent nomenclature during charging and discharging. The number of heat pipes in each module depends on the module length as well as the distance between heat pipes. For the specified module length of 2 m (and depth of 0.6 m), and the
transverse and longitudinal pitches of Fig. 2.4, each module includes 364 heat pipes. In order to deal with the temporal variation of the HTF temperature, the problem is discretized in time with the average HTF temperature in each module assumed to be invariant during each time step. The temperature distributions shown in Fig. 2.5 will be discussed in more detail in Sec. 3.4.
2.3.1 The overall charging-discharging process
Charging begins with a PCM temperature of Ti = 300°C for all PCM units, which is below the melting point of all of the PCMs. The duration of the charging process is specified, taking into account the availability of solar irradiation. In this work charging periods of 4, 8 and 12 hours are considered.
Depending on the local temperature difference between the PCM and the HTF, as well as the duration of charging, the onset of melting might occur in the vicinity of the heat pipes or adjacent to the channel walls. The PCM may completely melt and achieve temperatures in excess of the melting point,
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provided sufficient heat transfer occurs. Therefore, at the end of the charging period, the PCM in a specific module can be entirely solid, two-phase, or completely molten.
In contrast to charging, the duration of the discharging process is not specified. Rather, the discharge period continues until all of the energy stored during charging is recovered. Depending on the equilibrium condition in each module after the charging process, the discharging process may start with subcooling of the solid PCM, solidification, or cooling of superheated liquid PCM. Except cooling of superheated liquid PCM, during which the natural convection effects exist, other stages of heat transfer during discharge are conduction dominated. If the discharge process in a module starts with cooling of superheated PCM, it may be followed by solidification and subcooling of solid PCM, respectively, if sufficient heat transfer occurs.
It should be noted that the equilibrium conditions of the LHTES after the end of the charging period are imposed as the initial condition for the discharging mode. The thermal equilibrium implies that all of the PCM, as well as all of the channel walls, heat pipes and HTF in each module reach a thermal equilibrium. The conservation of thermal energy in each module is applied to determine the equilibrium state.
2.3.2 Thermal network heat transfer model
The thermal network model of Shabgard et al. (2010) is extended to quantify the heat transfer between the HTF and PCM, either through the channel walls or through the heat pipes. Extensions of the model included here are the simulation of heat transfer associated with sensible energy storage (recovery) in (from) the subcooled solid- or superheated liquid PCM, heat pipes, HTF channel walls, as well as in the HTF residing within the LHTES. (Sensible energy storage in the exterior walls of the PCM unit is
neglected.) Also in the model, the heat pipes, PCM, and HTF channel walls are included through use of a network of thermal elements characterized by both a thermal resistance and a thermal capacitance. Heat transfer between the HTF and the heat pipes or channel wall is simulated by a thermal resistance.
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A schematic of the thermal network is shown in Fig. 2.6. The thermal pathways through the HTF channel and through the heat pipes are designated as HTF-ch-PCM and HTF-HP-PCM, respectively. The numerical model includes discretization in both space and time (Shabgard et al., 2010). At each time step the thermal network model is applied to each module to compute the amount of heat transfer through the channel walls, as well as through the heat pipes. For the heat pipes, the heat transfer associated with one heat pipe is first determined and then multiplied by the number of heat pipes in the module. This heat rate is then added to the heat transfer associated with the channel to calculate the total heat transfer in each module.
The various thermal elements, TE, shown in Fig. 2.6 are described in detail in (Shabgard et al., 2010). The full model description is lengthy, and will not be repeated here. In short, thermal elements TE1
through TE4 account for heat transfer in the HP evaporator wall, evaporator wick, condenser wick, and
condenser wall, respectively. Thermal elements TE5 and TE6 are associated with axial conduction in the
wall and wick of the adiabatic section of the HP. Elements TE7 through TEn are affiliated with the PCM surrounding the HP where the temperature distribution within the PCM is accounted for by discretizing the PCM into (n – 6) elements. Thermal element TEn+1 is associated with conduction through the channel
wall, while elements TEn+2 – TEn+m account for conduction within the PCM in the vicinity of the channel
wall.
Melting and solidification of the PCM occurs at its solid-liquid interface. In reality, the interface forms a surface (or multiple surfaces) of complex shape that evolves with time. Because the thermal element model is not capable of capturing this complex topography (and prediction of the topography with a detailed 3D model is not plausible), several additional numerical approximations are made.
First, the PCM within each module is subdivided into two regions. The first (planar) region is adjacent to the channel wall and its thermal response is assumed to be due exclusively to heat transfer to or from the wall. The second region is composed of the remaining PCM; its thermal response is due
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exclusively to the heat pipes. Second, the volume ratio of the two regions is assumed to be inversely proportional to the steady-state thermal resistances of the two thermal pathways of Fig. 2.6. Specifically, a total thickness of the first planar region can be assumed, and the steady-state thermal resistance of pathway HTF-ch-PCM can be determined. Also, the steady-state thermal resistance of pathway HTF-HP- PCM can then be calculated, assuming conduction heat transfer in an annular region of maximum
theoretical radius about an individual heat pipe (see Fig. 2.4) where
r
max=S
Tsin( )π
3
π
. Calculationof the volume of the annular region by using rmax accounts for the PCM located in between neighboring
annuli. The total thickness of the planar region is adjusted until the ratio of the steady-state conduction resistances of the two regions is equal to the volume ratio of the two regions. Once the volume ratio of the two PCM regions is determined, the transient melting or solidification process is simulated with heat transfer in each subregion decoupled from the heat transfer in the other subregion.
2.3.3 Natural convection heat transfer
During melting, the effect of natural convection in the molten PCM is accounted for through use of correlations available in the literature. An empirical correlation for natural convection in a vertical annulus with a concentrically-located hot vertical tube is adopted to evaluate the natural convection in the melt region surrounding the heat pipes (Thomas, 1993). For the melt region adjacent to the bottom HTF channel, a correlation for natural convection in a horizontal cavity heated from below is employed
(Bergman et al., 2011). Details regarding the application of single phase natural convection correlations in phase change problem can be found in (Shabgard et al., 2010). The solidification process is assumed to be conduction dominated, and thermal contact resistances at the PCM – channel wall, and the PCM – HP interfaces are neglected. Neglecting natural convection during solidification is considered to be a good assumption, considering the small degree of superheat in the liquid PCM.
2.3.4 Energy balance for HTF
Neglecting heat conduction in the HTF flow direction, the quantity that links neighboring modules is the HTF temperature. As suggested in Fig. 2.5, a linear temperature profile is assumed for the HTF
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temperature in each module. During charging, the HTF temperature decreases as it flows through the module. In addition, the average HTF temperature increases with time as the PCM warms, as shown in Fig. 2.5a. During discharging (Fig. 2.5b) the opposite trends exist. The HTF temperature variation during a time step is determined from an energy balance over the HTF in each module:
2⁄ ∆
2⁄ ∆
,"
2⁄ #
2⁄ .
(1)
In Eq. (1) 2⁄ and 2⁄ are the average inlet and outlet HTF temperatures, respectively, over the time increment ∆t. The expressions 2⁄ and 2⁄ are the average HTF temperatures at the new and old times, respectively. The upstream module outlet
temperature, , is set equal to for the downstream module. The parameters, Qch and QHP are the
amount of thermal energy transfer due to the active HTF channels and one heat pipe, respectively. These heat transfer quantities, Qch and QHP, are obtained by applying the thermal network model with
2⁄ . After calculating Qch and QHP, Eq. (1) is solved for . The solution methodology is
then applied to the downstream module until heat transfer throughout the entire unit is predicted. Once predictions are achieved for an upstream unit, the solution methodology is then applied to the downstream unit or units.