# Comparison of Heat Transfer Enhancement Techniques in Latent Heat Storage
## 2 Methodology
### Definition of Proposed KPIs
In this case, the effectiveness of the heat transfer (\(\epsilon_{avg}\)) is defined by the relation of the actual average (over the discharge time) temperature difference between inlet (\(\overline{T_{HTF,In}}\)) and outlet (\(\overline{T_{HTF,Out}}\)), and the theoretical maximum temperature difference achievable, with respect to the phase change temperature (\(T_{PC}\)) as shown in Equation (4) [43; 45]. The \(T_{PC}\) values were reported by the individual studies, and are usually obtained through differential scanning calorimetry (DSC) measurements:  
\[\epsilon_{avg}=\frac{\overline{T_{HTF,In}}-\overline{T_{HTF,Out}}}{\overline{ T_{HTF,In}}-T_{PC}} \tag{4}\]  
This relation allows access to the average product \"U\(\cdot\)A\", or the average heat rate capacity of the heat exchange geometry in the core of the unit. This quantity can be considered independent of the operation conditions, but remains an intrinsic characteristic of the heat exchanger geometry, design, and material combination (PCM and heat exchanger materials). During the solidification process, the conductive resistance between the HTF and the solidifying (liquid) PCM increases as solid PCM builds up around the HEX structure surface. This also generates a changing solid-liquid PCM heat transfer surface throughout the process. With this in mind, it can be especially handy to consider the product U\(\cdot\)A averaged through time, since both the heat transfer surface, and heat transfer coefficient, vary throughout the discharge process with the state of charge of the unit. Even though the main required information pertaining to the geometry, materials, and amounts are uniformly available, how and which experimental results are readily displayed in literature remains very dependent on the authors and the focus of the studies. Some additional considerations to the definition of the pair of the previously discussed KPIs are suggested for an even representation with the proposed KPIs:  
The average HTF outlet temperature (\(\overline{T_{HTF,Out}}\)) was preferably estimated by fitting a polynomial function to the reported data and calculating a mean function value between the beginning of the discharge process up to an arbitrary point. For practical purposes, and considering that once a high degree of solidification is attained the power sharply decreases, a 90% solidification or melting is considered as a standard for a completed process and thus is defined as discharge time (\(t_{Disch}\)). For the few cases in which outlet temperature or power profiles were not provided, \(\overline{T_{HTF,Out}}\) can be approximated as shown in Equation (5) derived from the simplified steady-flow thermal energy equation [45]:  
\[\overline{T_{HTF,Out}}=\overline{T_{HTF,In}}+\frac{\dot{Q}_{avg}}{\dot{m}_{ HTF}\cdot c_{p,HTF}} \tag{5}\]Knowing \(t_{Disch}\) conveniently allows for indirectly representing the average output power-to-capacity ratio as it represents the inverse of the time required to achieve a certain amount of PCM solidification, and thus the average discharge power (\(\dot{Q}_{avg}\)) can be approximated using Equation (6) [46] and the energy balance based on material properties and temperature levels:  
\[\frac{\dot{Q}_{avg}}{E_{st,90}}=\frac{1}{t_{Disch.}} \tag{6}\]  
The energy associated with the defined standard degree of solidification (\(E_{st,90}\)) is estimated assuming the complete contribution of the sensible heat from container (\(m_{Cont}\):\(c_{p,Cont}\)) and heat exchanger materials (\(m_{HEX}\):\(c_{p,HEX}\)) and PCM (\(m_{PCM}\):\(c_{p,PCM}\)) from the initial temperature of the unit \(T_{init}\) up to the phase change temperature of the PCM \(T_{PC}\) in addition, to 90% of the latent contribution from the PCM, as shown in Equation (7):  
\[E_{st,90}=(m_{HEX}\):\(c_{p,HEX}+m_{Cont}\):\(c_{p,Cont}+m_{PCM}\):\(c_{p,PCM})\cdot(T_{init}-T_{PC})+(90\%\cdot m _{PCM}\cdot\Delta t_{PC}) \tag{7}\]  
Using these considerations and assumptions allows for the calculation of the proposed NHTTC with minimal representative information.