# Comparison of Heat Transfer Enhancement Techniques in Latent Heat Storage
## 2 Methodology  
### Definition of Proposed KPIs  
The main focus of this study is to allow the simplified and quick evaluation of the heat transfer capabilities in a LHES unit regardless of scale and operating conditions. Achieving this goal requires the usage of the most readily available information able to represent a highly transient process through averaged properties. The methodology proposed in this study uses this information adapted around the (\(\varepsilon\)-NTU) method. Two KPI are proposed representing both the heat transfer performance of a LHES unit as well as the degree of compactness and an indication of the energy density attainable by the system. Regarding the degree of compactness, the volume fraction of the major contributor to the storage capacity of the LHES unit, PCM to total volume of the unit, (\(\Phi_{PCM}\)) is suggested as an indicator of the energy density attainable by the system. \(\Phi_{PCM}\) provides an indication of the compactness degree attainable, but also information on the required trade-off of PCM storage volume for heat exchanger material to achieve certain heat transfer performance, and it is the ratio of PCM volume in the storage (\(V_{PCM}\)) to the total outer volume of the unit (\(V_{TOT}\)). See Equation (1)  
\[\Phi_{PCM}=\frac{V_{PCM}}{V_{TOT}} \tag{1}\]  
The total volume (\(V_{TOT}\)) was calculated considering the geometry of the outermost layer of the unit while excluding the additional volume used for insulation. The container wall thickness as well as additional volume dedicated to manifolds (flow development) and the like were all taken into account. From a heat exchanger perspective, an LHES unit can be regarded as a heat exchanger, the performance of which is defined by a static heat sink or source, or an analog case of a heat exchanger operating as a boiler or condenser. The average NTU (\(NTU_{avg}\)) represents the added effects of the heat exchanger tubes and growing layer of solid through the discharge process. It is defined as the ratio of the heat transfer rate capacity of the heat exchanger (product of the overall heat transfer coefficient (U) and the heat exchange surface (A)), and the heat capacity rate of the HTF (\(\dot{n}_{HTF}\cdot c_{p,HTF}\)) [45]. Additionally, the average NTU can be easily estimated as it is directly related to the average effectiveness (\(\varepsilon_{avg}\)) of the heat exchange, and under the assumptions of a phase change, similarly to boiler/condenser operation, the heat exchanger effectiveness relations can be simplified [45] as described in Equation (2):  
\[NTU_{avg}=\frac{U\cdot A}{\dot{n}_{HTF}\cdot c_{p,HTF}}=-ln(1-\varepsilon_{ avg}) \tag{2}\]This equation can be rearranged and divided by the total volume \(V_{TOT}\) to calculate the normalized heat transfer performance coefficient (NHTPC) as shown below in Equation (3):  
\[\mathbf{NHTPC}=\frac{\mathbf{U\cdot A}}{\mathbf{V}_{TOT}}=\frac{-ln(1-\epsilon_ {avg})\cdot\dot{m}_{HTF}\cdot c_{p,HTF}}{V_{TOT}} \tag{3}\]  
Thus, the proposed KPI on the heat transfer side, NHITPC, can be seen as the product of the overall heat transfer coefficient (U) and the heat exchange surface (A) normalized by the total volume of the LHES unit (\(V_{TOT}\)). Considering the transient nature of the solidification process due to the increasing resistance and changing surface, it is useful to express an average U\(\cdot\)A for the whole process. Dividing this product by the total volume of the LHES unit (\(V_{TOT}\)) excluding insulation enables comparison of the heat transfer behavior regardless of the final dimensions, operation conditions, and overall scale. Where \(\epsilon_{avg}\) represents the average heat exchanger effectiveness during discharge, \(\dot{m}_{HTF}\) and \(c_{p,HTF}\), the mass flow rate and specific heat capacity of the HTF, respectively, and finally \(V_{TOT}\) the total outer volume of the container without considering any insulation.