# Investigation on an Improved Household Refrigerator for Energy Saving of Residential Buildings
## 4 Mathematical Model  
The simplified mathematical model is built for the original and the modified freezer of the BCD-111 refrigerator. The following hypotheses are adopted. * The temperature fluctuation of air is neglected. * The walls of refrigerator have the same heat conductive coefficient. * Rectangular straight fin efficiency is adopted for aluminum plates [20]. * The heat transfer is steady-state and 1D. * The temperatures at evaporating and condensing sections of heat pipes are considered uniform (T\({}_{HP}=T_{con}=T_{eval}\)) since the internal two-phase thermal resistance is far less than the external convective thermal resistance. * The external surface temperature of side freezer walls is given as a boundary condition due to the influence of condenser of the BCD-111 refrigerator. The measured boundary conditions are utilized in following methods:  
Boundary condition 1: the surrounding air temperatures are given for the original freezer walls (or the evaporating sections of heat pipes in fresh food compartment);  
Boundary condition 2: the average outside wall temperatures are given for the improved freezer walls. Based on the above hypotheses and boundary conditions, the thermal analyses of the original and improved refrigerator are conducted using iterative computations. The heat fluxes at half of the topside freezer wall of the improved refrigerator are calculated based on the boundary condition 1, in which the heat conversion coefficient is calculated through self iteration:  
\[Q_{t,out}=\frac{T_{amb}-T_{HP}}{\frac{1}{\alpha_{t,out}}+\frac{(\beta-X_{t,in })}{\lambda}}A_{t} \tag{1}\]  
\[Q_{t,in}=\frac{T_{HP}-T_{f}}{\frac{1}{\alpha_{t,in}}+\frac{X_{t,in}}{\lambda} }A_{t} \tag{2}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{3}{c}{**Operating States of the Tested Refrigerator**} \\ \hline \hline
**Ambient Temperature [°C]** & **Temperature Gear** & \(T_{f}\) [°C] & \(T_{f\mu}\) [°C] & \(T_{s,out}\) [°C] \\ \hline
25.0 & C & \(-\)17.7 & 4.7 & 27.0 \\ \hline
17.0 & C & \(-\)16.0 & 5.2 & 18.1 \\ \hline
20.0 & C & \(-\)16.5 & 5.0 & 21.3 \\ \hline
31.0 & C & \(-\)20.2 & 2.9 & 32.6 \\ \hline
25.0 & A & \(-\)19.6 & 2.1 & 27.0 \\ \hline
25.0 & B & \(-\)19.3 & 2.6 & 27.0 \\ \hline
25.0 & D & \(-\)17.2 & 5.0 & 26.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Boundary conditions under different temperature gears and ambient temperatures [19]. where \(Q_{t,out}\) and \(Q_{t,in}\) are outside and inside heat fluxes, respectively; \(T_{amb}\), \(T_{HP}\), and \(T_{f}\) are ambient temperature, heat pipe temperature, and freezer temperature, respectively; \(\alpha_{t,out}\) and \(\alpha_{t,in}\) are the convective heat transfer coefficients of ambient and inside air, respectively; \(\delta\) is the wall thickness; \(X_{t}\) is the ratio of wall thickness inside of heat pipes to the whole topside wall thickness; and \(A_{t}\) is the topside heat transfer area. The heat fluxes at left or right freezer walls of the improved refrigerator are calculated based on the boundary condition 2. \[Q_{s,out}=\frac{T_{s,out}-T_{HP}}{\frac{(\delta-X_{s,in})}{\lambda}}A_{s} \tag{3}\]  
\[Q_{s,in}=\frac{T_{HP}-T_{f}}{\frac{1}{\alpha_{s,in}}+\frac{X_{s,in}}{\lambda}}A _{s} \tag{4}\]  
where \(Q_{s,out}\) and \(Q_{s,in}\) are outside and inside heat fluxes, respectively; \(T_{s,out}\) is the outside surface temperature of side freezer wall; \(\alpha_{t,out}\) and \(\alpha_{t,in}\) are the convective heat transfer coefficients of ambient and inside air, respectively; \(X_{s}\) is the ratio of wall thickness inside of heat pipes to the whole side wall thickness; and \(A_{s}\) is the side heat transfer area. For the condensing section of heat pipes, the heat is released to the topside and the side freezer walls.