## 2 Materials and Methods--Modelling and Simulation of Nuclear Power Plants (NPPs)
### Logarithmic Mean Temperature Difference (LMTD) for Precooler Design  
For a generic heat exchanger such as the precooler, the heat duty defines the amount of heat transferred by a known hot fluid quantity to a known cold fluid quantity for a given time. This is expressed as:  
\[q=h\cdot A_{pc}\cdot\Delta T_{log_{man}} \tag{30}\]  
whereby \(q\) is the heat duty and defines the thermal rating, \(h\) is the overall heat transfer coefficient which is dependent on the material conductivity, geometry, flow properties, specific heat capacity and the individual convection heat transfer, and conductivity of the hot and cold fluids, \(A_{pc}\) is the exchange area of the precooler, and \(\Delta T_{log_{man}}\) is the Logarithmic Mean Temperature Difference (LMTD). The LMTD is interchanageble with the NTU method adopted for the recuperator and is the temperature driving force of the heat transfer process because it is the difference in the hot and cold fluid. More heat is transferred as a result of a favourable LMTD depending on the design of the precooler (co-current or counter current), if the assumption remains that there are constant flow rates and fluid thermal properties. For a precooler of co-current or counter-current flow, if the heat transfer is set to occur along a specific axis (\(x\)) from point A to B, for two fluids called 1 and 2, with their temperature along \(x\) known as \(T_{1}(x)\) and \(T_{2}(x)\), then the exchange at that localised point (\(x\)) is proportional to the temperature delta:  
\[q(x)=\frac{h(T_{1}(x)-T_{2}(x))}{D}=\frac{h(\Delta T(x))}{D} \tag{31}\]  
whereby \(D\) is the distance between the two fluids. The heat that leaves the fluid causes a temperature gradient according to [PERSON]'s law:  
\[\frac{dT_{1}}{d\ x}=k_{a}(T_{1}(x)-T_{2}(x))=-k_{a}\Delta T(x) \tag{32}\]  
\[\frac{dT_{2}}{d\ x}=k_{b}(T_{2}(x)-T_{1}(x))=k_{b}\Delta T(x) \tag{33}\]  
Thus, the summation of Equations (32) and (33) becomes:  
\[\frac{d\Delta T}{d\ x}=\frac{d\ (T_{2}-T_{1})}{d\ x}=\frac{dT_{2}}{d\ x}- \frac{dT_{1}}{d\ x}=k\Delta T(x) \tag{34}\]and considers that \(k=k_{a}+k_{b}\). The full exchange is derived by integrating the local heat transfer from point \(A\) to \(B\):  
\[q=\int_{A}^{B}q(x)dx=\frac{h}{D}\int_{A}^{B}\Delta T(x)dx=\frac{h}{D}\int_{A}^{B }\Delta T\;dx \tag{35}\]  
with the exchange area \(A_{pc}\) being point \(A\) to \(B\) (length) multiplied by the distance of the internal pipe \(D\):  
\[q=\frac{hA_{pc}}{(B-A)}\int_{A}^{B}\Delta T\;dx=\frac{hA_{pc}\int_{A}^{B} \Delta T\;dx}{\int_{A}^{B}dx} \tag{36}\]  
substituting \(x\) for the temperature difference \(\Delta T\) in Equation (36) gives:  
\[q=\frac{hA_{pc}\int_{\Delta T(A)}^{\Delta T(B)}\Delta T\;\frac{dx}{d\Delta T}d (\Delta T)}{\int_{\Delta T(A)}^{\Delta T(B)}\frac{dx}{d\Delta T}d(\Delta T)} \tag{37}\]  
and substituting the relationship for delta \(T\) (as derived in Equation (34)) in Equation (37) yields:  
\[q=\frac{hA_{pc}\int_{\Delta T(A)}^{\Delta T(B)}\frac{1}{k\Delta}d(\Delta T)}{ \int_{\Delta T(A)}^{\Delta T(B)}\frac{1}{k\Delta T}d(\Delta T)} \tag{38}\]  
thus from integration and in relation to Equation (30), \(q\) is expressed as:  
\[q=h\cdot A_{pc}\cdot\Delta T_{log_{mean}}=h\cdot A_{pc}\cdot\frac{\Delta T(B) -\Delta T(A)}{ln\frac{\Delta T(B)}{\Delta T(A)}} \tag{39}\]  
Using the portion of Equation (39) relating to \(\Delta T_{log_{mean}}\) and with reference to Figure 4, the co-current and counter-current flow configurations are defined as follows:  
\[q(x)=\frac{h(T_{1}(x)-T_{2}(x))}{D}=\frac{h(\Delta T(x))}{D} \tag{31}\]0  
\[q(x)=\frac{h(T_{1}(x)-T_{2}(x))}{D}=\frac{h(\Delta T(x))}{D} \tag{31}\]1  
The tool was used to match the DP conditions of known NPPs in the literature in order to verify its functionality. The matched results were considered satisfactory for the purpose of this study.