# Modeling and Key Parameter Interaction Analysis for Ship Central Cooling Systems
## 3 Calculation Model  
### Mathematical Model  
The total power consumption of the central cooling system consisted of the power consumption of the system's seawater pumps and fresh water pumps. Based on the principle of energy conservation, the shaft power of a single pump is expressed by Formula (1) [4]:  
\[P_{-}p=\frac{p\cdot g\cdot Q\cdot H}{3.6\times 10^{6}\eta} \tag{1}\]  
where \(P_{-}p\) is the shaft power of a single pump, in kW; \(\rho\) is the fluid density (seawater or fresh water), in kg/m\({}^{3}\); \(g\) is the acceleration due to gravity, in m/s\({}^{2}\); \(Q\) is the seawater or fresh water flow rate, in m\({}^{3}\)/h; \(H\) is the pump head, in m; and \(\eta\) is the pump efficiency.  
The total power consumption of the system was the sum of the shaft powers of all relevant pumps, as shown in Formula (2):  
\[P\_total=\sum_{i=1}^{n}P\_(p,i) \tag{2}\]  
where \(P\_total\) is the total power consumption of the system, in kW; \(P\_(p,i)\) is the shaft power of the \(i\)th pump, in kW; and \(n\) is the total number of seawater and fresh water pumps in the system.  
The pressure loss in the system's piping consisted of frictional losses along the length and local losses. The frictional loss along the length is calculated according to Formula (3):  
\[h_{f}=\lambda\frac{L}{d}.\frac{v^{2}}{2g} \tag{3}\]  
where \(h_{f}\) is the frictional loss along the length, in m; \(\lambda\) is the friction resistance coefficient; \(L\) is the pipe length, in m; \(d\) is the internal diameter of the pipe, in m; \(v\) is the flow velocity, in m/s; and \(g\) is the gravitational acceleration, in m/s\({}^{2}\).  
Local resistance loss was generated when fluid flowed through pipe fittings (such as elbows, valves, etc.) and can be calculated using Formula (4):  
\[h_{a}=\xi\frac{v^{2}}{2g} \tag{4}\]  
where \(h_{a}\) is the local resistance loss, in m; \(\xi\) is the local resistance coefficient; and \(v\) is the flow velocity, in m/s.  
The total resistance loss was the pressure loss from both the pipeline and fittings and can be expressed as Formula (5):  
Figure 1: Schematic diagram of the central cooling system.  
\[\Delta H=h_{f}+h_{a} \tag{5}\]  
where \(\Delta H\) is the total head loss, in m.  
To accurately describe the heat transfer process of the system, the main heat balance governing equation was established, as shown in Formula (6):  
\[Q=mc_{\text{p}}(T_{\text{in}}-T_{\text{out}}) \tag{6}\]  
where \(Q\) is the heat transfer rate, in kW; \(m\) is the mass flow rate of the cooling medium, in kg/s; \(c_{\text{p}}\) is the specific heat capacity of the cooling medium at constant pressure, in kJ/(kg\(\cdot\)K); \(T_{\text{in}}\) is the inlet temperature of the cooling medium, in \({}^{\circ}\)C; and \(T_{\text{out}}\) is the outlet temperature of the cooling medium, in \({}^{\circ}\)C.  
The calculation formula for the flow characteristics of a three-way valve is shown in Formula (7):  
\[Q=K\cdot(\frac{V}{V_{max}})^{\alpha} \tag{7}\]  
where \(Q\) is the flow rate, in m\({}^{3}\)/s; \(K\) is the flow coefficient; \(V\) is the valve opening; \(V_{max}\) is the maximum flow rate, in m\({}^{3}\)/s; and \(\alpha\) is the opening exponent, dimensionless, reflecting the relationship between the valve flow rate and opening.  
Detailed calculations of pressure drop in the equipment, pump head calculations, and the specific characteristics of the three-way valve were all based on standard fluid dynamics principles and equipment specifications. Specific details can be found in the cited literature [19].