# Multi-Domain Modeling and Analysis of Marine Steam Power System Based on Digital Twin
## 2 Mathematical Model of Marine Steam Power System Equipment
### Mathematical Model of Condenser  
In order to establish the mathematical model of the condenser, the condenser is divided into a steam area and a cooling water area [14].  
1. Steam area:  
(1) Mass conservation equation:  
\[\frac{dM_{s}}{dt}=\frac{d(\rho_{s}V_{s})}{dt}=q_{s}-q_{air}-q_{c} \tag{4}\]  
where:  
\(M_{s}\)--The mass of steam, kg;  
\(\rho_{s}\)--The density of steam, kg/m\({}^{3}\); \(V_{s}\)--The volume of steam, m\({}^{3}\);  
\(q_{s}\)--The amount of steam discharged into the condenser, kg/s;  
\(q_{air}\)--The amount of air discharged into the condenser, kg/s;  
\(q_{c}\)--The amount of condensate outlet flow, kg/s.  
(2) Energy conservation equation:  
\[\frac{d(\rho_{s}V_{s}h_{s})}{dt}=q_{s}h_{s}-q_{air}h_{air}-q_{c}h_{c}-Q_{c} \tag{5}\]  
where:  
\(h_{s}\)--The enthalpy of steam, kJ/kg;\(h_{air}\)--The enthalpy extracted by steam jet ejector, kJ/kg;  
\(h_{c}\)--The enthalpy of condensate, kJ/kg;  
\(Q_{c}\)--The condensation of steam releases heat, kJ/s.  
\[Q_{c}=\alpha_{c}A_{t}(T_{s}-T_{t}) \tag{6}\]  
where:  
\(\alpha_{c}\)--The condensation heat release coefficient between steam and cooling tube, kW/(m^2.^oC);  
\(A_{t}\)--The external wall area of cooling tube, m\({}^{2}\);  
\(T_{s}\)--The temperature of steam, K;  
\(T_{t}\)--The temperature of cooling tube, K.  
2. Cooling water area:  
(1) Convective heat transfer of cooling water:  
\[Q_{w}=\alpha_{w}A_{s}(T_{t}-T_{w}) \tag{7}\]  
where:  
\(Q_{w}\)--The convective heat transfer of cooling water, kJ/s;  
\(\alpha_{w}\)--The convective heat transfer coefficient between cooling water and cooling tube, kW/(m^2.^oC);  
\(A_{s}\)--The heat exchange area of cooling pipe, m\({}^{2}\);  
\(T_{w}\)--The temperature of cooling water, K.  
(2) Energy conservation equation of cooling tube wall:  
\[M_{t}c_{t}\frac{dT_{t}}{dt}=Q_{c}-Q_{w} \tag{8}\]  
where:  
\(M_{t}\)--The quality of cooling tube, kg;  
\(c_{t}\)--The specific heat capacity of cooling tube, kJ/(kg.^oC).  
(3) Mass conservation equation of cooling tube:  
\[M_{w}c_{w}\frac{dT_{w}}{dt}=Q_{w}-q_{w}c_{w}(T_{f2}-T_{f1}) \tag{9}\]  
where:  
\(M_{w}\)--The quality of cooling water in cooling tube, kg;  
\(c_{w}\)--The specific heat capacity of cooling water, kJ/(kg.^oC);  
\(q_{w}\)--The flow of cooling water, kg/s;  
\(T_{f2}\)--The outlet temperature of cooling water, K;  
\(T_{f1}\)--The inlet temperature of cooling water, K.  
(4) The overall heat transfer coefficient:  
\[K=K_{0}\beta_{t}\beta_{m}\phi_{c} \tag{10}\]  
where:  
\(K\)--The overall heat transfer coefficient of condenser, kW/(m^2.^oC);  
\(K_{0}\)--The basic heat transfer coefficient, kW/(m^2.^oC);  
\(\beta_{t}\)--The correction coefficient of cooling water inlet temperature;  
\(\beta_{m}\)--The correction coefficient of cooling tube material and wall thickness;  
\(\phi_{c}\)--The cleanliness coefficient of cooling tube.  
(5) Logarithmic mean temperature difference:  
\[\Delta T_{\rm{In}}=\frac{T_{f2}-T_{f1}}{\ln\big{(}\frac{T_{s}-T_{f1}}{T_{s}-T_{f 2}}\big{)}}=\frac{\Delta T}{\ln\frac{\Delta T+\delta_{t}}{\delta_{t}}} \tag{11}\]  
where:  
\(\Delta T\)--The rise in temperature of cooling water, \(\Delta T=T_{f2}-T_{f1}\), K;  
\(\delta_{t}\)--The heat transfer end difference of condenser, K.  
(6) Heat transfer end difference:  
\[\delta_{t}=\frac{\Delta T}{e^{\frac{\Delta T_{s}}{\Delta 1816\sim\delta_{\rm{Its}}}}-1} \tag{12}\]