# DynamicWaterWaterExchanger


## Copyright © EDF 2002 - 2026  
## ThermoSysPro Version 4.2  
This component model is documented in Sect. 9.6.1 of the ThermoSysPro book.   
# Dynamic water water exchanger   
   
The plate heat exchanger is composed of thousand sheets separated from each other by a small space where fluids flow. The plates exhibit a wavy surface to create a turbulent flow that generates better heat transfers. This type of exchanger is widely used in the food industry as it can easily be taken apart for cleanup.  

This model represents a single-phase counter-flow thermal exchange between the hot fluid and the cold fluid. The two fluids are separated by a wall through which heat transfer takes place by conduction. Convection transfers the heat between each fluid and the wall.  

Following assumptions are made:  
- the flow in each mesh cell is single-phase.  
- the energy accumulation in the wall is neglected.  
- the phenomenon of longitudinal heat conduction in the wall and in the fluid is neglected.  
- the pressure and specific enthalpy are assumed constant in each mesh cell.  

For a steady-state plate heat exchanger, see [static water water exchanger](ThermoSysPro.WaterSteam.HeatExchangers.StaticWaterWaterExchanger).  

## Modelica component model  

The equations mentioned below are implemented in the component *DynamicWaterWaterExchanger*, located in the *WaterSteam.HeatExchangers* sub-library.   
   
![ThermoSysPro.WaterSteam.HeatExchangers.DynamicWaterWaterExchanger.svg](./icons/ThermoSysPro.WaterSteam.HeatExchangers.DynamicWaterWaterExchanger.svg)  

## Nomenclature  

| Symbol| Description| Unit| Definition| Modelica name |  
| :-------------------------------- | :------------------------------------------------------------------------------------------ | :------------------------------------------- | :----------------------------------------------------------------------------------------------------------------------- | :----------- |  
| $c_{p, c, i}$| Cold fluid specific heat in thermal cell $i$| $\mathrm{J} / \mathrm{kg} / \mathrm{K}$|| prof[i].cp |  
| $c_{p, \mathrm{h}, i}$| Hot fluid specific heat in thermal cell $i$| $\mathrm{J} / \mathrm{kg} / \mathrm{K}$|| proc[i].cp |  
| $D_{\mathrm{h}}$| Hydraulic diameter| $\mathrm{m}$|| - |  
| $e_{\mathrm{m}}$| Wall thickness| $\mathrm{m}$|| emetal |  
| $h_{\mathrm{c}, i}$| Cold fluid specific enthalpy in thermal cell $i$| $\mathrm{J} / \mathrm{kg}$|| Hmf[i] |  
| $h_{\mathrm{c}, i: i+1}$| Cold fluid specific enthalpy in hydraulic cell $i: i+1$| $\mathrm{J} / \mathrm{kg}$|| Hcf[i] |  
| $h_{\mathrm{h}, i}$| Hot fluid specific enthalpy in thermal cell $i$| $\mathrm{J} / \mathrm{kg}$|| Hmc[i] |  
| $h_{\mathrm{h}, i: i+1}$| Hot fluid specific enthalpy in hydraulic cell $i: i+1$| $\mathrm{J} / \mathrm{kg}$|| Hcc[i] |  
| $K_{\mathrm{c}, i}$| Convective heat exchange coefficient for the cold fluid in thermal cell $i$| $\mathrm{W} / \mathrm{m}^{2} / \mathrm{K}$ || hf[i] |  
| $K_{\mathrm{h}, i}$| Convective heat exchange coefficient for the hot fluid in thermal cell $i$| $\mathrm{W} / \mathrm{m}^{2} / \mathrm{K}$ || hc[i] |  
| $m_{\mathrm{c}, i: i+1}$| Cold fluid mass flow rate in hydraulic cell $i:$ $i+1$| $\mathrm{kg} / \mathrm{s}$|| Qcf[i] |  
| $m_{\mathrm{h}, i: i+1}$| Hot fluid mass flow rate in hydraulic cell $i:$ $i+1$| $\mathrm{kg} / \mathrm{s}$|| Qcc[i] |  
| $N$| Number of hydraulic cells| $-$|| N + 1 |  
| $N_{\mathrm{c}}$| Number of channels of each fluid| $-$| $\left(N_{\mathrm{p}}-1\right) / 2$| - |  
| $N_{\mathrm{p}}$| Number of plates| $-$|| nbp |  
| $P_{\mathrm{c}, i}$| Cold fluid pressure at the outlet of thermal cell $i$| $\mathrm{Pa}$|| Pcf[i + 1] |  
| $P_{\mathrm{h}, i}$| Hot fluid pressure at the outlet of thermal cell $i$| $\mathrm{Pa}$|| Pcc[i + 1] |  
| $P r_{\mathrm{c}, i}$| Prandtl number of the cold fluid in thermal cell $i$| $-$| $\frac{\mu_{\mathrm{c}, i} \cdot c_{p, \mathrm{c}, i}}{\lambda_{\mathrm{c}, i}}$| - |  
| $P r_{h, i}$| Prandtl number of the hot fluid in thermal cell $i$| $-$| $\frac{\mu_{\mathrm{h}, i} \cdot c_{p, \mathrm{h}, i}}{\lambda_{\mathrm{h}, i}}$| - |  
| $R e_{c, i:i+1}$| Reynolds number of the cold fluid in hydraulic cell $i: i+1$| $-$| $\frac{4 \cdot m_{\mathrm{c}, i: i+1}}{\pi \cdot D_{\mathrm{h}} \cdot \mu_{\mathrm{c}, i} \cdot N_{\mathrm{c}}}$| - |  
| $R e_{\mathrm{h}, i: i+1}$| Reynolds number of the hot fluid in hydraulic cell $i: i+1$| $-$| $\frac{4 \cdot \dot{m}_{\mathrm{h}, i: i+1}}{\pi \cdot D_{\mathrm{h}} \cdot \mu_{\mathrm{h}, i} \cdot N_{\mathrm{c}}}$ | - |  
| $S_{\mathrm{p}}$| Plate area| $\mathrm{m}^{2}$|| Sp |  
| $T_{\mathrm{c}, i}$| Temperature of the cold fluid in thermal cell $i$| $\mathrm{K}$|| Tmf[i] |  
| $T_{\mathrm{h}, i}$| Temperature of the hot fluid in thermal cell $i$| $\mathrm{K}$|| Tmc[i] |  
| $T_{\mathrm{w}, \mathrm{c}, i}$ | Wall temperature of for cold fluid in thermal cell $i$| $\mathrm{K}$|| Tmf[i] |  
| $T_{\mathrm{w}, \mathrm{h}, i}$ | Wall temperature for the hot fluid in thermal cell $i$| $\mathrm{K}$|| Tmc[i] |  
| $U_{i}$| Global heat transfer coefficient for thermal cell $i$| $\mathrm{W} / \mathrm{m}^{2} / \mathrm{K}$ || K[i] |  
| $V_{\mathrm{c}}$| Cold fluid volume| $\mathrm{m}^{3}$|| Vf |  
| $V_{\mathrm{h}}$| Hot fluid volume| $\mathrm{m}^{3}$|| Vc |  
| $\Delta A_{i}$| Heat exchange surface for thermal cell $i$| $\mathrm{m}^{2}$| $\frac{s_{\mathrm{p}} \cdot\left(N_{\mathrm{p}}-2\right)}{N-1}$| dS |  
| $\Delta W_{i}$| Thermal power released by the hot fluid to the cold fluid for thermal cell $i$| $\mathrm{W}$|| dW[i] |  
| $\lambda_{\mathrm{c}, i}$| Cold fluid thermal conductivity in thermal cell $i$| $\mathrm{W} / \mathrm{m} / \mathrm{K}$|| lambdaf[i] |  
| $\lambda_{\mathrm{h}, i}$| Hot fluid thermal conductivity in thermal cell $i$| $\mathrm{W} / \mathrm{m} / \mathrm{K}$|| lambdac[i] |  
| $\lambda_{\mathrm{m}}$| Metal thermal conductivity| $\mathrm{W} / \mathrm{m} / \mathrm{K}$|| lambdam |  
| $\Lambda_{\mathrm{c}}$| Friction pressure loss coefficient of the entire length of the exchanger for the cold fluid | $\mathrm{m}^{-4}$|| p_Kf |  
| $\Lambda_{\mathrm{h}}$| Friction pressure loss coefficient of the entire length of the exchanger for the hot fluid| $\mathrm{m}^{-4}$|| p_Kc |  
| $\mu_{\mathrm{c}, i}$| Cold fluid dynamic viscosity in thermal cell $i$| $\mathrm{Pa} \mathrm{s}$|| muf[i] |  
| $\mu_{\mathrm{c}, i: i+1}$| Cold fluid dynamic viscosity in hydraulic cell $i: i+1$| $\mathrm{Pa} \mathrm{s}$|| muf[i] |  
| $\mu_{\mathrm{h}, i}$| Hot fluid dynamic viscosity in thermal cell $i$| $\mathrm{Pa} \mathrm{s}$|| muc[i] |  
| $\mu_{\mathrm{h}, i: i+1}$| Hot fluid dynamic viscosity in hydraulic cell $i: i+1$| $\mathrm{Pa} \mathrm{s}$|| muc[i] |  
| $\rho_{\mathrm{c}, i}$| Cold fluid density in thermal cell $i$| $\mathrm{kg} / \mathrm{m}^{3}$|| rhof[i] |  
| $\rho_{\mathrm{h}, i}$| Hot fluid density in thermal cell $i$| $\mathrm{kg} / \mathrm{m}^{3}$|| rhoc[i] |  


## Governing equations  

The following set of equations must be completed by the state equations for the following water and steam properties $c_{p, c}, c_{p, h}, \rho_{c}, \rho_{h}, \mu_{c}, \mu_{h}, \lambda_{c}$ and $\lambda_{h}$.  

### Steady-state mass balance equation (hot fluid)  


- Validity domain:   
   
 $\forall \dot{m}_{\mathrm{h}, i:i+1}$  

- Mathematical formulation:   
   
 $$\dot{m}_{\mathrm{h}, i-1:i}-\dot{m}_{\mathrm{h}, i:i+1}=0$$  

- Comments:   
   

### Steady-state mass balance equation (cold fluid)  


    
    

- Validity domain:   
   
 $\forall \dot{m}_{\mathrm{c}, i:i+1}$  

- Mathematical formulation:   
   
 $$\dot{m}_{\mathrm{c}, i-1:i}-\dot{m}_{\mathrm{c}, i:i+1}=0$$   

- Comments:   
   



### Dynamic energy balance equation (hot fluid)  


    
    

- Validity domain:   
   
 $\forall \dot{m}_{\mathrm{h}, i:i+1}$  

- Mathematical formulation:   
   
 $$\frac{V_{\mathrm{h}}}{N-1} \cdot \rho_{\mathrm{h}, i} \cdot \frac{\mathrm{d} h_{\mathrm{h}, i}}{\mathrm{d} t}=\dot{m}_{\mathrm{h}, i-1:i} \cdot h_{\mathrm{h}, i-1:i}-\dot{m}_{\mathrm{h}, i:i+1} \cdot h_{\mathrm{h}, i:i+1}-\Delta W_{i}$$  

- Comments:   
   
 The fluid is assumed to be incompressible.  


### Dynamic energy balance equation (cold fluid)  


    
    

- Validity domain:   
   
 $\forall \dot{m}_{c, i:i+1}$  

- Mathematical formulation:   
   
 $$\frac{V_{\mathrm{c}}}{N-1} \cdot \rho_{\mathrm{c}, i} \cdot \frac{\mathrm{d} h_{\mathrm{c}, i}}{\mathrm{d} t}=\dot{m}_{\mathrm{c}, i-1:i} \cdot h_{\mathrm{c}, i-1:i}-\dot{m}_{\mathrm{c}, i:i+1} \cdot h_{\mathrm{c}, i:i+1}+\Delta W_{i}$$  

- Comments:   
   
 The fluid is assumed to be incompressible.  







    
    

### Heat exchanged between the hot and cold fluids  

- Validity domain:   
   
 $\forall T_{\mathrm{h}, i}$ and $\forall T_{\mathrm{c}, i}$  

- Mathematical formulation:   
   
 $$\Delta W_{i}=U_{i} \cdot \Delta A_{i} \cdot\left(T_{\mathrm{h}, i}-T_{\mathrm{c}, i}\right)$$  

- Comments:   
   
 The global heat transfer coefficient $U_{i}$ between the two fluids is given by $\frac{1}{U_{i}}=\frac{1}{K_{\mathrm{h}, i}}+\frac{1}{\lambda_{\mathrm{m}}}+\frac{1}{K_{\mathrm{c}, i}}$.   
    

### Momentum balance equation (hot fluid)  

- Validity domain:   
   
 $\forall \dot{m}_{\mathrm{h}, i:i+1}$  

- Mathematical formulation:   
   
 $$P_{\mathrm{h}, i+1} = P_{\mathrm{h}, i}-\frac{\Lambda_{\mathrm{h}}}{N} \cdot \frac{\dot{m}_{\mathrm{h}, i:i+1} \cdot \lvert \dot{m}_{\mathrm{h}, i:i+1} \rvert}{N_{\mathrm{c}}^{2} \cdot \rho_{\mathrm{h}, i}}$$  

- Comments:   
   
 Only pressure losses due to friction are taken into account. The friction coefficient $\Lambda_{\mathrm{h}}$ can be directly provided by the user or computed using a correlation.  


### Momentum balance equation (cold fluid)  


    
    

- Validity domain:   
   
 $\forall \dot{m}_{\mathrm{c}, i:i+1}$  

- Mathematical formulation:   
   
 $$P_{\mathrm{c}, i+1}=P_{\mathrm{c}, i}-\frac{\Lambda_{\mathrm{c}}}{N} \cdot \frac{\dot{m}_{\mathrm{c}, i:i+1} \cdot \lvert \dot{m}_{\mathrm{c}, i:i+1} \rvert}{N_{\mathrm{c}}^{2} \cdot \rho_{\mathrm{c}, i}}$$  

- Comments:   
   
 Only pressure losses due to friction are taken into account. The friction coefficient $\Lambda_{\mathrm{c}}$ can be directly provided by the user or computed using a correlation.  


## References   
   
El Hefni, Baligh and Bouskela, Daniel (2019). [Modeling and Simulation of Thermal Power Plants with ThermoSysPro](https://link.springer.com/book/10.1007/978-3-030-05105-1), sect. 9.6.1. Springer Nature Switzerland AG.