# DynamicTwoPhaseFlowPipe


## Copyright © EDF 2002 - 2026   
## ThermoSysPro Version 4.2  
This component model is documented in Sect. 9.4.2 of the ThermoSysPro book.   
# Dynamic two phase flow pipe  

This component is a tube bundle, one of the three parts composing a shell-and-tube heat exchanger.  
The component represents a two-phase fluid flowing inside the bundle and exchanging heat with the wall.  
The conduction inside the wall is represented in the associated model [Heat exchanger wall](ThermoSysPro.Thermal.HeatTransfer.HeatExchangerWall).  

The model of the tube bundle relies on several assumptions:  
- the fluid is homogeneous in each mesh cell (i.e. same velocity for both liquid and steam phases),  
- mass accumulation is considered in each mesh cell,  
- the inertia of the fluid is taken into account,  
- the longitudinal heat conduction in the metal wall and in the fluid is neglected,  
- the thermophysical properties are calculated via the average pressure and enthalpy in each cell.  

## Modelica component model  

The equations mentioned below are implemented in the component *DynamicTwoPhaseFlowPipe*, located in the *WaterSteam.HeatExchangers* sub-library.  
This component has 3 connectors:  
- C1: fluid inlet,  
- CTh: thermal connector,  
- C2: fluid outlet.  

![ThermoSysPro.WaterSteam.HeatExchangers.DynamicTwoPhaseFlowPipe.svg](./icons/ThermoSysPro.WaterSteam.HeatExchangers.DynamicTwoPhaseFlowPipe.svg)  

## Nomenclature  


| Symbol          | Description                                                         | Unit                          | Definition                                | Modelica name |  
|---------------- |------------------------------------------------------------------ |----------------------------- |------------------------------------------------------ | :----------- |  
| $ A $ | Internal cross section of the pipe bundle | $ \mathrm{m^2} $ | $ N_t \cdot \pi \cdot D^2 / 4 $ | A |  
| $ D $ | Internal diameter of one pipe | $ \mathrm{m}$ | | D |  
| $ D_t $ | Sum of the internal diameter of the pipes in the bundle | $ \mathrm{m} $ | $ N_t \cdot D  $ | - |  
| $ h_{c,i} $ | Convective heat exchange coefficient  in thermal cell *i* | $ \mathrm{W/m^2/K} $ || hi[i] |  
| $ h_i $ | Fluid specific enthalpy in thermal cell *i* | $ \mathrm{J/kg} $ | | h[i] |  
| $ h_{i:i+1} $ | Fluid specific enthalpy of the mass flow  $ \dot{m}_{i:i+1} $ at the boundary between thermal cells *i* and *i+1* | $ \mathrm{J/kg} $ | | hb[i] |  
| $ L $ | Pipe length | $ \mathrm{m} $ | | L |  
| $ \dot{m}_{i:i+1} $ | Mass flow rate crossing the boundary between cells *i* and *i+1*,  oriented positively from *i* to *i+1* | $ \mathrm{kg/s} $ | | Q[i] |  
| $ N $ | Number of hydraulic cells equal to the number of  thermal cells plus one | - | | N + 1 |  
| $ N_t $ | Number of pipes in parallel | - | | ntubes |  
| $ P_i $ | Fluid pressure in thermal cell *i* | $ \mathrm{Pa} $ | | P[i] |  
| $ T_i $ | Fluid temperature in thermal cell *i* | $ \mathrm{K} $ | | T1[i] |  
| $ T_{w,i} $ | Wall temperature in thermal cell *i* | $ \mathrm{K} $ | | Tp1[i] |  
| $ T_m $ | Melting temperature of the metal tubes | $ \mathrm{K} $ | | - |  
| $ u_i $ | Specific internal energy in thermal cell *i* | $ \mathrm{J/kg} $ | | - |  
| $ V_i $ | Volume of thermal cell *i* | $ \mathrm{m^3} $ | $ A \cdot \Delta x_1 $ | - |  
| $ z_i $ | Inlet altitude of hydraulic cell *i:i+1* | $ \mathrm{m} $ | | - |  
| $ z_{i+1} $ | Outlet altitude of hydraulic cell *i:i+1 | $ \mathrm{m} $ | | - |  
| $ \alpha $ | Corrective term for the heat exchange  coefficient $ h_{c,i} $ for each thermal cell | - | | - |  
| $ \Delta A_i $ | Internal heat exchange area for thermal cell *i* | $ \mathrm{m^2} $ | $ \pi \cdot D \cdot \Delta x_1 $ | dSi |  
| $ \Delta W_i $ | Thermal power exchanged in the water side  for thermal cell *i* | $ \mathrm{W} $ | | dW1[i] |  
| $ \Delta x_1$ | Length of a thermal cell | $ \mathrm{m} $ | $ L/(N-1) $| dx1 |  
| $ \Delta x_2$ | Average length of a hydraulic cell | $ \mathrm{m} $ | $ L/N $ | dx2 |  
| $ (\Delta P)_{i:i+1}^a $ | Advection pressure loss in hydraulic cell *i:i+1*  when fluid flow from *i* to *i+1* | $ \mathrm{Pa} $ | | dpa[i] |  
| $ (\Delta P)_{i:i+1}^f $ | Friction pressure loss in hydraulic cell *i:i+1*  when fluid flow from *i* to *i+1* | $ \mathrm{Pa} $ | | dpf[i] |  
| $ (\Delta P)_{i:i+1}^g $ | Gravity pressure loss in hydraulic cell *i:i+1*  when fluid flow from *i* to *i+1* | $ \mathrm{Pa} $ | | dpg[i] |  
| $ \zeta $ | Corrective term for the friction pressure loss for each cell| - | | dpfCorr |  
| $ \Lambda_{i:i+1} $ | Friction pressure loss coefficient in cell *i:i+1* | - || khi[i] |  
| $ \rho_i $ | Fluid densitiy in thermal cell *i* | $ \mathrm{kg/m^3} $ | | rho1[i] |  
| $ \rho_{i:i+1} $ | Fluid densitiy in hydraulic cell *i:i+1* | $ \mathrm{kg/m^3} $ | | rho2[i] |  
| $ \Phi_{lo}^2$ | Lockhart and Martinelli corrective coefficient | || Xtt[i] |  


## Governing equations  

The tube bundle is modeled as $N_t$ instances of the same tube.  


### Dynamic mass balance equation  

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

- Mathematical formulation:  

$$ A \cdot \left[ \left(\frac{\partial \rho_i}{\partial P_i} \right)_h \cdot \frac{\mathrm{d} P_i}{\mathrm{d} t} + \left(\frac{\partial \rho_i}{\partial h_i} \right)_P \cdot \frac{\mathrm{d} h_i}{\mathrm{d} t} \right] \cdot \Delta x_1 = \dot{m}_{i-1:i} - \dot{m}_{i:i+1} $$  

- Comments:  

This equation uses the expansion of the derivative of the density using $ P $ and $ h $ as state variables:  

$$ \frac{\partial \rho_i}{\partial t} = \left(\frac{\partial \rho_i}{\partial P_i} \right)_h \cdot \frac{\mathrm{d} P_i}{\mathrm{d} t} + \left(\frac{\partial \rho_i}{\partial h_i} \right)_P \cdot \frac{\mathrm{d} h_i}{\mathrm{d} t} $$  



### Static mass balance equation  

- Validity domain:   
   
 $ \forall \dot{m}_{i:i+1} $ and the fluid is incompressible  

- Mathematical formulation:  

$$ 0 = \dot{m}_{i-1:i} - \dot{m}_{i:i+1} $$  

- Comments:  

This equation uses $ \frac{\partial \rho_i}{\partial t} = 0 $ as the fluid is incompressible.   


### Energy balance equation  

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

- Mathematical formulation:  

$$ A \cdot \left[ \left(h_i \cdot \frac{\partial \rho_i}{\partial P_i} - 1 \right) \cdot \frac{\mathrm{d} P_i}{\mathrm{d} t} + \left(h_i \cdot \frac{\partial \rho_i}{\partial h_i} + \rho_i \right)_P \cdot \frac{\mathrm{d} h_i}{\mathrm{d} t} \right] \cdot \Delta x_1$$ 
 $$ = \dot{m}_{i-1:i} \cdot h_{i-1:i} - \dot{m}_{i:i+1} \cdot h_{i:i+1} + \Delta W_i $$  

- Comments:  


### Incompressible flow energy balance equation  

- Validity domain:   
   
 $ \forall \dot{m}_{i:i+1} $ and the fluid is incompressible  

- Mathematical formulation:  

$$ A \cdot  \left(\rho_i \cdot \frac{\mathrm{d} h_i}{\mathrm{d} t} - \frac{\mathrm{d} P_i}{\mathrm{d} t} \right) \cdot \Delta x_1 = \dot{m}_{i-1:i} \cdot h_{i-1:i} - \dot{m}_{i:i+1} \cdot h_{i:i+1} + \Delta W_i $$  

- Comments:  

This equation is derived from the latter with $ \frac{\partial \rho_i}{\partial P_i} = \frac{\partial \rho_i}{\partial h_i} = 0 $ as the fluid is incompressible.  


### Simple incompressible flow energy balance equation  

- Validity domain:   
   
 $ \forall \dot{m}_{i:i+1} $ and the fluid is incompressible  

- Mathematical formulation:  

$$ A \cdot  \rho_i \cdot \frac{\mathrm{d} h_i}{\mathrm{d} t} \cdot \Delta x_1 = \dot{m}_{i-1:i} \cdot h_{i-1:i} - \dot{m}_{i:i+1} \cdot h_{i:i+1} + \Delta W_i $$  

- Comments:  

This equation neglects the pressure derivative in the latter.  


### Dynamic momentum balance equation  

- Validity domain:   
   
 $ \forall \dot{m}_{i:i+1} $ and the fluid is incompressible  

- Mathematical formulation:  

$$ \frac{1}{A} \cdot \frac{\mathrm{d} \dot{m}_{i:i+1}}{\mathrm{d}} \cdot \Delta x_2 = P_i - P_{i+1} - (\Delta P)_{i:i+1}^a - (\Delta P)_{i:i+1}^f - (\Delta P)_{i:i+1}^g  $$ with:  

$$ (\Delta P)_{i:i+1}^a = \frac{\dot{m}_{i:i+1} \cdot  | \dot{m}_{i:i+1} |}{A^2} \cdot \left( \frac{1}{\rho_{i+1}} - \frac{1}{\rho_i} \right) $$ 
 $$(\Delta P)_{i:i+1}^f = \zeta \cdot \frac{\Phi_{lo}^2 \cdot \Lambda_{i:i+1} \cdot \Delta x_2 \cdot \dot{m}_{i:i+1} \cdot | \dot{m}_{i:i+1} |}{2 \cdot D \cdot A^2 \cdot \rho_{i:i+1}} $$ 
 $$ (\Delta P)_{i:i+1}^g = \rho_{i:i+1} \cdot g \cdot (z_{i+1} - z_i) $$  

- Comments:  

By default, the flow is considered turbulent (i.e. *Re* > 2300).  


### Static momentum balance equation  

- Validity domain:   
   
 $ \forall \dot{m}_{i:i+1} $ and the fluid inertia can be neglected.  

- Mathematical formulation:  
 $$ P_i - P_{i+1} = (\Delta P)_{i:i+1}^a + (\Delta P)_{i:i+1}^f + (\Delta P)_{i:i+1}^g $$ 
 $$ (\Delta P)_{i:i+1}^a = \frac{\dot{m}_{i:i+1} \cdot  | \dot{m}_{i:i+1} |}{A^2} \cdot \left( \frac{1}{\rho_{i+1}} - \frac{1}{\rho_i} \right) $$ 
 $$(\Delta P)_{i:i+1}^f = \zeta \cdot \frac{\Phi_{lo}^2 \cdot \Lambda_{i:i+1} \cdot \Delta x_2 \cdot \dot{m}_{i:i+1} \cdot | \dot{m}_{i:i+1} |}{2 \cdot D \cdot A^2 \cdot \rho_{i:i+1}} $$ 
 $$ (\Delta P)_{i:i+1}^g = \rho_{i:i+1} \cdot g \cdot (z_{i+1} - z_i) $$  

- Comments:  

As the fluid inertia is neglected, it is assumed that $ \frac{1}{A} \cdot \frac{\mathrm{d} \dot{m}_{i:i+1}}{\mathrm{d}t} \cdot \Delta x_2 \approx 0 $.  


### Convective heat exchanged between the fluid and the wall  

- Validity domain:   
   
 $ \forall T_{w,i}, T_i $  

- Mathematical formulation:  

 $$ \Delta W_i = \alpha \cdot h_{c,i} \cdot \Delta A_i \cdot (T_{w,i} - T_i) $$  

-  Comments:  

## 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. 9.4.2. Springer Nature Switzerland AG.