.ThermoSysPro.WaterSteam.HeatExchangers.DynamicTwoPhaseFlowPipe

Information

## 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](modelica://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.  

![modelica://ThermoSysPro/UsersGuide/Documentation/ThermoSysPro.WaterSteam.HeatExchangers.DynamicTwoPhaseFlowPipe.svg](modelica://ThermoSysPro/UsersGuide/Documentation/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.  
    

Revisions