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.

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.

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:

\[ \begin{align}\begin{aligned} \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) \end{aligned}\end{align} \]

\[(\Delta P)_{i:i+1}^f = \zeta \cdot \frac{\Phi_{lo}^2 \cdot \Lambda_{i:i+1} \cdot \Delta x_2 \cdot \dot{m}_{i:i+1} \cdot | \dot{m}_{i:i+1} |}{2 \cdot D \cdot A^2 \cdot \rho_{i:i+1}} \]

\[ (\Delta P)_{i:i+1}^g = \rho_{i:i+1} \cdot g \cdot (z_{i+1} - z_i) \]

• 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, sect. 9.4.2. Springer Nature Switzerland AG.