Information
## Copyright © EDF 2002 - 2026
## ThermoSysPro Version 4.2
This component model is documented in Sect. 14.4 of the ThermoSysPro book.
# Two phase cavity
The two-phase cavity is a reservoir used to separate water from steam and store the separated phases.
It can be a vertical or horizontal cylinder.
The component is divided in the desuperheating and condensation zones, located in the upper part, and the subcooled zone, located in the lower part.
The pipes inside the cavity are divided in three categories:
- the pipes drowned in the liquid, labeled Pipes 1,
- the pipes immersed in steam, labeled Pipes 2,
- the U-tubes completely immersed in steam, labeled Pipes 3.
The TwoPhaseCavity component represents the dynamics of the thermal hydraulic
phenomena of the fluids inside the cavity.
The following thermal exchanges are taken into account:
- between the fluids and the cooling fluid flowing in the tube bundle,
- between the fluid and the wall,
- between the cavity and the ambient environment,
- between the fluid phases (condensation and vaporization).
Following assumptions are made:
- pressure losses are not taken into account in the cavity,
- the liquid and vapor phases are not necessarily in thermal equilibrium, but always in pressure equilibrium.
## Modelica component model
The equations mentioned below are implemented in the component *TwoPhaseCavity*, located in the *WaterSteam.Volumes* sub-library.
This component has 7 connectors:
- Cv: steam input,
- Ce: water input,
- Cl: water output,
- Cth1: thermal port,
- Cth2: thermal port,
- Cth3: thermal port,
- yLevel: water level output.
## Nomenclature
| Symbol| Description| Unit| Definition| Modelica name |
| :-------------------------------------- | :----------------------------------------------------------------------------------------------------------- | :-------------------------------------------- | :--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | :----------- |
| \\(A\_{l}\\)| Cross-sectional area of the liquid phase in the cavity| \\(\mathrm{m}^{2}\\)| For a vertical cavity: \\(\pi \cdot \mathrm{R}^{2}\\). <br/> For a horizontal cavity: <br/> \\(\left\(\frac{\pi}{2}-\theta\right\) \cdot R^{2}\\)| Al |
| \\(A\_{l \mathrm{w}}\\)| Contact surface between the liquid phase and the cavity wall| \\(\mathrm{m}^{2}\\)| For a vertical cavity: <br/> \\(2 \cdot \pi \cdot R \cdot z\_{l}+A\_{l}\\).<br/> For a horizontal cavity: <br/> \\(\(\pi-2 \cdot \theta\) \cdot R \cdot L+2 \cdot A\_{l}\\).| Alp |
| \\(A\_{\mathrm{vl}}\\)| Heat exchange surface between the vapor phase and the liquid phase| \\(\mathrm{m}^{2}\\)| For a vertical cavity:<br/> \\(A\_{\mathrm{l}}\\).<br/> For a horizontal cavity:<br/> \\(2 . R \cdot L . \cos \(\theta\)\\)| Avl |
| \\(A\_{\mathrm{vw}}\\)| Contact surface between the vapor phase and the cavity wall| \\(\mathrm{m}^{2}\\)| For a vertical cavity: <br/> \\(2 \cdot \pi \cdot R \cdot\left\(L-z\_{l}\right\)+A\_{l}\\).<br/> For a horizontal cavity: <br/> \\(\(\pi+2 \cdot \theta\) \cdot R \cdot L\\) \\(+2\left\(\pi \cdot R^{2}-A\_{l}\right\)\\).| Avp |
| \\(A\_{\mathrm{wa}}\\)| Internal cavity surface| \\(\mathrm{m}^{2}\\)| \\(A\_{\mathrm{vw}}+A\_{\mathrm{lw}}\\)| Ape |
| \\(c\_{\mathrm{p}, 1}\\)| Specific heat capacity of the liquid phase in the cavity| \\(\mathrm{J} / \mathrm{kg} / \mathrm{K}\\)|| prol.cp |
| \\(c\_{\mathrm{p}, \mathrm{v}}\\)| Specific heat capacity of the vapor phase in the cavity| \\(\mathrm{J} / \mathrm{kg} / \mathrm{K}\\)|| prov.cp |
| \\(c\_{\mathrm{p}, \mathrm{w}}\\)| Specific heat capacity of the cavity wall| \\(\mathrm{J} / \mathrm{kg} / \mathrm{K}\\)|| cpp |
| \\(C\_{\text{cond }}\\)| Condensation coefficient with inverse time| \\(\mathrm{s}^{-1}\\)|| Ccond |
| \\(C\_{\text{evap }}\\)| Evaporation coefficient with inverse time| \\(\mathrm{s}^{-1}\\)|| Cevap |
| \\(\mathrm{COP}\_{l}\\)| Corrective term for the heat exchange coefficient for Pipes 1 \(desuperheating zone\)| \\(-\\)|| COPl |
| \\(\mathrm{COP}\_{\mathrm{v}}\\)| Corrective term for the heat exchange coefficient for Pipes 2 and Pipes 3 \(condensation and subcooled zones\) | \\(-\\)|| COPv |
| \\(D\_{\mathrm{e}}\\)| Pipe external diameter, for one pipe| \\(\mathrm{m}\\)|| Dext |
| \\(D\_{\mathrm{h}}\\)| Cross-sectional equivalent diameter| \\(\mathrm{m}\\)| For a square step: <br/> \\(\frac{4 . \mathrm{S}\_{\mathrm{L}}^{2}}{\pi \cdot D\_{\mathrm{e}}}-D\_{\mathrm{e}}\\).<br/> For a triangular step: <br/> \\(\frac{2 \cdot \mathrm{S}\_{\mathrm{L}} \cdot \mathrm{S}\_{\mathrm{T}}}{\pi \cdot D\_{\mathrm{e}} \cdot \frac{\alpha}{120}}-D\_{\mathrm{e}}\\) | DH |
| \\(D\_{\mathrm{s}}\\)| Shell internal diameter| \\(\mathrm{m}\\)|| DIc |
| \\(g\\)| Acceleration due to gravity| \\(\mathrm{m} / \mathrm{s}^{2}\\)|| g |
| \\(h\\)| Specific enthalpy of the fluid in the cavity \(liquid or vapor\)| \\(\mathrm{J} / \mathrm{kg}\\)|| - |
| \\(h\_{\text{conv}j, i}\\)| Convective coefficient of heat transfer by condensation between the vapor and the tube bundle for Pipes *j*| \\(\mathrm{W} / \mathrm{m}^{2} / \mathrm{K}\\)|| hcond2 |
| \\(h\_{\text{drain,i }}\\)| Specific enthalpy at the drain inlet| \\(\mathrm{J} / \mathrm{kg}\\)|| hcond3 | Ce.h | |
| \\(h\_{\mathrm{fg}}\\)| Latent energy at the cavity pressure| \\(\mathrm{J} / \mathrm{kg}\\)|| vsat.h - lsat.h |
| \\(h\_{l}\\)| Specific enthalpy of the liquid phase in the cavity| \\(\mathrm{J} / \mathrm{kg}\\)|| hl |
| \\(h\_{l, \text{ drain, }, \mathrm{i}}\\)| Specific enthalpy of the liquid at the drain inlet| \\(\mathrm{J} / \mathrm{kg}\\)|| Ce.h |
| \\(h\_{l,0}\\)| Specific enthalpy of the liquid at the outlet \(outgoing condensate\)| \\(\mathrm{J} / \mathrm{kg}\\)|| Cl.h |
| \\(h\_{l}^{\text{sat }}\\)| Saturation enthalpy of the liquid in the cavity| \\(\mathrm{J} / \mathrm{kg}\\)|| lsat.h |
| \\(h\_{\mathrm{v}}\\)| Specific enthalpy of the vapor phase in the cavity| \\(\mathrm{J} / \mathrm{kg}\\)|| hv |
| \\(h\_{\mathrm{v}, \text{ drain, } i}\\)| Specific enthalpy of the vapor at the drain inlet| \\(\mathrm{J} / \mathrm{kg}\\)|| Ce.h |
| \\(h\_{\mathrm{v}, \mathrm{i}}\\)| Specific enthalpy of the steam at the inlet, coming from the steam turbine| \\(\mathrm{J} / \mathrm{kg}\\)|| Cv.h |
| \\(h\_{\mathrm{v}}^{\mathrm{sat}}\\)| Saturation enthalpy of the vapor in the cavity| \\(\mathrm{J} / \mathrm{kg}\\)|| vsat.h |
| \\(K\_{\mathrm{corr}}\\)| Corrective term for the heat exchange coefficient between the liquid and the steam| \\(-\\)|| - |
| \\(K\_{l \mathrm{w}}\\)| Convective heat exchange coefficient between the liquid and the wall| \\(\mathrm{W} / \mathrm{m}^{2} / \mathrm{K}\\)|| Klp |
| \\(K\_{\mathrm{vl}}\\)| Convective heat exchange coefficient between the liquid and the vapor in the cavity| \\(\mathrm{W} / \mathrm{m}^{2} / \mathrm{K}\\)|| Kvl |
| \\(K\_{\mathrm{vw}}\\)| Convective heat exchange coefficient between the vapor and the wall| \\(\mathrm{W} / \mathrm{m}^{2} / \mathrm{K}\\)|| Kvp |
| \\(K\_{\mathrm{wa}}\\)| Convective heat exchange coefficient between the wall and the ambient| \\(\mathrm{W} / \mathrm{m}^{2} / \mathrm{K}\\)|| Kpa |
| \\(L\\)| Cavity length| \\(\mathrm{m}\\)|| L |
| \\(L\_{\mathrm{t}}\\)| Total pipe length| \\(\mathrm{m}\\)|| - |
| \\(L\_{j}\\)| Total length of Pipes 1| \\(\mathrm{m}\\)|| L1 |
| \\(L\_{j}\\)| Total length of Pipes 2| \\(\mathrm{m}\\)| \\(L\_{2}=L\_{l}\\)| L2 |
| \\(L\_{j}\\)| Total length of Pipes 3| \\(\mathrm{m}\\)|| L3 |
| \\(L\_{\mathrm{c}}\\)| Distance between two plates in the shell \(support plate spacing in the cooling zone\)| \\(\mathrm{m}\\)|| Lc |
| \\(N\_{\mathrm{t}}\\)| Number of pipes in a vertical row \\(\(\text{ tube bank }\)\\)| \\(-\\)|| NbTubV |
| \\(N\_{l}\\)| Number of Pipes 1| \\(-\\)|| NbTub1 |
| \\(N\_{2}\\)| Number of Pipes 2| \\(-\\)|| NbTub2 |
| \\(N\_{3}\\)| Number of Pipes 3||| NbTub3 |
| \\(N\_{\mathrm{s}}\\)| Number of segments for Pipes 1 and Pipes 2| \\(-\\)|| Ns |
| \\(N\_{\mathrm{s}\_{3}}\\)| Number of segments for Pipes 3| \\(-\\)| \\(2 \cdot N\_{s}\\)| Ns3 |
| \\(M\_{\mathrm{w}}\\)| Mass of the wall cavity| \\(\mathrm{kg}\\)|| Mp |
| \\(\dot{m}\_{l, \mathrm{o}}\\)| Mass flow rate of the outgoing condensate| \\(\mathrm{kg} / \mathrm{s}\\)|| Cl.Q |
| \\(\dot{m}\_{\mathrm{v}}\\)| Mass flow rate of the incoming vapor| \\(\mathrm{kg} / \mathrm{s}\\)|| Cv.Q |
| \\(\dot{m}\_{\text{drain, } \mathrm{i}}\\) | Mass flow rate at the drain inlet| \\(\mathrm{kg} / \mathrm{s}\\)|| Ce.Q |
| \\(m\_{\text{cond }}\\)| Condensation mass flow rate inside the cavity| \\(\mathrm{kg} / \mathrm{s}\\)|| Qcond |
| \\(\dot{m}\_{\text{evap }}\\)| Evaporation mass flow rate inside the cavity| \\(\mathrm{kg} / \mathrm{s}\\)|| Qevap |
| \\(P\\)| Cavity pressure| \\(\mathrm{Pa}\\)|| P |
| \\(P\_{\mathrm{b}}\\)| Fluid pressure at the bottom of the cavity| \\(\mathrm{Pa}\\)| \\(P+\rho\_{l} \cdot g \cdot z\_{l}\\)| Pfond |
| \\(P r\_{l}\\)| Prandtl number of the liquid phase| \\(-\\)| \\(\frac{\mu\_{l} \cdot c\_{\mathrm{pl}}}{\lambda\_{l}}\\)| Prl |
| \\(P r\_{\mathrm{v}}\\)| Prandtl number of the vapor phase| \\(-\\)| \\(\frac{\mu\_{\mathrm{v}} \cdot c\_{\mathrm{pv}}}{\lambda\_{\mathrm{v}}}\\)| - |
| \\(Q\_{\mathrm{s}}\\)| Surface mass flow rate in the shell| \\(\mathrm{kg} / \mathrm{s} / \mathrm{m}^{2}\\) | \\(\frac{\dot{m}\_{l, \mathrm{o}}}{D\_{\mathrm{s}} \cdot L\_{\mathrm{c}} \cdot \left\( \frac{S\_{\mathrm{L}}-D\_{\mathrm{e}}}{S\_{\mathrm{L}}} \right\)}\\)| QS |
| \\(R\\)| Radius of the cavity cross-sectional area| \\(\mathrm{m}\\)|| R |
| \\(R e\_{l}\\)| Reynolds number of the condensate flowing between the drowned tubes| \\(-\\)| \\(\frac{Q\_{s} \cdot D\_{h}}{\mu\_{l}}\\)| Rel |
| \\(R e\_{l, w}\\)| Reynolds number of the condensate flowing against cavity wall| \\(-\\)|| - |
| \\(R e\_{v, 1}\\)| Reynolds number of the vapor flowing against the free surface of the condensate| \\(-\\)|| - |
| \\(R e\_{v, w}\\)| Reynolds number of the vapor flowing against the cavity wall| \\(-\\)|| - |
| \\(S\_{\mathrm{L}}\\)| Longitudinal step| \\(\mathrm{m}\\)|| PaSL |
| \\(S\_{\mathrm{T}}\\)| Transverse step| \\(\mathrm{m}\\)|| PasT |
| \\(T\_{\mathrm{a}}\\)| Ambient temperature| \\(\mathrm{K}\\)|| Ta |
| \\(T\_{l}\\)| Liquid temperature in the cavity| \\(\mathrm{K}\\)|| Tl |
| \\(T\_{\mathrm{w}}\\)| Wall temperature of the cavity| \\(\mathrm{K}\\)|| Tp |
| \\(T\_{\mathrm{w}1, i}\\)| Wall temperature for Pipes 1| \\(\mathrm{K}\\)|| Tp1 |
| \\(T\_{\mathrm{w}2, i}\\)| Wall temperature for Pipes 2| \\(\mathrm{K}\\)|| Tp2 |
| \\(T\_{\mathrm{w}3, i}\\)| Wall temperature for Pipes 3| \\(\mathrm{K}\\)|| Tp3 |
| \\(T\_{\text{sat }}\\)| Saturation temperature in the cavity| \\(\mathrm{K}\\)|| lsat.T, vsat.T |
| \\(T\_{\mathrm{v}}\\)| Vapor temperature in the cavity| \\(\mathrm{K}\\)|| Tv |
| \\(u\\)| Fluid specific internal energy| \\(\mathrm{J} / \mathrm{kg}\\)| \\(h-\frac{P}{\rho}\\)| - |
| \\(V\\)| Volume of the cavity| \\(\mathrm{m}^{3}\\)| \\(V\_{l}+V\_{v}\\)| V |
| \\(V\_{l}\\)| Volume of the liquid phase in the cavity| \\(\mathrm{m}^{3}\\)| \\(V\_{l}=A\_{l} \cdot z\_{l}\\)| Vl |
| \\(V\_{\mathrm{v}}\\)| Volume of the vapor phase in the cavity| \\(\mathrm{m}^{3}\\)|| Vv |
| \\(W\_{1 \mathrm{t}}\\)| Total power exchanged from liquid or vapor to Pipes 1| \\(\mathrm{W}\\)|| W1t |
| \\(W\_{2 \mathrm{t}}\\)| Total power exchanged from liquid or vapor to Pipes 2| \\(\mathrm{W}\\)|| W2t |
| \\(W\_{3 \mathrm{t}}\\)| Total power exchanged from liquid or vapor to Pipes 3| \\(\mathrm{W}\\)|| W3t |
| \\(W\_{4 \mathrm{t}}\\)| Total power exchanged for steam desuperheating| \\(\mathrm{W}\\)|| W4t |
| \\(W\_{\mathrm{vl}}\\)| Power exchanged from the vapor to the liquid| \\(\mathrm{W}\\)|| Wvl |
| \\(W\_{\mathrm{lw}}\\)| Power exchanged from the liquid to the cavity wall| \\(\mathrm{W}\\)|| Wpl |
| \\(W\_{\mathrm{vw}}\\)| Power exchanged from the vapor to the cavity wall| \\(\mathrm{W}\\)|| Wpv |
| \\(W\_{\mathrm{aw}}\\)| Power exchanged from the ambient environment to the cavity wall| \\(\mathrm{W}\\)|| Wpa |
| \\(x\_{\mathrm{v}}\\)| Vapor mass fraction in the vapor phase| \\(-\\)|| xl |
| \\(X\_{\mathrm{vo}}\\)| Vapor mass fraction in the vapor phase from which the liquid starts to condensate| \\(-\\)|| Xvo |
| \\(x\_{l}\\)| Vapor mass fraction in the liquid phase| \\(-\\)|| xv |
| \\(X\_{\mathrm{lo}}\\)| Vapor mass fraction in the liquid phase from which the liquid starts to evaporate| \\(-\\)|| Xlo |
| \\(x\_{\mathrm{mv}}\\)| Vapor mass fraction at the inlet of the drain| \\(-\\)|| proe.x |
| \\(z\_{l}\\)| Liquid level in the cavity| \\(\mathrm{m}\\)| \\(V\_{l} / A\_{l}\\)| zl |
| \\(\alpha\\)| Average bend angle \(pipes triangular step\)| \\(\circ\\)|| Angle |
| \\(\lambda\_{l}\\)| Thermal conductivity of the liquid| \\(\mathrm{W} / \mathrm{m} / \mathrm{K}\\)|| kl |
| \\(\lambda\_{\mathrm{v}}\\)| Thermal conductivity of the vapor| \\(\mathrm{W} / \mathrm{m} / \mathrm{K}\\)|| - |
| \\(\Delta S\_{\text{ext }1}\\)| Heat exchange surface for each segment of Pipes 1| \\(\mathrm{m}^{2}\\)| If \\(j=1,2\\): <br/> \\( \pi \cdot D\_{\mathrm{e}} \cdot L\_{j} \cdot N\_{j} / N\_{\mathrm{s}}\\). <br/> If \\(j=3\\): \\( \pi \cdot D\_{\mathrm{e}} \cdot L\_{j} \cdot N\_{j} / N\_{\mathrm{s}_3}\\). | Surf_ext1 |
| \\(\Delta S\_{\text{ext }2}\\)| Heat exchange surface for each segment of Pipes 2| \\(\mathrm{m}^{2}\\)| If \\(j=1,2\\): <br/> \\( \pi \cdot D\_{\mathrm{e}} \cdot L\_{j} \cdot N\_{j} / N\_{\mathrm{s}}\\). <br/> If \\(j=3\\): \\( \pi \cdot D\_{\mathrm{e}} \cdot L\_{j} \cdot N\_{j} / N\_{\mathrm{s}_3}\\). | Surf_ext2 |
| \\(\Delta S\_{\text{ext }3}\\)| Heat exchange surface for each segment of Pipes 3| \\(\mathrm{m}^{2}\\)| If \\(j=1,2\\): <br/> \\( \pi \cdot D\_{\mathrm{e}} \cdot L\_{j} \cdot N\_{j} / N\_{\mathrm{s}}\\). <br/> If \\(j=3\\): \\( \pi \cdot D\_{\mathrm{e}} \cdot L\_{j} \cdot N\_{j} / N\_{\mathrm{s}_3}\\). | Surf_ext3 |
| \\(\rho\_{l}\\)| Density of the liquid in the cavity| \\(\mathrm{kg} / \mathrm{m}^{3}\\)|| rhol |
| \\(\rho\_{\mathrm{v}}\\)| Density of the vapor in the cavity| \\(\mathrm{kg} / \mathrm{m}^{3}\\)|| rhov |
| \\(\mu\_{l}\\)| Dynamic viscosity of the liquid in the cavity| \\(\mathrm{kg} /\(\mathrm{m} \mathrm{s}\)\\)|| mul |
| \\(\mu\_{\mathrm{IT}}\\)| Dynamic viscosity of the liquid at the wall temperature| \\(\mathrm{kg} /\(\mathrm{m} \mathrm{s}\)\\)|| mult |
| \\(\mu\_{\mathrm{v}}\\)| Dynamic viscosity of the vapor in the cavity| \\(\mathrm{kg} /\(\mathrm{m} \mathrm{s}\)\\)|| - |
| \\(\theta\\)| Chord angle of the liquid in the horizontal cavity,| \\(\mathrm{rad}\\)| \\(\arcsin \left\(\frac{R-z\_{l}}{R}\right\)\\)| theta |
## Governing equations
### Dynamic mass balance equation for the liquid phase
- Validity domain:
\\(\forall \dot{m}\\) and \\(0<V\_{l}<V\\)
- Mathematical formulation:
$$ \rho_{1} \frac{\mathrm{d} V_{1}}{\mathrm{d} t}+V_{1} \cdot\left[\left(\frac{\partial \rho_{1}}{\partial P}\right)_{h} \cdot \frac{\mathrm{d} P}{\mathrm{d} t}+\left(\frac{\partial \rho_{1}}{\partial h}\right)_{P} \cdot \frac{\mathrm{d} h_{1}}{\mathrm{d} t}\right] =-\dot{m}_{1, \mathrm{o}}+\left(1-x_{\mathrm{mv}}\right) \cdot \dot{m}_{\mathrm{drain}, \mathrm{i}} \\ + \dot{m}_{\mathrm{cond}} - \dot{m}_{\mathrm{evap}}$$
- Comments:
### Dynamic mass balance equation for the steam phase
- Validity domain:
\\(\forall \dot{m}\\) and \\(0<V\_{\mathrm{v}}<V\\)
- Mathematical formulation:
$$\rho\_{\mathrm{v}} \cdot \frac{\mathrm{d} V\_{\mathrm{v}}}{\mathrm{d} t}+V\_{\mathrm{v}} \cdot\left[\left\(\frac{\partial \rho\_{\mathrm{v}}}{\partial P}\right\)\_{h} \cdot \frac{\mathrm{d} P}{\mathrm{d} t}+\left\(\frac{\partial \rho\_{\mathrm{v}}}{\partial h}\right\)\_{P} \cdot \frac{\mathrm{d} h\_{\mathrm{v}}}{\mathrm{d} t}\right]=\dot{m}\_{\mathrm{v}}+x\_{\mathrm{mv}} \cdot \dot{m}\_{\mathrm{drain}, \mathrm{i}}+\dot{m}\_{\mathrm{evap}}-\dot{m}\_{\mathrm{cond}}$$
- Comments:
### Dynamic energy balance equation for the liquid phase
- Validity domain:
\\(\forall \dot{m}\\) and \\(0<V\_{l}<V\\)
- Mathematical formulation:
$$ V_{l} \cdot\left[\left(\frac{P}{\rho_{l}} \cdot\left(\frac{\partial \rho_{l}}{\partial P}\right)_{h}-1\right) \cdot \frac{\mathrm{d} P}{\mathrm{d} t} +\left(\frac{P}{\rho_{l}} \cdot\left(\frac{\partial \rho_{l}}{\partial h_{l}}\right)_{P}+\rho_{l}\right) \cdot \frac{\mathrm{d} h_{l}}{\mathrm{d} t}\right] \\ =-\dot{m}_{1, \mathrm{o}} \cdot\left(h_{1, \mathrm{o}}-\left(h_{l}-\frac{P}{\rho_{l}}\right)\right)+\dot{m}_{\mathrm{cond}} \cdot\left(h_{l}^{\mathrm{sat}}-\left(h_{l}-\frac{P}{\rho_{l}}\right)\right) \\ \quad -\dot{m}_{\mathrm{evap}} \cdot\left(h_{\mathrm{v}}^{\mathrm{sat}} -\left(h_{l}-\frac{P}{\rho_{l}}\right)\right) \\ \quad +\left(1-x_{\mathrm{mv}}\right) \cdot \dot{m}_{\mathrm{drain}, \mathrm{i}} \cdot\left(h_{\mathrm{l}, \mathrm{drain}, \mathrm{i}}-\left(h_{l}-\frac{P}{\rho_{l}}\right)\right)+W_{\mathrm{vl}}-W_{\mathrm{lw}}-W_{1 \mathrm{t}}$$
- Comments:
The value of \\(h\_{l, \text { drain }, i}\\) is given by:
$$ h_{1, \text{ drain }, \mathrm{i}}=\left\{\begin{array}{ll} h_{\text{drain, } i} \text{ for } x_{\mathrm{mv}}=0 \\ h_{1}^{\mathrm{sat}} \text{ for } x_{\mathrm{mv}}>0 \end{array}\right.$$
### Dynamic energy balance equation for the vapor phase
- Validity domain:
\\(\forall \dot{m}\\) and \\(0<V\_{\mathrm{v}}<V\\)
- Mathematical formulation:
$$ V_{\mathrm{v}} \cdot\left[\left(\frac{P}{\rho_{\mathrm{v}}} \cdot\left(\frac{\partial \rho_{\mathrm{v}}}{\partial P}\right)_{\mathrm{h}}-1\right) \cdot \frac{\mathrm{d} P}{\mathrm{d} t}+\left(\frac{P}{\rho_{\mathrm{v}}} \cdot\left(\frac{\partial \rho_{\mathrm{v}}}{\partial h_{\mathrm{v}}}\right)_{P} +\rho_{\mathrm{v}}\right) \cdot \frac{\mathrm{d} h_{\mathrm{v}}}{\mathrm{d} t}\right]\\ =\dot{m}_{\mathrm{v}} \cdot\left(h_{\mathrm{v}, \mathrm{i}}-\left(h_{\mathrm{v}}-\frac{P}{\rho_{\mathrm{v}}}\right)\right)-\dot{m}_{\mathrm{cond}} \cdot\left(h_{1}^{\mathrm{sat}}-\left(h_{\mathrm{v}}-\frac{P}{\rho_{\mathrm{v}}}\right)\right)\\ \quad+\dot{m}_{\mathrm{evap}} \cdot\left(h_{\mathrm{v}}^{\mathrm{sat}}-\left(h_{\mathrm{v}}-\frac{P}{\rho_{\mathrm{v}}}\right)\right)\\ \quad+x_{\mathrm{mv}} \cdot \dot{m}_{\mathrm{drain}, \mathrm{i}} \cdot\left(h_{\mathrm{v}, \mathrm{drain}, \mathrm{i}}-\left(h_{\mathrm{v}}-\frac{P}{\rho_{\mathrm{v}}}\right)\right) \\ \quad-W_{\mathrm{vl}}-W_{\mathrm{vw}}-W_{2 \mathrm{t}}-W_{3 \mathrm{t}}-W_{4 \mathrm{t}}$$
- Comments:
The value of \\(h\_{\mathrm{v}, \text { drain, } i}\\) is given by:
$$ h_{\mathrm{v}, \mathrm{drain}, \mathrm{i}}=\left\{\begin{array}{ll}h_{\mathrm{drain}, \mathrm{i}} \text{ for } x_{\mathrm{mv}}=1 \\ h_{\mathrm{v}}^{\mathrm{sat}} \text{ for } x_{\mathrm{mv}}<1\end{array}\right.$$
### Energy accumulation in the wall
- Validity domain:
\\(T\_{\mathrm{w}}<\\) melting temperature of the tubes metal
- Mathematical formulation:
$$M\_{\mathrm{w}} \cdot c\_{\mathrm{p}, \mathrm{w}} \cdot \frac{\mathrm{d} T\_{\mathrm{w}}}{\mathrm{d} t}=W\_{\mathrm{lw}}+W\_{\mathrm{vw}}+W\_{\mathrm{aw}}$$
### Power exchanged from the liquid to Pipes 1 \(subcooled\)
- Validity domain:
\\(\forall T\_{l}\\) and \\(\forall T\_{\mathrm{w} 1, \mathrm{i}}\\)
- Mathematical formulation:
$$W\_{l \mathrm{t}}=\Delta S\_{\mathrm{ext} 1} \cdot \sum\_{i=1}^{N\_{\mathrm{s}}} h\_{\mathrm{conv} 1, i} \cdot\left\(T\_{l}-T\_{\mathrm{w} 1, i}\right\)$$
- Comments:
The power is exchanged by convection from the liquid to the pipes
### Power exchanged from the vapor to Pipes 2
- Validity domain:
\\(\forall T\_{\mathrm{v}}\\) and \\(\forall T\_{\mathrm{w} 2 \mathrm{i}}\\)
- Mathematical formulation:
$$W\_{2 \mathrm{t}}=\Delta S\_{\mathrm{ext} 2} \cdot \sum\_{i=1}^{N\_{\mathrm{s}}} h\_{\mathrm{cond} 2, i} \cdot\left\(T\_{\mathrm{v}}-T\_{\mathrm{w} 2, i}\right\)$$
- Comments:
The power is exchanged by convection from the vapor to the pipes
### Power exchanged from the vapor to Pipes 3
- Validity domain:
\\(\forall T\_{\mathrm{V}}\\) and \\(\forall T\_{\mathrm{W} 3 \mathrm{i}}\\)
- Mathematical formulation:
$$W\_{3 \mathrm{t}}=\Delta S\_{\mathrm{ext} 3} \cdot \sum\_{i=1}^{N\_{\mathrm{s}}} h\_{\mathrm{cond} 3, i} \cdot\left\(T\_{\mathrm{v}}-T\_{\mathrm{w} 3, i}\right\)$$
- Comments:
The power is exchanged by convection from the vapor to the pipes
### Power exchanged for desuperheating of the vapor
- Validity domain:
\\(\forall \dot{m}\_{\mathrm{v}}\\)
- Mathematical formulation:
$$ W_{4 \mathrm{t}}=\left\{\begin{array}{ll}\dot{m}_{\mathrm{v}} \cdot\left(h_{\mathrm{v}, \mathrm{i}}-h_{\mathrm{v}}^{\text{sat }}\right) \text{for } h_{\mathrm{v}, \mathrm{i}}>h_{\mathrm{v}}^{\text{sat }} \\ 0 \text{for } h_{\mathrm{v}, \mathrm{i}}<h_{\mathrm{v}}^{\text{sat }} \end{array}\right.$$
- Comments:
The power is exchanged from the vapor to the pipes
### Power exchanged from the vapor to the liquid
- Validity domain:
\\(\forall T\_{\mathrm{v}}\\) and \\(\forall T\_{l}\\)
- Mathematical formulation:
$$W\_{\mathrm{vl}}=K\_{\mathrm{vl}} \cdot A\_{\mathrm{p}} \cdot\left\(T\_{\mathrm{v}}-T\_{l}\right\)$$
- Comments:
The power is exchanged by convection from the vapor to the liquid at the interface between the two phases.
### Power exchanged from the liquid to the cavity wall
- Validity domain:
\\(\forall T\_{l}\\) and \\(\forall T\_{\mathrm{w}}\\)
- Mathematical formulation:
$$W\_{\mathrm{lw}}=K\_{\mathrm{lw}} \cdot A\_{l} \cdot\left\(T\_{l}-T\_{\mathrm{w}}\right\)$$
- Comments:
The power is exchanged by convection from the liquid to the cavity wall.
### Power exchanged from the vapor to the cavity wall
- Validity domain:
\\(\forall T\_{\mathrm{v}}\\) and \\(\forall T\_{\mathrm{w}}\\)
- Mathematical formulation:
$$W\_{\mathrm{vw}}=K\_{\mathrm{vw}} \cdot A\_{\mathrm{v}} \cdot\left\(T\_{\mathrm{v}}-T\_{\mathrm{w}}\right\)$$
- Comments:
The power is exchanged by convection from the vapor to the cavity wall.
### Power exchanged from the ambient to the cavity wall
- Validity domain:
\\(\forall T\_{\mathrm{a}}\\) and \\(\forall T\_{\mathrm{w}}\\)
- Mathematical formulation:
$$W\_{\mathrm{aw}}=K\_{\mathrm{aw}} \cdot A\_{\mathrm{e}} \cdot\left\(T\_{\mathrm{a}}-T\_{\mathrm{w}}\right\)$$
- Comments:
The power is exchanged by convection from the ambient to the cavity wall.
### Condensation mass flow rate
- Validity domain:
\\(\forall x\_{\mathrm{v}}\\) close to \\(X\_{\mathrm{vo}}\\)
- Mathematical formulation:
$$\dot{m}\_{\text {cond }}=\max \left\(C\_{\text {cond }} \cdot \rho\_{\mathrm{v}} \cdot V\_{\mathrm{v}} \cdot\left\(X\_{\mathrm{vo}}-x\_{\mathrm{v}}\right\), 0\right\)$$
- Comments:
### Evaporation mass flow rate
- Validity domain:
\\(\forall x\_{l}\\) close to \\(X\_{\mathrm{lo}}\\)
- Mathematical formulation:
$$\dot{m}\_{\mathrm{evap}}=\max \left\(C\_{\text {evap }} \cdot \rho\_{l} \cdot V\_{l} \cdot\left\(x\_{l}-X\_{\mathrm{lo}}\right\), 0\right\)$$
- Comments:
### Convective heat transfer coefficient in zone 1 corresponding to the drowned tubes
- Validity domain:
\\(100<\mathrm{Re}\_{l}<10^{6}\\)
- Mathematical formulation:
$$h\_{\text {conv1 }}=\frac{\lambda\_{l}}{D\_{\mathrm{e}}} \cdot 0.36 \cdot \mathrm{COP}\_{l} \cdot \operatorname{Re}\_{l}^{0.55} \cdot \mathrm{Pr}\_{l}^{0.33} \cdot\left\(\frac{\mu\_{l}}{\mu\_{\text {IT }}}\right\)^{0.14}$$
- Comments:
### Convective heat transfer coefficient in zones 2 and 3 corresponding to the condensation zone
- Mathematical formulation:
$$ h_{\text{cond} 2}=\left\{ \begin{array}{ll}1,13 . \mathrm{COPv} \cdot\left[\frac{g \cdot \rho_{l}\left(\rho_{l}-\rho_{v}\right) \lambda_{l}^{3} \cdot h_{f g}}{L_{2} \cdot \mu_{l}\left(T_{s a t}-T_{w 2}\right)}\right]^{0,25} \text{for vertical cavity} \\ 0,728 \cdot \mathrm{COPv} \cdot\left[\frac{g \cdot \rho_{l}\left(\rho_{l}-\rho_{v}\right) \lambda_{l}^{3} \cdot h_{f g}}{N t_{n} \cdot \mu_{l}\left(T_{s a t}-T_{w 2}\right) D_{e}}\right]^{0,25} \text{for horizontal cavity} \\ \end{array}\right.$$
$$ h_{\text{cond} 3}=\left\{ \begin{array}{ll}1,13 . \mathrm{COPv} \cdot\left[\frac{g \cdot \rho_{l}\left(\rho_{l}-\rho_{v}\right) \lambda_{l}^{3} \cdot h_{f g}}{L_{3} \cdot \mu_{l}\left(T_{s a t}-T_{w 3}\right)}\right]^{0,25} \text{for vertical cavity } \\ 0,728 \cdot \mathrm{COPv} \cdot\left[\frac{g \cdot \rho_{l}\left(\rho_{l}-\rho_{v}\right) \lambda_{l}^{3} \cdot h_{f g}}{N t_{n} \cdot \mu_{l}\left(T_{s a t}-T_{w 3}\right) D_{e}}\right]^{0,25} \text{for horizontal cavity } \end{array}\right.$$
## 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. 14.4. Springer Nature Switzerland AG.
