Abstract
The present chapter introduces the governing equations, whose structure is important to ensure the robustness of the library, and which are of two fundamental kinds: the dynamic mass, energy, and momentum conservation equations that compute the average physical state inside the control volumes, and the equations that compute the average physical state on the boundaries of the control volumes. Both kinds of equations, complemented with the closure equations for the computation of pressure losses or heat exchange coefficients, are necessary to compute the full thermal hydraulic state of the system. The main simplifications used for the efficient simulation of complex systems are presented and fully justified.
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Notes
- 1.
In principle, the partial derivative with respect to time in (4.2) can be moved out of the integral only if the integration volume Va is constant (i.e., does not vary in time). However, if the integration volume is not constant, one can consider a constant volume V′a sufficiently large so as to contain Va at all times and consider the fluid density ρ′ that is equal to the fluid density ρ inside Va and equal to zero outside of Va and inside V′a. Then
\(\int\limits_{{V_{a} }} {\rho \cdot g \cdot {\text{d}}V} = \int\limits_{{V_{a}^{'} }} {\rho^{\prime} \cdot g \cdot {\text{d}}V}\)
and
\(\int\limits_{{V_{a} }} {\frac{\partial }{\partial t}(\rho \cdot g) \cdot {\text{d}}V} = \int\limits_{{V_{a}^{'} }} {\frac{\partial }{\partial t}(\rho^{\prime} \cdot g) \cdot {\text{d}}V}\)
As V′a is constant, the time derivative can be moved out of the integral
\(\int\limits_{{V_{a} }} {\frac{\partial }{\partial t}(\rho \cdot g) \cdot {\text{d}}V} = \int\limits_{{V_{a}^{'} }} {\frac{\partial }{\partial t}(\rho^{\prime} \cdot g) \cdot {\text{d}}V} = \frac{\text{d}}{{{\text{d}}t}}\int\limits_{{V_{a}^{'} }} {\rho^{\prime} \cdot g \cdot {\text{d}}V} = \frac{\text{d}}{{{\text{d}}t}}\int\limits_{{V_{a} }} {\rho \cdot g \cdot {\text{d}}V}\)
Control volumes varying in time are in particular to be found in tank models or two-phase cavity models (cf. Sects. 14.5.2 and 14.4.3).
Abbreviations
- \(\left[ G \right]\) :
-
Unit of quantity \(G\)
- \(A_{b{:}a}\) :
-
Flow cross-sectional area of boundary \(b{:}a\) (m2)
- \(A_{{{\text{g}},x}}\) :
-
Cross-sectional area of the gas phase at coordinate \(x\) along the tube (m2)
- \(A_{k,x}\) :
-
Cross-sectional area of phase \(k\) at coordinate \(x\) along the tube (m2)
- \(A_{{{\text{l}},x}}\) :
-
Cross-sectional area of the liquid phase at coordinate \(x\) along the tube (m2)
- \(A_{x}\) :
-
Flow cross-sectional area at coordinate \(x\) (m2)
- \(b{:}a\) :
-
Boundary between volumes \(b\) and \(a\) (–)
- \(c_{p}\) :
-
Specific heat at constant pressure (J kg−1 K−1)
- \(c_{v}\) :
-
Specific heat at constant volume (J kg−1 K−1)
- \(C_{0}\) :
-
Profile parameter for the drift-flux model (–)
- \(C_{0,x}\) :
-
Profile parameter (or flux concentration parameter, or distribution coefficient) (–)
- \(C_{\text{cond}}\) :
-
Time constant for the condensation rate (s−1)
- \(C_{\text{evap}}\) :
-
Time constant for the evaporation rate (s−1)
- \({\text{d}}P_{{{\text{f}},x}}\) :
-
Pressure loss along \({\text{d}}x\) at coordinate \(x\) along the tube (Pa)
- \(D_{{{\text{H}},x}}\) :
-
Tube hydraulic diameter at coordinate \(x\) along the tube (m)
- \(D_{x}\) :
-
Tube diameter at coordinate \(x\) along the tube (m)
- \(g\) :
-
Specific extensive quantity (\(\left[ G \right]\).kg−1)
- \(g\) :
-
Gravity constant (m s−2)
- \(G_{x}\) :
-
Total value of specific extensive quantity \(g\) in the tube from the origin up to coordinate \(x\) \((\left[ G \right])\)
- \(h_{b{:}a}\) :
-
Average specific enthalpy over \(A_{b{:}a}\) (J kg−1)
- \(h_{{{\text{bo}},x}}\) :
-
Heat transfer coefficient due to boiling at coordinate \(x\) along the tube (W m−1 K−1)
- \(h_{{{\text{g}},a}}^{0}\) :
-
Specific enthalpy of the liquid phase characteristic of the energy transfer due to mass transfer to the gas phase in volume \(a\) (J kg−1)
- \(h_{{{\text{l}},a}}^{0}\) :
-
Specific enthalpy of the liquid phase characteristic of the energy transfer due to mass transfer to the liquid phase in volume \(a\) (J kg−1)
- \(h_{\text{in}}\) :
-
Fluid specific enthalpy at the inlet (J kg−1)
- \(h_{\text{out}}\) :
-
Fluid specific enthalpy at the outlet (J kg−1)
- \(h_{n}\) :
-
Pump head (m)
- \(\bar{h}_{n}\) :
-
Dimensionless pump head (–)
- \(h_{x}\) :
-
Average specific enthalpy over \(A_{x}\) (J kg−1)
- \(h_{{{\text{g}},x}}\) :
-
Average specific enthalpy of the gas phase over \(A_{{{\text{g}},x}}\) (J kg−1)
- \(h_{{{\text{l}},x}}\) :
-
Average specific enthalpy of the liquid phase over \(A_{{{\text{l}},x}}\) (J kg−1)
- \(h_{{{\text{g}},x}}^{0}\) :
-
Specific enthalpy of the gas phase characteristic of the energy transfer due to mass transfer to the liquid phase at coordinate \(x\) along the tube (J kg−1)
- \(h_{{{\text{l}},x}}^{0}\) :
-
Specific enthalpy of the liquid phase characteristic of the energy transfer due to mass transfer to the gas phase at coordinate \(x\) along the tube (J kg−1)
- \(h_{{{\text{w}},x}}\) :
-
Heat transfer coefficient at coordinate \(x\) along the tube (W m−1 K−1)
- \(j_{x}\) :
-
Average volumetric flux (or superficial velocity) over \(A_{x}\) (m s−1)
- \(j_{{{\text{g}},x}}\) :
-
Average superficial velocity of the gas phase over \(A_{{{\text{g}},x}}\) (m s−1)
- \(j_{{{\text{l}},x}}\) :
-
Average superficial velocity of the liquid phase over \(A_{{{\text{l}},x}}\) (m s−1)
- \(J\) :
-
Rotational inertia (kg m2)
- \(J(b \to a)\) :
-
Total thermal diffusion through boundary \(b{:}a\), oriented positively from \(b\) to \(a\) (W)
- \(k\) :
-
Polytropic exponent (–)
- \(k\) :
-
Thermal conductivity (W K−1 m−2)
- \(k_{b{:}a}\) :
-
Average thermal conductivity over \(A_{b{:}a}\) (W K−1 m−2)
- \(k_{x}\) :
-
Average thermal conductivity over \(A_{x}\) (W K−1 m−2)
- \(L\) :
-
Tube length (m)
- \(L_{a}\) :
-
Length of volume \(a\) (m)
- \(L_{x}\) :
-
Average latent heat over \(A_{x}\) (J kg−1)
- \(\dot{m}_{a} ({\text{g}} \to {\text{l}})\) :
-
Condensation mass flow rate from the gas phase to the liquid phase in volume \(a\) (kg s−1)
- \(\dot{m}_{a} ({\text{l}} \to {\text{g}})\) :
-
Evaporation mass flow rate from the liquid phase to the gas phase in volume \(a\) (kg s−1)
- \(\dot{m}_{\text{cond}}\) :
-
Condensation mass flow rate (kg s−1)
- \(\dot{m}_{\text{evap}}\) :
-
Evaporation mass flow rate (kg s−1)
- \(\dot{m}_{\text{g}} (b \to a)\) :
-
Mass flow rate of the gas phase through boundary \(b{:}a\), oriented positively from \(b\) to \(a\) (kg s−1)
- \(\dot{m}_{\text{l}} (b \to a)\) :
-
Mass flow rate of the liquid phase through boundary \(b{:}a\), oriented positively from \(b\) to \(a\) (kg s−1)
- \(\dot{m}_{\text{g}} (x)\) :
-
Mass flow rate of the gas phase through \(A_{x}\) (kg s−1)
- \(\dot{m}_{\text{l}} (x)\) :
-
Mass flow rate of the liquid phase through \(A_{x}\) (kg s−1)
- \(M\) :
-
Molar mass of the mixture (kg mol−1)
- \(M_{\text{g}}\) :
-
Molar mass of the gas phase (kg mol−1)
- \(M_{\text{l}}\) :
-
Molar mass of the liquid phase (kg mol−1)
- \(\vec{n}(b \to a)\) :
-
Orientation of boundary \(b{:}a\), positively from \(b\) to \(a\)
- \(\vec{p}\) :
-
Fluid specific momentum (m s−1)
- \(p_{x}\) :
-
Average fluid specific momentum over \(A_{x}\) (m s−1)
- \(P\) :
-
Fluid pressure (Pa)
- \(P_{a}\) :
-
Average pressure in volume \(a\) (Pa)
- \(P_{x}\) :
-
Average pressure over \(A_{x}\) (Pa)
- \(q\) :
-
Volumetric flow rate (kg m−3)
- \(\bar{q}\) :
-
Dimensionless volumetric flow rate (–)
- \(r(x)\) :
-
Hyperbolic function (–)
- \({\Re }\) :
-
Universal gas constant (J mol−1 K−1)
- \(R_{\text{g}}\) :
-
Specific gas constant (J kg−1 K−1)
- \(S\) :
-
Entropy (J K−1)
- \(\text{sgn} (x)\) :
-
Sign function (–)
- \(T_{{{\text{g}},x}}\) :
-
Temperature of the gas phase at coordinate \(x\) along the tube (K)
- \(T_{{{\text{l}},x}}\) :
-
Temperature of the liquid phase at coordinate \(x\) along the tube (K)
- \(T_{\text{f}}\) :
-
Friction torque (N m)
- \(T_{\text{h}}\) :
-
Hydraulic torque (N m)
- \(\bar{T}_{\text{h}}\) :
-
Dimensionless hydraulic torque (–)
- \(T_{\text{m}}\) :
-
Motor torque (N m)
- \(u\) :
-
Fluid specific internal energy (J kg−1)
- \(u_{a}\) :
-
Average specific internal energy inside volume \(a\) (J kg−1)
- \(u_{x}\) :
-
Average specific internal energy over \(A_{x}\) (J kg−1)
- \(\vec{v}\) :
-
Fluid velocity (m s−1)
- \(v_{x}\) :
-
Average velocity over \(A_{x}\) (m s−1)
- \(v_{{{\text{g}},x}}\) :
-
Average velocity of the gas phase over \(A_{{{\text{g}},x}}\) (m s−1)
- \(v{}_{{{\text{l}},x}}\) :
-
Average velocity of the liquid phase over \(A_{{{\text{l}},x}}\) (m s−1)
- \(V_{{{\text{g}}j,x}}\) :
-
Drift velocity of the gas phase over \(A_{x}\) (m s−1)
- \(v_{{{\text{gl}},x}}\) :
-
Average slip velocity between the gas phase and the liquid phase over \(A_{x}\) (m s−1)
- \(v_{x} ({\text{g}} \to {\text{l}})\) :
-
Velocity of the gas phase entering the liquid phase at coordinate \(x\) along the tube (m s−1)
- \(v_{x} ({\text{l}} \to {\text{g}})\) :
-
Velocity of the liquid phase entering the gas phase at coordinate \(x\) along the tube (m s−1)
- \(V(a)\) :
-
Set of volumes \(b\) neighboring volume \(a\)
- \(V_{a}\) :
-
Volume of control volume \(a\) (m3)
- \(V_{{{\text{g}},a}}\) :
-
Volume of the gas phase in volume \(a\) (m3)
- \(V_{{{\text{l}},a}}\) :
-
Volume of the liquid phase in volume \(a\) (m3)
- \(\dot{W}_{a}\) :
-
Total heating power received by volume \(a\) (J s−1)
- \(\dot{W}_{a} ({\text{g}} \to {\text{l}})\) :
-
Heat flow from the gas phase to the liquid phase in volume \(a\) (W)
- \(\dot{W}_{a} ({\text{l}} \to {\text{g}})\) :
-
Heat flow from the liquid phase to the gas phase in volume \(a\) (W)
- \(\dot{W}_{\text{F}} (b \to a)\) :
-
Work produced by the forces acting on the boundary between \(b\) and \(a\) (W)
- \(\dot{W}_{{{\text{g}},a}}\) :
-
Heat flow received from the wall into the gas phase in volume \(a\) (W)
- \(\dot{W}_{{{\text{l}},a}}\) :
-
Heat flow received from the wall into the liquid phase in volume \(a\) (W)
- \(x_{{{\text{g}},a}}\) :
-
Vapor mass fraction in the gas phase of volume \(a\) (–)
- \(x_{{{\text{l}},a}}\) :
-
Vapor mass fraction in the liquid phase of volume \(a\) (–)
- \(x_{{{\text{g}},0}}\) :
-
Set point for the vapor mass fraction in the gas phase (–)
- \(x_{{{\text{l}},0}}\) :
-
Set point for the vapor mass fraction in the liquid phase (–)
- \(x_{x}\) :
-
Quality at coordinate \(x\) along the tube (–)
- \(x_{{{\text{v}},x}}\) :
-
Vapor mass fraction at coordinate \(x\) along the tube (–)
- \(z_{a}\) :
-
Altitude of volume \(a\) (m)
- \(z_{x}\) :
-
Tube altitude at coordinate \(x\) (m)
- \(\alpha_{a}\) :
-
Average void fraction over volume \(a\) (–)
- \(\alpha_{x}\) :
-
Average void fraction over cross-sectional area \(A_{x}\) (–)
- \(\gamma\) :
-
Specific heat ratio (–)
- \(\gamma_{x} ({\text{g}} \to {\text{l}})\) :
-
Mass transfer rate from the gas phase into the liquid phase at coordinate \(x\) along the tube (kg s−1 m−3)
- \(\gamma_{x} ({\text{l}} \to {\text{g}})\) :
-
Mass transfer rate from the liquid phase into the gas phase at coordinate \(x\) along the tube (kg s−1 m−3)
- \(\Delta P_{\text{f}} (a \to b)\) :
-
Pressure loss due to friction between volumes \(a\) and \(b\) oriented positively from \(a\) to \(b\) (Pa)
- \(\varepsilon_{x}\) :
-
Tube roughness at coordinate \(x\) along the tube (m)
- \(\eta_{\text{h}}\) :
-
Hydraulic efficiency (–)
- \(\theta_{x}\) :
-
Tube angle with the horizontal line at coordinate \(x\) (rad)
- \(\lambda_{a{:}b}\) :
-
Friction coefficient at boundary \(a{:}b\) (–)
- \(\lambda_{x}\) :
-
Friction coefficient at coordinate \(x\) along the tube (–)
- \(\mu_{x}\) :
-
Fluid viscosity at coordinate \(x\) along the tube (kg m−1 s−1)
- \(\mu_{{{\text{g}},x}}\) :
-
Viscosity of the gas phase at coordinate \(x\) along the tube (kg m−1 s−1)
- \(\mu_{{{\text{l}},x}}\) :
-
Viscosity of the liquid phase at coordinate \(x\) along the tube (kg m−1 s−1)
- \(\xi_{a{:}b}\) :
-
Pressure loss coefficient at boundary \(a{:}b\) (–)
- \(\xi_{x}\) :
-
Pressure loss coefficient at coordinate \(x\) along the tube (–)
- \(\pi_{{{\text{l}}:{\text{g}},x}}\) :
-
Interfacial liquid–gas perimeter at coordinate \(x\) along the tube (m)
- \(\pi_{{{\text{w}},x}}\) :
-
Wetted perimeter of the wall at coordinate \(x\) along the tube (m)
- \(\pi_{{{\text{w}}:{\text{g}},x}}\) :
-
Wetted perimeter of the wall for the gas at coordinate \(x\) along the tube (m)
- \(\pi_{{{\text{w}}:{\text{l}},x}}\) :
-
Wetted perimeter of the wall for the liquid at coordinate \(x\) along the tube (m)
- \(\rho\) :
-
Fluid density (kg m−3)
- \(\rho_{b{:}a}\) :
-
Average fluid density over \(A_{b{:}a}\) (kg m−3)
- \(\rho_{{{\text{g}},a}}\) :
-
Average density of the gas phase in volume \(a\) (kg m−3)
- \(\rho_{{{\text{l}},a}}\) :
-
Average density of the liquid phase in volume \(a\) (kg m−3)
- \(\rho_{{{\text{g}},x}}\) :
-
Average density of the gas phase over \(A_{x}\) (kg m−3)
- \(\rho_{{{\text{l}},x}}\) :
-
Average density of the liquid phase over \(A_{x}\) (kg m−3)
- \(\tau_{{{\text{l}}:{\text{g}},x}}\) :
-
Interfacial friction from the liquid phase acting on the gas phase at coordinate \(x\) along the tube (N m−2)
- \(\tau_{{{\text{w}}:{\text{g}},x}}\) :
-
Friction from the wall acting on the gas phase at coordinate \(x\) along the tube (N m−2)
- \(\tau_{{{\text{w}}:{\text{l}},x}}\) :
-
Friction from the wall acting on the liquid phase at coordinate \(x\) along the tube (N m−2)
- \(\tau_{{{\text{w}},x}}\) :
-
Friction stress from the wall acting on the fluid at coordinate \(x\) along the tube (N m−2)
- \(\varphi_{{{\text{w}},x}}\) :
-
Heat flux from the wall at coordinate \(x\) along the tube (W m−2)
- \(\varphi_{{{\text{w}}:{\text{g}},x}}\) :
-
Heat flux received from the wall into the gas phase at coordinate \(x\) along the tube (W m−2)
- \(\varphi_{{{\text{w}}:{\text{l}},x}}\) :
-
Heat flux received from the wall into the liquid phase at coordinate \(x\) along the tube (W m−2)
- \(\varphi_{x} ({\text{g}} \to {\text{l}})\) :
-
Heat flux from the gas phase into the liquid phase at coordinate \(x\) along the tube (W m−2)
- \(\varphi_{x} ({\text{l}} \to {\text{g}})\) :
-
Heat flux from the liquid phase into the gas phase at coordinate \(x\) along the tube (W m−2)
- \(\Phi _{g} (b \to a)\) :
-
Flow of specific extensive quantity \(g\) through boundary \(b{:}a\), oriented positively from \(b\) to \(a\) (\(\left[ g \right]\).s−1)
- \(\Phi _{g} (x)\) :
-
Flow of specific extensive quantity \(g\) at coordinate \(x\) along the tube, oriented positively with increasing \(x\) (\(\left[ g \right]\).s−1)
- \({\nabla }T(b \to a)\) :
-
Average value of the temperature gradient over \(A_{b{:}a}\) oriented positively from \(b\) to \(a\) (K m−1)
References
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El Hefni, B., Bouskela, D. (2019). Governing Equations. In: Modeling and Simulation of Thermal Power Plants with ThermoSysPro . Springer, Cham. https://doi.org/10.1007/978-3-030-05105-1_4
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