Capacity formulation:

Below are the general equations for a non-adiabatic constant volume. It is assumed that all the mixture (non condensable plus main fluid in liquid, gas or two phase conditions) is at only one temperature.

Mass conservation equation

where� �is the volume of the component, is the massflow entering and leaving the volume and ρ is the density of the fluid in the component.

Energy conservation equation

where�� �is the volume of the node i, is the massflow and ρ is the density of the node, u is the internal energy, h the enthalpy, �is the velocity of the fluid, �the slope of the pipe and g gravity.

 

Energy conservation equation

 

where �and u are the fluid mixture (including two phase flow) density, and the total energy respectively; mi,hi and mj,hj are the mass and enthalpy flows at port number i/j calculated at the connected resistive type components.

 

Pressure, temperature and quality calculation

The above conservation equations enable calculating the derivatives of the mixture density and mixture energy. These variables can be integrated, so they are known at any time.

Assuming thermodynamic equilibrium, the conservation equations are always valid even if the fluid conditions are liquid, vapor or homogeneous two phase flow. Then, the complete thermodynamic state (partial pressures, temperature, quality �) can be calculated using the pure fluid thermodynamic routines:

CRYO_FL_state_vs_ru(fluid, eos, rho, u-0.5*vel**2, phase, rho_f, rho_g, P, T, Tsat, h_f, h_g,x, alpha, cp, cp_f, cp_g, drho_dp, drho_dh, vsound, visc, visc_f, visc_g, cond, cond_f, cond_g, sigma, ier)

Inputs are: fluid and eos with the fluid name and its type; rho and u are the mixture density and the mixture energy (dynamic variables).

All the arguments from phase (liquid, vapor or two-phase) to ier (the error code) are outputs: x, alpha are the quality and the void fraction respectively; drho_dp, drho_dh are thermo derivatives;

Both actual and saturated (liquid and vapor) properties are returned. So, cp, cp_f, cp_g are the mixture, saturated liquid and saturated vapor heat capacities respectively; h_f, h_g are the saturated liquid and vapor enthalpies, rho_f, rho_g the saturated liquid and vapor densities, etc.

 

Initialization

The component Capacity and all its child components can be initialized by means of the variable 'init_condition'. If 'init_condition' is Gas then the state variables ρ and υ are calculated calling the function CRYO_PF_prop_vs_pT with the initial temperature (To) and pressure (Po) defined by the user. In the case that 'init_condition' is TwoPhases then the temperature in the tank is the saturation temperature for the pressure defined by the user. The void fraction is calculated as function of the initial level of liquid defined by the user:

And density and quality are calculated as follows:

The internal energy is calculated calling the function CRYO_PF_prop_vs_Px with the pressure and the quality previously calculated.