Heat removal in multicomponent distillation condenser with partial condensation

[danielnl]

Hi everyone, I'm new here and I'm a little worried because I'm not able to explain some facts about this topic.

I'm trying to model and simulate a condenser that is in the top of a distillation column for a control project. The inlet stream is a multicomponent ideal mixture and it can be assumed that is a saturated steam at a given pressure (and temperature, of course). In the condenser, depending on the coolant flow, a part of the vapor will condense and the rest will be aspired by a vacuum system.

Assuming equilibrium in the outlet streams (condensed liquid and aspired vapour), I can obtain the mass flow of each stream and also its mole fractions (by solving the mass balances and using the equilibrium relations).

But when I try to obtain the heat that must be removed, I think that something is wrong:
The Energy balance:

F*H_F - L*h_L - V*H_V = Q
Being: F=inlet vapour, V=outlet vapour and L=outlet liquid and x and y the respective mole fractions.

If I define the reference state as a subcooled liquid at T_ref (As its usual for a sigle component system), the specific enthalpy can be defined as:

h_L = sum(j in mixture; x_L(j)*Cp(j)*(T_eq - T_ref) )

H_V = sum(j in mixture; y_L(j)*Cp(j)*(T_eq - T_ref) + y_L(j)*DH_vap(j) )

H_F = sum(j in mixture; y_F(j)*Cp(j)*(T_eq* - T_ref) + y_F(j)*DH_vap(j) )

being T_eq the outlet temperature (equilibrium) and T_eq* the dew temperature of the inlet mixture.

For a single component system, when all the vapour is cooled, then Q=F*DH_V, something expectable because the heat that must be removed is equal to the heat of condensation. But in muticomponent systems, T_eq isn't equal to the dew temperature and if all the vapour is condensed, the heat that must be removed is equal to:

F*sum(j in mixture; y_F(j)*Cp(j)*(T_eq* - T_eq) + y_F(j)*DH_vap(j) ) = Q

Which is different to the enthalpy of condensation of the mixture, then I'm not sure whether the last expression is correct or I'm doing somerhing wrong... may be I'm stucked in something obvious but I can't see what is

If anybody can help me I'll be very gratefull.
Thanks in advance

PS.: The liquid and the vapour have ideal behavior

 

[danielnl]

Nobody can help me?

I'was reading about this topic and the only thing that I've found is to use directly the enthalpy with the correction for the ideal gas state:

H = H_ideal_gas - H_correction,
being the correction for the enthalpy:

H_correction = $$\int_{0}^{P}(T (\frac{\partial V}{\partial T})_{P} - V)dP$$

and use an EOS to obtain the partial derivatives...

But I'm not sure if this will work in a dynamic simulation problem, so I was thinking into define a pseudo Cp, such as:

Cp_aux(j)= $$\frac{DH(j)}{T_{dew} - T_{bubble}}, j \in mixture$$

What do you think guys?

Thanks in advance!