Modelling 1D-Evaporation

Deranda · January 26, 2022, 08:10

Dear all,

I need some help, cause I am looking at my set of equations and thats jsut driving me crazy, that I cannot solve such a "simple" problem numerically.

So what is my problem:
Let us consider we have a simple pipe that is heated (or cooled) and the source term can be assumed to be $\dot{Q}$ =k*A* $\Delta$ T (k is the overall heat transfer coefficient, A is surface area and $\Delta$ T is the temperature difference driving the heat transfer.
Within the pipe there is a refrigerent of which we have a huge database, so thermophysical properties such as spec. enthalpy, density, temperature and vapour quality are well known depending on the pressure and spec. internal energy.
We want to calculate the evaporation process along the pipe numerically.

For the solution, I am writing a 1D matlab code. Along the length, the pipe is devided into several segments. Each segment has a volume, internal energy, temperature, vapour density, liquid density, vapour quality and volume fraction.

Initially, the pressure is set to ambient pressure and the pipe is filled with fluid slightly below the saturation line (T=T_sat - 0.5K).
On the one end of the pipe, a mass flow boundary condition is set and fluid enters the pipe at T=T_sat-0.5K.

I assume that the heat transfer takes place during an isobaric stage. So the pressure will be constant or in other words: the pressure will not affect the thermophysical properties. Thus a change in velocity can only be introduced by thermal effects (thermal expansion/contraction or evaporation/condensation).
In my model, I will use the volume fraction to differ between liquid and vapour phase. The volume fraction can be directly calculated if densities (liquid and vapour phase) and the vapour quality q is known.

$\alpha = \stackrel{q}{\overline{q+\stackrel{\underline{\rho_{G}}}{\rho_{L}}+q*\stackrel{\underline{\rho_{G}}}{\rho_{L}}}}$

on the first face, the mass flow rate is given by the boundary condition. The mass flow rate on the second face is determined by the following equation:
$\dot{m}^{f_{2}}=\dot{m}^{f_{2}}_{G}+\dot{m}^{f_{2}}_{L}$
And the mass flow rates of each phase are calculated from their mass balance (Volume, mass, volume fraction and densities belong to the segment, mass flow rates belong to the adjacent faces):
$\stackrel{\underline{dm_{G}}}{dt}=V*[\alpha*\stackrel{\underline{d\rho_{G}}}{dt} + \stackrel{\underline{d\alpha}}{dt}*\rho_{G}]=\dot{m}^{f_{1}}_{G}-\dot{m}^{f_{2}}_{G}+\dot{m}_{evaporation}$
and:
$\stackrel{\underline{dm_{L}}}{dt}=V*[(1-\alpha)*\stackrel{\underline{d\rho_{L}}}{dt} - \stackrel{\underline{d\alpha}}{dt}*\rho_{L}]=\dot{m}^{f_{1}}_{L}-\dot{m}^{f_{2}}_{L}-\dot{m}_{evaporation}$

So adding gas and liquid mass flow rate at face 2 results in the overall mass flow rate on face 2:
$\dot{m}^{f_{2}}=\dot{m}^{f_{21}}+V[\stackrel{\underline{d\alpha}}{dt}(\rho_{L}-\rho_{G})-\alpha*\stackrel{\underline{d\rho_{G}}}{dt}-(1-\alpha)*\stackrel{\underline{d\rho_{L}}}{dt}]$

The energy balance is formulated as:
$\stackrel{\underline{dU}}{dt}=m\stackrel{\underline{du}}{dt}+\stackrel{\underline{dm}}{dt}u=\dot{H}^{f_{1}}-\dot{H}^{f_{2}}+\dot{Q}$

Enthlpy flow rates are determined by an upwind scheme using the mass flow rates provided by the previously shown euqations.

Does this set of euqations looks solvable to you or do I have some major mistakes or wrong assumptions

thanks a lot and sorry for the long post

January 26, 2022, 08:10	Modelling 1D-Evaporation	#1
Deranda Member Nils Join Date: Nov 2015 Posts: 59 Rep Power: 10	Dear all, I need some help, cause I am looking at my set of equations and thats jsut driving me crazy, that I cannot solve such a "simple" problem numerically. So what is my problem: Let us consider we have a simple pipe that is heated (or cooled) and the source term can be assumed to be $\dot{Q}$ =kA $\Delta$ T (k is the overall heat transfer coefficient, A is surface area and $\Delta$ T is the temperature difference driving the heat transfer. Within the pipe there is a refrigerent of which we have a huge database, so thermophysical properties such as spec. enthalpy, density, temperature and vapour quality are well known depending on the pressure and spec. internal energy. We want to calculate the evaporation process along the pipe numerically. For the solution, I am writing a 1D matlab code. Along the length, the pipe is devided into several segments. Each segment has a volume, internal energy, temperature, vapour density, liquid density, vapour quality and volume fraction. Initially, the pressure is set to ambient pressure and the pipe is filled with fluid slightly below the saturation line (T=T_sat - 0.5K). On the one end of the pipe, a mass flow boundary condition is set and fluid enters the pipe at T=T_sat-0.5K. I assume that the heat transfer takes place during an isobaric stage. So the pressure will be constant or in other words: the pressure will not affect the thermophysical properties. Thus a change in velocity can only be introduced by thermal effects (thermal expansion/contraction or evaporation/condensation). In my model, I will use the volume fraction to differ between liquid and vapour phase. The volume fraction can be directly calculated if densities (liquid and vapour phase) and the vapour quality q is known. $\alpha = \stackrel{q}{\overline{q+\stackrel{\underline{\rho_{G}}}{\rho_{L}}+q\stackrel{\underline{\rho_{G}}}{\rho_{L}}}}$ on the first face, the mass flow rate is given by the boundary condition. The mass flow rate on the second face is determined by the following equation: $\dot{m}^{f_{2}}=\dot{m}^{f_{2}}_{G}+\dot{m}^{f_{2}}_{L}$ And the mass flow rates of each phase are calculated from their mass balance (Volume, mass, volume fraction and densities belong to the segment, mass flow rates belong to the adjacent faces): $\stackrel{\underline{dm_{G}}}{dt}=V[\alpha\stackrel{\underline{d\rho_{G}}}{dt} + \stackrel{\underline{d\alpha}}{dt}\rho_{G}]=\dot{m}^{f_{1}}_{G}-\dot{m}^{f_{2}}_{G}+\dot{m}_{evaporation}$ and: $\stackrel{\underline{dm_{L}}}{dt}=V[(1-\alpha)\stackrel{\underline{d\rho_{L}}}{dt} - \stackrel{\underline{d\alpha}}{dt}\rho_{L}]=\dot{m}^{f_{1}}_{L}-\dot{m}^{f_{2}}_{L}-\dot{m}_{evaporation}$ So adding gas and liquid mass flow rate at face 2 results in the overall mass flow rate on face 2: $\dot{m}^{f_{2}}=\dot{m}^{f_{21}}+V[\stackrel{\underline{d\alpha}}{dt}(\rho_{L}-\rho_{G})-\alpha\stackrel{\underline{d\rho_{G}}}{dt}-(1-\alpha)*\stackrel{\underline{d\rho_{L}}}{dt}]$ The energy balance is formulated as: $\stackrel{\underline{dU}}{dt}=m\stackrel{\underline{du}}{dt}+\stackrel{\underline{dm}}{dt}u=\dot{H}^{f_{1}}-\dot{H}^{f_{2}}+\dot{Q}$ Enthlpy flow rates are determined by an upwind scheme using the mass flow rates provided by the previously shown euqations. Does this set of euqations looks solvable to you or do I have some major mistakes or wrong assumptions thanks a lot and sorry for the long post

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