Two-phase flow with evaporation

Cooperation with: M. Dreyer (Universität Bremen, ZARM), M. Adamov, M. Lozano Avilés (Technische Universität Berlin)

Supported by: BMBF-DLR: ``Treibstoffverhalten in Tanks von Raumtransportsystemen -- Comportement des Ergols dans les Réservoirs''

Description: The prediction of the dynamic behavior of liquids with free capillary surfaces in partly filled containers is of importance for the construction of space vehicles using liquid propulsion. In the case of cryogenic propellants, this behavior is influenced to a large extent by thermal effects, such as evaporation. As a first step for simulations, a model for two-phase flow with evaporation has been developed.

To model the situation mathematically, balance equations for mass, momentum, and energy in the bulk of the phases and on the phase boundary, respectively, were established, assuming gas and liquid to be incompressible Newtonian fluids and using the Boussinesq approximation to model buoyancy. In the bulk of both phases this leads to the Navier-Stokes equations with convection-diffusion equations for heat and vapor:

$\displaystyle \rho$ ( $\displaystyle \partial_{t}^{}$ u + u^. $\displaystyle \nabla$ u) - $\displaystyle \mu$ $\displaystyle \Delta$ u + $\displaystyle \nabla$ p	=	$\displaystyle \rho$ g - $\displaystyle \rho$ $\displaystyle \beta_{T}^{}$ ( $\displaystyle \vartheta$ - $\displaystyle \vartheta_{0}^{}$ )g,	(1)
$\displaystyle \nabla$ ^.u	=	0,	(2)
$\displaystyle \rho$ c_p( $\displaystyle \partial_{t}^{}$ $\displaystyle \vartheta$ + u^. $\displaystyle \nabla$ $\displaystyle \vartheta$ ) - $\displaystyle \lambda$ $\displaystyle \Delta$ $\displaystyle \vartheta$	=	$\displaystyle \boldsymbol\tau$ : ( $\displaystyle \nabla$ u),	(3)
$\displaystyle \partial_{t}^{}$ $\displaystyle \rho_{v}^{}$ + u^. $\displaystyle \nabla$ $\displaystyle \rho_{v}^{}$ - $\displaystyle \varsigma$ $\displaystyle \Delta$ $\displaystyle \rho_{v}^{}$	=	0.	(4)

[ $\displaystyle \rho$ (u^.n - u)]	=	0,	(5)
[ $\displaystyle \rho$ u (u^.n - u) - Tn]	=	- ( $\displaystyle \sigma$ ( $\displaystyle \nabla_{S}^{}$ ^.n)n - $\displaystyle \sigma_{T}^{}$ $\displaystyle \nabla_{S}^{}$ $\displaystyle \vartheta$ ),	(6)
[ $\displaystyle \rho$ c_p $\displaystyle \vartheta$ (u^.n - u) - $\displaystyle \lambda$ $\displaystyle \partial_{{\mathbf{n}}}^{}$ $\displaystyle \vartheta$ ]	=	j $\displaystyle \Lambda$ ,	(7)
$\displaystyle \rho_{v}^{}$ (u_g^.n - u) - $\displaystyle \varsigma$ $\displaystyle \partial_{{\mathbf{n}}}^{}$ $\displaystyle \rho_{v}^{}$	=	j.	(8)

Further conditions are needed to determine the evaporation rate j. Thus, we add the assumption that the temperature $\vartheta$ is continuous and is always equal to the saturation temperature $\vartheta_{{eq}}^{}$ given by the partial pressure of vapor $\psi$ ( $\vartheta$ , $\rho_{v}^{}$ ) on the phase boundary. Note that in a precise physical sense this assumption is contradictory to a non-vanishing evaporation rate, since it is the statement of equilibrium. However, the difference $\vartheta$ - $\vartheta_{{eq}}^{}$ is negligible in many practical cases. This assumption yields a Dirichlet boundary condition for the temperature in both phases that is sufficient to solve the heat transport equation (3). Equation (7) may then be used to calculate j. This approach results in a fully coupled two-phase flow problem.

In general, viscosities in gas are much smaller than in liquid and thus T_gn $\ll$ T_ln. Using this assumption, one may decouple the flow problems from both phases by neglecting the shear stresses from the gaseous phase on the boundary. The flow from the gaseous phase does not have any direct influence on the shape of the free surface then. The flow problems in the two phases are only weakly coupled by the temperature and the mass flux.