In contrast to the liquid-vapor interface, the solid heating wall and consequently the solid-fluid interface is not moving or being deformed during the simulation. Furthermore, only the heat conduction equation (2.12) needs to be solved in the solid domain. For these two reasons it is convenient to use a separate mesh for the solid domain. The energy equation in the fluid domain and the energy equation in the solid domain are solved separately on their individual meshes and coupled by the common boundary at the solid-fluid interface (see Figure 4.8). In the following the subscripts F and S are used to denote quantities on the fluid and solid side of the solid-fluid interface, respectively. The classical conjugate heat transfer problem between two domains is solved by applying the thermal equilibrium, i.e. continuity of temperature and heat flux. For any face f on the solid-fluid interface, the following equations are applied.
TF,f = TS,f (4.39)
qF,f = qS,f (4.40)
4.3 Conjugate heat transfer between solid and fluid 46
Solid Fluid
Temperature calculated boundary condition Heat flux boundary condition calculated
Table 4.1:The temperature field and heat flux field at the solid-fluid interface are imposed as boundary conditions or calculated depending on the domain (solid or fluid).
The following boundary conditions are chosen: For the solid domain, a fixed value for the normal temperature gradient (corresponding to a particular heat flux at the face f) is imposed on the solid-fluid interface while a fixed value for the temperature is used as boundary condition for the fluid domain (see Table 4.1). In the numerical model, the classical coupling of the energy equations is implemented in the following iterative procedure.
• Step 1
The temperature field at the solid-fluid interface of the solid domain (calculated in the last time step or iteration loop) is imposed as a boundary condition on the solid-fluid interface of the fluid domain.
• Step 2
The heat flux field at the solid-fluid interface of the fluid domain (calculated in the last time step or iteration loop) is imposed as a boundary condition on the solid-fluid interface of the solid domain.
• Step 3
The energy equation is solved in the fluid domain taking into account the boundary condition that is imposed on the solid-fluid interface during step 1. Hereby, the fluid side heat flux at the solid-fluid interface is calculated.
• Step 4
The energy equation is solved in the solid domain taking into account the boundary condition that is imposed on the solid-fluid interface during step 2. Hereby, the solid side temperature field at the solid-fluid interface is calculated.
• Step 5
The newly calculated temperature field on the solid side of the solid-fluid interface is compared to the temperature field on the fluid side. If the maximum of the magnitude of the difference between the two fields is larger than a certain value6, another correction loop is performed beginning with step 1. Otherwise, the solution is converged and the temperature fields are kept.
InOpenFOAMit is also possible to directly couple the energy equations in the two domains by combin-ing the two matrices which result from the FVM treatment of the energy equations into a scombin-ingle matrix.
The explicit, iterative coupling which is applied here has the major drawback of a limited time step size.
However, the time step is typically much more limited by the transport of the volume fraction field than by the solution of the energy equation. Therefore the temperature fields are typically converged after less than five iterations and do not change much from one time step to another. The advantage of the explicit, iterative coupling is the straightforward implementation and the possibility to directly manipu-late the conjugate heat transfer. The latter is used for coupling the subgrid scale model for the contact line evaporation to the CFD simulation. After the reconstruction of the liquid-vapor interface, the faces which are adjacent to the solid-fluid interface are subdivided into three groups.
• Group 1:Faces on the liquid side of the liquid-vapor interface without 3-phase contact line
• Group 2:Faces on the vapor side of the liquid-vapor interface without 3-phase contact line
6 In the present thesis, an absolute temperature difference of10−4K is used as convergence criterion.
4.3 Conjugate heat transfer between solid and fluid 47
liquid vapor
reconstructed interface
fluid cells
solid cells thermal coupling at solid-fluid interface Group 2
= = 0 qS qF
Group 1
= q q T
S F
S=TF
Group 3
= +
q q q
T
S F Δ cl,f
S=TF
solid cell with increased
source term due to Δqcl,f
Figure 4.8:Illustration of the solid and fluid domains and the thermal coupling across the solid-fluid interface.
• Group 3:Faces containing a segment of the 3-phase contact line
These groups and the corresponding coupling conditions are shown in Figure 4.8. For group 1 and group 2, the classical thermal coupling as described above can be applied. However, in nucleate boiling conditions the heat transfer from the wall into the vapor is usually negligible compared to the heat transfer into the liquid. Therefore, the above mentioned coupling is only applied for those parts of the solid-fluid interface which are in contact to liquid (group 1). The heat transfer at those parts of the solid-fluid interface which are in contact to vapor (group 2) is assumed to be zero. Even though the use of the VOF method enables the solution of the energy equation in both phases, the temperature of the vapor is assumed to be constant and equal to the bulk saturation temperature. The consequence is that the heat transfer in the vapor phase is decoupled from the heat transfer in the solid. The faces which contain a segment of the 3-phase contact line (group 3) are treated in a particular way, i.e. the classical coupling which is described above is modified. While Eq. (4.39) is directly used to ensure continuity of the temperature field, the heat flux at a particular face f of group 3 on the solid side of the solid-fluid interface is modified by the heat transfer correction from the contact line evaporation model (according to Eq. (4.38)) .
qS,f =qF,f +∆Qcl,f ~Sf
=qF,f + ∆qcl,f (4.41)
Hence, the heat flux is discontinous over the faces on the solid-fluid interface which contain a segment of the 3-phase contact line. The heat flux which is transported out of the solid domainqS,f is higher than the heat flux which is transported into the fluid domainqF,f. Nevertheless, the energy is conserved as the difference between the two heat fluxes directly intensifies the local rate of evaporation. The difference is equal to the heat transfer correction of the subgrid scale model for contact line evaporation. Due to the very small local film thickness, the heat that is transferred at the 3-phase contact line does not contribute to the heating up of the liquid but is almost immediatly consumed by the evaporation. Therefore, the heat flux correction according to Eq. (4.38) directly enhances evaporation locally by contributing to the sharp source term field described in section 4.1.1. The sharp source term field ρ˙0 is increased by the term∆Qcl,f/ ∆hVVcell
in a cell cif it has a face f on which a segment of the 3-phase contact line has been reconstructed. This local increase of the sharp source term field is performed prior to the smearing, cropping and scaling of the source term field that is described in section 4.1.2.
4.3 Conjugate heat transfer between solid and fluid 48
liquid
vapor vapor liquid vapor liquid
t= 0 s
T
x
t= 0.1 s t= 1.1 s
5 K 0.4 mm
Figure 4.9:Sucking interface problem: Sketch of the problem setup with interface positions (dashed lines) and temperature profiles (solid curves) at different instances. The fluid is water at p=1013 mbar, the wall superheat is 5 K.