[cig-commits] commit 2399 by bangerth to /var/svn/dealii/aspect

[dealii.demon at gmail.com]

Revision 2399

Read through Juliane's section on phase transitions.

U   trunk/aspect/doc/manual/manual.tex
U   trunk/aspect/doc/manual.pdf


http://www.dealii.org/websvn/revision.php?repname=Aspect+Repository&path=%2F&rev=2399&peg=2399

Diff:
Modified: trunk/aspect/doc/manual/manual.tex
===================================================================
--- trunk/aspect/doc/manual/manual.tex	2014-04-02 13:21:24 UTC (rev 2398)
+++ trunk/aspect/doc/manual/manual.tex	2014-04-02 15:03:03 UTC (rev 2399)
@@ -6016,14 +6016,23 @@
       temperature is $T_2 = 1107.39 \, 	ext{K}$.}
   \label{fig:latent-heat-benchmark}
 \end{figure}
-It tests if the latent heat production when material crosses a phase transition is calculated correctly according to the laws of thermodynamics. The material model defines two phases in the model domain with the phase transition approximately in the center. The material flows in from the top due to a prescribed downward velocity, and crosses the phase transition before it leaves the model domain on the bottom. As initial condition, the model uses a uniform temperature field, however, upon the phase change, latent heat is released. This leads to a characteristic temperature profile across the phase transition with a higher temperature in the bottom half of the domain. For steady-state one-dimensional downward flow, we have to solve the equation
-
+It tests whether the latent heat production when material crosses a phase
+transition is calculated correctly according to the laws of thermodynamics. The material
+model defines two phases in the model domain with the phase transition
+approximately in the center. The material flows in from the top due to a
+prescribed downward velocity, and crosses the phase transition before it leaves
+the model domain at the bottom. As initial condition, the model uses a uniform
+temperature field, however, upon the phase change, latent heat is released. This
+leads to a characteristic temperature profile across the phase transition with a
+higher temperature in the bottom half of the domain. For steady-state
+one-dimensional downward flow, we have to solve the equation
 egin{gather*}
-rac{\partial T}{\partial y} = T rac{\Delta S}{c_p} rac{\partial X}{\partial y} + rac{\kappa}{v_y} rac{\partial^2 T}{\partial y^2}
+rac{\partial T}{\partial y} = 
+T rac{\Delta S}{c_p} rac{\partial X}{\partial y} 
++ rac{\kappa}{v_y}
+rac{\partial^2 T}{\partial y^2}.
 \end{gather*}
-
 The first term on the right-hand side of the equation describes the latent heat produced at the phase transition: It is proportional to the temperature T, the entropy change $\Delta S$ across the phase transition divided by the specific heat capacity and the derivative of the phase function X. If the velocity is smaller than a critical value, and under the assumption of a discontinuous phase transition (i.e. with a step function as phase function), this latent heating term will be zero everywhere except for the one point $y_{tr}$ where the phase transition takes place. This means, we have a region above the phase transition with only phase 1, and below a certain depth a jump to a region with only phase 2. inside of these one-phase regions, we can solve the equation above (using the boundary conditions $T=T_1$ for $y 
ightarrow \infty $ and $T=T_2$ for $y 
ightarrow -\infty $) and get
-
 egin{align*}
 T(y) =egin{cases}
 T_1 + (T_2-T_1) e^rac{v_y (y-y_{tr})}{\kappa}, & y>y_{tr}\
@@ -6045,60 +6054,83 @@
 
 In addition, we have tested the approach exactly as it is described in