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Heat transfer
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The main article is for now Modeling with discrete exterior calculus but the geometry of the model is incomplete...
The amount of heat exchanged between two rooms is proportional to:
- the temperature difference,
- the conductance of the wall,
- the length of the wall that separates them,
- and, inversely, to the distance between the centers of mass of the rooms.
But there is more!
The amount of heat depends on the angle at which it passes through the wall.
Suppose $a$ is the wall (edge) and $b=a^*$ is the pipe between the centers (its Hodge dual). Suppose as vectors they are: $a=(x,y),b=(u,v)$. Then $c=(y,-x)$ is perpendicular to $a$ and has the same length.
Then the heat flow is proportional to the cosine of the angle between $b$ and $c$. To take items 3 and 4 above into account we have the "adjustment coefficient": $$A=\frac{|<c,b>|}{\|b\|^2},$$ where $|<c,b>|$ is the dot product. Note that when the geometry is Euclidean, $a$ and $b$ are perpendicular and equal in length. In that case, $A=1$.
This derivation follows Diffusion...
The preservation of the material in cell $\sigma$ is given by $$d_t U(\sigma,t)=−\int_{∂\sigma} *F(·,t),$$ where $d_t$ is the exterior derivative with respect to time (just the difference since the dimension is $1$) and $*$ is the Hodge duality. The flow $F$ through face $a$ of cell $\sigma$ is proportional to the difference of amounts of material in $\sigma$ and the other adjacent to $a$ cell. So, $$F(a)=-kd_x(*U)(a^*,t).$$ Substitute: $$d_t U(\sigma,t)=\int_{∂\sigma} *kd_x(*U)(a^*,t).$$ Integration is summation: $$d_t U(\sigma,t)=\sum \{*kd_x(*U)(a^*,t):a,∂\sigma =\sum a\}.$$