Cooling Effect of a Duct with a Cooling Liquid

Compute the cooling effect of a duct with a cooling liquid in an axisymmetric cross section of a pipe.

Neumann values prescribe the flux over the boundary edge:

In[1]:=

Ω=ImplicitRegiony≥0&&

2

x-

3

2

+

2

y

≥

1

25

&&

2

x

+

2

y

≥1&&

2

x

+

2

y

≤4&&y≤xTan

π

8

,{x,y};

Visualize the simulation region:

In[2]:=

ShowGraphicsCircle[{0,0},2],Circle[{0,0},1],

circles

,RegionPlot[Ω],

...



Out[2]=

Set up the heat transfer model variables

vars

:

In[3]:=

vars={T[x,y],{x,y}};

Specify heat transfer model parameter material:

In[4]:=

pars="Material"

tungsten

ELEMENT

;

Specify boundary condition parameters for a heat flux

q

of

1000W/

2

m

:

In[5]:=

pars["BC"]="HeatFlux"1000;

Specify Dirichlet boundary conditions to set temperatures inside the pipe and outside:

In[6]:=

Γ

D

={HeatTemperatureCondition

2

x

+

2

y

1,vars,pars,"SurfaceTemperature"200,HeatTemperatureCondition

2

x

+

2

y

4,vars,pars,"SurfaceTemperature"15};

Specify a flux into the cooling duct in the pipe:

In[7]:=

Γ

N

=HeatFluxValue

2

x-

3

2

+

2

y



1

25

,vars,pars,"BC";

Solve the Laplace equation:

In[8]:=

uif=NDSolveValue[{HeatTransferPDEComponent[vars,pars]

Γ

N

,

Γ

D

},T,{x,y}∈Ω];

Plot the solution:

In[9]:=

cp=ContourPlotuif[x,y],{x,y}∈Ω,



Out[9]=

In[10]:=

ShowGraphicsCircle[{0,0},2],Circle[{0,0},1],

circles

,

...

,cp

Out[10]=

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