Thermal Distribution in an Optical Fiber with Heat Source

The power efficiency of fiber lasers is often determined by the core temperature distribution in the fiber. The optical fiber is modeled as a doped inner core of radius and a cladding region of thickness . The length of the optical fiber is . The thermal conductivity of the inner core is taken to be and the cladding region is . The heat density within the inner core is taken to be , where is a suitable absorption coefficient. The temperature within the inner core is denoted by and the temperature in the cladding as . The thermal boundary conditions on the optical fiber are as follows. The outer surface of the cladding is subject to Newton's law of cooling with a specified heat transfer coefficient :
.
Newton's law of cooling is also assumed for the ends of the fiber containing cladding. Thus for the cladding region:
at and .
At we specify a heat flux for the inner core:
at .
At we assume Newton's law of cooling for the inner core:
at .
This Demonstration finds the temperature distribution both in the cladding and core region for user-set values of the Biot number , thermal conductivities ratio , dimensionless heat generation , and dimensionless absorption coefficient . In order to obtain this distribution, a finite difference scheme is applied as described in depth in Details.

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DETAILS

A schematic of the problem geometry is shown in the first snapshot.
Mathematical formulation
For computational convenience we introduce the following dimensionless coordinates:
, , , , , and .
Thus the thermal conduction equations for the inner core and cladding are:
for and ,
for and .
The quantity is a dimensionless heat generation term defined as:
,
while is the dimensionless absorption coefficient for the light source entering the core region.
Boundary conditions
The boundary conditions for the core region are:
BC1: ,
BC2: , 0<ξ<ξ1, at ,
BC3: ,
Here is a Biot number and is a ratio of thermal conductivities for the cladding and core regions. Typically the conductivity of the cladding is much smaller than that of the core so that .
At the core/cladding interface we must have continuity of heat flux and temperature. These conditions are:
BC4: ,
BC5: ,
For the cladding region, we have Newton's law of cooling at all exposed surfaces:
BC6: , , at ,
BC7: , ξ1<ξ<1, at ,
BC8: , ξ1<ξ<1, at .
Finite difference formulation
We consider a uniform grid in the and directions. Let be the number of grid points in the direction, be the number grid points in the direction for the core, and be the number of grid points in the direction for the cladding region.
We discretize the domain into equal segments and use central differences for the Laplacian operator. The result is:
.
However at there is a singularity. Using L'Hopital's rule we have at :
.
Thus the Laplacian at (i.e. ) is given by
,
where is a fictitious node. Then noting that at , (because of symmetry). A central difference approximation of the derivative at gives , so on eliminating the fictitious node we have:
.
Boundary conditions
As a rule we use central differences for derivatives at the boundary.
Consider BCs in direction:
For BC 1 we require that:
BC1:
For BC2 we have:
BC2: for .
For BC3 we have:
BC3: for .
Similar finite difference formulas can be derived for the other BCs. However, for the interface conditions we use first-order differences. This allows us to readily couple the two grid boundaries.
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