Heat Transfer Boundary Conditions

In summary: Thus, we can write$$\frac{1}{r}\frac{d}{dr}\left(rk\frac{dT}{dr}\right)=-\int_{r_{1}}^{r_{2}}{(2\pi r'\phi(r')dr')\delta(r'-r)\frac{q}{k}}=\frac{q}{k}(2\pi r\phi(r))$$where phi is the fraction of heat generated in the graphite at radius r. In summary, the composite cylindrical wall of a high temperature, gas cooled nuclear reactor with a thorium fuel element encased in graphite and gaseous helium flowing through an annular coolant channel can be analyzed using the heat balance equation $$\frac{1}{
  • #1
luca131
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Homework Statement


A high temperature, gas cooled nuclear reactor consists of a composite cylindrical wall for which a thorium fuel element (Ka=57 W/m*K) is encased in graphite (Kb= 3 W/m*K) and gaseous helium flows through an annular coolant channel. Consider conditions for which the helium temperature is T=600k and the convection coefficient at the outer surface of the graphite is h= 2000 W/m^2*K. If thermal energy is uniformly generated in the fuel elemnt at a rate of q=10^8 W/m^3 what are the temperaturesT1 and T2 at the inner and outer surfaces, respectively, of the fuel element?

Homework Equations


1/r d/dr (r dT/dr)+q/k=0

The Attempt at a Solution


Not really sure how to derive this formula. The boundary conditions I have chosen was
at r=0 dT/dr=0 because the max temperature should be in the center. My second boundary condition is the T(r0)= Ts. If someone could help derive this formula that would be great been trying to derive the temperature profile for sometime now.
If someone could so me a crude derivation of the temperature profile that would be great.
 
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  • #2
If you do a heat balance on the section of one of the layers between r and r + delta r, you obtain: $$\left[-2\pi r k\frac{d T}{dr}\right]_{r+dr}-\left[-2\pi r k\frac{dT}{d r}\right]_{r}=(2\pi r dr)q$$or$$\frac{1}{r}\frac{d}{dr}\left(rk\frac{dT}{dr}\right)=-q$$Heat is generated only in the thorium fuel element, and the temperature of the surrounding helium is 600 K, although this is not the outer temperature of the graphite (because of the thermal boundary layer between the outer surface and the bulk helium).
 

1. What are heat transfer boundary conditions?

Heat transfer boundary conditions refer to the set of conditions or parameters that determine the transfer of heat between two or more materials or systems. These conditions include temperature, heat flux, and convection, among others.

2. Why are heat transfer boundary conditions important?

Boundary conditions play a crucial role in determining the behavior of heat transfer in a system. They help in predicting the temperature distribution, heat flow, and other important parameters, which are essential for designing and optimizing heat transfer processes.

3. What are the different types of heat transfer boundary conditions?

There are three main types of heat transfer boundary conditions: Dirichlet boundary conditions, Neumann boundary conditions, and Robin boundary conditions. Dirichlet boundary conditions specify the temperature or heat flux at the boundary, while Neumann boundary conditions specify the heat flux. Robin boundary conditions combine both temperature and heat flux parameters.

4. How do you determine the appropriate boundary conditions for a heat transfer problem?

The appropriate boundary conditions for a heat transfer problem depend on several factors, including the geometry of the system, the type of material, and the desired outcome. Generally, the boundary conditions should be chosen to reflect the physical behavior of the system and should be consistent with the laws of thermodynamics.

5. Can heat transfer boundary conditions change during a heat transfer process?

Yes, heat transfer boundary conditions can change during a heat transfer process. For example, in a transient heat transfer process, the boundary conditions may vary with time as the system reaches a steady-state. Additionally, changes in external factors such as temperature or heat flux can also alter the boundary conditions.

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