Heat transfer: Temperature distribution inside a sphere submerged in a fluid

  • #1
happyparticle
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Homework Statement
What is the temperature distribution inside a sphere and fluid
Relevant Equations
$$\kappa \nabla^2 T = 0$$

$$T(r,\theta) = \sum_{m=0}^{\infty} = (A_m r^m + \frac{B_m}{r^{m+1}}) P_m(cos \theta)$$
Consider a sphere of thermal diffusivity ##\kappa_2## is submerged in an incompressible and stationary fluid of thermal diffusivity ##\kappa_1##.

The fluid is held between 2 large plates ( at T_0 for the top plate and T_1 for the bottom plate).

What is the stationary temperature distribution inside the sphere and in the fluid ?

Since ##\frac{\partial}{\partial t} = 0## and we have symmetry in ##\phi## coordinates, the heat conduction equation becomes:


##\kappa \nabla^2 T = 0##

With the following solution in spherical coordinates :

$$T(r,\theta) = \sum_{m=0}^{\infty} = (A_m r^m + \frac{B_m}{r^{m+1}}) P_m(cos \theta)$$.

Also, since ##T \rightarrow \infty## at ##r \rightarrow 0##

B must be 0.

This is as far as I can go. I'm not sure what are the boundary conditions to find the value of A and m.
 
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  • #2
Is the temperature distribution axisymmetric (a function of r only)? Please provide a diagram of the situation. From your description, I am unable to visualize the system.
 
  • #3
You need one solution inside the sphere and a separate solution outside the sphere. Since [itex]T[/itex] is specified on each of the plates, you must start by finding the outer solution. Having done so, the inner solution is found from the requirement that [itex]\kappa\frac{\partial T}{\partial r}[/itex] must be continuous on the surface of the sphere.

For the outer solution, if the plates are at [itex]z = r \cos \theta = \pm h[/itex], then we have the conditions [tex]\begin{split}
\sum_{m=0}^\infty B_m \frac{1}{r^{m+1}}P_m\left(\frac{h}{r}\right) &= T_0 \\
\sum_{m=0}^\infty B_m \frac{1}{r^{m+1}}P_m\left(-\frac{h}{r}\right) &= T_1.\end{split}[/tex] Exploiting the parity of the [itex]P_m[/itex], we can add and subtract these to get one condition for even [itex]m[/itex]
and one condition for odd [itex]m[/itex]: [tex]
\begin{split}
\sum_{m=0}^\infty \frac{B_{2m}}{r^{2m+1}} P_{2m}\left( \frac hr \right) = \frac{T_0 + T_1}{2} \\
\sum_{m=0}^\infty \frac{B_{2m+1}}{r^{2m+2}} P_{2m+1}\left( \frac hr \right) = \frac{T_0 - T_1}{2}
\end{split}[/tex] The left hand sides are power series in [itex]r^{-1}[/itex], and by comparing coefficients of powers of [itex]r^{-1}[/itex] it should be possible to determine the [itex]B_m[/itex].

EDIT: I' not sure if it's [itex]T[/itex] or [itex]\kappa \frac{\partial T}{\partial r}[/itex] which is continuous across the surface of the sphere. Also it may be necessary to include [itex]P_0[/itex] and [itex]rP_1[/itex] in the outer solution, since far from the sphere the temperature should approximate [itex]C + Dr\cos \theta[/itex].
 
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  • #4
In order to relate ##A## to ##B## and solve the system we really need to know about the boundary conditions across the sphere. Both in terms of ##T## and ##\frac{\partial T}{\partial R}##. I have no domain experience for this type of thermodynamics problem BUT it is very similar to electrostatics (Laplace's equation shows up everywhere doesn't it?).

I'm going to assume ##T## is continuous but ##\frac{\partial T}{\partial r}## is not continuous and it's related to ##\kappa## of each medium across the boundary.

I don't know what that relationship is but if we did this problem would be a lot easier to solve.
 

FAQ: Heat transfer: Temperature distribution inside a sphere submerged in a fluid

What is heat transfer and why is it important in this context?

Heat transfer is the process by which thermal energy moves from one physical system to another. In the context of a sphere submerged in a fluid, understanding heat transfer is crucial for predicting how the temperature distribution within the sphere changes over time, which can affect material properties, chemical reactions, and overall system performance.

How does the temperature distribution inside a sphere change when submerged in a fluid?

The temperature distribution inside a sphere submerged in a fluid is influenced by several factors, including the thermal conductivity of the sphere material, the fluid properties, and the temperature difference between the sphere and the fluid. Initially, the temperature inside the sphere will be uniform, but as heat transfers from the fluid to the sphere or vice versa, the temperature will vary, typically resulting in a gradient from the surface to the center of the sphere.

What are the governing equations for heat transfer in this scenario?

The governing equations for heat transfer in a sphere submerged in a fluid are typically based on the heat conduction equation (Fourier's law) and can be described using partial differential equations. In steady-state conditions, the heat conduction equation in spherical coordinates can be used, while transient conditions may require the unsteady heat conduction equation. Additionally, convection at the fluid-sphere interface must be considered, often described by Newton's law of cooling.

What factors affect the rate of heat transfer between the sphere and the fluid?

The rate of heat transfer between the sphere and the fluid is influenced by several factors, including the temperature difference between the sphere and the fluid, the thermal conductivity of the sphere material, the convective heat transfer coefficient of the fluid, and the flow characteristics of the fluid (laminar vs. turbulent). Surface area and the presence of any insulating layers can also play a significant role.

How can the temperature distribution be experimentally measured or simulated?

Temperature distribution inside a sphere can be experimentally measured using thermocouples or infrared thermography. For simulations, computational methods such as finite element analysis (FEA) or computational fluid dynamics (CFD) can be employed to model heat transfer in the system. These simulations can provide insights into temperature profiles under various conditions and help validate experimental results.

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