Electrical (Joule) heating in my kitchen oven. I was in my kitchen recently and realized that the small oven we have has two cylindrical metal rods from where the thermal energy comes from when we turn on the oven thanks to the conversion of electrical energy into thermal energy. I took some photographs to show what it looks like (on the right). I would like to know the temperature profile inside one of these rods. I am assuming both of them work basically the same. So, the same profile would work for both. I haven’t measured the dimensions of the rods, but I can easily do it once I have the model. So you can say that the metal rod has a radius R and a length L. The material from which the rod is made of is some metal alloy with a thermal conductivity represented by the symbol krod. The little oven doesn’t have a fan, so it works with natural convection. I just bought a thermal camera that I can use to know the temperature of surfaces. So, I can also easily know the temperature in the outer surface of the rod, Tout. The rod is solid of course. And it can be assumed to have a uniform composition. The rod is of course stationary, at rest. Thermal variations in volume are negligible. I would like to have an initial steady-state model considering that the energy losses by radiation are negligible. The rod is obviously losing some heat in the form of light, but I think it is fair to assume this to be a small fraction of the energy coming in. The sides of the rods are insulated so that the thermal energy is transferred by conduction most significantly in the radial direction. I checked the manual to find that the internal heat generation term can be assumed to be 𝑆𝑒 = 𝛽 ∗ 𝑟, where the term 𝛽 has appropriate units so that the heat generation term has dimensions of energy per volume.
c) (1 points) Select a version of the Equation of Energy that you can establish as an appropriate partial differential equation (PDE) to use as initial step towards models for heat conduction at steady state considering a heat generation term.
d) (2 points) Explain how you can use the some of the information mentioned in the problem statement to cancel out terms in your PDE to simplify it to an ODE. Explain the cancellation and write down your governing differential equation (GDE) for heat transfer by conduction through the wall of tube, which is the equation that you propose solving for an initial simplified model.
e) (2 points) Establish appropriate boundary conditions.
f) (4 points) Obtain a 1D model for the appropriate vector component of the heat flux.
g) (4 points) Obtain a 1D model for that describes the temperature variations inside
your control volume.
h) (4 points) Show proof that your profiles are correct,
i. (0.5 points) Evaluate the boundary conditions.
ii. (0.5 points) Check the sign of the momentum diffusion flux.
iii. (0.5 points) Check the sign of the velocity vector component.
iv. (0.5 points) Check dimensions.
i) (2 points) Obtain an equation for the heat flow rate through the control volume.
j) (5 points) Obtain an equation for the average temperature rise.
Images for GDE
- There Is Another Problem With Different Questions So Read Carefully To Answer Please Electrical Joule Heating In My 1 (37.2 KiB) Viewed 34 times
- There Is Another Problem With Different Questions So Read Carefully To Answer Please Electrical Joule Heating In My 2 (53.14 KiB) Viewed 34 times
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