MATLAB for Breakfast Evolution of heat distribution across bottom of frying pan on spiral-shaped - PowerPoint Presentation

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MATLAB for Breakfast

Evolution of heat distribution across bottom of frying pan on spiral-shaped electric Coil stovetopTim Nickell – EPS 109, Fall 2013 – Prof. Burkhard Militzer

Context: Spiral Stovetop

Type of electric-coil stovetop

Made from high-resistance

nichrome

alloy (~80% nickel, ~20% chromium)

High electrical resistivity allows coil to rapidly heat up as electric current passes through it

Goal of simulation: to model heat distribution across bottom surface of aluminum frying pan sitting on this type of burner

Coding: methods, techniques, etc.

Main method = variant of the 2D time-dependent heat equation (includes internal heat source): Adapted this equation to 2D, using form from Lab/HW 8 (discretized form of PDE): Pan is composed of aluminum  use the density, heat conductivity, and specific heat of Al to derive kappa (heat diffusivity = k/rho*cp )

Conditions and Approximations

Approximations:Pan is extremely thinHeat loss to air via convection = negligible on short time scales (for rough modeling purposes)Lip of pan = excluded – only looking at the bottom surface of pan (2D)Spiral burner = very thin (in the simulation, approximated to width of pixel)

Source term, Q: spiral-shape (like shape of burner) Boundary conditions (bottom of pan): insulating  shaped like a circle (see below)cooling via convection < heat transfer through pan aluminumFor both spiral and circle BC, method from Lab 2 used (if statement with radius r):

“And now, our feature presentation:”

Bon Appétit! (Thanks for listening!)

You can run my code by:

1. Setting up source distribution pattern (spiral)

First, use

i

, j nested loop with an n = ~201x201 matrix (e.g., for I = 1:n; for j = 1:n)

Next, use method from Lab 2 to get a radius in terms of

i

, j (same one for making circles)

Then, come up with theta in terms of x(

i

), r (two loops, one for top half, one for the other)

Finally, set matrix = 1 (or whatever source temp) for all matrix(

i,j

) with r close to the

r-value

for theta at that point (solve for

rnew

(theta), set equal to r-old)

2. Running the PDE as described above

3. Insulating boundary conditions (eight loops!)