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MATLAB for Breakfast EVOLUTION OF HEAT DISTRIBUTION ACROSS BOTTOM OF FRYING PAN ON SPIRAL-SHAPED ELECTRIC COIL STOVETOP TIM NICKELL – EPS 109, FALL 2013 – PROF. BURKHARD MILITZER

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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

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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*c p )

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Conditions and Approximations Approximations: 1.Pan is extremely thin 2.Heat loss to air via convection = negligible on short time scales (for rough modeling purposes) 3.Lip of pan = excluded – only looking at the bottom surface of pan (2D) 4.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 aluminum For both spiral and circle BC, method from Lab 2 used (if statement with radius r):

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“And now, our feature presentation:”

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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!)

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