# My digester

by Gorkemakinci
Tags: digester
 Homework Sci Advisor HW Helper Thanks P: 9,783 Here goes... Taking the digester to be a uniform solid sphere radius R, surface temp fixed at 0 (we only care about relative temperatures), some initial internal profile. At time t, radius r, temperature = T(r, t). Diffusion equation in polar is δ/δr(r^2.δT/δr) = k.r^2.δT/δt some constant k (based on conductivity and specific heat). This has solutions of the form sin(a.r).exp(-λt)/r where λ = (a^2)/k A general solution will be a linear combination of these. The T(R, t) = 0 condition means a = n.π/R for some integer n (making sin(a.R) = 0). So T(r, t) = Ʃc$_{n}$.sin(n.π.r/R).exp(-λ$_{n}$t)/r where λ$_{n}$ = (n.π/R)$^{2}$/k The final step is to find the sequence c$_{n}$ s.t. T(r, 0) matches the initial profile (standard Fourier analysis). If you are interested in the core temperature at time t, i.e. at r=0, you need to take the limit of sin(a.r)/r as r tends to 0. This is simply a, giving: T(0, t) = Ʃc$_{n}$.(n.π/R).exp(-λ$_{n}$t) But you can't avoid having to supply the c$_{n}$. Over time, the dominant term will become the one with the smallest λ$_{n}$, i.e. the n=1 term: T(0, t) ~ c$_{1}$.(π/R).exp(-λ$_{1}$t) for large t HTH