Problem 1. Obtain the USS concentration profile for the diffusion of solute outside an infinitely long cylinder (neglect end effects) with convection at the surface. Consider the diffusion is in the radial direction. (Check Chapter 4 Hines and Maddox) The cylinder is drying out so you can consider that at time t=0 the initial concentration is CA=C0 at t=0. At times greater than zero, the concentration at the surface of the cylinder (r=R) should be such that "diffusion should equal convection at the surface", therefore: NAr=DrCA=kcC=kc(CABCAB) at r=R Where CA=KCAB and CA=KCAB Also consider that inside the cylinder there should be symmetry in the concentration profile, therefore rCA=0 at r=0 Show that the solution is equal to: CA0CACACA=n=1(n2+2)J0[n]2J0[n(r/R)]exp(R2n2Dt) where =KDkc Hint: You can define a variable f=T(t)R(r) and use separation of variables. Use orthogonality rules for Bessel Functions When n are the roots of this equation: nJ1(n)J0(n)=0 then the orthogonality rule you should use is: 0RrJi(nr)Jk(nr)dr=2n2ik(R22+R2n2i2)Ji2(nR)