1 0 0 5 0 0 0 5 1 0 3 2 1 0 1 2 3 2 A Very Narrow Pipe Of Length L And Cross Sectional Area A Is Filled With Wa 1 (81.67 KiB) Viewed 42 times
1 0 0 5 0 0 0 5 1 0 3 2 1 0 1 2 3 2 A Very Narrow Pipe Of Length L And Cross Sectional Area A Is Filled With Wa 2 (55.33 KiB) Viewed 42 times
1.0. 0.5 $ 0.0 -0.5 -1.0 -3 -2 -1 0 1 2 3 2. A very narrow pipe of length L and cross-sectional area A is filled with water. At time t=0, a certain amount of salt, of total mass M, is placed at one end the pipe, reaching a length d. The pipe is then sealed at both ends. We wish to know the concentration of salt, C, at any time. Since the pipe is narrow, the variations of concentration of salt over the cross section of the pipe can be neglected, and the concentration is given by C = C(x, t). The concentration satisfies the 1-D diffusion equation: ac =D at ar2 The initial concentration profile is ac C(3,0) = { = M Ad 0, 0<x<d d<r<L and since the pipe is closed at both ends: ac(0,t) ar ac(L,t) дах 0
a) Start with a solution of the form C(x,t) = X(2)T(t), replace in the diffusion equation and show that the problem reduces to two ordinary differential equations: X"(x) = \X(x); T'(t) = \DT(t) where l is an arbitrary constant. b) We seek time-decaying solutions, show that implies X = -k2, where k is an arbi- trary constant. c) Solve the two ODEs and show that after applying the two boundary conditions: C(x,t) = ancos = пт L -x)e-(E){Dt ) n=0 d) Find C(2, t) so that it satisfies the initial concentration profile. e) Show that your expression for C(x, t) satisfies the equilibrium profile C(x, t0) expected from conservation of mass.
Join a community of subject matter experts. Register for FREE to view solutions, replies, and use search function. Request answer by replying!