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1、Exact solution of the diffusion-convection equation in cylindrical geometry,Oleksandr Ivanchenko, Nikhil Sindhwani, and Andreas A. LinningerLaboratory for Product and Process Design, M/C 063University of Illinois at ChicagoChicago, IL 60607-7000, USA,1,NS, LPPD,LPPD seminar: 2nd September 2009,Exact
2、 solution of the diffusio,Problem Formulation,NS, LPPD,2,C(r,t)=CS/C0q= Qin/h, or the radial conductive velocity field.,Problem FormulationNS, LPPD2C(,NS, LPPD,3,Boundary conditions:,Convection-diffusion equation in radial co-ordinates:,Initial conditions:,NS, LPPD3Boundary conditions:C,General solu
3、tion, Separation of variables.,NS, LPPD,4,Substituting this in the convection-diffusion equation.,Temporal solution.,i used because there could be more than one that satisfy the equation above.i has to be a positive number for the solution to be stable.,General solution, Separation o,NS, LPPD,5,Radi
4、al solution:,where, =1-V0/D, also,By using Forbenius power series method of solution,This can also be written by using the definition of a Bessel function of the First Kind and order .,NS, LPPD5Radial solution:where,NS, LPPD,6,Using , we can find the roots of the bessel function equation at r=L.,Whe
5、re, si, are the roots of the bessel function. Now, is known and we can write the final analytical solution,NS, LPPD6Using,NS, LPPD,7,By using the initial condition, we can find out Ai, by fourier-bessel decomposition:,NOTE: This solution fails when f(r)=0. This is the case when the domain is empty i
6、nitially.,NS, LPPD7By using the initial,Advanced solution: by decomposing steady state and dynamic parts.,NS, LPPD,8,Steady state part and dynamic part of the solution are separated.Boundary Conditions:Initial Condition:,Advanced solution: by decompos,NS, LPPD,9,Steady state solution:,At t=0:,This g
7、ives the initial condition for dynamic part of the solution, which can be replaced in the general solution to give:,NS, LPPD9Steady state solution,NS, LPPD,10,Final Solution:,NS, LPPD10Final Solution:,NS, LPPD,11,Solution for D=0.01 and =1,NS, LPPD11Solution for D=0.01,NS, LPPD,12,N=20,N=60,Effect o
8、f number of terms used, more terms, smoother solution.,NS, LPPD12N=20N=60Effect of nu,NS, LPPD,13,Solution in Non-dimensional form: =Dt/L2 , =r/L, P=Vo/D,Steady state solution,NS, LPPD13Solution in Non-dime,NS, LPPD,14,Effect of Peclet Number on solution trajectories:,NS, LPPD14Effect of Peclet Num,Development of Convection,NS, LPPD,15,Development of ConvectionNS, L,Thank You,NS, LPPD,16,Thank YouNS, LPPD16,
