1、Exact solution of the diffusion-convection equation in cylindrical geometryOleksandr Ivanchenko,Nikhil Sindhwani,and Andreas A.LinningerLaboratory for Product and Process Design,M/C 063University of Illinois at ChicagoChicago,IL 60607-7000,USA1NS,LPPDLPPD seminar:2nd September 2009Problem Formulatio
2、nNS,LPPD2()()CV rCD Ct u rC(r,t)=CS/C0q=Qin/h,or the radial conductive velocity field.00()2();()22ooooVV rrqV rVV rrrrrNS,LPPD32021VCCCCDtrrrrrBoundary conditions:Convection-diffusion equation in radial co-ordinates:0(0,)()(,)0C rtCfiniteC rL t(,0)()C r tf rInitial conditions:General solution,Separa
3、tion of variables.NS,LPPD4 ,C r tR rT t2211;TRqRD TtRrrr Substituting this in the convection-diffusion equation.Temporal solution.()expiiT tDti 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.0TD TNS,LPPD5Radial so
4、lution:00 RRRorr RRrRr()0R L 202/20(1)1-()with=222!(1)nniinnirVR rDnvwhere,=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.(),iiR rrJrNS,LPPD6()0R L Using ,we can find the roots of the b
5、essel function equation at r=L.()=0;iiiisR LLJLLWhere,si,are the roots of the bessel function.Now,is known and we can write the final analytical solution0(,)expiiiiC r tA rJrDtNS,LPPD7By using the initial condition,we can find out Ai,by fourier-bessel decomposition:0220()LiiLisf rJrrdrLAsJrrdrLNOTE:
6、This solution fails when f(r)=0.This is the case when the domain is empty initially.Advanced solution:by decomposing steady state and dynamic parts.NS,LPPD8Steady state part and dynamic part of the solution are separated.Boundary Conditions:Initial Condition:dynss,=,t+C r tC rC r(0,)1 (,)0C rtC rL t
7、(,0)()0C r tf rNS,LPPD900202/1()1VDssVDVCCCDrrrrrrC rL Steady state solution:dynssdynssss,0=0=,0+,0()C rC rC rC rC rfor dynamic partf rC r At t=0:This gives the initial condition for dynamic part of the solution,which can be replaced in the general solution to give:NS,LPPD1000/02201LVDiVDiLisrJrrdrL
8、LAsJrrdrL Final Solution:00/0(,)exp1VDiiiVDirC r tA rJrDtLNS,LPPD11Solution for D=0.01 and=1NS,LPPD1200.10.20.30.40.50.60.70.80.91-0.200.20.40.60.81r N=20N=60Effect of number of terms used,more terms,smoother solution.NS,LPPD13Solution in Non-dimensional form:=Dt/L2,=r/L,P=Vo/D 202/2/2011;2(,)expPiPiiiVCCCCqPPDDCAJ()1PssC Steady state solutionNS,LPPD14Effect of Peclet Number on solution trajectories:0/PVDDevelopment of ConvectionNS,LPPD1500.10.20.30.40.50.60.70.80.9100.10.20.30.40.50.60.70.80.91r/L IC,t=0.00s t=0.01s t=0.10s t=1.00s t=10.00s t=100.00s0CCrLThank YouNS,LPPD16
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