ON THE DANCKWERTS BOUNDARY CONDITIONS AND COMMENTS ON"THE AXIAL DIFFUSION MODELTHE REACTION OF FIRST ORDER AND RESIDENCE TIME DISTRIBUTION IN CLOSED MODEL"BY GAO WEIYUE
Xiao Wende (UNILAB Research Center of Chemical Reaction Engineering,East China University of Chemical Technology,Shanghai 200237)
The Primitive exit boundary condition: uC_0 intergral from n=0 to ∞ E(t)e~(Rt)dt(uCD(dC)/(dx))_(x=L~)(1) was proposed by Danckwerts(1953).Because it is undetermined,Danckwerts came to the final condition: (dC)/(dx)=0,x=L (2) Recently,based on Eq.(1),Gao Weiyue(1989)concluded that Eq.(2)is not rea sonable,and that there is a critical Peclet Number,Pe',for the reaction when the system Peclet Number is less than Pe',the conversion is smaller than CSTR's. According to the flux balance equation: (uCD(dC)/(dx))x=L_=(uCD(dC)/(dx))x=L_+ (3) and material balance equation: uC_0uC_0 integral from n=1 to ∞ E(t)e~(kl)dt=uC_0(uCD(dC)/(dx))x=L_+(4) this work obtains Eq.(1),Only by replacing L with an any point X~*Eq.(1), Eq.(3)and(4)are satisfied the same way.So it will be no meaning employing Eq.(1)as the exit boundary condition. There are two critical mistakes in Gao′s paper.One appeared in the inequality group. C(L)≤0 (5a) (dC)/(dx)x=≤(5b) (11.a.b.in Gao′s paper).This group means concentration and its gradient are contineous at exit which is contradictious to prerequisite of discontinuity at exit.The other appeared in the solving process where a largest L was adopted.So the results by Gao did not hold for the systems of definite L. To obtain the correct boundary condition,a general reactor of inflow section,reaction section and outflow section is essential.And to verify Eq.(2),the continuity of concentration at exit has been shown If one denys the continuity,he must obtain: C_x=L_C_x=L_+ Practically,Ineq.(6)may cause the system unstable At steady state,diffusion disappears outside point L.But inside L,due to molecular diffusion,C_x=L_+ must decrease,Cx_=Lincrease,and finally C_x=Land C_x=L_+ must equal with each other.Then C is contineous at exit when the system reaches steady state.So this work concludes that the Danckwerts Boundary Condition is both reasonable and correct.


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