IntroReactionDiffusion [read]

6 n 3 5 sol out18 3 2 findroot log 1 a a n 1 a n 1

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: IntroReactionDiffusion.nb 1 In[19]:= Plot Evaluate 1, Axes 0.6 n A 1 n 1 Φ2 Ξ 1 . sol , n1 Ξ, 0, 1 , PlotStyle Thick, Frame True, FrameLabel Style "Ξ z ∆", 14 , Style " ΧA cA C", 14 , AspectRatio 1 False, PlotRange 0, 0.6 0.6 0.5 Out[19]= ΧA cA C 0.4 0.3 0.2 0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Ξ z∆ This plot shows that the concentration at the catalytic surface Ξ=1 is finite xA 1 another simulation with a much larger value of In[20]:= n 3; Φ 50 ; sol Out[22]= 0.0664. Let us run FindRoot Log 1 A 0.00773113 A n 1 A n 1 Φ2 Log 1 0.6 n 1 , A, 0.2 7 IntroReactionDiffusion.nb 1 In[23]:= Plot Evaluate 1 n1 Ξ, 0, 1 , PlotStyle A 1 Thick, Frame Style "Ξ FrameLabel 0.6 n n 1 Φ2 Ξ 1 . sol , True, z ∆", 14 , Style " ΧA cA C", 14 , AspectRatio 1 0.6 0.5 cA C 0.4 0.3 Out[23]= ΧA 8 0.2 0.1 0.0 0.0 0.2 0.4 0.6 Ξ 0.8 1.0 z∆ In this case the surface concentration is almost zero: xA ∆ 0.0077 . Summary The previous calculations show that in the limit Φ2 , the dimensionless flux A 0. To understand 2 this limit we should recognize that for fixed AB and ∆, the limit Φ implies that k and consequently A k C 0. This suggest that in this limit we should scale the variables differently. Recall A we found a limiting value of A when Φ , which is given by (39) C AB Log 1 A n1∆ When Φ is finite the expression for Log 1 A c0 A (50) n1 C A is (see 43) n1 n1 ∆ Log 1 A kC C c0 A n1 (51) C AB Now to better understand the limiting process, let us introduce a new scaling for A that does not involve k. Equation (49) suggests the following scaling: A ∆ C AB . With this definition (50) becomes Log 1 A n1 Φ2 This equation can be rearranged to give A n1 Log 1 c0 A C n1 (52) IntroReactionDiffusion.nb 1 Log 1 A n1 c0 A 1 n1 A Log 1 C 9 (53) n1 2 n1 Φ The first term on the RHS of (52) is the limiting flux when Φ2 0. Equation (52) shows that the limiting value is an upper bound for A as the last term on the RHS is always positive. We can illustrate how In[24]:= n A 0 varies with Φ by solving (52) with cA C 0.6, n 3 as shown below 3; FluxEqn Ψ_ : Chop 1 . FindRoot A n In[26]:= 1 Log 1 A 0.6 n 1 1 A Log 1 n 1 Ψ 2 n 1 , A, ListPlot Table i, FluxEqn i , i, 0.1, 15, 0.1 , Joined True, PlotRange Frame True, PlotStyle Thick, FrameLabel Style "Φ", 14 , Style " A ", 14 .1 0, .4 , 0.4 A 0.3 0.2 Out[26]= 0.1 0.0 0 2 4 6 8 10 12 14 Φ 0 When cA C In[27]:= Out[27]= 0.6, n 3, the limiting value for A is FluxEqn 10 0.39034 The previous remarks illustrate the importance of choosing appropriate scaling variables for the dependent variables when examining limiting processes. If the scaling variables become infinite in a limiting process, the true nature of how the dependent variable behaves will be obscured. References The above problem is discussed in various textbook on mass transfer, though often the underlying assumptions are not carefully stated. Rarely is the species jump condition invoked. A good reference is R. B. Bird, W. E. Stewart, E. N. Lightfoot, Transport Phenomena, 2nd Edition John Wiley, 2005...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online