Instantaneous Point-source Solution - IfH

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Section 4: Interpretation of the similarity solution 18 4. Interpretation of the similarity solution Figure 4 shows the solution (35) in non-dimensional space. The animation also plots this solution, confirming that our solution is valid for a Fickian diffusion process. Comparing (35) with the Gaussian probability distribution reveals that (35) is the normal bell-shaped curve with a standard deviation, σ, of width σ 2 = 2Dt. (36) The concept of self similarity is now also evident: the concentration profile shape is always Gaussian. By plotting in non-dimensional space, the profiles also collapse into a single profile; thus, profiles for all times t > 0 are given by the result in the figure. [PgUp] [PgDn] [Back]
Section 5: Point-source solution with advection 19 5. Point-source solution with advection The final step is to substitute the coordinate transformation for the moving reference frame into the point source solution (35). In the moving coordinate system, this solution is ( ) M C(ξ, τ) = A √ 4πDτ exp − ξ2 . (37) 4Dτ Our coordinate transformation was ξ = x − (x 0 + ut) and τ = t. Substituting into (37) gives: ( M C(x, t) = A √ 4πDt exp − (x − (x 0 + ut) 2 ) . (38) 4Dt Figure 5 shows the schematic behavior of this solution for three different times, t 1 , t 2 , and t 3 . A second animation shows this solution in action. [PgUp] [PgDn] [Back]
Page 1 and 2: 1 UNIVERSITY OF KARLSRUHE Institute
Page 3 and 4: Section 1: Review of the advective
Page 5 and 6: Section 1: Review of the advective
Page 7 and 8: Section 2: Moving coordinate system
Page 9 and 10: Section 3: Similarity solution to t
Page 17: Section 3: Similarity solution to t
Page 21 and 22: Figures and tables 21 Table 1: Mole
Page 23 and 24: Figures and tables 23 Table 3: Mole
Page 25 and 26: Figures and tables 25 Figure 3: Def
Page 27 and 28: Figures and tables 27 Figure 4: Sel
Page 29: References Fischer, H. B., List, E.

Instantaneous Point-source Solution - IfH

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