Unformatted text preview: d discretization schemes for
.
∂z ' Exercise 15.29 (continued)
Analysis: (continued)
(3) Use of a spreadsheet with a finitedifference method where backward differences are used for
∂φ
and an Euler predictorcorrector method is used for the time derivatives, as discussed
∂z '
by D. D. Fry, Chem. Eng. Education, Fall 1990, pp 204207.
The PDESOL program is particularly convenient and easy to use, and was applied to this
exercise. The solution method selected from the many options provided by PDESOL was the
same as that described for TSA. Thus, the method of lines was used with Eqs. (15127) and (15∂φ
128). The
derivative was approximated by a 5point, biased, upwind, finitedifference
∂z '
approximation, and a stiff integrator, LODES, was used on the resulting system of ODEs with
time derivatives of φ and ψ. To obtain accuracy, 121 grid points were used over the 30 ft length
for the spatial domain in z'. For the time domain, an arbitrary 800 minutes was chosen for the
final time of desorption, with an output as a function of z' every 50 minutes. The initial
conditions for φ and ψ, given in the table above were inputted as text files, which were
interpolated by PDESOL to obtain values for the 121 grid points.
The resulting PDE program equations were as follows, where the symbols φ and ψ were replaced
by c and q, respectfully, and z' was replaced by x, with xL =0 and xU = 30 ft.
c_x = dxu(c,1)
'Uses the 5point, biased, upwind finite difference approximation
c_t = 98.5*c_x  206*(c  q)
q_t = 0.206*(c  q)
[email protected] = 0
[email protected] = lintc("c_init.txt",x) 'Calls the file c_init.txt for the initial values of c
[email protected] = lintc("q_init.txt",x) 'Calls the file q_init.txt for the initial values of q
The form of the two .txt files were as follows, using the above table:
c_init.txt
q_init.txt
x
0
1
2
3
4
etc.
etc.
15
30 c at t = 0
0.050
0.090
0.150
0.235
0.343
etc.
etc.
1.000
1.000 x
0
1
2
3
4
etc.
etc.
15
30 q at t = 0
0.044
0.081
0.137
0.217
0.321
etc.
etc.
1.000
1.000 Exercise 15.29 (continued)
Analysis: (continued)
The results of the calculations are given in the following tables and graphs, in increments of 50
minutes, where by 400 minutes the fronts have almost left the bed, indicating that the desorption
time is less than 400 minutes. Exercise 15.29 (continued)
Analysis: (continued)
Profiles for ψ = q/qF* during desorption: Exercise 15.29 (continued)
Analysis: (continued)
Plots of profiles for ψ = q/qF* and profiles for φ = c/cF during desorption: Exercise 15.29 (continued)
Analysis: (continued)
To determine from the above profiles the time for 90% of the benzene to be desorbed, compute
the average value of loading, ψ, for each of the desorption times, by numerical integration, using, ψ avg = 0.5ψ z '= 0 + 29
z ' =1 ψ at z ' + 0.5ψ z '= 30
30 Then, from above, the loading in lb benzene = ψavg (9.27 lb benzene at sat'n/ft3 )(47.1ft3)
From above, the desired value is 35.8 lb benzene. Using the above table of loadings as a
function of time and location i...
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 Spring '11
 Levicky
 The Land

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