Unformatted text preview: t of temperatures from the energy balance
equations as in the sumrates method, a good procedure to simultaneously compute a new set of
total vapor flow rates and a new set of temperatures from a combination of the energy balance,
sum rates, and equilibrium equations. This procedure is as follows:
(a) Combine the energy balance equation (105) with the total mole balance (106) to eliminate
variables in Lj so as to obtain the following equation in terms of Vj and Tj , assuming that mole
fraction compositions have just been calculated:
H j = H j V j , Tj = V j +
V j +1 + j
m =1 j −1
m =1 Fm − U m − Wm − V1 hL j −1 + V j +1hV j+1 + Fj hFj −
(1) Fm − U m − Wm − V1 + U j hL j − V j + Wj hV j − Q j = 0 Instead of using the sumrates equation (1033) in terms of total liquid flow rates, use the
following similar equation in terms of the total vapor flow rates: V j( k +1) = V j( k ) C
i =1 yi , j Combining this equation with equation (10  2)
V j( k +1) = V j( k ) C
i =1 Ki , j xi , j Writing this equation as a sums function, S j , gives:
S j = S j V j , Tj = V j( k +1) − V j( k ) C
i =1 Ki , j xi , j = 0 (2) Equations (1) and (2) are the indexed equations for computing new sets of Vj and Tj , where in
Eq. (1), Vj = Vj(k+1) and all Tj implicit in the Kvalues and enthalpies are Tj(k+1). Exercise 10.15 (continued)
Analysis: (continued)
(b) The truncated Taylor's series expansions for Eqs. (1) and (2) are: 0 = H (j k ) + 0= S (k )
j + (k ) ∂H j ∆Tj −1 + ∂Tj −1
(k ) ∂S j ∆V j + ∂V j ∂H j (k ) ∆Tj + ∂Tj ∂S j
∂Tj ∂H j
∂Tj +1 (k ) ∆Tj +1 + (k ) ∆Tj (4) The derivatives for Eq. (3) are: ∂H j = Vj + ∂Tj −1
∂H j
∂H j = V j +1 ∂Tj +1
∂V j Fm − U m − Wm − V1 m =1 = V j +1 + ∂Tj ∂H j j −1 j Fm − U m − Wm − V1 m =1 ∂hV j+1
∂Tj +1 = hL j −1 − hV j ∂H j
∂V j +1 = hV j+1 − hL j The derivatives for Eq. (4) are:
∂S j
∂V j
∂S j
∂Tj =1
= −V j( k ) C ∂Ki , j i =1 ∂Tj xi , j ∂hL j −1
∂Tj −1
∂hL j
∂Tj − V j + Wj ∂hV j
∂Tj ∂H j
∂V j (k ) ∆V j + ∂H j
∂V j +1 (k ) ∆V j +1 (3) Exercise 10.15 (continued)
Analysis: (continued)
(c) Consider a simple case of 3 equilibrium stages. The set of 3 equations each for Eq. (3) and
Eq. (4), will form a block tridiagonal matrix structure if the equations and correction variables
are ordered as shown below. More details on this method are given by J. F. Tomich, AIChE
Journal, 16, 229 (1970).
The following is the incidence matrix for the Jacobian matrix: ∆V1 ∆T1 S1 x x H1 x S2
H2 ∆V2 ∆T2 x x x x x
x x
x H3 x ∆T3 x x x S3 ∆V3 x x x Exercise 10.16
Subject:
Effect of interlinking streams of a thermally coupled distillation system on the
Jacobian matrix structure.
Given: Thermally coupled system of Fig. 10.31. Solution of the equations by the NewtonRaphson method.
Find: Whether the matrix equation retains a block tridiagonal structure.
Analysis: Consider the system shown below, where the stages in the first column are numbered
from 12 to 15 and the stages in the second column are numbered from 1 to 11 and from 16 to 19.
The component material balance equations, similar to Eq. (1058, around stages 4, 12, 11, and
16, respectively are:
M i ,4 = li ,4 (1 + s4 ) + υi ,4 − li ,12 − li ,3 − υi ,5 = 0
M i ,12 = li ,12 + υi ,12 − li ,4 s4 − υi ,13 = 0
M i...
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 Spring '11
 Levicky
 The Land

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