Chemical Engineering Science 58 (2003) 5121 – 5124
www.elsevier.com/locate/ces
Shorter Communication
A new formulation of the Kremser equation for sizing mass exchangers
Uday V. Shenoya , Duncan M. Fraserb;∗
a Department
b Department
of Chemical Engineering, Indian Institute of Technology, Bombay 400 076, India
of Chemical Engineering, University of Cape Town, Private Bag, Rondebosch 7701 South Africa
Received 16 July 2002; received in revised form 18 August 2003; accepted 19 August 2003
1. Introduction
The Kremser equation is the classical method for determining the number of stages, N , in counter-current mass
exchange units, when both the operating line and the equilibrium line are linear. The traditional form in which this
equation is expressed is as follows (Treybal, 1981; Hallale
& Fraser, 1998):
(yin −mxin −b)
1
1
ln
1−
+
(yout −mxin −b)
A
A
N=
for A = 1;
ln A
(1)
N=
yin − yout
yout − mxin − b
for A = 1;
(2)
where A is the absorption factor [A = L=(mG)]. The notation
employed is shown in Fig. 1.
environment, both the form of this equation and the singularity it contains lead to diFculties. Szitkai, Lelkes, Rev, and
Fonyo (2001, 2002) proposed ways of handling the singularity when using available solvers for optimising MENs, in
order to obviate the use of conditional statements. For each
exchanger to be sized, their formulations involve the introduction of a number of restrictions and arbitrary intervals,
as well as binary variables. Msiza (2002) also proposed a
diGerent formulation that further reduces the problem to no
restrictions and only one continuous variable per exchanger
to be sized.
In this short communication we propose a new way of
formulating the Kremser equation that leads to a ratio of
two logarithmic mean terms. While this form of the equation
has a similar diFculty to those discussed above when the
components of the logarithmic means are identical (which
happens when A = 1, as before), we will show that this is
readily overcome by using one of the proposed logarithmic
mean approximations.
yin
Operating line
(Slope = L/G)
∆y
y
Equilibrium line
y* = mx + b
y*out
∆y*
yout
y*in
2. New formulation
∆y1
∆y2
The linear equilibrium relation
∗
y = mx + b
(3)
and the material balance equation
G(yin − yout ) = L(xout − xin )
xin
x
xout
Fig. 1. y–x plot for counter-current mass exchanger.
When using this equation for sizing mass exchangers, particularly when optimising mass exchange networks (MENs)
in a mixed integer non-linear programming (MINLP)
∗ Corresponding author. Tel.: +27-21-650-2515;
fax: +27-21-689-7579.
E-mail address: dmf@chemeng.uct.ac.za (D. M. Fraser).
0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2003.08.007
(4)
may be substituted in Eq. (1) to give the following alternative form for the Kremser equation when A = 1 (McCabe,
Smith, & Harriott, 2001):
∗
∗
)=(yout − yin
))
ln((yin − yout
for A = 1:
N=
(5)
∗
∗
ln((yin − yout )=(yout − yin ))
This may be re-arranged to give
∗
∗
ln((yin − yout
)=(yout − yin
))
N=
∗ ) − (y
∗
(yin − yout
out − yin )
∗
∗
(yin − yout ) − (yout
− yin
)
×
∗ − y ∗ )) ;
ln((yin − yout )=(yout
in
(6)
5122
U. V. Shenoy, D. M. Fraser / Chemical Engineering Science 58 (2003) 5121 – 5124
which is no more than simply:
mean temperature diGerence when the stream mean
heat capacities (and hence the temperature diGerences)
are equal.
Chen (1987) proposed Nrst:
∗
∗
− yin
))
log–mean((yin − yout ); (yout
∗
∗ ))
log–mean((yin − yout ); (yout − yin
N=
for A = 1:
(7)
Referring to Fig. 1, this may be expressed as
log–mean(Jy; Jy∗ )
N=
log–mean(Jy1 ; Jy2 )
for A = 1;
(8)
where Jy1 and Jy2 are the driving forces at either end of
the mass exchanger, Jy is the diGerence in compositions of
the rich stream and Jy∗ is the diGerence in the equilibrium
compositions across the exchanger.
In this form, when A = 1, it leads to Jy = Jy∗ and also
to Jy1 = Jy2 . In this case, Eq. (8) reduces to:
N=
Jy
Jy1
for A = 1:
(9)
Eq. (8) leads to numerical diFculties when A = 1, i.e. if the
slope of the operating line (L=G) and the slope of the equilibrium line (m) are identical. While this is unlikely to arise
in practice, it causes diFculties in mathematical programming models and therefore it is safer to avoid this possible
problem by using one of the proposed approximations for
the logarithmic mean (Underwood, 1970; Paterson, 1984;
Chen, 1987).
3. Approximations to the logarithmic mean
Underwood (1933, 1970) proposed the following form
for calculating the logarithmic mean temperature diGerence, where the individual temperature diGerences are 1
and 2 :
3
1=3
1 1=3
(10)
Underwood = 2 ( 1 + 2 ) :
Underwood pointed out that this approximation is accurate to about 1% even when the ratio 1 = 2 is as large
as 27.
Paterson’s (1984) logarithmic mean approximation has
the following form:
Paterson
=
1
3 AM
+
2
3 GM ;
(11)
where AM and GM are, respectively the arithmetic and
geometric means of 1 and 2 . Paterson indicated that
the accuracy of this approximation was within 1% for a
ratio of 1 = 2 equal to 10 (which he considered a large
ratio for temperature diGerences in a heat exchanger). This
approximation has been used (Colberg & Morari, 1990;
Shenoy, 1995) when carrying out rigorous area targeting
for heat exchanger networks using a non-linear programming (NLP) transshipment model, in order to avoid the
diFculties associated with singularities in the logarithmic
Chen1
=
1=3 2=3
AM GM
(12)
and then a modiNcation of the Underwood approximation
which is as follows:
Chen2
= [ 12 (
0:3275
1
+
0:3275 1=0:3275
)]
:
2
(13)
The Nrst Chen approximation has been used in optimisation models for heat exchanger network synthesis (Yee &
Grossmann, 1990). Chen did not calculate accuracies for
these two methods, but showed that his second method was
more accurate than Paterson’s over a range of 1 = 2 from
1.5 to 10. As pointed out by Paterson (1987), this latter
equation is slightly less accurate than Underwood’s for
1 = 2 ratios around 1.5, but more accurate when they are
around 10.
The generalised form of both the Underwood approximation (Eq. (10)) and the second Chen approximation (Eq.
(13)), when used to simplify Eq. (8), leads to the following
elegant expression for the number of stages:
1=n
Jy n + Jy∗ n
N=
;
(14)
Jy1n + Jy2n
where n = 1=3 according to Underwood and n = 0:3275
according to Chen.
Note that all the approximations were originally proposed
in the context of using these means for the logarithmic mean
temperature diGerence in heat exchanger design calculations,
where the temperature diGerences seldom vary by more than
an order of magnitude. This is not the case in mass exchanger
design, where the composition diGerences may vary by more
than two orders of magnitude.
4. Results and discussion
We have applied our formulation of the Kremser equation
(Eq. (8)) to 71 diGerent mass exchangers in eight diGerent
networks designed by Hallale (1998), using both the true
logarithmic mean and each of the proposed approximations.
The worst results (those with the largest deviations from the
true value) are shown in Table 1.
The second last row in Table 1 also shows a case where
the slopes of the operating and equilibrium lines are equal,
in which case all the results are exact, but the alternative
form of the Kremser equation (Eq. (2)) had to be used to
obtain the true number of stages (because A = 1:00 exactly).
In all the other 58 cases examined the errors were less than
those in the third last row of Table 1.
It is notable that Chen’s Nrst approximation performed
considerably worse than all the others, almost always giving
U. V. Shenoy, D. M. Fraser / Chemical Engineering Science 58 (2003) 5121 – 5124
5123
Table 1
Comparison of N calculated by diGerent approximations of the Kremser equation
Nlog–mean
NPaterson
Error
(%)
NChen1
Error
(%)
NUnderwood
Error
(%)
NChen2
Error
(%)
Jy=Jy∗
Jy1 =Jy2
1.879
4.945
12.176
12.088
18.944
3.016
8.295
1.771
13.150
6.651
3.079
12.463
6.000
1.617
4.366
10.829
11.009
17.596
2.844
7.979
1.725
12.877
6.526
3.030
12.266
6.000
−13:97
−11:71
−11:06
−8:93
−7:12
−5:69
−3:81
−2:60
−2:08
−1:88
−1:59
−1:58
0.00
2.639
6.695
16.233
15.300
22.937
3.520
9.247
1.907
14.008
7.044
3.235
13.092
6.000
40.42
35.38
33.32
26.57
21.08
16.72
11.48
7.66
6.52
5.92
5.06
5.05
0.00
1.738
4.649
11.492
11.551
18.285
2.933
8.146
1.750
13.024
6.593
3.056
12.372
6.000
−7:51
−5:99
−5:62
−4:44
−3:48
−2:74
−1:80
−1:22
−0:96
−0:87
−0:73
−0:73
0.00
1.757
4.712
11.649
11.692
18.484
2.958
8.209
1.758
13.098
6.628
3.070
12.434
6.000
−6:53
−4:72
−4:33
−3:28
−2:43
−1:92
−1:04
−0:78
−0:40
−0:34
−0:29
−0:23
0.00
22.65
2.85
1.52
1.47
1.26
3.83
1.53
6.41
1.25
1.54
2.45
1.25
1.00
352.22
176.55
157.33
108.00
76.00
57.16
34.76
26.88
19.50
17.89
15.85
15.60
1.00
Average of the
absolute error (%)
1.55
4.67
an overestimation of the number of stages. The other approximations almost always gave an underestimation of
the number of stages. As may be seen in the last row of
Table 1, Chen’s second approximation performed best of
all, followed by Underwood’s (average error about 50%
higher) and then Paterson’s (average error about three times
larger).
The last two columns of Table 1 show the ratios of the
composition diGerences and the ratios of the driving forces.
The ratios of the driving forces are seen to be much larger
than the ratios of the composition diGerences, due to close
approaches to equilibrium. The error in using the approximations is a strong function of the ratio of the driving forces,
Jy1 =Jy2 , and not such a strong function of the ratio of the
composition diGerences, Jy=Jy∗ . The largest errors in using the approximations occur at large ratios of the driving
forces.
It should also be noted that the overall errors are smaller
than might be expected from the individual logarithmic mean
approximation errors. This is because the equation uses a
ratio of two approximations, each of which is generally an
over estimation of the true logarithmic mean. For example,
in the worst case given in the Nrst row of Table 1, for the
Underwood approximation, the numerator is 1.2% too high
and the denominator is 9.4% too high, whereas the ratio is
7.5% too low.
5. Conclusion
In order to avoid singularities in process synthesis and optimisation models, it is recommended that
mass exchangers be sized using the new formulation
of the Kremser equation as given by Eq. (8) in conjunction with either the Underwood or the second Chen
0.76
0.53
approximation (in other words, Eq. (14) with n = 1=3
or n = 0:3275).
Acknowledgements
The authors wish to acknowledge the Nnancial support
from the University of Cape Town Visiting Scholar’s Fund
that made this work possible.
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