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remarkable convergences properties. The shell-side pressure drop
equation is based on the Bell–Delaware method, and the model
for the tube-side includes the estimation for end effects. However,
this algorithm does not explicitly take into account some of the
geometric and operational constraints regularly imposed for ex-
changer design, and it only considers the pressure drops as optimi-
zation variables. Therefore, sub-optimal design solutions are
typically obtained.
Chaudhuri and Diwekar [8] used simulating annealing for the
optimal design of heat exchangers, and developed a command
* Corresponding author. Tel.: +52 461 611 7575x139; fax: +52 461 611 7744.
E-mail address: arturo@iqcelaya.itc.mx (A. Jiménez-Gutiérrez).DPT ¼ KTAðhTÞ
n
ð2Þ
Eq. (2) accounts for pressure drops in straight tubes and in tube
ends. For turbulent flow on the shell-side flow
DPS ¼ KSAðhSÞ
m ð3Þ
Eq. (3) is based on the Bell–Delaware method. The compact formu-
lation was obtained after an analytical treatment of the original
equations, and therefore it has the same degree of applicability as
the original Bell–Delaware method. The definitions of KS, KT, m
and n, and how these parameters depend on the geometric param-
eters of the exchanger and the fluid physical properties, along with
the design algorithm used in this work, are shown by Serna and
Jimenez [10]. The main steps of the design algorithm are included
in Appendix A.
2.2. Constraints for a feasible design
The design of a heat exchanger involves a number of constraints.
For convenience, the constraints may be classified into operating
constraints and geometric constraints. Some of the operating con-
straints are maximum allowable pressure drops and velocities for
both sides of the exchanger. Geometric constraints include maxi-
mum shell diameter, maximum tube length, minimum and maxi-
mum ratio of baffle spacing to shell diameter, and minimum and
maximum ratio of cross-flow area to area in a window.
The maximum pressure drops (DPT,max, DPS,max) depend on the
external pumps, and set upper bounds for the operating pressure
drops
DPT 6 DPT;max ð4Þ
DPS 6 DPS;max ð5Þ
Upper bounds on the velocities for both the tube-side and the shell-
side prevent erosion and flow-induced tube vibration, while lower
bounds prevent fowling. These constraints are considered in the
model,
vt;min 6 vt 6 vt;max ð6Þ
vs;min 6 vs 6 vs;max ð7Þ
Sinnott [1] recommends velocities for liquids from 1 to 2.5 m s1
on
the tube-side and 0.3–1 m s
1
on the shell-side.
Upper limits of the shell diameter and the tube length are part
of the primary geometrical constraints
Ds 6 Ds;max ð8Þ
LTT 6 LTT;max ð9Þ
As shown in Example 3, the algorithm can also be restricted to the
use of standard dimensions for these variables.
For baffles, close spacings leads to higher heat transfer coeffi-
cients but at the expense of higher pressure drops. On the other
hand, wide baffle spacings result in bypassing and reduced cross-
flow, with a decrease in the heat transfer coefficient. Therefore,
constraints on Rbs are set
Rbs;min 6 Rbs 6 Rbs;max ð10Þ
Typical values of Rbs,min and Rbs,max are 0.2 and 1.0, respectively.
Bounds on the ratio of cross-flow area to area in a window, Sm/
Sw, are also considered
ðSm=SwÞmin 6 Sm=Sw 6 ðSm=SwÞmax ð11Þ
Typical values are 0.8 and 1.4 for lower and upper bounds,
respectively.
2.3. Objective function
The objective function consists of minimizing the total annual