1. Presented By
Allen Meshach
Archana Yadav
Ashwin John Oommen
Edison P Kuriakose
George T Cherian

2. The Problem
 Given is a process that involves the following set of hot
and cold streams
Stream Fcp ( kW/K ) Tin (K) Tout (K)
H1 20 700 420
H2 40 600 310
H3 70 460 310
H4 94 360 310
C1 50 350 650
C2 180 300 400

3.  The following utilities are available for satisfying
heating and cooling requirements:
Utility Temperature( Cost($/KW-yr) Maximum
K) available(kW)
Fuel 750 120 infinity
HP steam 510 90 1000
LP steam 410 70 500
Cooling water 300-325 15 infinity

4. Q. The goal is to predict the minimum utility cost for a
heat exchanger network that has a minimum
temperature approach of 10K. Stream H1 is not to be
allowed to exchange any heat with stream C1.

5. Formulation
 We are asked to find a way of exchanging heat among
the process streams and the utilities so that the target
temperatures are met for the process streams and the
total utility cost is minimized.
 The data of the problem are displayed in the table 1
below, where heat contents of the hot and cold
processing streams are shown at each of the
temperature intervals which are based on the inlet,
and highest and lowest temperatures given.

7. The flow of heat can be expressed
by the Heat Cascade Diagram

8.  The usefulness of the heat cascade diagram in Fig. 1 is that
it can be modeled as a transshipment problem which we
can formulate as a linear programming problem.
 In terms of the transshipment model hot streams are
treated as source nodes, and cold streams as destination
nodes.
 Heat can then be regarded as a commodity that must be
transferred from the sources to the destinations through
some intermediate “warehouses” which correspond to the
temperature intervals which guarantee feasible heat
exchange.

9.  When not all of the heat can be allocated to the
destinations (cold streams) at a given temperature
interval, the excess is cascaded down to lower
temperature intervals through the heat residuals.

10.  The transshipment model for predicting the minimum
utility cost given an arbitrary number of hot and cold
utilities, and hot and cold streams, can be formulated
as follows.
Hk = {i | hot stream i supplies heat to interval k} (1)
Ck = {j | cold stream j demands heat from interval k} (2)
Sk = {m | hot utility m supplies heat to interval k} (3)
Wk = {n | cold utility n extracts heat from interval k} (4)
S = set of hot utilities (5)
W = set of cold utilities (6)

12.  The minimum utility cost for a given set of hot and
cold processing streams can be formulated as the
following LP

13.  In the above formulation it is easy to impose upper
limits on the heat loads that are available from some of
the utilities (eg. Maximum heat from high pressure
steam).
 On the other hand, it is not possible to use it as it is in
order to enforce constraints that exclude matches
between any given pair of hot and cold streams.
 This could be due to the fact that the streams are too
far apart, or because of other operational
considerations such as control, safety or start-up.
 For this reason, Papoulias and Grossmann (1983) have
formulated an expanded transshipment model which
we present next.

14.  The condensed LP transshipment model in (7) to (11) implicitly
assumes that any given pair of hot and cold streams can exchange heat
since there is no information as to which pairs of streams actually
exchange heat.
 In order to develop an LP formulation where we do have that
information, we can consider within each temperature interval, a link
for the heat exchange between a given pair of hot and cold
streams, where the cold stream is present at that interval, and the hot
stream is either also present, or else it is present in a higher
temperature interval.
 For this reason, within an interval k we will define the variable Qijk to
denote the heat exchange between hot stream i and a cold stream j.
 Whereas in the condensed LP transshipment model we assigned a
single overall heat residual Rk exiting at each temperature interval, in
the expanded transshipment model we will assign individual heat
residuals Rik, Rml for each hot stream i and each hot utility m that are
present at or above the temperature interval k.
 Fig. 3 illustrates that above ideas for an interval k where we consider a
hot stream i and a cold stream j.

17. The new expanded LP Transhipment Model is the following

18. Some Assumptions
 In order to forbid a match between hot i and cold j we
need to set Qijk=0 for all intervals k. Alternatively, we
simply delete these variables from the formulation.
 In order to impose a match between hot stream i and
cold stream j, we specify that the total heat exchange,
which is the sum of Qijk over all intervals, lies within
some specified lower and upper bounds

19. Result
 GAMS is used to find the solution in this problem.
 The input file for GAMS consist of three parts. The
data are entered in the first section of the program.
 The second part uses heat contents of the streams.
Here we make use of the LOOP statement of GAMS in
order to repeatedly execute a set of commands.
 Finally, the third part of the program defines and
solves the model.

20. The optimal solution shown in the GAMS output file
indicates that by disallowing the match between stream
H1 and stream C1 the minimum utility cost of 570,000 $
per year can be achieved with the following loads for
the utilities:

21. Thank You