transshipment lp model for minimizing the utility cost in a heat exchanger network

8/18/2019 Transshipment LP Model for Minimizing the Utility Cost in a Heat Exchanger Network

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Transshipment LP Model for Minimizing

the Utility Cost in a Heat Exchanger Network

Nikolaos V. Sahinidis and Ignacio E. GrossmannDepartment of Chemical Engineering Carnegie Mellon University

1.

Problem Statement

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 420H2 40 600 310H3 70 460 310H4 94 360 310C1 50 350 650C2 180 300 400

The following utilities are available for satisfying heating and cooling requirements:Utility Temperature(K) Cost($/KW-yr) Maximum available(kW)Fuel 750 120

HP steam 510 90 1000LP steam 410 70 500

Cooling water 300-325 15

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.

2.

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 Table 1, 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.

Thot Tcold Heat ContentsIntervalH1 H2 H3 H4 C1 C2

700 6901 2000 3000

600 590

2 1800 3600 4500

510 500

3 1000 2000 2500

460 450

4 800 2000 3500 2500

410 400

5 2000 3500 2500 9000

360 3506

310 3002000 3500 4700 9000

Table 1: Heat contents of process streams

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The flows of the heat contents can be represented in the heat cascade diagram of Fig. 1. Here

the heat contents of the hot streams are introduced in the corresponding intervals, while the

heat contents of the cold streams are extracted also from their corresponding intervals. The

variables R 1 to R 5 represent heat residuals, while QFuel, QHP, QLP, QW represent the heating and

cooling loads respectively.

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. 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.

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. First, we

consider that we have K temperature intervals that are based on the inlet temperatures of the

process streams and of the intermediate utilities whose inlet temperatures fall within the range

of the temperatures of the process streams. We assume that the intervals are numbered from the

top to the bottom. We can then define the following index sets:

H k = {i | hot stream i supplies heat to interval k } (1)

C k = { j | cold stream j demands heat from interval k } (2)

S k = {m | hot utility m supplies heat to interval k } (3)

W k = {n | cold utility n extracts heat from interval k } (4)

S = set of hot utilities (5)

W = set of cold utilities (6)

When we consider a given temperature interval k, we will have the following known

parameters and variables (see Fig. 2):

Known parameters:C

jk

H

ik QQ , heat contents of hot streams i and cold stream j in interval k

nm cc , unit costs of hot utility m and cold utility n

Variables:W

n

s

m QQ , heat loads of hot utility m and cold utility n

k R heat residual exiting interval k

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Fig. 1: Transshipment model for heat flows

k H i

H

ik Q k C j

C

jk Q

k S m

S

mQ k W n

W

nQ

Fig. 2: Heat flows in interval k.

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The minimum utility cost for a given set of hot and cold processing streams can then be

formulated as the following LP (Papoulias and Grossmann, 1983):

(11) 0

(10) 1,2,3...Kk 0

(9) 0 ,0Q

(8) 1,2,3...Kk

..

(7) min

0

Sm

1

K

k

W

n

Ck j

C jk

Hk i

H ik

Wk n

W n

W m

S mk k

W

n

W n

n

S

m

S m

m

R R

R

Q

QQQQ R R

t s

QcQc

The above model is known in the literature as the condensed transshipment model. Its objective

function represents the total utility cost, while the K equations are heat balances around each

temperature interval k. If R k =0 at the optimal solution, then the temperature interval kcorresponds to a pinch point.

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 modelwhich we present next.

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, alink 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. Likewise, we can define similar variables for the

exchange between process streams and utilities. Whereas in the condensed LP transshipment

model we assigned a single overall heat residual R k exiting at each temperature interval, in the

expanded transshipment model we will assign individual heat residuals R ik , R ml 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.

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H ik Q

C jk Qijk Q

1, k i R

k i R ,

Fig. 3: Heat exchange between hot and cold stream for expanded transshipment model.

Note that using this representation, we allow to a cold stream j to exchange heat in interval k not only with a hot stream which is available in that interval, but also with a hot stream i which

is available at a higher interval simply because hot stream i is transferring heat to interval k

through its residual Rik . Let us now define some notation:'k H = {i | hot stream i supplies heat to interval k or at a higher interval} (12)

'k S = {m | hot utility m supplies heat to interval k or at a higher interval} (13)

The index sets C k , W k are defined as in (2) and (4), respectively.

We also have the following new variables (see Fig. 4):

Qijk exchange of heat of hot stream i and cold stream j at interval k (14)

Qmjk exchange of heat of hot utility m and cold stream j at interval k (15)

Qink exchange of heat of hot stream i and cold utility n at interval k (16)

Rik heat residual of hot stream i exiting interval k (17)

Rmk heat residual of hot utility m exiting interval k (18)

H

ik Q

S

mQ

C

jk Q

W

nQ

Fig. 4: Heat flows in interval k for expanded transshipment model.

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The variables , and the parameters are identical to the ones of the

condensed transshipment model. In contrast to the compact LP transshipment model (7) to (11)where we simply did and overall heat balance around each temperature level, in this case wehave to consider balances at the following points within each temperature interval (points A, B,C and D of Fig. 4): a)for the hot process and utility streams at the interval nodes that relate theheat content, residuals and heat exchanges, b) for the cold process and utility streams at the

destination nodes than relate the heat content and heat exchanges. In this way the expanded LPtransshipment model is the following:

W

n

S

m QQ , nmC

jk

H

ik C C QQ ,,,

(25) 0

(24) 0,,,,,,

(23) ...3,2,1

(22) ...3,2,1

(21) ...3,2,1

(20) ...3,2,1

..

(19) min

0

'

''

'1,

'1,

iK i

W

n

S

mink mjk ijk mk ik

k

W

n

H i

ink

k

C

jk S m

mjk H i

ijk

k

C j

S

mmjk k mmk

k

H

ik

W j

ink

C j

ijk k iik

S m

W

n

W n

n

S

mm

R R

QQQQQ R R

W nK k QQ

C jK k QQQ

S mK k QQ R R

H iK k QQQ R R

t s

QcQc

k

k k

k

k k

The size of this LP is obviously larger than the one in (7) to (11). The importance of the

expanded formulation is however the fact that we can easily specify constraints on the

matches:

a) 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.

b) 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:

(26) 1

K

k

U

ijijk

L

ij QQQ

We can also simply specify a fixed value for the sum in (26).

The expanded transshipment model (19) to (26) can be extended to an MILP transshipment

model which, for a given utility consumption, can rigorously predict the number of fewest

units in the network, as well as the stream matches that are involved in each unit, and the

amount of heat that they must exchange. This is simply done by introducing 0-1 variables for

each potential match as discussed in Papoulias and Grossmann (1983). Finally, the design of

heat exchanger networks with several factors (utility cost, number of units, heat exchanger area)

being considered simultaneously has also been addressed in the literature. An MINLP modeladdressing this problem is presented in another study (Yee and Crossmann, 1991)

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3. Result

The input file for GAMS consist of three parts. The data are entered in the first section of the

program. Notice that the temperature intervals need not be entered since they are calculated in

the second part of the program along with the 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.

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:

fuel 3900 kW

high pressure stream 500 kWcold utility 3800 kW

Also, from the solution we can see that there is no pinch point for this problem since there are

nonzero residuals for all the temperature intervals.

4. Suggested exercises

(1) It is interesting that by imposing constraints on matches the minimum utility cost may or

may not change. Consider the following (un)restricted cases for the given problem:(a)

No restrictions for heat matches.

(b) Stream H1 is not allowed to exchange heat with stream C2.

(c) Stream H1 is not allowed to exchange heat with either stream C1 or stream C2.

(2) Show that the expanded LP transshipment model in (19)-(25) reduces to the LP model

(7)-(11) if no constraints on matches are imposed.

5. References

Papoulias, S.A., and I.E. Grossmann, “A Structural Optimization Approach in Process

Synthesis – Part II: Heat Recovery Networks”, Computers and Chemical Engineering, 7(6),

707-721(1983).

Yee, T.F., and I.E. Grossmann, “Simultaneous Optimization Model for Heat Exchanger

Network Synthesis”, this Volume (1991).

transshipment lp model for minimizing the utility cost in a heat exchanger network

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