Chapter_8

8-i

8.1 INTRODUCTION ..................................................................................................................................................... 2

8.2 PROBLEM DEFINITION AND DATA EXTRACTION ............................................................................................. 4

8.2.1 Mass-Transfer Problem..............................................................................................................................4

8.2.2 Corresponding Concentration Scales and Minimum Approach Concentrations............................5

8.2.3 Capacity Flowrates...................................................................................................................................12

8.3 MINIMUM EXTERNAL MSA DUTY WITHOUT MASS INTEGRATION ............................................................ 15

8.3.1 Limiting Concentration Profiles.............................................................................................................15

8.3.2 Minimum External MSA Duty.................................................................................................................17

8.4 MINIMUM EXTERNAL MSA DUTY WITH MASS INTEGRATION ................................................................... 19

8.4.1 Process-Stream and Process-MSA Composite Curves.......................................................................19

8.4.1.1 Graphical Approach: Process-Stream and Process-MSA Composite Curves ...................................... 19

8.4.1.2 Tabular Method: Concentration-Interval Diagram (CID) .................................................................... 30

8.4.2 Minimum External MSA Duty.................................................................................................................38

8.4.2.1 Graphical Method: Process-Stream and Process-MSA Composite Curves ......................................... 38

8.4.2.2 Tabular Method: Concentration -Interval Diagrams (CIDs) ................................................................ 42

8.4.3 Utility Placement: Grand Composite Curve........................................................................................47

8.5 DESIGN TOOLS: REPRESENTING MASS-EXCHANGE NETWORKS................................................................. 52

8.5.1The Grid Diagram......................................................................................................................................52

8.5.2 The Mass-Content Diagram....................................................................................................................53

8.6 PRELIMINARY MASS-EXCHANGE NETWORK DESIGN ................................................................................... 57

8.6.1 Pinch Subnetworks....................................................................................................................................63

8.6.2 Minimum Number of Mass-Exchange Units.........................................................................................66

8.6.3 Maximize Exchanger-Mass Loads.........................................................................................................66

8.6.4 Capacity-Flowrate Rule for Match Feasibility....................................................................................71

8.6.5 Matches Away from the Pinch ................................................................................................................77

8.6.6 Stream Splitting .........................................................................................................................................81

8.7 NETWORK EVOLUTION ...................................................................................................................................... 88

8.8 SUMMARY............................................................................................................................................................ 92

NOMENCLATURE ....................................................................................................................................................... 95

8-ii

REFERENCES .............................................................................................................................................................. 97

8-1

Chapter 8: Mass Integration through Pinch Technology: Analysis and Synthesis of Mass-

Exchange Networks

8-2

8.1 Introduction

This chapter presents the basic principles for analyzing systems of contaminant-rich

process streams. These process streams require treatment to reduce their contaminant

concentrations to target levels in unit operations requiring mass-separating agents (MSAs). We

develop strategies to use first process MSAs available at little cost from within the process. If

necessary, we then use external MSAs purchased or available from the process at higher costs.

We also present an example to compare the maximum and minimum external MSA without and

with mass integration, respectively.

This chapter begins by defining the mass-transfer problem between rich and lean streams

and discussing the thermodynamic relationships and data required (Section 8.2). Next, we

analyze the system without allowing integration between rich process streams and process MSAs

to identify the maximum duty of external MSAs (Section 8.3). Finally, we minimize external

MSA requirements for an integrated system that maximizes integration between rich process

streams and process MSAs (Section 8.4). For these analyses, we introduce the limiting

concentration profile, the process-stream and process-MSA composite curves, the concentration-

interval diagram, and the application of the pinch concept to mass integration.

Sections 8.5 to 8.7 turn to designing networks of mass-exchange units (e.g., extractors

and strippers) and explains how to construct a network that will meet targets for process and

external MSAs with the fewest number of mass-exchange units. Before developing mass-

8-3

exchange networks, this chapter describes the tools we use in designing networks and introduces

a new concept, the mass-content diagram (Section 8.5). From there, we will turn to the mass-

exchange network. Constructing a network for this system is a two-step process:

1. Design a preliminary mass-exchange network guaranteed to transfer the contaminant

from process streams to MSAs (Section 8.6).

2. Simplify the preliminary network to reduce the number of mass-exchange units

through an evolutionary process (Section 8.7).

8-4

8.2 Problem Definition and Data Extraction

8.2.1 Mass-Transfer Problem

This section introduces the analysis of systems of contaminant-rich and contaminant-lean

streams, and the data required through an illustrative example. Table 8.1 lists the stream data for

the two rich process streams of Example 8.1 taken from a dephenolization problem (El-Halwagi,

1997). Here, Gi is the flowrate of process stream i, and ysupplyi and ytarget i are the supply and target

concentrations (i.e., mass fractions) of the contaminant, respectively.

Table 8.1. Rich-stream data for Example 8.1.

Stream

i

Gi

(kg/s)ysupplyi ytargeti

R1 2.0 0.050 0.010

R2 1.0 0.030 0.006

Table 8.2 shows the stream data for two process MSAs available at little cost for

Example 8.1. In this case, Lj refers to the upper bound on the flowrate of process MSA j, and

xsupplyj and xtarget j are the supply and target concentrations of the contaminant, respectively.

However, the concentration scales, y (for contaminant-rich process streams) and xj (for MSA j)

are not equivalent. Below, we introduce the concept of corresponding concentration scales to

relate the MSA-concentration scales (xj) to the rich-stream concentration scale (y).

8-5

Table 8.2. Process-MSA data for Example 8.1.

Process MSA

j

Lj

(kg/s)xsupplyj xtargetj

S1 5.0 0.005 0.015

S2 3.0 0.010 0.030

8.2.2 Corresponding Concentration Scales and Minimum Approach Concentrations

The concept of corresponding concentration scales defines quantitative relationships

between the concentrations of all streams (process streams and MSAs) and incorporates

thermodynamic and other constraints into the stream data (El-Halwagi, 1997). For this example,

Equation 8.1 defines a linear relationship between the process-MSA scales, xj, and the rich-

stream concentration scale, y.

jjxay = (8.1)

We incorporate thermodynamic and other constraints into the stream data by including a

minimum approach concentration (MAC) for mass transfer, e j. In order for mass transfer to occur

from a process stream, at concentration y, to MSA j, the concentration of the contaminant in the

MSA, xj, must be e j below that defined in Equation 8.1. Equation 8.2 is the equivalent rich-

stream concentration y that just allows mass transfer to a MSA at a concentration xj.

8-6

( )jjj xay e+= (8.2)

Similarly, by rearranging Equation 8.2, we find the lean stream concentration, xj, that will

just allow mass transfer from a rich process stream to MSA j:

jj

j ay

x e-= (8.3)

Equation 8.4 is a more general form of the linear equilibrium expression given in

Equation 8.1 including the equilibrium constant bj.

( ) jjjj bxay +e+= (8.4)

Rearranging Equation 8.4, we group the constant term, ajbj, with the minimum approach

concentration, e j.

( )jjjjj baxay e++= (8.5)

Thus, we can easily incorporate more general linear equilibrium relationships into the analysis.

Figure 8.1 illustrates the feasibility of mass transfer between a process stream (at

concentration y) and MSA j (at concentration xj). The solid line corresponds to the case defined

by Equation 8.1 without a MAC. At point A, no driving force for mass transfer exists between a

8-7

rich process stream at concentration yA and a MSA at concentration xAj. The dashed line

corresponds to feasible mass transfer defined by Equations 8.2 or 8.3 where the MAC shifts the

line e j to the right along the xj axis. In this case, the driving force for mass transfer between a rich

process stream at concentration yA and MSA j at concentration xBj is at its minimum due to

thermodynamic or other constraints.

8-8

xj

y

AyA

xAj

ej

xBj

EquilibriumLine

Feasible Region

Feasible Line

Figure 8.1. Feasible mass transfer and the minimum approachconcentration (MAC) (El-Halwagi, 1997).

8-9

Equations 8.2 and 8.3 allow us to consider MSAs on the same concentration scale as the

contaminant-rich process streams. Table 8.3 lists the thermodynamic data, aj, for process MSAs,

S1 and S2. Table 8.4 is the process MSA stream data following conversion to the rich process-

stream-concentration scale, y, according to Equation 8.2 with a global minimum approach

concentration, e j, equal to 0.001.

Table 8.3. Equilibrium data for process MSAs S1 and S2 of Example 8.1.

Process

MSA jaj

S1 2.00

S2 1.53

Table 8.4. Process MSA stream data for Example 8.1. Concentrations

shifted to the corresponding process-stream scale (y) with a minimum

approach concentration of 0.001.

Process

MSA j

Lj

(kg/s)ysupplyj ytargetj

S1 5 0.0120 0.0320

S2 3 0.0168 0.0474

For Example 8.1, a third MSA (S3) is available for purchase at an unlimited flowrate. We

refer to this as an external MSA. The equilibrium relationship of Equations 8.2 and 8.3 apply

with a coefficient aS3 = 0.04 and a MAC, eS3 = 0.001. Equations 8.2 and 8.3 are:

8-10

( )001.0x04.0y S3 +=

001.00.04

yxS3 -=

Table 8.5 lists the supply and target concentrations of external MSA S3 with respect to its

concentration scale xS3 and the rich-stream concentration scale y (Equation 8.2).

Table 8.5. Corresponding concentration scales for the external MSA, S3.

Concentration

Scale

Supply

Concentration

Target

Concentration

xS3 0.100 0.200

y 0.00404 0.00808

8-11

0.0

0.1

0.2

0.3

0.4

0.5

0.049

0.099

0.149

0.199

0.249

0.064

0.130

0.195

0.260

0.326

y

Figure 8.2. Corresponding concentration scales for process streams (y), process MSAs (x and x)and external MSA (x ) of Example 8.1.

1 2

S3

2.499

4.999

7.499

9.999

12.499

0.0012.00

yxS1 -= 0.0011.53

yxS2 -= 0.0010.04

yxS3 -=

8-12

8.2.3 Capacity Flowrates

We introduce a useful term for evaluating the mass load of contaminant removed from

process streams or the capacity of MSAs to accept this mass load, called the capacity flowrate

(El-Halwagi, 1997). In particular, we determine the mass load of contaminant transferred from

process streams and to MSAs from Equations 8.4 and 8.5, respectively.

yGm ii D=D (8.5)

jjj xLm D=D (8.6)

Recall, Equation 8.3 defines thermodynamic relationships between lean concentration

scales, xj, and the process-stream concentration scale, y. Substituting for Dxj in Equation 8.6 and

simplifying gives the mass load of contaminant transferred to MSA j in terms of the modified

flowrate, ja

L

, and the process stream concentration scale, y.

yaL

mj

j D

=D (8.7)

8-13

We call this modified flowrate, ja

L

, the capacity flowrate of MSA j. Note that the

capacity flowrate of process stream i is simply the available flowrate, Gi. From this point

forward, we shall use the capacity flowrate to simplify calculations.

Equations 8.8 and 8.9 demonstrate the relationship between the capacity flowrates and

driving forces for heat and mass transport, respectively. In both cases, the transport of heat or

mass is the product of a capacity flowrate and a driving force.

( )( )( ) TCm

force drivingflowratecapacity Q

p D==D

&(8.8)

( )( )

yaL

force drivingflowratecapacity m

j

j

=

=

(8.9)

Table 8.6 lists the shifted stream for MSAs with capacity flowrates rather than actual

flowrates. For now, we have not determined the flowrate of external MSA S3 and its capacity

flowrate is considered unlimited. Sections 8.4.2 and 8.4.3 determine the duties and capacity

flowrates, respectively, of external MSAs.

8-14

Table 8.6. Shifted stream data for the MSAs of Example 8.1 with capacity

flowrates. Concentrations shifted to the corresponding process-stream scale

(y) with a minimum approach concentration of 0.001.

MSA

jja

L

(kg/s)

ysupplyj ytargetj

S1 2.5 0.0120 0.0320

S2 1.961 0.0168 0.0474

S3 - 0.00404 0.00808

8-15

8.3 Minimum External MSA Duty without Mass Integration

Now that we can consider process streams, process MSAs and external MSAs on the

same concentration scale, we can evaluate the systems potential for integration. In other words,

we maximize the use of process MSAs to minimize the duty on external MSAs. However, it will

be useful to first identify the minimum duty of external MSAs prior to considering integration of

process MSAs. For this, we introduce a plot of contaminant-mass load versus concentration

called the limiting concentration profile.

8.3.1 Limiting Concentration Profiles

Figure 8.3 shows a plot of concentrations within process stream i and MSA j versus

contaminant-mass load transferred representing a unit operation. The process stream and external

MSA enter the unit at inlet concentrations, yini and yinj, respectively, and leave the unit at outlet

concentrations, youti and youtj, respectively. In this case, we represent the unit operation as a

countercurrent contact between the streams. The process stream enters the unit from the right of

the diagram and exists to the left. Conversely, the external MSA enters from the left of the

diagram and exists to the right. Rearranging Equations 8.5 and 8.7 gives an inverse relationship

between the capacity flowrates of the process stream and MSA, respectively, and the slope of

lines on the limiting concentration profile:

8-16

yinR1

youtR1yinS3

youtS3

0.05

0.08

Figure 8.3. Limiting concentration profiles for process stream, i, and MSA j. Concentrations shifted to the process-stream scale, y.

8-17

slope1

yys

kgm

yskg

m

skg

Gouti

ini

i =-

D

=D

D

=

(8.10)

slope1

yys

kgm

yskg

m

skg

aL

outj

injj

=-

D

=D

D

=

(8.11)

8.3.2 Minimum External MSA Duty

Because the supply and target concentrations for the external MSAs (ysupplyj and ytarget j)

are clearly defined, we can find the minimum capacity flowrate of external MSA S3, 3Sa

L

,

required by the operation represented in Figure 8.3 based on the mass load of contaminant

transferred from the process stream to the external MSA.

( ) ( )[ ]( )supplyjtargetjtargeti

supplyii

j y-yyykg/sG

kg/saL -=

(8.12)

For process stream R1 and external MSA S3 of Example 8.1, Equation 8.12 gives:

( )[ ]( )

( )[ ]( )

kg/s 802.190.004040.00808

0.010.05kg/s 2.0

y-y

yykg/sGaL

supplyS3

targetS3

targetR1

supplyR1R1

S3

=-

-=

-=

8-18

The actual flowrate required of MSA S3, LS3 = 0.80 kg/s, is simply the product of the

required capacity flowrate, S3a

L

= 19.802 kg/s, and the equilibrium coefficient, aS3 = 0.04.

Table 8.7 lists the similar results for process stream R2 and the total capacity flowrate and actual

flowrate of the external MSA S3 required for Example 8.1 without mass integration of process

MSAs.

Table 8.7. Minimum flowrates of the external MSA S3 for Example 8.1

without integrating process MSAs. aS3 = 0.04.

Process

Stream

i

S3aL

(kg/s)

LS3

(kg/s)

R1 19.802 0.80

R2 5.941 0.24

Total 25.743 1.04

8-19

8.4 Minimum External MSA Duty with Mass Integration

This section integrates process MSAs to minimize the use of external MSAs. First, we

construct the key tool of pinch technology, the concentration composite curve. Then, we use the

curve to analyze systems of contaminant-rich and lean streams for identifying the minimum

external MSA duties.

8.4.1 Process-Stream and Process-MSA Composite Curves

We can represent the integrated system of rich process streams and MSAs either

graphically or tabularly.

8.4.1.1 Graphical Approach: Process-Stream and Process-MSA Composite

Curves

First, let us consider the graphical method, called the concentration composite curve.

Constructing these curves for process streams and process MSAs is a five-step procedure:

First, plot all of the process streams involved on a single graph of contaminant

concentration versus mass load. The operations should be plotted head to toe so that while the

8-20

y-axis, which corresponds to concentration, is absolute, the x-axis, which represents the mass

load, is relative and one stream begins where the prior one ends.

1. Divide the y-axis into concentration intervals by drawing horizontal lines (shown as

dashed lines on Figure 8.3) at the target and supply concentrations for process

streams. Dashed horizontal lines mark the interval boundaries, denoted as y*k (where

k = 1, 2, 3, ). In Example 8.1, these intervals occur at 0.006, 0.010, 0.030 and 0.050

for process streams.

Figure 8.4 displays such a plot for the contaminant-rich process streams of Example 8.1.

2. Sum the mass loads of all rich process streams present in each concentration interval

and draw a new line across the interval corresponding to that sum.

For Example 8.1, the first concentration interval (from y*1 = 0.006 to y*2 = 0.010)

includes only 0.004 kg/s of mass load from process stream R2. However, the second

concentration interval (from y*2 = 0.010 to y*3 = 0.030) contains both 0.040 kg/s of mass load

from process stream R1 and 0.020 kg/hr from process stream R2. Figure 8.5 shows the result of

this procedure for the process streams of Example 8.1.

8-21

Figure 8.4. Graphical approach to the construction of the process-stream composite curve for Example 8.1.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.02 0.04 0.06 0.08 0.1 0.12

Mass Load (kg/s)

Con

cent

rati

on (

y)

R1

R2

mR1 = 0.08 kg/s mR2 = 0.03 kg/s

Dmtot = 0.104 kg/s

8-22

Figure 8.5. Cumulative process stream (dashed lines) for each interval constructed in Figure 8.4 for Example 8.1.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.02 0.04 0.06 0.08 0.1 0.12

Mass Load (kg/s)

Con

cent

rati

on (

y)

R1

R2

mR1 = 0.08 kg/s mR2 = 0.03 kg/s

Dmtot = 0.104 kg/sProcess-Stream Composite Curve

8-23

3. Construct the final rich-stream composite curve by eliminating the original stream

lines from the diagram and leaving only the sum of the mass loads within each

concentration interval.

Figure 8.6 shows the process-stream composite curve for Example 8.1.

8-25

Figure 8.6. Process-stream composite curve for Example 8.1.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.02 0.04 0.06 0.08 0.1 0.12

Mass Load (kg/hr)

Con

cent

rati

on (

y)

Dm tot = 0.104 kg/s

Process-Stream Composite Curve

8-26

4. Construct the process-MSA composite curve by repeating steps 1 through 4 for the

process MSAs. However, we must use the shifted stream data for process MSAs

reflecting the equilibrium relationships defined in Equation 8.2 to fix the

concentration-interval boundaries. We determine the mass load transferred to each

process MSA, mj from:

( ) ( )[ ]supplyjtargetjj

j yykg/saL

kg/sm -

= (8.13)

where ytarget j and ysupplyj are the target and supply concentrations, respectively, for

process MSA j on the process-stream concentration scale and ja

L

is the available

capacity flowrate of process MSA j.

For Example 8.1, Equation 8.13 gives a mass load of contaminant for process MSA S1 on

the y-concentration scale from 0.012 to 0.032 of:

( )[ ] ( )[ ] kg/s 0.0500.0120.032kg/s 2.5yykg/saL

m supplyS1targetS1

S1S1 =-=-

=

Figures 8.7 to 8.9 illustrate the construction of the process-MSA composite curve.

8-27

Figure 8.7. Graphical approach to the construction of the process-MSA composite curve for Example 8.1. Concentrations shifted to the process-stream scale (y) with a minimum approach concentraiton of 0.001.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.02 0.04 0.06 0.08 0.1 0.12

Mass Load (kg/s)

Con

cent

rati

on (

y)

S1

S2

mS1 = 0.05 kg/s mS2 = 0.06 kg/s

Dmtot = 0.11 kg/s

8-28

Figure 8.8. Cumulative process-MSA stream (dashed lines) for each interval constructed in Figure 8.7 for Example 8.1. Concentrations shifted to the process-stream scale (y) with a minimum approach concentration of 0.001.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.02 0.04 0.06 0.08 0.1 0.12

Mass Load (kg/s)

Con

cent

rati

on (

y)

S1

S2

mS1 = 0.05 kg/s mS2 = 0.06 kg/s

Dm tot = 0.11 kg/s

Process-MSA Composite Curve

8-29

Figure 8.9. Process-MSA composite curve for Example 8.1. Concentrations shifted to the process-stream scale (y) with a minimum approach concentration of 0.001.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.02 0.04 0.06 0.08 0.1 0.12

Mass Load (kg/s)

Con

cent

rati

on (

y)

Dmtot = 0.11 kg/s


8-30

8.4.1.2 Tabular Method: Concentration-Interval Diagram (CID)

We often prefer a tabular method for constructing the composite curves over graphical

method presented in Section 8.4.1.1, as it is more readily adaptable to computer programming.

As in the graphical method, the tabular method uses the concentration-interval boundaries

determined from the process-stream and shifted process-MSA data (such as Tables 8.1 and 8.6).

Here, we will use the y-concentration scale (Table 8.6) for the supply and target concentrations

of the process MSAs. In this case, we sort the concentrations in ascending order to form the

concentration-interval boundaries, y*k (where k = 1, 2, 3, ). Once we have defined the

concentration intervals, we can proceed as follows:

1. For a given concentration interval, k, calculate the mass load of contaminant to be

removed from each process stream i, within the interval, mi,k. From Equation 8.14, we

determine the mass load transferred from process stream i in interval k, mi,k, as:

( ) ( )[ ]*k* 1kiki, yykg/sGkg/sm -= + (8.14)

where y*k+1 and y*k are the upper and lower interval boundaries and Gi is the capacity

flowrate of process stream i. We can then calculate the total mass load transferred,

8-31

mk, in interval k, as the sum of the mass loads for each rich process stream i, in the

interval, mi,k:

( ) ( ) ( )-= +i

i*k

*1kk kg/sGyykg/sm (8.15)

2. With the intervals in ascending order, and calculate the cumulative mass load at the

end of each interval (i.e., at concentration-interval boundary k+1) by summing the

mass loads, mk, to the concentration-interval boundary k+1:

( ) ( )=D +k

k1k kg/smkg/sm (8.16)

Returning to Example 8.1, Table 8.8 lists the concentration intervals for the overall

system: y*1 = 0.006, y*2 = 0.010, y*3 = 0.012, y*4 = 0.01683, y*5 = 0.030, y*6 = 0.032, y*7 =

0.04743 and y*8 = 0.050. Tables 8.1 and 8.6 give the capacity flowrates Gi and Sja

L

for ith

process stream and jth process MSA, respectively.

The first interval, y*1 = 0.006 to y*2 = 0.010, has only one rich process stream, so the

calculations are simply:

( ) ( )( )( )

kg/s 0.004

kg/s 1.00.0060.010

kg/sGyymi

i*1

*22

=-=

-=

8-32

The second interval, y*2 = 0.010 to y*3 = 0.012, contains both rich process streams, so the

calculation for that interval is:

( ) ( )( )[ ]( )

kg/s 0.006

kg/s 1.02.00.010-20.01

kg/sGyymi

i*2

*32

=+=

-=

The cumulative mass load at the end of interval 2 (i.e., at the 3rd concentration-interval

boundary), Dm3, is 0.004 + 0.006 = 0.010 kg/s.

Table 8.8 lists the mass load of contaminant within each interval k, mk, as well as the

cumulative mass loads of contaminant at the end of each interval, Dmk+1. Note that the

cumulative mass load of contaminant for interval 1 is simply m1, the cumulative mass load of

contaminant for interval 2 is m1 + m2, the cumulative mass load of contaminant for interval 3 is

m1 + m2 + m3, and so on.

8-33

Table 8.8. Data required for the construction of the process-stream

composite curve for Example 8.1.

Interval

k

Concentration-

Interval Boundaries

(y*k y*k+1)

mk

(kg/s)

DD mk+1

(kg/s)

1 0.006 0.01 0.00400 0.00400

2 0.010 0.012 0.00600 0.01000

3 0.012 0.01683 0.01449 0.02449

4 0.01683 0.030 0.03951 0.06400

5 0.030 0.032 0.00400 0.06800

6 0.032 0.04743 0.03086 0.09886

7 0.04743 0.050 0.00514 0.10400

3. The process-stream portion of the CID is simply a tabular representation of these

data. For a system with i process streams, column 1 contains the concentration-

interval boundaries, in ascending order; columns 2 through i + 1 represent each of the

i process streams in the system with respect to their target and supply concentrations

(also in ascending order according to ytarget i and ysupplyi); column i + 2 contains the

mass load of contaminant transferred within each interval; and column i + 3 contains

the cumulative mass load of the system for process streams.

Table 8.9 is a partial CID for Example 8.1 including process streams. Note that a plot of

column 1 (concentration) versus column 5 (cumulative mass load) yields the process-stream

composite curve of Figure 8.6.

8-34

Table 8.9. Partial CID for Example 8.1 including data for the process-

stream composite curve.

Concentration

(y*i)

R1

2.0 kg/s

R2

1.0 kg/s

Mass Load

Removed

(kg/s)

Cumulative

Mass Load

(kg/s)

0.00600 0.00000

0.00400

0.01000 0.00400

0.00600

0.01200 0.01000

0.01449

0.01683 0.02449

0.03951

0.03000 0.06400

0.00400

0.03200 0.06800

0.03086

0.04743 0.09886

0.00514

0.05000 0.10400

8-35

4. We generate the data for constructing the process-MSA composite curve in a similar

manner. However, we calculate the mass load of contaminant transferred within each

interval through Equation 8.15.

( )

-= +

j j

*k

*1kk s

kgaL

yym (8.17)

where y*k+1 and y*k are the concentration-interval boundaries of interval k on the

concentration scale associated with process streams and ja

L

is the capacity flowrate

of the jth process MSA

Returning to Example 8.1, we apply Equation 8.14 to find the mass load of contaminant

transferred to process MSAs in the 3rd concentration interval:

( )

( )

( )

kg/s 0.012075s

kg5.201200.001683.0

skg

aL

yy

skg

aL

yym

S1

*3

*4

j j

*k

*1kk

=

-=

-=

-= +

For the fourth interval, Equation 8.17 determines the mass load transferred to both

process MSAs as:

8-36

( )

( )[ ]

kg/s 0.05875skg

961.15.201683.003000.0

skg

aL

aL

yym2S1S

*4

*54

=

+-=

+

-=

5. We calculate the cumulative mass load column for process MSAs in the identical

manner as the cumulative mass load column for rich process streams.

Table 8.10 shows that two additional columns are added to the CID of Table 8.9 to

illustrate the relative position of each process MSA j with respect to its target and supply

concentrations, ytarget j and ysupplyj. A plot of column 5 versus column 1 gives the process-stream

composite curve and a plot of column 9 versus column 1 gives the process-MSA composite

curve.

8-37

Table 8.10. CID for Example 8.1 including data for the process-stream and process-MSA composite curves.

Concentration

(y*i)

R1

2.0 kg/s

R2

1.0 kg/s

Mass Load

(kg/s)

Cumulative

Mass Load

(kg/s)

S1

2.5 kg/s

S2

1.961 kg/s

Available

Capacity

(kg/s)

Cumulative

Capacity

(kg/s)

0.00600 0.00000 0.00000

0.00400 0.00000

0.01000 0.00400 0.00000

0.00600 0.00000

0.01200 0.01000 0.00000

0.01449 0.01208

0.01683 0.02449 0.01208

0.03951 0.05875

0.03000 0.06400 0.07082

0.00400 0.00892

0.03200 0.06800 0.07975

0.03086 0.03025

0.04743 0.09886 0.11000

0.00514 0.00000

0.05000 0.10400 0.11000

8-38

8.4.2 Minimum External MSA Duty

Once we have constructed either composite curves or a CID for a given system, the final

step is to determine the minimum utility targets (external MSA duty) using the concept of the

pinch. The concept of the pinch concentration is critical, as above that concentration, we do not

use external MSAs. Again, we proceed either graphically or tabularly.

8.4.2.1 Graphical Method: Process-Stream and Process-MSA Composite Curves

Once we have established the process-stream and process-MSA composite curves for the

system, we can readily obtain a target for the duty of external MSAs by adjusting the process-

MSA composite curve to the left on the mass load axis until it just touches the process-stream

composite curve at the pinch point:

1. Slide the process-MSA composite curve to the left of the diagram until it just touches the

process-stream composite curve at the pinch point.

2. The horizontal distance at the bottom left represents the excess mass load of contaminant

in the process streams that is not removed by process MSAs. This excess must be

transferred to external MSAs.

8-39

3. The horizontal distance at the top right represents the excess capacity of process

MSAs to remove mass load from process streams. In this case, if any excess exists,

we wish to eliminate it by either reducing the flowrate or target concentration of a

process MSA. It is not necessary to replot the process MSA composite curve, as

minor adjustments above the pinch concentration will not affect the utility target

below the pinch concentration.

Returning to Example 8.1, Figure 8.10 presents the both composite curves on the same

plot. In the figure, we see an excess mass load of rich streams (lower left) that requires

0.01242 kg/s of contaminant mass load be removed by external MSAs. In addition, we see an

excess capacity of the process MSAs to remove contaminant (top right) equal to 0.01842 kg/s.

To eliminate this excess, we choose to reduce the capacity flowrate of process MSA S2

according to:

kg/s 359.101683.004743.0

kg/s 0.01842kg/s 961.1

yykg/s 0.01842

aL

aL

supplyS2

targetS2S2

adj

S2

=-

-=

--

=

(8.18)

In other words, we decrease the actual flowrate of process MSA S2 from 3.0 kg/s to

2.0793 kg/s.

( ) kg/s 2.0793kg/s 1.3591.53aL

aLadj

S2S2

adjS2 ==

=

8-41

Figure 8.10. Process-stream and process-MSA composite curves for Example 8.1 with the minimum external MSA duty. Concentrations shifted to the process-stream scale (y) with a minimum approach concentration of 0.001.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Mass Load (kg/hr)

Con

cent

rati

on (

y)


Pinch Concentration,

y*4 = 0.01683

External MSA Duty,0.01242 kg/s

Excess Capacity of Process MSAs,0.01842 kg/s


Possible Mass Integration

8-42

8.4.2.2 Tabular Method: Concentration -Interval Diagrams (CIDs)

Identifying the pinch concentration on a CID involves essentially the same principle as

identifying the pinch concentration on a composite curve. To accomplish this, we add three

columns to the CID of Table 8.10. The process is straightforward, again more readily adaptable

to computer programming than the graphical method:

1. In each concentration interval, evaluate the net mass load of contaminant to be

transferred as the difference between the available mass load from process streams

(Equation 8.15) and the capacity of the process MSAs (Equation 8.17) as:

( ) ( ) ( )[ ]

( ) [ ]( ) ( )

--=

---=

+

++

j jii

*k

*1k

j

*kj,

*1kj,j

ii

*k

*1kk

kg/saL

kg/sGyy

xxkg/sLkg/sGyym

(8.19)

2. Cascade the net mass load to be removed starting with zero at the highest

concentration-interval boundary (bottom).

3. Place the negative of the minimum (most negative) value from the cascaded mass

load column at the bottom concentration-interval boundary in the final column of the

CID. Once again, cascade the net mass load to be removed starting with that value at

the highest concentration-interval boundary (bottom right). The pinch concentration is

located where zeros are found in this column. The minimum external MSA duty and

8-43

the excess process-MSA capacity are found at the top and bottom of the last column,

respectively.

Table 8.11 shows the final CID, including the minimum external MSA duty, for

Example 8.1. We identify the pinch concentration at y*4 = 0.01683 by a zero in the last column.

Again, we see that the minimum external MSA duty is 0.01242 kg/s (top right) and the excess

capacity of process MSAs is 0.01842 kg/s (bottom right). Table 8.12 is the CID for Example 8.1

with a reduced capacity flowrate of process MSA S2 (1.359 kg/s) to eliminate the excess

capacity (0.01842 kg/s) of process MSAs.

kg/s 359.101683.004743.0

kg/s 0.01842kg/s 961.1

yykg/s 0.01842

aL

aL

supplyS2

targetS2S2

adj

S2

=-

-=

--

=

Notice from Table 8.12 that the excess capacity (bottom right) is now zero.

8-45

Table 8.11. CID for Example 8.1 including the minimum external MSA duty.

Concentration

(y*k)

R1

2.0 kg/s

R2

1.0 kg/s

Mass Load

(kg/s)

Cumulative

Mass Load

(kg/s)

S1

2.5 kg/s

S2

1.961 kg/s

Available

Capacity

(kg/s)

Cumulative

Capacity

(kg/s)

Net

Mass Load

(kg/s)

Cascaded

Mass Load

(kg/s)

Adjusted

Mass Load

(kg/s)

0.00600 0 0 -0.00600 0.01242

0.00400 0 0.00400

0.01000 0.00400 0 -0.01000 0.00842

0.00600 0 0.00600

0.01200 0.01000 0 -0.01600 0.00242

0.01449 0.01208 0.00242

0.01683 0.02449 0.01208 -0.01842 0

0.03951 0.05875 -0.01924

0.03000 0.06400 0.07082 0.00082 0.01924

0.00400 0.00892 -0.00492

0.03200 0.06800 0.07975 0.00575 0.02416

0.03086 0.03025 0.00061

0.04743 0.09886 0.11000 0.00514 0.02356

0.00514 0 0.00514

0.05000 0.10400 0.11000 0 0.01842

8-46

Table 8.12. CID for Example 8.1 after reducing the capacity flowrate of process MSA S2 to eliminate the excess

capacity of process MSAs.

Concentration

(y*k)

R1

2.0 kg/s

R2

1.0 kg/s

Mass Load

(kg/s)

Cumulative

Mass Load

(kg/s)

S1

5.0 kg/s

S2

1.359 kg/s

Required

Capacity

(kg/s)

Cumulative

Capacity

(kg/s)

Net

Mass Load

(kg/s)

Cascaded

Mass Load

(kg/s)

Adjusted

Mass Load

(kg/s)

0.00600 0 0 0.01242 0.01242

0.00400 0 0.00400

0.01000 0.00400 0 0.00842 0.00842

0.00600 0 0.00600

0.01200 0.01000 0 0.00242 0.00242

0.01449 0.01208 0.00242

0.01683 0.02449 0.01208 0 0

0.03951 0.05082 -0.01131

0.03000 0.06400 0.06290 0.01131 0.01131

0.00400 0.00772 -0.00372

0.03200 0.06800 0.07062 0.01503 0.01503

0.03086 0.02097 0.00989

0.04743 0.09886 0.09159 0.00514 0.00514

0.00514 0 0.00514

0.05000 0.10400 0.09159 0 0

8-47

8.4.3 Utility Placement: Grand Composite Curve

The grand composite curve is a graphical representation of the excess mass load available

within each concentration interval. In intervals where a net mass-load surplus exists, we cascade

that mass to lower concentration intervals and use external MSAs to remove the remaining

contaminant at low concentrations.

For Example 8.1, Figure 8.11 plots the adjusted cascaded mass load (last column in

Table 8.12) versus the y-concentration scale to give the grand composite curve. In the figure,

region A represents mass transfer from process streams to process MSAs (i.e., process-to-process

mass transfer) and region B requires an external MSA or utility stream (i.e., process-to-utility

mass transfer).

8-49

Figure 8.11. Grand composite curve for Example 8.1. Concentrations shifted to the process-stream scale (y) with a minimum approach concentraiton of 0.001.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

Mass Load (kg/s)

Con

cent

ratio

n (y

)


y*4 = 0.01683

"Nose" or "Pocket":Self Sufficient Process-to-Process-MSA Mass Transfer


No Excess Capacity of Process MSAs

8-50

Figure 8.12 illustrates the application of external MSA S3 as a utility for Example 8.1. In

the figure, S3 must remove a mass load of 0.01242 kg/s of contaminant and is placed according

to its supply and target concentrations, ysupplyS3 and ytarget S3, respectively, on the process-stream

scale (y). The capacity flowrate required, 3Sa

L

, of external MSA S3 is given by:

( ) ( ) kg/s 1053.0.004040.00808kg/s 0.01242

yykg/s 0.01242

aL

supplyS3

targetS3S3

=-

=-

=

The actual flowrate of external MSA S3 is 0.1242 kg/s, as aS3 = 0.04.

8-51

Figure 8.12. Grand composite curve for Example 8.1 including the external MSA S3 as a utility stream. Concentrations shifted to the process-stream scale (y) with a minimum approach concentration of 0.001.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

Mass Load (kg/s)

Con

cent

ratio

n (y

)


y*4 = 0.01683

External MSA S3,0.01242 kg/s y

supplyS3 = 0.00404

ytargetS3 = 0.00804

8-52

8.5 Design Tools: Representing Mass-Exchange Networks

We represent mass-exchange networks in several ways. Two common methods are the

grid diagram and the mass-content diagram. To illustrate these tools, let us look at a preliminary

mass-exchanger network for Example 8.2 - a simple three-unit system.

8.5.1The Grid Diagram

The most common representation scheme is the grid diagram, in which each mass-

exchange unit is represented as a vertical line connecting two streams. In a grid diagram:

Horizontal lines at the top of the diagram represent process streams. These streams flow

from the left (rich side) to the right (lean side) of the diagram.

Horizontal lines at the bottom of the diagram represent process and external MSAs.

These streams flow from the right (lean side) to the left (rich side) of the diagram.

Vertical lines represent mass-exchange units. Each line connects a rich process stream

and a process or external MSA. We indicate the mass load of the unit (kg/s) within the

circles connecting the lines, and also show the inlet and outlet concentrations of the rich

process stream and the MSA.

8-53

Bold vertical dashed line(s) indicate the position of any pinch points for the system.

Figure 8.13 shows the grid diagram for our three-unit example.

As the next section makes clear, grid diagrams are an invaluable tool for designing and

representing networks for mass integration. We divide the grid diagram into subproblems across

the regions defined by the pinch points. Within these regions, we apply simple design rules to

achieve the minimum duties of process and external MSAs as well as the minimum number of

mass-exchange units.

8.5.2 The Mass-Content Diagram

We introduce mass-content diagrams as a new tool for designing and representing mass-

exchange networks. These diagrams provide an alternative to grid diagrams and give a unique

visualization of each mass-exchange unit in the network. In a mass-content diagram:

We represent each rich process stream with a box on the rich side (above the x-axis), and

each lean stream (both process and external MSAs) with a box on the lean side (below the x-

axis). We label each corresponding pair of rich and lean boxes with the same letter.

8-54

0.1 R12.0 kg/s

y = 0.4y = 0.2

0.1

S21.0 kg/s

S12.0 kg/s

R12.0 kg/s

y = 0.3

y = 0.35

y = 0.35

y = 0.05

0.3

0.3y = 0.35

S21.0kg/s

Figure 8.13. Grid diagram of a preliminary mass-exchange network for Example 8.2.

8-55

The top and bottom of a box on the rich side correspond to the supply and target

concentrations of a rich process stream, respectively. The width of the box, on the relative x-axis

represents the capacity flowrate of the rich process stream. Therefore, the area of the box

corresponds to the mass load of contaminant removed.

The bottom and top of a box on the lean side correspond to the supply and target

concentrations of a MSA, respectively. Once again, the width of the box, on the relative x-axis

represents the capacity flowrate of the MSA. Therefore, the area of the box corresponds to the

mass load of contaminant accepted.

Figure 8.14 shows the mass-content diagram for Example 8.2.

8-56

Mass Load (kg/s)

Con

cent

rati

on (y

)

1.0 2.0

0.1

0.2

0.3

0.4

0.5

2.0 kg/sy = 0.2

B

Ay = 0.35

y = 0.4

0.0

0.1

0.2

0.3

0.4

0.5

2.0 kg/s

1.0 kg/sy = 0.05

y = 0.35y = 0.35y = 0.3A

B

Figure 8.14. Mass-content diagram for Example 8.2. Concentrations shifted to the process-stream scalewith minimum approach concentrations.

3.0 4.0Lean Side(MSAs)

Rich Side(Process Streams)

Capacity Flowrate

8-57

8.6 Preliminary Mass-Exchange Network Design

This section presents a method for designing preliminary mass-exchange networks that

meet minimum targets for external MSAs as determined through the analysis in Section 8.3.

First, we examine the details of designing a simple preliminary mass-exchange network for a

new example. We first employ the shifted stream data to incorporate a minimum approach

concentration into the design, and later adjust the concentrations to reflect the true approach

concentrations within each mass-exchange unit on the appropriate concentration scales.

We introduce Example 8.3 as a tutorial for designing preliminary mass-exchange

networks. Tables 8.13 and 8.14 present the shifted stream data for the three process streams and

three MSAs, respectively, of Example 8.3. Here, MSAs S1, S2 and S3 are shifted according to

Equation 8.2 with equilibrium coefficients, aj, of 1.0, 2.0 and 1.2, respectively, and approach

concentrations of e j = 0.001. Tables 8.15 and 8.16 give the CIDs for the example before and after

reducing the flowrate of process MSA S2 to eliminate the excess capacity of process MSAs.

Figures 8.15 and 8.16 show the process and MSA composite curves corresponding to the CIDs

presented in Tables 8.15 and 8.16, respectively. From Table 8.16 and Figure 8.16, we see that the

system is pinched at a concentration, y*pinch, equal to 0.5 (mass fraction) and requires a capacity

flowrate of the external MSA S3 equal to 0.0552 kg/s or an actual flowrate of 0.0662 kg/s.

8-58

Table 8.13. Rich-stream data for Example 8.3.

Stream

i

Gi

(kg/s)ysupplyi ytargeti

R1 5.0 0.75 0.45

R2 2.0 0.70 0.59

R3 7.0 0.50 0.30

Table 8.14. Shifted stream data for the MSAs of Example 8.3 with capacity

flowrates. Concentrations shifted to the corresponding process-stream scale

(y) with a minimum approach concentration of 0.001.

MSA

jSa

L

(kg/s)

ysupplyj ytargetj

S1 2.5 0.0120 0.0320

S2 1.961 0.0168 0.0474

S3 - 0.00404 0.00808

8-59

Table 8.15. CID for Example 8.3 including the minimum external MSA duty.

Concentration

(y*k)

R1

5.0 kg/s

R2

2.0 kg/s

R3

7.0 kg/s

Mass Load

(kg/s)

Cumulative

Mass Load

(kg/s)

S1

6.0 kg/s

S2

4.0 kg/s

Available

Capacity

(kg/s)

Cumulative

Capacity

(kg/s)

Net

Mass Load

(kg/s)

Cascaded

Mass Load

(kg/s)

Adjusted

Mass Load

(kg/s)

0.2410 0.0000 0.0000 -0.0480 0.0960

0.0000 0.3540 -0.3540

0.3000 0.0000 0.3540 0.3060 0.4500

1.0500 0.9000 0.1500

0.4500 1.0500 1.2540 0.1560 0.3000

0.6000 0.3000 0.3000

0.5000 1.6500 1.5540 -0.1440 0.0000

0.4500 0.9000 -0.4500

0.5900 2.1000 2.4540 0.3060 0.4500

0.0840 0.1200 -0.0360

0.6020 2.1840 2.5740 0.3420 0.4860

0.6860 0.5880 0.0980

0.7000 2.8700 3.1620 0.2440 0.3880

0.0050 0.0060 -0.0010

0.7010 2.8750 3.1680 0.2450 0.3890

0.2450 0.0000 0.2450

0.7500 3.1200 3.1680 0.0000 0.1440

8-60

Table 8.16. CID for Example 8.3 after reducing the capacity flowrate of process MSA S2 to eliminate the excess

capacity of process MSAs.

Concentration

(y*k)

R1

5.0 kg/s

R2

2.0 kg/s

R3

7.0 kg/s

Mass Load

(kg/s)

Cumulative

Mass Load

(kg/s)

S1

6.0 kg/s

S2

2.588 kg/s

Available

Capacity

(kg/s)

Cumulative

Capacity

(kg/s)

Net

Mass Load

(kg/s)

Cascaded

Mass Load

(kg/s)

Adjusted

Mass Load

(kg/s)

0.2410 0.0000 0.0000 0.0960 0.0960

0.0000 0.3540 -0.3540

0.3000 0.0000 0.3540 0.4500 0.4500

1.0500 0.9000 0.1500

0.4500 1.0500 1.2540 0.3000 0.3000

0.6000 0.3000 0.3000

0.5000 1.6500 1.5540 0.0000 0.0000

0.4500 0.7729 -0.3229

0.5900 2.1000 2.3269 0.3229 0.3229

0.0840 0.1031 -0.0191

0.6020 2.1840 2.4300 0.3420 0.3420

0.6860 0.5880 0.0980

0.7000 2.8700 3.0180 0.2440 0.2440

0.0050 0.0060 -0.0010

0.7010 2.8750 3.0240 0.2450 0.2450

0.2450 0.0000 0.2450

0.7500 3.1200 3.0240 0.0000 0.0000

8-61

Figure 8.15. Process-stream and process-MSA composite curves for Example 8.3 with the minimum external MSA duty. Concentrations shifted to the process-stream scale (y) with a minimum approach concentration of 0.001.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Mass Load (kg/s)

Con

cent

rati

on (

y)



y*4 = 0.5


Excess Capacity of Process MSAs,0.144 kg/s

Process-MSA Composite CurvePossible Mass Integration

8-62

Figure 8.16. Process-stream and process-MSA composite curves for Example 8.3 after eliminating the excess capacity of process MSAs. Concentrations shifted to the process-stream scale (y) with a minimum approach concentration of 0.001.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Mass Load (kg/s)

Con

cent

ratio

n (y

)



y*4 = 0.5


Excess Capacity of Process MSAs Eliminated


8-63

8.6.1 Pinch Subnetworks

The nature of the pinch allows us to divide the design problem into subnetworks defined

by the pinch concentration(s). Recall that no contaminant should be transferred across the pinch.

Beginning at the pinch and working away form the pinch, we select matches according to the

design rules presented in Sections 8.6.2 and 8.6.6 to satisfy the stream data.

Figure 8.17 is a grid diagram for Example 8.3. At this point, we have not identified mass-

exchange units. However, the problem is divided into two subnetworks above and below the

pinch at a shifted concentration (y) of 0.5.

8-65

Pinchy = 0.5

R15 kg/s

Figure 8.17. Grid diagram for designing a preliminary mass-exchange network for Example 8.3 divided intotwo subnetworks above and below the pinch concentration. Concentrations shifted to the process-streamscale (y) with a minimum approach concentration of 0.001.

S16 kg/s

S30.05515 kg/s

S30.05515kg/s

S22.589 kg/s

S16 kg/s

R15 kg/s

R22 kg/s

R37 kg/s

R37 kg/s

R22 kg/s

8-66

8.6.2 Minimum Number of Mass-Exchange Units

Eulers graph theory identifies the theoretical minimum number of mass-exchange units

from the number of contaminant-rich process streams and MSAs. For systems where the pinch

divides the design in to two separate components (see Section 8.6.1), the number of units is:

( ) ( ) Pinch theBelowSRPinch theAboveSRunits 1NN1NNN -++-+= (8.20)

where NR and NS are the number of rich process streams and MSAs, respectively.

For the three rich process streams and three MSAs of Example 8.3, Equation 8.20 gives

the minimum number of units as:

( ) ( )( ) ( )6

122122

1NN1NNN

Pinch theBelowPinch theAbove

Pinch theBelowSRPinch theAboveSRunits

=-++-+=

-++-+=

8.6.3 Maximize Exchanger-Mass Loads

To minimize the number of mass-exchange units, we maximize the contaminant

transferred in each unit by first identifying the total mass load of contaminant to be transferred

from the process stream and the total capacity for contaminant of the MSA. Second, we choose

the lesser of the two to maximize the mass load of contaminant transferred in the unit. Equations

8.21 and 8.22 give the mass load of contaminant to be removed from process streams and the

8-67

capacity of MSAs, respectively, above the pinch concentration. Here, y*pinch is the pinch

concentration.

( )*pinchsupplyiiremovedi yyskg

Gs

kgm -

=

(8.21)

( )*pinchtargetjj

capacityj yys

kgaL

skg

m -

=

(8.22)

Similarly, Equations 8.23 and 8.24 give the mass loads of contaminant to be removed

from process streams and the capacity of MSAs, respectively, below the pinch concentration.

( )targeti*pinchiremovedi yykgGkgm -

=

ss(8.23)

( )supplyj*pinchj

capacityj y-y

kgaLkg

m

=

ss(8.24)

Figures 8.18 and 8.19 illustrate the two possible matches between process stream R1 and

MSA S1 above the pinch concentration for Example 8.3. In the figures, 1.2500 (Figure 8.18)

and 1.2060 (Figure 8.19) kg/s of contaminant are transferred from process stream R1 or to

MSA S1 according to Equations 8.21 and 8.22, respectively.

( )s

kg 1.25000.500.75

skg

5.0m removedR1 =-

=

8-68

( )s

kg 1.20600.500.701

skg

6.0m capacityS1 =-

=

Thus, the unit can feasibly transfer 1.2060 kg/s of contaminant as illustrated in Figure

8.19, but not as much as 1.2500 kg/s of contaminant as depicted in Figure 8.18.

8-69

Pinchy = 0.5

R15 kg/s

R22 kg/s

Figure 8.18. Grid diagram of an infeasible match between process stream R1 and MSA S1 for Example 8.3 abovethe pinch concentration Concentrations shifted to the process-stream scale (y) with a minimum approachconcentration of 0.001. Mass loads in kg/s.

S16 kg/s

1.2500

S30.05515 kg/s

S22.589 kg/s

S16 kg/s

R15 kg/s

R22 kg/s

y = 0.5y = 0.751.2500

y = 0.5y = 0.7083

R37 kg/s

R37 kg/s

8-70

Pinchy = 0.5

R15 kg/s

R22 kg/s

Figure 8.19. Grid diagram of a feasible match between process stream R1 and MSA S1 for Example 8.3 above thepinch concentration. Concentrations shifted to the process-stream scale (y) with a minimum approachconcentration of 0.001. Mass loads in kg/s.

S16 kg/s1.2060

S30.05515 kg/s

S22.589 kg/s

S16 kg/s

R15 kg/s

R22 kg/s

y = 0.5y = 0.74121.2060

y = 0.5y = 0.701

R37 kg/s

R37 kg/s

8-71

8.6.4 Capacity-Flowrate Rule for Match Feasibility

With more than one possible match between a process stream and a MSA available, we

select stream matches according to their capacity flowrates. Figure 8.20 depicts a mass-exchange

unit operating just above the pinch concentration on limiting concentration profiles (i.e., a plot of

concentration versus mass load of contaminant transferred). In the figure, Gi and ja

L

are the

capacity flowrates of the process and MSA streams, respectively, yini and youti are the inlet and

outlet concentration of the process stream on the process-stream scale (y), respectively, and yini

and youti are the inlet and outlet concentrations of the MSA on the process-stream scale (y),

respectively. The mass loads of contaminant transferred from the process stream (Equation 8.25)

and to the MSA (Equation 8.26) are:

( )outiiniii yyGm -= (8.25)

( )injoutjj

j yyaL

m -

= (8.26)

The mass load of contaminant transferred from the process stream (Equation 8.25) equals

the mass load of contaminant removed by the MSA (Equation 8.26):

( ) ( )injoutjj

outi

inii yya

LyyG -

=- (8.27)

8-72

yini

y = y = 0.05in outj i

youtj

0.08

Figure 8.20. Mass-exchange unit operating just above the pinch concentration.

Pinch

8-73

Operating just above the pinch concentration, both the process-stream inlet concentration,

youti, and the MSA inlet concentration, yinj, equal the pinch concentration, y*pinch, and Equation

8.27 becomes:

( ) ( )*pinchoutjj

*pinch

inii yya

LyyG -

=- (8.28)

Furthermore, for feasible mass transfer, the process-stream inlet concentration, yini, must

be greater than or equal to the MSA outlet concentration, youtj. Thus, the capacity flowrate of the

MSA, ja

L

, must be greater than or equal to the capacity flowrate of the process stream, Gi.

Above the pinch concentration, we use the following simple rule for selecting stream matches:

ij

GaL

Through a similar analysis, we find that for feasible matches below the pinch

concentration, the capacity flowrates of process streams must be greater than or equal to the

capacity flowrates of MSAs.

A general rule for matching a process stream to a MSA is that the capacity flowrates of

streams leaving the pinch concentration (i.e., process MSAs above the pinch concentration or

process streams below the pinch concentration) must be greater than or equal to the capacity

8-74

flowrate of streams approaching the pinch concentration (i.e., process streams above the pinch

concentration or MSAs below the pinch concentration).

For Example 8.3, Figure 8.21 illustrates the capacity-flowrate rule on two limiting

concentration profiles below the pinch concentration. In the figure, the solid lines represent

process streams R1 and R3, and the dashed lines represent MSA S1. Recall that when we use the

process-stream scale (y), the capacity flowrate is equal to the inverse of the slope of the line

representing the limiting concentration profile. Figure 8.21a illustrates the case where the

capacity flowrate of the process stream is less than that of the MSA (i.e., against the capacity-

flowrate rule). Here, the MSA is always above the process stream and mass transfer is infeasible.

However, in the case of Figure 8.21b, the capacity flowrate of the process stream (R3) is less

than that of the MSA (i.e., in agreement with the capacity-flowrate rule) and we see the streams

diverge from left to right and mass transfer between the streams is always feasible.

8-75

y = 0.3outR3

y = y = 0.5in ou tR3 S1

y = 0.241inS1

1.4

Pinch

(b)

1.0

Feasible Mass Transfer

y = 0.189outR1

y = y = 0.5in outR1 S1

y = 0.241inS1

1.554

Pinch

(a)

1.0

Infeasible Mass Transfer

Figure 8.21. Concentration versus mass load for matches between (a) process stream R1and MSA S1 and (b) process stream R1 and MSA S2.

8-76

A simple and effective technique for identify matches with respect to the capacity

flowrates of streams entering and leaving the pinch is the tick-off table. Table 8.17 lists the

capacity flowrates of the three process streams and three MSAs of Example 8.3 above (left) and

below (right) the pinch concentration. In the table, we match streams above the pinch by drawing

lines from a MSA to a process stream (i.e. right to left) such that the line always points to a

process stream with a lower capacity flowrate. Conversely, below the pinch, we draw lines to

identify matches from a process stream to a MSA (i.e., from left to right), such that the line

always points to a MSA with a lower capacity flowrate. We do so until each stream entering the

pinch (i.e., process streams above the pinch concentration and MSAs below the pinch

concentration) has been matched with a stream leaving the pinch.

For Example 8.3, we match MSA S1 to process stream R1 above the pinch concentration

and process stream R3 to MSA S1 below the pinch concentration. We note that MSA S2 (above

the pinch concentration) and process stream R1 (below the pinch concentration) both leave the

pinch and are not required to follow the capacity-flowrate rule for stream matching at the pinch.

8-77

Table 8.17. Tick-off table for Example 8.3.

Above the pinch Below the Pinch

Stream

i

GRi

(kg/s)Sia

L

(kg/s)

GRi

(kg/s)Sia

L

(kg/s)

1 5.0 6.0 5.0 6

2 - 2.589 - -

3 - - 7.0 -

8.6.5 Matches Away from the Pinch

Once we have identified the matches between process streams and MSAs near the pinch

concentration, the design problem is relaxed. In other words, away from the pinch concentration,

we have greater latitude in selecting stream matches. It is at this point where we are likely to

generate alternative designs for mass-exchange networks. Here, we may consider other factors

like physical location and stream compatibility to reduce network complexity or operational

hazards. However, by reducing the excess capacity of process MSAs above the pinch

concentration, we tightened the design problem and must take care to insure match feasibility at

the highest concentration intervals. Figure 8.22 shows a grid diagram of the complete

preliminary mass-exchange network for Example 8.3. Figure 8.23 displays a mass-content

diagram representing the same network for Example 8.3.

8-79

1.4000

0.0960

Pinchy = 0.5

R15 kg/s

R22 kg/s

y = 0.01280

Figure 8.22. Grid diagram of a complete preliminary mass-exchange network for Example 8.3. Concentrations shifted to the process-stream scale (y) with minimum approach concentrations of 0.001. Mass loads in kg/s.

y = 0.45

1.4000

0.1540

0.1540S1

6 kg/s1.2060

0.2200

0.0440 0.2200

0.0960S3

0.05515 kg/sS3

0.05515kg/s

S22.589 kg/s

S16 kg/s

R15 kg/s

R22 kg/s

y = 0.3y = 0.5

y = 0.5y = 0.7412y = 0.75

y = 0.59

1.2060

y = 0.70

0.0440

y = 0.5y = 0.701

y = 0.5y = 0.585

y = 0.602

y = 0.5y = 0.2667

y = 0.241

y = 0.0972y = 0.1692

y = 0.5

R37 kg/s

R37 kg/s

8-80

Mass Load (kg/s)

Con

cent

rati

on (

y)

10.0 20.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

5.0 kg/s

2.0 kg/s

7.0 kg/s

y = 0.45

y = 0.75

y = 0.5

y = 0.3

D

B

A

C

E y = 0.4692y = 0.5

y = 0.7412

y = 0.7

y = 0.59

F

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

6.0 kg/s

2.589 kg/s

0.05515 kg/s

y = .0972

y = 0.1692

y = 0.5

y = 0.585y = 0.602

y = 0.701

y = 0.5

y = 0.2667y = 0.241

A

B

C

D

EF

Figure 8.23. Mass-content diagram of a complete preliminary mass-exchange network for Example 8.1.Concentrations shifted to the process-stream scale (y) with minimum approach concentrations of 0.001.Mass Loads in kg/s.

8-81

8.6.6 Stream Splitting

For some problems, we may not be able to strictly follow the capacity-flowrate rule

(Section 8.6.3) for stream matching without segmenting streams. To illustrate this situation, we

return to Example 8.1.

Figure 8.24 shows the streams and their capacity flowrates for Example 8.1. Table 8.18

lists the capacity flowrates of process streams and MSAs above and below the pinch for

Example 8.1 for the tick-off matching procedure. Above the pinch concentration, there are two

feasible matches - process stream R1 to MSA S1 (2.0 kg/s is less than 2.5 kg/s) and process

stream R2 to MSA S2 (1.0 kg/s is less than 1.359 kg/s). However, we are not so fortunate below

the pinch concentration. Table 8.18 shows that matches between process stream R1 and MSA S1

(2.0 kg/s is less than 2.5 kg/s) and process stream R2 and MSA S1 (1.0 kg/s is less than 2.5 kg/s)

are infeasible .

Table 8.18. Tick-off table for Example 8.1.

Above the pinch Below the Pinch

Stream

i

(MCp)Hi

(kW/ C)

(MCp)Ci

(kW/ C)

(MCp)Hi

(kW/ C)

(MCp)Ci

(kW/ C)

1 2.0 2.5 2.0 2.5

1.0 1.359 1.0 -

2 - - - -

8-83

Pinchy = 0.01683

R12 kg/s

R21 kg/s

Figure 8.24. Grid diagram for designing a preliminary mass-exchange network for Example 8.1.

S12.5 kg/s

S33.1038 kg/s

S33.1038 kg/s

S21.3590 kg/s

S12.5 kg/s

R12 kg/s

R21 kg/s

S21.3590 kg/s

8-84

How do we supply the necessary MSAs below the pinch? We split the available MSA

and use a portion of its capacity flowrate to remove contaminant from each process stream.

Figure 8.25 illustrates one method for splitting MSA S1 to accomplish the mass transfer below

the pinch. We supply a capacity flowrate of 1.6667 kg/s of MSA S1 to process stream R1 and

0.8333 kg/s to process stream R2. We distribute the capacity flowrate of MSA S1 to the process

streams in proportion to the capacity flowrates of the process streams:

skg

6667.11.02.0

1.0skg

2.5

GGG

aL

aL

R2R1

R2

S1R2S1

=

+

=

+

=

skg

0.83331.02.0

1.0skg

2.5

GGG

aL

aL

R2R1

R1

S1R2S1

=

+

=

+

=

The outlet concentration for both units are equal because we split the capacity flowrate of

MSA S1 proportional to the mass loads of each unit.

Figures 8.26 and 8.27 illustrate grid and mass-content diagrams, respectively, of

complete preliminary mass-exchange networks for Example 8.1.

8-85

0.00805 0.00561

Pinchy = 0.01683

R12 kg/s

R21 kg/s

y = 0.01280

Figure 8.25. Grid diagram of a preliminary mass-exchange network for Example 8.1 below the pinch concentration featuring stream splitting. Concentrations shifted to the process-stream scale (y) with minimum approachconcentrations of 0.001. Mass Loads in kg/s.

y = 0.01

y = 0.006y = 0.01280

0.00805

0.004025

0.004025

S12.5 kg/s

0.0068

0.00561

0.0068

S33.1038 kg/s

S33.1038 kg/s

S21.3590 kg/s

R12 kg/s

R21 kg/s

y = 0.01280y = 0.01683

y = 0.01683

S21.3590 kg/s

y = 0.01683

y = 0.01683

y = 0.012

y = 0.012

0.8333 kg/s

1.6667 kg/s

y = 0.00404y = 0.00804

y = 0.00804 y = 0.00404

1.4025 kg/s1.7012 kg/s

S12.5 kg/s

8-86

0.00805 0.00561

Pinchy = 0.01683

R12 kg/s

R21 kg/s

y = 0.01280

Figure 8.26. Grid diagram of a complete preliminary mass-exchange network for Example 8.1 .Concentrations shifted to the process-stream scale (y) with minimum approachconcentrations of 0.001. Mass Loads in kg/s.

y = 0.01

y = 0.006y = 0.01280

0.00805

0.004025

0.004025

S12.5 kg/s

0.0068

0.037925

0.00561

0.0068

S33.1038 kg/s

S33.1038 kg/s

S21.3590 kg/s

R12 kg/s

R21 kg/s

y = 0.01280y = 0.01683

y = 0.01683y = 0.03579y = 0.05

y = 0.01683y = 0.03

0.01317

S21.3590 kg/s

y = 0.01683

y = 0.03

y = 0.02652

0.028742

y = 0.032

y = 0.01683

y = 0.04743 y = 0.01683

y = 0.01683

y = 0.01683

y = 0.012

y = 0.012

0.8333 kg/s

1.6667 kg/s

y = 0.00404y = 0.00804

y = 0.00804 y = 0.00404

1.4025 kg/s1.7012 kg/s

0.01317

0.037925

0.028742

S12.5 kg/s

8-87

Mass Load (kg/s)

Con

cent

rati

on (y

)

1.0

0.01

0.02

0.03

0.04

0.05

0.06

2.0 kg/s

1.0 kg/s

y = 0.01

y = 0.3

y = 0.006

D

B

A

C

E

y = 0.03579

y = 0.05

y = 0.0128

y = 0.01683

F

0.01

0.02

0.03

0.04

0.05

0.06

2.5 kg/s

1.359 kg/s

3.1038 kg/s

y = .0.00804

y = 0.00404

y = 0.01683

y = 0.02652

y = 0.04743

y = 0.0.12y = 0.01683

y = 0.032A

BC

D E

5.04.03.02.0

y = 0.01683

Gy = 0.0128

Stream Splitting

F G1.4025 kg/s 1.7012 kg/s

1.667 kg/s 0.833 kg/s

Figure 8.27. Mass-content diagram of a complete preliminary mass-exchange network for Example 8.1.Concentrations shifted to the process-stream scale (y) with minimum approach concentrations of 0.001.Mass Loads in kg/s.

8-88

8.7 Network Evolution

In this section, we present a guideline for optimizing preliminary mass-exchange

networks by identify loops within preliminary designs and shifting mass loads away from small,

inefficient units to create fewer, larger, more cost-effective units. We begin by relaxing the

restrictions on preliminary networks and allowing individual exchangers to operate below

minimum approach concentration and/or transfer mass across the pinch.

Figure 8.28 illustrates a grid diagram for the complete preliminary mass-exchange

network for Example 8.3 with a loop (bold dashed line) between mass-exchange units. To

identify the loop, we begin at unit C and proceed toward the bottom of the diagram to MSA S1.

Following, the line representing MSA S1, we reach unit E and follow the unit toward the top of

the diagram to process stream R1. Proceeding toward the left of the diagram along process

stream R1, we return to unit C.

8-89

1.4000

0.0960

Pinchy = 0.5

R15 kg/s

R22 kg/s

y = 0.4692

Figure 8.28. Grid diagram of a complete preliminary mass-exchange network for Example 8.3 with aloop highlighted with bold dashed lines. Concentrations shifted to the process-stream scale (y) withminimum approach concentrations of 0.001. Mass loads in kg/s.

y = 0.45

1.4000

0.1540

0.1540S1

6 kg/s1.2060

0.2200

0.0440 0.2200

0.0960S3

0.05515 kg/sS3

0.05515kg/s

S22.589 kg/s

S16 kg/s

R15 kg/s

R22 kg/s

y = 0.3y = 0.5

y = 0.5y = 0.7412y = 0.75

y = 0.59

1.2060

y = 0.70

0.0440

y = 0.5y = 0.701

y = 0.5y = 0.585

y = 0.602

y = 0.5y = 0.2667

y = 0.241

y = 0.0972y = 0.1692

y = 0.5

R37 kg/s

R37 kg/s

- mD

- mD

+ mD

+ mD

A

B

C

D

E F

8-90

In Figure 8.28, we ignore the pinch concentration and shift a mass load, Dm, across

exchangers to optimize the network design constrained only by feasible mass transfer (i.e.,

positive driving forces) and other practical guidelines (e.g., minimum and maximum unit sizes).

By doing so, we leave the mass balance over each stream, and the minimum external MSA duty,

unchanged, while opening a degree of freedom in the final network design. The shifting of mass

loads across units within the loop can continue until the smallest unit in the loop is eliminated

(unit E, 0.1540 kg/s).

Figure 8.29 shows a simplified mass-exchanger network for Example 8.3 after

eliminating unit E (0 kg/s). However, mass transfer in unit C has become infeasible (i.e., youtR1

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