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Available online at www.sciencedirect.com

Computers and Chemical Engineering 32 (2008) 2203–2216

Selection of optimal feed flow sequence for a multipleeffect evaporator system

R. Bhargava a, S. Khanam b,∗, B. Mohanty a, A.K. Ray c

a Department of Chemical Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, Indiab Department of Chemical Engineering, National Institute of Technology Rourkela, Rourkela 769008, India

c Department of Paper Technology, Indian Institute of Technology Roorkee, Roorkee 247667, India

Received 12 January 2007; received in revised form 3 October 2007; accepted 22 October 2007Available online 11 December 2007

bstract

A nonlinear model is developed for a SEFFFE system employed for concentrating weak black liquor in an Indian Kraft Paper Mill. The systemncorporates different operating strategies such as condensate-, feed- and product-flashing, and steam- and feed-splitting. This model is capable ofimulating a MEE system by accounting variations in τ, U, Qloss, physico-thermal properties of the liquor, F and operating strategies.

The developed model is used to analyze six different F including backward as well as mixed flow sequences. For these F, the effects of variationsf input parameters, T0 and F, on output parameters such as SC and SE have been studied to select the optimal F for the complete range of operatingarameters. Thus, this model is used as a screening tool for the selection of an optimal F amongst the different F.

An advantage of the present model is that a F is represented using an input Boolean matrix and to change the F this input matrix needs toe changed rather than modifying the complete set of model equations for each F. It is found that for the SEFFFE system, backward feed flow

equence is the best as far as SE is concerned.

2007 Elsevier Ltd. All rights reserved.

rator s

dw

po(Laa(tTs

eywords: Screening tool; Optimal feed flow sequence; Flat falling film evapo

. Introduction

An energy audit shows that MEE House of an Indian paperndustry consumes about 24–30% of its total energy (Rao &umar, 1985) and thus any measure to cut down this tax-

ng energy bill by proper selection of operational strategy willmprove the profitability of the plant. Keeping this fact in

ind, since last few decades researchers have tried to developperating strategies for MEE systems, which would consumeinimum amount of live steam and in other words would pro-

ide maximum SE to the system. These operating strategiesnclude flow sequences of feed for operation, feed-, product-nd condensate-flashing, and feed- and steam-splitting. How-

ver, one of the easiest ways to increase SE of the system is toperate the MEE system with optimal F. However, screeninghe optimal F, out of the feasible ones, is time consuming and if

∗ Corresponding author. Tel.: +91 9938185505; fax: +91 661 2462999.E-mail addresses: shabinahai@gmail.com,

khanam@nitrkl.ac.in (S. Khanam).

erMvst

l

098-1354/$ – see front matter © 2007 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2007.10.012

ystem; Steam economy

one experimentally becomes an arduous task. One reasonableay of doing it is to use simulation.For the analysis of MEE system many investigators have

roposed mathematical models. A few of these were devel-ped by Kern (1950), Itahara and Stiel (1966), Holland1975), Radovic, Tasic, Grozanic, Djordjevic, and Valent (1979),ambert, Joye, and Koko (1987), Mathur (1992), Bremfordnd Muller-Steinhagen (1994), El-Dessouky, Alatiqi, Bingulac,nd Ettouney (1998), El-Dessouky, Ettouney, and Al-Juwayhel2000) and Bhargava (2004). These models were not used forhe selection of optimal F. However, for this purpose Harper andsao (1972) developed a model for the optimization of a MEEystem considering forward- and backward-F. Their work wasxtended by Nishitani and Kunugita (1979) to propose an algo-ithm for generating non-inferior F amongst all possible F of a

EE system. The non-inferior F was based on the constraints ofiscosity of liquid and the formation of scale and/or foam. They

uggested that if an optimal F is required, one should examinehe set of non-inferior F.

All these mathematical models are generally based on a set ofinear/nonlinear equations and accommodate effects of varying

2204 R. Bhargava et al. / Computers and Chemic

Nomenclature

a coefficient of Eq. (14)a1–a4 coefficients of cubic polynomial (Eq. (9))aij element of the inverse matrix of [Y–I]A heat transfer area of an effect (m2)b coefficient of Eq. (14)bij a parameter used in Table 8c coefficient of Eq. (14)CPP specific heat capacity of black liquor (J/(kg K))C1–C5 constant in mathematical modelCO condensate flow rate (kg/s)d coefficient of Eq. (14)F feed flow rate (kg/s)F feed flow sequenceh specific enthalpy of liquid phase (J/kg)H specific enthalpy of vapor phase (J/kg)I identity matrixk iteration numberL liquor flow rate (kg/s)MEE multiple effect evaporatorn number of total effectsns number of effects supplied with live steamP pressure of steam and vapor bodies of evaporator

(N/m2)PI performance index as defined in Eq. (23)Qloss heat loss from an evaporator (W)SC steam consumption (kg/h)SE steam economySEFFFE septuple effect flat falling film evaporatorT vapor body temperature of an effect (K)U overall heat transfer coefficient (W/(m2 K))V rate of vapor produced in an evaporator (kg/s)x mass fraction of solids in liquorY flow fraction matrix

Greek charactersα a parameter as defined by Eq. (25)β a parameter as defined in Eq. (34)δ difference between same variable over two suc-

cessive iterations� difference between two parameterθ a parameter as defined in Eq. (29)τ boiling point rise (K)Υ a parameter as defined in Eq. (40)Υ ′ a parameter as defined in Eq. (39)Υ ′′ a parameter as defined in Eq. (41)

Subscriptsavg average of inlet and outlet conditionse exit conditionF feedi effect numberL black liquorV vapor0 steam

pTom

oIselebcmtOarstdtts

htcissias

2

SPdTwafcuS

2

irsf7

al Engineering 32 (2008) 2203–2216

hysical properties of vapor/steam and liquor with temperature.hese models either use fixed value of U for different effectsr variable U based on empirical equations or mathematicalodels.It is a fact that set of model equations developed for a given

perating strategy does not work when the strategy is changed.n fact, it compels one to restructure and reformulate the wholeet of governing equations to address the new operating strat-gy. For example, if operating strategy such as flow sequences,iquor splitting and conditions of flashing is changed, the gov-rning equations, which describe this operating strategy muste changed (Mathur, 1992). The above fact further adds diffi-ulty in simulating all the operating strategies through a givenodel without changing the set of its governing equations. On

he other hand Stewart and Beveridge (1977) and Ayangbile,keke, and Beveridge (1984) developed generalized cascade

lgorithm in which the model of an evaporator body is solvedepeatedly to address the different operating strategies of a MEEystem. With the change in operating strategy the sequence ofhe solution of the model equation of the effect and the inputata to it changes. The solution strategy, employed for solvinghe model, automatically selects the above sequence based onhe input data file where the investigator describes the operatingtrategy.

The modeling technique proposed by Ayangbile et al. (1984)as been further modified and improved in the present worko account for feed sequencing, feed- and steam-splitting, andondensate-, feed- and product-flashing. Besides the above, themproved model also considers variations in τ, U and Qloss fromurfaces of different effects and physico-thermal properties ofteam/vapor, condensate and liquor, which were not accountedn the work of Ayangbile et al. (1987). This model is useds a screening tool for the selection of optimal F for a MEEystem.

. Problem statement

The MEE system selected for above investigation is aEFFFE system that is being operated in a nearby Indian Kraftaper Mill for concentrating weak black liquor. The schematiciagram of a SEFFFE system with backward F is shown in Fig. 1.he first two effects of it are considered as finishing effects,hich require live steam and the seventh effect is attached tovacuum unit. This system employs feed- and steam-splitting,

eed- and product-flashing along with primary and secondaryondensate flashing to generate auxiliary vapor, which are thensed in vapor bodies of appropriate effects to improve overallE of the system.

.1. Operating parameters and their variations

Typical operating parameters of a SEFFFE system are givenn Table 1 and these are considered as a base case. Further, the

esults of base case values are used for comparison with otheruggested cases of operation involving different F. It is seenrom Table 1 that steam temperature going into effect no. 1 is◦C cooler than the steam into effect no. 2. This is an actual

R. Bhargava et al. / Computers and Chemical Engineering 32 (2008) 2203–2216 2205

Fig. 1. Schematic diagram of SEFFFE system.

Table 1Base case operating parameters for the SEFFFE system

S. no. Parameter(s) Value(s)

1 Total number of effects 72 Number of effects being supplied live

steam2

3 Live steam temperatureEffect 1 140 ◦CEffect 2 147 ◦C

4 Black liquor inlet concentration 0.1186 Liquor inlet temperature 64.7 ◦C7 Black liquor feed flow rate 56,200 kg/h8 Last effect vapor temperature 52 ◦C9 Feed flow sequence Backward

10 Feed flash tank positionAs indicatedin the Fig. 1

11 Product flash tank position12 Primary- and secondary-condensate

sTfsTpopic

TR

S

Table 3Geometrical parameters of the flat falling film evaporators

S. no. Parameter Value

1 Area of effect nos. 1–7 540, 540, 660, 660, 660, 660 and 690 m2

2 Size of lamella 1.5 m (W) × 1.0 m (L)34

2

eatiIvo

due to variation in typical physico-thermal properties of blackliquor with the change in temperature and concentration of liquorduring evaporation. In the present investigation various feasibleF, shown in Table 4, are studied in order to obtain the optimal F,

Table 4Feasible F in SEFFFE system

flash tank position

cenario and thus it has been taken as it is during simulation.he plausible explanation is the unequal distribution of steam

rom the header to these effects leading to two different pres-ures in the steam side of these effects. The data presented inable 2 shows the variation in operating parameters of Indianaper mill. The above information is used to study the effectf variation of operating parameters on SE. The geometricalarameters for this system are shown in Table 3. While study-

ng the different F, geometrical parameters are considered to beonstant.

able 2anges of operating parameters of a SEFFFE system

. no. Parameters Variation in value

1 Steam temperatureEffect 1 120–160 ◦CEffect 2 127–167 ◦C

2 Liquor flow rate 56,200–78,680 kg/h

S

a

b

cdef

Type of lamella Pas-Axero Dimpled Plate TypeMaterial of construction SS 2333

.2. Feasible F

F represents the movement of feed in a MEE system. Gen-rally, F is of three types—forward, backward and mixed. Forforward F feed first enters into first effect where evaporation

akes place. Then concentrated feed exits first effect and entersnto second effect. This sequence follows up to the last effect.n a MEE system effects are numbered according to the flow ofapor/steam through the system and this numbering methodol-gy is used for specifying a flow sequence.

The SEFFFE system generally uses backward- and mixed-F

eq. no. F Remarks

7 → 6 → 5 → 4 → 3 → 2 → 1 Backward F, feed to seventheffect

Backward F, feed splitsequally among sixth andseventh effects

6 → 7 → 5 → 4 → 3 → 2 → 1 Mixed F, feed to sixth effects5 → 6 → 7 → 4 → 3 → 2 → 1 Mixed F, feed to fifth effect4 → 5 → 6 → 7 → 3 → 2 → 1 Mixed F, feed to fourth effect3 → 4 → 5 → 6 → 7 → 2 → 1 Mixed F, feed to third effect

2206 R. Bhargava et al. / Computers and Chemical Engineering 32 (2008) 2203–2216

weo(ia

3

f

3

Sd

L

O

V

C

L

O

L

w

�

E

V

Table 5Data for the determination of τ

xi τi (K)

0.0767 0.600.091 0.700.106 0.800.13 1.100.169 1.400.244 2.30000

T

h

w

4

tcw

τ

Tto1a

T

a

wp(w

a

a

a

a

a

Fig. 2. Block diagram of an evaporator.

hich yields highest possible SE. In all these F steam is splittedqually and fed in first two effects. These F are selected basedn the flow sequences of MEE systems reported in the literatureRay & Singh, 2000) as well as that have been employed inndustry. However, the present model can be used to simulateny flow sequence.

. Mathematical model

The complete model for the SEFFFE system is developed inollowing sections.

.1. Model for an effect

By taking mass and energy balances over ith effect of aEFFFE system, shown in Fig. 2, following equations can beeveloped.

Overall mass balance around evaporation section:

i+1 = Li + Vi (1)

verall mass balance around steam chest:

i−1 = COi−1 (2)

omponent mass balance:

i+1 xi+1 = Lixi = LFxF (3)

verall energy balance:

i+1hLi+1 = LihLi + ViHVi + �Hi (4)

here

Hi = UiAi(Ti−1 − TLi ) (5)

nergy balance on steam/vapor side:

i−1 = �Hi

HVi−1 − hLi−1

(6)

3

a

.369 4.30

.462 6.20

.47 6.40

he correlations for enthalpy (hL) for black liquor is given as

L = CPP(TL − C5) J/kg (7)

here CPP = C1(1–C4x) and TL = T + τ.The values of coefficients C1, C4 and C5, used in Eq. (7), are

187, 0.54 and 273, respectively.For the development of a correlation for τ of black liquor,

he functional relationship is taken from well-established TAPPIorrelation (Ray, Rao, Bansal, & Mohanty, 1992). For ith effecthere concentration of black liquor is xi, τ is given as

i = C3(C2 + xi)2 (8)

he correlation of τ is obtained by fitting equation (8) againsthe data of a nearby paper mill as given in Table 5. The valuef C3 is computed as 20, using value of C2 as 0.1 (Ray et al.,992). This correlation gives a maximum error of 6% with anverage error of 3.4%.

Combining Eqs. (1)–(8) and eliminating Vi, xi, hi, �Hi andLi one gets following cubic polynomial in terms of Li:

1L3i + a2L

2i + a3Li + a4 = 0 (9)

here coefficients a1, a2, a3 and a4 are functions of input liquorarameters and other known parameters like heat transfer areaAi) and overall heat transfer coefficient (Ui) of the effect forhich it is being used.The expression for coefficients a1, a2, a3 and a4 are

1 = HVi − C1Ti − C1C22C3 + C1C5 (9a)

2 = Li+1hLi+1 + UiAi(Ti−1 − Ti − C3C22)

+Li+1xi+1(C1C4Ti − 2C1C2C3 + C1C3C22C4

−C1C4C5 − Li+1HVi ) (9b)

3 = (Li+1xi+1)2(2C1C2C3C4 − C1C3)

−2C2C3UiAiLi+1xi+1 (9c)

nd

4 = (C1C3C4Li+1xi+1 − C3UiAi)(Li+1xi+1)2 (9d)

.2. Model for liquor flash tank

For a liquor flash tank, shown in Fig. 3, a similar cubic models developed for an effect, presented in Eq. (9), is proposed to

R. Bhargava et al. / Computers and Chemical Engineering 32 (2008) 2203–2216 2207

cf

a

a

a

a

Ictflila

Olktrr

3

y(aam

C

a

V

Table 6Cross-correlation coefficients between the parameters U, delta T (�T), Xavg andFavg

OHTC delta T Xavg Favg

OHTC 1delta T −0.96999 1XF

3

rdf

pdcphQ

Q

wtssitd

ehcc

ftin1bo1n

onba

Fig. 3. Block diagram of a liquor flash tank.

alculate Le for a known value of L1. The modified expressionsor constants a1–a4 in Eq. (9) are described below:

1 = HVe − C1Te − C1C22C3 + C1C5 (10a)

2 = L1hL1 + L1x1(C1C4Te − 2C1C2C3

+C1C22C3C4 − C1C4C5) − L1HVe (10b)

3 = (L1x1)2(2C1C2C3C4 − C1C3) (10c)

4 = (L1x1)3C1C3C4 (10d)

t should be noted here that in the present investigation only oneubic equation with different expressions of coefficients is usedo model an effect as well as liquor flash tank (feed and productash tanks). Hence, in this case only one cubic equation solver

s used to predict the exit liquor flow rate from an effect andiquor flash tank which reduces the length of computer programs well as burden of computation.

The cubic equation, Eq. (9), is solved to get its real root(s).ut of the real roots only one root, which has a value equal or

ess than F, is selected for further processing. Once this root isnown, other parameters like exit liquor-concentration, liquor-emperature, vapor produced (Vi) and the quantity of vaporequired (Vi−1) are computed using Eqs. (3), (4), (1) and (6),espectively.

.3. Model for condensate flash tank

Material and energy balances over condensate flash tankields following relation to determine exit condensate flow rateCOj), for a known condensate mass flow rate, COi, entering attemperature, Ti, with specific enthalpy, hi, and being flashed

t temperature, Tj to generate vapor of amount, Vj. The overallass and energy balance give

Oj = COi(HVj − hi)

HVj − hj

(11)

nd

j = COi − COj (12)

e

avg 0.928086 −0.92326 1

avg −0.81645 0.875356 −0.90815 1

.4. Empirical correlations for U and Qloss

The present model for a SEFFFE system uses empirical cor-elations for the prediction of U and Qloss from each effect. Forevelopment of these empirical correlations the data is collectedrom a Indian Kraft Paper Mill employing SEFFFE system.

A simplified empirical model for Qloss from different effects,iping, heat exchangers, flash tanks, etc. of a SEFFFE system iseveloped based on the correlation of heat transfer for naturalonvection (Nu = C(Gr·Pr)n) (Coulson & Richardson, 1996) andlant data. The power law equation, which is a modified form ofeat transfer due to natural convection, is proposed to calculateloss as

loss = 1.9669 × 103(�t)1.25 (13)

here �t is the temperature difference exists between tempera-ure of effect and that of atmosphere. Predictions from Eq. (13)how an error limit of −33 to +29%. In the present SEFFFEystem the average Qloss is of the tune of 5.8% of total energynput to the system through steam and liquor. It appears thathe present Qloss is at a higher side in the plant may be due toegraded insulation.

To develop the empirical correlation for U of an effect, param-ters �T across the steam chest of an effect, xavg and Favgave been considered as these parameters show a high level ofross-correlation amongst themselves. The statistical analysis ofross-correlation between these parameters is shown in Table 6.

Further, a plot of calculated values of U of all seven effects forour different sets of plant data is shown in Fig. 4. The analysis ofhis figure clearly shows that the trend of U for first two effects,.e. effect nos. 1 and 2 is totally different than the trend of effectos. 3–7. The values of U are substantially low for effect nos.and 2 which is primarily due to the higher concentration of

lack liquor in first two effects (43–53%). In fact, in the vicinityf 48% solid concentration the scale formation starts (Suren,995). This phenomenon causes U to fall drastically in effectos. 1 and 2.

Therefore, two different empirical correlations are developed,ne for effect nos. 1–2 and the other for effect nos. 3–7. Theormalized Power law equation, shown in Eq. (14), is used foroth the correlations using divisors 2000 W/(m2 K), 40 ◦C, 0.6nd 25 kg/s as these are higher than the respective highest valuesncountered in the plant data:

U

2000= a

(�T

40

)b(xavg

0.6

)c(

Favg

25

)d

(14)

2208 R. Bhargava et al. / Computers and Chemical Engineering 32 (2008) 2203–2216

Fig. 4. Profiles of U from plant data.

Table 7Value of coefficients of Eq. (14)

Effect no. a b c d % Error band Eq.

13

TeaSTa

F(

FE

3

Agffl

wT

y

wei

and 2 0.0604 −0.3717−1.227 0.0748 −11.32 to 7.25 (14a)–7 0.1396 −0.7949 0.0 0.1673 −11.75 to 8.20 (14b)

he computed values of U from plant data, for all the sevenffects, are used to estimate the unknown coefficients a, b, cnd d of Eq. (14) using constrained minimization technique ofigma Plot—a scientific data analysis and plotting software.he estimated coefficients for both the correlations (Eq. (14a)nd (14b)) are given in Table 7.

Fig. 5 has been plotted to show the extent of fitting of the U.rom this figure it is clear that the correlations, Eq. (14a) and14b), predict the U data within an error limit of +10%.

ig. 5. Comparison of U from plant data for all effects and those predicted fromq. (14a) and (14b).

Tcb

Fig. 6. Block diagram of an evaporator for cascade simulation.

.5. Development of generalized model for a MEE system

A MEE system consisting of n effects has been modeled byyangbile et al. (1984). This model is improved to develop aeneralized model which can account different flow sequencing,eed- and steam-splitting, and condensate-, feed- and product-ashing.

The modified block diagram of ith effect is shown in Fig. 6,hich accommodates any flow sequencing and liquor splitting.he black liquor feed rate to ith effect can be expressed as

oiLF +n∑

j = 1

j �= i

yjiLj (15)

here yoi is the fraction of the feed (after feed flash), whichnters into ith effect and yji is the fraction of black liquor whichs coming out from jth effect and enters into the ith effect.

Total mass balance around ith effect gives

yoiLF +n∑

j=1

yj,iLj = Li + Vi or

n∑j=1

yj,iLj − Li = Vi − yoiLF (16)

his equation, developed for ith effect and shown in Eq. (16),an be represented for all n effects by a Matrix equation as givenelow:⎡

⎢⎢⎢⎢⎢⎢⎢⎣

y11 − 1 y21 y31 ... yn1

y12 y22 − 1 y32 ... yn2

y13 y23 y33 − 1 ... yn3

......

... ......

y1n y2n y3n ... ynn − 1

⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎣

L1

L2

L3

...

Ln

⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢

V1 − yo1LF

V2 − yo2LF

⎤⎥⎥⎥

= ⎢⎢⎢⎢⎣V3 − yo3LF

...

Vn − yonLF

⎥⎥⎥⎥⎦

emic

i

[

o

L

wi−

s

L

S

y

o

∑

o

[

w

L

o

L

U

L

C

x

Ft(L

ocvs(ccVfi

P

Tecgv

fab

vd

w

R. Bhargava et al. / Computers and Ch

.e.:

Y − I]L = V (17)

r

= [Y − I]−1V = AV (18)

here A is the inverse of the matrix [Y − I]. As the value of yjj

s equal to 0, the diagonal elements of matrix [Y − I] become1.From Eq. (18), the exit liquor flow rate, Li, from ith effect as

hown in Fig. 6, can be written as

i =n∑

j=1

aijVj − LF

n∑j=1

yojaij (19)

imilarly, component mass balance around ith effect provides:

oiLFxF +n∑

j=1

yjiLjxj = Lixi

r

n

j=1

yjiLjxj − Lixi = −yoiLFxF

r

Y − I]LX = −Y0LFxF

here

X = [ L1x1 L2x2 L3x3 . . . Lnxn ]

r

X = [Y − I]−1(−Y0LFxF) = A(−Y0LFxF) (20)

sing Eq. (20), the total solids coming out of ith effect is

ixi = −⎛⎝ n∑

j=1

yojaij

⎞⎠ LFxF (21)

h

δ

Fig. 7. The schematic diagram of vapor flow

al Engineering 32 (2008) 2203–2216 2209

ombining Eqs. (19) and (21):

i =(∑n

j=1yojaij

)LFxF(∑n

j=1yojaij

)LF − ∑n

j=1aijVj

(22)

or the development of a general model of an evaporator sys-em, mathematical model for ith effect as given by Eqs. (9) and9a)–(9d) is generalized by replacing the inlet liquor flow term,i+1, by expression given in Eq. (15).

The model for an evaporator, shown in Fig. 6, is solved tobtain Li, xi, Vi and Vi−1. The vapor required in ith effect steamhest, Vi−1, is re-designated as Vbi. Whereas, the actual pool ofapor available for ith effect steam chest consists of several vaportreams at same operating pressure, namely vapor produced ini−1)th effect and the vapor produced by feed-, product- andondensate-flashing and is designated as Vi−1. This has beenlearly shown in Fig. 7. The amount of vapor denoted by Vi−1 andbi should be equal for an exact solution. An index called “per-

ormance index (PI)” is defined as a measure of the differencen the values of Vi−1 and Vbi:

I =∑ (

Vi−1 − Vbi

Vbi

)2

(23)

he summation term shown in Eq. (23) is for ‘ns + 1’ to ‘n’ffects, where ns effects are fed with live steam and in presentase is equal to two. The summation of Vbi for first ns effectsives total SC, and summation of Vi from first ns effects is theapor to (ns + 1)th effect, as shown in Fig. 7.

The operating parameters Ti, TLi , Vi and xi of ith effect areunctions of vapor body pressure, Pi. If variations of parametersre within a small range then one can always approximate it toe a linear one.

As proposed by Ayangbile et al. (1984), it is assumed thatapor produced in an effect, Vi, is a function of temperatureifference (driving force, �Ti) responsible for heat transfer:

here �Ti = Ti−1 − TLi (24)

ence

Vi = ∂Vi

∂(�Ti)(δ�Ti)

in a MEE system consists of n effects.

2 hemic

d

δ

Ts

δ

w

s

δ

F

δ

Co

δ

w

θ

D

a

δ

a

δ

F(t

V

Ot

�

Ap

δ

C

a

(

w

ei

oe

D

w[Ei

a

β

γ

γ

γ

wt

�ae

(

4

210 R. Bhargava et al. / Computers and C

efining ∂Vi/∂(�Ti) = αi:

Vi = αiδ(�Ti) = αi(δTi−1 − δTLi ) (25)

o determine change in pressure, Pi, with a perturbation inaturation temperature, Ti, one can get

Ti = ∂Ti

∂Pi

δPi = γiδPi (26)

here γ i = (∂Ti/∂Pi).The black liquor temperature is functions of pressure and

olid concentration and hence

TLi = ∂TLi

∂Pi

δPi + ∂TLi

∂xi

δxi = γ ′i δPi + γ ′′

i δxi (27)

rom Eq. (22), one can write:

xi =∑n

j=1(yojaij)LFxF∑n

j=1aijδVj[∑nj=1(yojaij)LF − ∑n

j=1aijVj

]2 (28)

ombining Eqs. (27) and (28), the following expression can bebtained:

TLi = γ ′i δPi + θi

n∑j=1

aijδVj (29)

here

i = γ ′′i

∑nj=1(yojaij)LFxF[∑n

j=1(yojaij)LF − ∑nj=1aijVj

]2

efining �Vi=Vi−1−Vbi, for i=n, n−1, . . . , ns+1 (30)

nd for kth iteration, changes in Vi−1 and Vbi are defined as

Vi−1 = Vki−1 − Vk−1

i−1 (31a)

nd

Vbi = Vkbi − Vk−1

bi (31b)

or the solution of MEE system, the vapor available fromi − 1) th effect should be equal to vapor required in ith effect,hat is

ki−1 ≈ Vk

bi (32)

n combining Eqs. (30)–(32) and rearranging, following equa-ion is formed:

Vi = δVbi − δVi−1 (33)

s vapor required for heating in an effect is a function of vaporroduced in that effect, on linearization, one gets:

Vbi = ∂Vbi

∂Vi

δVi = βiδVi (34)

ombining Eqs. (25), (26), (29), (33) and (34), one gets

δVi − βi+1 δVi+1 = −�Vi+1,

where i = 1, 2, . . . , (n − ns) (35)

trSi

al Engineering 32 (2008) 2203–2216

nd

1 + αiθiaii)δVi + αiθi

i−1∑j=1

aijδVj

+ αiθi

n∑j=i+1

aijδVj − αiγiδPi−1 + αiγ′i δPi = 0 (36)

here i = 1, 2, . . ., (n − ns + 1).Eqs. (35) and (36) form a set of (2(n − ns) + 1) linear algebraic

quations with same number of unknowns, namely, �Vi, where= 1, 2, . . ., n − ns + 1 and �Pi, where i = 1, 2, . . ., n − ns.

Matrix representation of [2(n − ns) + 1] linear algebraic setf equations given by Eqs. (35) and (36) is given in followingquation form:

Z = E

here the size of the coefficient matrix D is2(n − ns) + 1][2(n − ns) + 1]. Similarly, the size of Z and

matrices is 1[2(n − ns) + 1] each. These matrices are detailedn Table 8 for the SEFFFE system.

αi, βi, γ i and γ ′i for different iteration number, denoted by k,

re defined as

αi = Vki

�T ki

, k = 1;

αi = Vki − Vk−1

i

�T ki − �Tk−1

i

, k = 2, 3, . . . (37)

i=Vkbi

V ki

, k=1; βi= Vkbi−Vk−1

bi

V ki − Vk−1

i

, k=2, 3, . . . (38)

′i = ∂TLi

∂Pi

= ∂(Ti + BPRi)

∂Pi

= ∂Ti

∂Pi

(39)

i = ∂Ti−1

∂Pi−1(40)

′′i = ∂TLi

∂xi

= ∂(Ti + BPRi)

∂xi

= 2C3(C2 + xi) (41)

here derivatives, γ ′i and γ i, given by Eqs. (39) and (40), respec-

ively.By solving Eqs. (35) and (36), one can obtain the values of

Pi, where i = 1, 2, . . ., (n − ns). The values of �Pi, so obtained,re used to modify the pressure of all the effects except lastffect—the pressure of which is kept at a fixed value.

This process is to be repeated iteratively till desired precision5 × 10−6) in the value of PI is obtained.

. Solution of the model

For the solution of the model, developed in the present inves-

igation, a computer program is developed in FORTRAN 90 andun on Pentium IV machine, using Microsoft FORTRAN Powertation 4.0 compiler. The flow chart of the algorithm is shown

n Fig. 8.

R. Bhargava et al. / Computers and Chemical Engineering 32 (2008) 2203–2216 2211

Table 8Matrix representation of set of linear algebraic equations

Where bij = αiaijθi.

Fig. 8. Flow chart for solution of the model.

2 hemical Engineering 32 (2008) 2203–2216

fuw

tartbie

stetpBserreSa

Ioit

Ft

5

td

bvfctlperror of 23% between plant data and simulation result. However,for the similar MEE system the published model (Bremford &Muller-Steinhagen, 1994) reported a maximum error of 43.43%

212 R. Bhargava et al. / Computers and C

For the generalized cascade algorithm the equation developedor an effect is solved repeatedly for different effects dependingpon the sequence of computation determined by the selected Fhich in turn is decided by the Boolean matrix B.For the SEFFFE system, order of the matrix is 8 × 8, where

he first column denotes the feed stream and subsequent columnsre source effects 1–7 and first 7 rows are sink effects and lastow is product stream. A unit value of element bij indicateshat liquor exiting from (j − 1)th effect enters ith effect. Theackward F is represented by the Boolean matrix, given below,n which element b13 = 1 shows that liquor exits from secondffect and enters first effect:

If a liquor stream exiting from (j − 1)th effect is splitted forupply to more than one effect then it will be depicted by morehan one entry of unit value in the jth column. Similarly, if a sinkffect (ith) receives liquor from more than one source effecthen, in the ith row of matrix B there will be more than onelace, where unit value will exist. This is clearly represented byoolean matrix for flow sequence, b, given below; in which feed

tream is splitted to enter in effect nos. 6 and 7, thus unit valuexists at two places in the first column and sixth and seventhows. Further, these splitted streams enter into fifth effect givingise to two entries of unit value in fifth row and seventh andighth columns, which represent effect nos. 6 and 7, respectively.imilarly, Boolean matrices for condensate and liquor flashingre drawn:

t should be noted that in the present investigation to select theptimal F, one has to re-configure the Boolean matrix only whichs an input data file to the computer program and need not derivehe model equations for each F separately.

Fd

ig. 9. Comparison between solid concentration in liquor from plant data andhat predicted by model.

. Validation of model

To establish the reliability of the present model it is requiredo compare the results predicted from simulation with the plantata obtained from a nearby paper mill.

Figs. 9 and 10 have been plotted to show the comparisonetween data of the mill for concentration of black liquor andapor body temperature of different effects with that obtainedrom model, respectively. Predicted results show that the liquoroncentration match within an error band of −0.2 to +0.4%, andhe vapor temperature of different effects match within a errorimit of −0.26 to +1.76%. The present model computes the tem-erature difference (�T) for each effect with a maximum relative

ig. 10. Comparison between vapor temperature of different effects from plantata and that from model.

emical Engineering 32 (2008) 2203–2216 2213

fiw

6

usiflrproeid

eaooaaaflrrtoaetu

6

trs

6

FwiSfl‘

htite

fT

oc

om(eltsniain case of U�T is also observed for SC.

Fig. 13 shows the effect of T0 on value of SE for differentflow sequences a, b, c, d, e and f. The figure clearly indicatesthat SE gradually decreases with the rise in steam temperature.

R. Bhargava et al. / Computers and Ch

or the prediction of temperature difference in each effect. Thus,t appears that the present model predicts the plant data fairlyell in comparison to the published model.

. Selection of optimal F

A SEFFFE system employed in a Paper Mill can be operatednder different operating strategies, which include different flowequences as detailed in Table 4. The SEFFFE system consideredn the present work includes feed-, product- and condensate-ashing as well as feed and steam splitting. Out of the above-eferred operating strategies, selection of proper F, which willrovide maximum SE, is the aim of the present study. In fact, ineal situation, the best F is not only decided by SE but also basedn other parameters such as foaming of feed, scaling of tubes,tc. Nevertheless, grading of F based on SE has always been anmportant step towards finalization of best flow sequence whichepends on SE data and operating convenience.

To screen the optimal F amongst the feasible F, it is nec-ssary to study the effect of variation of input parameters T0nd F on output parameter, SC and SE. It will help to select anptimal F, which will remain as optimal for complete range ofperating parameters given in Table 2. It is well-known fact thatchange in input parameters triggers a complex chain reaction

ffecting almost all output parameters, including, vapor bodynd liquor temperatures from first to sixth effect, almost allash vapor fractions, heat transfer coefficients, rate of evapo-ation in each effect, Qloss from each effect and amount of heatejected to atmosphere with condensate. It appears that SE ishe single most prominent parameter to evaluate the efficiencyf the system as it varies with variation in operating parametersnd geometrical parameters as well. SE also directly affects theconomy of the operation. Nevertheless, the study of variablesowards the variation of SC with input parameter offers betternderstanding.

.1. Effect of variations of T0 and F on SC and SE for all F

An investigation based on the present model has been under-aken to study the effect of steam temperature, T0, and feed flowate, F, on SC and SE, which is carried out for different flowequences a, b, c, d, e and f as defined in Table 4.

.1.1. Effect of steam temperatureFig. 11 shows the effect of steam temperature, T0, on SC for

, a, b, c, d, e and f and shows that the value of SC increasesith the increase in steam temperature for all the flow sequences

nvestigated. However, for a given value of steam temperature,C for different F is markedly different. In fact, SC for backwardow sequence is minimum whereas, SC for the flow sequencef’ is maximum followed by e, d, c, b and a in decreasing order.

The effect of T0 on SC is attributed to the decrease in latenteat of condensation of the steam with the increase in the satura-

ion temperature. Further, an increase in the steam temperature,ncreases the temperature difference between steam and liquor,hus provides conducive environment to pump more heat in to theffect causing more evaporation. The combined effect, of above

Fig. 11. Effect of T0 on SC for feed sequence a–f.

actors, is such that the value of SC increases with increase in0.

In the present system live steam is used in first two effectsf the SEFFFE system. When any operating strategy like F ishanged it alters the SC of these two effects.

The variation in the values of SC for F, a–f, for a given valuef T0 can be explained as follows: the SC for a flow sequenceainly depends on values of two factors: U and driving force

�T) available for transfer of heat across the steam chest of anffect as the heat transfer area of all the effects are constant. Asive steam is fed to effect nos. 1 and 2, the product of abovewo factors (U�T) are computed for these effects for all flowequences and are shown in Fig. 12. From this figure, it can beoted that values of (U�T) in first two effects are decreasingn the following order of flow sequences: f, e, d, c, b and a. Asreas of first two effects are constant similar order as observed

Fig. 12. Values of (U�T) in first two effect for six FFSs.

2214 R. Bhargava et al. / Computers and Chemic

Taa

tetobv

afio

sfntdaIms

Ft

lsbfoSo

6

Ua

iSoi

SS

icoptteaiet

tfsmth

Fig. 13. Effect of T0 on SE for feed sequence a–f.

he maximum SE is observed for F ‘a’ followed by c, b, d, end f in decreasing order. However, the values of SC for F, bnd d, are close to each other.

For a given F the value of SE mainly depends on total watero be evaporated and live steam consumed to carry out the abovevaporation. To show the variation of SC, SE and total evapora-ion with change in F, Fig. 14 is drawn in which fractional valuesf these parameters for F, b–f, with respect to ‘a’ are plotted forase case operating conditions, given in Table 1. The fractionalalue of SC for flow sequence, b with respect to a is defined as

amount of SC for flow sequence, b

amount of SC for flow sequence, a

Similarly, fractional values of other parameters are obtainednd shown in Fig. 14. The dotted line in this figure shows theractional values of SC, SE and total evaporation for F ‘a’ whichs obviously unity. This line is used for the sake of comparisonf other F with that of ‘a’.

Fig. 14 shows that the total water evaporated for flowequences, d, e and f is slightly more (within + 1%) than thator ‘a’. For F ‘c’ fractional value of total water evaporated isearly unity whereas for F ‘b’ it is less by 2.32% in comparisono ‘a’. As a result, the product concentration, xP, is different forifferent F. However, the fractional values of SC for F, ‘b’–‘f’

re greater than unity and continuously increase from F, ‘b’–‘f’.t is clear from Fig. 14 that the increase in the value of SC is muchore than increase in total evaporation for F, d–f in compari-

on to ‘a’. Hence, the values of SE for these F are considerably

ig. 14. Total evaporation, SC and SE for flow sequences ‘b’–‘f’ with respecto ‘a’.

t5btr

Sfld‘aSda‘

Si

al Engineering 32 (2008) 2203–2216

ess. It is also noted from Fig. 14, that total evaporation for flowequence, ‘c’ is equal to that of ‘a’, whereas, SC for ‘c’ is greatery 1.2% in comparison to ‘a’. The cumulative effects of theseactors lower the value of SE for flow sequence ‘c’. It is furtherbserved from Fig. 14, that for F ‘b’ total evaporation is less andC is slightly more than that for ‘a’. As a result of it the valuef SE is lowest for F ‘b’.

.1.2. Effect of feed flow rateTable 9 shows the effect of variation of feed flow rate on

�T, total evaporation, SC and SE for different flow sequences–f.

Table 9 shows that the value of SC increases with the increasen black liquor feed rate. For lowest value of feed flow rate, theC is highest for F, f followed by e, d, c, b and a in decreasingrder. However, at highest flow rate the F f, e, d, c, a and b aren order of decreasing value of SC.

Further, Table 9 indicates that for a flow sequence the value ofE decreases with increase in feed flow rate and the maximumE is observed for the F ‘a’ followed by c, b, d, e and f.

When feed flow rate is increased the U of different effectsncrease. This in turn increases heat flow to each effect andauses the temperature of each effect to increase due to formationf higher amount of vapor in each effect, which subsequentlyressurizes the vapor body to some extent. This rise in tempera-ure will have a reverse effect on heat transfer rate as it decreaseshe driving force �T. However, the overall effect will be gov-rned by the product U�T. As the behaviors of first two effectsffect the SC most, U�T values of only these effects are reportedn Table 9. From the simulated results, shown in Table 9, it isvident that U�T increases with increase in feed flow rate andhus the value of SC also increases with feed flow rate.

Table 10 shows the percent variation in the values of U�T,otal evaporation, SC and SE with change in feed flow raterom 56,200 to 78,680 kg/h (a 40% increase) for different flowequences. This table presents the data of Table 9 in a differentanner. Table 10 clearly shows that the values of U�T in first

wo effects increase with increase in feed flow rate resulting inigher SC in all F. However, the average value of U�T of firstwo effects increases by 27% when the feed rate is changed from6,200 to 78,680 kg/h for F ‘a’ and the value of U�T increasesy 41.7% for F ‘f’ for the same variation in feed rate. Due tohis, the percent increase in SC for different F with feed flowate is in the following order: b > (a = c) > (d = e) > f.

As can be seen from Table 10, total evaporation as well asC increase for all the flow sequences with the increase in feedow rate. However, the cumulative effect is such that the SEecreases with the increase in feed flow rate. For example, fora’ when feed rate is increased from 56,200 to 78,680 kg/h themount of total water evaporated increases by 12.6% whereasC rises by 27.44%. The cumulative effects of above parametersecrease the SE by 11.64%. Similar behavior is seen for other Flso. Further, it is observed that the flow sequences ‘a’, ‘b’ and

c’ almost behave similarly as far as SE is concerned.

From the above investigations, it is clear that for the presentEFFFE system, backward F is found to be the optimal F as

t offers highest value of SE for complete operating range of

R. Bhargava et al. / Computers and Chemical Engineering 32 (2008) 2203–2216 2215

Table 9Values of U�T, total evaporation, SC and SE for minimum and maximum feed flow rate

Parameter(s) a b c d e f

Feed flow rate = 56,200 kg/hU�T (W/m2)

1 3236.3 3240.4 3285.8 3380.1 3600.8 4131.32 6062.5 6140.6 6128.4 6276.9 6635.6 7357.1

Total evaporation (kg/h) 43916.6 42897.3 43916.4 43961 44076.4 44270.9

SC (kg/h) 8776 8863 8881.2 9100 9622 10742.6SE 5.00 4.84 4.94 4.83 4.58 4.12

Feed flow rate = 78,680 kg/hU�T (W/m2)

1 4035.6 3919.4 4116.7 4317.7 4757.1 6133.42 7852.6 7928.1 7921 8139.6 8597.2 9924.8

Total evaporation (kg/h) 49450.3 47876.8 49468.6 49592.4 50278.2 51521.7

SC (kg/h) 11,184 11154.5 11322.3 11702.2 12512.7 14954.3SE 4.42 4.29 4.37 4.24 4.02 3.45

Table 10Percent variations in U�T, total evaporation, SC and SE when the feed flow rate in increased by 40%

Parameter(s) a b c d e f

U�T (W/m2)1 24.70 20.95 25.29 27.74 32.11 48.462 29.53 29.11 29.25 29.67 29.56 34.90

TSS

pso

7

a

1

2

3

4

R

A

B

B

C

E

E

H

H

I

KL

M

N

otal evaporation (kg/h) 12.60 11.61C (kg/h) 27.44 25.85E −11.64 −11.32

arameters. It is interesting to note that the present SEFFFEystem employed in a nearby Indian Kraft paper industry alsoperates under backward F to achieve maximum SE.

. Conclusions

The salient conclusions drawn from the present investigationsre:

. The empirical correlation of U for a SEFFFE system predictsthe plant data within ±10%.

. The present model computes the liquor concentration profileand vapor body temperature profile of different effects withinmaximum error limit of ±2%.

. The model developed in the present paper can work as aneffective screening tool for the selection of optimal F.

. Out of the different F investigated for SEFFFE system, thebackward F was found to be optimal leading to highest valueof SE.

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