General calculation method for stagewise models of mass and heat transfer operations

Chemicnl Engineering Science, Vol. 44, No. 2, pp. 343 Printed in Britain.

352,

1989.

000%2509/89 S3.OOt0.00 0 1989 Pergamon Press plc

Great

GENERAL CALCULATION MODELS OF MASS AND

Department

METHOD FOR STAGEWISE HEAT TRANSFER OPERATIONS

HARRY V. NORD6Nt and MARTTI A. PEKKANEN of Chemical Engineering, Helsinki University of Technology, 02150 Espoo, Finland (Received 12 January 1988; accepted 21 June 1988)

Abstract-The paper presents a new calculation method for steady-state stagewise mass and heat transfer operations. The method is restricted to linear equilibrium relationships. The ideal stage model used is a new one and equations obtained are quite simple which makes the treatment of complicated flow sheets possible. For practical calculations the equations are presented in matrix form. Equations are derived for various side streams and for piecewise linear equilibrium relationship. The calculation method can also be applied to continuous contact operations if the number of ideal stages N can be expressed by the equivalent number of transfer units No,. This makes it applicable to heat transfer, too. A very general flow rate transformation is presented by means of which constant flow rates may be obtained.

are transformed in the beginning of the calculations, the transformed equilibrium line can be used for the whole study. If a constant is subtracted from all of the concentrations, as in eq. (I), all solute mass balances hold using the transformed concentrations.

1. INTRODUCTION

The concept of an equilibrium or ideal stage is an extremely useful tool in the field of chemical engineering. Models based on cascades of ideal stages can be used for very many mass and heat transfer operations, whether stagewise or not. The most well known calculation methods are the ones of Kremser (1930), Souders and Brown (1932), Horton and Frankling (1940), Edminster (1943) and Smith and Brinkley (1960). In addition, several other equations analogous to these early absorption or stripping factor equations have been presented. The new calculation method to be presented is an attempt to develop a general and versatile absorption factor method of calculation, the features of which are, e.g. the model used is a new one and offers simple modelling possibilities for different unit operations, the equilibrium relationship need not be y.* = WCC”,but must only be linear or piecewise linear, the equations used hold even if there are certain side streams which may he useful in practical calculations and a flow rate transformation can be used to obtain constant flow rates. Because of the generality and versatility of the new method, it should offer advantages in industrial practice and education as it eliminates the need for several different methods for calculation.

2.

3. A SINGLE

The treatment units that

-have a linear equilibrium relationship same line for the whole unit and Nonform to the model in Fig. 1.

yb=y,--u,

EQUILIBRIUM

where u = b/( I-

m)

rates, but the concentrations yi is always

equal

must not be changed,

to the concentration

X0

Xl

x2

xN-l

XN

xb

Fig. 1. Countercurrentstagewise model of a unit.

should be addressed. 343

i.e.

of the V phase,

a

xa to whom correspondence

which is the

f$jij&$~~b

(1)

can be used to obtain an equilibrium line &* = mx:, i.e. to eliminate the constant b. If all of the concentrations

?Author

to single

3.1. The model The fundamental model used in the calculation method is presented in Fig. 1. In the context of this paper, a unit is an entity of calculational and not of physical significance. A unit can represent a physical mass or heat transfer device, a part of a device or several physical devices. The unit in Fig. 1 consists of ideal stages and interstreams. Normally, streams labelled with L and V represent phases with greater density and lower density, respectively, e.g. the liquid and vapour flows in distillation. The interstreams Vi and Li can be useful, e.g. in leaching, washing, distillation and stripping. In distillation, for example, one could specify Vi = L, and get a simple model of the top of a distillation column. Interstreams Vi and Li can be. given negative flow

The calculation method to be presented is applicable if the equilibrium relationship is linear (or linearized). If the equilibrium line is y,* = mx, + b, then the following concentration transformation x:,=x,-u,

UNIT

in this section is restricted

344

HARRY

V.NORDI?.N

and MARTTI

and xi is equal to the concentration of the L phase. If this were not so, the interstreams Vi and Li could not be combined with the first and last ideal stages, respectively. It is a convention of the model in Fig. 1 used for stagewise calculations that changes in the flow rates of the interstreams Vi and Li affect the flow rates in streams L,, V,,Land V,+i only. It can be seen that if all of the other quantities remain the same, the flow rates Vi and Li can be chosen freely. Another useful consequence of this convention and ofthe requirement yi =yU and xi=xb is that concentrations are affected only in the streams LO and V,, i. 3.2. Basic equations Consider the model for a single unit presented in Fig. 1. The equilibrium relationship Yz = mx, + b for an ideal stage n can be manipulated using the definition of C to give the basic difference equations of the new method ~“Y”=S”(v”+1Y”+1--)+

V,b,

v,Y:,=S”(v”+,Y:,+,-C’)

(2a) (2b)

where C is the net solute flow rate cocurrent to the Y phase. As eqs (2) state, in fact, only that the outgoing streams are in equilibrium, they hold for stages 1 and N even if the interstreams Vi and Li are combined with them. Each combined stage is still equivalent to one ideal stage, but the stripping factors S1 and S, are replaced by S, = m V,/L, and S, = m r/,/L,. Thus for the first and last combined stage, respectively, v,Y‘z=S,(v,Y,-C)+

va’,b,

VNYYN= S,( VhYb- C) + V&P V,Yb = &( v,y;,

v*Y; -

(34

V-y, = I Vayb - JC + Kb,

(W

v,y: = Ivhy; - JC’.

(W

where I, J and K are parameters

n

“=2

calculated

as follows

s”=;g fi1 s,,

(54

.l=s,+s,s,+s,s,s,+

+S,S,S, K=

1

. . .

b”

+s,s,s, . . . s,_,s,_,

. . . s,-Is,,

v,+s,v,+s,s,v,+ . . . SN-ZVN-1 +s,s,s:, . . . SN_lVnr.

Vl &I

N

/

’

(6a)

sN-l-1

J=S&SN-‘+S

a

s-i

(6W If all of the stripping i.e. S,=Sb=SR=S

factors are equal in magnitude,

J=S-.

I=SN,

P-1

(7)

s-1

The equation for K can be simplified when the internal flow rates V,,, are constant. Then V,, 1 = V and S,=S for all II, but not necessarily V,,= V,= V or S, = Sr,= S, and eq. (5~) becomes K=

P-1 v,-. S-l

(8)

When S, = S eq. (8) can be used with eqs (6) and when with eqs (7). S ==&=S,,=S The concentrations can be eliminated from eq. (4a) using the following method that can be used for other equations as well: apply transformation (1) with arbitrary u to all concentrations and use eq. (4a) to get V,(y.-u)=IV,(y,-u)-J(C-DDu)+K[(m-l)u+b]. where D is the net flow rate cocurrent to the V phase. This equation is subtracted from eq. (4a), the result divided by u and the following important equation obtained V,=IV,-JD+K(l-mm),

(9

which can be used to relate I, J and K. If the equilibrium line is yz = x, (m =l, b = 0), eq. (9) reduces V,=IV,-JD.

W)

Using eqs (3) and eqs (2) for n E [2, N - 1] recursively, the difference equations for the whole transfer unit in Fig. 1 and the corresponding equation in transformed concentrations are obtained

I=&&

vaLN

z=s,s~sN-~=--s

to

C’),

= S,( v,y; - C’).

N-l

A. PEKKANEN

(W

. . . +s,s,s,

3.3. Calculation of the number of ideal stages N and the flow rate daflerence D If the internal stripping factor is constant, i.e. S, = S for all n, eqs (4b) and (6) give

v*yb-(* +g)c]. Vdb-&C’=S.S.S”2[ (11) If the equilibrium line is the diagonal, i.e. y.* =x,, it can be shown (see Section 4.5) that for a unit in Fig. 1 it holds that V,L(Y.

If the internal stripping factor is constant, i.e. S, = S for all n, V,+, for all n can still vary from stage to stage, and the equations for Z and J become

-xc,) = 1 V&,(Y*

- %).

(12a)

Now, if S, is constant, eq. (6a) can be used in eq. (12a) which leads to v,&,(Y,-

(5c)

(10)

Equations

%) = hsbs

N-2V,,~b(J'b-Xb).

(12b)

(11) and (12) can be used to calculate

-N, when all of the flow rates and end concentrations are given,

345

Stagewise models of mass and heat transfer

-D

(usually iteratively), when the flow rates of one phase, N and three end concentrations are given and -the limiting flow rates, when the operating and equilibrium lines intersect (when ZV= co)

3.4. Calculation of the limiting flow rates The treatment is restricted to cases in which the internal flow rates are constant, i.e. V,, 1 = V and L, = L for all n and applies for both absorbers and strippers. When N = 00, eq. (11) shows that y,,,, 1 =mx,+b for S>l and that yl=mxO+b for S-=zl. Thus the cases to consider are Calculation of the minimum internal flow rate Lmln when V, Li, Vi, x,, yb and y, are given. In this case S>l and ~~+~=yz=~x~+b (y,=y,*=mx,+b if Li = 0).

L

.

m,n

L,)y* - Vb

( V-

X*=

vm-

yN+l

Li

V(Y,-YY,+,)-

=

=mxb+b

VAY,--xJ

(134

%I-%

-Calculation of the minimum internal flow rate Vmi” when L, Li, Vi, x,, yb and xb are given. In this case St1 and yII=yg=mx,,+b (y,=y,*=mx,+b if Vi = 0). y =(LII y

. = nl,n

Vi)mx, + Lb L-

XO=l(ya-b) m

Vim

L(~~-xb)-Li(yb-xb)

Wb)

Y.-Y,

3.6. Equations for the flow rates Equations for the flow rates can be derived e.g. by the method used in the derivation of eq. (9); for example, eq. (2a) gives V,=S,(V*+,

--D)+

V”(1 -m)_

(18a)

Since the flow rates do not depend on m, it can be chosen freely. Choosing nz= 1 gives V.=+‘.,,-D). n

(1W

The method used is a general one for obtaining equations for flow rates. These equations are sometimes useful for calculating flow rates or other parameters in the system. For example, using this method and the substitution I?I= 1, eq. (4a) gives V.=ZV,-

JD.

(19)

Equation (19) holds for any equilibrium curve, but when calculating the parameters Z and J, e.g. from eqs (Sj-(lO) the substitution made in the derivation of eq. (19) must be taken into account, i.e. S, must be replaced by VJL,. Equation (19) shows that V,, V*, S,, S,, S, and D are interrelated, which must sometimes be taken into account

4. A

SINGLE

UNIT:

EXTENSIONS

This section deals with single units which -have a linear equilibrium relationship that is the same line for the whole unit and -need not strictly conform to the model in Fig. 1. In the context of this paper

3.5. Calculation of the outgoing concentrations When all of the flow rates and incoming concentrations are known, eqs (4) can be used with the definition of C to obtain equations for the outgoing concentrations. Hence V,y,,(I + J ) = Z V,y, + J&,x, + Kb,

(Ida)

L,x,(l+J)=(l+J-Z)V,y,+L,X,-Kb,

(14b)

V&(1

+ J)=ZV,y;+

J&,x;,

(1W

L,X;(l+J)=(l+J-Z)VV,y;+L,,X;.

(l=)

If S, = S, = S, = S = constant eqs (15) become V,yh(P”

- l)=P(S-

1)Vby6:+(SN+1-S)L,x:, (16a)

L,x;(P+

l-

l)=(SN-l)Vby;+(S-1)&x:.

(16bf

Using eq. (10) in eqs (14) it follows that, when m= 1 and b=O,

VJYa - Yb)=

v,

IV,z

v

6

_

vb-

Lb(xb

-

Yb)

=

zv

L Lh Lb

b

- Ya),

(174

a

+,(xa-Yb)-

_L

a

(17W

-an

interstream

is a stream

flowing between streams

V,+ , and Lj for the same j, i.e. a stream connecting the phases between any two ideal stages (or at the ends of the unit), -a recirculation is a stream flowing between streams Vj and V,, between Lj and L, or between Vj and L, for any j and k, i.e. a stream not confined between any two ideal stages and -a side stream is a stream connecting the unit to the environment, that is not any of the four terminal streams. (In practical calculations a recirculation must often be treated as two side streams.) If there are side streams in the unit, the parameters C and D have different values at each part of the unit separated by a side stream, e.g. C, and D, at the righthand side end of the unit.

4.1 Equations that do not include L,, i.e. eqs (l)-(ll) and (18)-(19), are valid even if there are arbitrary side streams in the first ideal stage of the unit if C and D are replaced by Cb and D,, respectively,

346

HARRYV.

NORDBN

andM~~rrr

4.2 Equation (4a) used with eqs (5)-(7) is valid even if there are side streams with zero concentrations in the unit, because the total mass balances have not been used in the derivation. 4.3 Equations (12) and (17) rium line y.* =mx,+ b if constant i.e. V,= V, = V,, replaced by L/m and x is

are valid even for equiliball of the flow rates are 1 = V for all n, and if L is replaced by y*.

4.4 Equations (4) used with eqs (9) and (10) and equations derived in the same manner are valid even if there are side streams, recirculations and interstreams in the unit, if -the side streams, recirculations and interstreams have concentrations that are either the same as or in equilibrium with the conerentration of one of the streams with which they are connected, and if -C and D are replaced by C, and D,, respectively. ofeq. (12) 4.5. The derivation If the equilibrium line is the diagonal, i.e. y.* =x,, the following equations can be written for a unit V,Y, =ZV,Y,

- JC,

where I, J, P and Q are parameters describing the unit. The unit need not conform to the model in Fig. 1 and hence I, J. P and Q are calculated either from eqs (5H7) or equations to be presented in Sections 5.2, 5.4 and 5.5. The method used to derive eq. (9) is used to eliminate the concentrations from eqs (20), and the equations so obtained are presented with eqs (20) in a single matrix equation

Taking determinants of both sides gives V,y,D,

- V,C,=

UQ + Jp I( V,Y,& - V,G),

(22a)

A.PEKKANEN

reducing the degrees is to set the requirement that the internal flow rates V,, 1 and L, must be constant, i.e. V n+l = V and L, = L for all n. Further, if the equilibrium line is Y* =x, the degrees of freedom can be reduced still by one if the number of design variables are kept unchanged. This can be achieved, for example, by requiring that the constant internal flow rates are equal to the terminal flow rates at either end of the model e.g. V= V, and L = L, (i.e. L,=O).The simplified models thus obtained can usually be made equivalent to the original models conforming to Fig. 1.

5. SEVERALUNITS

In this section the method is applied to the calculation of stagewise, countercurrent and steady-state transfer systems with side streams, interstreams and recirculations and with an equilibrium line that may change in the system. Side streams and interstreams are treated explicitly, whereas recirculations are treated implicitly as one outgoing and one incoming side stream. In practice, transfer systems are sometimes too complicated to be represented by means of a single unit presented in Fig. 1, and have thus to be divided into several units. In order to accommodate the calculation method, this can be done according to the following rules: Aach part of the system having: (a) no side streams and (b) a constant equilibrium line constitutes a unit, --each part of the system (e.g. an ideal stage or a stream between units) that is connected to a side stream constitutes a unit and ---each part of the system that contains interstreams which cannot be combined into some other unit, constitutes a unit of its own. Thus a system with a curved equilibrium relationship can be calculated if the equilibrium relationship is first approximated by one, two or more straight lines, the intersections of which then divide the system into units, each with an equilibrium line of its own.

and ~,L,(Y,

- A) =

IZQ+ JP 1VtA,b’, - ~1.

C=bl

If there are no side streams in the unit, P= 0 and Q = 1, and thus V&LAY,-

x,) = Z VbL(Yb--

&.).

5.1. The matrix equation To make practical calculations as straightforward as possible the equations are presented io matrix form. The general form of matrix equation used for a unit is

(12a)

Due to their derivation eqs (22) are very general ones, and are valid even if there are side streams, interstreams and recirculations with concentrations that are the same as the concentration of one of the streams with which they are connected. 4.6. The degrees offreedom The model in Fig. 1 has, for practical purposes, too many degrees of freedom. One convenient way of

(23)

Stagewise models of mass and heat transfer where I, J, K, P, Q and R are parameters describing the unit and A4 and A are square matrices. If Kb=O the third row can be deleted from this equation as well as the third column of the matrix M. The matrix A has non-zero elements only if there are incoming side streams in the system. In all other cases A = 0. *When A =O, parameters P, Q and R can be interrelated with equations that are analogous to eq. (9) derived for I, J and K. Thus D,=Pr/,+QD,+R(l-mm).

(24)

Equation (24) is derived in the same manner as eq. (9) and has thus the same validity, e.g. it is valid for unchanging and straight equilibrium relationships only. 5.2. Units with no side streams If there are no side streams and no interstreams that cannot be combined with the ends of the unit, the unit corresponds to Fig. 1. For such a unit P = R = 0, Q = 1, and matrix A = 0, and hence eq. (23) reduces to

Y= -If

347

V, V.-t

Vj/rn’

Kc&

J-O,

m

(28)

the interstream Li is in equilibrium with L, and thus has a concentration xi=mxb + h, N = 0, and Vi=O, then mL, z=l+L’

b

mL, J=Lv

K=L,.

(29)

b

Thus interstreams can be handled as calculation units consisting of ideal stages in series. 5.3. Units with incoming side streams It is assumed that the flow rates and concentrations of the side streams are given. If the side streams are connected to one ideal stage in a cascade of ideal stages, the ideal stage constitutes a unit (Fig. Pa) (N = 1) described by the following equation

(30) If the unit consists of one ideal stage only, S = m VnjLb and the equation is

(26)

In the general case, I, J and K are calculated using eqs (S)-(lO).

where S = m V,,/L, . If the side streams are connected directly to streams between ideal stages, a unit without ideal stages (N =0) is formed (Fig. 2b). The unit is described by the following equation, where the matrix M’ is a unit matrix, and is thus left out of the equation

Interstreams only. If there are no side streams, and if the unit consists of interstreams only, then the unit corresponds to Fig. 1 with N =O. Equations (25) is then valid for the I, J and K given below.

--If the concentrations of the interstreams are the same as that of the stream from which they originate, then K =O.

(27a)

If, further, &=O: I=$,

J=O,

K=O,

W’b)

5.4. Units with outgoing side streams It is assumed that the flow rates of the side streams are known but the concentrations are not. If there are no incoming side streams the matrix A = 0. If the side streams are connected to one ideal stage in a cascade of ideal stages, the one ideal stage constitutes a unit (Fig. 3a). If the concentrations of the side streams are the same as the concentrations of departing streams of the ideal stage, i.e. y,=y, and x,= xb, the equation for the unit is

and if V(=O: I+,

J+

b -If

b

K = 0.

(27~1

the interstream Vi is in equilibrium with V, and thus has a concentration L,=O,

then

yi =‘(y, m

- b), N = 0, and Fig. 2. Incoming side streams.

348

HARRY

V.

NORDEN

and MAR-IV

A.

PEKKANEN

eqs (25H29) and (32)-(34). If there are a total of A4 units m in series each described by a matrix M, in eq. (23) (A, = 0 for all m) then A = 0 for the system, and the matrix M is M=

fj

m=1

M,,

A=O,

(35)

-Fig. 3. Outgoing side streams.

S

-s

-!+-~

++$+l

L1 0

(I

Vab -

V*Y*

V,b

G

b 0

1

ii

where S=mV,/L,. If the side streams are connected directly to streams between ideal stages, a unit without ideal stages (N = 0) is formed (Fig. 3b). If the concentrations of the sicle streams are the same as the concentrations of departing streams, i.e. y,=y, and x,= xb, then

1

1 (32)

--In special cases, parameters calculated diretly as follows I=I,I,I,I,

I, J and K

can be (36a)

.. .)

J=J,+I,J,+I,I,J,+I,I,I,J,.

. .,

WW

Kb=K,b,+I,K,b,+I,I,K,b, +I,I,I,K,b,.

If the side streams are connected directly to streams between ideal stages, and if the concentrations of the side streams are in equilibrium with the concentrations of departing streams, i.e. y,=$y.

- b) and

x, = mx, + b, then

where I=

5.5.

System

V‘?

. .

(3W

Equations (36) are valid if there are no side streams but only recirculations and interstreams and different equilibrium hnes in the units. The recirculations and interstreams must have concentrations that are either the same as or in equilibrium with the concentration of one of the streams with which they are connected. -A

more complicated case is presented in Fig. 4. In the system in Fig. 4, y, is in equilibrium with y,, x, is given, and yi is equal to y, and y,. Because of the concentration (not because of the direction of flow), the interstream Vi constitutes a unit. The system has thus six units, and the flows are numbered accordingly. Each unit is described by means of eq. (23), where the matrices M,, M,, M,, A,, A, and A, are obtained from eq. (29, matrices M, and A, from eq. (34), M, and A, from eq. (31), and matrices M, and A, from eqs (25) and (27b). A,=0 for all units except for unit 4 (incoming side stream) and M, is a unit matrix. The equation for the system is derived by using eq. (23) with the appropriate matrices recursively for each unit in the system

V, + Vs/rn’ equation

All of the matrix equations presented relate properties at one end of a unit to properties at the other end. Whole systems consisting of units in series can now be calculated using these equations recursively, to obtain an equation for the whole system -The simplest example is a series of units without incoming side streams, that are thus described by

v3 -

--, L, -

1

l-1

* Lr -

v,

v5 I

3 L3

L,

l -

v6

5

"b

v, L,

Lb

1Ls Fig. 4. A transfer

system

with

side streams.

Stagewise models of mass and heat transfer

+M,M2M,A,

(37)

349

If this is the case, the equilibrium relationship is the diagonal Y* = X. Equations (12) (17) and (38H39) can now be used to calculate heat transfer whether stagewise or differential contact, if, in the latter case, the heat capacity flow rates V,,, and L, for all n are constant. In direct contact heat transfer K, in eq. (39) denotes the overaH heat transfer coefficient. For indirect contact eq. (39) reads No,=?,

6. OTHER

6.1. Calculation of differential contact transfer units Equations dcrivcd for stagewise transfer units can also be used for differential contact transfer units, if -the internal flow rates are constant, i.e. V,+ 1= I’, L, = L for all n and -the equilibrium relationship is linear in mole fractions. When the above conditions are fulfilled, both the equilibrium and operating lines are linear, and the number of ideal stages N can be replaced by the number of transfer units No, using the following relationship N=N,-

S-l

In S ’

or

(40)

APPLICATIONS

P=exp[N,,,(S-

I)].

K aA’h

No,, is the number of overall mass transfer units for the internal part of the unit in Fig. 1, where the flow rates are constant. If the concentrations are expressed in mole fraction or temperature coordinates, and if the equilibrium relationship is linear, eq. (38) holds exactly, if the corresponding internal total flow rates are constant. It holds approximately, however, for most other coordinates, e.g. for mole ratio, mass fraction or mass ratio coordinates or for coordinates z? and JJof Section 6.3, if Nou in eq. (38) is replaced by NOy -OSln( V1/VN+l), where V, and I’,+, are the total molar Aow rates (Norden, 1988). This requires, however, that reasonably cascade in 2 in transpart of the is usually

6.2. Calculation of heat transfer units The general calculation method can be applied heat transfer, if --x and Y denote temperatures and -L. and V denote heat capacity flow rates. CLS14:2-5

(Y, -A) - (Ya - &) VY, - Y,) = No, I’ ln Y,--x, Yb-Xb

=

v(Y, -Y,)

Nay

W

-

to

(414

41.9

= No, v(Y, - -%) = No, vb’, -

(38)

No, = L_ V

-the mass transfer ratio z = d( Vy)/d V is constant for the internal part of the Fig. 1, giving constant flow rate 9 and formed coordinates, and that _-OS -c(z- y)/(z-Y*) -C2 for the internal cascade in Fig. 1, which requirement fulfilled.

where k is the overall heat transfer coefficient and A is area of the heat transfer surface. For indirect contact it is usually true that V, = Vb= V, = V and L,= L,=L, = L thus giving also S, = S, = S, = S. If this is the case, and since y* = x, eqs (12) and (38) can be used to obtain an equation for mass or heat transfer in a differential contact transfer unit

for

xbh

G=

1.

(41b)

Equations (41) can be used even when the equilibrium line is Y* = m + b, if x, and xb are replaced by yz and yb+ and L by L/m, respectively. 6.3. Flow rate transformation The requirement that the internal flow rates V.+ 1 and L, are constant, is a very severe one. It can in some cases be met by a suitable choice of concentration coordinates: -for equimolar (equimass) mass transfer concentrations are expressed in mole (mass) fractions, i.e. xi=n,/Zni (wi=mfimi) to obtain constant total molar (mass) flow rates, -for one transferred component concentrations are expressed in mole (mass) ratios to obtain constant inert molar (mass) flow rates. In general, if the mass transfer ratio z=d( Vy)/dV =d(L.x)/dL (z = N,/(N, + NB) in the binary case and t= N,/XN, in the multicomponent case) is constant (but not necessarily equal to cu or 1 as in the molar cases above) in a steady-state, countercurrent unit, the following transformations can be used to obtain a linear operating line in the respective coordinate frame: &LF,

pcykz, 2-Y &=xk, Z--x

(42a)

350

HARRY V. NORD~N -zk+_O

v= v-

zk/k’ ’

L=Lp

zk+9 zk/k’ ’

and MARTTI A. PEKKANEN

y’zp

zk+g’ x=&

(42ti)

*uantities connected with the V phase are replaced by the corresponding L phase quantity: e.g. V+L, yax, y*+x*, S-+A and K,+K,, -the subscripts are changed a*b

where Vand L are the total molar flow rates and x and y are the mole fractions of component A. If z, k and k’are constant, Pand 2 are constant, and the operating line is straight. This is an extremely useful result, because now the only physical requirement is that z be constant, which is usually more or less the case at least in binary absorption, desorption, distillation and stripping. The parameters k and k’can be given any constant value. The transformed variables V, J?, j and 9 have physical significance, however, only with a proper choice of k and k’, e.g.
6.4. Choice of concentration coordinates Concentrations in the equations of this paper can be expressed in: (1) mole fraction coordinates, (2) temperature coordinates, (3) mole ratio, mass fraction or mass ratio coordinates, (4) the generalized coordinates of eq. (42) or coordinates as used in, e.g., 15) enthalpy-temperature water-air operations.

O&N+l,

1~9N,2+N-l,

3eN-2,.

n-l*n+l, -the

equilibrium

line is changed:

m+m-

I, 6-+ -b/m.

The substitutions produce new equations, which might prove valuable in many cases. This is in contrast to the fact that either phase can be designated as the L or V phase, since the method is completely symmetrical with respect to the phases. 6.6. Calculation of units not operating countercurrently In some cases a countercurrent model may not be the best one for the description of a physical heat or mass transfer operation. Heat transfer literature contains a vast amount of equations and solutions for equipment that do not operate either counter- or cocurrently. For heat transfer, the equilibrium is y,* =x,, and the heat capacity flow rates are usually constant. Hence this data can be used in the calculation of differential contact (or countercurrent stagewise) mass or heat transfer units, if the equilibrium relationship is a straight line y.* = mx, + b if V, = Vb = V” = V and L, = L, = L, = L. The following substitutions should be made: one heat capacity flow rate is replaced by V and its temperature by y; the other heat capacity flow rate is replaced by L/m and its temperature by y*; if data from differential contact units are used for stagewise units, eq. (38) is used to relate N and No,,. A cocurrent differential contact case can usually be calculated using the formulas for the corresponding countercurrent case, if negative values are substituted for the flow rates of either phase, but preferably of the L phase, because this preserves non-negative values for Nov. Stagewise cocurrent cases are of little interest since after one ideal stage no further mass (or heat) transfer takes place for constant V and L. 7. EXAMPLES

The first two are equivalent with a proper choice of the units for flow rates, and are used in this paper. For stagewise devices, all five are equally valid for all the equations presented. For differential contact devices, however, the conditions presented for the exactness of eq. (38) used in the calculations of differential contact devices must be taken into account.

6.5. Calculations on the L phase side The equations have been derived on the V phase side. The equations on the L phase side can be obtained with the following substitutions:

7.1. Example 1 A countercurrent differential contact stripper is used to remove solute from water. The water feed enters the top of the column, and the stripping steam is provided by a reboiler. The water feed is 8 kmol/s, and the solute concentration of the feed is 0.01 in mole fractions. The equilibrium relationship can be approximated by y* =5.0x. The reboiler produces a steam flow rate of 3 kmol/s, and it is assumed that at the top of the column the cold water feed condenses 2 kmol/s of the steam. The outgoing concentrations are to be calculated when the number of transfer units in the packing section is 8.11.

351

Stagewisemodels of mass and heat transfer Solution. It is assumed that the internal flow rates are constant. The stripper fully conforms to the model in Fig. 1, and the standard choice is made for the phases. Hence, the water feed is L, = 8 kmol/s, the condensation at the top is Vi = 2.0 kmol/s, the stear from the reboiler is Li= 3.0 kmol/s and V, =O. The other flow rates are: L, = L = L, + Vi = 10.0 kmol/s, L, = L - L, = 7.0 kmol/s, V, + i = Y= Li = 3.0 kmol/s and V, = V- Vi = 1 .O kmol/s. The stripping factors are s,=---

5x1 10

5x3 S=-.-_= 10

zO.5

5x3 IL5

L,&, = L,x, = 0.8 kg/s P&J?,= V,y, = 0 kg/s. Equations (17) are used for the solution, and the parameter L is calculated from eq. (6a), noting that in this case Sb =S I=S

a

s”-Lf

v

v

0

8.5

z

= 2.260

&,R,=0.7344

15

kg/s,

S”=-=1.7

The equivalent N in the stripper is calculated from eq. (38)

The desired quantities are then calculated

N=N,~=8.11=L0.0 In 1.5 and thus the total N including the reboiler is 11. J is calculated from eq. (6b)

1.51°-

J=o.5;1.5p+o.5 1.5-l

1

J

1+J

L,x, = 0.0792 kmol/s

1 LbXb = -I_+. l+J

= 0.000809 kmol/s

Xb = 0.00012.

0.6

and

0.4

Zo=L,z-xn ~ v,=

z

zto ; --;

.

k=,k’=

C

c D

I J k k k

1.0.

4xj (kg/s) = 1.667 kg/s

2.0 kg/s

= 0.4667 kg/s

&,+ v--6=0.8

b

K

.

p, = P= V*F=

A A A,,A

h

Now the transformation gives

Z

a A’

1

Solution. The flow rate transformation is used to obtain constant internal flow rates: the dependance between L and x leads to the following choice of parameters

~b=~=Lz--x=

Xb = - =xb = 0.03847, z+&,

y0 = 0.079

___ kg/s =0.6-0.4X

d(Lx) z=-=-=1.5 dL

R,= 0.03948,

kg/s

V, = 1.29 kg/s Lb = 1.71 kg/s.

NOTATION

7.2. Example 2 Outgoing concentrations and flow rates are to be calculated in a countercurrent leaching device when N =9.5, the equilibrium line is yz=xn, the concentrations are in mass fractions, L, = 1.0 kg/s, x, =0.8, V, = 2.0 kg/s, yb = 0 and when the underflow departing from an ideal stage depends on the concentration of the underflow as follows L

y,= -=0.5695, =$a e+P,

= 97.85.

The concentrations are obtained from eqs (14)

V,Yil =-

y^,=O.9180,

I% KX L m m 2 M Mi

specific area between phases in eq. (39), m- ’ cross-section of mass or heat transfer unit in eq. (39), m2 heat transfer area in eq. (40), m2 matrix in eqs (23H37) absorption factor = L,,/m V, and = L/m V, dimensionless constant of the equilibrium line, dimensionless net solute flow rate cocurrent to V phase, usually kmol/s (or kJ/s): C, = V,, 1 y, + I -L,% c,V,ya-L,&, Cb= vbyb--bxb C after concentration transformation of eq. (l)=C--Du net flow rate cocurrent to V phase, usually D,= V, kmol/s (or kJ/s K): D,= V,,, -I,“, -L,, D,= T/,--L, height of mass or heat transfer unit in eq. (39), m parameter, dimensionless parameter, dimensionless parameter of transformation in eq. (42) parameter of transformation in eq. (42) overall heat transfer coefficient in eq. (40), kJ/s m2 K parameter, usually kmoI/s (kJ/s K) overall mass transfer coefficient of V phase, kmol/m2 s overall mass transfer coefficient of L phase, kmol/m2 s flow rate of L phase, usually kmol/s (kJ/sK) slope of the equilibrium line, dimensionless unit m, m E Cl, M] mass of component i, kg number of units, dimensionless matrix in eqs (23)-(37) molar mass of component i, kg/kmol

352 n ni N N

OY

Ni

; R s,,s

u

V W xv

x Y z

Y

HARRYV. NORDBNand MARTTIA. ideal stage n, n E [ 1, N] number of moles of component i, kmol number of ideal stages, dimensionless number of overall transfer units of V phase, dimensionless molar fltix of component i, kmol/m’ s parameter, dimensionless parameter, dimensionless parameter, usually kmol/s &J/s K) stripping factor =mV,JL. and = mV/L, dimensionless of the concentration transconstant formation in eq. (1) = b/(1 - m), dimensionless flow rate of V phase: usually kmol/s W/s W mass fraction, dimensionless mole fraction, dimensionless concentration of L phase, usually mole fraction (or K) concentration of fi phase, usually mole fraction (or K) mass transfer ratio, dimensionless

PEKKANEN

Subscripts the entrance end of L phase of a unit the entrance end of V phase of a unit ideal stage n, PIE [0, N] unit M, m E [0, M] interstream side stream

; n m i S

Superscripts 1

*

value after transformation value after transformation equilibrium value

in eq. (1) in eq. (42)

REFERENCES

Edminster, W. C., 1943, Znd. Engng Chem. 35, 8374339. Horton, G. and Franklin, W. B., 1940, Znd.Engng Chem. 32, 1384-1388.

Kremser, A., 1930, Nat! Petroleum News 22, 42. Nordkn, H. V., 1988, Chemical Engineering II. Lectures at Helsinki University of Technology. Smith, B. D. and Brinkley, W. K., 1960, A.Z.Ch.E. J. 6, 446450. Soude&M. and Brown, G. G., 1932, Znd. Engng Chem. 24, 519-522.