Modeling of packed absorbers at unsteady state operation—IV

Chemical Engineering Science, 1970, Vol. 25, pp. 1283-1296. Pergamon Press. Printed in Great Britain.

Modeling of packed absorbers at unsteady state operation - IV

RONALD MCDANIEL and C. D. HOLLAND

Department of Chemical Engineering, Texas A & M University, College Station, Texas 77843, U.S.A.

(First received 15 Jaunary 1970; accepted 20 January 1970)

Abstract-The use of field tests in the modeling of a packed absorber at unsteady state operation is demonstrated. Calculational procedures developed for this purpose are presented and demonstrated by means of numerical results.

THE PACKED absorber described previously [lo] was used to make the field tests employed in the formulation of the unsteady state model. This column, located at the Zoller Gas Plant, Refugio, Texas, was used to absorb propane and heavier components from a natural gas stream. The column had an inside diameter of 36 in. and a 23-ft packed section which contained 4900 lb of metallic Pall Rings.

After the calculational procedures have been developed, they are used in conjunction with the results of field tests in the formulation of the unsteady state model.

Again the concept of a mass transfer section [2,6,7,10,12] is employed as well as the concept of perfect heat transfer sections[5]. In the following analysis, the column is divided into elements and the streams entering and leaving each element are identified as indicated in Fig. 1.

DEFINITIONS OF THE MASS AND HEAT TRANSFER SECTIONS

The mass and heat transfer sections for unsteady state operation are defined such that each element of packing Az~ of the packed column becomes a perfectly mixed section; that is,

Yji = EjiKjixji

Xi = xjf, (Zj < 2 5 Zj+l) (1)

Yi = Yjb (Zj 5 Z < Zj+l)

where Xji and y,, are again the mole fractions of component i in the vapor and liquid streams leaving the jth element of packing. For nonideal

Fig. 1. Sketch of a typical packed absorber.

solutions, the quantity Kji is preceded by an activity coefficient yji*

The heat transfer section is defined by

Tjv = ejTjL, Tj” = Tjs

TV = Tj’y (Zj 5 Z < Zj+t)

TL = TjL9 (ZJ < Z 5 Zj+t)*

(2)

This definition supposes that the temperatures of

1283

R. MCDANIEL and C. D. HOLLAND

the vapor and liquid phase are uniform but may differ over each element of packing. Also, the temperature of the packing is taken to be equal to that of the liquid in each element. Throughout the remainder of the development, perfect heat transfer sections are assumed; that is,

The liquid holdups may be stated in terms of the vapor rates by use of Eq. (4); that is,

TrV= TjL= Tjs= Tj, (3)

Similarly, the molal holdup of component i in the vapor phase is given by

As a consequence of the definition of the mass transfer section, the vapor and liquid component- flow rates are related in the usual way,

where

Iji = AjiVj(, or Va = Sjil*i (4)

FUNDAMENTAL RELATIONSHIPS

Let the total mass of packing contained in the element Azj be denoted by &!j. Since the bulk density of the packing is constant, it follows that

(3

where pb = mass of packing per unit VOhUne of bed;

S = internal cross-sectional area of the column.

As a consequence of the definition of the mass transfer section, relatively simple relationships exist between the holdups in the vapor and liquid phases and the vapor and liquid flow rates. Since xi = x, (zj < z_ s z~+~), it follows that the molal holdup of component i in the liquid phase is given by

Uj”

Uh = Uj”Xji = z lji ( > where uj”i = moles of component i in the liquid

phase in the element of packing AZ at any time t;

UjL = total molal holdup in the liquid phase in the element of packing Azj at any time t.

ujv 4 = U*%, = F v*i ( ) (8’

where ~1 and Ujv are the component and total holdups in the vapor phase in the element of packing Azj.

From the definition of a perfect heat transfer section, it is seen that the heat contents of the liquid and vapor phases in the element Azj at time tare given by

UjLh* = i h,uk = UjL i h X. izl ** j’ i=l

(9)

Uj’Hj = i H~u% = Ujv 5 Hjiyj, (10) i=l i=l

Since the temperature of the packing is taken to be constant and equal to Tj over the element Azj, it follows that the heat content of the packing contained in the jth element at any time t is given by

where hjs is the heat content of the packing in B.t.u.‘s per unit mass of packing, and JZi is the mass of packing contained in the elementj.

COMPONENT-MATERIAL BALANCES

When the concept of the mass transfer section is employed in the description of a packed column, the equations so obtained are of the same form as those which describe a column with plates. For any mass transfer section j, the material balance for component i over the time period from t, to t,+l is given by the following integral-difference equation

1284

Modeling of packed absorbers at unsteady state operation- IV

I &I+1

b [ vj+l.*+ lj-l,i--ji -fjildr= (uE+u$dItn+,

- (u:;+uh)(t,. (11)

For each element Azj, Eq. (11) may be trans- formed to algebraic form by use of the implicit method [6,8] to give

where

When the b’s, u$s, and U$S are stated in terms of the vjls by use of Eqs. (4), (7), and (g), the following matrix equation is obtained for each component i.

enthalpy and total material balances are presented.

ENTHALPY AND TOTAL MATERIAL BALANCES

The enthalpy balance for each section j (1 S j 5 N) over the time period from c, to t n+1 is given by

I bt+1

tn [L’j+lIfj+l+Lj-lhj_1_VjHj_Ljhj] dt

= ( Uj”Hj + U/hj + Mjhjs) ltntl

-(UjVHj+ UjLh,+Ajhjs)It,. (14)

When the integral appearing on the left-hand side of Eq. (14) is approximated by use of the implicit method, the following result is obtained.

where = &[Uj"Hj+, Uj'hj +Jli;h,] -A

pji= [l+A;,(l+~jL/~)+~jV/~],(l SjS N+l); X [UjVHj+ UjLhj +J!jhjs]O. (15)

rj”= Uj”/(LJt);TjV= Uj”/(VjAt); The expression for the total material balance

Pji = (TIVj+~,i+fj_~,~-Vji-Iji]“+ &M+ 4” for the jth element is obtained by summing each side of Eq. (11) over all components. Reduction

+Cji;Cji=0,(2(=j~N_l);Cli=l,i; of the expression so obtained to algebraic form by use of the implicit method gives

cNi = VN+l,i.

Thus, for a known set of Eji)S and sets of T* Lj, and Vj, Eq. (13) is readily solved for the corres- =~[((/;y+UjL)-(~,~+U~~)o]. (16)

ponding set of component-flow rates by use of Gaussian elimination[6,8] or the recursion For the case where the holdups are taken to be formulas based on this method. Next the constant with respect to time, the expression

1285

obtained by summing Eq. (11) over all components is given by

s "+' [vj+l+~~_,-vj-Lj]dt=O. (17) &I

Since this expression is equal to zero for all choices oft, and t,+l, it follows that the integrand is identically equal to zero for all t; that is,

Vj+l+Lj_l-Vj-Lj=O (18)

for all t. The relationships given by Eqs. (1) through (18) are utilized in the calculational procedure that follow. The procedure presented makes use of the Newton-Raphson method for the determination of temperatures from the enthalpy balances [13].

In general, certain steps of the procedure presented are carried out by the process known as iteration or successive iteration[ 1,5]. The sufficient conditions for the convergence of the procedure of successive iteration are much more restrictive than those for the Newton-Raphson procedure [ l] where it is applied to each variable. However, when the Newton-Raphson procedure is applied to only a selected group of variables and the remaining variables are determined by iteration, then the procedure inherits some of the same restrictions on convergence possessed by iterative procedures. On the other hand, when the Newton-Raphson procedure is applied to each variable, convergence is assured provided among other things the initial set of assumed values of the variables is close enough to the solution set. The picking of such a set of assumed values of the variables could prove difficult to achieve.

CALCULATIONAL PROCEDURE: DETERMINATION OF THE TEMPERATURES

FROM ENTHALPY BALANCES

This procedure was first proposed by Sujata [ 131 and later used by Friday et al. [3] for absorbers at steady state operation. Actually, this procedure constitutes a modification


of the calculational procedure originally proposed by Greenstadt et al.[4] wherein each variable was found by use of the Newton- Raphson procedure. A summary of most of the procedures which had been proposed prior to 1966 has been presented elsewhere [6]. Also described in this reference is a combination of the o-method, &-method, and the constant composition method for the solution of unsteady state distillation problems. However, this combination of methods was not used because of the difficulty of applying it to the particular unsteady state absorber problem. This difficulty arises when T,,, TN+lr Lo, and V,,, are specified (as was the case for the field test) rather than To, I/,, TN+I, Lo, and VN+I. More recently, Osborne [ 111 developed a calculational procedure for the solution of unsteady distillation problems which was based on the use of the Range-Kutta-Gill predictor.

In the procedure that follows, supposed that the following specifications have been made for all time t 2 0.

Speci$cations: TN+~, VN+,, {yN+l,i}l To, Lo, {Xl)i}y Uj”(1 SjS N), Ut(1 SjS N), *Aj (1 5js N),andN.

Before the presentation of the details of the calculational procedure which was developed for the primary purpose of analyzing the results of the field tests made on the absorber at the Zoller Gas Plant, a brief discussion of the holdup specifications is in order. In packed absorbers, the total holdup of vapor and liquid over the entire column is relatively small, and the distribution between the vapor and liquid phases is not generally known. This distribution was determined from the experimental results as described in a subsequent section. In this determination, the total holdup of vapor and the total holdup of liquid in the packing were divided into parts ( Ujv and U,“) , which were taken to be fixed for all time t 2 0.

In the determination of the temperatures by use of enthalpy balances, Eq. (15) was restated in functional notation as follows.

1286

Gj = ‘Vj+lHj+l +Lj_lhj_l - VjHj- Ljhj

+a[~j+lHj+,+Lj-lhl-l-T/jHj-Ljh,]’

-&[Uj”Hj+ UjLhj+Mjhj,]

+ &[ Uj”HjV + UjLhj + Jlt,h,] 7

(1SjSIV). (19)

The total flow rates { Vj} and {Lj} and the enthalpies {Hj} and {h,} appearing in the Gj functions were evaluated in the following manner. First, suppose that the values of all variables at the beginning of the time period under consideration are known. The immediate problem to be solved consists of finding that set of values of all variables (evaluated at the end of the time period under consideration) which satisfy all of the equations simultaneously.

Any one trial for the time period under consideration is initiated by the assumption of a complete set of temperatures and total flow rates at the end of the time period under consideration. Then Eq. (13) is solved for the vapor rates for each component, and the corresponding liquid rates are then computed by use of Eq. (4). The rates so obtained are identified by the subscript “~a”. Next the total flow rates and mole fractions at the end of the time period under consideration are computed as follows:

vj= i: (Vji)ca,Lj= i (lji)ca (20) i=l i=l

Yji = (cVji)ca , Xji = Jiji)c. (21)

zl Cvji)ca ,z, (bi>ca-

The Vj’s and Lj’s computed by use of Eq. (20) were used in the Gj functions, and the mole fractions obtained by Eq. (21) were used to compute the enthalpies H, and h, appearing in the Gj functions as follows:

Hj = i Hjiyji, hj = i hjixji. i=l i=l

(22)


The temperatures were found by use of the Newton-Raphson method which consists of the successive iteration of the Newton-Raphson equations. For the kth trial of this iterative procedure, these equations take the form:

AkATk = Ck (23)

where

ac,ac, dT1 dTz

0 0 0 . . ..o

G G aG o aT, dT, aT,

0 0 . . . .

0 ac,ac,ac, 0 0 aT, aT, aT, ‘***

A,= . . . . .._.........................

0 aGN+ aG,_, aGN_l

. . . . . . . . . O--- aT,_, aT,_, aT,

0 . . . . . . . . . 0 () aGN ac,v aT,_, dTN

Gc = [(---G,) (+d . . . (-GN)lT; ATI,= [AT~AT~...AT~]T;

A Tj = Tjk- Tj,/,_l.

The elements of Ak and Ck are of course evaluated at the temperatures assumed to make the kth trial. The partial derivatives appearing in A, may be evaluated by use of the following formulas which are obtained in an obvious manner by partial differentiation of Gj.

If the system contains single phase light

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components (components which appear only in the gas phase) and single phase heavy components (components which appear only in the gas phase) minor modifications of the above equations are required. The component-material balances for the single phase light “L” and single phase heavy “H” components are given

by

flow rates of each component i in the liquid by use of Eq. (4).

(3) Compute the total flow rates from the results of Step (2) as follows:

Vj= 5 (Vjf),,, (1 Zj s N) i=l

Lj = Lo + V*+, -VI, (1 ZjS N).

tljL=V~,lj~=O, (1 sjr N+l) (24)

and

(4) Compute the mole fractions by use of Eqs. (20) and (2 1).

VjH=O,Ij~=I~,(O~j~:). (25)

(5) Solve for the temperatures at the end of the time step by successive iteration of Eq. (23).

Also, since it has been shown for distillation columns with plates[6] that the inherited error remains bounded for all choices of At > 0, provided

(6) If the difference between each temperature of the set {Tj} found in Step (5) and the corresponding value assumed in Step (2) is less than some preassigned number, the solution set of the variables at the end of the given time step has been found.

a value of I_L = O-6 (used in the treatment of distillation columns [6] was successfully employed in the analysis of absorbers.

(7) If the conditions of Step (6) are not satis- fied, the procedure is repeated by returning to Step (2) and by using the most recent set of total flow rates found in Step (3) and temperatures found in Step (5).

Furthermore, it should be noted that the steady state solution at the conditions of the upset may be obtained at the end of the first time step by choosing At sufficiently large such that the right-hand sides of Eqs. (12) and (15) are for all practical purposes equal to zero. It is of course supposed that the column is initially at steady state and that an upset occurs at time t = 0 +. Under the conditions stated, it is evident that Eqs. (12) and (15) reduce to the component- material balances and energy balances, respectively, for steady state operation.

After convergence has been achieved for the given time period under consideration, the procedure is repeated for the next time period. The characteristics of this procedure are discussed in a subsequent section.

ANALYSIS OF THE RESULTS OF THE FIELD TESTS

To summarize, the steps of the proposed calculational procedure are as follows.

(1) Find the initial steady state solution as described above.

(2) Fix p for all time periods and choose a AT for the first step. Assume a set of temperatures and total flow rates at the end of the time period under consideration. Solve for the component flow rates {uji} for each component i by use of Eq. (13). Compute the corresponding

The series of field tests used in the modeling of the packed absorber at unsteady state operation consisted of two steady state field tests and one unsteady state field test. A sketch giving the details of the mechanical construction of the absorber used to make the field tests described herein has been presented previously [lo]. After a brief description of the experimental procedures employed, the use of the results so obtained in the modeling of the column is presented.

Initially at time t = 0, the column was at steady state operation, and at time t = 0+, the lean oil rate was changed abruptly from its initial steady state value of 156.655 lb-moles per hr to 194.7’14 lb-moles per hr. The temperatures recorded are

1288

Modeling of packed absorbers at unsteady state operation-IV

given in Table 1. The times given in this table are only approximate because it took about two minutes to record all of the temperatures. The times do correspond, however, to the precise times at which the temperatures of the lean oil, lean gas, rich oil, and rich gas were observed. At the instant of the upset, the temperature of the lean oil increased immediately from -1mO”F to 2.5”F.

Several samples of the inlet gas were taken before and after the field tests. Since the upset had no effect on the composition of this stream, the analyses were averaged to obtain the results presented in Table 2. The flow rate and composition of the rich gas were determined by making a simple flash calculation on the inlet gas at the temperature in the space below the rich oil draw-off tray [lo] and at the column pressure. Several samples of the lean oil were also taken during the upset. No significant differences in the compositions could be detected; hence the analysis were averaged to obtain the results given in Table 2.

Two samples were taken from both the lean gas and rich oil streams prior to the upset. Two more samples were taken from these streams two hours after the upset. For the first 30 min after the upset, samples were taken every five

Table 2. Feed analyses for the unsteady-state test

Component

Carbon dioxide Nitrogen Methane Ethane Propane i-Butane n-Butane i-Pentane n-Pentane Hexane Heptane Octane Nonane Decane

Lean oil Inlet gas (mole %) (mole %)

0.0 0606 0.0 0.176 0.0 86407 0.0 6643 0.0 3.329 0.0 1.027 0.0 0.788 0.056 O-271 0.075 0.213 0.788 0.289

11.396 O-208 39.941 0.043 3 1.883 0.001 15.861 0@002

minutes from the two streams. For the next hour, samples were taken every 10 min. A five minute interval was about the shortest time in which samples could be taken manually. The analyses of the lean gas are presented in Table 3. The complete sets of flow rates and product distributions at the initial and final steady state operating conditions are presented in Tables 4 and 5. As discussed previously [lo], the flow rates of each component in the rich oil at the initial and final steady states were obtained by material balance.

Table 1. Observed temperatures for the unsteady-state test

Packing depth (ft)

0.0 (Lean oil) O-0 (Lean gas) 2.0 6.0

10.0 14.0 18.0 21.5 21.5t 23.0$ 23.0 (Rich oil) 23.0 (Rich gas)

0

-1.0 26.0 22.0 31.5 28.0 235 19.4 13.0 13.0

-2.5 20.0

2.0

Temperatures (“F) at the cumulative time (mitt) indicated

5 10 20 30

2.5 2.5 2-5 2.5 26.0 27.0 27.0 27.0 24.0 25.0 25.0 25.0 33.0 34.0 34.0 34.0 32.0 32.0 33.0 33.0 25.4 27.2 28.4 29.0 20-O 21.0 22.0 22.5 13-o 13.5 13.5 14.0 13.0 14.0 14.0 14.0

-2.5 -2.5 -2.5 -2.5 20.0 21.0 21-o 21.0

2.0 2.0 2.0 2.0

120

2.5 27.0 25.0 34.0 34.0 29.4 22.5 14.0 14.0

-2.5 21.0

2.0

tThis thermowell was contained in a V-shaped trough. *This thermowell was located in the vapor space below the liquid dray-off tray.

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Table 3. Lean-gas analyses for the unsteady-state test

Mole % at the cumulative time (mitt) indicated

Component 0 5 10 15 20 25 30 120

Carbon dioxide 0.572 0.569 0.569 0.569 O-569 0.569 0.569 0.569 Nitrogen 0.165. 0.161 0.161 0.161 0.161 0.161 0.161 0.161 Methane 93.652 94-044 94.063 94.059 94.056 94.059 94.057 94.059 Ethane 4.936 4.723 4.719 4.723 4.727 4.723 4.725 4.724 Propane 0.595 0.432 0.419 0.419 0.418 0.419 0.417 0.418 i-Butane 0.015 0.008 0.006 0.006 o+tO6 0.006 0.006 OGI6 n-Butane 0.002 0.001 0.001 0.001 OXtO oXtO 0.001 0.001 i-Pentane O@Ol 0.001 O+Kll 0.001 0.001 0.001 OWl 0.001 n-Pentane 00ll oGo1 O+IOl o@Ot 0.001 O+Ol O+lOl 0.001 Hexane 0.003 0.003 oXKl3 0.003 o+nl3 oGO3 0.003 0.003 Heptane 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 Octane 0.030 0.030 0.030 0.030 0.030 0*030 0.03 1 0.030 Nonane 0.010 0.010 0.010 0.010 0.010 0.010 0.011 0.010 Decane 0.002 0.002 o+tO2 0.002 0032 0.002 o+Ml2 0.002

Table 4. Initial steady state of the unsteady-state test

Component

Carbon dioxide Nitrogen Methane Ethane Propane i-Butane n-Butane i-Pentane n-Pentane Hexane Heptane Octane Nonane Decane

Lean oil

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.08732 0.11779 1.23424

17.85307 62.56989 49.94669 24.84636

Flow rates (lb-moles/hr)

Rich gas Lean gas

14.6563 1 13.036 4.61737 3.755

2233.06000 2133.470 158.75030 112.445 66.12759 13.560 15.82934 0.347 10.20640 0.035 2.2997 0.027 1.41099 0.027 0.86719 0.071 0.26689 0.342 0.02486 0.682 0.00023 0.235 0~00003 0,047

Rich oil

I.620 0.863

99.624 46.307 52.568 15.483 10.172 2.360 1.502 2.031

17.777 61.913 49.712 24.799

Product distributions

~‘&!i

0.1243 x lo0 0.2298 x loo 0.4669 x 10-I 0.4118 x 100 0.3877 x 10’ 04468 x 102 0.2918 x IO3 0.8633 x lo* 0.5539 x 102 0.2876 X lo2 0.5185 x lo* 0.9077 x 102 0.2115 x 103 0.5234 x 10”

Total 156.65530 2508.11300 2278.059 386.709

Column pressure = 722 psia; T0 = -1 W’F, T N+l = 2.O”F; total mass of packing = 4900 lb; heat capacity of packing = 0.12 B.t.u./lb “R.

In the analysis of the field test at unsteady state 4 and 5 represent the initial and final steady operation which follows, the transient values of states of the unsteady field test. The result of the component flow rates (or compositions1 of these two steady state field tests were used to only one stream were needed, and those of the determine two sets of E,‘s and the two sets of &‘s lean gas were used. and /3N+I’~ (see Table 6) by use of Modification

(1) of the procedure for the determination of the Use of the results offield tests in the modeling of the packed absorber at unsteady state

E,,'s[lo]. In the determination of these efficien- ties, the optimum value of N determined

operation previously [lo] was employed. Also, the K-data The two steady state field tests given in Tables of Table 1 of Reference[9] was employed. The

1290


Table 5. Final steady state of the unsteady-state test

Component Lean oil

Flow rates (lb-moles/hr)

Rich gas Lean gas Rich oil

Product distributions

WV”

Carbon dioxide 0.0 14.65643 12.810 1846 0.1441 x 100 Nitrogen 0.0 4.61741 3.630 0.987 0.2719 x 10” Methane 0.0 2233.07900 2118.147 114-948 0.5427 x 10-l Ethane 0.0 158.75160 106.372 52.231 0.4924 x 10” Propane 0.0 66.12817 9.408 56.72 1 O-6029 x 10’ i-Butane 0.0 15.82948 0.146 15.684 0.1075 x 103 n-Butane 0.0 10.20649 0,012 IO.194 0.8231 x lo3 i-Pentane 0.10853 2.29999 0.027 2.381 0.8667 x lo* n-Pentane 0.14641 1.41100 0.027 1.531 0.5712 x lo* Hexane 1.53409 0.86719 0.070 2.332 0.3351 x 102 Heptane 22.19035 0.26689 0.337 22.120 0.6557 x lo2 Octane 77.77081 0.02486 0.671 77.124 0.1149x 103 Nonane 62.08090 OWO23 0.231 61.850 0.2674 x lo3 Decane 30.88263 0~00003 0.047 30.836 06615 x 103

Total 194.71370 2508.13500

Column pressure = 722 psia; To = 2.5”F, TN+, = 2.O”F.

2251.919 450.929

Table 6. Vaporization efficiencies for the unsteady-state field test

Initial steady Final steady Geometric state state mean

Component Ei Ei Ei.m

Carbon dioxide 2.1945 2.2998 2.2465 Nitrogen o+I491 0.0501 0.0496 Methane 1.3768 1.4032 1.3899 Ethane 09093 0.9823 0.945 1 Propane 0.7842 0.8293 0.8064 i-Butane 0.9128 0.9263 0.9196 -n-Butane 0.9511 0.9665 0.9588 i-Pentane 0.9315 0.9734 0.9522 n-Pentane 0.9375 0.9504 0.9439 Hexane 0.5988 0.63 10 0.6147 Heptane 0.4984 0.5244 0.5112 Octane 06479 0.6753 06615 Nonane 0.5985 0.6228 0.6105 Decane 0.4984 0.5257 0.5119

Pl I.0791 1.0948 1.0869 PS 0.9267 0.9134 0.9200

enthalpies employed consisted of those given in Although the two steady state field tests Tables 2 and 3 of Reference [9]. These enthalpies presented herein were made about one year were adjusted as required to place the column after those presented in Reference [ 101, the in approximate energy balance as described agreement of the {Ei,m} and {PI m, pBLm} in Table previously[lO]. 6 with those under Modification (1) of Table 6

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of Reference [lo] is seen to be good in view of the possibility that changes in the lean oil could have occurred during the interval of time between the two sets of tests. The efficiencies presented in Table 6 were used in the model for the unsteady state operation of the column.

Vj+l,* + lj_l,i - Vji - lji = z (28)

Vj+,Hj+, +Lj_lhj-I-VjHj-Ljhj= dt

(29)

The liquid and vapor holdups in the column were estimated by using a free space of 156 ft3. (This space was 96 per cent of the total volume occupied by the packing.) The molar volume of the liquid was taken to be I.9035 ft3 per lb-mole and the molar volume of the vapor was taken to be 378.7 ft3 per lb-mole. The molar volumes were computed on the basis of the lean gas and the rich oil streams at the conditions at which they left the absorber. Since the total volume of the vapor and liquid must be equal to the total volume of the free space, it follows that the total molar holdups U,r and UL must satisfy the following relationship

respectively. If UjL is constant, and if

then

UjL = UL/N

and

d( UjLhj) _ UL dhj _-- dt N dt’

Since UL is independent of time, change of variable is permissible

378*7U,“+ 19035UL = 156. (26)

In addition to the free space in the packing, there was approximately 17.7 ft3 of space above the packing. Since this space contained only vapor, it follows that the total molar holdup UV of the vapor is given by

d(U/hj) _ 1 dh. 3

dt Nd(t/UL)’

WV = UPv+ 17~71378.7. (27)

Equation (26) would permit a maximum liquid holdup of 81.89 lb-moles, and Eqs. (26) and (27) permit a maximum vapor holdup WV of O-459 lb-moles.

The parameter UL was determined from the results of the unsteady state run by use of a procedure based upon the following considera- tions.

These equations imply that if the transient values of any variable were plotted vs. t/Us then the resulting plot would constitute a generalized relationship which would hold for every choice of UL. (For the absorber used to make the field tests, the plot would constitute an approximation because the vapor holdups and the change in the heat content of the packing were not negligible.) Thus, the fractional response of any component may be regarded as a function of t/UL. In particular, let the fractional response of the mole fraction of propane in the lean gas be defined by

If the initial steady state, the conditions of the upset, and the holdups are specified for an existing column, then the transient values of the variables may be calculated. For the case where the vapor holdup and the change in the heat content of the packing in each section are negligible, then Eqs. (11) and (14) may be reduced to the following differential equations.

Fractional Yl -Y1I =- response Yl Yl’

F- (35)

where yl’ is the mole fraction at the initial steady state, ylF is the mole fraction of propane at the final steady state, and y1 is the mole fraction of propane in the lean gas at any time t. The mole

du;

d ( UiLhj)

duk _ UL dxji -_-- dt N dt

the following

du; _ 1 dxji -=-- dt N d(t/UL)

(30)

(31)

(32)

(33)

(34)

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Modeling of packed absorbers at unsteady state operation - IV

fraction of propane was selected for consideration because it exhibited a significant variation with respect to time as shown in Table 3.

To obtain a set of values for the fractional response vs. t/UL, a liquid holdup UL = 20 lb- moles was arbitrarily selected. The corresponding vapor holdup U,” was O-3 11 lb-moles. On the basis of the steady state model found previously [lo], eight mass transfer sections (N = 8) were employed. The corresponding holdups for each mass transfer section were as follows:

Uk=25 (1 SjS 8);

ul” = 0.0855, Uj” = O-0389 (2 S j 5 8).

In the solution of the corresponding unsteady state problem, a At = 0.1 min was used for the first 30 time steps, a At = 0.5 min for the next 14 time steps, a At = 1.0 min for the next five time steps, and a At = 5 min for the last three time steps. The K-data presented in Table 1 of Reference[9] were used. Enthalpy data

given in Table 2 and corrected by use of the CIv’s, Ct’s, and E = O-256 (see Table 3 of Reference [9]) were used. From the transient solution so obtained, the fractional response of propane vs. tlUL as shown in Table 7 and Fig. 2 was obtained. From the experimental results presented in Table 3, the experimental response at t = 5 min is seen to be equal to O-9209. From

Fig. 2. Fractional response of the mole fraction of propane in the lean gas (predicted on the basis of UL = 20 lb moles).

Table 7. Fractional response of the mole fraction of propane in the lean gas (predicted on the basis of U L = 20 lb-moles) P

I/ iJPL, mitt/lb-mole Fractional response t/UPL, min/lb-mole

Fractional response

0.0 0.0 0.105 0.7558 0405 0.0581 0.110 0.7674 0.010 0.1511 0.115 0.7790 0.015 0.2267 0.120 0.7906 0.020 0.2848 0.125 0.8023 0.025 0.3430 0.130 0.8139 0.030 0.3895 0.135 0.8255 0.035 0.4302 0.140 0.8372 0.040 0.4651 0.145 0.8430 0.045 0.5000 0.150 0.8546 0.050 0.5290 0.175 0.8895 0.055 OS581 0.200 0.9186 0.060 0.5892 0.225 0.9418 0.065 0.6104 0.250 0.9593 0.070 O-6337 0.275 0.9709 0.075 0.6511 0.300 0.9825 0.080 0.6744 0.325 O-9883 0.085 0.6918 0.350 09941 0.090 0.7093 0.375 0.9941 0.095 0.7267 0400 1woO 0.100 0.7383 0.425 103Otl

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Fig. 2, the value of t/UL corresponding to a fractional response of O-9209 is O-2024. Thus, the next predicted value of UL is given by

5.0 uL = 0.2024

- = 24.7 lb-moles.

The corresponding value of UPv given by Eq. (26) is O-288 lb-moles. On the basis of these values of the holdups UL and .!I,“, the following distribution of holdups is obtained

U/=3.0875 (1 ~jr 8);

UIv = 0.0825, U,” = O-0359 (z S j S 8).

The unsteady state problem again was solved by use of the new distribution of holdups. At the end of t = 5 min, a fractional response of O-9186 for propane was obtained. This value was considered to be close enough to the observed value. The results of this transient solution are presented in Tables 8, 9, and Fig. 3. A compari- son of the calculated and observed fractional responses for each component at t = 5 min is presented in Table 8. A plot of the calculated mole fraction of propane in the lean gas vs. time

Table 8. Fractional responses of the mole fraction of each component in the lean gas at five minutes after the upset

Component Calculated responset Experimental response

Carbon dio Nitrogen Methane Ethane Propane i-Butane n-Butane i-Pentane n-Pentane Hexane Heptane Octane Nonane Decane

lxide 1 .oo 1.00 1 .oo 1.00 0.96 0.96 1.00 1.00 0.92 0.92 0.75 0.78 1.00 1.00 1.00 1.00 1.00 l+O 1.00 1.00 l+Kl 1.00 1.00 1 .oo 1.00 1.00 1.00 1.00

tThese values were calculated using UL = 24.7.

Table 9. Typical results for a given time stept

Temperatures (“F) Vapor flow rates (lb/moles/hr)

Trial No. Trial No.

Stage No. 1 9 16 1 9 16

27.341 24.681 24.678 2242.79 2261.02 2260.55 30.386 29.997 30.019 2313.94 2354.39 2353.90 30.078 30.534 30.563 2346.37 2373.48 237240 28.850 29602 29.625 2371.12 2380.43 2378.86 27.032 27.986 27.997 2393.25 2386.98 2385.25 24.740 25.813 25.806 2413.00 2394.54 2392.87 21.995 22.792 22.783 2430.93 2404.89 2403.42 19.330 18.089 18.061 2449.93 242 1.98 2420.84

tThese results were calculated for the first time step (At = 0.1 min) on the basis of UL = 24.7 lb-moles. The criteria for convergence were 1 (AVJV,) 1 S 0.001 and IATjj 5 0.001.

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Fig. 3. Transient values of the mole fraction of propane in the lean gas (predicted on the basis of CJL = 24.7 lb-moles).

is presented in Fig. 3. The results presented in Table 9 are typical of the rate of convergence of Procedure 1 for a given time step.

Acknowledgment-The support of this project by Humble Oil and Refining Company, Hunt Oil Company, the National Science Foundation, and Texaco Inc. is gratefully acknow- ledged.

NOTATION

Aji absorption factor for component i and mass transfer section j; defined by Eq. (4)

c total number of components Cji constant defined below Eq. (13)

ej heat transfer efficiency; defined by Eq. (2)

Eji vaporization efficiency for component i and mass transfer section j

F total feed rate lb-moles/hr; F = vlv+l+ LO

Gj enthalpy balance function; defined by Eq. (19)

hi enthalpy of pure component i in the liquid phase, B.t.u./lb-mole

h, enthalpy of the packing in element j, B.t.u./lb

Hi enthalpy of pure component i in the vapor, B.t.u./lb-mole

Kji value of K for component i at the temperature of mass transfer section j

/ii flow rate at which component i in


the liquid phase leaves the jth mass transfer section, lb-moles/hr

Lj total flow rate at which the liquid phase leaves the jth mass transfer section, lb-moles/hr

tidj total mass of packing in the jth mass transfer section

N total number of mass transfer sections

Pji constant defined below Eq. (13) S internal cross-sectional area of the

column Sfi stripping factor for component i and

mass transfer section j; defined by Eq. (4)

t time in consistent units; t, denotes the beginning and tn+l the end of any given time period under consideration At = t,+l - t,

Tj temperature of mass transfer section j

u;, US holdups of component i in mass transfer section j in the liquid and vapor phases, respectively

UjL, Uj” total holdups in the liquid and vapor phases, respectively, in the jth mass transfer section

Uji molar flow rate at which component i in the vapor phase leaves the jth mass transfer section, lb- moles/hr

Vj total flow rate at which vapor leaves the jth mass transfer section, lb-moles/hr

xji mole fraction of component i in the liquid phase leaving the jth mass transfer section

yji mole fraction of component i in the vapor leaving the jth mass transfer section

z depth of packing, measured from the top of the column; Azj is equal to the length of the jth mass transfer section

Greek symbols p a weight factor used in the implicit

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Pb

PJi 7j

Subscripts

ca H

i

: L S

Superscripts

F


method for the approximation of value of an integral bulk density of the packing, lb/fP constant appearing in Eq. (13) constant defined below Eq. (13)

I initial value L liquid S packing V vapor

calculated value single phase heavy component component number mass transfer section number iteration number single phase light component packing

final steady state

Mathematical symbols

5 xi sum over all components from i = 1 i=l through i = c

1-4 set Of all X(‘S, x1, xs, . . . ,x,

[xl-G&IT transpose of the row matrix which is

equal to the column matrix x1

il X2 X3 X matrix

REFERENCES

[I] CARNAHAN B., LUTHER H. A. and WILKES J. 0.. Applied Numerical Methods, Wiley, New York 1969. [2] BASSYONI, A. A., MCDANIEL R. and HOLLAND C. D., Chem. Engng Sci. 1970 25 437. [3] FRIDAYJ.R.andSMITHB.D.,A.I.Ch.E.JI196410698. [4] GREENSTADT J., BARD Y. and MORSE B., Ind. Engng Chem. 1958 50 1644. [5] HOLLAND C. D., Multicomponent Distillation. Prentice-Hall, Englewood Cliffs 1963. [6] HOLLAND C. D., Unsteady State Processes with Applications in Multicomponent Distillation. Prentice-Hall,

Englewood Cliffs. [7] HOLLAND C. D. and McMAHON K. S., Chem. Engng Sci. 1970 25 43 1. 181 LAPIDUS L.. Dieitul Comoutarion for Chemical Eneineers. McGraw-Hill. New York 1962. i9] MCDANIEL R., Ph.D. Dissertation: Texas A & M University, College Station, Texas 1970.

[IO] MCDANIEL R., BASSYONI A. A. and HOLLAND C. D., Chem. Engng Sci. 1970 25 633. [l 11 OSBORNE A., Ph.D. Dissertation, University of Delaware, Newark, Del. (1968). [12] RUBAC R. E., MCDANIEL R. and HOLLAND C. D., A.I.Ch.E. .I1 1969 15 568. [ 131 SUJATA A. D., Hydrocarb. Process. Petrol. Rejin. 196 140 137. [14] TETLOW N. J., GROVES D. M. and HOLLAND C. D., A.I.Ch.E. J11967 13 476.

RCsnmC-On demontre l’usage de tests reels darts le modelage dun amortisseur gami en itat instable. Les methodes de calcul developpees a cet effet sont prtsentees et demontrees au moyen des resultats numeriques.

Znsammenfassung- Die Ausfihrung praktischer Versuche zur Formulierung des Modells eines Fiillkiirper-Absorptionss5ule unter nicht station&en Betriebsbedingungen wird beschrieben. Die fur diesen Zweck ausgearbeiteten Berechnungsverfahren werden dargelegt und mit Hilfe numerischer Ergebnisse veranschaulicht.

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Modeling of packed absorbers at unsteady state operation—IV

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