THERMAL FEEDBACK IN COUNTERCURRENT EXCHANGE COLUMNS HERBERT A. POHL' Explosives Department, E. I. du Pont de Nemours & Go., Inc., Wilmington, Del.
A significant variation of temperature may occur in certain processes involving countercurrent mass transfer when only small amounts of heat are generated within the equipment. Contributing causes include heats of vaporization of the solvent and solution of the solutes, the small enthalpy change owing to the exchange reaction, Joule-Thompson effects in liquid and gas, and changes in the potential energy of the streams. A theoretical treatment of the effects is given, and the computed temperature profiles are compared with observed temperatures in large columns used for enriching natural water b y deuterium isotope exchange with gaseous H2S.
5' to 7' c. below liquid entrance temperatures were observed in the first stage "cold towers" of the dual-temperature isotope exchange process [the "GS process" described by Bebbington and Thayer ( 7 ) ] . These low temper,atures were of concern because of the possibility of the formation of an ice-like hydrate of HzS a t 27.4' C. at a \vorking pressure of 250 p.s.i. ( 3 ) . Furthermore, owing to the influence of temperature on the isotope exchange equilibrium, small temperature changes in the cold towers may have an important influcnce on the process yield. A fall in liquid temperature there of only 10' C. could incrase the production rate by about 25%. Improving the efficiency of the process, therefore, requires a close look at the causes of the temperature changes taking place in the columns. The observed temperature changes in countercurrent towers containing a large number of trays, where both incoming streams have nearly the same temperature, are caused by a magnification of small heat effects arising from the countercurrent action. None of the heat effects is large and: as a result, enthalpy changes of both gas and liquid owing to several different sources Iiave to be considered carefully. T h e straightfonvard application of familiar thermodynamic principles would be expected to lead to a n explanation of the observed effects. \Vith the publication of complete enthalpy data ( 3 ) for the system .tlzS-H20-HDO-HDS this is possible, and this article presents the results of such detailed computations. N E X P E C T E D TEEMPERAI'URE DROPS O f
enrich the liquid with H D O by taking advantage of a slightly favorable equilibrium in the chemical reaction : HzO(1) f HDS(g) d HDO(1)
+ HZS(g)
for which the heat of reaction is AH0 = -593 cal. per gramatom of D exchanged. If H L and HG represent molar enthalpies, P.E., and P.E., represent potential energies, and L and G are the constant molar flows, the energy balance is: L[dHL
+ ~ ( P . E . L ) =] G[dHo + ~ ( P . E . G ) ]
(1)
Changes in kinetic energy are neglected. Furthermore, the rate of enthalpy exchange between the phases can be represented by: L[dHL
+ ~(P.E.L)]
=
UA(t - T)dn
(2)
where U is determined empirically. The total number of trays is assumed to be so great that n can be regarded as a continuous variable. Computation of the functions t ( n ) and T(n) proceeds by relating the enthalpy values HL and HG to temperature, pressure, and composition in each phase, followed by solution of the pair of equations. Enthalpy Values. T h e following standard states for the evaluation of enthalpies are chosen: for H2S and HDS the gas is a t a standard temperature, To, and zero pressure; for HzO and H D O the liquid is a t TOand 1 atm. pressure. T h e enthalpy of the gas is a function of the temperature, t, the pressure, P, and the water vapor mole fraction, y . Its differential is:
Energy Balance i n Countercurrent Flow
Consider an adiabatic, countercurrent contactor in steady flow. The liquid consists of a mixture of Hz0 and H D O plus dissolved H2S; the gas contains principally H2S and water vapor plus traces of HDS. T h e object of the operation is to
Note that the water vapor mole fraction is taken equal to the equilibrium value a t the temperature of the gas, which is a function of the temperature. As a result, the ordinary heat capacity of the gas is augmented by a term that depends on the heat of vaporization of water, A, the net coefficient of dt in the above equation being:
Present address, Plastics Laboratory, Princeton University, Princeton, N. J. VOL. 1 NO. 2 M A Y 1 9 6 2
73
Because the enthalpy must be represented very accurately, it is necessary to include the term proportional to the pressure change per tray-Le., the Joule-Thompson effect:
temperature, the augmented heat capacity of the liquid becomes :
where pQ is the Joule-Thompson coefficient of the gasprincipally HZS-at the temperature and pressure in the tower. This term, ordinarily neglected in energy balance computations such as these, makes a major contribution to the temperature changes in isotope exchange towers; of the various terms, it is the one with the right algebraic sign to produce a temperature fall in the gas as it flows upward through the tower. The differential of gas enthalpy per mole becomes: dHG
=
The coefficient of the pressure differential may be found from the specific volume of the liquid and its coefficient of thermal expansion :
(s) T,D.S
The term proportional to d 0 is evaluated from the heat of the exchange reaction, as follows: Based on one gram-atom of oxygen, the number of atoms of deuterium is 2 0 ; the moles of H D O are also equal to 2 0 . The moles of HsO equal 1-20. Thus, in terms of the molar enthalpies a t the standard temperature, the enthalpy of a mole of liquid is:
C*,G X d T - MGC,GX d P
The liquid enthalpy is a function of its temperature, t ; the fraction of deuterium in the mixture of deuterium and hydrogen combined with oxygen, 0; the pressure; and the mole fraction of dissolved hydrogen sulfide, S. The differential is given by :
The first and second terms can be combined if it is assumed that the liquid is always saturated with HzS. Adding them together and using the exothermic molar heat of solution of HZS, Q, to evaluate the derivative of solubility with respect to
=V-T($)
+
(1 - 2 D ) H 0 ~ , o 2 D ( H 0 ~ o o=) H'H?o
+ 2D(IHO)
where AH0 represents the standard enthalpy change in the exchange reaction per gram-atom of D. The partial differential coefficient needed in Equation 6 is:
The differential of liquid enthalpy becomes, finally :
d D , cal./gram-mole HnO (10)
-
k
1
1 60
POSITION BY PLATE NUMBER, n CASE 3 , 7 i I 0 0 0 , L / Q * 0 . 4 9 8 4 7
Figure 1 . Calculated temperature profile in liquid and gas streams of a countercurrent tower Entering gas and liquid temperatures equal, y = 1
74
I&EC FUNDAMENTALS
0 CASE I
,7
i
0 922813,
L/G
E
0 46
Figure 2. Calculated temperature profile in liquid and gas streams of a countercurrent tower Entering gas and liquid temperature equal, y = 0.923
CASE 5 , 7
Figure 3. Calculated temperature profile in liquid and gas streams of a countercurrent tower Liquid enters 5’ C. cooler than entering gas, */ =
liquid enters 8’ C. cooler than entering gas, y = 0.903
Differential Equations for Temperature Distribution
Dividing each side of Equation 1 by dn, introducing the enthalpy expressions, and letting g = - dP/dn, the observed negative pressure gradient per tray, one obtains the differential equation:
(2)
!c*~~ /
0 . 9 0 2 9 0 5 , L/G:0.48
Figure 4. Calculated temperature profile in liquid and gas streams of a countercurrent tower
0.903
The energy balance also contains the potential energies. These are easily expressed by using the molecular weights of liquid and gas and Az, i:he vertical spacing between adjacent trays in the column:
L
i
T h e first of these is positive for a gas below its Joule-Thompson inversion temperature; the second is small and positive because the potential energy term exceeds the first term. For a gas composed principally of H2S a t 250 p.s.i. and 30’ C. and a liquid that is almost pure water ( 3 ) , a plate spacing Az = 18 inches, and a pressure drop of 0.2 inch of H20 per tray, the numerical values are: Bo
=
$0.1041
BL
=
-0.00236
+ 0.0365
+ 0.0193
=
+0.1406 cal./(gram-mole) (plate)
=
+0.0169 cal./(gram-mole) (plate)
T h e order of the terms on the right is the same as that used in Equations 14 and 15>respectively, indicating that the potential energy term is the major one for the liquid; it is less important but not negligible for the gas. I n terms of these symbols the energy equation takes the final form :
- [VL - T ( g ) , ] g f
where 7 = LC*,,/GC*,,, the ratio of heat capacity flows. Similarly, the heat exchange rate equation becomes : It is convenient to introduce new symbols to represent the constant quantities in each bracket: Bo
+ Az(gL/gc)Mo
= PG(:~G~
(14)
T h e group UA/LC*p, (=N*) may be interpreted as the “number of heat transfer units” per actual tray. Solution of Equations for T e m p e r a t u r e Profiles. NEGLIGIBLE E X C H A N G E REACTIOS. In the first of the several towers that VOL.
1 NO. 2 M A Y 1 9 6 2
75
4
J Y
e 3.
5. W
P c
3
3 POSITION
BY PLATE
NUMBER, n
Figure 5. Observed liquid temperatures in isotope exchange tower a t dilute end of cascade
Figure 6. Observed liquid temperatures in isotope exchange tower at dilute end of cascade
Liquid enters and emerges a t same temperature
Liquid enters
are used in series in the GS process for producing deuteriumenriched water, the heat effect owing to the exchange reaction is minor because the changes in D even over the 70 trays of the tower are small. Under these circumstances, it is instructive to drop the term representing the enthalpy contribution of the reaction from Equations 16 and 17. Then the properties of the solution and, in particular, the influence of the JouleThompson cooling of the gas can be studied more simply. I t is found, as shown later, that competing effects in the energy balance between heat exchange between phases and the pressure-differential of enthalpy can cause minima in the liquid temperature under some circumstances. Such minima have been observed in operating towers. Solutions of Equations 16 and 17 can easily be obtained for any values of the boundary temperatures of the liquid and gas streams at the top and bottom of the tower, t ( Y ) and T(O), respectively, and of the important parameter, y. However,
4.3'
C. cooler than it emerges
solutions for the special case y = 1 are simplest and \+ill bring out the effects of special interest. \ m e n y = 1. Equation 16 has the solution:
T - T ( 0 ) = t - ( 0 ) f [ ( B L / C * ~ L-) (BG/C*pG)]n (18) where t ( 0 ) represents the temperature a t which the liquid leaves the tower. Introducing this equation into Equation 17, one obtains for y = 1 :
+
dt/dn = [(BG/C*~G -) ( B L / C * , L ) ] ~ V * ~[ t ( O ) - T(O)l.V*
-
( B L / C * , L ) (19)
Evidently, Equation 19 permits the possibility of having dt,'dn = 0 a t some value of n in the range 0 < n < S. In fact, this is what is observed when the detailed solutions of Equations 16 and 17 are carried out, as illustrated in Figures 1 to 4. The conditions applying in the calculations illustrated are listed in Table I.
Table I. Conditions Used for Calculating Tower Temperature Profiles N = 70 actual trays; UA = 85,000 p.c.u./(hr.)(tray)(' C.)
Case i\?O.
1
3 4 5
76
Flow Rate, Lb. MoleslHr. G L
13,000 13,000 13,000 13,000
5,980 6,480 6,240 6,240
I&EC FUNDAMENTALS
Heat Capacities, Pressure and Height Coeficients, P.c.u./(Lb. Mole)( a C.) (P.c.u./Lb. Mole) C*pC C*PL PL BT-
10.56 10.56 11.04 11.04
21.19 21.19 20.78 20.78
0.0517 0.0517 0.0474 0,0474
0,1386 0,1386
0.1387 0.1387
Relatioe Heat Transfer, y
0.9228
1 .oooo
0.9029 0.9029
Heat Transfer Units per Tray, N* 0.672 0.620 0.655 0,655
L\70.
Gas Inlet Temp.,
Lipid Inlet Temp.,
?(O),
t(O,V,
C. 34
34
40 40
C. 34 34 35 32
II
40
10
39
9
8
38 7
9
9" I1
w
2 37
s
T 0 5
n
2 4
36
2
3s
I
34
C
Figure 7. Observed liquid temperatures in isotope exchange tower at dilute end of cascade Liquid enters
4..9' C. cooler than it emerges
Comparison of Figureis 1, 2, and 3 shows the effect of decreasing the relative thermal capacity factor, y. As indicated in Table I, this is accomplished by changing the liquid flow and,'or the heat capacities. As the heat capacity transport into the column by one stream is increased, the position of the minimum temperature r,hifts toward the end of the column where this stream leaves. Comparison of Figures 3 and 4 shokvs the effect of incrasing the average temperature difference between the fluids by lowering the inlet fluid temperature while holding the inlet gas temperature constant. As the temperature difference increases, the minimum temperature tends to disappear. Figures 5, 6, and 7 S ~ O M .typical temperature distributions observed in the cold dilute-end towers of the GS process by Keller (2). Qualitatively, the same behavior was shown as in the results of the calculations. When the inlet and outlet liquid temperatures were equal, as in Figure 5, a pronounced minimum in the profile was observed; as the inlet liquid temperature \vas reduced, the minimum disappeared as expected. O n the other hand, the observed temperatures approached their minima more sharply than did the calculated temperatures. The explanation for this lies in the omission so far of the heat effects from the i:;otope exchange reaction. HEATO F ISOTOPE EXCHANGE ISCLUDED. If a n expression were introduced for the rate of mass exchange of the isotopic compounds between phases, another differential equation would become available which could be solved simultaneously with the energy balance and the energy rate equation to determine profiles of both temperature and deuterium concentration, D. M'hile this is not difficult in principle, the extra
Figure 8. Observed deuterium/hydrogen ratios in isotope exchange columns a t two temperatures Curves correspond to Equation 20 with constants determined to fit ccmand 140 position on plates 0,70,
mathematical work involved tends to obscure the effects producing the temperature variations. An approximate method of estimating the energy term from the heat of the reaction is, therefore, employed. This involves the assumption that, despite the variable liquid temperature in the column. the distribution of isotope concentration, D ( n ) , can be found from the Kremser formula since the quotient of equilibrium distribution ratio for deuterium between vapor and liquid by the ratio of flows is constant. Under this assumption:
where D ( Y ) represents the deuterium-hydrogen atom fraction a t the top of the tower; a is the change in D per actual tray; a is the enrichment factor, equal to 1.079 for the reaction used; and E is the plate efficiency. Both a and E have to be found from observations of the distribution of D in operating to\vers. Measurements a t the top the middle, and the bottom of towers having N = 140 gave D = 0.0007, 0.0055. and 0.105, respectively. Solution of Equation 20 for these values gave a = 0.00188 and E = 0.57. Using these values, the entire D-distribution over 140 trays could be plotted, as sho\vn in Figure 8. Based on Equation 20, one obtains: a
E lna
d D / d n = -exp. [ E lna (,V - a ) ] a - 1
When this expression is introduced into Equations 16 and 17, they remain linear equations in the temperatures t and T , and a solution can be effected by standard methods. VOL. 1 N O . 2 M A Y 1 9 6 2
77
equations presented here make it possible to study the consequences of such changes in boundary-value temperatures on the internal tower characteristics. Nomenclature
D per actual tray
a A B C,
= interfacial surface area per tray in column = coefficient of dn in energy equation, p.c.u./lb. mole = heat capacity a t constant pressure p.c.u./(lb. mole)
C*,
= apparent heat capacity a t constant pressure, allowing
D
=
E G
= =
= change in
(" C.) 40
g
=
gL
=
go
=
H
= =
AH0
L M N* AT n
= = = = =
P
= = P.C.U.= Q = R =
P. E.
--
140
120
100
-
WSlTlON
LIP
I ~~
BO 60 40 BY P L A T € NUMBER, n
20
0
c GAS
Figure 9. Calculated temperature distributions in isotope exchange tower a t concentrated end of cascade Assuming a = 1.079, L/G - 0.46 AHo = - 5 9 3 cal./mole D
Figure 9 shows the results of calculations for a 140-tray tower into which liquid and gas streams were fed a t t ( N ) = 35" C. The measured values of D were those used above. Using L = 2088 lb. mole per hour of water, G = 4539 lb mole per hour of HzS, C*,, = 20.87 p.c.u./(lb. mole)(' C.), and C*, = 10.92 p.c.u./(lb. mole)(' C.), one obtains y = 0.8795. Furthermore, based on an average temperature of 37' C. and 250 p.s.i. operating pressure, published values ( 3 ) of enthalpy and the observed pressure drop across 140 trays of 12 p.s.i. correspond to B L = 0.0484 p.c.u./(lb. mole H20) and B, = 0.1387 p.c.u./(lb. mole H2S),as discussed above. An empirically satisfactory value of UA = 25,000 p.c.u./(hour) (tray) (" C.) corresponded to N* = 0.546 heat transfer units per actual tray. Figure 9 is based on these numerical values; it indicates a temperature maximum, because of the predominating influence of the heat source accompanying isotope exchange, in good agreement with observations. Futher detailed discussion and an alternate, somewhat more rigorous approach are available (4, 5). Conclusions
With means available for computing temperature profiles inside adiabatic isotope exchange towers, it becomes possible to consider ways of improving process efficiency by suitable introduction of heaters or coolers to counteract undesirable heat effects. Such devices might conceivably be installed inside the towers at points where undesired effects are most pronounced; but it is easier to install them a t the entrances to towers or between successive towers in a cascade. Then the 78
l&EC FUNDAMENTALS
S
=
T
=
t U UA
= = =
V
=
y Az a
= = =
y
=
X
=
p,
=
for enthalpy of condensable vapor or soluble gas a t saturation, p.c.u./(lb. mole) (" C.) fraction of hydrogen-plus-deuterium in liquid that is deuterium plate efficiency gas flow rate, lb. moles/hr. - dP/dn, observed negative pressure gradient per tray local gravitational acceleration, ftJsec.2 gravitational conversion constant: (lb. mass) (ft.)/ (lb. force) (sec.2) enthalpy, p.c.u./lb. mole standard heat of the isotope exchange reaction, p.c.u./ 1b.-atom of deuterium liquid flow rate, lb. moles/hr. molecular weight of gas or liquid number of heat transfer units per actual tray total number of actual trays in column number of actual trays, counting from bottom of column pressure in consistent units potential energy, p.c.u./lb. mole enthalpy to raise 1 lb. H 2 0 1O C., analogous to B.t.u. latent heat of solution of H1S in water, p.c.u./lb. mole gas constant mole fraction of dissolved H2S in liquid a t saturation gas temperature " C. ; occasionally absolute temperature, ' K . liquid temperature, C. heat transfer coefficient heat transfer coefficient at interface between gas and liquid, p.c.u./(hr.) (tray) (" C.) molar volume of liquid, cu. ft./lb. mole mole fraction of water vapor in gas at saturation vertical spacing between trays, ft. separation factor for isotope exchange heterogeneous reaction ratio of heat capacity flows of the streams, LC*pL/ GC*~G latent heat of vaporization of water, p.c.u./lb. mole Joule-Thompson coefficient of gas, " C.,'atm.
Subscripts
L G
= liquid stream = gas stream
Acknowledgment
The author expresses appreciation for helpful discussions with D. F. Babcock, B. S. Johnson, TV. R. Keller, W. E. Winsche, V. R. Thayer, and H. L. Hull, all of E. I. du Pont de Nemours 8L Co. Literature Cited
(1) Bebbington, W. P., Thayer, V. R . , Chem. Eng. Progr. 5 5 , 70 (1959). (2) Keller, W. R., E. I. du Pont de Nemours & Co., Inc., private rnmmiiniratinn. Julv ..~1953. (3) Pohl, H. A., J. C h i m Eng. Data 6, 515 (1961). (4) Princeton Univ. Plastics Lab. Tech. R e p . 61A, . . Pohl, H. -4., April 1, 1961. (5) Pohl, H. A., Zbid., 61B. RECEIVED for review April 7, 1961 ACCEPTEDJuly 24, 1961 Work supported by U. S. Atomic Energy Commission under Contract AT(07-2)-1. Article written with approval of U. S. Atomic Energy Commission and E. I. Du Pont de Xemours & Co., Inc., at Princeton University under Signal Corps Contract DA-36-039~~-78105. _ _ . . . . . . _ ~ ~ . - . . . ~ . ~ ~ I
~
