I
DONALD R. OLANDER Department of Chemical Engineering, University of California, Berkeley, Calif.
of Direct Contact Cooler-Condensers Adequate tower sizing for systems other than air-water generally requires a more precise treatment than the standard enthalpy potential method. Numerical solution of the design equations is outlined here T H E design of a direct-contact coolercondenser for a particular operation usually begins Icith these d a t a :
Flo\.c. rate: temperature, and vapor content of the feed gas Inlet temperature of the coolant (assumed to be the same substance as the condensable vapor) Desired vapor content or temperature of the exit gas T h e coolant flon. rate may or may not br specified; if not, it can be chosen (Lvithin suitable limits) to minimize process costs. The most important design parameter (and the most difficult to determine) is the height or total interfacial area of the unit. Because of sharply changing temperatures and compositions and the interaction benveen heat and mass transfer phenomena \vithin the column, no single equation can predict tower height as a function of the end conditions, system geometry, and fluid properties. Rather, as sho\cn by Colburn and Hougen ( 9 ) : the anal?sis requires a point-by-point determination of the conditions as a function of height-an approach tantamount to solving the relevant differential equations numerically (usuall) by the Euler method). This report presents design procedures free from all unnecessary simplifying
assumptions and suggests a method for estimating the relevant heat and mass transfer coefficients in direct-contact cooler-condensers. T h e method is not intended to replace standard techniques for sizing cooling tolcers or other water-air humidification equipment. Rather, it is designed to extend the well-known concepts of simultaneous heat and mass transfer to more exotic systems. A mixture of a condensable vapor and an inert carrier gas entering a coolercondenser follows one of two courses, depending upon the relative magnitudes of simultaneous mass and heat transfer to the coolant liquid. T h e mixture may become superheated, or it may become supersaturated, with fog or droplets of condensed vapor appearing in the gas phase. Either condition can be predicted, although there is no guarantee that a mixture initially superheating or fogging will remain so throughout the tower. Only the incremental analysis can trace the actual history of the vapor phase during its journey u p the column. T h e analysis presented here is limited to superheated vapors-Le.? to systems entirely free from condensation in the bulk of the vapor phase.
INLET COOLANT
Subject
Kef.
Design equations for both direct contact and externally cooled condensers Numerical solution of differential design equations; treatment limited by simplifications Enthalpy potential method Extension of Merkel's method (16) to account for liquid phase heat transfer resistance: restricted to vapor systems for which Schmidt and Prandtl numbers are about equal Discussion of approximate relations for design purposes; system of simplified equations is internally inconsistent Exact relations which do not suffer from logical contradictions in old method
(2-5)
Consider a saturated feed gas which becomes superheated a t its initial contact with the coolant liquid and remains so throughout the column (see diagram). Of the nine external variables, G, Ho, t o , t l T . and either HT or tr (depending on whether the unit is to be primarily a cooler or a condenser) are known. For superheated mixtures, HT and tT are not related b) the saturation curve? and one of the remaining four variables must be specified to render the problem determinate. Generally the inlet coolant flow rate: LT:is selected, although in some cases t L , might be specified. I n addition to the restriction on coolant rate imposed by the physical operating limits of the column, L T cannot be so small that ti, is greater than to. Thus. the system is reduced to three unknowns? LO?tl,, and either HT or t T . From the over-all material and enthalpy balances:
(Lo/G) - ( L d G ) =
- HT
(1)
where s is the molar humid heat. de-
LT
G
tiT
I-*
H, tT I -
1
Literature Background
Material and Enthalpy Balances
EXIT GAS
(??)
Schematic column shows external variables relating to material and enthalpy balances
(16'1
(17) Z
I
(28)
(19)
OUTLET COOLANT
Ho SATURATED FEED GAS
'0
Lo ti0 VOL. 53, NO. 2
FEBRUARY 1961
121
define a gas phase mass transfer coefficient: .V.d = k,(P - PL) (5) Equation 5 contains no reference to a liquid phase mass transfer resistance, as coolant and condensable vapor are the same. Combining Equations 4 and 5: GdH = -k,(p - p..)avdr (6) An enthalpy balance over the liquid phase (up to but not including the interface proper) contained in the same volume element yields (79) :
fined as: s = C.vc
+ HC,
(3)
A third equation relating tT and HT is required. Unfortunately, these external variables are determined solely by the composite of heat and mass transfer characteristics a t every point within the column. T h e final relationship is thus the result of the point-bypoint calculations through the column. Therefore, before beginning the column calculations, a n end condition must be assumed and Lo and ti, obtained from Equations 1 and 2. Once the incremental column calculations are completed (to a height Z a t which either HI o r t I equals the specified value), the calculated value of the remaining external variable must be compared to the initial assumption. If these do not agree, a new assumption must be made and the calculations repeated. Once the external variables have been specified, the differential material and enthalpy balances, which a r e the basis for the incremental calculation within the column, can be utilized.
GdHCl(t, - ti)
GdHC,(t - I , )
A differential material balance a t any point within the column jields: -NAu,&
(4)
where Q, dz is the interfacial area available for mass transfer in a volume element of unit cross sectional area and thickness dz. For well-irrigated packing, ab is assumed to apply to both heat and mass transfer. T h e mass flux from the vapor to the coolant can be used to
G (sdt
+ [Ci(t - ti) f XtldH)
.4t
BLOWING NEGATIVE
0 .I 0.1
TT:,N:, D. CONSTANT
2
.4
(9)
T h e heat transfer coefficient in Equation 8 is a n "apparent" coefficient, and
a
a
a -----A
%
EQUATION (11) .2i
=
LCidti
, , ,,
.,,
, , , ,
2
4
a
.6 .E 1.0
a =-
6
', II
NACV
Mickley and others (78) measured heat transfer coefficient between a porous flat plate and main stream of air flowing parallel to it for various flow rates of air sucked or blown through the wall. At least for the case in which bulk phase and injected gas are identical, Equations 1 1 and 1 2 satisfactorily express the effect of NaC,/h, on h , INDUSTRIAL A N D ENGINEERING CHEMISTRY
h, = ( a / l - e-a)h,
(11)
where a =
AV.dC$/hQ
(12)
By convmtion, -VAis positive for transfer from gas to liquid. This derivation neglects turbulent heat transfer and the possibility that a convective mass flux through the boundary layer might alter its thickness. If the mass flux wcre sufficiently large. the region ofprincipal heat transfer resistance would not resemble (hydrodynamically) its counterpart in the simple heat transfer situation. Also, variations of thermal conductivity and diffusivity with temperature and composition are neglected. Despite these seemingly insurmountable difficulties, the Ackermann correction has been shown by Mickley and others (18) to be a satisfactory method for predicting the first order effect of simultaneous mass transfer on the heat transfer coefficient for turbulent flow (see graph). Assuming the .4ckermann correction to apply as well to conditions adjacent to the coolant-vapor interface within a cooler-condenser, Equation 11 can be introduced into Equation 8 which, with Equation 4,becomes:
The factor a, (e5- 1) includes not only the Ackermann correction to the heat transfer coefficient but the enthalpy contribution of the transferring vapor (the term on the right of Equation 8) as well. Equation 13 has also been derived in a slightly different form (8).
810
hg Effect of simultaneous mass transfer on heat transfer coefficient
122
The boundary conditions reflect known temperatures a t the interface and outer film boundary. Assuming that the true heat transfer coefficient, h, can be identified with k ' I :
(8)
Heat transfer coefficients, h, and hl, as previously derived (79), were denoted by hi and h;. T h e primes are omitted in this analysis. T h e signs in Equations 4, 6. 7 , and 8 stem from the folloiving conventions: Heat and mass fluxes are considered positive if they occur from the vapor toward the liquid; z is measured from the bottom of the column. T h e terms on the left of Equations 7 and 8 represent sensible heat fluxes a t the liquid and vapor sides of the interface, respectively. An enthalpy balance over all of the material in the volume element yields:
I
SUCTION P O S I T I V E
(7)
A similar enthalpy balance over the associated gas phase in the volume element (again not including the interface itself) gives:
Differential Material and Enthalpy Balances
d L = GdH
+
hia,(t, - ti)dz = -LCidti
h,a,(t - t , ) d t = -Gsdt
differs from that which would be a p plicable in the absence of simultaneous mass transfer for the same geometry and flow conditions. This true heat transfer coefficient, h,, can be related to the apparent coefficient by a degenerate form of the energy equation, first proposed by Ackermann (7). T h e derivation assumes a truly stagnant film adjacent to the interface, in which the sensible heat flux is governed by Fourier's law. An enthalpy balance over a differential slice within the film parallel to the interface yields :
Heat and Mass Transfer T h e most difficult design parameters to estimate. and the most critical, a r e the heat and mass transfer coefficients. Because of changing temperature and composition, they are functions of posi-
C O N TACT COO LER-CO ND E N SE R S tion within the tower. However. heat and mass transfer mechanisms are similar; thus. when the value of one is known, the other may be calculated from one of the mass-momentum and heat-momentum transfer analogies. Equipment for direct contact cooling often resembles that used for the conventional mass transfer operations of absorption or distillation. More information is available pertaining to mass transfer characteristics of such equipment than to heat transfer properties. Furthermore, mass transfer correlations generally include the effect of the physical properties of the components, and performance may be predicted for systems other than the classic water and air. Thus. k , (or its equivalent) can be determined from the appropriate correlation for simple absorption-Le., mass transfer without simultaneous heat transfer-and h, from one of the analogies; h , can be obtained in a similar manner. Unfortunately, there are a number of contacting devices in use (wood slat cooling towers and perforated plate and cascade towers) for which reliable heat or mass transfer correlations are not available.
Correlations for Determining Heat and Mass Transfer Coefficients Subject
Summary of gas phase correlations obtained from humidification or dehumidification experiments Correlation of heat transfer coefficients for water-air system in square, wood grid-packed towers; mass transfer coefficient predicted by Chilton-Colburn analogy ( 7 ) Correlations for mass transfer behavior of spray towers Data for packed towers Correlation for packed towers
Ref. (25)
(13)
(12, 15)
t h e binary gas a t t h e approximate mean t e m p e r a t u r e of t h e system, and consider it constant t h r o u g h o u t t h e column. This gross simplification is justified as follows : Humidification or dehumidification involves transfer of condensable vapor through an insoluble carrier gas, so k , of Equation 5 pertains to the movement of one component through a stagnant medium. According to the StephanMaxwell relation for molecular diffusion, the film model predicts: k , = D.~T/RTIPB.U (14) Because the binary gas phase diffusion coefficient is relatively insensitive to composition, Equation 14 suggests that k@BM is independent of vapor concentration. Consequently, this group has been incorporated into the definition of the height of a gas phase transfer unit in an attempt to minimize its concentration dependence:
However, Equation 1 4 is based solely on molecular diffusion, while the concept of HQ is primarily designed for analyzing transport in turbulent flow systems. The propriety of retaining the first power relation between P B . ~and k , in turbulent transfer processes has been questioned by Shiilman and Delaney ( 2 4 ) . They suggest that since the PBZf term arises from the molecular contribution to the total transport, then its relative importance should, by Equation 14, be the same as the diffusivity. Or. if k , varies as the 2 / 3 power of the Schmidt group, then the same exponent should apply tOfiB.1.I. However, consider purely turbulent transfer of component A through stagnant B! as in the region beyond the laminar sublayer and buffer region in pipe flow. Because component A exhibits a significant concentration gradient in the turbulent core for systems of low Schmidt number, so must component B. However, B has no net motion perpendicular to the wall, so some mechanism must be assigned to the existence of a concentration gradient of component B down which it does not move. By analogy to molecular diffusion, the turbulent transport equation for component B should be of the form :
(11)
(24)
There are many correlations describing mass transfer behavior of the same equipment, but they contain different constants, exponents, and variables, and they often yield radically different numerical answers for the same systems and equipment. In view of the substantial number of uncertain sources flom which to obtain mass transfer coefficients for a particular design. it seems overly optimistic to proceed, as Bras (5) has done, by computing h, (or k,) a t each point in the toxver. Rather. the following approach is suggested: 4 Select a generous value for t h e height of t h e gas-phase mass transfer unit HG a p p r o p r i a t e to t h e flow rates, t h e n a t u r e of t h e column a n d its packing, a n d t h e Schmidt n u m b e r of
For the case of stagnant B transfer: the last term on the right is required to keep NB = 0 and still maintain dps/dy # 0. Consequently, eddy diffusivity is most properly defined in terms of the velocity of each component relative to the average velocity of the mixture. This is tantamount to accepting the applicability of PEW to the first power for turbulent transfer as well as molecular diffusion. Cairns and Roper ( 6 ) have also presented data indicating k~(fiB.bf/$T)-~”’ to be independent of vapor concentration level. However, the exponent is quite small, and for all but very high humidities (fiB.+f/pT)-o‘l’ can be considered unity (whenfie.w/PT = 0.5, fiBw/$T-0,17 = 1.1). Sherwood and Pigford (27) have quoted data from three sources indicating that HG varies little with temperature as welli.e., 0.2 to 0.7% increase per ’ F. Furthermore, Ha, if not completely independent of velocity (or GM), is much more so than k,. Many studies (27, 24, 25) indicate k , to be proportional to the 0.6 to 0.8 power of mass velocity, while HG
varies as the 0.4 to 0.2 power. Except for large velocity changes through the tower arising from significant depletion of a high humidity vapor stream, HQ can be considered independent of flow rate. The value of HG selected, however, depends on average flow rates through the tower. Thus, despite substantial changes in temperature, vapor composition, and flow rate, a constant Ha is reasonable for estimating mass transfer resistance. Moreover, HGneed not be evaluated from the sparse data on dehumidification or humidification experiments. I n principle, the more plentiful data on isothermal absorption of dilute solutes in systems of constant distribution coefficient can be utilized directly. For these systems, the logarithmic mean driving force used to calculate k, from experimental data is theoretically correct. T h e extension of correlations based on dehumidification and humidification experiments, even to other concentration levels of the same system, is of dubious reliability; the mass transfer coefficients have generally been computed on a mean driving force basis, which is not valid for the radically changing temperatures and compositions prevalent in cooler-condensers. There are wide variations in mass transfer results reported in the literature, as shown by Wilke (29, Figure 13) who summarized experimental results of a number of gas-phase controlled mass transfer studies. All investigators employed 1-inch Raschig rings as packing and air as the inert carrier gas, yet the scatter of the results is appalling. For design purposes, it seems safest to use the data of Dwyer and Dodge (70), van Krevelen and others ( 7 4 , Vivian and Whitney (27), or Fellinger ( 7 7 ) . T h e effect of liquid flow rate must be obtained from the individual correlations or data. Having obtained a n appropriate HG, now select a n equally appropriate analogy to obtain t h e gas phase heat transfer coefficient. From a n investigation relating mass transfer coefficient geometry, flow rate, and possibly, the Schmidt number, a jDtype correlation has been developed. Experiments on the heat transfer behavior of the same system (using the Prandtl instead of the Schmidt number) should yield a jH-type correlation. The tlvo together constitute an empirical analogy. An analogy is often referred to as a “psychrometric ratio,” although it is no more than a convenient regrouping of the dimensionless terms embodied in the analogy. T h e well-known relation of Chilton and Colburn (7) is one of the most commonly used, but is by no means the only analogy available:
VOL. 53, NO. 2
FEBRUARY 1961
123
O r , Lvith the psychrometric ratio defined as:
Equation 16 can be rewritten to yield: (18)
i3 = (Pr/Sc)*'3
Wilke and Lynch ( 2 9 ) have proposed the relation: = 0.91 (Pr/Sc)'
*
(19)
which is valid for packed towers down to Pr;Sc = 0.4. T h e exponent of 0.5 seems more in accord with available data on mass transfer in packed towers (25, 29)> although more recent data support the traditional *)'8 ( 2 3 ) . The situation regarding the estimation of hl is in much the same state. T h e height of a liquid phase mass transfer unit can be defined by: H L = L/kia,pi
prevent their use in predicting a heat transfer coefficient for the liquid, provided that no properties of the imaginary t1,ansferring species appear in the final result. T h e most commonly accepted correlation of liquid phase-controlled mass transfer in a packed tower is that of Sherwood and Holloway (20):
(20)
where kl for the transfer of a hypothetical solute is defined by : -v.4 = i ; l ( C l - 6.) (21 1 One must be careful in applying Equation 21 to the cooler-condenser considered herein; for the liquid phase, consisting solely of one substance, cannot exhibit a concentration gradient. However, the nebulous physical meaning of Equations 20 and 21 does not
where n and CY are characteristic of the particular packing. Assume the availability of an analogy relating the Stanton numbers for heat and mass transfer of the type: (kq)-(Sci)h =
T h e reciprocal of the left-hand term of Equation 24 is the height of a liquid phase heat transfer unit. Unlike HG,it
0 Vapor phase is superheated at all times. 0 Mass transfer correlations based on isothermal absorption of a dilute gas are applicable to the radically changing temperature and composition conditions of a cooler-condenser. 0 Height of the gas-phase transfer unit i s constant throughout the column. 0 Appropriate heat and mass transfer analogies are available. 0 Interfacial areas per unit volume are equal for heat and mass transfer. 0 Both phases move through the tower in slug (or piston) flow. 0 Tower operation i s adiabatic. 0 There i s no enthalpy change upon mixing the condensable vapor and the carrier gas. 0 Carrier gas i s completely insoluble in the coolant. 0 The Ackermann correction can be directly incorporated into the vapor phase enthalpy balance. Liquid entrainment i s negligible. 0 Thermodynamic and transport properties of the two phases can b e adequately esiimated. 0 All heat capacities are constant over the range of temperatures in the tower.
O f these assumptions, the first six are the most serious. The unreliability of available mass transfer correlations for simple absorption has been discussed (29),and the piston flow assumption has recently been found inadequate for most liquid-liquid extraction design in packed towers (26).
INDUSTRIAL AND ENGINEERING CHEMISTRY
Design Equations Assuming that the transport properties of the carrier gas-vapor mixture are known or can be estimated a t each point in the column and that a n initial set of external variables has been calculated (e.g.. by assuming saturation of the exit vapor), the equations to be used in the incremental analysis a r e :
(23)
Equations 20. 22, and 23 can be combined to eliminate all vestiges of the hypothetical solute, provided only that the exponent b in Equation 23 is 2 . T h e Chilton-Colburn analogy, which results if b is set equal to 3 in Equation 23. is therefore not acceptable. For b = 1, however. the result is
Assumptions in the Design Method
1 24
(&) (Prljb
may be a strong function of temperature and therefore should be evaluated locally. I n addition. it is doubtful whether SIickley's assumption of constant hl k, (77) is a valid design criterion for the general case.
(1
Humidity Gradient. Because GI[ H)G, Equations 6 and 15 yield :
+
The humidity gradient of Equation 25 tacitly assumes that the chan5e in vapor content of the gas phase stems from a diffusive mass transfer process and not from bulk condensation or fogging. For saturated vapors, the humidity gradient must follow the temperature profile (via the saturation curve if there is no kinetic restriction to fog formation) a n d is denoted by dH*,'dz. Gas Phase Temperature Gradient.
Additional Assumptions in the Enthalpy Potential Method The enthalpy driving force method of analyzing simultaneous heat and mass transfer operaiions i s perhaps the most commonly employed of all the techniques available. In addition to those listed in the table on the left, its assumptions include the following: All external variables can be fixed in an unspecified manner (by assuming saturation of the exit vapor).
.p=
=
1.
0 Heat transfer resistance in the liquid phase i s neglected.
Mickley's graphical approach (17)replaces this assumption by one requiring a finite but constant h l / k , ratio. 0 pT - p L = p T - p (Le., pB.II= p r and the partial pressure driving force can b e replaced by the humidity driving force). 0 a/(ea - 1 ) = 1 (i.e., both the Ackermann correction and the enthalpy contiibution of the diffusing vapor are NAC,). neglected; h , = h, and h, 0 s = s,. 0 l / G i s constant, equal to l o / G . 0 ds/dz = 0 (i.e., the humid heat of the gas phase is constant throughout the tower).
>
0 C,(t - t l ) p*> the system is fogging, in which case further calculation with this or any other existing design method is fruitless.) A s the liquid phase heat transfer resistance is. in general, not negligible, a trial and error method is required for determining the interface temperature ti.
Assume t , = t l . Calculate p , from Equation 29. Calculate pB.,I = ( p - p z ) In J P T - pt P T - P). Calculate a from Equation 26. Calculate dH,'d: and d t d z from Equations 25 and 27, respectively. Calculate ti from Equation 28. Wht:n the value of t i computed in the last step agrees with the initial assumption, then the trial and error sequence
has been completed. (This will transpire on the first attempt only if h i is very large.) Once ti has been determined (and the corresponding temperature and huniidity gradients calculated in the fifth step above), the conditions at z f Az are obtained Goni: H (at r f 1 (at t
12)
+ Az)
= H (at z ) = t (at z )
4- (dH/dz)
Jz
+ (dt/dt)
+
T h e terms L and tl a t z Az can be obtained from Equations 1 and 2, with the 0 subscripts removed. T h e heat of vaporization in Equation 2 is a t the inlet coolant liquid wmperature, Lvhile Equation 28 requires the use of A,. Thus, the variation of X rvith temperature is of no direct significance in the design method, except in the relarivrly minor effect embodied in the heat transfer resistance of the liquid phase, Because the heat capacities have been assumed constant, A, = Xtir (CL - cri ( t - L l T ) . 0 T h e procedure is repeated for the next increment by returning to the first step Lvith new values of t , t L . p , and L . 0 \\;hen the specified end condition ( t T or H T ) has been attained, the corresponding condition ( H , or t T ) which has resulted from the incremental calculations is compared to the value assumed for the completion of the overall material and enthalpy balances. If the nvo do not agree? then the entire set of column calculations must be performed again: with the assumed end condition required for the over-all balances taken as the result of the previous incremental computations.
Results and Interpretation Several typical dehumidification operations have been analyzed on a digital computer by the numerical technique described here and, for comparison, by t h e enthalpy potential method. The systems investigated were : condrnsation of CCla from HCI %as in a packed tower at inlet humidities of 0.08 and 0.5: condensation of benzene from air, also in a packed tower; and condensation of water \rapor from CO:! in a grid tower, the example presented by Bras (5). The most important consideration pc>rtaining to the mechanics of solving the simultaneous differential equations is the size of t h e interval required; for low inlet humidities ( < 0 . 1 ) , no significant alteration of the calculated tower height iva3 observed for Az less than -0.1 foot. For high-humidity feed vapors an interval of -0.01 foot \vas required. These values, however, are merely rough guides. T i v o to five trials were required for convergence to the correct exit vapor conditions. For all of the systems examined, the assumption that the gas phase remained just at saturation throughout the towcr provrd inadequate; the exit vapor w a s invariably below its dew point, sometimes by as much as 6' F., dcspite psychrometric ratios which varied from 0.5 to 1.2 for t m three combinations considered. T h e tcndency of each of these diverse systrms to VOL. 53, NO. 2
FEBRUARY 1961
125
supersaturate clearly indicates that a satisfactory design method cannot avoid the phenomenon of fog formation. As expected, the enthalpy potential method gave results which were radically different from the exact solution for high inlet vapor humidities, even though the psychrometric ratio was close to unitye.g., a 50% discrepancy in the calculated tower height for the water vapor-COn system with HO= 5. For low inlet humidities ( 1). For Pr/Sc < 1, the two fi curves were always within 9 % of each other and the choice of the psychrometric ratio expression was of no consequence. The effect of liquid film resistance, as inferred from simple absorption data, proved to be insignificant for packed columns. For the combination of cy and n which predicts the poorest liquid-phase heat transfer coefficient by Equation 22 (1.5- or 2-inch rings), the benzene-air system exhibited differences between t , and ti as large as 7’ F. at ti = 110’ F. However, the tower height calculated under these conditions differed by only 3% from the height for the same system with this resistance ignored. Although liquid film resistance appears to be negligible in packed towers, other types of contacting equipment, particularly those with high gas flooding velocities, may be appreciably affected by poor liquid phase heat transfer.
= bulk fog index, Eq. 30 a n d 31 G , G,M = molar flow rates of carrier gas
fB
h*
h,
’F.
h,
= vapor
H
=
HG
=
HL
=
k
=
a
= constant in Ackermann cor-
a
= interfacial area for heat and
rection, Eq. 12
A, B
=
b
=
G
=
C
=
mass transfer, sq. ft./cu. ft. constants in vapor pressure relation, Eq. 29 exponent o n Schmidt group in Eq. 23 liquid concentration, lb. moles/cu. ft. heat capacity a t constant pressure, B.t.u./lb. mole-
’F.
CP
D 126
= heat
capacity a t constant pressure of vapor phase, B.t.u./lb. mole-’ F. = gas phase diffusion coefficient, sq. ft./hr.
phase heat transfer coefficient, B.t.u. hr.-sq. ft.-” F. molar humidity, lb. moles vapor per lb. mole carrier gas height of a gas phase mass transfer unit defined bv Eq. 15, ft. height of a liquid phase mass transfer unit defined bv Eq. 20, ft. thermal conductivity of the gas phase, B.t.u. hr.-ft.-
’F.
= gas phase mass transfer co-
=
1
L
= =
M
=
n
=
n,, &YB= P
=
R
= = =
s
=
PT
Pr
efficient, lb. moles/’hr.-sq. ft.-atm. liquid phase mass transfer coefficient, ft./hr. fictitious film thickness, ft. molar flow rate of the coolant liquid per unit cross-sectional area of tobver, Ib. moles ihr.-sq. ft. molecular weight constant in Eq. 22 gas phase mass transfer rate, 1b.-mole,’hr.-sq. ft. partial pressure of condensable vapor: atm. total pressure, a t m . Prandtl number gas constant = 0.73 cu. ft.atm. ‘lb.-mole-’ R . humid heat, defined by Eq. 3, B.t.u.;lb.-mole carrier gas-’ F. Schmidt number temperature, O F. temperature, R. distance from interface. ft. height from bottom of tower, ft. total height of tower. ft. constant in Eq. 22 psychrometric ratio defined by Eq. 17 eddy diffusivity, sq. it. hr. heat of vaporization, B.t.u. lb. mole molar density, Ib. moles cu. ft. viscosity, Ibs. hr.-ft. I
sc t
=
z
= = = =
Z
=
a:
= =
T
P Nomenclature
a n d total vapor stream. respectively, per unit crosssectional area, lb. moles/ hr.-sq. ft. = gas phase heat transfer coefficient in the absence of simultaneous mass transfer, B.t.u./hr.-sq. ft.-’ F. = liquid phase heat transfer coefficient, B.t.u./hr.-sq.ft.-
E
=
x
=
P
=
!J
=
Subscripts B.ZI
= log mean of interface and
1
= = = = =
1
Lvc
0 t
T z:
INDUSTRIAL AND ENGINEERING CHEMIS#TRY
bulk values interface liquid phase noncondensable carrier gas bottom of tower a t a particular temprrature = top of tower = condensable vapor
Superscript
*
= saturation value
Acknowledgment T h e author acknowledges the encouragement and assistance of R. W. Lundeen a n d the Western Division of the Dow Chemical Co., where much of the basic work in this study was undertaken, a n d the helpful comments of T. H. Chilton.
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