rapidly, and suddenly the hydrocarbon potential resulting from chemisorption becomes dominant. The 25 to 50% increase in cell resistance within the first few hours of operation on hydrocarbon fuel occurs at the same time that performance declines. Formation of an oxide film on the active electrode surface probably increases resistance a t the electrode-electrolyte interface and decreases electrode activity. Both factors contribute independently to lower performance. Restoration of cell conductivity and performance occurs within minutes after reintroduction of hydrogen fuel. This behavior is consistent with an oxide film which would be rapidly reduced by hydrogen. Cathode Polarization
Dilution of the oxygen feed with nitrogen increased cathode polarization for both types of electrode. The Type AA-1 cathode was less affected (Figure 5). A mixture of 10 cc. S T P per minute of oxygen and 30 cc. S T P per minute of nitrogen resulted in higher polarization a t all currents than the equivalent amount of oxygen, but was similar (within experimental error) to air, a t a rate which would supply 10 cc. S T P per minute of oxygen (see also Table V). Porous carbon cathodes were more affected by the oxygen dilution (Figure 6). Polarization increased much more rapidly for both the oxygen-nitrogen mixture and air than for a n equivalent amount of pure oxygen. Air and the oxygennitrogen mixture had the same cathode polarization curve. T h e porous carbon cathode was very susceptible to concentration polarization resulting from the accumulation of inert diluents and cathode reaction products near the electroactive sites. Conclusions
Concentration polarization is a problem when the phosphoric acid paste electrolyte fuel cell is operated on dilute hydrogen or dilute oxygen. I t is a more serious problem with the porous carbon electrodes than with the Type AA-1 electrodes. T h e superior reliability in ext.ended operation of the porous
carbon electrodes (Mather and Webb, 1968), however, suggests that these electrodes be further considered. A thinner, more open pore structure of porous carbon should lead to improved performance on dilute feeds. With carbonaceous fuels major problems are “open-circuit” activation polarization and electrode oxidation, leading to even greater activation polarization with both types of anodes. T o obtain efficient utilization of such fuels the present activation polarization of about 0.4 volt must be lowered. A more active electrocatalyst, preferably one which is less easily oxidized in the presence of carbonaceous fuels, will be required. At present the phosphoric acid paste electrolyte single fuel cell unit has been shown to be operable on methanol (steamreformed a t the fuel cell operating temperature) and air a t 45 mw. per sq. cm. This unit should be attractive as a component of a very simple, convenient, portable power source. Ac knowledgment
We are indebted to R. M. Suggitt for many helpful discussions and to 0. B. Purdy for assistance with experimental work. literature Cited
Breiter, M. W., Gilman, S., J . Electrochem. Sac. 109, 622 (1962). Gilman, S., Breiter, M. W., J . Electrochem. Sac. 109, 1099 (1962). Grubb, W. T., Michalske, C. J., J . Electrochem. Sac. 111, 1015 (1964).
Haldeman, R. G., Colman, W. P., Langer, S. H., Barber, W. A., Adoan. Chem. Ser., No. 47, 106-15 (1965). Mather, W. B., Webb, A. N., IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 7,11 (1968). Niedrach, L. W., J . Electrochem. Sac. 109, 1092 (1962). Oswin, H. G., Hartner, A. J., Malaspina, F., Nature 200 (4903), 256 (1963). Petroleum Week 10 (18), 67, 70 (May 6, 1960). Poirier, A. R., Briese, J. A., Chem. Eng. Progr. 62 ( 5 ) , 81 (May 1966). Schlatter, M. J., Advan. Chem. Ser., No. 47, 292-317 (1965). Thacker, R., Electrochem. Tech. 3, 312 (1965). Webb, A. N., Mather, W. B., Suggitt, R. M., J . Electrochem. Sac. 112, 1059 (1965). RECEIVED for review March 2, 1967 ACCEPTEDAugust 17, 1967
VAPOR FLOW LIMITATIONS IN A MELTER-CONDENSER P. L. T. B R I A N , K. A. S M I T H , AND L. W . PETRI Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass. 02139
processes for water desalination show promise of producing potable water a t a lower cost than that achievable by distillation processes in small and medium size desalting plants (Barduhn, 1965, 1967; Brian, 1967). Processes are currently being developed based upon vacuumflash freezing and the use of a n immiscible refrigerant, such as butane, in direct contact with the freezing brine. An important step in these processes is the melting of the washed ice crystal product by direct contact condensation of the compressed refrigerant vapor, low pressure water vapor in the case of vacuum-flash freezing, and butane vapor a t approximately atmospheric pressure in the case of the butane freezing process REEZING
(Barduhn, 1965; Brian, 1967). Most of the research and pilot plant development work on the freezing processes has focused upon the crystallization and washing steps, with relatively little attention to the ice melting step. T h e meltercondenser units in the freezing pilot plants have operated well, with over-all temperature driving forces between 1’ and 2” F., and therefore, this unit operation has not received the attention given to the freezing and washing steps. But the scale-up of the freezing processes to large sizes and the optimization of the processes with respect to temperature driving forces and ice crystal sizes will require a much better understanding of the ice melting step than is available (Brian, 1967). VOL. 7
NO. 1
JANUARY
1968
21
A simple model is analyzed for the melting of a drained bed of ice crystals b y direct contact with a condensing vapor. Only the outermost 0.01 inch of the ice bed is accessible to water vapor at approximately 5 mm. of mercury pressure. For butane vapor at atmospheric pressure, all condensation occurs in the outermost inch of the ice bed. In both cases, the inner regions of the ice bed are not available for ice melting. Thus the design and scale-up of a melter-condenser unit in a freeze-desalination plant must b e based upon ice bed surface area, not total volume. The analysis is formally identical to that of Thiele for diffusion and first-order surface reaction in porous catalysts.
Melter-Condenser Pilot Units
VAPOR S U P P L Y
I n the 100,000-gallon per day vacuum flash freezing pilot plant of Colt Industries (Barduhn, 1965; Brian, 1967), the washed and drained ice crystal product is sent down a chute to the top of a dumped bed of ice crystals in the melter-condenser. A rake rotates slowly across the top of the bed, distributing the ice from the chute over the surface of the bed. The compressed water vapor, a t a pressure of approximately 5 mm. of mercury, is directed onto the top of the bed of ice crystals, and product water obtained from the melting ice and the condensing vapor is drained down through the bed of ice crystals and through holes in the bottom support. The meltercondenser in the 55,000-gallon per day butane freezing plant of Blaw-Knox Co. was similar to the Colt design, employing direct contact between the compressed butane vapor and a dumped, drained bed of ice crystals. I n contrast, the 15,000gallon per day butane freezing pilot plant of Struthers Scientific and International Corp. employs a column fitted with splash trays for contact of the compressed butane vapor with a slurry of the ice crystals in a n excess of water (Barduhn, 1965; Brian, 1967). This paper presents an analysis of the design employed by Colt and Blaw-Knox--namely, the direct contact of the condensing vapor with a dumped, drained bed of ice crystals. T h e over-all rate of heat transfer between the condensing vapor and the melting ice in such a system will be limited either by the thermal resistance of the water film surrounding the ice particles in the packed bed or by the accessibility of the inner regions of the ice bed to the vapor, which must flow in through the bed from the outside, or by both of these limitations, assuming that the system is free of noncondensable gases. T h e purpose of this paper is to examine, in a preliminary way, the extent to which the inner regions of the ice bed are accessible to the vapor flowing in from the outside. This is an important consideration because it determines the fraction of the ice bed which is effective in the melting-condensing step, and thus it has an important influence upon melter-condenser design and scale-up. Theoretical Model. Figure 1 is a diagram of an idealized model of a melter-condenser. A dumped, drained bed of ice crystals of height L is supported by a false bottom and contained by the side walls of the vessel. The vapor to be condensed is supplied to the top of the ice bed a t a pressure, Po,which corresponds to a vapor condensation temperature, To*,which is greater than the ice melting temperature of 32' F. The vapor flows through the voids in the packed bed and condenses on the ice crystals. The liquid produced by the condensing vapor and the melting ice drains down through the bed and out the bottom. The vapor flow is driven by a pressure gradient and is assumed to obey Darcy's law
V,
=
-B
($)
Inertial effects are neglected throughout the analysis, and for the examples considered here this is admissible. The local rate of heat transfer between the condensing vapor and the melting ice is described by a heat transfer coefficient, h, which represents the thermal conductance of the liquid film which 22
IbEC PROCESS DESIGN A N D DEVELOPMENT
l l l l l
.DUMPED, DRAINED BED OF I C E CRYSTALS
FALSE BOTTOM
___
Figure 1.
-
WATER OUT
Idealized model of a melter-condenser
surrounds each ice crystal and insulates it from the vapor. A quasisteady material balance on the condensing vapor within the ice bed yields the differential equation
dx
(p
B,;)
=
(T* - 32)
The left-hand side of Equation 2 represents the rate of change of the vapor flow, and the right-hand side represents the local rate of condensation of vapor expressed as the heat transfer rate divided by the latent heat of condensation, A. The local rate of heat transfer is given as the product of the heat transfer coefficient, the ice crystal surface area per unit of gross bed volume, and the temperature driving force expressed as the difference between the vapor condensation temperature a t the local pressure and the ice melting point of 32' F. For small departures of T* from 32' F., the vapor pressure curve for the condensing vapor can be linearized with little error:
T*
- 32 N p(P - P E )
(3)
where p is the reciprocal of the slope of the vapor pressure curve and PE is the vapor pressure a t 32' F. Often, p is most readily evaluated from the Clausius-Clapeyron equation,
P"
T/pX
(4)
Employing the linearization of Equation 3 together with the following definitions of dimensionless variables,
- P* y = P~-
(5)
Equation 2 can be written as
,dzY = ( p B h ) YhaOL2 =92Y
(7)
In transforming Equation 2 to Equation 7, the authors assumed that the vapor density, p , and the permeability of the ice bed to vapor flow, B, are constant. Furthermore, in integrating Equation 7, they assumed that the volumetric heat transfer coefficient, ha, is also constant. These assumptions are not exact. T h e vapor density will vary somewhat as the pressure drops within the bed. Furthermore, both the ice crystal size and the liquid holdup within the voids of the ice bed may vary with position, and these will affect the heat transfer coefficient and the permeability of the bed to vapor flow. But the liquid drainage rate will generally be an order of magnitude lower than the liquid flooding rate for the ice bed, and thus the liquid holdup in the bed will probably not be much greater than the static holdup, which may be of the order of 20% of the void volume. Thus the variations in p , B, and ha would be expected to be small. Neglecting these variations simplifies the analysis considerably and will surely not invalidate the qualitative conclusions. T h e boundary conditions for Equation 7 are At
Z=O,
(8)
Y = l
Equation 9 results from the assumption that there is a liquid seal a t the bottom of the melter-condenser and hence that the vapor flow rate is zero a t x = L. T h e solution to Equations 7, 8, and 9 is
Differentiating Equation 10 and inserting the pressure gradient into Equation 1 yield the superficial vapor velocity a t any position. Evaluating this vapor velocity a t the top of the ice bed yields one important result of this analysis Go = (To*- 32)
tanh
+
PA
where Go is the mass rate of condensation of the vapor in the entire bed divided by the horizontal cross-sectional area of the bed. Since the bed height, L , enters only as a factor in 9 and since the hyperbolic tangent is within 1% of its asymptotic value of unity for 4 values greater than 2.5, it follows that bed depths corresponding to 6 greater than about 2.5 are of little utility . Another result of interest is the effectiveness factor of the bed Go
__---__
rhaL(T,*
Effectiveness of a Melter-Condenser in Vacuum Flash Freezing
T o examine the implications of Equations 11 and 12, a sample calculation is made for a melter-condenser employing low pressure water vapor, such as that in the vacuum-flash freezing plant of Colt Industries. The ice crystals produced in the various freezing pilot plants have been platelets several tenths of a millimeter in diameter. Permeability measurements for brine flow through packed beds of these ice crystals, made in conjunction with an analysis of the wash column operation, have been translated into equivalent spherical particle diameter by the Carman-Kozeny equation. Results for the Colt pilot plant (Brian, 1967) have indicated a diameter of approximately 0.2 mm. Assuming a value of 0.2 mm. for the equivalent spherical diameter of the ice crystals, the permeability of the ice bed to the flow of vapor can be calculated from the Carman-Kozeny equation
(9)
al:
tl=
This problem is completely analogous to the problem of diffusion in a porous catalyst with a first-order chemical reaction; (Satterfield and Sherwood, 1963; Thiele, 1939). T h e variable 4 in the present analysis is the Thiele modulus for vapor flow and condensation within a packed bed. This idealized model of vapor flow in a melter-condenser has been used previously (Orcutt, 1966) in a computer simulation of an entire butane freezing plant (Orcutt, 1965). Orcutt did not, however, examine the implications of this model and draw the conclusions which are drawn in this paper.
- 32)l
tanh $ - __
9
T h e effectiveness factor, 7, is defined as the rate of vapor condensation in the bed divided by the condensation rate which would be obtained if the pressure of the vapor throughout the bed were equal to its external value, Po. Thus p is a measure of the fraction of the heat transfer surface within the ice bed which is accessible to the vapor.
The porosity of the ice bed, e, is assumed to be 0.5, and the viscosity, p, is taken to be that of saturated water vapor a t 32" F. The effect of the liquid holdup in the voids of the ice bed upon the vapor permeability is neglected, and this would be expected to result in a calculated value of B somewhat greater than the true value. The ice crystal surface area per unit of gross bed volume, a, is taken to be that of 0.2-mm. diameter spheres packed a t a void fraction of 0.5. The heat transfer coefficient, h, is approximated as the thermal conductance of a water film 0.01 mm. thick. This water film, spread over the surfaces of 0.2-mm. diameter spheres packed a t a void fraction of 0.5, would correspond to a water holdup of approximately 33% of the void volume. The value of p is taken to be the average slope of the vapor pressure curve for water between 32" and 33' F. The temperature driving force, To* - 32, is assumed to be 1' F., and the bed height, L, is taken to be 1 foot for the first example. Using these values, the superficial mass velocity of the vapor a t the top of the bed, Go, and the bed effectiveness factor, 7, are calculated from Equations 11 and 12. Table I summarizes the assumed and calculated values. The results of the calculations are startling. The Thiele modulus, #, is calculated to be 3220, and the effectiveness is only 0.03'%. Indeed, if the bed height, L, were only 0.1 inch, the Thiele modulus would be 26.8 and the effectiveness factor would be only 0.037. So inaccessible are the interior regions of the ice bed to the vapor that a bed only 0.1 inch high would still be less than 470 effective. The condensation rate per unit of bed cross-sectional area, Go. is computed to be 13.2 pounds per hour per square foot for a bed depth of 1 foot. = 2.68, and the hyperbolic For a bed depth of 0.01 inch, tangent of @ is 0.99. Thus, Equaiion 12 shows that a bed 0.01
+
VOL. 7
NO. 1
JANUARY
1968
23
Implications to Design and Scale-up Table 1.
h =
Assumed and Calculated Values for Flash Freezing Example 0.2mm. To* - 32 = 1’F. 0.5 L = 1 ft. 4570 sq. ft./cu. ft. B = 25.5 ft.4/(lb.,)(hr.) 10,000 B.t.u./(hr.)(sq. ft.) 4 = 3220
/3 = p = X =
(’ F.) 271’ F./p.s.i. 0.000303 lb.,/cu. 1076 B.t.u./lb.,
p =
1.69 X 10-7 (Ib.f)(sec.)/sq. ft.
d = E
=
a =
Go = 13.2 lb.m/(hr.)(sq. ft.) 7 = 0,00031 ft.
go e 0.5 P
inch high can condense 99% as much vapor as a bed of infinite height. A number of simplifying assumptions were made in this analysis, and the calculated results must be regarded as only approximate. I n particular, the effective depth is only one particle layer deep (0.2 mm. y 0.01 inch), and this raises questions about the applicability of Darcy’s law. But it is difficult to visualize the simplifying assumptions causing the computed result to be off by many orders of magnitude, and the essential conclusion that only the outermost fraction of a n inch of the ice bed is accessible for vapor condensation seems inescapable. Furthermore, the computed value of Go is of the right order of magnitude. Equation 11 shows that Go is directly proportional to the over-all temperature driving force, and thus the calculated value of Go would be 26.4 pounds per hour per square foot for a n over-all temperature driving force of 2’ F. I n the Colt Industries’ vacuum-flash freezing pilot plant a t the Office of Saline Water Test Station, Wrightsville Beach, N.C., plant production rates of approximately 100,000 gallons per day of fresh water have been achieved (Brian, 1967) with an over-all temperature driving force in the melter-condenser between 1’ and 2’ F. The melter-condenser has a cross-sectional area of approximately 120 square feet and an ice bed height of some 5 feet. Thus the fresh water production rate is approximately 290 pounds per hour per square foot of melter-condenser cross-sectional area. Since approximately pound of water vapor is condensed per pound of fresh water product, this corresponds to a vapor condensation rate of about 40 pounds per hour per square foot of melter-condenser cross-sectional area. Therefore, the observed condensation rate in the Colt pilot plant is higher than the value calculated in this analysis by a factor of 1.5 to 3, depending upon whether 1’ or 2’ F. is assumed for the over-all temperature driving force. Considering the fact that the ice bed surface in the melter-condenser is not smooth and could easily have 2 sq. feet of ice bed surface per square foot of column cross-sectional area, the agreement between the calculated and observed vapor condensation rates is impressive. With the simplified method of calculation adopted here, Go is independent of ice particle size. T h e permeability of the ice bed varies as the square of the ice particle diameter, as shown in Equation 13. But the heat transfer coefficient, h, is assumed to vary inversely as the particle diameter, and the particle surface area per unit volume, a, varies inversely as the particle diameter. Equation 11 shows, therefore, that the effects of particle diameter upon B, h, and a cancel and leave Go independent of ice particle diameter as long as @ remains greater than 2.5. On the other hand, the effectiveness factor is dependent upon particle size. With the present simplifications, 4 is inversely proportional to d2, and, a t low values of q , q is directly proportional to d2. 24
I&EC PROCESS DESIGN A N D DEVELOPMENT
The significance of these calculations to the design and scale-up of the melter-condenser unit for a vacuum flash freezing process is obvious. The vapor can penetrate effectively only a fraction of a n inch into the ice bed, and thus the design and scale-up of the melter-condenser unit must be based upon the surface area of contact between the ice bed and the vapor, not upon gross volume. Likewise, in thinking about the type of equipment which will accomplish the ice melting step with the least cost, these calculations offer a great deal of perspective. Obviously the need is not to provide ice particle surface area but rather ice bed surface area in contact with the vapor. Any mechanism for distributing the ice throughout the vapor or the vapor throughout the ice should increase the effectiveness. This might take the form of rakes which dig down deeply into the ice bed and cut vapor paths into it, or other mechanical means for creating vapor channels which would promote contact of the ice bed with the vapor. Of course, there may also be vapor flow limitations in these channels due to wall friction. Furthermore, while inertial effects were negligible for flow within the ice bed, this need not be the case for flow in large channels. Calculations indicate that acceleration effects will ultimately predominate over wall friction effects, and thus the vapor velocity in the channel cannot exceed that velocity a t which one kinetic head is equal to the over-all pressure driving force. I n this special case, the over-all pressure driving force is the total head of the supplied vapor minus PE. For water vapor with a n over-all pressure driving force equivalent to a temperature driving force of 1’ F., the limiting velocity a t the channel entrance corresponds to about 370 pounds mass per hour per square foot, which is 28 times the value of 13.2 pounds mass per hour per square foot shown in Table I. Assuming that vapor channels into the ice bed will occupy no more than 50% of the bed area, this acceleration effect places a limit of 14-fold on the possible increase in Goachievable by use of vapor channels. A network of open channels appears to form throughout the ice bed in the melting chamber of the Colt pilot plant (Brian, 1967; Johnson, 1966). Experiments were performed in which the normal operation was interrupted and a door was opened in the side of the melter chamber. Photographs of the ice bed through that door, extending several feet beneath the surface of the ice bed, showed the presence of a network of open channels, several inches in diameter, which appeared to run more or less uniformly throughout the volume of the ice bed. T h e mechanism by which these channels are formed is not now understood, and indeed it is conceivable that they were formed during the breaking of the vacuum within the chamber rather than during the operation of the melter. On the other hand, they might have been formed by “fingering” of the vapor flow, if excessive ice melting a t one point were to decrease the bed porosity a t that point and this were to feed back to increase the melting rate a t that point even further. Another possibility is that the water, being slightly above 32’ F., melted away the channels as it flowed down through the ice bed. But whatever the mechanism by which these channels were formed, if they are present during the operation of the unit, most of them are probably sealed off from the vapor above the ice bed. I n view of the agreement between the value of Go computed here and that observed in the pilot plant, it seems unlikely that such channels are contributing greatly to the condensation rate. Furthermore, in view of the inability of the vapor to penetrate appreciably into the ice bed, it is easy to see how the rake a t the surface could seal off such channels.
Butane Freezing Process
T h e implications of Equations 11 and 12 to the meltercondenser in a butane freezing process will now be examined by means of a second example calculation. T h e permeability, B , will not be very different from that for water vapor, but the vapor density will be much higher for the butane vapor a t approximately atmospheric pressure than it was for the low pressure water vapor. Although the liquid phase permeating through the ice bed is a mixture of water and butane in the present case, it is likely that most of the liquid holdup will be the wetting liquid, water, and hence the heat transfer coefficient is assumed to be the same value as that used with water vapor condensation. T h e assumed and calculated values are summarized in Table 11. In this case, the Thiele modulus is calculated to be 35.6, and the effectiveness factor for a bed of ice 1 foot high is 0.028. For a bed height of 1 inch, the Thiele modulus would be 2.97, and the effecriveness factor would be 0.34. T h e vapor condensation rate in a bed 1 inch high would be more than 99% of the condensation rate in an infinitely deep bed. T h e calculated value of the vapor condensation rate, Go, is 7800 pounds per hour per square foot. This is almost 600 times the value calculated for water vapor condensation, but the heat transfer rate for butane condensation is only approximately 100 times that for water vapor condensation because of the six-fold difference in latent heats of condensation. T h e Reynolds number based upon the particle diameter and the vapor velocity a t the top of the ice bed was computed to be 280. This value is so high that Darcy’s law and the Carman-Kozeny equation would not be expected to apply; rather, the vapor flow in the ice bed would be expected to be turbulent. Therefore this analysis, when applied to butane vapor condensation, probably predicts too high a vapor condensation rate and also too high a n effectiveness factor, because the actual resistance to vapor flow in the ice bed would be higher than that predicted by Darcy’s law and the Carman-Kozeny equation. But it can still be concluded that essentially all of the vapor condensation takes place within 1 inch of the top of the ice bed. While the vapor penetration into the bed is substantially greater than in the case of low pressure water vapor, the accessibility of the inner regions of the ice bed to the vapor is still limited, and the conclusions regarding the necessity of providing ice bed surface area in contact with the vapor are applicable to butane vapor condensation also.
crystals limit the rate of melting of the ice by direct contact condensation of the vapor. T h e results indicate that vapor flow restrictions are limiting, the vapor having access to only the outermost 0.01 inch of ice bed for the case of low pressure water vapor condensation and only the outermost inch of the bed for atmospheric pressure butane vapor condensation. This severe inaccessibility of the inner regions of the ice bed to the condensing vapor has apparently not been recognized previously, and yet it is obviously very important to the design and scale-up of the melter-condenser unit in a freeze-desalination plant. Nomenclature
= ice crystal surface area per unit of gross bed volume, sq. ft./cu. ft. B = ice bed permeability to vapor flow, f ~ ~ / ( l b .(hr.) ,) d = equivalent spherical diameter of ice crystals for CarmanKozeny equation, mm. or ft. Go = mass rate of vapor condensation per unit of bed crosssectional area, lb.,/(hr.) (sq. ft.) h = heat transfer coefficient from condensing vapor to melting ice crystal, B.t.u./(hr.) (sq. ft.)(” F.) L = height of ice bed, ft. P = pressure of vapor, p.s.i.a. or 1b.f/sq. ft. P E = saturation pressure of vapor a t 32’ F., p.s.i.a. or lb.f/sq. ft. T* = saturation temperature of the vapor a t pressure P, ” F. or O R. = superficial velocity of the vapor in the x-direction, ft./hr. x = distance down from top of ice bed, ft. Y = ( p - pE)/(po - PE) XlL a
v,
z =
GREEKLETTERS @
=
e
=
7
X
= =
p
=
p
=
Q
=
inverse slope of vapor pressure curve for condensing vapor, ” F./p.s.i. or (” F.)(sq. ft.)/lb., void fraction of ice bed = 1.0 volume fraction ice crystals effectiveness factor, defined in Equation 12 latent heat of condensation of vapor, B.t.u./lb., or ft.-lb. f/l b., viscosity of vapor, (lb.f)(sec.)/sq. ft. or (1b.f)(hr.)/sq. ft. density of vapor, lb.,/cu. ft. Thiele modulus dhapL2/pBX
-
SUBSCRIPT a
=
conditions a t top of ice bed, x = 0
literature Cited Conclusions
A simple model has been used to analyze the extent to which vapor flow restrictions within a packed, drained bed of ice
Table II. d = e = u =
h =
Assumed and Calculated Values for Butane Freezing Example 0.2mm. To* - 32 = 1 ” F. 0.5 L = 1 ft. 4570 sq. ft./cu. ft. B = 26.9 ft.4/(lb.f)(hr.) 10,000 B.t.u./(hr.)(sq. ft.) @ = 35.6
( ” F.) 3” F./p.s.i. p = 0.17 lb.,/cu. ft. X = 165 B.t.u./lb., p = 1 . 6 X 10-7 (lb.f)(sec.)/ sq. ft.
6
‘v
Go = 7800 lb.,J(hr.)(sq. 7 = 0.028 dGa -_
ft.)
Barduhn. A. J.. Chem. Em. Prom. 63.98 (1967). Barduhn; A. J.; First Intgrnatikal Symposium on Water Desalination, Washington, D. C., Oct. 3-9, 1965, Paper SWD/88. Brian, P. L. T., A m . Sod. Mech. Ener. Paper 67-UNT-10, Underwater Technology Conference, Himptdn, Va., April 30-May 3, 1967. Johnson, IValter, Colt Industries Pilot Plant, 0.S.T.V. Test Station, Wrightsville Beach, N.C., private communication, Dec. 8, 1966. Orcutt, J. C., “Optimization Studies on a Freezing Process,” A.1.Ch.E. Houston Meeting, Feb. 7-11, 1965. Orcutt, J. C . , Research Triangle Institute, Durham, N.C., private communication, July 6, 1966. Satterfield, C. N., Sherwood, T. K., “The Role of Diffusion in Catalysis,” Addison-Wesley Press, Readin , Mass., 1963. Thiele, E. W., Znd. Eng. Chem. 31, 916 (19397.
- 280
RECEIVED for review May 18, 1967 ACCEPTEDSeptember 27, 1967
CI
Work supported by the Office of Saline \Vater, U. S. Department of the Interior, Washington, D. C.
VOL. 7
NO. 1
JANUARY 1 9 6 8
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