ENGINEERING, DESIGN. AND EQUIPMENT more different particles through ,P. or P through two or more different particles, has become a significant factor. Example
I n this example the distribution of sizes amon@;particles will be assumed to be g ( r ) = k4r4e-k~
This distribution is illustrated in ~ value of g ( r ) in Equation 3
be the expected number of particles of radius r(r1 < r < r z ) that will hit a particle of radius R and random initial position. With d r ) and c given by Equations 15 and 16, one obtains from Equation 6 13440 9 (Icr,kR) = k V ( r R)3(r - f 2 ) e - k ’ (17i
+
(*”
i 1. L~
6720sC = v k 4
Figure 4 shows how 4 ( k r , k R ) varies with kr for the case when k T using ~ ~ this ~, ~~ R = ~ 2. T h a t is, it shows the espected distribution of size? among the particles that do the hitting. The total area under is equal to the ordinate of the curve from kr = 2 t o kr = the H / ( v k L ) curve in Figure 2 a t kEz = 2. (16)
After substituting these values for y ( ~ and ) c in Equations 6, 9, 12, and 14, one obtains for H / ( o k L ) , h / ( v k L ) , F / v and ,f/v> the results shown in Figures 2 and 3. The expected Eize distribution among the particles which hit a particular size particle may be obtained, if desired. For example, consider the particles hitting particles of radius R. Let
Acknowledgment
This study of collisions during sedimentation was suggeBted bgr Henry Seaman. T h e author wishes to thank John W. Odle and R. T. KnaPP for Constructive criticisms. RECBIVED for review March 8, 1964
ACCEPTED Ianuary 11. 1853
Barrier Systems in Thermal Diffusion Columns J. C. TREACYl
AND
R. E . RICH
Departmenf of Chernicol Engineering, University o f Nofre Dame, Nofre Dome, Ind.
A
T PRESENT, the Clusius column is the only gas separation
apparatus in industrial use that utilizes the thermal diffusion effect. Clusius columns consist of long, cooled outer containers with heated wires or other surfaces placed along the column axis. The thermal diffusion effect operates horizontally, tending (with few exceptions) to concentrate the lighter molecules at thg hot surface. Thermal siphoning moves the partially separated gases to the column ends. Light gas can be removed from the top and heavy gas from the bottom of the column. I n order to be effective, Clusius columns must be long, have small hot-cold surface clearance, and utilize large temperature differences. With equipment 20 feet or more in length, built with only fractions of an inch of hot-cold surface clearance, considerable care must be exercised in construction, power requirements are high, and gas must spend a considerable time in moving through the very considerable length. The work described in this article presents an attempt to utilize barrier systems between hot and cold surfaces instead of small clearances between hot and cold surfaces. Several of the objectionable features of thermal diffusion separation are thus eliminated. Brewer and Bramley ( 1 )suggested the value of baffles attached to the hot surface in enhancing separation effected by thermal diffusion columns. Donaldson and Watson ( 3 ) reported that spacers used for positioning of hot wires had the effect of increasing separations. This effect was attributed to a possible advantageous effect of controlled turbulence. Barriers (horizontal and vertical sections suspended from the ends of the column into the gas space, but not attached to either hot or cold surface) are considered to be of more value than baffles, inasmuch as they allow flow at both hot and cold surfaces. Thus, they do not require molecules to move long distances while diffusing and do not limit separation rates as seriously as do baffles. I
Deceased.
1544
Apparatus i s designed to test requirements for precisely constructed column
This nork was partially a t e d of barrier effectiveness with respect to reducing constructional requirements. Thus, in construction of the apparatus precise requirements were intentionally abandoned. Apparatus. The column was constructed from 2-inch water pipe closed at the ends with couplers and plugs. The column was 23 inches long and was cooled by a n open spray of tap water at 17” f 1” C. The hot surface was made from l/c-inch water pipe with a borosilicate glass tube insert containing a coil of 20 feet of Sichrome wire (1.75 ohms per foot). One end of the Nichrome wire was grounded while the other was led from the equipment through rubber insulation t o a rheostat and the 110-volt alternating current line. Spacer plates used in conjunction with the end-plug shape served to center the hot element when it was introduced into the column. Taps cloped with stopcocks were provided a t the ends of the column for removal of samples and for connection to vacuum and to a manometer. A mid-column t a p served for introduction of feed taken from tankage or a premixing vessel. Runs were made with the open column and with the following types of barrier systems placed between hot and cold surfaces: I . Horizontal barriers, consisting of 1/64-inch metal plates (for certain runs asbestos was used), 1 T / 8 inches in outside diameter, with a 3s/6,-inch hole bored in the center. These were suspended from the ends of t h e column and were not attached t o the hot or cold walls. An annulus of area 0.06 square inch was between the horizontal barrier and the hot surface, and an annulus of area 0.60 square inch remained between the barrier and cold wall. The 1/4-inch pipe was inserted through the 39/@-inch holes.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47 No. 8
ENGINEERING, DESIGN. AND EQUIPMENT 2. Vertical barriers, made from electrical conduit of diameter 0.75 and 1.50 inches extending the full length of the column. These barriers were perforated with various numbers of different size holes. They were supported a t the ends only so that their axes were coincident with the column axis. Either one or two vertical barriers were used a t one time. 3. Compound barriers, made by cutting the vertical into sections of length suitable for insertion between horizontals, or by bending rectangles of special copper screen into short cylinders for insertion between the horizontals.
Procedure. The component gases t o be separated only had to he stable, readily separable in the column, and easily analyzed. Methane and nitrogen were used throughout this work. h'either the C.P. methane (Matheson Co.) nor the oil-pumped nitrogen (Ohio Chemical Co.) gave detectable differences in composition a t either end of the column when they were allowed to remain in the column under operating conditions for prolonged time intervals. For batch operatlon, the two gases were mixed in a large carboy. The column was brought to operating temperature, evacuated, filled with gas mixture, re-evacuated, and then refilled. The operation was then allowed t o proceed for the desired time. Analysis was made of gas samples by the gas density method. Identical bulbs of known volume were evacuated and then connected to the upper and lower column taps. Stopcocks were simultaneously opened and a short time was allowed for achievement of temperature equilibrium. The bulbs were weighed on an analytical balance. With the known volume, temperature, pressure, and air bouyancy, the ideal gas law was used to compute mean molecular weight and mole per cent methane in the Qamples. It is estimated that compositions could be determined t o =k 0.3 mole yo. For flow operations, the column was evacuated. One tank valve was slightly cracked until the rate of manometer fall indicated the desired flow rate. The flow rate of the second component was similarly adjusted. Flow rate was checked after runs and was found t o show variations of 5 to 10%. After a volume of gas equivalent to about two column volumes had swept through the column into the room through the open end taps, analysis was quickly made by attaching sample bulbs to the end taps and taking samples for analysis as in the batch runs. Samples were taken at slightly above atmospheric pressure. rlfter time was allowed for temperature equilibrium to be attained, these samples were bled to the atmosphere, thus ensuring that precise values of sample pressure were known. The method of sampling was one that introduced disturbance. Thus the results may be somewhat lower than actually obtaining, especially in flow operation. The introduction of barrier systems made precise measurement of the hot surface temperature almost impossible. For engineering use, power input is of more significance than any other column variable. These considerations led to the decision to conduct all work a t constant power input. The power input was determined by noting cooling water rate and temperature rise, and was 3400 calories per minute. Temperatures noted in this article are estimations based on convectional heat transfer rate (by assuming the hot surface to be a long cylinder) and radiant heat transfer rate (by assuming emmissivity to be 0.8). Such estimation probably gives fairly precise temperatures for the open column, but no doubt indicates temperatures higher than actually present when barriers were employed. The pressure was 1 atmosphere (ambient barometric pressure of 745 =k 10 mm.). By using the heat transfer equation
a A
= 0.25
+ (0.173 X
lo8)0.8 ( T t - T:)
Th was estimated to be 560" K. for the open column. Thus, the average column temperature, T , was (560 290)/2 - 425' K. I n reporting separations, the difference in mole percentage methane between the top and bottom of the column was used
+
August 1955
AT-270" K.
O
/
I d
I
1.2
0.8
0.4
I
0.0
0.4
LOGiOTIME,
Figure 1.
Batch separation
1
0.8
I
1.2
HR,
as function of time
Horizontal barriers spaced 18/ft. Vertical barriers perforated 20'70 of original area
A. B. C.
1 vertical, 1.5-inch diom. 2 verticals, 0.75-, 1.5-inch diam. 1 vertical, 0.75-inch diam.
rather than the usual log separation ratio, In Q = In [ ( X A T / X , B ) / ( X B T I X b B ) ] . This was done for reasons of clarity. With 'the feed compositions used (betM-een 30 and 70 mole yomethane) and the separations noted, X T - X B = s is related t o Q, Q = (100 s / l O O - s ) ~ within , a few pel cent error.
+
Striking increases in rate of separation are also result of barrier systems
Comparison of the present work with that reported by Drickamer and coworkers ( 4 ) has been made. The latter work was done using carbon dioxide-propane in a column 244 cm. long, using a hot-cold wall semigap spacing of 0.15 em. and an average temperature of 380" K. This column seemed to give results that are in agreement with theory and offers a good basis for comparison. Drickamer and coworkers reported a final equation In Q = (2.6/p2)(1
+ O.495/p4)-l
which fits the over-all equation In Q = (CluT4L/w4p2)(1
+ C2T10/p4w6)-1
where C1 and CZ are constants involving diffusivities, viscosity, and density, T is the average column temperature, L is the column length, w is the semigap spacing between hot and cold surfaces, p is the pressure in atmospheres, and a is the thermal diffusion ratio, D T / D . By substitution of the proper ratios, In Q for the open column used in the present work can be calculated. Present gap spacing is greater by a factor of 0.95/0.15, T is greater (as estimated from heat transfer equations) by a factor of 425/380, and length is less by a factor of 244/58. The constant, CI, is greater for methanenitrogen than for carbon dioxide-propane by a factor of about 2; a is given as 4.10F4 for carbon dioxide-propane and is estimated
INDUSTRIAL AND ENGINEERING CHEMISTRY
1545
ENGINEERING, DESIGN, AND EQUIPMENT to be of the order of 0.05 for methane-nitrogen (9). The very much larger gap spacing reduces the CzTlO/pW to negligible proportions, and In Q = 0.13, for the present column. This corresponds to a separation of 4.5 mole % between the top and the bottom of the present column. Considering the approximations made in estimating the thermal diffusion ratio for methanenitrogen, this result is in excellent agreement with theory and decreases rapidly if attempts were made to shorten the column. KO separation was detected in a 12-inch column, and separations about half as efficient as those obtained in the 2-foot column were obtained with an IS-inch column.
'q
AT-27O0K.
rg
20
Figure
40 60 80 F L O W R A T E CC/MIN.AT
2.
Separations
under
1.20
Kx)
STAND, COND.
flow
conditions
Horizontal barriers spaced 18/ft. Vertical barriers perforated 20% of original area
A. 6.
C.
D. E.
F.
1 vertical, 1.5-inch diam. 2 verticals, 0 . 7 5 , 1.5-inch diam. 1 vertical, 0.75-inch diam. Horizontals only Open column Vertical only, 1.5-inch diam.
Asbestos horizontal plates were used in place of the metal ones. Asbestos horizontal barriers showed no advantage over the open column. Comparisons of separations noted after 24 hours are given in Table I.
Table 1.
Separations with Open Column and with Horizontal Barriers (XT - XB)"
after
24 Hr.
4.5
Open column Metal barriers, plates/inch 1.0 1.5
10.0 10.9 10.6
2.0
Asbestos barriers, plates/inch 1 .o
4 2 4.4
1.5
a
Average of four or more runs.
It was found that feed compositions within the limits 35 to 65 mole % ' methane gave separations that were identical (within limits of error in analysis). Feed charges in runs reported in this article are confined to this range. Compound vertical-horizontal metal barriers substantially increase separation efficiency
Separations effected as functions of time for batch operation are shown in Figure 1. All types of barriers increased separation efficiency and rate of separation, but compound types are in-
1546
dicated to be much superior to either vertical or horizontal barriei s used separately. Large diameter vertical segments proved to be more effective than small diameter segments, and single vertical systems were somewhat more effective than double systems. Figure 2 shows separations under flow conditions as functions of flow rate. Observations of the effectiveness of barriers made with batch operation were generally substantiated by flow data. As flow increased, separation increased to a maximum near flow rates of 30 cc. (standard conditions) per minute, and then fell off rapidly. The size of the holes used in perforating vertical barriers was not found to affect separation. However, the percentage of original barrier area devoted to perforations seemed to be of considerable importance. Separation fell off seriously if the percentage of perforation was less than 15 or greater than 25 (Figure 3). Curve A of Figure 1 represents data for hole sizes ranging from 1/84 to 9/64 inch, but with 20% perforation. Horizontal plate spacing affected separation. As the number of plates per inch increased, separation increased, came to a maximum, and then began to decrease (Figure 4). Limited data indicate that separation increases u ith increased hot surface temperature, as predicted by theory. Separation fell off the observed value of 4.5 f.0.5 mcle It is difficult t o compare the column with barriers in an analogous manner. With barriers, the concept of gap spacing loses much of its meaning, but no doubt is reduced by the horizontal barriers, leading to the enhanced separations noted. Barriers do not enhance separation through effusion. Hole sizes were large and did not affect separation, even when varied 10-fold in vertical and compound barrier systems. It is postulated that vertical barriers act solely to limit convectional flow in a horizontal &* plane. D eo r e a s i n g the percentage per10 20 30 foration increases sep70VERTICAL PERFORATION aration by limitation Figure 3. Separation as function of convection to a certain point, where of percentage perforation further reduction re24-hour separation with 1 vertical barrier, duces area for dif1.5-inch diam. fusion more seriously than it reduces turbulent remixing, hence the maxima shown in Figure 3. Asbestos horizontal barriers do not show the effects noted with metal horizontals. Thus, a t least in this case, a theory of controlled turbulence cannot be used to explain the results. Asbestos horizontals of the same dimensions ought t o produce similar limitations t o flow, etc., as metal horizontals, but they did not affect separation. The only significant difference between these materials is heat conductivity. It is postulated that metal horizontals, since they are a t nearly the same temperature throughout the metal, create a region of almost constant temperature between the hot and cold surfaces. At constant temperature, density differences due to temperature changes are a t a minimum and turbulent remixing is held to a minimum. The regions between horizontal plates may be looked upon as perfectly permeable buffer zones between the rising gas stream a t the hot wall and the falling stream in the cold wall annulus. An optimum plate spacing occurs. As plate spacing decreases turbulent remixing is reduced, but, a t the same time, for operation a t constant power input, hot surface temperature is reduced. This is a consequence of addition of conducting material. Reduction in temperature reduces separation; reduction in turbulence increases separation. The two effects bring
fzzl
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 8
ENGINEERING, DESIGN, AND EQUIPMENT about a maximum in separation at about 1-inch plate spacing in t h e present column (Figure 4).
-0
I 2 HORIZONTAL PLATES PER INCH
Figure
4.
Separation as function of plate spacing
24-hour separation with horizontal barriers
appreciable separations. Considering not only separation but also the number of batch runs that could be made, columns with compound barriers are 70 time8 more effective than columns with no barriers. The enhanced separation rates allow flow operation to be used. The cclumn without barriers gave only one indication of separation, and that indication was hardly beyond t h e limit of experimental precision. With barriers, appreciable flows could be accomodated, and separations greater than those from batch operations were attained. It is difficult even to conceive of long columns with close clearances acting BS other than fluid flow devices under flow operation. With further work t o evaluate the effect of diverse barrier parameters, thermal diffusion utilizing “waste” heat may yet become a more common industrial operation. Summary
Systems of barriers introduced between hot and cold surfaces of a Clusius-Dickel column have been found to b e effective in
The data taken under flow conditions substantiate the fcregoing hypothesis. T h e 10-fold ratio of outer to inner annulus area for flow of gas necessitates a flow of gas across the column in a horizontal direction if the pressure drop is to be constant. T h e addition of feed gas at the outer wall augments the downflowing outer stream and reduces cross-plate flow to a pcint where the turbulence involved in introducing the stream counterbalances t h e reduction in cross-plate flow. Thus, flow separations increase and are almost double those that can be achieved under static conditions at 30 cr. per minute and then fall off rapidly to zero. No doubt annulus areas are important parameters t,hat determine the shape and position of curves, as shown in Figure 2. The-e parameters are a t present being investigated. Even more striking than improvement in separation efficiency is the increase in rate of separation. Only after 0.5 hour did the open rolumn shorn any separation a t all, whereas after only 4 to 5 minutes (the minimum that could be used to feed column, take samples, etc. ), columns with barriers had already produced very
increasing efficiency of batch separation, rate of separation, and separation under flow conditions. Barrier systems enable columns with wide spacing between hot and cold surfaces to be used, with attendant reduction in power requirements and construction cost. Compound hcrizontal-vertical barriers were most effective, increasing separation Pfficiency sixfold over open column separation and increasing separation rate tenfold. Only metal barriers were found to be effective. literature cited (1)
Brewer, A. K., and Bramley, A , U. S. Patent 2,258,594 (Oct. 14, 1942).
(2) Chapman, S., and Cowling, T. G., “Mathematical Theory of
Nonuniform Gases,” pp. 223, 254, Cambridge Univ. Press, Cambridge, Eng., 1939. (3) Donaldson, J., and Watson, W. W.,Phys. Rev., 82, 909-13 (1951).
(4) Drickamer, H. G., O’Brien, V. J., Breese, J. C . , and Ockert, C. E., J . Chem. Phvs., 16, 122 (1948). RECEIVED for review March 3, 1964.
ACCEPTED March 1 1 , 1955.
Natural Convection to Cold Cylinders ROBERT LEMLICH
AND
CHARLES SHARN’
Department o f Chemical Engineering, University o f Cincinnati, Cincinnati 2 I , Ohio
N
URIEROUS studies involving heat transfer by natural convection from a single long hot horizontal cylinder to a large
expanse of surrounding fluid have been made. The results of most of these investigations have been successfully correlated in dimensionless form b y McAdams ( 2 ) . However, a search of the literature revealed no information for natural convection occurring in the reverse direction-that is, from a large expanse of ssrrounding fluid to a colder cylinder. Accordingly, the purpose of this investigation was t o examine the effect of such an “inverse” temperature and velocity profile on the coefficient of natural convective heat transfer. Four cold cylinders, each of a different diameter, were subjected to unsteady-state natural convection with room air. Control runs for warm cylinders were also conducted. The results for the two opposite directions of heat transfer were compared with each other and with the correlation of McAdams. 1
Present address, 1735 North 26th St., East St. Louis, Ill.
August 1955
Expressions for heat transfer are presented
The rate of heat transfer by convection from a warm body to cooler air is given by
where Q is the heat transferred from the body, 7 is the time, h is the coefficient of heat transfer, A is the surface area, and At is the absolute value of the temperature difference driving force between the surface and the bulk fluid. Previous investigation ( 3 ) has shown that, for a warm cylinder of moderately large diameter in natural convection to room air under a moderate At, h = K(At)1’4
where K is a constant for the particular system.
INDUSTRIAL AND ENGINEERING CHEMISTRY
(2)
For a cylinder
1547
