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equilibrium cell. A detailed discussion of the data reduction procedure is given by Rivas (1978). Typical Results Table I gives some typical results. Table I1 compares some experimental Henry’s constants with previously published results. Figure 4 shows Henry’s constants for ethane, carbon dioxide, and hydrogen sulfide in propylene carbonate. The accuracy of these Henry’s constants is approximately f l % . A more complete discussion of solubilities of natural-gas components in “physical” and “chemical” solvents (and their mixtures) is to be published shortly.
Acknowledgment The authors are grateful to the National Science Foundation for financial support and to the Venezuelan Fund for Research in Hydrocarbons (FONINVES) for a fellowship to O.R.R. Literature Cited Cukor, P. M., Prausnitz, J. M., Ind. Eng. Cbem. Fundam., 10, 638 (1971). Reid, R. C., Prausnitz, J. M., Sherwood, T. K., “The Properties of Gases and Liquids”, 3rd ed, Section 3-16, McGraw-Hill, New York, N.Y., 1975. Rivas, 0. R., Dissertation, University of California, Berkeley, Calif., 1978.
Received f o r review June 23, 1978 Accepted April 2, 1979
A Modified Differential Refractometer for Continuous Liquid-Phase Residence Time Distribution Studies Patrick L. Mills and Milorad P. DudukoviE’ Chemical Reaction Engineering Laboratory, Washington University, St. Louis, Missouri 63130
Modifications to a commercially available differential refractometer to eliminate baseline instability in trickle-bed reactor liquid phase dynamic studies are described. Substantial improvement of continuously measured tracer concentrations in the liquid effluent is obtained with the modified instrument when compared to the original version. More reliable and accurate dynamic studies may be performed as a result.
Introduction Sensitive detectors that can continuously monitor the liquid effluent from a given reactor configuration are of prime importance in residence time distribution studies. There exists no liquid phase equivalent to the thermal conductivity or flame ionization detectors of gas chromatography. These latter detectors are commonly employed in gas phase dynamic studies in gas-liquid reactors (Kunugita et al., 1962; Hochman and Effron, 1969; Niiyama and Smith, 1976, 1977; Misic and Smith, 1971; Komiyama and Smith, 1975a,b; Joosten et al., 1977; Ramachandran and Smith, 1978). Conductivity cells can be used to detect certain ionic solutes in aqueous carrier liquids as indicated by Shah et al. (1978), but are not applicable to totally organic systems. Detectors based on ultraviolet absorption are excellent provided the selected compounds absorb in the UV range (Furusawa and Smith, 1973; Midoux and Charpentier, 1973; Suzuki and Kawazoe, 1974; Miller and Clump, 1970; Komiyama and Smith, 1974). Fluorimetric, IR absorption, heat of sorption, polarographic, radioactive, and solute transport detectors have been less widely used (Sater and Levenspiel, 1966; Schugerl, 1967; Ostergaard and Michelsen, 1969, 1970). Our dynamic studies in trickle-bed reactors have been performed using various organic carrier liquids and a wide variety of tracers, some of which do not absorb in the UV range. We felt that the choice of a differential refractometer (Schwartz et al., 1976) to continuously monitor the reactor response would be more suitable than any of the above detectors. The main reason is that it will respond to a single solute whose concentration is changing in a given carrier liquid provided the refractive indices of the carrier and solute are different. It should be noted that the carrier liquid may be composed of a mixture of totally 0019-7874/79/ 1018-0292$01.OO/O
miscible liquids. This suggests that the refractometer is not restricted to binary systems but only to detection of a single solute in a given carrier liquid that has a uniform base refractive index. One of the problems associated with the particular refractometer employed (Waters R-403) was the baseline instability. This characteristic, however, was not restricted to this particular refractometer since several others were tested and exhibited a similar behavior. This difficulty was mainly attributed to the temperature coefficient of refractive index being on the order of to 104/OC. This means that if a sensitivity of lo-’ RI units is required, then the temperature of the liquid entering the detector cell must be constant within “C. The present model relied solely on an internal heat exchanger for thermostatting the liquid stream and contained no external provisions for equilibrating the liquid to a constant temperature prior to entering the optical block. Therefore, it was decided to modify the instrument for our studies. It should be possible to perform similar modifications to other commercially available units as well. While the emphasis here is on a laboratory scale refractometer, the general principles are still applicable to process refractometers which can be expected to be plagued with similar difficulties since the design principles are the same. Deininger and Halaiz (1970) modified a commercially available differential refractometer by first removing the original 0.5 mm i.d. x 1.8 m long inlet tubes which were cast into the optical block. New tubes were installed which had an elliptical cross section at fixed distances while the points in between were circular. The entire tube assembly was coiled to induce secondary flow and thus approach plug flow behavior. However, the tubes were considerably shorter than the original version. As one would have 0 1979 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
expected, the band broadening was less than the original but sensitivity to temperature fluctuations was increased. Such a situation is contrary to what would be desired in a detector for engineering type tracer studies. Brooker (1971) performed modifications to a UV flow cell detector that allowed stable operation at temperatures between 70 and 80 "C. This was carried out by installing 10 ft of 0.51 mm 0.d. X 0.25 mm i.d. stainless steel tubing in a water jacket placed between the column effluent and the flow cell. The flow cell itself was encased in a copper jacket which had heat exchanger coils soldered to its body. Tap water was utilized as a cooling source. It was shown that the base line stability was substantially improved over the previous arrangements where thermostatting was not utilized. The use of a more accurately controlled water source through the jacket would probably have resulted in a more substantial increase in stability as opposed to the use of tap water. Although Brooker did not mention it, the fact that the inlet tube was coiled probably resulted in secondary flow which in essence increased the radial mass transport. The net result was that plug flow behavior probably was present in the tubing and consequently the concentrations measured at the detector did not have to be corrected for peak broadening or other external column effects. Such phenomena have been studied by Koutsky and Alder (1964) for various coil geometrical variables. Based on the above studies it was decided to modify our tracer detection system so that optimum performance could be achieved. The first modification consisted of installation of a coil of capillary tubing. This was located between the outlet of the liquid effluent from the trickle-bed column and the entrance to the optical block in a thermostatted water bath. The effluent liquid from the coil was then routed to the optical block in which the detector was situated. The optical block was located on the outside of the thermostatted bath, and open to the laboratory air. Although the baseline was improved, substantial testing showed that temperature extremes of k3 O F in the laboratory caused both short and long term baseline drift. The optical block was then modified such that it could be immersed in the thermostatted bath itself. In this fashion the liquid effluent exiting from the coil could be joined directly to the tube leading into the optical block to the detector thus eliminating the problem of external temperature fluctuations. Description of the Modifications Coil Design. The selection of the proper coil was based on preliminary pressure drop and heat transfer calculations for a selected set of coil geometrical variables and a hexane volumetric flow rate of 60 cm3/min a t 25 "C. The coil design was based on this solvent since this was the carrier liquid in our tracer experiments. The given flow rate would be the maximum total flow rate to the reactor; above this rate the exiting liquid from the reactor would be divided into two streams with constant flow maintained to the refractometer. The friction factor correlation of Ito (1959) was used to estimate the pressure loss. For the given flow rate the critical Reynolds number (Ito, 1959) was found to be 5575 when a coil diameter of 5.5 cm and an inner tube diameter of 1.016 mm was assumed. The actual Reynolds number was found to be 2604, indicating that the flow was still laminar. Ito's correlation for laminar flow resulted in a coiled tube to straight tube friction factor ratio of 2.3. The pressure loss per unit length was then found to be 2.8 X lo3 dyn/ (cm2cm) by the usual Fanning formula (Perry et al., 1969). A safe reactor back pressure was found to be about 4.2 X lo5 dyn/cm2, which resulted in a coil length
293
of about 150 cm. The total inlet tubing length was then 186 cm which included both the coil and the inlet tubing to the optical block of the refractometer. Repeating the above calculations for an inner tube diameter of 0.508 mm resulted in a pressure drop per unit length of 6 X lo4 dyn/ (cm2 cm), which was too excessive for the available reactor back pressure. The heat transfer characteristics of the above coil were examined next. The hexane was assumed to enter the coil at some fixed temperature and it was desired to examine how accurately this would have to be maintained such that the temperature difference between the outlet liquid from the coil and the bath fluid in which the coil was immersed would be within a certain tolerance. In actuality, all experiments performed to date have been performed at laboratory temperature (-22 "C) such that the hexane entering the coil has equilibrated to this condition. The mean inside-film heat transfer coefficient was estimated using the correlation of Mori and Nakayama (1967) for curved pipes and found to be 0.017 cal/(cm* s "C). The outside film coefficient was estimated using the correlation of Chilton, Drew, and Jebens (1944) to be about 0.27 cal/(cm2 s "C). The heat transfer by conduction through the tube wall was calculated to be 1.1 cal/(cm2s "C). The overall heat transfer coefficient was then calculated to be 0.010 cal/(cm2 s "C). As expected, the heat-transfer resistance inside the coil is controlling. Equating the expressions relating the total heat transfer rate to the product of the overall heat transfer coefficient, coil external surface area, and log-mean temperature difference yields the following general expression
where ATi = Tb - T i is the difference between the bath temperature Tb and the inlet liquid to the coil Ti, AT, = Tb- Tois the difference between the bath temperature Tb and the outlet liquid from the coil To,and K = U&,/ Q,,phcp. Inserting the values A, = 74.8 cm2,cp = 0.54 cal/(g "C), Ph = 0.657 g/cm3 with the above values of U, and Qh yields K = 2.1. Thus to maintain the liquid entering the optical cell of the refractometer to within 0.01 "C of the bath temperature requires that the inlet liquid be maintained within 0.08 "C. This could be improved by using a longer coil consistent with pressure drop limitations. On the other hand, this analysis does not take into account the heat transferred to the connecting tubing between the end of the coil and the entrance of the optical cell ( 36 cm) and consequently represents a conservative estimate. Including this contribution to the above calculations shows that the liquid entering the coil must be maintained within 0.14 "C of the bath temperature. In the time required to perform a tracer experiment, the deviation of the carrier liquid temperature as a result of the usual room temperature fluctuations is within this last bound. Rebuilding the Refractometer. The optical block of the refractometer used consisted of a 4.1 kg aluminum casing in which the flow-through cell and photodetector are situated at locations A and B as indicated on the top view Figure l a and the side view Figure lb. The inlet tubes on the original instrument leading to the flow-through cell entered at location C and were bent to follow channel D where they connected to the cell situated a t location A. These were held in position by lead. The exit lines from the cell followed channel E and exited a t location F. Provision was made for thermostatting the block by
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I
1
8 $30
&
e Gm i d
s 5
s1010
0
5
10
15
23
25
30
TIME, MIN.
Figure 2. Effect of optical block submersion on baseline stability a t an attenuation of 8X: A, after submersion; B, before submersion.
A i .
&&I
Figure 1. Refractometer optical block.
connection to the external tubes at G. The inlet tubes to the optical cell were not in direct contact with the thermostatted liquid. It was noticed that by squeezing the inlet tubes between two fingers at the block entrance a quite noticeable deflection of the recorder could be obtained at all flow rates. This was a direct indication of the sensitivity of the instrument to outside temperature effects and of the inadequacy of the block to thermostat the liquid in the tubes properly. The modifications began by removing the outside cover attached to the casing and carefully removing all components, leaving only the aluminum casing and inlet tubes. The casing was then placed in an oven and maintained at 400 "C for a period of 0.5 h. The lead-like material which filled the channel where the tubes were embedded melted and was easily removed while the block was still hot. The tubes were then pulled through the channel at location C and straightened such that they were protruding beneath the optical cell at location H. Fittings were installed on the ends of the tubes so the exit side of the coil and the inlet to the reference side of the cell could be readily connected. The opening where the shaft for the optical zero emerged from the block (location K) was tapped with a 1/4 N P T so that a copper pipe could be connected. The pipe was of sufficient length that it protruded through the thermostatted bath top, thus preventing liquid from entering the block when it was submersed. The old shaft was replaced by a longer version which resided in the copper pipe and from which the optical flat could be easily adjusted from the bath top. Stainless steel plates of the appropriate size were fabricated to cover the optical screw gear assembly on the bottom of the casing, the mirror adjustment screw access holes on the back of the block, and optical adjustment mirror shaft opening. A cover for the photodetector and light source (location B) was machined from round stock, hollowed out, and provided with a 1 / 4 NPT on the circumference to accomodate a copper pipe through which the wires could be routed to the bath top. The screw terminal originally located at position J was fastened to the bath top and the wires attached. All sealing of the various covers was made using GE RTV silicone. This has
proved to be satisfactory since inspection of the material after 5 months of submersion has indicated that it is still in excellent condition. The entire optical block was supported by two brackets whose ends were fastened to the cover of a Tronac CTB-405 water bath. A coil support hanger was located in close proximity of the inlet tube to the casing for simple connection. The outlet tube of the flow through cell as well as the reference side tubes were routed through the bath top. Testing A series of tests were performed to evaluate the performance of the detector at room temperature since the dynamic studies in the trickle-bed reactor are performed a t the same condition. The particular temperature controller used in the water bath was a Tronac Model PI'C-40. The control heater was a 0.635 mm 0.d. stainless steel tube having a wall thickness of about 0.381 mm. A thin silicone rubber liner is installed inside the tube through which a cooling stream maintained at a fixed temperature below the bath temperature flows continuously from an auxiliary refrigerated bath. The silicone liner provides a heat transfer resistance between the heat surface on the outside and the cooling water flowing on the inside. The variation of the bath temperature about the set point was measured by a 1-mV strip-chart recorder which was driven by a Tronac Model 120 temperature bridge. A temperature sensitivity of about &0.0005 "C can be determined. The temperature of the bath is usually well within fO.OO1 "C variation. Early attempts to stabilize the baseline consisted of employing the coil only without submersion of the optical block as briefly described earlier. This was successful to a certain degree but operation was very critically dependent on the maintenance of a stable room temperature which at times was difficult. The baseline stability so obtained at an attenuation of 8X is compared to that after the block was modified is given Figure 2 for a flow rate of 26 cm3/min. It is evident that the noise level was substantially reduced as a result of the submersion. The response of the coil and detector to a series of injections made on the inlet side of the coil is illustrated in Figure 3. These were made using a Perkin-Elmer six-port sampling valve with an injection loop volume of 0.606 cm3 and solutions of 0.1 vol % tetralin in hexane. The carrier in this case was technical grade hexane which is utilized in a typical trickle-bed tracer experiment. It is seen that the reproducibility is very good. The maximum relative error of the peak heights was 1.4% while the standard deviation was 0.74. A typical trickle-bed reactor response to a pulse of nonadsorbing tracer (heptane) is given in Figure 4. The bed was packed with porous 20-28 mesh Alcoa F-1 alumina granules as used by Schwartz et al. (1976). Three injections were made at the same flow rate and the mean residence
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
295
p n l Run2 ~
80
0
2
1
3 4 TIME, MlN.
5
6
F i g u r e 3. Use of the refractometer to examine the reproducibility of the tracer injection system. Conditions: hexane flow rate at 26 cm3/min; T = 23 "C; tracer, 0.1 vol% tetralin in hexane; attenuation, 8X.
TIME, SEC.
Figure 5. Trickle-bed reactor response to a pulse of nonadsorbing tracer before the refractometer modifications.
2 I 1
0
2
0.4
1
3
TIME, MIN.
F i g u r e 4. Use of the refractometer to continuously measure the trickle-bed reactor response to a pulse of nonadsorbing tracer. Conditions: hexane flow rate at 26.4 cm3/min; T = 23 "C; tracer, 0.07 g/cm3 heptane in hexane; attenuation, 8X.
time and variance were calculated by using discrete points from the experimental curves and entering in a computer program which extrapolates the tail to infinity by an exponential-decay equation as suggested by Curl and McMillan (1966) and implements the usual formulas for calculating the absolute moments. The calculated mean residence times were found to be 60.8,61.1, and 61.2 s while the variances were calculated to be 0.1092, 0.1095, and 0.110 min2. This reproducibility is excellent for pulse inputs and is attributed to the reliability of the measured tails. Some curves from Schwartz (1975) are given in Figure 5 which illustrates the difficulty of using the instrument without modifications. This difficulty is caused by baseline drifts which affect the tail of the curves. In repeat injections, the tail extends below the baseline in the first run while in the second it extends above the baseline to such an extent that extrapolation of the tail is difficult. The mean residence time and variance characteristics of the coil and detector were determined at hexane volumetric flow rates between 7.8 and 56.0 cm3/min. These were obtained by locating the pulse injection valve directly at the inlet of the coil. The results of the mean residence time measurements are illustrated in Figure 6. The slope of Figure 6 corresponds to the volume of the coil and detector plus fittings as determined by the tracer experiments. The measured volume of 3.43 cm3 as determined by weighing the amount of water present when the tubing, detector, and associated fittings are drained is within 3.7% of the tracer based value. The variance data were correlated to obtain the following relationship between measured variance d and volumetric flow rate through the tubing Q = 2710Qh-1'89
(2) This relation is useful for determining the variance associated with the coil, detector, and fittings on the outlet U '
.Experimental Results -Fitted Equation
0
2
4
6
R E C I P R W VOLUMETRIC FLOW RATE
X
Iz lo2), MN/CU CM.
Figure 6. Experimentally based mean residence time characteristics of the coil.
side of the column. The system variance can readily be corrected using the above equation to obtain the true variance of the reactor when tracer experiments are performed with hexane as a carrier liquid. The contribution of the detector cell itself to the variance was not determined separately. However, the manufacturer's specifications (Waters, 1971) indicate that the maximum 0 tubing dispersion of the cell and their original ( ~ 3 cm) is 100 ILL a t 1 cm3/min flow. This volume corresponds to approximately 36 s2 of peak broadening and is based on the peak width which occurs at 0.6 of the maximum peak height. The original tubing has a volume of 240 p L and an i.d. of 0.1 cm while the cell is a mere 10 pL. This suggests that the contribution of the tubing to the peak broadening is probably more significant than that of the cell. Also, the flow rates employed through the cell are usually a minimum of one order of magnitude higher (>lo cm3/min) than that given above which would reduce the peak broadening of the cell to an amount which would be insignificant with respect to that of the coil. Conclusions and Recommendations It has been shown that baseline stability for the given differential refractometer can be obtained only if proper temperature control in maintained. This is best accomplished by installation of a coiled capillary tube to thermostat the inlet liquid to the refractometer and by total immersion of both the coil and the optical block in a precisely controlled temperature bath. Characterization
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of the selected coil can be easily performed for application to dynamic studies in trickle-bed reactors. Acknowledgment These modifications were performed as part of a trickle-bed reactor study supported by National Science Foundation Grant ENG 73-08284-A02. This support is greatly appreciated. Nomenclature A, = external surface area of the coil, cm2 c = specific heat of hexane, cal g-' O C - ' = constant appearing in eq 1 ( = U&/Q@hCp), dimensionless Qh = volumetric flow rate of hexane through the coil, cm3m i d Tb = water bath temperature, "C Ti = temperature of the liquid entering the coil, "C To = temperature of the liquid exiting from the coil, "C U, = overall heat transfer coefficient of the coil, cal cm-2,s-l
2
oc-1
Greek Letters = liquid density of hexane, g cm-3 Subscripts b = denotes bath c = denotes coil h = denotes hexane i = denotes inlet o = denotes exit Ph
L i t e r a t u r e Cited Brooker, G., Anal. Chem., 43(8), 1095 (1971). Chilton, T. H.. Drew, T. B., Jebens, R. H., Ind. Eng. Chem., 38, 510 (1944). Curl. R. L.. McMillan. M. L.. AIChE J.. 12f4). 819 (1966). Deihger,'G., Hallsz, I., J: Chromatigr. 'Sci., 8, 499 (1970). Furusawa, T., Smith, J. M.. Ind. Eng. Chem. Fundam., 12(3), 360 (1973). Hochman, J. M., Effron, E., Ind. Eng. Chem. Fundam., 8(1), 63 (1969). Ito, H., J. Bas. Eng., Trans. ASME, 81, 123 (1959). Joosten, G. E. H., Schilder. J. G. M., Janssen, J. J., Chem. Eng. Sci., 32, 563 ( 1977). Komiyama, H., Smith, J. M., AIChE J., 21, 664 (1975a). Korniyama, H., Smith, J. M., AIChE J.. 21, 670 (1975b). Komiyama, H., Smith, J. M. AIChE J., 20(6), 1110 (1974). Koutsky, J. A., Adler, R. J., Can. J . Chem. Eng., 43, 239 (1964). Kunugita, E., Otake, T., Yoshii, K., Kagaku Kogaku, 28, 672 (1962). Midoux, N.. Charpentler, J. C., Chem. Eng. Sci., 28, 2108 (1973). Miller, C. 0. M., Clump, C. W., AIChE J . , 16(2), 169 (1970). Misic, D. M., Smith, J. M., Ind. Eng. Chem. Fundam., 10, 380 (1971). Mori, Y., Nakayama, W., Int. J . Heat Mass Transfer, 10, 681 (1967). Niiyama, H., Smith, J. M., AIChE J., 22(6). 961 (1976). Niiyama, H., Smith, J. M., AIChE J., 23(4), 592 (1977). Ostergaard, K., Adv. Chem. Eng., 7, 71 (1968). Ostergaard, K., Michelsen, M. L., Can. J . Chem. Eng., 47, 107 (1969). Peny, R. H., Chilton, C. H., K m b k k ,S. D., Ed., "Chemical Engineers' Handbook", 4th ed, McGraw-Hill, New York. N.Y., 1969. Rarnachandran, P. A., Smith, J. M., Ind. Eng. Chem. Fundam., 17, 17 (1978). Sater, V. E., Levenspiel, O., Ind. Eng. Chern. Fundam., 5, 86 (1966). Schugerl, K., Proceedings of the International Symposium on Fluidization, p 782, 1967. Schwartz, J. G., Weger, E., DudukoviC, M. P., AIChE J., 22, 894 (1976). Schwartz, J. G., DSc. Thesis, Washington University, College of Engineerlng and Applied Science, Aug 1975. Suzuki, M., Kawazoe, K., J,; Chem. Eng. Jpn., 7(5), 346 (1974). Waters Associates, Inc., Instruction Manual: Series R-400 Differential Refractometers", Table 1, p 4, Framingham, Mass., 1971.
Received f o r review July 18, 1978 Accepted February 7, 1979
COMMUNICATIONS An Explicit Equation for Friction Factor in Pipe A single explicit equation correlating friction factor, pipe roughness, diameter, and Reynolds number for transition and turbulent flow regions is proposed with the same accuracy a s the implicit Colebrook equation.
The estimation of friction factor for a Newtonian fluid flowing through a pipe has long been known. Usually it is done by using the Moody (1944) friction factor chart which is made up of the following equations. For laminar flow with Re < 2100, the Hagen-Poiseuille equation gives fD
=
64
where f D is the Darcy friction factor which is four times the Fanning friction factor f f , i.e. f D = 4ff; Re is the Reynolds number = Dup/,uL.For fully developed turbulent flow in smooth pipe with 3000 < Re < 3.4(10)6,Prandtl's universal law of friction for smooth pipes which has been verified by Nikuradse (1932) is
- 1- - 2.0 log (Re&)-
&
0.8
(2)
For fully developed turbulent flow in rough pipes with (D/t)/(Rev'/ff) > 0.01, Von Karman (1930) proposed 0019-7874/79/1018-0296$01.00/0
--
- 2.0 log (D/t)
6
+ 1.74
(3)
where t is the roughness of the pipe. For transition flow in which the friction factor varies with both Reynolds number and c / D , the equation universally adopted is due to Colebrook (1939) 2.5226
1
(4)
This equation is valid up to a value of (D/t)/Red/ff) = 0.01. Although most fluid flow problems are calculated by this Moody chart, it is sometimes more convenient to use equations. As a matter of fact, the Colebrook equation covers not only the transition region but also the fully developed flow regions for smooth and rough pipes. Inasmuch as 6 0, eq 4 reduces to eq 2 and as Re w , eq 4 becomes eq 3. In other words, the Colebrook equation
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