Newton, R. H., lnd. Eng. Chem., 27, 302 (1935). Nielsen, A., “An Investigation on Promoted Iron Catalysts for the Synthesis of Ammonia”, 3d ed., Jul. Gjellerups Forlang, Copenhagen, 1968. Nielsen, A,, Kjaer, J., Hansen, B., J. Catal., 3, 68 (1964). Ozaki, A.. Tavlor. H. S..H. S.. Boudart. M.. Proc. Rov. SOC.London. Ser. A. 250. 47 (1960): Sidorov, I. P., Livshits, V. D., Zh. Fiz. Khim., 21, 1177 (1947) Sholten, J. J. F., Thesis, Delft, the Netherlands, 1959.
Temkin, M. I., Zh. Fiz. Chim., 24, 1312 (1950). Temkin, M. I., Morozov, N. M., Shapatina. E. N.. Kinet. Katal., 4, 260,565 (1963). Temkin, M. I., Pyzhev, V., Acta Physicochim. (URSS), 12, 327 (1940).
Receiued for reuiew November 3, 1975 Accepted September 21, 1976
Experimentally Determined Effect of Artificially Roughened Surfaces on Hydraulic Loss Coefficients Robert H. McFetrldge Westinghouse Electric Corporation, Pressurized Water Reactor Systems Division, Pittsburgh, Pennsylvania 75230
George E. Klinzing” Chemical and Petroleum EngineeringDepartment, University of Pittsburgh. Pittsburgh, Pennsylvania 1526 7
Experimental data were obtained for the flow of water between a “smooth” and an artificially roughened plate for the purpose of determining the effect of artificially roughened surfaces of moderate to large relative roughness (i.e., 0.04 < d D < 1.05) on hydraulic loss coefficients. The experiment was designed to obtain data for various depths and widths of squaretooth and sawtooth shaped serrations for flow gaps of 0.012, 0.030, and 0.050 in. between the “smooth” and artificially roughened plates with pressure differentials ranging from 10 to 130 psi. The reduced experimental data indicate that artificially roughened surfaces significantly increase the hydraulic loss coefficient which results in decreased flow rates between two parallel plates. Based on observations, the experimental data can be correlated to various physical dimensions of the test plate configurations.
Introduction One of the most interesting problems in the field of fluid mechanics deals with the prediction of frictional losses and form losses for turbulent flow in pipes, annuli, and between parallel plates. Numerous experimental studies have been performed which correlate frictional losses for randomly distributed and uniformly distributed roughnesses (Idel’chik, 1966). The correlations have been developed, however, for geometries in which the ratio of roughness to the hydraulic diameter of the channel is relatively small (Le., c/D I0.05). For this type of geometry, energy losses are primarily due to shear stress effects a t the walls of the channel. Kinetic energy losses associated with secondary flows due to small changes in channel flow area are minimal. For the case of a uniformly distributed roughness, as the relative roughness of the flow channel becomes increasingly larger, kinetic energy losses increase as a result of secondary flows. If the relative roughness is sufficiently large, the form losses of the flow channel can be predicted using correlations for contraction and expansion losses (McAdams, 1954; Rouse, 1950). Obviously, there is a transition region of relative roughness for which neither form loss predictions or frictional loss predictions will adequately correlate the measured data. As a result of the inability to obtain generalized correlations of loss coefficients for moderate relative roughness, hydraulic systems have been historically designed to avoid this region. A few exceptions to the rule do exist, such as systems in which it is desirable to augment heat and mass transfer or to minimize leakage flows through small annuli. For years it has been desirable to minimize leakage flows in small annuli, such as 176
Ind. Eng.’Chem., Process Des. Dev., Vol. 16, No. 2, 1977
the clearance between a turbine shaft and the housing, to increase the efficiency of the apparatus. With the exception of the use of labyrinth seals employed in the design of rotating turbomachinery (e.g., steam turbines), leakage flows have been controlled by imposing extremely “tight” tolerances which significantly increase the cost of a hydraulic system. Therefore, if it were possible to effectively employ uniformly distributed moderate roughness, such as the machining finish on components or small grooves, manufacturing and fabrication cost could be decreased. As mentioned above, labyrinth seals have been employed on steam turbine shafts in order to decrease the leakage between the shaft and housing, resulting in an increased efficiency of the turbine. Various correlations are available for the design of labyrinth seals (Vermes, 1961). These empirical correlations, however, have been developed for the flow of compressible fluids. Pressure losses incurred by a compressible fluid are converted into an adiabatic temperature increase of the gas. For an incompressible fluid, such as water, the magnitude of the temperature increase becomes insignificant due to the intensive properties of the fluid. I t is probable, therefore, that the labyrinth seal correlations developed for compressible fluids are not applicable to incompressible fluids. The objective of this experimental work was one of obtaining hydraulic loss coefficient data in the range of moderate relative roughness (Le., 0.04 < t/D < 1.05) for the flow of water between a “smooth” and a “grooved” plate. The experimental data are compared to various existing empirical equations in order to verify or extend the range of the correlations. If the data can be correlated either to existing relationships or ob-
-
TO GLOBE TURBINE METER
uu
P L A T E S 10-14 0-0
*-A
Figure 2. Test section configuration.
Table I
PLATES 15A-15D
Figure 1. Test plate geometry.
served trends, the results can be used to “relax” manufacturing and fabrication tolerances in the design of systems in which leakage flows are undesirable. This relaxation of tolerances will decrease the manufacturing and fabrication cost of the system. Experimental data are obtained for each of the “serrated” plates for clearances between the plates of 0.012, 0.030, and 0.05 in. with pressure losses ranging from 10 to 130 psi. Experimental Apparatus The experimental apparatus consisted of a Hewlett-Packard Model T140B oscilloscope, a Foxboro Model WHKC-421 turbine meter, a Marsh 150 psi pressure gauge, a 2-in. globe valve, the test plates shown in Figure 1and Table I, a fixture to secure the test plates as shown in Figure 2, and a thermometer. Figure 2 depicts the general layout of the experimental apparatus. The turbine meter, employed to measure test section flow rate, was located upstream of the globe valve in order to avoid turbulence effects associated with flow downstream of a valve. The globe valve was utilized in the test section to facilitate the adjustment of flow rates and pressure losses. After passing through the globe valve, the flow enters into a large plenum where any turbulence associated with the valve is minimized. The flow turns upward and enters into the fixture which secures the test plates. The flow area of the test plate fixture is sufficiently large relative to the flow area between the test plates that the dynamic head of the flow in the fixture is essentially zero. Since the dynamic head at this plane is essentially zero and all pressure losses from this point through the test section exit are a result of the test plates, the pressure loss across the plates is equal to the gauge pressure of the test plate fixture plenum. The test plates are secured in the fixture by bolts. The bolts prevent the gap between the plates from changing. Flexible gaskets were also employed to prevent undesirable leakage around the test plates. Figure 1 shows the plates which were tested. I t should be noted that the desired clearance between the “serrated” and “smooth” plate was controlled by machining the desired value into the
Width,
Flat,
Plate
in.
in.
1 2
0.035
0.035
0.080 0.124
0.040 0.040 0.033 0.040 0.040 0.035 0.041
3 4 5 6 7 8 9
10 11
12 13 14
15A
15B 15C 15D
0.033 0.080
0.125 0.035 0.076
0.120 0.090 0.080 0.090 0.090 0.085
0.040
0.032 0.045 0.035 0.035 0.035
Depth, in. 0.012 0.012
0.012 0.017 0.021 0.021
0.035 0.034 0.033 0.012 0.020
0.030 0.040 0.050
0.000 0.012 0.030
0.050
smooth plate. This eliminates the necessity of using various sizes of shim stock to obtain the desired clearance and results in a significant reduction in “set up” time. Experimental Results The experimental data obtained for each of the test configurations were the serration depth, serration width, and serration land or flat of each plate tested, the flow gap between the test plates, the measured pressure drop across the test plates, and the measured turbine meter voltage. The water temperature for all test data was 62 OF. Based upon the following estimated errors, a differential error analysis was performed yielding: (a) pressure drop, f l . O psi; (b) flow gap, fO.001 in; (c) water temperature, f 2 . 0 “F; (d) turbine meter voltage, &*/?division on oscilloscope scale which corresponds to 1-5 mV. It should be noted that the dimensions of the test plates were determined using an optical comparator and molds of the “as built” plates. Minor variations, &0.002 in., were observed in the serration depth, width, and flat for a given plate. These deviations, however, could not be incorporated in a macroscopic analysis of the data. In addition to the differential error analysis, checks were made to assure the reproducibility of data. These checks indicated that the data were reproducible within the limits of the estimated errors presented above. Ind. Eng. Chem.. Process Des. Dev., Vol. 16, No. 2 , 1977
177
Table 11. Explanation of Plotting Symbols Used in Figures 3 and 4 P l a t e Number 0.1
-1
0
0
0
oo
0 0 0
1
1.0
I
8
I
1
I
I
I
I
(
!
I
1
I ,-, .1
I
I
.4
I I ( 1 0 1
I
I
8
4
.6
I
I I I O
6
10
I
'
20
Wt d?yF Ia t: *
Figure 4. Modified Gaddis-Worley friction factor correlation for squaretooth0.03 and 0.05-in."flow gap": - - -, original Gaddis-Worley
correlation; -, modified Gaddis-Worley correlation.
As a result of the extensive amount of experimental data obtained and the volume of mathematical calculations involved, all data were reduced through the use of a computer program. Due to the number of test configurations involved, each geometry was assigned a unique plotting symbol. A summary of the symbols utilized is presented in Table 11. Figure 3 shows a typical representation for the relationship between friction factor and Reynolds number for the data. It can be seen that the friction factor is dependent on the Reynolds number. The measured friction factors did not follow a monotonic function over the range of Reynolds numbers investigated. This finding is in line with other data of friction factor vs. Reynolds number for uniformly distributed roughnesses. Based on the differential error analysis, the uncertainties associated with the experimentally determined loss coefficients, K = (AP/pV2),for the three different size plates were f20% for the 0.012-in. flow gap, f 1 0 % for the 0.030-in. flow gap, and *7% for 0.050-in. flow gap. The dynamic head of the jet in the test section was calculated to be a maximum of 0.02 psi and thus negligible in reference to the overall pressure drop. Comparison of Measured Data w i t h Existing Empirical Correlations Comparisons were made between the measured loss coefficients and various empirical correlations; the Colebrook (1939) equation which predicts the Moody friction factor chart; the Gaddis-Worley (1973) correlation of friction factor 178
0
0
0
0
+A
5
B
h
ll
b
7
0
B 0
a
8
0
8 Q 4
4 4
n
10
0
11 12
0
h
B
0 0
13
0
14
0
Ip
15
a
8
9
L @
a
1
~
O0.4. 4
.03
1.15:
6
9 I
3.339
8 A
3 4
I
0.312
a 0 A h
2
I
F l o w Gap ( Inches)
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 2, 1977
data for various artificial roughness waveforms; an empirical equation relating the friction factor to the Reynolds number of the form f = (constant) (Re)n ; and equations which are used to predict form loss coefficients for contractions and expansions. Most of the data obtained when used to calculate the Reynolds number, relative roughness, and friction fell outside of the range of the limits 3000 < Re < D/t/O.Oldf which are defined by the Colebrook equation. The Gaddis-Worley correlation uses a number of functional groups for analysis such as a cross-correlation coefficient, R ]2(0), which is dependent on the groove shape, the serration land to serration pitch ratio, and T/X.The squaretooth data did show the proper trend for the 0.030 and 0.050-in. flow gaps; however, the sawtooth data did not correlate by this technique. A least-squares regression of the 0.030 and 0.50-in. flow gap squaretooth data was performed and a correlation coefficient of 0.95 resulted. If the 0.012-in. flow gap squaretooth data are included in the least-squares analysis, the correlation coefficient is reduced to 0.86. Due to the scatter in the 0.012-in. flow gap square-tooth data, it was not included in the correlation. Based on the least-squares analysis of the 0.030 and 0.050-in. flow gap squaretooth data, the Gaddis-Worley correlation for the range of WIDTH'/FLAT(D) > 0.35 was modified as follows
I t can be seen from Figure 4 that the modified GaddisWorley equation better correlates the 0.030 and 0.050-in flow gas squaretooth data than the original equation for this particular experimental work. It is interesting to note that in the vicinity of WIDTH'/FLAT(*D) C= 0.35 a significant change is noted in the value of FUNC. Gaddis-Worley also indicate a significant change in the value of FUNC at this point as shown in Figure 4. Winkel (1923) experimentally measured friction factors for flow in concentric annuli in which the inner tube has rings on its surface. The clearance between the rings and the outside tube was small. Although no general correlations could be developed from Winkel's investigation, he was able to correlate the friction factor for individual geometries as a function of Reynolds number. The geometries in this investigation are similar to those tested by Winkel, except that the flow is between parallel plates rather than in concentric annuli. An attempt was made to correlate the measured friction factor data as a function of
O0 00
/I
0.8
-
0.6
-
$0.4
-
0.6
-m < 0.1
-
J 0
0.2
0.4 0.6 Calculated
0.8
1.0
Figure 5. (Qser/Qsmouth)acrual VS. (Qsei/Qsmoath)calculated for waretooth and D.05O-h flow gaps. plates; @.@12,0.@3@, Reynolds number. A least-squares regression was performed for each test configuration. The results of the analysis showed a scatter in the correlation coefficient, thereby indicating that the friction factor was not necessarily a monotonic function with respect to Reynolds number. Comparisons of the measured total loss coefficient to those predicted considering the serrations to be a series of contractions and expansions were also considered. The total measured loss coefficient, however, could not be predicted via this method of analysis. Summarizing the above discussion, none of the existing correlations investigated was capable of adequately representing all the data from this experimental work. I t was possible, however, to represent the squaretooth data for the 0.030 and 0.050-in. flow gap by a modified form of the Worley correlation. The data obtained for the 0.012-in. flow gap squaretooth configuration exhibited trends significantly different from the 0.030 and 0.050-in. squaretooth configurations. This possibly indicates the presence of another flow regime if the gap is sufficiently small. Correlations Related to the P l a t e Geometry Based on the preceding discussion, it is apparent that none of the existing formulas correlates all of the experimental data. During the course of the data reduction, it was observed that various relationships between the physical dimensions of the plates and the ratio of the volumetric flow rate for a serrated plate to the volumetric flow rate obtained for the “smooth” plate exist. The concept of utilizing the geometrical configuration of the roughened plates (i.e., serration depth, serration width, and serration pitch) and the flow gap was investigated to determine the combinations of geometric parameters which best correlated the ratio of “serrated” to “smooth” volumetric flow rates. Utilizing the squaretooth plate data for the 0.012,0.030, and 0.050-in. flow gaps the following equation was found to be representative of the data to f30%.
As mentioned previously, the 0.012-in. flow gap squaretooth data exhibited trends substantially different than the square-tooth data for the larger flow gaps. It was deemed appropriate, therefore, to develop a correlation for the 0.030 and 0.050-in. squaretooth data and a separate correlation for the 0.012-in. squaretooth data. Employing the 0.030 and 0.050-in. square-tooth data the following correlation represented the data to f 5 % . Qser Qsmooth
= 0.746
EPTH DEPTH + 0.828 (DPITCH ) - 0’030 (CAP)(3) ~
02
0
0.4
0.6
0.8
I
Calculated
The 0.012-in. flow gap squaretooth data, by itself, can be correlated per the following equation to f 2 0 % as
(4) Figures 5 and 6 graphically depict the comparison of the measured data to the correlations presented in eq 2 and 3. This method of correlation was also employed with respect to the sawtooth plate data. For the sawtooth plates the ramp angle of the serration must be accounted for. The ramp angle was incorporated into the analysis as a correlating parameter via the following function DEPTH () 16’ f ( 0 ) = -for tan-’ (-) 8 - 16 WIDTH 1 B
f ( 0 ) = - for tan-’
(5)
The ramp angle of 16’ is the angle at which a minimum value of volumetric flow was observed in the measured data. I t is worthy of noting that a 16’ angle is approximately the angle a t which flow would be expected to expand freely from one area to a larger area without flow separation. Utilizing the function, f ( 8 ) in conjunction with the other geometric parameters, the following correlation with an error of f 1 2 % was developed for the 0.030 and 0.050-in. flow gap sawtooth data
Figure 7 depicts a comparison of the measured data to the correlation presented in eq 6. Based on the above discussion it is apparent that the geoInd. Eng. Chem., Process Des. Dev., Vol. 16, No. 2 , 1977
179
metric parameters of the artificially roughened plates can be employed to successfully correlate the volumetric flow ratios measured from the experimental work. Nomenclature D = hyraulic diameter, 4AIS DEPTH = serration depth f = friction factor FLAT = serration land FUNC = Gaddis-Worley correlation functional group ( f ) = Gaddis-Worley predicted friction factor GAP = flowgap g, = gravitational constant PITCH = serration pitch, flat width Q = flow rate R 12(0) = Gaddis-Worley cross correlation coefficient Re = Reynolds number, D ( u ) p / p S = wettedperimeter T = serration land, flat V = velocity WIDTH = serration width
+
Greek Letters root mean square roughness over one X 8 = ramp angle of sawtooth serration X = serration pitch, PITCH p = viscosity p = density Subscripts ser = refers to serrated data smooth = refers to smooth data e =
L i t e r a t u r e Cited Colebrook, C. F., J. lnst. CivilEng. (London), 11, 138 (1938). Gaddis, E. C., Jr., Worley, F. L., Preprint, Proceedings of the 66th Annual AlChE Meeting, Philadelphia, Pa., Nov 1973. Idel'chik, I. E., "Handbook of Hydraulic Resistance," p 128, Israel Program for Scientific Translation, Jerusalem, 1966. McAdams, W., "Heat Transmission," pp 126-136, 159, McGraw-Hill, New York, N.Y., , 1954. Rouse, H., "Engineering Hydraulics," pp 412-415, Wiley, New York, N.Y., 1950. Vermes, G., ASME Trans., J. Eng. Power 83, 161 (1961). Winkel, R., Z.Angew. Math. Mech., 3, 251 (1923).
Receiued for reuieu! November
13, 1975
Accepted December 30,1976
Infinite-Dilution Activity Coefficients of Ethylene in Solvent-High-Density Polyethylene Mixtures David C. Banner'' Chemical Engineering Department, Texas Tech University, Lubbock, Texas 79409
Norman F. Brockmeier Amoco Chemicals Corporation, Naperville, lllinois 60540
Infinite-dilution activity coefficients of ethylene have been determined experimentally by gas-liquid chromatography in mixtures of n-hexane-high-density polyethylene (HDPE) at 143.2 and 157.3 O C at 69 atm, and isooctane-HDPE at 157.5, 200.5, and 237 OC at 69 atm. A correlation is developed for the infinite-dilution activity coefficient of ethylene based on a straightforward extension of the corresponding-states theory of polymer sohtions to three-component mixtures of nonpolar substances, one of which is a polymer. We show how the correlation may be used, with a minimum of data, to provide a useful tool for the design engineer to correlate and predict data for ternary solutions containing one polymer component.
Introduction In many polymer processing operations a molten polymer exits the manufacturing process containing one or more volatile components. Previous studies of highly concentrated polymer solutions have tended to consider only binary, solvent-polymer solutions. Further, the most accurate practical model of polymer solutions, the corresponding-states theory of polymer solutions, has only been developed to consider two-component solutions (assuming that polymer polydispersity is neglected). We take here an initial step toward consideration of polymer solutions containing two low-molecular-weight components and a polymer, a t temperatures greater than the polymer melting range. The solutions we consider experimentally and theoretically are n-hexaneAddress correspondence t o t h i s a u t h o r a t Chemical Engineering Department, Texas A & M University, College Station, Texas 77843.
180
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 2, 1977
high-density polyethylene (HDPE) and isooctane-HDPE solutions containing trace amounts of ethylene. We show how to measure ethylene activity coefficients at essentially infinite dilution and how to treat such activity coefficients using a three-component version of the corresponding-states theory of polymer solutions. Throughout the theoretical development, we assume that the solution components are nearly nonpolar. Experimental Section Preparation of Carrier Gas. Figure 1shows the apparatus used for preparing the carrier gas and controlling its flow through the GC columns. It is basically the same design as that used for low pressure work described by Brockmeier et al. (1972). The important changes include the replacement of all glass and copper tubing with 316 stainless steel tubing, Y4-in. 0.d. in most cases. The pressures a t the inlets and outlets of
