Ind. Eng. Chem. Res. 1987, 26, 1254-1259
1254
Mashelkar, R. A.; Devarajan, G. V. Trans. Inst. Chem. Eng. 1977, 55, 29. Mishra, P.; Gupta, S. N. Ind. Eng. Chem. Process Des. Deu. 1979, 18, 137. Mori, Y.; Nakayama, W. Znt. J . Heat Mass Transfer 1965, 8, 67. Mujawar, B. A.; Rao, M. R. Znd. Eng. Chem. Process Des. Deu. 1978, 17, 22. Oliver, D. R.; Asghar, S. M. Trans. Znst. Chem. Eng. 1975,53, 181. Oliver, D. R.; Asghar, S. M. Trans. Znst. Chem. Eng. 1976,54, 218. Prandtl, L. In Boundary-Layer Theory;7th ed.; Schlichting, H. Ed.; McGraw-Hill: New York, 1979; p 627. Pratt, N. H. Trans. Znst. Chem. Eng. 1947, 25, 163. Ranade, V. R.; Ulbrecht, J. J. Chem. Eng. Commun. 1983,20, 253. Rogers, G. F. C.; Mayhew, Y. R. Int. J . Heat Mass Transfer 1964, 7, 1207.
Schlichting, H. Boundary-Layer Theory, 7th ed.; McGraw-Hill: New York, 1979; p 600. Seban, R. A.; McLaughlin, E. F. Znt. J . Heat Mass Transfer 1963, 6, 387. Shenoy, A. V.; Renade, V. R.; Ulbrecht, J. J. Chem. Eng. Commun. 1980, 15, 269. Skelland, A. H. P. Non-Newtonian Flou and Heat Transfer; Wiley: New York, 1967; p 276. Tarbell, J. M.; Samuels, M. R. Chem. Eng. J . 1973, 5, 117. von Karman, T. NACA T N 1092, 1946. White, C. W. Proc. R. Sac. London, Ser. A 1929, A123, 645. White, C. W. Trans. Znst. Chem. Eng. 1932, 10, 66.
Received for review January 21, 1986 Accepted December 31, 1986
COMMUNICATIONS Improved Apparatus for Measurement of Specific Surface Areas of Powders An account is given of a new electronic permeameter designed for measurement of the specific surface areas of powders by observing gas flow rates and pressure drops across beds of the solid particles. T h e gas permeametry results were examined by using the Carman-Arne11 equation, a two-term equation consisting of a viscous flow term (the Carman-Kozeny expression) and a slip flow term (the Knudsen expression). T h e results obtained from this new flow apparatus are reproducible. Regression analysis produced high correlations between gas flow rate and mean pressures which are statistically valid at a significance level of 0.1%. The estimated fractional standard errors of the specific surface were only a few percent. The equation (Carman and Arnell, 1948) relating the transport of a gas through a compacted bed of powder under nonturbulent flow conditions to the surface area of the powder is made up of two terms
Rather than solve this equation directly for Sv,the calculation can be carried out by evaluating two separate terms and combining them to give Sv.
Equation 2 and the first term in eq 1 apply to viscous (or laminar) flow~andgive the contribution to the surface area of the powder due to this streamline flow. This equation was first developed by Kozeny (1927) and later modified by Carman (1938), who adopted a value of 5 for the constant of proportionality. The second term applies to molecular (or Knudsen) flow and relates to a surface which includes surface cracks and fissures as well as the voids between the particles of which the bed is composed.
0888-5885/87/2626-1254$01.50/0
The combined equation was proposed by Carman and Arnell (1948), who introduced a “slip” coefficient in the second term. This coefficient corrects for the nature of the molecule-surface interaction: A fraction of the molecules striking the surface wall will be specularly reflected; i.e., if the capillary walls are smooth, molecules striking them at any angle rebound at the same angle with the Same average velocity and with the component of velocity perpendicular to the wall reversed. The surfaces of packed beds of powder are not smooth, and molecules striking them rebound in any direction, i.e., diffuse reflection. For any powder, as the inlet pressure falls, the contribution of the viscous flow term to measured surface area will decrease and the contribution of the molecular flow term will increase. Further, the finer the powder, the greater the contribution of the slip flow term in the combined equation. Combining eq 1-3 gives the quadratic in S v having the solution
These equations have to be evaluated. Instrumentations in Gas Permeametry A large number of gas permeameters have been designed for measuring the flow rate and pressure drops across packed beds. They can be classified into two basic types according to the way in which flow resistance is measured. With constant-flow-rate permeameters, gas is passed through the bed of powder at a constant volume rate of 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1255 flow. The entry or exit of the apparatus is set a t atmospheric pressure, and the pressure drop is measured by a manometer. With constanbvolume permeameters, on the other hand, a fixed volume of gas is passed through the bed of powder by creating a pressure difference across the bed and allowing the gas to flow from the high-pressure side until a predetermined decrease in the pressure difference is attained.
Constant-Flow-RatePermeameters Lea and Nurse (1947) designed a gas permeability apparatus which was connected to a capillary manometer and a flow meter. Dry gas was forced through the sample bed under pressure or by suction, enabling flow rates and pressure drops to be determined. The simplicity of the method is readily apparent and has been found to be quite reliable for regular and irregular particles, provided they have no internal surfaces and are above 10 pm in diameter. Gooden and Smith (1940) modified the Lea-Nurse apparatus by incorporating a self-calculating chart which enabled specific surface to be read off directly, using the relationship (Allen, 1974), Sv = 6/cl,,, where d,, is the average particle diameter. This greatly reduces the time required per sample. A commercial instrument has been developed from the Gooden-Smith method, called the Fisher subsieve sizer. This uses a standardized computation procedure, and the average particle diameter of a powder is read directly from one of a family of curves on a chart incorporated in the instrument, a t porosities, t, within the range 0.80-0.40. The resulting figure for this diameter varies with the porosity of the powder, inadequate compression of the sample yielding a high result. Constant-Volume-RatePermeameters A different geometrical approach to gas permeability measurements was taken by Rigden (1947). In this procedure, the time taken for the pressure drop to fall by a given amount is measured rather than the rate of flow under a steady pressure. This is also the basis of the method developed by Blaine (1943). The essentials in this method are that the powder is compacted into a bed in a cell and the two ends of the cell are connected to the two ends of a U-tube containing an oil of vapor pressure. The oil is displaced initially by drawing up on one side; as it returns to its equilibrium level, it forces gas through the powder bed. The pressure differential thus diminishes as flow proceeds. The time for the oil to travel between two marks on the U-tube limb is measured, and the specific surface is then calculated. The Griffin surface-area apparatus is a commercial version of the Rigden device. Basically, gas is caused to flow through a bed of powder by the pressure of oil, displaced from equilibrium in two chambers which are connected to the permeability cell and to each other in a U-tube fashion. The oil is brought to the start position by using a rubber hand pump with the tap open to the atmosphere. The taps are then rotated so that the oil manometer rebalances by forcing air through the powder bed. Timing is from start to height hA(for fine powders) or hB (for coarse powders), the height differences denoting different flow volumes. Experimental Techniques Flow Apparatus Designed for the Investigation. Since the two important parameters that have to be measured in permeametry are flow rate and pressure drops, two types of electronic measuring devices, operating under
# 7
Q?..
thonnal
-1
ih't, Iran.-
l
-
Figure 1. Simple line flow diagram of the experimental assembly.
Figure 2. Multitron pressure transducer.
constant-gas-flow-rate conditions, have been developed. Figure 1shows the simple flow line and general layout of the assembly diagrammatically. In order to convert the nonelectrical physical parameters of pressure drop across the packed bed into electrical signals, two sensitive Multitron absolute-pressure transducers were used to provide a rapid response system. The transducer, based on the linear variable differential transformer (LVDT) principle, consists of primary and secondary windings. Displacement of a moveable magnetic core from its central zero position produces an output (at carrier frequency) which is proportional in magnitude to the core displacement. The range of pressure measurements for the two pressure transducers (inlet P1and outlet Pz) is 0-1 atm (0-760 mmHg). The gas flow rate is constant at dV/dt = 4.167 x m3/s. The prime sensing source of the pressure transducer is a precision capsule element made by welding together two annuler corrugated-metal nesting diaphragms. Any pressure changes in the packed bed will cause the pressure sensitive elements to deflect. This causes a movement of the ferromagnetic core and provides an ac signal proportional to the displacement of the sensing element. (See Figure 2.) This signal is then transmitted to an electronic transmitter which provides the excitation, demodulation, amplification, and conversion needed by a voltage-based two-channel pen recorder. A typical graphical output from the two pen recorders is shown in Figure 3. It can be seen from the graphical output in Figure 3 (red, P1,and black, P2) that P1 = 0.131 V 99 mmHg, Pz = 0.103 V 78 mmHg, and AP = 21 mmHg. Now, from the same Figure 3, the amplified pressuredrop pen recorder gives the true value of AP = 21 mmHg. It can be seen, therefore, that this pressure-drop device provides a more accurate account than a manometric fluid in terms of accuracy and reproducibility of the pressure
1256 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 Table I. Determination of Specific S u r f a c e Area a n d Carman-Arne11 Constant Data: Dicalcium Phosphate Dihydrate [Ca(H2PO,),] L = 7.1986 X m dV/dt = 4.167 X 10" m3/s w = 7.4701 X kg V = 20 x IO4 m3 p s = 2420 kg m-3 A = 1.246 X m2 m/s [ ( ~ R ~ ' ) / ( T M )=] ' 231.74 /~ t = 480 s t = 0.6559 g = 9.81 m s-' N s m-* 1 = 186 X
volts 0.932 0.838 0.738 0.632 0.530 0.430 0.330 0.225 0.120 0.0231
P, in mHg N 0.7083 0.6369 0.5609 0.4803 0.4028 0.3268 0.2508 0.1710 0.0912 0.01 76
m-2 X 9.4498 8.4973 7.4833 6.4079 5.3740 4.3600 3.3461 2.2814 1.2167 0.2348
volts 0.902 0.808 0.708 0.603 0.502 0.403 0.304 0.203 0.102 0.016
P , in mHg N 0.6855 0.6141 0.5381 0.4583 0.3815 0.3063 0.2310 0.1543 0.0775 0.0122
U ,N m-2 X m-2 X 9.1457 8.1972 7.1863 6.1144 5.0898 4.0865 3.0819 2.0876 1.0340 0.1628
P I /
w/
0.3041 0.3001 0.2970 0.2935 0.2842 0.2735 0.2642 0.2438 0.1827 0.0720
in m2/m3x lo6
P,N m-'
(dV/dt), m3/s x 10-3 129.4830 117.9882 104.9929 90.9769 78.7947 66.4282 52.7752 38.9934 27.7503 13.5891
x 10-4 9.2977 8.3472 7.3348 6.2612 5.2319 4.2233 3.2140 2.1595 1.1253 0.1988
SK
SM
sv
1.7847 1.7715 1.7603 1.7472 1.7162 1.6793 1.6436 1.5673 1.3412 0.8056
0.1766 0.1938 0.2178 0.2514 0.2902 0.3443 0.4333 0.5865 0.8241 1.6829
1.8752 1.8710 1.8726 1.8774 1.8674 1.8603 1.8745 1.8878 1.8151 2.0064
From eq 5
A regression analysis on the 10 experimental points gives slope = 12.6524 X
lo-"
std error of slope = 0.0801
X
intercept = 12.2133
X
correlation coeff, r = 0.9998
lo-',
Therefore,
S"2 = From the table, Sv = 1.8808 X
= 1.8800
44'3601
12.6524 X
X
lo6 m2/m3
lo6 m2/m3, with a standard deviation of s = 0.0483 X lo6 m2/m3. 6ko - 1.8800 X lo6 X 12.2133 X lo-* = o.1710 _ k
1.3373
ob Figure 3. Typical graphical output from the two pen recorders.
drop across the packed bed of powder.
Experimental Results and Discussion The outlet pressures and the pressure drops across packed beds of powders were determined from data collected while a constant and known amount of gas passed through the bed. The data were determined by using the Carman-Arne11 equation in the form PI dV a p dt ----p+A t3 1 - 8( %) A - t2 (5) k p L (1 - e ) 2 S v 2 3 k LSv 1 - C
i
1
3
i
4
P INm2x1Q'1
Q
i
a
9
ro
Figure 4. Evaluation of surface area and the Carman-Arne11 constant.
From the slope of the graph of the left-hand function, y, against P, a value of the specific surface area, Sv,can be found; the intercept yields a value of 6 k o / k . Thus, (7)
and
[ yz]1'2
or, alternatively,
These values were obtained by using a regressional analysis. Typical results obtained from the electronic gas permeameter designed are shown in Tables I-IV and as graphs in Figures 4 and 5.
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1257 Table 11. Determination of Specific Surface Area and Carman-Arne11 Constant Data: Dicalcium Phosphate Dihydrate [Ca(H,P0d)21 L = 6.5940 dV/dt = 4.167 X m3/s w = 7.4701 X kg A = 1.246 X m2 V = 20 x lo4 m3 pa = 2420 kg m-3 c = 0.6243 t = 480 s = 231.74 m/s [(~R?")/(TM)]'/~ N s m-2 p = 186 X g = 9.81 m s - ~ (Pi/W/ fl,N P1in P2 in in m2/m3 x (d V/ dt), m-2 x P , N m-2 volts mHg N m-2 X volts mHg N m-2 X lo4 m3/s X 10" lo4 x 10-4 SK SM 0.960 0.858 0.755 0.651 0.550 0.440 0.340 0.232 0.130 0.040
0.7296 0.6521 0.5738 0.4947 0.4180 0.3344 0.2584 0.1763 0.0988 0.0304
9.7340 8.7000 7.6554 6.6001 5.5768 4.4614 3.4475 2.3521 1.3182 0.4056
0.907 0.805 0.703 0.600 0.501 0.394 0.297 0.195 0.104 0.028
0.6893 0.6118 0.5340 0.4560 0.3807 0.2994 0.2257 0.1482 0.0790 0.0210
9.1964 8.1624 7.1244 6.0837 5.0791 3.9945 3.0112 1.9772 1.0540 0.2802
75.4493 67.4347 60.0754 53.2274 46.6918 39.8172 32.9263 26.1435 20.7908 13.4779
0.5376 0.5376 0.5310 0.5167 0.4977 0.4669 0.4363 0.3749 0.2642 0.1254
9.4652 8.4312 7.3899 6.3419 5.3279 4.2279 3.2293 2.1646 1.1861 0.3429
2.0957 2.0922 2.0752 2.0424 1.9987 1.9281 1.8530 1.7026 1.4133 0.9438
0.3581 0.4007 0.4497 0.5076 0.5787 0.6786 0.8206 1.0335 1.2995 2.0047
lo4 SV 2.2824 2.3021 2.3122 2.3119 2.3088 2.2970 2.3082 2.2960 2.2053 2.3791 ~~
From eq 5
--=A€' d t
sv2
A regression analysis on the 10 experimental points gives slope = 6.6212 X
lo-''
std. error of slope = 0.07167
intercept = 11.7509 X lo-* correlation coeff, r = 0.9995
X
Therefore,
S"2 = From the table, Sv = 2.3003
X
Sv = 2.2995
35*0101
6.6212 X lo-''
X
lo6 m2/m3, with a standard deviation of s = 0.0422 X 6ko - 2.2995 _ k
X
lo6 X 11.7509 X
lo6 m2/m3 lo6 m2/m3,
= o,2230
1.2116
Table 111. Determination of Specific Surface Area and Carman-Arne11 Constant Data: Dicalcium Phosphate Dihydrate [Ca(H2P0,),] L = 6.280 X m dV/dt = 4.167 X m3/s w = 7.4701 X kg A = 1.246 X lo4 m2 V = 20 x lo4 m3 pa = 2420 kg m-3 t = 480 s t = 0.6055 [(2R?")/(?rM)]1/2= 231.74 m/s = 186 X lo-' N s m-2 g = 9.81 m s-2 p,N U'i/W/ P, in P2 in in m2/m3 x lo6 m-2 x (dV/dt), P , N m-2 volts mHg N m-2 X lo4 volts mHg N m-2 X lo4 lo-' m3/s X loT8 X lo4 SK SM SV 0.984 0.880 0.780 0.670 0.562 0.450 0.350 0.244 0.142 0.075
0.7478 0.6683 0.5928 0.5092 0.4271 0.3420 0.2660 0.1854 0.1079 0.0570
9.9768 8.9229 7.9089 6.7935 5.6982 4.5628 3.5489 2.4735 1.4396 0.7605
0.921 0.818 0.719 0.610 0.505 0.396 0.301 0.202 0.110 0.049
0.6999 0.6217 0.5464 0.4640 0.3838 0.3009 0.2288 0.1535 0.0836 0.0372
9.3378 8.2945 7.2898 6.1905 5.1205 4.0145 3.0526 2.0479 1.1154 0.4963
0.6390 0.6284 0.6191 0.6030 0.5777 0.5483 0.4963 0.4256 0.3242 0.2642
65.0599 5.91689 53.92327 46.9461 41.1016 34.6766 29.7970 24.2178 18.5034 11.9947
9.6573 8.6087 7.5993 6.4920 5.4093 4.2886 3.3007 2.2607 1.2775 0.6284
From eq 5
A regression analysis on the 10 experimental points gives slope = 5.6686 X
intercept = 10.3973 X
std. error of slope = 0.0942 X
correlation coeff, r = 0.9990
Therefore, sv2
From the table, Sv = 2.3317
X
=
30.4382 5.6686 X
Sv = 2.3172
X
lo6 m2/m3,with a standard deviation of s = 0.1051 X 6ko - 2.3172 X lo6 X 10.3973 X _ k
1.1394
lo6 m2/m3 lo6 m2/m3. = o.2115
2.1256 2.1044 2.0845 2.0516 2.0014 1.9402 1.8362 1.6856 1.4496 1.2628
0.3704 0.4073 0.4527 0.5133 0.5863 0.6950 0.8088 0.9951 1.3024 2.0091
2.3188 2.3179 2.3231 2.3242 2.3160 2.3186 2.2846 2.2551 2.2404 2.6182
1258 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987
Table IV. Determination of Specific Suface Area and Carman-Arne11 Constant Data: Dicalcium Phosphate Dihydrate [Ca(H2P04),] L = 5.634 X lo-' m dV/dt = 4.167 X m3/s w = 7.4701 X kg I/ = 20 x lo4 m3 A = 1.246 X m2 p s = 2420 kg m-3 t I(
0.991 0.885 0.785 0.680 0.571 0.451 0.351 0.250 0.159 0.080
0.7532 0.6726 0.5966 0.5168 0.4339 0.3428 0.2668 0.1900 0.1208 0.0608
= 0.5603 = 186 X
10.0489 8.9736 7.9596 6.8949 5.7889 4.5735 3.5595 2.5349 1.6117 0.8112
t = 480 s 9.81 m s?
N s m-2
0.901 0.797 0.701 0.599 0.492 0.379 0.281 0.192 0.110 0.047
[(2RT)/(?rM)]'/*= 231.74 m / s
P =
0.6848 0.6057 0.5330 0.4552 0.3739 0.2880 0.2136 0.1459 0.0836 0.0357
0.9126 0.8926 0.8485 0.8218 0.8005 0.7311 0.7097 0.5884 0.4963 0.3349
9.1363 8.0810 7.1111 6.0731 4.9884 3.8424 2.8498 1.9465 1.1154 0.4763
45.8840 41.8922 38.0897 34.9611 30.1341 26.0672 20.8996 17.9519 13.5320 10.0934
9.5926 8.5273 7.5353 6.4840 5.3886 4.2079 3.2046 2.2407 1.3636 0.6437
2.1271 2.0989 2.0426 2.0035 1.9673 1.8691 1.6587 1.6436 1.4768 1.1748
0.4266 0.4672 0.5007 0.5598 0.6495 0.7509 0.9365 1.0903 1.4464 1.9392
2.3511 2.3455 2.3082 2.3029 2.3187 2.2819 2.1918 2.2768 2.3676 2.4929
From eq 5 --=-
AP dt
sv2
A regression analysis on the 10 experimental points gives
slope = 3.9932 X std. error of slope = 0.0754
intercept = 8.4085 X
X
10-8
correlation coeff, r = 0.9990
lo-'*
Therefore, S"2 =
21'6397 3.9932 X
Sv = 2.3279
From the table, Sv = 2.3237 X lo6 mZ/m3,with a standard deviation of s
_-
: E I
100-
23E =
0.6055
r = 0.9990
Figure 5. Evaluation of surface area and the Carman-Arne11 constant.
As can be seen from Tables I-IV, values of the variables which may affect the results obtained are listed at the head of each experimental sheet. Columns 1-3 contain the chart recorder readings for the inlet pressure (PIin volts) and the conversions into meters of mercury and Newtons per square meter. Columns 4-6 give the final pressures, Pz. The pressure drop across the packed bed is given in column 7. Multiplying the constant-gas-flowrate by the ratio, PI/@, gives the values contained in column 8. The mean pressure, P, is shown in column 9. Values of the calculated SK,SM,and Svusing eq 2-4 are given in columns 10-12. It is generally observed from all the graphs evaluating the surface areas of the powders, at decreasing porosities, t = 0.6559-0.5603, that the flow rate of gas through the packed beds is a linear function of the mean pressure.
lo6 m2/m3
= 0.0773.
6ko - 2.3279 X lo6 X 8.4085 X k 0.9757
=0
X
lo-'
= o.2006
A regression analysis on the 10 experimental points from Figure 4, a t all pressures, gives a slope of 12.6524 X with a standard error of the slope of 0.0801 X 10-l2. The line intercepts the y axis at a value of 12.2133 X loTs,with a correlation coefficient of r = 0.9998. From the regression analysis, Sv = 1.8800 X lo6 m2/m3. The tabulated data for the same 10 values give Sv= 1.8808 X lo6 m2/m3with a standard deviation of s = 0.0483 X lo6 m2/m3,indicating good agreement between calculated and measured surfaces. The results for SKand SMwere as expected. For SK, as the inlet pressure falls, the contribution of the viscous flow term to measured surface area will decrease, and the contribution of the molecular flow term, S M , will increase. The Carman-Kozeny equation applies in the viscous flow region, but as the powder becomes finer this equation breaks down. The controlling factor is the ratio of hydraulic diameter and mean free path of the gas molecules, and as this approaches unity, "slip flow" occurs. This effect increases with increasing fineness, decreasing porosity, and decreasing pressure. In order to account for the enhanced flow that arises due to this breakdown, a second term is introduced into the equation, and this term includes an unknown "constant" (6ko/k),the coefficient of slip. It is not intended to discuss this unknown constant further, but from Tables I-IV, a general constancy in the value of 6ko/k was observed for the powder tested, except a t a porosity of t = 0.6559, where a slightly lower value of 0.1710 was obtained, probably due to surface irregularities causing channeling in the nonhomogeneous bed. A statistical analysis of all the results obtained at different porosities shows that the correlation coefficients and the parameters for the other regression lines are statistically very significant a t the significance levels tested. It
I n d . E n g . C h e m . R e s . 1987, 26, 1259-1262
should be noted further that the tables of the distribution of Student's t function with 8 degrees of freedom (u = n - 2) gives at 5% (5% is the compliment of 95% level of confidence) significance level, t = 2.3060, 1%significance level, t = 3.3554, and 0.1% significance level, t = 5.0413. The high correlations obtained from these calculations indicate good experimental technique and instrument design. Conclusions Dicalcium phosphate is a porous powder, and from the study, Sv,the surface areas, generally increased with decreasing porosity, which is in agreement with theory. In order to carry out the investigation, as electronic gas permeameter operating under constant-gas-flow-rate conditions, was designed. Inlet and outlet pressures were measured by using pressure transducers, and a particularly useful feature was the incorporation of an electronic gauge to amplify the difference between these two pressures, i.e., the pressure drop. The amplified pressure-drop results are reproducible, as is evident from the high correlations found between gas flow rates and mean pressures. Acknowledgment I thank Dr. T. Allen and Dr. N. G. Stanley-wood, both of the Department of Chemical Engineering, University of Bradford, England, for their useful criticism and advice, especially Dr. T. Allen, during the preparation of this paper. Nomenclature A = cross-sectional area of permeameter cell d,, = surface volume mean diameter dV/dt = volume flow rate through the powder bed of the gas g = acceleration due to gravity (9.81 m s - ~ ) k = aspect factor (Kozeny constant, equal to 5.0) ko = shape factor 6ko = variable in Carman-Arne11 equation
1269
L = powder bed length &f= molecular weight of the gas (29 kg/kmol for air) P = mean gas pressure in the bed PI = pressure at bed inlet P2 = pressure at bed outlet AP = pressure drop across the packed bed R = universal gas constant (8.314 kJ/kmol) Sv = volume specific surface SK = surface area using viscous flow term S M = surface area using molecular flow term Sv2 = powder surface area as given by the Carman-Arne11 equation at a range of pressures t = temperature, K W = mass of powder in the bed e = powder bed porosity Greek Symbols = viscosity of air (186 X
/I
p,
N s m-2)
= powder density
Literature Cited Allen, T. Particle Size Measurements, 2nd ed.; ChaDman and Hall: London, 1974. Blaine. R. L. Bull. Am. Soc. Test. Material 1943. 123. 51. Carman, P. G. J. SOC.Chem. Ind., London, Trans. 1938, 57, 225. Carman, P. G.; Arnell, J. C. Can. J.Res., Sect. A 1948, 126A, 129. Gooden, E. L.; Smith, C. M. Ind. Eng. Chem. Anal. Ed. 1940,12,479. Griffin Surface Area of Powder Apparatus; Griffin & George Ltd.: New York, 1986 (available in England under the above name). Kozeny, J. Ber. W e n . Akad. 1927, 136A, 271. Lea, F. M.; Nurse, R. W. Trans. Inst. Chem. Eng. Symp. 1947,25, 47.
Rigden, P. J. J. SOC.Chem. Ind. 1947, 66, 130. Technical literature on Fisher subsieve sizer, Fisher Scientific Co., New York.
Godwin-Joseph I. Igwe Department of ChemicallPetrochemical Engineering University of Science & Technology Port Harcourt, Nigeria Received for review March 11, 1985 Accepted October 27, 1986
Relative Gain Array for Units in Plants with Recycle In most previous studies, the relative gain array (RGA) has been used for assessing the degree of interaction between variables in isolated process units (e.g., single distillation columns, etc.). T h e presence of recycle loops in a process can have a significant effect on the gain matrix of a unit and thus on the RGA. This is demonstrated by considering a simple example of a plant with recycle. It is found that the RGA's for isolated units cannot be relied upon to give a correct measure of even the steady-state interactions in the flow sheet. Most units in a chemical plant can be described as multiinput, multioutput (MIMO) systems. For the initial screening of control systems for MIMO processes, the most common practical approach has been that of designing a network of single-input,single-output (SISO) control loops. The main advantage of such an approach to process control is that only one manipulated variable is used to control one controlled variable. The main problem associated with a control policy implemented through SISO loops is the interaction between the different control loops. Bristol(l966) introduced the concept of a Relative Gain Array (RGA) in order to assess the static interaction between the SISO control loops. Loops that exhibit large interactions are then dropped from further consideration. However, in previous studies (Shinskey, 1979,1981;Jafarey et al., 1979; Georgakis and Papadourakis, 1982), the RGA has been calculated for individual process units only. The 0888-5885/87/2626-1259$01.50/0
presence of recycle loops in a process can have a significant effect on the gain matrix of a unit and thus on the RGA. This is demonstrated by considering a simple example of a plant with recycle. It is found that RGA's for isolated units cannot be relied upon to give a correct measure of even the steady-state interactions in the flow sheet. This observation opens up new research opportunities for calculating and interpreting interaction measures in systems with recycle. RGA for Units in Complete P l a n t s For some units (i.e., distillation columns) there are approximate, analytical expressions giving the RGA as a function of the design parameters of the units (Jafarey et al., 1979, Georgakis and Papadourakis, 1982). However, a unit placed in a complete chemical plant can become coupled with the rest of the units in the plant via recycle 0 1987 American Chemical Society
