Table I. Estimated Ratios of Induced Charge Force Parameters to Coulombic Force Parameter
* 0.117 1.17 11.7
Values correspond to the calculations by George and Poehlein. Nomenclature I S that of Kraemer and Johnstone (except for K l ( I * ) ) .Also included are ratios of gravitational to inertial parameters for the three values of Kv/KI
1.0 10 1.0
x x 10-4 x lo-'
Kr K I
K I I",/Ki
0.20 0.02 0.064
1 . 4 X lo-? 1 . 4 x lo-. 4.6 x
G/*
8.9 X IO-' 8.9 X 1 0 P 8.9 X 10-j
tion) for t h e image charge forces using George and Poehlein's values for particle and collector size and assuming the same fraction of maximum charge for particle and collector. In this table, t h e ratio of charged particle image force parameter to coulombic force parameter is
two sets of results. From Figure 4b it can be seen that for constant \If, the efficiencies calculated by George a n d Poehlein are generally higher t h a n ours. especially for large 1' ' a n d large - K E . T h e trend in differences is more obvious in Figure s a , where their results are much higher than ours whenever t h e effect of particle inertia begins to dominate over the influence of t h e electric force. In all cases. we are able t o explain qualitatively t h e differences in terms of the inadequate criteria described above for determining whether a particle t h a t has passed t h e collector will ultimately be collected. Our results presented here and those for several other electrostatic force and flow situations will be presented in complete form in the near future.
Achnocc iedgment We would like to t h a n k H. F. George a n d G . W. Poehlein for several helpful discussions and for providing additional unpublished information.
the ratio of charged collector image force parameter to coulombic force parameter is
Note A d d e d in Proof: Further d a t a provided by Poehlein indicate t h a t -KI. should indeed replace E S in Figures 4 and 5 , a s we h a d assumed, and t h a t some entries in Table I should be adjusted for certain computer runs.
and the ratio of the next higher order mutual interaction term (involving Q1&) to the coulombic term is
Literatitre Cited
The only noncoulombic term of possible significance in Table I is for t h e charged collector image force, which falls off with distance as r-5; b u t for 9 = 0.1, t h e limiting trajectory lies far away from the collector. and t h e term should be negligible. Consequently. we would expect that the method of images and the coulombic formula would give nearly identical results. Finally, tabulated results in the thesis indicate that gravitational forces were included in the calculation of particle trajectories in the electrostatic case, whereas we did not include them. T h e appropriate parameters are listed in Table I. T h e only case where the influence of gravity could be significant is for t h e low value of 9.In this case the electrostatic force dominates when G is small, as can be inf'erred from the analytical result for \I, = 0 and G > 0. E = -4K~;,'(1 - G ) . Figure 4b presents the results of our calculations in the same form as Figure 4. Included for comparison are t h e calculated results of George and Poehlein. T h e values of \I/ shown are t h e closest available to those used by George and Poehlein. T h e mechanism of interception was not included either in our calculations or those by George and Poehlein for the electrostatic force case. In Figure 5a we have presented our results in the same form as Figure 5 and have reproduced t h e curves from Figure 5 for comparison. Clearly, fundamental differences exist between these
(1) Nielqen, Kenneth A., M S thesis, Iowa S t a t e University, 1974. 1974 (2) George, H e r m a n F., MS thesis, Department of Chemical E n gineering, Lehigh University, 1972. (3i'Kraemer. H . F . . .Johnstone. H. F., Ind. Eng. Chem., 45 (12). 2426-34 (1955). 14) W h i m l e . F. J . W.. Chalmers. .J. A , . Quart. ./. Ro\. M e t e o r o l Soc., 'id, 103-19 (1944).
Received for rerieu March 11, 1974. Accepted M a y 20, 1974. This work mas supported by the Engineering Research Institute of Ioua State Unicersity.
Kenneth A. Nielsen James C. Hill"
Department of Chemical Engineering a n d Nuclear Engineering a n d Engineering Research Institute Iowa State University Ames. Iowa 50010
SIR: Xielsen and Hill clearly demonstrate that the convergence criteria used in our calculations were not adequate for the computation of some particle trajectories.
Gary Poehlein Department of Chemical Engineering Lehigh University Bethlehem, Pa. 18015
Volume 8 . Number 8 , August 1974
769
