and the expected results led to further investigation of the procedures, assumptions, and equations used for the simulation. We would like to offer the following comments. A. I n the equation for the heat transfer coefficient h3RADI for radiation from the inner kiln wall to the solid material, the “correction factor, f,” is introduced to account for “the radiant energy that is absorbed by the gas.” If that is truly its purpose, a corresponding factor should also appear in the gas-to-inner kiln wall heat transfer calculation: any radiqnt energy absorbed by the gas will surely affect the temperature of the gas. But the wall is heated only by the gas, and therefore cannot be a t a higher temperature than the gas. Thus radiant heat cannot be transferred from the wall t o the higher temperature gas; the heat goes in the qther direction, and is accounted for in the gas-to-inner wall calculation. (Only the net heat transfer from gas to wall is important; if the wall does in fact re-radiate to the gas, this only reduces the net heat transfer from gas to wall by radiation, and is otherwise of n o consequence.) I n any case, because of the relatively small difference in temperature between the wall and gas in this region, radiant heat transfer between wall and gas is practically negligible. B. Instead of the correction factor f for the purpose stated, the wall-to-solid radiation coefficient should take into account the re-radiation which takes place. I n addition to the emissivity of the material, E,, which the author’s equation includes, the net radiant transfer must also depend on the emissivity of the wall and the relative areas of wall and material between which this transfer takes place. Eckert and Drake (1959) give a n equation which accounts for these factors. Using the author’s notation,fE, should be replaced by the expression
;{
+A=
(& -
I)}-’
C. For the wall-to-solid convection coefficient, the handbook expression 0.05 Go.67 is multiplied by a factor of 5, as “proposed by Imber and Paschkis” (1962). Actually, they fpund that computations “based upon data for two rotary kilns” indicated that the wall-to-solid boundary conductance is apsroximately five times that for the gas-to-wall coefficient, and suggest that “this should be used only as a rule of thumb.” O n this basis, the author multiplies an empirical convection coefficient,
0.05 G0.67,which is based on the assumption that “at any point the bed temperature approaches the wall temperature” (Perry, 1963) by a factor of five to obtain a conduction coefficient. D. T h e author compares his simulation results with those obtained by using two handbook equations, and concludes that the simulation results are “far superior to the results obtained from existing correlations.” T h e first handbook equation, U, = 0.12 Go.46/0 (Perry, 1950), is found to be in error by 795% and 900% for two examples. I n the 1963 edition of this handbook this equation has been eliminated; this fact in itself would lead one to believe it suspect. Perry (1963) presents the equation mentioned above, U, = 0.05 G0.87 as the “convection heat transfer from gas to biick,” based on the assumption mentioned previously; Sass finds that errors of 31% and 71% result from the use of this equation. Not only is the above assumption invalid for these examples, but to use the equation as the author did to calculate the required length of the kilns requires a further assumption: that all the heat lost by the gases is transferred from the gases to the kiln was by convection. By including termsfor radiant transfer from gas to solid and gas to wall in his heat transfer coefficients, the author admits this latter assumption is isvalid. O n e can oqly conclude that the comparisons of the author’s simulation results with these “existing correlations” are not valid comparisons. literature Cited
Eckert, E. R. G., Drake, R. M., “Heat and Mass Transfer,” p. 405, McGraw-Hill, New York, 1959. Imber, M., Paschkis, V., Intern. J . Heat Mass Transfer 5,, 623-38 ” (1962). Perry, J. H., ed., “Chemical Engineer’s Handbook,” 3rd ed., p. 831, McGraw-Hill, New York, 1950. Perry, J. H., ed., “Chemical Engineer’s Handbook,” 4th ed., section 20, p. 23, McGraw-Hill, New York, 1963. DESIGNDEVELOP. 6, 532-5 Sass, A., IND.ENG.CHEM.PROCESS (1967).
Vaughn A . Kaiser James W . Lane Projmatics, Inc. Woodland Hills,Calif.
SIMULATION O F T H E HEAT TRANSFER PHENOMENA I N A ROTARY K I L N
SIR: Based on the comments by Kaiser and Lane concerning the paper titled “Simulation of the Heat Transfer Phenomena in a Rotary Kiln,” it is evident that they are well versed in the area of heat transfer in kilns, but it is unfortunate that they have not published their findings since this would have saved me and many others a great deal of work. I n the absence of this help, I was forced to use a n admittedly very simplified approach to this problem since no reasonably accurate design equations were available in the literature. I clearly stated that the development is based on “many assumptions” and that I expect it to be modified as more iqformation becomes available. I would like to offer my comments as follows on the specific questions which were raised: 1. I n the comment on the shape of the solid temperature profile, Kaiser and Lane state, “one cannot expect the tempera-
ture of the solid material to increase by some llOOo F. in 15 feet of kiln length.” This is not a valid question since it is based on intuitive feel rather than heat transfer analysis. T h e residence time of the material in that section of kiln was approximately 1 hour (based on 12Yo fill) and this was more than adequate to d o the heating job. As a point of interest, our main reason for undertaking the simulation was to determine kiln operating conditions which would result in the steepest solid temperature profile a t the hot end. We were heating a material which oxidized readily above 1000° F. and we therefore wanted to minimize its residence time a t excessive temperatures. 2. Kaiser and Lane state, “the data points can be better fit with a straight line than with the curved profile resulting from the simulation (Figure 3); this fact indicates that flame radiation exists, and the assumption (of no flame radiation) is not VOL 7
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valid.” Again this is conjecture. Linearity of the gas temperature profile may be related to flame radiation, but is not solely controlled by it. I t is interesting to note that in Figure 2 of the paper, the predicted gas temperature is linear even with the assumption of no flame radiation. 3. The comments concerning the uncertainties in some of my assumed heat transfer coefficients are true, and I clearly admit this in my paper. However, I think some of their comments are in error. You notice they state “the ‘correction factor,f’ is introduced to account for ‘the radiant energy that is absorbed by the gas.’ ” If that is truly its purpose, a corresponding factor should also appear in the gas-to-inner kiln wall heat transfer calculation.” Although the gas temperature is higher than the wall temperature, the wall will still radiate to
the colder solid. This radiation will definitely he “filtered” by the absorbing gas species in the combustion products. I n the radiation of the gas to the wall, this filtering is not present. 4. The comment that my comparison of the simulation results to those predicted by the handbook correlations is meaningless is true in one sense. The correlations are not really valid for the cases studied. However, the comparison was made to emphasize strongly the danger of using these correlations in the absence of better information.
Allan Sass
U. S. Steel ,4pplied Research Laboratory Monroeville, Pa. 75746
Correction
CRYSTALLIZATION OF CALCIUM SULFATE FROM PHOSPHORIC ACID I n this article by A. B. Amin and M. A. Larson [IND.ENG. CHEM.PROCESS DESIGN DEVELOP.7, 133 (1968)], the following error has been discovered. T h e conversion factor from base 10 logarithms to natural logarithms was misapplied. All growth rates reported should be divided by 5.3, the square of the conversion factor, and the slope given in Figure 2 should be multiplied by 5.3. T h e slopes in Figures 5 and 6 are unaffected by this correction, but they should be shifted to the left a distance equal to log 5.3. This error does not alter the nucleation kinetic expression obtained nor the conclusions drawn.
SIMULATION OF HEAT-TRANSFER PHENOMENA IN A ROTARY KILN In this article by Allan Sass [IND.ENG. CHEM.PROCESS DESIGNDEVELOP.6 , 532 (1967)], Equation 3 is incorrect and should read
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I & E C PROCESS D E S I G N A N D DEVELOPMENT
