ENGINEERING. DESIGN, AND PROCESS DEVELOPMENT Rear roll tighter on the right Rear roll tighter on the left Front roll tighter on the right Front roll tighter on the left Apron nip tob tight for feed nip setting Apron nip too loose for feed nip setting 7. Both nips too tight 8. Both nips too loose
1. 2. 3. 4. 5. 6.
These 8 types of misadjustment and 72 combinations of them are possible. Thus there is a total of 80 ways in which a fourpoint adjustment mill can be out of adjustment. I n contrast, the floating roll mill can have only four types of misadjustment 1. Mill tighter on the right 2. Mill tighter on the left 3. Mill too tight 4. Mill too loose And four combinations of them are possible so there is a total of 8 ways in which the floating roll mill can be out of adjustment. This is only a tenth as many possibilities for misadjustment as there is with conventional mills; consequently, the floating roll mill is much easier to put and to keep in proper adjustment. This marked relative ease of adjustment has been amply borne out by experience. Floating roll mill has been laboratory and production tested
Laboratory Scale Tests. Three small roll mills have been converted to the floating roll arrangement a t the institute. These mills are a Kent 4 X 8 inches, a Brasington one point 6 X 14 inches, and a Fritsch 2.5 X 5 inches. Each mill has a different gearing arrangement and is a different size. The most marked change created in these mills by the conversion was the ease of adjustment. Laboratory mills must be particularly easy to set since very small batches are normally run, and little time is available before the entire batch has passed through the mill. When a four-point adjustment mill was used, as many as two or three passes were required before the mill was properly set. The amount of milling which the batch received was poorly defined. A floating roll mill, however, can be set within the first pass, and the batch can be milled almost entirely a t the final setting. Furthermore, the training of new laboratory personnel in the proper operation of a mill is no longer a major problem. Floating roll mills have been used in this laboratory for over 2 years, and a four-point mill would no longer be considered. The single point mechanism used in the Brasington mill is excellent when it is properly aligned and adjusted, but this adjustment requires skill and is not permanent. The mechanism tends
to drift slowly from proper adjustment so that eventually a good setting can no longer be obtained with the single point adjustment. After this mill was converted to a floating roll mill by merely releasing the center roll bearing blocks, much of the usual difficulty apparently disappeared, and this mill is now much more reliable. A comparison was made in this laboratory of the dispersion effected by the floating roll mill and a standard roll mill. Although it was difficult to make a rigorous comparison, the results indicate that the floating roll mill behaves essentially like a fourpoint adjustment mill a t optimum setting. To date a four-point adjustment mill properly set has not out performed an otherwise similar floating roll mill. Additional information has been obtained on this point by subsequent work on production mills. Use in Plant Production. Since the initial announcement of the floating roll mill to the-members of the Printing Ink Institute, a considerable number of production mills have been converted to operate on the floating roll principle. The returns from a recent questionnaire provide considerable information on the adoption of the floating roll mill by the industry. The 21 replies indicate that 15 companies have converted 80 mills in sizes to 16 X 40. The reports show that these mills are easier to adjust, and production has increased an average of 15% with a maximum of 30%. The training of mill hands was considerably simplified. Actually in some cases a better dispersion was obtained as well as increased production. These reports from the industry confirm the simplicity of adjusting the floating roll mill. They show clearly that the floating roll mill is practical for production use, and that it has important advantages over the conventional four-point adjustment mill as used. The generally increased production with the floating roll mill is ample evidence that the imposed condition of equal spreading forces in the two nips is nearly optimum for a three roll mill. This conclusion was also reached on the basis of laboratory work where no setting of a four-point adjustment mill was found that gave results superior to those obtained on a floating roll mill. It should be emphasized that the conversion of most types of present three roll mills to floatin8 roll mills takes only a short time and can usually be handled readily by ink plant personnel. Literature cited (1) Brasington, C. P., U. S. Patent 2,254,512(September 1941). (2) Shurts, R. B., and Rosa, Prisco. Natl. Paint, Varnish Lacquer Assoc., Sci. Sect. Circ. 759, October 1952. (3) Vasel, G. A., U. S.Patent 1,788,964(January 1931). ACCEPTED October 6. 1954. RECEIVED for review December 16, 1963. Presented a t the 124th Meeting, ACS, Chicago, Ill., September 1953.
(DIs PERSION ST uDIES)
Correlation of Floating Roll Mill Variables ALBERT
C. ZETTLEMOYER,, JAMES H. TAYLOR,
AND
LOUIS MAUS, Jr.'
National Prinfing Ink Research Insfifote, Lehigh University, Bethlehem, Pa.
IIV
T H E first paper ( 2 ) published here a method of correlation for roll mills was presented, and in the second (3) an improved and simplified three roll mill-the floating roll mill-was described. The initial correlation of roll mill variables was derived from 1 Present address, Depar6ment of Cheniical Engineering, Lehigh Univeraity, Bethlehem, P a .
April 1955
data taken on an experimental two roll mill operated under conditions simulating individually both a feed nip and an apron nip as found on a conventional three roll mill. The present study was designed to extend the correlation work t o apply to a floating roll mill as described in the second paper. On the experimental used in this work the two types of nips operated simultaneously with the apron roll fixed and a take-off knife in use. The
INDUSTRIAL AND ENGINEERING CHEMISTRY
703
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT relationships resulting from this study, then, are applicable to the complete three roll floating roll mill as it is used commercially. The correlating equations that were obtained in the initial paper in this series (2)may be expressed as Power required for the apron nip
WU,3LP pg = K [ g ]
uf2
-118 '
[z]] , [
D~~~
-1.15
+ 1)1'3( R +
= = w = ua =
-a13
(2)
(3)
K = (R)(R g
3 W = 2.58 upLp [G]
Dtu,p [F]
where
P Power required for the feed nip
-1l3
L = R = Dt = k
p p
= =
=
power consumed by mill gravitational constant width of rolls rim velocity of feed roll roll diameter roll speed ratio total clearance of two nips fraction of material from feed nip adhering to center roll density of material being milled viscosity of material being milled -pg
wu,3Lp
-
Power
NO.
ua2 _ - Froude N o .
D tB
gp = Reynolds KO. J !
Total clearance was used because it was the quantity which could be measured with the experimental arrangement. Three roll mill equation i s tested
t-
0
UNPIOYENTCD POLYBUTENE
0
IRON BLUE IN V A R N I S H
---LINES
PREDtCTED EY DERIVED EQUATION
I IO2 2
Figure 1.
4
e
a 19
a
4
6
0
ITL
2
Comparison of measured data with derived equation at each roll speed ratio
Consider the application of these equations to a floating roll mill, I f this mill has a constant roll speed ratio, R, then Equations 1 and 2 might be written for each of the two nips, and the sum of the two equations might be expected to apply to the more complex case of two nips acting in sequence, T o facilitate the summation of these two equations, the following simplifying assumptions have been made:
1. The total power consumed by the mill is the sum of that required by the apron nip and by the feed nip. 2. The value of the exponent for the term ut/u, may be taken as 4/3 for both the apron nip and the feed nip. This change of exponent was found to have a very small effect on the deviation of the plotted data in the first of this series. The to the fact that the insignificance of this effect is due maximum range of this velocity term is from one to two. With these assumptions the derivation of the three roll mill equation was carried out by summing Equations 1 and 2, by rearrangement, by using a material balance and standard algebra, and introducing roll speed ratio and total clearance
704
The mill used in this work was a 4 x 8 floating roll mill with the apron roll fixed, equipped with a take-off knife, so that the experimental runs were essentially the same as actual operation runs. This mill was equipped with hydraulic inserts a t the two screw jack controls. The mill was operated a t roll speed ratios of 3:l and 2 : l and a t various speeds. Materials. Both Newtonian and non-Newtonian materials were studied. The Newtonian materials were polybutene oils furnished by the Oronite Chemical Co. These oils are nondrying and in this study were not pigmented. Five runs were also made with an ink that consisted of 50y0 Iron Blue in No. 0 transparent lithographic varnish. This ink was non-Newtonian in behavior a t low shear rates, but a t the rates of shear encountered on the mill, the stress-r.p.m. relationship could be considered essentially linear (1). This material differed in density from the polybutenes and made possible the inclusion of the density variable which was untested in the earlier work ( 2 ) . By means of a precision rotational viscometer, viscosity-temperature curves were prepared for each of the substances milled. Mill Operation. The procedure employed in operation of the roll mill was
1. Before making a test the mill was allowed to operate with no material in the nips and with the rolls out of contact for apProximately 5 minutes in order to allow the lubricant to approach operating temperature. Readings of the wattmeter were taken a t the end of the warm-up period to determine the no-load power reauirements of the apparatus. 2. With the mill stopped and end plates in place, the rolls
E 2$yzgi{ey2it ~ ~ t ~ t~ ~~ $ ~ &~' $ ~ ~t T i li $: The knife was placed in regular operating position on the apron
ro:; The roll setting was made by tightening the Screw jacks until the desired initial pressure was read on both right and left hydraulicpressuregages. 4. The was started and '"ling water was adjusted to The give approximate'y equal temperatures on a'' three mill was allowed to run until temperature equilibrium was obtained. 5. The mill speed was determined by measuring, with a stop watch, the time required for the feed roll to make a definite number of revolutions. The speeds of the other rolls were then known from the gear ratios. 6. The power input to the three-phase motor was measured by the two-wattmeter method. The power output of the
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Vol. 47, No. 4
~
f
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT motor was then obtained from performance data furnished by the manufacturer. 7. The operating pressure of the mill was read from the hydraulic pressure gages on the adjusting cylinders. 8. The mill was stopped'momentarily, and the rest pressure on the hydraulic gages was read. The difference between this and the operating pressure was used to obtain the total clearance by the technique previously described (I). '3. With the mill running temperatures were measured with a surface contact thermocouple in the bank, a t the center of the feed and center rolls, and in the material removed from the apron roll by the knife. 10. The mill' was stomed, the feed bank was refilled. the mill setting pressure was changed, and the same procedure was repeated for each additional test. 11. At the end of a series of tests the mill was cleaned, the rolls were separated, and a second no-load power determination was made. No-load power is the power required by the mechanism of the mill and is subtracted from the motor output to give the power actually consumed by the roll-fluid system. This technique was used to gather data for a wide variety of operating conditions. Roll speed ratios of 2 : l and 3 : l were used for several speeds over the pressure range from 200 to 800 pounds of force on each end of the mill. These conditions resulted in a wide range of clearances with the various materials used and amply spanned the reasonable operating conditions for this mill. In order to test the three roll equation developed by addition of the apron nip and feed nip two roll equations, the three roll mill data were plotted as
I,[
[A-
The value of the Reynolds number exponent was slightly different for the two roll speed ratios studied. Earlier results ( 9 ) indicate that, if sufficient data were taken and experimental error were reduced, this exponent would probably have a constant value for all roll speed ratios. In order to determine the expbnent with greater accuracy, tests should be run over a larger range of Reynolds number. Because of the interdependence of the variables involved, however, practical roll mill operation is limited to the Reynolds number range investigated. In order to obtain a single correlation that would include the factor of roll speed ratio, it was necessary to select a single value for the exponent of Reynolds number that would represent the behavior of the mill at ratios of two and three. The value of was selected in order to compare the results with the derived equation.
Table 1.
Constants for Correlation Equation
Roll Speed Ratio a 9.2 30 42 2 -0.64 34.6 10.6 98 149 48.5 3 -0.77 a KO = Value of K calcuIated a t experimental value of a . b K p = Value of K calculated from center point of data msurning a -%/a.
Value of K calculated from relationship K = R(R 1)"a (R f k)1/8 ( 1 . 7 8 where it is assumed that k = 0.75. C
versus Dtuap
on logarithmic coordinates. The plots obtained are shown in Figure 1 and the constants derived from the best straight lines through those points are given in Table I. Figure 1 also shows the theoretical lines calculated from the derived equation.
Kt
=
+
=
5%
4- 2 . 5 8 )
The comparison given in Table I shows that, for both ratios tested, the values of the constant K were less than predicted by the derived equation. This variation is also shown in Figure 1 where the experimental data are compared with lines predicted by the derived equation. The difference is probably due to the factor of apron nip loading. Maus, Walker, and Zettlemoyer ( 2 ) observed that small changes in the loading of the two roll mill in the apron nip condition caused significant changes in the power consumption. The belief that this difference between predicted and observed three roll behavior was due to apron nip loading was strengthened by the observation: The dependence of K on roll speed ratio R was
where
X Y
2
' JRON BLUE
0
IO2
April 1955
I
IN V A R N I S H
2-1 ROLL SPEED R A T I O
,
,
,
I
, , I ,
-
= =
constant characteristic of apron nip constant characteristic of feed nip
The values of X and Y can be found by the method of simultaneous equations using the observed values of K in Table I for 2:1 and 3:1 ratios. The value of k has been found experimentally t o be about 0.75 although this factor has a very small effect in this expression. This calculation gives the values of 1.08 and 2.56 for X and Y , respectively. The value of Y is in good agreement with the value of 2.58 found from two roll mill feed nip studies, but the value of X deviates much more from the value of 1.78 found from the two-roll mill apron nip studies. This result indicates that the apron nip of the three roll mill consumed about 61% of the power consumed by the apron nip in the two roll studies. Final Correlation. For the three roll mill with take-off knife the equation found to correlate the data a t roll speed ratios of two and three was
INDUSTRIAL AND ENGINEERING CHEMISTRY
90s
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Figure 2 shows the agreement of this relationship with the experimental data. Although in this investigation the roll dimensions were not varied, the complete agreement in form between this correlation and that derived from two roll studies indicates that the conclusions drawn from either of these studies might be applied to the other. In the two roll study, roll dimensions were varied, and the variation can be expressed according to Equations 1 and 2. In this study, the density was varied and the variation can be expressed according to Equation 6. Since the equations are the same in form and are a result of derivation and experimental confirmation, conclusions drawn from one case can be applied t o the second case. The difference in the constant of proportionality seems to be directly related to mill loading alone and in no way serves to describe the hydrodynamics of the roll mill beyond mill loading. Because of the large number of quantities measured in this work, the cumulative experimental error was quite large. Among the most important of these errors was the measurement of the total clearance where the problems were the same as those described in the first paper ( 8 ) . Another point a t which an important error may have developed was in the determination of the power consumed by the material on the rolls from the power input to the motor. There was also a question as to whether the temperatures measured to get the viscosity of the material are actually close to those prevailing in the nips where the viscosity is of greatest importance. The correlations obtained are considered to be very good in view of these possible errors. Conclusions
A 4 X 8 inch, three roll, laboratory printing ink mill with floating center roll was studied in an effort to correlate the operating variables, power consumption, mill speed, roll speed ratio, nip clearance, and viscosity. Roll dimensions and density were included in the correlation but not varied in this investigation.
A general correlation equation was derived by summation of separate equations for the feed and apron nips alone which were developed in a previous paper. This equation is
+ 1)1'S(R + k)1'3
where K = ( R ) ( R
Best fit with the measured data was obtained when
+ 1)1/3(R+ k)Ij3
K = (R)(R
This study, in confirmation of earlier. work, has shown that three roll mill operation may be treated as a fluid flow phenomenon. The principles of fluid dynamics that have been employed with great success in the study of numerous chemical engineering operations may now be applied to the improvement of roll mill design and operation. Acknowledgment
The authors wish to acknowledge the combined support of the National Printing Ink Research Institute and the Troy Engine and Machine Co. which made this work possible. literature cited (1) Lower, G. W., Walker, W. C., and Zettlemoyer, A. C., J. Colloid Sci., 8 (l),116-29 (1953). (2) Maus, L., Walker, W. C., and Zettlemoyer, A. C., IND.ENG. CHEM.,47, 696 (1955). (3) Maus, L., Zettlemoyer, A. C., and Gamble, E., I b i d . , 701 (1955). RECEIVED for review December 16, 1953. ACCEPTED October 6, 1954. From a dissertation submitted by James H. Taylor to the Graduate School of Lehigh University in partial fulfillment of the requirements for the degree of Master of Science, June 1952.
Design Equations f o r .
Continuous Stirred-Tan k Reactors Transients of Second-Order Chemical Reactions F. s. ACTON
AND
L. LAPIDUS
Princeton University, Princeton,
I
N. J.
N RECENT years the development of design equations describing the behavior of a chemical reaction in a series of continuous homogeneous stirred-tank reactors has reached a status of maturity. Since the contributions of Denbigh ( 1 , 2 ) both the steady-state operation and the transient approach to the steady state have been considered ( 4 , 6-8, 1 2 ) . Because of the algebraic nature of the steady-state equations it has been shown by Eldridge and Piret (3') and Jones ( 5 ) that the effluent from any reactor may be calculated independently of the order of the chemical reaction. By combining graphical and numeriEal techniques, the system of equations can always be obtained. While experimental data to check the theoretical equations have been few, the agreement between theory and experiment has been excellent. The transient approach to steady state, a condition of utmost physical importance, can be treated by present methods only when certain simplified chemical reactions are occurring (9). For any reaction other than first order or those equivalent to first order, the differential equations become nonlinear. The
706
case of a second-order chemical reaction has only been touched and no solution has appeared in literature. This problem is not unique to stirred reactors. In many chemical engineering operations the transition from a first-order reaction to one of higher order results in nontractable differential equations (11). The equations presented in this article aid in the design of systems involving transient continuous stirred-tank reactors with a second-order chemical reaction. Solutions of the proposed equations are presented in closed form for the first reactor. For the reactors beyond the first, a set of approximate equations are proposed that allow the effluent concentrations to be calculated within the limits of engineering accuracy. Contrary to the current trend of using an electronic computer to evaluate all interesting combinations of the physical parameters, the computer is used in this work as an intermediate tool t o suggest analytical approximations. This technique deserves emphasis because of the saving in time and money and the more compact form of the results.
INDUSTRIAL AND ENGINEERING CHEMISTRY
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