Correlation of Roll Mill Variables - Industrial & Engineering Chemistry

A NEW FLOATING ROLL MJLL developed from studies, has

. . . shown

Dispersion Studies

these

15% average

Correlation of Roll Mill Variables

production increase

,

. . reduced

the possible out-of-adjustment points from 80 to 8

LOUIS MAUS, Jr.I, WILLIAM C. WALKER, AND ALBERT C. ZETTLEMOYER Nafional Printing Ink Research Institute, Lehigh University, Bethlehem, Pa.

T

HE basic unit operation in the production of printing inks, paints, and many plastic products involves the dispersion of pigments and fillers in vehicles and binders. Chief among the machines used for printing inks and paints are roll mills, colloid mills, and ball mills. For over half a century the three roll mill, essentially in its present form, has been one of the most important machines used to produce a thorough dispersion of pigments in an oil medium. I n spite of its extensive use, investigations of the fluid mechanics of the three roll mill have not appeared in the literature. As might be expected, then, there have been no significant changes in the basic design of three roll mills. Metallurgical advances and water cooling have been incorporated in the roll design, and the roll speed has been increased, but until recently the basic design of roll mills has remained the same. Ultimate objectives of the Printing Ink Institute project in dispersion studies concerning roll mill operation are the establishment of the most economical operating conditions for production mills for various types of dispersions, the development of useful relationships between the performance characteristics of laboratory and production mills, and the development of new and better mill designs. Since these objectives cannot be attained without a fundamentally sound knowledge of the behavior of roll mills, an engineering study of the variables in three roll mill operation was initiated. For this initial study the situation was simplified as much as possible. Only two rolls (or one nip) were considered a t a time, and the material used was a simple, unpigmented oil. The variables investigated and correlated were roll speed, roll speed ratio, clearance, viscosity, roll diameter, roll length, and power input. Three roll mill has a fixed center roll

Plan and side views of the three roll mill in its conventional form are diagramed in Figure 1. Three hollow, water-cooled steel rolls of the same size are mounted parallel in a horizontal plane on a fixed bed. Shafts extend from the ends of each roll and rotate in bearing blocks. The bearing blocks of the center roll are fixed to the frame of the mill, while the bearing blocks of the two outside rolls slide in horizontal traeks toward and away from the fixed center roll. Four screw jacks a t the corners of the mill bear against the bearing blocks of the two end rolls and force these rolls against the center roll. The rolls are geared together with spur gears in such a way that contacting rolls move in the same direction in the nips but with different speeds. Thus the direction of rotation alternates from one roll to the next as indicated in Figure 1. The speeds of the rolls increase from back 1 Present address, Department of Chemical Engineering, Lehigh University, Bethlehem, Pa.

696

to front in a ratio such as 1:2:4or 1:3:9 with a usual top speed of 350 r.p.m. on the fast roll. These rolls are known as the feed, center, and apron rolls, respectively. When the mill is in operation, a premixed paste of pigment and oil is placed between the center and feed rolls and between the end plates which prevent it from flowing over the ends of the rolls. The material is forced through the feed nip, where it, is subjected to shear, and the bulk of it is carried on the under side of the center roll of the apron nip. The material is subjected t o much more severe shearing aetion in the apron nip, and most of it leaving this nip adheres to the top of the apron roll from which it must be scraped by a take-off knife and delivered to the apron as product. Thus, the ink or other material progresses through the mill to a great extent by its tendency to follow the faster roll as it leaves a nip. There are two distinctly different types of nip-the feed nip with a large bank of material rolling above it and the apron nip with no noticeable accumulation a t its entrance. I n this investigation, the two types of nips were studied independently with an experimental mill made up of only two rolls. It1 the experimental mill the feed nip condition and the apron nip condition were simulated by appropriate use of end plates and by placing on the rolls the approximate amount of ink fourid tit these nips on a normally operating mill. Roll mill dispersion involves fluid flow

Fluid flow in the roll mill is analogous to flow in pipes and ducts, agitation, drag of boats, and other problems that have received careful analysis. The variables governing fluid flow are known from studies of these situations and from operations dependent on fluid flow, such as countercurrent heat transfer. I n the analysis of such situations the variables are usually arranged irk certain dimensionless groups, and fluid flow phenomena can be discussed better in terms of these groups than in terms of t h e individual variables. Mill variables expressed in terms of mass, length, and time units are given in Table I. The symbols selected to represent the various variables are the fluid flow analogs in common use in chemical engineering practice. I n a mill the material in process is subjected to shear. This shearing action is directly analogous to that which occurs during the flow of fluids in pipes and ducts (fluid friction). Thus powerinput to a mill is a function of the viscosity and density of t h e material, just as power in the fluid flow analogy is related to these fluid properties. Power input in the mill can also be expressed as a function of the length of path through which the shear i s taking place a t any moment. This corresponds in a pipe flow analogy to the length through which flow takes place. Since actual measurement of the length of the nip in a roll mill is difficult

INDUSTRIAL AND ENGINEERING CHEMISTRY

Vol. 47 No. 4

ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT

MILL FLOW PATTERNS Table 1.

Symbol Definitions m = mas8 = length = time

1

BANK FLOW

t

Symbol

Definition

Units

pe W

Power input/unit of mill width

ml -

D

Clearance

U8

Rim speed of slower roll

1 1

W

Rim speed of faster roll

Ut

us

O P

Gravitational constant Dinmeter of roll Density of fluid

m -

P

Viscosity of fluid

m -

L

+ ut

18

’

t

1 t 1 t

It2 2 1s tl

to determine, the path length has been assumed to be proportional to the roll diameter in this study. Figure 2 represents a cross-sectional view of the material in the bank of a mill. The arrows indicate the direction of the flow. The clearance between the rolls is magnified so that the flow pattern in the nip can be shown. Most of the material carried toward the nip by the rolls is rejected and sent upward. This rejection produces the cylindrical shape of the bank which is really a gravitational wave. In order to correlate flow conditions in which a gravitational wave exists, the gravitational constant, g, must be included in the equations. At this point a dimensional analysis may be carried out in the usual manner ( d ) , based on the relationship

RWON

SLOW ROL

Figure 2.

‘I

ROLL

Flow patterns encountered in bank of three roll mill (nip greatly exaggerated)

The assumption here is that the power term depends only on the variables included and that the relationahip is a simple logarithmic one. One solution of this analysis, in which the exponents h, j , and k are retained, is

n

n

Frequently in fluid flow the ratio L I D is raised to unit exponent. If this assumption is made, and if a = h, and b = - k , Equation 2 becomes

-

F I

Figure 1. A. B.

Essential features of three roll mill

Feed roll (slow) Center roll C. Apron roll (fast) D. Apron

April 1955

E.

F. G. H.

End plate Screw jack Movable bearing blocks Fixed bearing black

The term D u t p / p is called the Reynolds number and describes flow conditions due to viscosity effects as differentiated from gravitational effects. It appears in fluid flow ( I ) , agitation (5), and ship drag ( 3 )equations, as well as in heat transfer equations ( a ) where heat is transferred t o moving streams of fluids. The term u?-/Dg is called the Froude number and describes energy dissipated due to formation of gravitational waves. This term also appears in agitation (6) and ship drag (3) equations. The term Pg/wut3Lpis called the Power number and appears in modified form in equations that define friction loss during flow in a pipe ( I ) , power consumed by agitation (6),and power to propel ships (3). All the groups operate as if they were individual variables in the examples listed, and each is a pure number whose significance is independent of the units used express the variables making up the number. The validity of Equation 3 and the soundness of the assumptions chosen rest on the demonstration that a, b, and k remain essentially constant over the range of experimental data.


697

ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Actual mill clearances are determined

Mill Modification. For this study a Kent 4 X 8 inch watercooled laboratory three roll mill was modified in the following manner; the take-off knife and apron roll were removed so that material that passed through the feed nip was recycled to the bank. Roll force was exerted on the movable bearing blocks by screw jacks through hydraulic systems so that the operating force could always be determined from Bourdon tube pressure gages as shown in Figure 3.

I

J

I

K

U

I

N Figure 3. 1. K. L.

M.

Essential features of experimental two roll mill

Slow roll Fast roll End plates (optional) Screw jacks

N.

P. Q. R.

Hydraulic insert Pressure gage Movable bearing block Fixed bearing block

This mill is equipped with a cam-type throwout device so that the mill can be opened at any time without changing the screw jack setting. Two sets of rolls were used in the test, one pair was 4 inches in diameter by 8 inches long, and the other was 6 inches in diameter by 5 inches long. Methods of Measurement. The measurements of clearance, power input, roll speeds, and viscosity bear some discussion. Clearance measurement is a very important and evasive variable which has not been handled adequately on such equipment in the past. The procedure used to measure actual mill clearances was 1. When the mill was not in motion, the screw jacks were tightened until the pressure gages read some initial value P,. 2. Then the mill was opened by use of the throwout device without changing the screw jack setting, and a shim of known thickness was inserted in the nip. 3. The mill waR then closed on the shim. The resultant opening of the nip produced an increase in pressure in the hydraulic system. If the final pressure indicated is designated as P j , then

698

Pj - Pi is a measure of the nip clearance. Thus, if the rolls are separated for any reason, the distance of separation can be determined from the pressure readings. When the mill is not in motion, fluid is forced from the nip, and the rolls move into contact. When the mill is in motion, the outward force of the fluid on the rolls opens the nip to a certain clearance, and this clearance can be measured as a function of P j - Pi. Clearance calibration curves were prepared with the function P,) as ordinate and the clearance as abscissa, because the resulting plot was almost linear and made interpolation reliable. Power input to the three-phase motor was determined by the two-wattmeter method. The manufacturer of the motor supplied some data on motor efficiencies at various loadings; other intermediate points on an efficiency-load curve were determined by La Grange's theorem (6). Since the motor on the mill was arranged to drive the rolls through a chain and sprockets, the speed of the mill could be altered by changing the size of the sprockets. The speed ratio of the rolls was varied by changing the gears that drive the rolls. Roll speed was determined with a revolution counter and a stop watch for one roll, and the speed of the other roll was calculated from the gear ratio. The materials used as fluids for the mill were two polybutene oils whose viscosities were approximately 200 and 700 poises at room temperature. Both of these substances exhibited Newtonian behavior when tested on a precision rotational viscometer over the pertinent temperature range and at rates of shear to 600 set.-' Viscosity-temperature curves were used to correct for change in viscosity due to temperature changes. Oil temperatures were determined by use of an Alnor thermocouple designed to read temperatures on roll faces. The temperatures were determined in the thin film of fluid on the roll surface, not in the mass of fluid in the bank. The mill was stopped for temperature determinations. I n the operational procedure for the apron nip (Pj - Pi)/(P/

+

1. The screw jacks were set so that the rolls were not in contact, and the no-load power was determined. This power was assumed to be equal to that necessary to overcome the friction in gears and bearings. This amount may have been in some error since no force was applied to the bearings as was the case when a fluid was passing through the nip. 2. The mill was closed and just enough material placed in t h e nip so that the mill would operate. This condition exists between the center and apron rolls of a three roll mill when it is running properly. No end plates were used, and as a result, there was a small amount of flow over the ends of the rolls. This overflow had to be picked up with a spatula and fed back to the mill so that the total load of oil could be kept constant. 3. Cooling water was fed to the rolls, and the mill was operated until temperature equilibrium was established. Several trials were required, but once the cooling water rates were established, slight further adjustment was necessary. If the difference in temperatures on the two rolls did not exceed 2" F., the system was considered a t equilibrium. 4. When equilibrium was reached, power and gage pressure were read and recorded while the mill was in operation. 5. The mill was stopped, and temperatures and gage pressure were read. 6. The mill was started again and the speed determined. This value was nearly constant from run t o run, but high loadings slowed the mill slightly. 7 . No-load power was redetermined after a sequence of 4 t o 8 runs.

For the feed nip all steps listed for apron nip were followed except for a change in Step 2: 2. End plates were placed in operating position and the mill inch of the top of the rollers. As in the was filled to within apron nip runs, material that ran over the end of the rolls had to be fed back with a spatula.

For infinite roll speed ratio studies only the center roll was in operation, and the rear roll was fixed. Temperatures were


Vol. 47, No. 4

ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT

0 FEED NIP 0 APRON NIP


I-

LO ?..

(u

Figure 4.

Correlation plots at roll speed ratio of 1

Figure


5.



t 0

E OJ

Figure 6.

y x l d

10.0


determined by releasing the fixed roll and turning this roll so that the thermocouple could be placed on the roll face a t the point that had just previously served as a nip. During these studies the variables were changed in the following ranges : Roll diameter, inches Roll length, inches Roll speed, r.p.m. Roll meed ratio Clearance, inches Viscosity, poises Force, lb. Power input, hp.

4-6

5-8 0-300 1--m 0 . 2 5 - 8 . 0 X 10-8 200-700 440-2060

Figure 7. Correlation plots at infinite roll speed ratio

sumption of the mill, then a t the apron nip a small error in loading will result in a large deviation in power reading. Since the amount of material used should be a function of the clearance, it was controlled by visual observation, and therefore some error was probable. The data seemed to support this supposition since there was greater scattering of points in this correlation than in the feed nip correlation. Preliminary Correlation. The experimental results a t constant roll speed ratios are plotted in Figures 4, 5, 6, and 7. Each figure represents a plot of the groups of Equation 3

0.08-0.73

Correlating equations are developed

Discussion of Errors. Two inherent errors in experimental procedure were recognized. The first of these lay in the determination of clearance. Clearance was measured indirectly as a function of two pressure readings on the 1000-pound gages of the hydraulic inserts. The gage scales were marked in increments of 10 pounds per square inch. For most runs the pressure differences, P f - Pi, were two figure numbers so that to obtain results to two significant figures the gages had to be read to 1 pound per square inch. Thus, the second figure was very questionable. The second inherent error was in the loading of the mill when the mill was operated as an apron nip. For the apron nip condition a small error in the volume of material used makes a much larger percentage difference in the size of the bank than would an equal error in the volume applied to the feed nip. If the size of the bank affects the power conApril 1955

on log-log coordinates.. Roll speed ratios of one, two, three, and infinity are represented by these figures, respectively. The constants and average deviations of the plotted results are shown in Table 11. By use of the average value of the exponent, a, the equation is (4)

where K is a function of the nip loading and roll speed ratio. Examination of Figures 4 and 7 for unit and infinite roll speed ratios indicates that much energy is consumed in maintaining the bank. Some of this energy overcomes the viscous forces in moving the fluid in the bank, and some maintains the configuration of the bank. The energy consumed in maintaining the bank is necessarily applied to the fluid in process and as a consequence produces


699

ENGINEERING. DESIGN. AND PROCESS DEVELOPMENT Table II.

Constants for Correlation Plots in Figures 4, 5, 6, and 7

General Equation

Figure Roll Num- Speed ber Ratio

Nip a Loading Apron -0.64 -0.70 Feed Apron -0.60 Feed -0.63 -0.75 Apron -0.71 Feed Apron -0.69 -0.67 Feed a Ka is equation constant when exponent b K p is equation constant if a = -2/3; a t geometric center of data plots.

Av. DeviaKpb tion, % 10 0.68 0.61 1.33 9 1.08 5 0.97 1.48 4 1.65 2.00 11 1.36 0.77 1.93 1.46 6 9 1.74 1.47 2.75 2.68 11 a has its tabulated value. these values were determined Koa

ably be represented by a Reynolds number only, without the Froude number. This condition is analogous t o fluid flow in an enclosed space. The significant velocity term in a Reynolds number as just described would include the difference in speed between the rolls, Au, but Au = 0 a t unit roll speed ratio where considerable power is consumed. Therefore, to express this lattjer condition some finite Reynolds number is required. The values ut and uf are always finite. Since ut was used in the Froude number because of its physical significance, u t was also used in the Reynolds number and Power number for simplicity in calculation.

RATIO e 1-1

...-, 2-1

some dimersion in the bank-that is. the material in the bank undergoes some preliminary dispersion before entering the nip. Treatment of Velocity Terms. For the correlation of the data a t any roll speed ratio for one of the types of nip it was necessary to introduce a new dimensionless group involving this factor. In this new group the use of the rim velocities of the two rolls involved is sufficient to characterize the speed and speed ratio. This group also must have finite values a t the boundary conditions of unit and infinite roll speed ratio. The dimensionless group that most logically fulfills the boundary conditions is ( u t u.)/u, = ut/uf. At infinite roll speed ratio, this term is one, and a t unit roll speed ratio, this term is two. Therefore, a t any condition this ratio always has a finite value. For the correlations a t constant roll speed ratios (Figures 4, 5, 6, and 7) ut was the velocity term used in each case. For any specific roll speed ratio ut is proportional to u/, and therefore these quantities may be used interchangeably a t constant roll speed ratio. Neither uBnor Au = uf - us can be used in corfelations because a t each boundary condition one or the other becomes equal to zero and the equation becomes meaningless.

tI

+

IM

.

I I ll.lll

RATIO 1-1

0

2-1

0

3-1

I IIIM

t=

=I

I Illllll

Lo

I I I IIII Ku)

D y x 103

Figure 9.

Over-all correlation plot for feed nip

Final Results. The function, u t / u / , was used successfully to correlate data taken a t various roll speed ratios in an equattion of the type

An examination of' the data reveals that B is a function of nip loading. Correlations using these values of 6 are shown in Figure 8 for the apron nip, and Figure 9 for the feed nip. In these figures the functions

are plotted on log-log coordinates. The constants of' the equations, determined from these plots, and the deviations :we shown in Table 111.

Table 111.

Constants of Plots Shown in Figures 8 and

9

General Equation

Figure 8.

Over-all correlation plot for apron nip

Mechanically, the gravitational wave is produced by the impact of two masses of fluid carried toward the nip by the rotating rolls. Since very little of the fluid carried by the rolls can escape through the nip, the majority must be diverted upward (Figure 2). Since this impact is the result of two velocities, the physical significanceof ut in the Froude number is obvious. The physical significance of u t in the Reynolds number may be questionable. I n the region directly inside the nip, beyond the point where fluid is diverted upward, the flow conditions can prob-

700

Nip Loading Kpb tion, % B Y K "/a 1.78 0 1 -0.879 1 65 Apron 1.45 2.58 8 7 -0.688 2 35 Feed 1.15 a K y is equation constant when exponent I has its tabulated value. b K p is equation constant if y = -2/3; these values were determined a t geometric center of data plots.

A comparison of Tables I1 and I11 shows that the broadening of the correlation to include all roll speed ratios has not appreciably increased the average deviation. Thus the mean of the average deviations has increased only from 8.1 in Table I1 to 8.9 in Table 111. Very little more can be expected of such a correlation.


Vol. 47, No. 4

ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Conclusionr

Fundamental engineering concepts of fluid flow have been applied to roll milling and correlating equations have been developed for a single nip with a nonpigmented oil. The variables correlated were viscosity, clearance, power input, roll speed, roll speed ratio, roll diameter, and roll length. The density of the fluid was constant throughout the tests. The term representing fluid density is included in the equation to suggest its possible relationship based on other fluid flow examples. Apron and feed nip loadings have been investigated. The correlating equations for various conditions are 1. For the apron nip or feed nip, a t constant roll speed ratio, the general correlating equation is

The values for K appear in Table 11. 2. For the apron nip, at any roll speed ratio, the correlating equation is

3. For the feed nip, a t any roll speed ratio, the correlating equation is

literature cited

Badger, W. L., and RIcCabe, W. L., “Elements of Chemical Engineering,” 2nd ed., Chap. 2, RlcGraw-Hill Book Co., New Y- o. r k~. 1936. ~ ~ , Ibid., Chap. 4. Langhanr, H. L., “Dimensional Analysis and Theory of Models,” Cham 2, I). 21, Wiles. New York, 1951. Murpcy, N. F., Bu1l:Virginia Polytech. Znst.,42, No. 6 (1949). Rushton, J. H., Costich, E. W., and Everett, H. J., Chem. Eng. Progr., 46, KO.8, 395 (1950). Worthington, A. G . , and Geffner, J., “Treatment of Experimental Data,” Chap. 1, p. 20, Wiley, New York, 1943. RECEIVED for review December 16, 1953. ACCEPTED October 6, 1954. Presented a t the 122nd Meeting of the ACS, Atlantic City, N . J , September 1952. From a dissertation submitted b y Louis Maus, ,Jr,, t o the Graduate School of Lehigh University in partial fulfillment of the requirements for the degree of doctor of philosophy, J u n e 1951.

(DISPERSION STUDIES)

Floating Roll Mill LOUIS MAUS, JR.~,ALBERT C. ZETTLEMOYER, AND ERNEST GAMBLE National Printing Ink Research Institute, Lehigh University, Befhlehem, Pa.

I

ix

N T H E preceding paper the operation of the conventional three roll paint or ink mill was described. This age-old design is capable of excellent dispersion performance but has basic difficulties of adjustment and control. The operation of a three roll mill has been an art, and a t least several months have been necessary for the training of a good mill hand. With the usual screw jacks there is no indication of how the mill is set, and the mill hand must judge the correctness of his adjustment by the appearance of the ink films on the rolls and apron and by the fineness-of-grind achieved. There are 80 different ways or combinations of ways in which a conventional four-point adjustment mill can be out of adjustment. This situation has led to the development of several schools of thought as to how a three roll mill should be set. The conflicts among these schools of thought have remained unresolved for many years because of the lack of basic information on roll mill behavior and lack of means for measuring mill conditions. The great needs in three roll mills today, then, are for a simple system for bringing a mill to proper adjustment and for a method of numerically evaluating the mill setting. I n recent years some progress. has been made toward satisfying these needs, but adoption by the industry has been very slow in most cases. Twenty years ago Vase1 (3)developed a mill with hydraulic elements in place of the conventional screw jacks so that a pressure gage indicated the forces exerted on the roll bearings. Thus, the operator could reproduce force conditions on the mill and could intelligently study optimum conditions. The Vasel mill could also be arranged so that the four hydraulic elements were interconnected and thus maintained a force balance 1 Present address, Department of Chemical Engineering, Lehigh University, Bethlehem, Pa.

April 1955

around the mill. This arrangement also permitted adjustment of the mill from a single point of control. The unified hydraulic system did not work as well as anticipated, however, because of unbalanced thrusts from the gears connecting the rolls. Gears are normally mounted on only one end of each roll so that their spreading forces tend to open one end of the mill and produce an unbalance. This tendency can be corrected with the four-point arrangement but not with the four elements tied in a single system. The problem was finally solved by mounting gears on both ends of the rolls. The Vasel mill was not accepted by the industry a t the time, but mills with four-point mechanical adjustment with hydraulic gages are now coming into use. A mechanical system of simplifying roll mill adjustment was designed by Brasington (1). This system brings both end rolls to the fixed center roll simultaneously by turning a single control wheel. The mechanism is usually designed to permit a clearance ratio of 2:1 between the feed and apron nips. The four corners of the mill have individual adjusting screws to bring the two outer rolls into proper relationship to one another and to the fixed center roll. Once this adjustment has been made, the mill can be opened and closed reproducibly by the single control wheel. In practice, however, each screw needs periodic readjustment. Since one oE the factors affecting machine adjustment is maintenance of the initial roll relationships, and since the same degree of skill is required to readjust as to set a conventional four-adjustment machine, frequent readjustment may be necessary. Not only does this one-point control simplify mill adjustment but also the mill is equipped with an indicator for setting tightness on an arbitrary scale. Although this indicator is helpful in reproducing settings, i t must be connected t o one of the adjusted points and if the operator does not maintain this reference adjustment, loss of the zero point occurs.


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Correlation of Roll Mill Variables - Industrial & Engineering Chemistry

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