Ind. Eng. Chem. Process Des. Dev. 1885, 2 4 ,
1005
1005-1009
Power Consumption in a Two-Roll Mill Yasuhlro Murakaml, Tsutomu Hlrose,+Glshi Chung, Tsuglo Tenda, and Yujl Aklmoto' Department of Chemlcal Engineering, Kyushu University 36, Fukuoka 8 12, Japan
A solution to the power consumption in a two-roll mill rotating at unequal speeds was derived based on the lubrication model analysis coupled with the boundary condition of the nonsymmetric pressure and compared with the measurement for Newtonian fluids. The experiment was carried out with the three pairs of rolls of diameter D = 8.8, 11.3, and 16.4 cm in the range of the minimum nip clearance H , = 40-610 pm, roll speed ratio Nr = 1-7.6, and fluid viscosity p = 2-1 1 Pa s. The theoretical dependence of the power on the bank width agrees well with the experimental dependence. The experimental resuits for the total power consumption and the contribution of the individual roll to it are well correlated by the dimensionless variables derived from the analysis. However, the value of the exponent of the Reynolds number is 10% smaller than the value of -1 which is expected in the theory.
Roll mills are widely used in industrial dispersing processes of highly viscous suspensions, such as paints, printing inks, cosmetics, and metal pastes for microcircuits. The suspension is forced to pass through the very close clearance formed by a pair of rolls rotating at different speeds, where the shear rate in the nip amounts to an order of 103-105 s-l. Due to high shear stresses exerted on the suspension, good dispersion of the highly viscous suspension can be expected to the extent which cannot be obtained in other conventional mixers. Examples of a three-roll mill and a two-roll mill which are usually used in the dispersing industry are shown in Figure 1. The power requirement is an important variable in the design and operation of a roll mill, but it is difficult to determine exactly since the power is consumed not only by fluid motion in the nips but also at the end plates and the scraper. In order to determine the net power consumed by fluid motion in the nips, it should be accurately separated from a gross power measured in a roll mill system. The power consumption in a roll mill has only been previously measured by Maus et al. (1955). Their experimental result was rewritten by Patton (1964) to a simpler form. Maus et al. studied the power input of a two-roll mill and a floating-type three-roll mill which has a fixed apron roll. In their experiments, unfortunately, the influence of the bank width is not clarified. The fluid dynamics in a pair of rotating rollers are studied in calendering, roll coating, and roll milling. Calendering is different from roll milling in the exit region since the fluid leaves off without wetting the rotating rollers in calendering, while in roll milling it continues to adhere each moving surface until it split some distance downstream of the nip. This adhesion leads to the negative pressure. The flow in a roll mill was studied by Hummel (1956) and Zettlemoyer and Taylor (1960), but their analyses are the same as in calendering and are not valid for roll milling. The flow in a roll mill is similar to that in a roll coater. The symmetric flow of the fully immersed roller pair in a viscous fluid was studied by Banks and Mill (1954) and other workers. But the flow in a roll mill and roll coater has a nonsymmetric pressure profile as pointed out by Myers and Hoffman (1961) and Schneider (1962), and the fully immersed condition cannot be used for actual flow. Recently, Greener and Middleman (1979) derived the nonsymmetric pressure distribution and coating
thickness in an equal roll speed system by applying the simple model of the film splitting region. Benkreira et al. (1981) also obtained the coating thickness in the case of unequal sized roll rotating at different speeds with the same model as Greener and Middleman by using the experimental data of the flux ratio through the nip. However, the power consumption and the nonsymmetric pressure distribution in an unequal speed system have not been obtained yet. In this paper, the power consumption in a pair of rolls rotating at different speeds is analyzed by the generalization of the model of the film splitting region proposed by Greener and Middleman. The net power consumption was measured with a two-roll mill for Newtonian fluids and the experimental results were correlated with the dimensionless parameters obtained by the lubrication model analysis, and the semiempirical equation of the power was determined. Theoretical Consideration For a steady isothermal lubrication flow of an incompressible Newtonian fluid between two rolls, a mathematical equation can be written in the form a - d2U -d = (1) dC; aq2 where dimensionless variables are defined as C; = x/(Rh0)'/' q = y / h o
'Department of Industrial Chemistry, Kumamoto University, Kumamoto 860, Japan. t Shoei Chemical Inc., 2-6-1, Nishi-Shinjuku, Shinjuku-ku, Tokyo 160-91, Japan.
f(E) =
in terms of the nomenclature given in Figure 2. The local velocity U is given as
where q R (=1+ E2/2) represents the roll surface and X is a dimensionless flow rate through the nip defined by X = Y2SnRUdq
(3)
-7R
The pressure distribution, a,is integrated with the inlet boundary condition a = 0 at the attaching point C; = toto get the equation a = f(O - f(C;o) (4) where (12 - 9X)C;(C;2+ 2) - 12xC; 4([2
+ 2)2
0198-4305/85/1124-1005$01.50/00 1985 American Chemical Society
+
(12 - 91)
4(2)'12 tan-'
(A)
1006
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985 End plates
,End plates
7 Mill base
Feed rol I
Mill base
Apron roll Center roll
Slower roll Faster roll
Three-roll m i l l
Two-roll mill
Figure 1. Schematic diagrams of a three-roll mill and a two-roll mill.
The first one is the assumption that the fluid leaves the roll surface at the point of the minimum pressure; Le., x = 0 at E, = [2(X - 1)]'/?+.This is applied to the flow in a calender. The second one is the assumption of the symmetric pressure distribution in which a = 0 a t [ = lo,f,. This leads the flow of fully immersed condition when the values of toand E, go to infinity. These boundary conditions are not applicable to the flow in a roll mill as mentioned previously. The last one leads to the nonsymmetric pressure distribution. It is assumed in the present work that the film splitting would take place at the first stagnation point from the nip; that is,
U = 0, d U = 0 a t = E, and q= 7 , da T = -ho/(rNca) a t [ = f ,
I
I
(10) (11)
where N, = pum(D/Ho)'/2/ris a modified capillary number. Equation 10 is reduced to the relationship that U = 0 at 17 = 0 for the equal roll speed as proposed by Greener and Middleman (1979), and in this respect eq 10 may be a generalized condition to the case of the unequal roll speed. The model proposed by Greener and Middleman for the radius of curvature in a roll coater is extended to use in this case; that is
"=
Figure 2. Nomenclature for roll mill analysis.
The power consumption for each roll is obtained by integrating the wall shear stress from the attaching point loto the equilibrium point 6, as
Np = 27.9(H0/0)1/2R;'
[
("-').I
A Nr* + 1
or
(5)
The individual power number Np and the individual Reynolds number Re in the above are defined as
r
-= 1 h0
+ 2 - 1/(Xf + A,) fS2
Thus, eq 11 becomes
Np = P/(pN2NmD4L) Re = pNDHo/p and A and B are The dimensionless roll film thicknesses Xf and A, can be calculated from the flow rate of each roll at f = as
The roll speed ratio Nr* is defined as Nr* = Nf/N, for the faster roll and Nr* = N,/Nf for the slower roll so that eq 5 can be applied to either roll. Then, P = Pf and N = Nf for the faster roll and P = P, and N = N, for the slower roll. The total power of two rolls is obtained from eq 5 as
[
Npz = 27.9(Ho/D)'izRe2-' 2A ( NNrr -+' ) 12 + 2 B ] ~
(8)
where the total power number N and the mean Reynolds number Rez are Np2= (Pf + f',)/pNm3D4L and Rez = PNmDHOlCL . L The relationship between X, E,, and Eo in eq 4, 6, and 7 is unknown yet. The attaching point Eo has the relationship with the bank width HB as
lEol
= [2(HB/HO - 1)11'2
(9)
and HBcan be controlled and measured very easily in the experiment. The dimensionless flow rate X and the equilibrium point [e can be determined by using the boundary condition in the film splitting region. Three methods have been proposed (Gaskell, 1950; Banks and Mill, 1954; and Greener and Middleman, 1979) to determine them.
where qR = 1 + Es2/2. The present analytical method differs from Benkreira's in that Xf and X, are evaluated experimentally in the latter but analytically by eq 15 and 16 in the former. The unknown parameters 5, and X are obtained by solving simultaneously eq 10 and 14 coupled with eq 2 and 4 and the equilibrium point E, is evaluated from the pressure distribution. Once the parameters X and 5, are determined, the power consumption can be calculated from eq 5 or 8 for experimentally determined value of ip A similar calculation is possible based on the experimental value of X but tois more favorable as a starting parameter than X because of easier measurement. The calculated result of eq 8 is shown in Figure 3 and compared with the results for the calendering and the symmetric flow condition. It should be noted that the power calculated for the calendering and symmetric flow condition goes to zero as HB/Ho goes to unity but has a finite value in the present theory. It is found from the calculation that the power changes only slightly with N,, greater than 50 independently of Nr. Experimental Section The schematic diagram of the experimental system is shown in Figure 4. A pair of steel rolls of diameter D =
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985 1007 Table I. Property Constants of Fluids in Equations 17 and 18
c, linseed oil varnish A, &-A) linseed oil varnish B,
E
i,i I
c3
c4
1.486
0.02841 0.9801 7.083 X
1.737
0.03021 0.9830 7.125 X
(L-B)
-
1 ; ; 2
c2
0.9976 0.02415 0.9779 8.221 X
I t ' 1 1 1 ' 1
10
1
I
I I ' ' 1 ' 1 '
100
I ' 1 1 1 1 1 '
linseed oil varnish C, (L-C)
I
low
N.0.833
S'
0.5 5
1 JJ
Slower roll
10
20
[Pas1
Figure 5. Influence of fluid viscosity p on torque Toexerted on roll by end plates.
Fasterroll Slower roll (Fixed) Faster roll
a ) Side view
b) Plane view
Figure 4. Schematic diagram of the experimental system.
11.3 cm and roll length L = 10, 15, and 20 cm was mostly used. A pair of steel rolls of D = 8.8 cm and L = 20 cm and a pair of stainless steel rolls of D = 16.4 cm and L = 20 cm were used in the measurement of the total power of two rolls. The minimum nip clearance Ho was forced wider when the fluid is supplied between the two rotating rolls, and therefore the accurate value of Ho had to be measured in the operating condition. The value of Hoin operation was determined by measuring the distance of the roll surfaces from the reference position by three dial gauges located across the rolls, as reported in the previous paper (Murakami et al., 1983). The bank width HB was obtained by measuring the angle 6 between the y axis and the attaching point where the bank contacted firstly with the roll as shown in Figure 4. It was found from preliminary experiments that the fluid temperature on the roll surfaces and in the vicinity of the minimum nip was slightly higher than that in the bulk of the bank. The maximum difference is about 0.5 OC which decreases the viscosity by about 4%. The temperature measured by thermocouple 1 in the vicinity of the minimum nip was considered to be the representative temperature for the power estimation since the flow behavior around these contributes to the power very much. For the fluid temperature used in evaluating the torque exerted by end plates, the temperature of the fluid adhering to both roll sides was measured by the thermocouples 2 and 3 in Figure 4. These temperatures and torques were measured at the same time. A strain gauge type torque transducer was used in torque measurement by attaching it to each roll shaft in case of unequal roll speed. The total torque of two rolls was measured by one torque transducer on a driving shaft. Linseed oil varnishes ( p = 2-11 Pa s) were used as Newtonian working fluids. The viscosity and the density of fluids were correlated to the fluid temperature 8 by the eq 17 and 18 where constants C1-C4 are listed in Table I. log p = c1- CZB (17) p = c3 - C48 (18) The experiment was carried out in the range of Ho =
A ] 312
0.25 ' 0.5
'
, , 01 5 7 8 1
N Is-'1
5
Figure 6. Influence of roll speed N on torque Toexerted on roll by end plates.
40-610 pm, N = 0.49-3.8 s-l, and Nr = 1-7.6. Results and Discussion Torque Exerted by the End Plates. The torque exerted on the rolls by the end plates Towas measured as follows, and the net torque in the roll nip was evaluated by subtracting the former from the measured gross torque. The end plates designed as shown in Figure 4 exerted the steady force against both sides of the rolls so that the torque by the end plates was independent of the nip clearance. Since the torque Tois the torque exerted when fluid exists only between rolls and end plates, To was measured after the bank fluid was all scraped off. The value of Tois considered to be a function of the roll speed N and the viscosity p and then may be expressed as
To = ENapB
(19)
where E is a constant which depends on only the force pressing the end plates against the roll. The values of To thus measured with various viscosities and roll speeds are shown in Figures 5 and 6. Figure 6 shows that To is independent of the nip clearance as expected. From the slopes of Figures 5 and 6, To is expressed as To = EN0.29P 056 ' (20) where the force pressing the end plates was kept at a given value. All the torque data shown in the following discussion are net torques in which Towas subtracted from measured gross torques. Total Power Consumption of Two Rolls with Equal Roll Speeds. Firstly, the total power consumed by two rolls rotating at the same speed was obtained. The following form is suggested for the total power by equating Nr = 1 in eq 8 in which B is a function of HB/Ho alone. Np2=
55.8B(Ho/D)1/2R,2-1
(21)
1008
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
51 1
1'
' ' ' t ' l l l
10
'
"
1 1 ' 1 1 1
1 oz
'
I
' ' I I j I I
1o3
4
z"
HE/ Ha
Figure 7. Dependence of total power of two rolls with equal roll speeds on H B / H o
101
I
I
I
10
I
I 50
I I I I I
I
I
i
100
c
i
500
t4/H.
Figure 9. Plots of power of each roll for determining the constant K , in eq 24.
40
g?30
p"2 z"
20 10
0
Figure 8. Correlation of the total power consumption at equal roll speeds.
In order to determine the function form of B, the experimental data are plotted against HB/Ho in Figure 7 together with the theoretical curve calculated from eq 8. The parameter N,, ranged over 500-6900 in this experiment. Since the calculated value changes with N , for N,, > 50 only slightly, the calculation in Figure 7 is carried out for N , = 500. The experimental dependence of Np2on HB/Ho agrees well with the present theory. The power consumption for calendering and symmetric flow are also calculated to compare with the experimental data. In the range of HB/Ho < 10, the power calculated for calendering and symmetric flow tends to deviate remarkably. It is found that the function B is proportional to (He/HO)O.l4 for HB/Ho> 10. Using this relationship, the power data of two rolls are plotted in Figure 8. All data can be well correlated by the correlation
Np2 = 36.8(Ho/o)1/2(HB/Ho)o'14R~z-0'~ (22) Equation 22 can be used to predict the total power consumption for two rolls rotating at the same speed over a wide range of Re2. The value of -0.90 in the exponent of the Reynolds number is 10% smaller than the value of -1 predicted in the theory, but it improves the fit to the present data considerably. In the present case of the roll mill, the deviation in the exponent may be caused by the complicated behavior of the free surfaces existing at the bank and the film splitting region as shown in Figure 2. Maus et al. (1955) proposed the empirical equation of the power in a two-roll mill. The end plates were not used in their experiment for the apron nip and their equation for the apron nip can be compared with the present data. Their equation can be rearranged for equal speed system as
in the SI base unit. For comparing this equation with the present data, it is assumed that HB/ Ho = 10 since the bank width is small in the case of the apron nip. The value of eq 23 ranges within two broken lines in Figure 8 when the present experimental conditions are substituted into eq
Figure 10. Correlation of the individual power consumption of each roll.
23. It is found that eq 23 predicts the comparable order of magnitude of the present data. Power Consumption of Each Roll. The power consumption of each roll is obtained as eq 5 and can be rearranged as follows so that it is reduced to the degenerate form eq 22 for Nr* = 1, N, =
Kz2
1
(24)
II
where K1 and K3 are constants to be determined and K2 must be equal to 36.8. The power consumption data of each roll are plotted in Figure 9 to determine the constant K3. The value of K,, i.e., the slope of a straight line in Figure 9 is found to be -0.14. The power data are plotted again in Figure 10 to determine the constant K1. The intercept of a straight line in Figure 10, i.e., K2/2, is equal to 18.4 as expected. The slope of the straight line leads to K1 = 61.0. Thus, the power consumption of each roll can be well correlated by N,, =
[
(Nr* - 1 ) Nr* + 1
(H0/D)1/2Re-0.90 61.0 -
(2)"'"]
+ 18.4
-
(25) where the roll speed ratio Nr* is N,IN, for the faster roll and N,/N, for the slower roll, respectively, as mentioned previously. Total Power Consumption of Two Rolls with Unequal Roll Speeds. Now, the total power consumption of two rolls rotating at different speeds can be obtained from eq 25 and is also derived as a similar formula to eq 8 as Np2
=
[
(Nr-l)l (H0/~)1/2Re2-O~90 122 Nr + 1
+ 36.8]4I):(
-
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
Figure 11.
Correlation
of the total power consumption at unequal
r o l l speeds.
Since the value of the exponent in the Reynolds number in eq 25 is not -1, the total power of eq 26 is a little different from the value calculated by eq 25. The deviation increases with increasing Nr but it is within only 6% for Nr C 7. So, the data of the total power consumption in this experiment are approximately correlated by eq 26 as shown in Figure 11.
Conclusions The power consumption of a pair of equal-sized rolls of Newtonian fluids was measured with various roll speed ratios over a wide range of the Reynolds number. A set of the dimensionless parameters obtained from the lubrication model analysis are used to correlate the results. Equations 26 and 25 are obtained for the total power consumption for two rolls and the individual power consumption of each roll, respectively. Theoretical dependence of the power on the bank width in which the nonsymmetric pressure distribution is considered agrees well with the experimental dependence, but the value of the exponent of the Reynolds number in these equation is 10% smaller than the value of -1 predicted in the theory. Nomenclature D = roll diameter, m E = proportional constant in eq 19 and 20 HB = bank width, m Ho= minimum nip clearance, m hf = faster roll film thickness, m h, = slower roll film thickness, m ho = half of the minimum nip clearance, m L = roll length, m N = roll speed, l / s N,, = modified capillary number ( p u m ( R / h 0 ) ' / ' / ~ ) N , = averaged roll speed ((Nf+ N , ) / 2 ) , l / s Np = individual power number of each roll (P/pN2NmD4L)
1009
Np2= total power number of two rolls ((Pf+ P,)/pNm3D4L) Nr = roll speed ratio (Nf/N,) Nr* = roll speed ratio ( N f / N ,for faster roll, N , / N f for slower roll) P = power, J/s p = pressure, Pa R = roll radius, m Re = individual Reynolds number (pNDHo/k) Re2 = mean Reynolds number (pNmDH0/~) r = radius of curvature at separation, m T = torque, N m To = torque exerted by end plate, N m U = 5 component of dimensionless velocity u = x component of velocity, m/s uf = linear faster roll speed, m/s u, = averaged linear roll speed, m/s us = linear slower roll speed, m/s x = primary flow coordinate, m y = cross-flow coordinate, m Greek Letters = surface tension, N/m
y
9 = dimensionless cross-flow coordinate 8 = temperature of fluid, "C h = dimensionless flow rate hf = dimensionless faster roll film thickness A, = dimensionless slower roll film thickness p
= viscosity of fluid, Pa s
7~
= dimensionless pressure
E,
= stagnation point
[ = dimensionless primary flow coordinate E , = equilibrium point where pressure becomes zero
to= attaching point between the bank and the roll
p = C$
density of fluid, kg/m3
= bank contact angle, deg
Subscripts f = faster roll s = slower roll
Literature Cited Banks, W. H.; MIII, C. C. Roc. R . SOC.Lonson, Ssr. A 1954, 223, 13, 414. Benkreira, H.; Edwards, M. F.; Wllklnson, W. L. Che" Eng. Sci. 1981, 36, 423. Gaskell, R. E. J . A@. M8Ch. 1950, Sept, 334. Greener, J.; Middleman, S. Ind. Eng. Chem. Fundam. 1970, 18, 35. Hummel, C. J . Oil Colour Chem. Assoc. 1956, 39, 777. Maus, L., Jr.; Walker, W. C.; Zettlemoyer, A. C. Ind. Eng. Chem. 1955, 4 7 , 696. Murakami, Y.; Hirose, T.; Chung, 0. Shikizai Kyokdshi 1983, 56, 590. Myers, R. R.; Hoffman, R. D. Trans. SOC.Rheol. 1961, 5 , 317. Patton, T. C. "Paint Flow and Pigment Dispersion"; Wiley: New York, 1964. Schneider, G. 8. Trans. SOC. Rheol. 1962, 6, 209. Zettlemoyer, A. C.; Taylor, J. H. Off. Dig., Fed. SOC.Paint Techno/. 1960, 32,648.
Receiued for review October 6, 1983 Revised manuscript received October 30, 1984 Accepted December 11, 1984
