428
Ind. Eng. Chem. Res. 1994,33, 428-435
Newtonian Viscosity of High Solids Kraft Black Liquors: Effects of Temperature and Solids Concentrations Abbas A. Zaman and Arthur L. Fricke' Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611
The Newtonian (zero shear rate) viscosities of four different softwood kraft black liquors from a four variable-two level factorially designed experiment for pulping slash pine were determined for solids concentrations up to 84% and temperatures up to 140"C (413.2K). Methods of measurement and estimation of zero shear rate viscosities from viscosity-shear rate data have been described and compared. The combination of the absolute reaction rates and free-volume concepts were used to express the relationship between the Newtonian viscosity and temperature. Attempts were made to obtain a generalized correlation for Newtonian viscosity as a function of temperature and solids concentrations. The results of this model and results of our previous empirical correlation have been compared and discussed.
Introduction Kraft black liquors are the by-products of chemical pulping in the pulp and paper industry. In kraft pulping, wood is chemically delignified by using a solution of sodium hydroxide and sodium sulfide. The products of the delignification are cellulose fibers and black liquor. Black liquor contains inorganic salts, extracted light organic compounds, carbohydrate derivatives, and organic constituents in the form of polymeric lignin. Interest in the rheological properties of black liquors comes from the fact that, in the pulp and paper industry, liquors are concentrated and burned to recover the inorganic chemicals as well as energy from the combustion of the organic compounds. In order to make the recovery process more energy-efficient,there is a continuous trend toward firing blackliquor at higher solids concentrations in the recovery furnace which leads to handling black liquors a t high viscosity. If the solids contents for firing were increased from 65% to 80% before combustion a t a steam economy of 411,the energy savings would be about 760 X lo9 J/day for a typical 1000 ton/day mill (Fricke, 1987). At a firing temperature of 130-140 O C (403.16-413.16 K) and concentrations up to 80-83 %, it can be expected that the liquors show Newtonian behavior up to shear rates of about 2000 s-1 (Zaman and Fricke, 1991). Therefore, knowledge of black liquor viscosity over a wide range of temperature and concentration is essential for improvement in design and operation of kraft recovery systems such as evaporators, concentrators, and recovery furnaces. Viscosity has an important effect on heat transfer during evaporation, on the capacity of the pumps, and on the size of the droplets in the recovery boiler system. Yet, to date, very little work has been done to develop appropriate correlations for Newtonian viscosity of black liquors. The purpose of work described here is (1)to study methods of measurement and estimation of the Newtonian viscosity of concentrated (>50%) black liquors and (2) to evaluate the utility of a fundamentally based model for correlating the zero shear rate viscosity of different black liquors as a function of temperature and concentration of nonvolatile components. These correlations can be used to estimate the Newtonian viscosity of the mill black liquors with the same experimental conditions at different temperatures and concentrations. Background Viscosity data for black liquors a t high solids concentrations (250%) have been reported in several papers,
but most of these data are restricted to temperatures below 100 "C (373 K) and concentrations below 65% solids. However, there are some data above these limits. Kim et al. (1981),Sandquist (1981),SBderhjelm (1986),Small (19841,Wight (19851,Stevens (19871,and Fricke (1985, 1987,1990)have reported viscosity data for concentrated (up to 77% solids) black liquors at different temperatures (up to 393 K) and shear rates. The most extensive data have been reported by Zaman and Fricke (1991)and Zaman (1993)for temperatures up to 140 "C (413K)and solids concentrations approaching 85%. They suggested an empirical method to correlate Newtonian viscosity data with temperature and solids concentrations. Black liquor a t low solids concentrations is a complex solution of lignin, low molecular weight organic compounds, and inorganic salts in water (Fricke, 1985,1987, 1990)with polymeric lignin as the main constituent (up to 50% by weight of the organic compounds). At high solids concentrations, black liquors behave as a polymer continuous material (water dissolved in solids) (Masse et al., 1986) and theories for polymer melts or plasticized polymers can be applied to concentrated black liquors. The lignin concentration and ita molecular weight is believed to have the main effect on the viscosity of black liquor. It has been reported that there was a big drop in the viscosity when the high molecular weight portion of lignin was removed by ultrafiltration from several black liquors at the same solids concentrations (SMerhjlem, 1986). Also, it is known that the viscosity of black liquors varies with the degree of delignifkation of the wood species (Wight, 1985;Fricke, 1987),which is due to the fact that the molecular weight distribution of the lignin is affected by the liquor composition which will be determined by the wood species and cooking conditions (Milanova and Dorris, 1989;Fricke, 1987). At high solids (>50%), black liquors can exhibit non-Newtonian behavior. In general, liquors behave as shear-thinning fluids. The degree of shear thinning increases with decreasing temperature or increasing solids concentrations. However, black liquors will show Newtonian behavior depending upon the solids concentrations, solids composition, and temperature (e.g., Zaman and Fricke, 1991).
Determination of the Newtonian (Zero Shear Rate) Viscosity Zero shear rate viscosities are of particular importance in normalizing the data and applying superposition
oaaa-5a8519412633-0428~04.5010 0 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 2, 1994 429 principles. For non-Newtonian fluids at sufficiently low shear rates, one can expect that the variation in structure will become insignificant, so that the dependency of viscosity on shear rate will disappear, unless the material has a yield stress (Dealy, 1982). The value of viscosity in this range is called the zero shear rate viscosity, which is equal to the shear rate independent viscosity for Newtonian fluids. In this study, Newtonian viscosities were determined through the use of a parallel plate viscometer (for 40 "C I T I 85 "C and % solids 1 65), an open cup coaxial cylinder viscometer (for 40 "C I T I90 O C and % solids I 65), and a pressure cell coaxial cylinder viscometer (for T 1 100 "C and % solids < 75%). In experimental work, however, it is not always possible to reach the very low shear rate regions of Newtonian flow of highly concentrated black liquors. In some cases, experimental data on viscosity versus shear rate must be extrapolated to zero shear rates (or shear stresses) in order to determine the zero shear rate viscosities.
system of this type is an appropriate instrument in cases where evaporation is a major problem. In a "couette" type viscometer the outer cylinder of radius R, is fixed, while the inner one of radius Ri rotates at an angular velocity w. The applied torque, T, which is required to turn the rotating cylinder or to hold the stationary cylinder in place, is measured. The basic assumptions for mathematical analysis of coaxial cylinder viscometer to yield the fundamental equation are (1)steady-state laminar flow, (2) incompressible fluid, (3) constant temperature, (4) negligible inertial effects, (5)no slip boundary conditions, and (6) negligible end effects. Using a cylindrical coordinate system, the velocity of the fluid can be assumed to be in the &direction only (Collyer and Clegg, 1988). Thus
V = (O,rd(r),O) (4) where d is the angular velocity of the fluid. The shear rate and the corresponding shear stress can be defined as
-du= r - =dd dr dr
Parallel Plate Viscometer The zero shear rate viscosity, t o , can be determined directly by measurement at very low shear rates by using a parallel plate viscometer. For black liquors, this system can be used at temperatures below 85 "C, where evaporation is not a major problem. However, this method is difficult and time consuming, especially for highly concentrated liquors. A Rheometrics RMS-800mechanical spectrometer with a solutions chamber was used in this study for parallel plate measurements. In this study 1.25- and 2.5-cmdiameter plates were used with gap heights of 0.05-0.15 cm, depending upon the solids concentration and temperature of measurement. In this work, the viscometer was operated with plate rotation at constant speed. For small gap height, H, the twisting component of the shear rate tensor, Yze,is a linear function of the radial position, r, as
ize = rwIH
dd dlog,r
(5)
T = T/2nr2L, (6) where Le is the effective length of the rotor and will be discussed later. The functional relationship between the shear stress and shear rate can be expressed as (Krieger and Elrod, 1953)
= duldr = g ( T ) Equations 5-7 can be combined to yield
(7)
2 dd g(T) = - (8) d log, T Equation 8 can be integrated, considering the following boundary conditions, to give at r = Ri:
T = T ~
and d = o
at r = R,:
T=T,
and d = O
(1)
The viscosity can be determined from
YR
where T = torque, R = plate radius, = shear rate a t the disk rim, and w = angular velocity. The shear rate at the rim, TR,is given by YR
= Rw/H
(3)
Details of derivation of these relations have been given by Bird et al. (1978). The derivative on the right side of eq 2 is the non-Newtonian correction to the stress (Connelly and Greener, 1985) and is the same as WeissenbergMooney-Rabinowitch correction in capillary flow (Bird et al., 1987). For Newtonian fluids this term is unity, but for shear-thinning and shear-thickening fluids, it will be smaller or greater than unity, respectively. Equation 2 is valid in the limit of negligible inertia effects and isothermal conditions.
Coaxial Cylinder Viscometer A Haake RV-12 viscometer with normal open cup and a custom-built closed cell coaxial cylinder unit were used in this work. A coaxial cylinder viscometer is generally used for shear viscosity measurements of fluids in flows which approximate a simple shearing motion. A closed
where e = RdRi. When eq 9 is differentiated with respect , followingdifference equation is obtained (Krieger to ~ ithe and Maron, 1952).
The above difference equation has been solved by various investigators (e.g., Krieger and Elrod, 1953; Krieger and Maron, 1954; Krieger, 1968; Code and Rad, 1973; Yang and Krieger, 1978)using different assumptions. However, in this work the following equation was used to calculate the shear rate, T(T~),a t the wall of the inner cylinder. The accuracy of this equation is better than 1% for values of e up to 1.75 (Calderbank and Moo-Yang, 1959)
where N is the rotational speed and ~i is the shear stress at the wall of the rotor and the viscosity will be equal to
430 Ind. Eng. Chem. Res., Vol. 33, No. 2, 1994 TI(+) =
T~/Y(TJ
-20 1
(12)
L
For a Newtonian fluid, it can be shown that (d log, N)/ (d log, ~ i = ) 1; therefore eq 11 can be written as
In the concentric cylinder viscometer,the equations have been derived on the basis of the infinite length of the cylinder (rotor) or no axial variation in the flow pattern. Since in practice the rotor has finite length, at the ends this cannot be the case. End effects may be expressed as the torque due to the cylinder ends and can be corrected by using an effective length, Le,instead of actual length of the rotor. In this work, Newtonian fluids of known viscosity were used to determine Le,as suggested by Highgate and Whorlow (1969). However, there is no satisfactory way to completely eliminate end effects for non-Newtonian fluids.
Zero shear rate viscosities cannot always be measured directly, and must be estimated by extrapolation in some cases. In order to estimate the zero shear rate viscosity, experimental data on viscosity versus shear rate must be extrapolated to zero shear rate (or shear stresses). Kataoka and Ueda (1968) have summarized several methods that can be used to estimate the zero shear rate viscosities. Methods used in this work are the fo.llowing: 1. Use plots of 1/11as a function of y2/3,which is based on the equation derived by Cross:
where 7- (Pad is the viscosity at infinite shear rate, (s-l) is the shear rate, and a is a constant. 2. When the data from a capillary viscometer are available, log(Q/?rR3~,)versus T,, where Q (m3/s) = flow rate, R = capillary radius (m) will be a straight line at low shear rates. For this range we can assume (Spencer and Dillon, 1948, 1949)
T,
:
-40
:
g
-50
:
e
-60
0
2
* ' A
A
0
0
-70 :
A
A
A
'*
A
ABAFX011.12
0
ABAFX013,14
A
ABAFX025.26
0
50
ABAFX043,44
60
70
90
60
% Solids Figure 1. Glass transition temperature as a function of solids concentration for different black liquors. Table 1. Comparison between Calculated and Measured Newtonian Viscosity of Different Black Liquors at Different Temperatures
Estimation of Zero Shear Rate Viscosities
L=& 4% irR37,
.30
(15)
(N/m2)is the shear stress at the wall of the capillary
andisequalto((R/Z)(AP/L)),whereAP(N/m2)=pressure drop along the capillary and L (m) = length of the capillary. Therefore, by plotting log(Q/irR3~,)as a function of T,, and extrapolating to zero T , (equivalent to very low shear rates), the value of (Q/7rR37,) appropriate for eq 15 can be determined and 70 estimated. The results using the two methods for liquors in our work do agree; however, method 1gives straight lines over a wider range of shear rate and this method appears to be better. Some of the results have been summarized in Table 1. To check the method, estimates of zero shear rate viscosity determined from capillary data were compared with results of direct measurement in some cases. The estimates agreed with direct measurements to within f8% or less. Glass Transition Temperature of Kraft Black Liquors Because of the particular importance of the glass transition temperature, TG,in modeling the zero shear
liquor ABAFX013,14 ABAFX013,14 ABAFXOl3,14 ABAFX013,14 ABAFX025,26 ABAFX025,26 ABAFX025,26 ABAFX043,44 ABAFX043,44 ABAFX043,44 ABAFX043.44
7% solids 76.01 76.01 76.01 76.01 81.05 81.05 81.05 76.28 76.28 76.28 76.28
temp, K 313.16 328.16 343.16 358.16 313.16 328.16 343.16 313.16 328.16 343.16 358.16
llo, Pa.s calcd measd 709664 700060 47883.77 49774.0 2073.12 2068.9 152.07 150.43 39134.51 40033 2104.74 2219 172.38 168.74 323481.4 308060 22879.35 22990 1191.44 1200 121.63 114.68
7% error -1.4 -3.8 0.2 1.1 -2.25 -5.2 2.2 5 -0.5 -0.71 6.0
rate viscosity data, it is worthwhile to discuss this. The freezing point and glass transition phenomena in kraft black liquors were studied by Fricke (1985) and Masse (1984,1986)using differential scanning calorimetry (DSC). They were able to illustrate that black liquor behaves qualitatively in a manner similar to polystyrene-benzene and poly(vinylpyrro1idone)-watersystems; Le., it behaves as a polymer-solvent system with water as the principal solvent (Fricke, 1987). They observed that there exists a particular concentration (-43 5%) at which the phenomenon occurring changes from freezing to glass transition. At lower solids, the black liquor solutions exhibit freezing of the liquor, while at high solids concentrations, the liquor undergoes a glass transition and there is no evidence of freezing. In this work, we were interested in the glass transition temperature of black liquors at solids concentrations higher than 50 % to model our Newtonian viscosity data. Measurements of TGfor four different black liquors were made using a differential scanning calorimeter. Figure 1 shows a composite plot of TG as a function of solids concentrations for all liquors. As could be expected, the glass transition temperature is a function of solids concentrations. There is a small variation in TG from liquor to liquor which is due to the lignin molecular weight and the amounts of the inorganics in the liquor. However, at higher solids concentrations, the differences are more significant. Comparison of the results of this work with those for polymer-solvent systems show that black liquor can be treated as a binary system of water and a compositionally complex solute with lignin as a main constituent. This was also mentioned earlier by Fricke (1985) and Masse (1984, 1986).
Ind. Eng. Chem. Res., Vol. 33, No. 2, 1994 431 ......
100000 A
84.14% Solid
.
A
53.12% Solld
100000
58.33% Solid
10000
-
A 0
65.4% Solid
k.
10
=
1 0.1
0.01 2.40
2.50
2.60
2.70
2.80
2.90
3.00
3.10
3 20
1/T x 1000 K-' Figure 2. Effect of temperature on zero shear rate viscosity of black liquor ABAFX011,12.
0
59.08% Solid
A
67.2% Solid
10000
g
1000
2
100
p.
10
2.30
2.50
2.70
2.90
3.10
3.30
1/T x 1000 K-' Figure 5. Effect of temperature on zero shear rate viscosity of black liquor ABAFX043,44.
oriented unit can move. Using this theory, the Newtonian viscosity of liquids and polymer solutions can be described as
70.05% Solid
cn
4
76.01% Solid
1
0.1 0.01 2.40
2.50
2.60
2.70
2.80
2.90
3.00
3.10
3.20
1/T x 1000 K-I Figure 3. Effect of temperature on zero shear rate viscosity of black liquor ABAFX013J4.
To = A exp(EIRT) (17) where R is the gas constant and E is the activation energy for flow and can be defined as
71.98% solid 0
where T is the absolute temperature and TO is the temperature at which the extrapolated conformational energy and free volume of the liquid become zero (Macedo and Litovitz, 1965). Parameter A is a constant, and parameter B is related to the internal barriers to rotation of the main chain bonds in the polymer molecule (Miller, 1963). Wight et al. (1981) and Fricke (1985, 1987) have expressed the viscosity data for black liquors using the theory of absolute reaction rates or Arrhenius relationships. This equation was derived theoretically by the application of the absolute rate processes to viscous flow and can be written as
75.7% solid
c 0.1 0.01
0 0 0 1 ~ " ' ~ ' " ' " ' " ~ " " " " ~ ' ~ 2.40 2.58 2.72 2.88
3.04
3.20
1/T x 1000 K-I Figure 4. Effect of temperature on zero shear rate viscosity of black liquor ABAFX025,26.
Influence of Temperature on Zero Shear Rate Viscosity The Newtonian viscositiesof four different blackliquors were determined at different solids concentrations using methods which were discussed earlier. The results are presented in Figures 2-5. Several viscosity-temperature relationships can be found in the literature for black liquors at high solids concentrations. Wennberg (1985) used the free-volume theory to model the viscosity data for black liquors. The free-volume approach takes into account the probability that there is an adjacent empty site into which a properly
Equation 17 predicts that a plot of log, qo as a function of 1/T should yield a straight line of slope EIR. Some investigators (e.g., Wennberg, 1985; Soderhjelm, 1986)have used empirical correlations to express viscositytemperature relationships for concentrated black liquors. Most of them have used Tx instead of T in eq 17 which, however, does not have a theoretical basis. From plots of log qo versus 1/T, it can be observed that, over narrow ranges of temperature, the plots are linear, and it seems that either eq 16 or 17 can be used to fit the experimental data. However, over wide ranges of temperature, the plots are nonlinear, and both equations fail to express the viscosity-temperature relationship. For this reason, many workers have used a combination of the absolute reaction rates and free-volumeconcepts to express the Newtonian viscosity of liquids and polymer solutions. The expression used with success for these polymer solutions and melts is
where B = EJR, E, is the activation energy for flow at constant volume, and A and C are constants.
432 Ind. Eng. Chem. Res., Vol. 33, No. 2, 1994 Table 2. Coefficients of Eq 19 for Different Black Liquors at Different Solids Concentrations liquor
% solids
ABAFXOl1,12 ABAFXO11,12 ABAFXOllJ2 ABAFXOl1,12 ABAFXOl 1,12 ABAFX013J4 ABAFX013,14 ABAFX013,14 ABAFX013,14 ABAFX025,26 ABAFX025,26 ABAFX025,26 ABAFX025,26 ABAFX025,26 ABAFX043,44 ABAFX043,44 ABAFX043,44 ABAFX043,44
55.84 63.49 72.67 75.36 84.14 55.22 59.10 67.20 70.05 57.68 64.85 71.98 75.70 81.05 58.33 65.40 72.89 76.28
B 2.526 X le7 2436.19 2.810 X 10-9 4233.92 4.101 X 10057.0 4.380 X lo-" 10154.0 1.840 X 12549.0 560.29 3.188 X 1O-g 4.158 X lo-' 1375.1 9.311 X lk13 8001.56 7.70 X 8788.95 6.70 X 1O-g 4210.123 8.70 X 10-g 4246.69 4.98 X 7732.1 8.265 X 7973.1 2.152 X 11891.0 1.877 X leB 4256.41 1.10 X 11044.0 5.44 X 14425.0 3.45 X 16701.0 A
C 0.825 1.188 0.1931 0.220 0.0102 2.202 2.40 1.161 0.896 0.2782 0.530 0.465 0.484 0.2475 1.369 0.446 0.0879 0.0882
2.00e+04
R2 0.998 0.999 0.999 1.0 1.0 1.0 1.0 1.0 0.999 1.0 1.0 1.0 0.999 1.0 0.999 1.0 1.0 1.0
The above equation was used to fit the zero shear rate viscosity data of our work as a function of temperature for different black liquors at high solids concentrations. TO is an adjustable parameter and is related to the glass transition temperature of the material (Ferry, 1980). The data were fitted with eq 19 using SAS statistics nonlinear data fitting. Two important factors must be considered regarding the values of B, which are related to the activation energy for flow: (1)B must be a positive number, as can be observed from the slopes of log 70as a function of 1/T plots, and (2) B should increase as the solids concentration of a particular black liquor increases. Considering these two factors, we varied TOas a function of TG for each liquor at different solids concentrations ( TGfor every liquor was determined as a function of solids concentrations). The best fits were obtained at TO N 1.32'~. The resultingvalues of A , B, and C are summarized in Table 2. As can be observed, R2 1 0.998 for all the liquors and the values of B, which are related to the activation energyfor flow, are positive and are an increasing function of the solids concentrations. The constants for each liquor are concentration dependent, and we tried to determine consistent relationships for constants as a function of solids concentrations. These results will be discussed later.
Results and Discussion The model based on the combination of the absolute rate and free-volume theories developed in this work is a four-constant model, A , B, C, and TOwith B related to the activation energy for flow and TOrelated to the glass transition temperature. The model yields highly accurate results and can be used to correlate the Newtonianviscosity of the concentrated black liquors at different temperatures at a fixed concentration. The constants are dependent upon the solids concentration, S. For different solids concentrations, it was found that A = exp(a, + b,SJ (20) for all liquors of one species at all concentrations. For the slash pine liquors used in this study (except liquor ABAFX043,44 at 72.89 and 76.28% solids), A may be written as
A = exp{11.824- 52.45683 (21) The value of A can be replaced in eq 19, and values of B and C can be redetermined for each liquor at different solids concentration.
.
ABAFX011,lZ
. . 1,00e+04 -
ABAFX013.14
0
:/ ABAFX043,44
*
0.00e+00
ABAFX025,26
A
.
liauor ABAFXOl1,12 ABAFXOl1,12 ABAFXOllJ2 ABAFX011,lP ABAFXO11,lP ABAFX013,14 ABAFX013J4 ABAFX013J4 ABAFXOl3,14 ABAFX025,26 ABAFX025,26 ABAFX025,26 ABAFX025,26 ABAFX025,26 ABAFX043,44 ABAFX043,44 ABAFX043,44 ABAFX043,44
O
'
'
"
.
'
.
'
'
'
'
'
'
'
% solids
B
C
55.84 63.49 72.67 75.36 84.14 55.22 59.10 67.20 70.05 57.68 64.85 71.98 75.70 81.05 58.33 65.40 72.89 76.28
3492.654 5054.503 8192.80 8913.41 11966.0 2688.134 3444.96 6131.19 7522.10 4056.74 5868.20 7699.20 8950.34 10680.0 3627.04 5526.45 8908.83 10250.0
0.5931 1.053 0.3312 0.2862 0.0117 1.676 1.970 1.396 1.017 0.298 0.30 0.468 0.404 0.3023 1.51 1.364 0.451 0.3495
'
'
R2 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
The new values of B, C, and R2for each liquor have been summarized in Table 3. Figures 6 and 7 represent B and C as a function of solids concentration for the slash pine liquors. As can be observed, B for each liquor may be written as
B = a2 + b2S
(22)
and the constant, C, for a single liquor can be fitted with a polynomial. Figure 6 shows that the values of the constant B at specificsolids concentrations for all of the slash pine liquors are very nearly equal, and one may assume that the dependency of B on the solids composition is negligible. Figure 7 shows that the constant C is a strong function of the solids concentration and solids composition. C can be fitted as a parabola, except for ABAFX025,26 for which C is nearly constant. Although complex, the model is sound and results are qualitatively in agreement with expectations. A t a given temperature, one would except the constant A to be dependent upon the polymer type and concentration. Since the type for liquors from one wood is the same, A should be a function of solids concentration only and should be the same function for all liquors, as is the case. Similarly, one should expect the activation energy for flow and hence the constant B to be only a function of the solids concentration for liquors from one wood, as is the case. The constant C for the free volume should be a function of the solids composition, and should vary for liquors from one wood, as is the case. It appears that the constant C
I I
2.50
t
-
2.00
A
ABAFXOll.12
0
ABAFX013.14
A
ABAFX025.26
1.00
-
0.50
-
100000
10000 e,
?
100
k*
ABAFX043,44
u
Ind. Eng. Chem. Res., Vol. 33, No. 2,1994 433 I /I
1000
-
1.50
1000000
10
0
=
1 0.1 0.01
A
0.80 50
70
60
80
0.89
0.97
1.06
1.15
1.24
1.32
1.41
1.50
90
% Solids Figure 7. C as a function of solids concentration for different black liquors with A considered as a universal constant.
S/(S+l) x 1/T x 1000 K-I Figure 8. Reduced plot for zero shear rate viscosity of black liquor ABAFX013,14. 1000000
Used in Eq 24 for Table 4. Values of go, gl,gr,and Fitting Zero Shear Rate Viscosities black liauor ABAFXO11,lPa ABAFX013,14 ABAFX025,26 ABAFX043,44
Pn
PI
Pz
6.313 22.125 12.802 9.572
-3.668 X lo4 -7.4027 X 104 -5.0235 X lo4 -4.8376 X 104
2.726 X lo7 4.961 X lo7 3.398 X lo7 3.667 X lo7
A
53.08% solid
100000
R2 0.998 0.998 0.997 0.998
a These values for liquor ABAFX011,12 are from our earlier work (Zaman and Fricke, 1991).
can be fitted as a parabola. One should note that C is nearly a constant for a liquor containing a high concentration of inorganics and a low concentration of lignin in the solids. Although complex, this model offers the potential for correlating viscosity for liquors from one wood with much less effort. Once relations for A and B are determined, C can be defined quite well with measurements for a new liquor at five to six conditions.
0
57.68% solid
A
64.85% solid
10000 1000 100
71.98% solid 0
10
*
75.7% solld 81.05% solid
0
=
1 0.1 0.01
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
S/(S+l) x 1/T x 1000 K-1 Figure 9. Reduced plot for zero shear rate viscosity of black liquor ABAFX025,26.
,
1000000 I
I
I
Comparison with Previous Work An empirical correlation was suggested in our earlier work (Zaman and Fricke, 1991)to correlate the zero shear rate viscosities to temperature and solids concentrations, which was
where S = mass fraction solids, T = temperature (K), and gi = empirical constants. In order to investigate the validity of this correlation for other black liquors, in this work, the zero shear rate viscosities for four different black liquors were fitted by
The values of go, gl,gz, and R2for each liquor are listed in Table 4. Figures 8-10 represent the zero shear rate viscosities of these liquors as a function of the reduction variable
The liquors used in this study are from a 24 factorial designed pulping experiment. The cooks were done at a liquor-to-wood ration of 411. The cooking conditions and kappa numbers for the liquors used in this study are summarized in Table 5. The liquors ABAFXO11,lB and
0.80
0.89
0.87
1.06
1.15
1.24
1.32
1.41
1.50
S/(S+l) x 1/T x 1000 K-' Figure 10. Reduced plot for zero shear rate viscosity of black liquor ABAFX043,44.
Table 5. Pulping Conditions and Kappa Numbers for Black Liquors Used in This Study cookno. ABAFXO11,lS ABAFX013,14 ABAFX025,26 ABAFX043,44
cooking time,h 0.6667 1.3333 1.3333 1.OOO
T, K 438.7 450.0 450.0 444.3
eff alkali, % 13 13 16 14.5
sulfidity, kappa % no. 20.0 107.0 20.0 40.2 35.0 18.5 27.5 51.1
ABAFX025,26 are extremes and the liquor ABAFX043,44 is the center point of the design. However, it can be expected that eq 24 will fit the experimental data for other liquors. Figure 11is a composite plot of zero shear rate viscosities as a function of the reduction variable.
434
Ind. Eng.Chem. Res., Vol. 33, No. 2, 1994
1000000 100000
-
0.80
ABAFX013.14
0.90
1.00
1.10
S/(S+l)
x
1/T
1.20
x
1.30
1.40
1.50
1000 K-'
Figure 11. Reduced plot for zero shear rate Viscosity for different black liquors. Table 6. Zero Shear Rate Viscosities for Liquors ABAFX013,14 and ABAFXO26,26 mass fraction solids .'2 K
Conclusions
70, Pa.8
calcd bv ea 19 calcd bv ea 24
measd
(a) Liquor ABAFX013,14
0.591 0.591 0.591 0.591 0.591 0.591 0.591 0.591 0.700 0.700 0.700 0.700 0.757 0.757 0.757 0.757 0.757 0.757 0.810 0.810 0.810 0.810 0.810
19.28 21.12 323.16 333.16 4.13 3.90 1.23 1.12 343.16 0.39 353.16 0.38 0.13 368.16 0.14 0.05 383.16 0.05 0.03 0.03 393.16 403.16 0.02 0.02 35821.00 313.16 32718.53 95.00 343.16 77.27 13.00 358.16 13.44 2.80 2.37 373.16 (b) Liquor ABAFX025,26 3060.68 313.16 3478.00 200.30 328.16 179.70 25.91 343.16 25.31 358.16 5.48 5.10 0.28 393.16 0.35 0.15 403.16 0.18 33056.11 313.16 35358.93 156.49 343.16 155.02 28.30 358.16 29.11 6.08 373.16 7.02 1.36 1.26 393.16
a single liquor are affected not only by the lignin molecular weight and lignin concentration, but also by other organic and inorganic components. In order to compare the results of eq 19 and eq 24, B and C for liquors ABAFX013,14 and ABAFX025,26 were fitted with polynomials and replaced in eq 19. Then the zero shear rate viscosities were calculated at different temperatures and solids concentrations using eqs 19 and 24 for these two liquors. The results are given in Table 6. Comparison of the results shows that both methods can be used to estimate the Newtonian viscosity of the liquors within f20%; however, eq 24 is easier to use and the degree of its accuracy can be increased by using higher order polynomials. Equation 19 gives highly accurate results (f5% error) for a single liquor at a specific concentration over a wide range of temperature without using ita generalized approach. The generalized form, however, affords the advantage of evaluating a new liquor with very limited experimental effort.
23.81 4.21 1.10 0.35 0.12 0.05 0.03 0.02 33470.00 86.55 13.80 2.13 3478.10 179.66 25.31 5.50 0.35 0.18 40033.0 168.44 33.10 6.60 1.56
The zero shear rate viscosities of liquors ABAFX013,14 and ABAFX043,44 are considerably higher than those of liquors ABAFXO11,lS and ABAFX025,26. The liquors can be categorized in two groups, A and B. Group A consist of liquors ABAFXO11,lS and ABAFX025,26, and group B consist of liquors ABAFX013,14 and ABAFX043,44. For every group, there is a region where the zero shear rate viscosities are very close together. Below this region, liquor ABAFX043,44 is more viscous than liquors ABAFX013,14, ABAFXO11,12, and ABAFX025,26 respectively. These regions represent lower solids concentrations and higher temperatures. Probably, the viscosity is dominated by the lignin molecular weight in this range. The weight average molecular weight of lignin in liquor ABAFX043,44 is higher than in liquors ABAFX013,014, ABAFXO11,lZ or ABAFX025,26 (Schmidl, 1992). Above this region, liquor ABAFXO13,14 is more viscous than liquors ABAFX043,44, ABAFX025,26, or ABAFX011,12. The lignin concentration in liquor ABAFX013,14is higher than in liquors ABAFX043,44, ABAFX025,26, or ABAFXO11,lP. In this region, the zero shear rate viscositiesare probably dominated by the lignin concentration in the liquors. However, the zero shear rate viscosities for
At high solids concentrations (>50 % 1, black liquor can be treated as a polymer continuous material and theories for polymer melts can be applied to concentrated liquors. The relationship between the Newtonian viscosity and temperature can be expressed by using a combination of the absolute reaction rates and free-volume concepts. This model can be used to extrapolate over wide ranges of temperature at a fixed concentration, and results are highly accurate. Equation 19 can be used to obtain a general correlation for viscosity of a single liquor as a function of temperature and solids concentration, but the method is difficult and time consuming. Further application of our previous empirical model has been investigated, and it appears that this model can be used to express the Newtonian viscosity of all liquors as a function of solids concentration and temperature. The zero shear rate viscosities can be estimated by using this model over wide ranges of temperature and concentration with very good accuracy. This is more practical, less time consuming, and easier to use for a single liquor, although it does not have a theoretical basis. The constants go, gl,gz ...can be correlated empirically with solids composition and pulping conditions for a single species which will require the determination (measurement) of the Newtonian viscosity for all the liquors of single species. As can be observed, the viscosity is not only a function of temperature and solids concentrations, but will vary from liquor to liquor due to differences in the solids composition and lignin molecular weight in different liquors.
Acknowledgment The authors are grateful for the financial support of the
U.S.Department of Energy through Grant No. DE-FG0285CE40740and of a large number of industrial firms, and for the assistance of Mr. M. P. Alazraki and Mr. A. Preston.
Nomenclature A = constant al = constant a2 = constant B = constant bl = constant b2 = constant C = constant E = activation energy for flow E, = activation energy for flow at constant volume
go, gl,and 82
= constants of eq 24
H = gap height L = length of the capillary Le = effective length of the rotor N = rotational speed
Q = volumetric flow rate, (m3/s) R = radius of the plate in eqs 2 and 3and radius of the capillary in eq 16, gas constant elsewhere Ri = radius of the inner cylinder R , = radius of the outer cylinder r = radius S = solid mass fraction T = torque in eqs 2 and 6, absolute temperature elsewhere, K TC = glass transition temperature To= reference absolute temperature, K u = velocity a = constant in eq 14
y = shear rate, s-l
= shear rate at the disk rim, s-l = Ro/Ri 71 = shear viscosity, Pa-s 70 = zero shear rate viscosity, Pa-s 7- = infinite-shear-rate viscosity, Pa-s T = shear stress, Pa ~i = shear stress at the wall of the inner cylinder T, = shear stress at the wall of the outer cylinder T, = shear stress at the wall of t h e capillary, Pa Q = angular velocity of the fluid, rad/s w = 27rN angular velocity, rad/s YR t
Literature Cited Bird, R. B.; Armstrong, R. C.; Hassager, 0. Dynamics of Polymeric Liquids: Fluid Mechanics, 2nd ed.; John Wiley and Sons: New York, 1987; Vol. 1. Calderbank, P. H.; Moo-Yang, M. B. Trans. Znst. Chem. Eng. (London) 1959,37,26-33. Code, R. K.; Raal, J. D. Rates of Shear in Coaxial Cylinder Viscometers. Rheol. Acta 1973,12,578-587. Collyer, A. A,; Clegg, P. W. Rheological Measurements; Elsevier Applied Science: New York, 1988; p 264. Connelly,R.W.; Greener,J. High-ShearViscometrywithaRotational Parallel-Disk Device. J. Rheol. 1985,29 (2), 209-226. Dealy,J. M. RheometersforMoltenPlastics; Van Nostrand Reinhold Company: New York, 1982. Ferry, J. D. Viscoelastic Properties ofPolymers, 3rd ed.; John Wiley and Sons: New York, 1980. Fricke, A. L. Physical Properties of Kraft Black Liquor: Interim Report-Phase ZI;DOE Report on Contract DG AC02-82CEQ0606, 1985. Fricke, A. L. Physical Properties of Kraft Black Liquor: Summary Report-Phase Z and ZI; DOE Report Nos. AC02-82CE40606and FG02-85CE40740,1987. Fricke, A. L. A ComprehensiveProgram to Develop Correlations for the Physical Properties of Kraft Black Liquors; Interim Report No. 2, December, 1990. Highgate, D. J.; Whorlow, R. W. End Effects and Particle Migration Effects in Concentric Cylinders Rheometry. Rheol. Acta 1969,B (2), 142-151. Kataoka, T.; Ueda, S. Flow Properties of Polyethylene Melt. J. Polym. Sci. 1968, 12, 939-953.
Ind. Eng. Chem.Res., Vol. 33, No. 2, 1994 435 Kim, H, K.; Co, A,; Fricke, A. L. Viscosity of Black Liquors by
Capillary Measurements. AIChE Symp. Ser. 1981,77 (No. 207). Krieger, I. M. Shear Rate in the Couette Viscometer. Tram. Soc. Rheol. 1968, 12 (l),5-11. Krieger, I. M.; Maron, S. H. Direct Determination of the Flow Curves of Non-Newtonian Fluids. J. Appl. Phys. 1962,23 (l),147-149. Krieger, I. M.; Elrod, H. Direct Determination of the Flow Curves of Non-Newtonian Fluids. 11. Shearing Rate in the Concentric Cylinder Viscometer. J. Appl. Phys. 1953,24 (2), 134-136. Krieger, I. M.; Maron, S. H. Direct Determination of the Flow Curves of Non-Newtonian Fluids. 111. Standardized Treatment of Viscometric Data. J. Appl. Phys. 1954, 25 (l),72-75. Macedo, P. B.; Litovitz, T. A. On the Relative Roles of Free Volume and Activation Energy in the Viscosity of Liquids. J.Chem. Phys. 1965,24 (l), 245-256. Masse, M. A. Thermal Analyses of Kraft Black Liquor. M.S. Thesis, University of Maine, Orono, 1984. Masse, M. A.; Kiran, E.; Fricke, A. L. Freezing and Glass Transition Phenomena in Polymer-Diluent Mixtures. Polymer 1986, 27 (April), 619-622. Milanova,E.; Dorris, G. M. Effects of Residual Alkali Content on the Viscosity of Kraft Black Liquors. Chem. Recover. 1989,101-118. Miller, A. A. Free Volume and Viscosity of Liquid Effects of Temperature. J. Phys. Chem. 1963,67,1031-1035. Sandquist, K. Rheological Properties of Black Liquor. Pulp Pap. Can. 1981,84 (21, 30-34. Schmidl, G. W. Molecular Weight Characterization and Rheology of Lignins for Carbon Fibers. Ph.D. Dissertation, University of Florida, Gainesville, 1992. Small,J. D. The Thermal Stability of Kraft Black Liquor at Elevated Temperatures. M.S. Thesis, University of Maine, Orono, 1984. Soderhjelm, L. Viscosity of Strong Black Liquor. Pap. Puu 1986,68 (9), 642-652. Spencer, R. S.; Dillon, R. E. The ViscousFlow of Molten Polystyrene. J. Colloid Sci. 1948, 4, 163-180. Spencer, R. S.; Dillon, R. E. The Viscous Flow of Molten Polystyrene 11. J. Colloid Sci. 1949,4, 241-255. Stevens, S. Rheology of Concentrated Kraft Black Liquor. M.S. Thesis, University of Florida, Gainesville, 1987. Wennberg, 0.Boiling Point Elevation and Viscosity of Black Liquor at High Solids Contents and High Temperatures. Znt. Chem. Recover. 1985,275-279. Wight, M. 0. An Investigation of Black Liquor Rheology Versus Pulping Conditions. Ph.D. Dissertation, University of Maine, Orono, 1985. Wight, M. 0.; Co, A.; Fricke, A. L. Viscosity of Black Liquor by Cone-and-Plate and Parallel-Disk Viscosmetry. AIChE Symp. Ser. 1981, 77 (No. 207). Yang, T. M. T.; Krieger, I. M. Comparisonof Methods for Calculating Shear Rates in Coaxial Viscometers. J. Rheol. 1978,22 (4), 413421. Zaman, A. A. An Investigation of the Rheological Properties of High Solids Kraft Black Liquors. Ph.D. Dissertation, University of Florida, Gainesville, 1993. Zaman, A. A.; Fricke, A. L. Viscosity of Black Liquor up to 130 "C and 84% Solids. Forest Products Symposium, AZChE Proceedings; AIChE New York, 1991; pp 59-77.
Received for review July 21, 1993 Revised manuscript received October 22, 1993 Accepted November 15, 1993. Abstract published in Advance ACS Abstracts, January 1, 1994. @
