Ind. Eng. Chem. Res. 1998, 37, 275-283
275
Vapor Pressure and Boiling Point Elevation of Slash Pine Black Liquors: Predictive Models with Statistical Approach A. A. Zaman,*,†,‡ T. W. McNally, and A. L. Fricke† Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611, and Engineering Research Center for Particle Science and Technology, University of Florida, Gainesville, Florida 32611
Vapor-liquid equilibria and boiling point elevation of slash pine kraft black liquors over a wide range of solid concentrations (up to 85% solids) has been studied. The liquors are from a statistically designed pulping experiment for pulping slash pine in a pilot scale digester with four cooking variables of effective alkali, sulfidity, cooking time, and cooking temperature. It was found that boiling point elevation of black liquors is pressure dependent, and this dependency is more significant at higher solids concentrations. The boiling point elevation data at different solids contents (at a fixed pressure) were correlated to the dissolved solids (S/(1 - S)) in black liquor. Due to the solubility limit of some of the salts in black liquor, a change in the slope of the boiling point elevation as a function of the dissolved solids was observed at a concentration of around 65% solids. An empirical method was developed to describe the boiling point elevation of each liquor as a function of pressure and solids mass fraction. The boiling point elevation of slash pine black liquors was correlated quantitatively to the pulping variables, using different statistical procedures. These predictive models can be applied to determine the boiling point rise (and boiling point) of slash pine black liquors at processing conditions from the knowledge of pulping variables. The results are presented, and their utility is discussed. 1. Introduction Knowledge of black liquor equilibrium vapor pressure and boiling point rise over wide ranges of temperature and nonvolatile solids concentration is of particular importance for the development and design of kraft recovery components. These properties vary considerably with the solids concentrations (and solids composition) and, therefore, become more critical at higher solids concentrations as more evaporation stages are required to achieve higher solids concentrations and to increase performance characteristics of the evaporation units. Accurate vapor pressure data and boiling point rise data are necessary for design and operation of evaporators for black liquor concentration to high solids. The present knowledge of the vapor pressure data and boiling point elevation of black liquors is incomplete due to the variation with pulping conditions, black liquor composition, solids concentration, and type of the wood species. Past research has not established methods that can be used to estimate vapor pressure or boiling point rise at processing conditions from knowledge of pulping conditions and/or black liquor composition. In this paper, we report results on equilibrium vapor pressure and boiling point rise-concentration-temperature relations for softwood slash pine black liquors based on data obtained for liquors derived from statistically designed pulping experiments that were conducted in a pilot scale digester with liquor circulation. The cooking variables were effective alkali, sulfidity, cooking temperature, and cooking time, and the cooks were * To whom correspondence should be addressed. Phone: (352) 392-6509. Fax: (352) 392-9513. E-mail: zaman@ eng.ufl.edu. † Department of Chemical Engineering. ‡ Engineering Research Center for Particle Science and Technology.
made at a liquor-to-wood ratio of 4/1. In all cases, the white liquor was adjusted to a causticizing efficiency of 85% and a reduction of 93% with Na2CO3 and Na2SO4. The details of the pulping procedure and experimental results have been reported in several earlier publications (Fricke 1987, 1990, 1993; Zaman et al., 1991). The purpose of this research was to develop a consistent set of vapor pressure and boiling point elevation data for slash pine black liquors. We were interested in the following: (1) the variation of vapor pressure and boiling point rise with increasing solids concentration, especially at high solids content where the solubility limit of some of the components in black liquor is exceeded; (2) the variation of these properties with solids composition and pulping variables; and (3) development of statistically based quantitative models for these variables as a function of the pulping conditions. A series of well-characterized liquors covering a wide range of pulping conditions were used to generate data for vapor pressure properties of black liquors. These data were used to study the effect of pulping variables on the boiling point rise of black liquors and to develop predictive models as a function of the cooking conditions. The results are presented, and their utility is discussed. 2. Overview and Background In an earlier paper (Stoy et al., 1992), we reported the results on the development of an apparatus (smallscale evaporator) that was successfully applied for direct determination of black liquor vapor pressures that were thermodynamically consistent with treatment of black liquor as a pseudobinary system. Boiling point data at different pressures were directly determined with this system over a wide range of concentration up to 85% solids, depending upon the viscosity of the liquor. It was observed that the boiling point elevation is pressure dependent at concentrations above about 20% solids.
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276 Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998
The boiling point rise varies from liquor-to-liquor due to the differences in the composition of the liquors, and this dependency is more significant at higher solids concentrations. Several other methods have been reported in the literature for measuring the vapor pressure of black liquors, and these are briefly reviewed here. Kobe and Sorenson (1939) have reported boiling point data for black liquors up to 63% solids at different pressures. Frederick et al. (1980) reported boiling point elevation data at atmospheric pressure for black liquors at solids concentrations up to about 65% solids. These researchers have used direct measurement techniques in which the liquor is heated continuously, the boiling point is recorded, and the resulting condensate is measured to determine the solids concentration of the remaining liquor in the vessel. The accuracy of their technique is questioned due to the fact that liquor and water vapor were probably not at equilibrium under the conditions used, and superheating of the liquor at higher solids concentrations was reported to be a problem. Some researchers (Arlt and Onken, 1982; Thomas et al., 1982) have tried to modify this technique by circulating the liquor to achieve equilibrium, but the method does not seem to be successful at high solids concentrations where the viscosity is the main limiting factor. Clay and Karnofski (1981) used a pilot scale flash evaporator to determine the boiling point elevation for black liquors dispersed in an immiscible oil. With this technique, they were able to measure the boiling point of black liquors for solids concentrations up to almost 100% solids. Although this method is potentially operable up to dry solids, superheating of the liquid phase appears to be a problem, especially in the case of two immiscible liquids, leading to an overestimate of boiling point elevation. Clay and Grace (1984) used a laboratory evaporator operating at total reflux and atmospheric pressure to determine the boiling point elevation for one black liquor dispersed in oil. All their experiments were conducted at atmospheric pressure, and oils of different volatility were used to alter the partial pressure of water vapor in the liquid-oil systems. Wennberg (1985) determined the boiling point for sulfate liquors at concentrations from 55% to almost 100% solids. In his procedure, the water was evaporated continuously from the liquor, the temperature of the liquor was recorded, and the evaporated water was condensed and measured to determine the solids concentration of the liquor in the evaporator. These results are suspicious due to the following: (1) the fact that the liquor may not be at thermal equilibrium in the absence of total reflux and (2) the liquor phase is superheated, especially at high solids concentrations where the convective heat transfer rate is decreased considerably by the very large increase in viscosity of the liquor at high solids concentrations. Some researchers (Szymonski and Grace, 1985; Robinson and Clay, 1986) have used equilibrium gravimetric techniques to determine the vapor pressure of black liquors. A known mass of black liquor was coated on a brass screen and then suspended over a saturated salt solution at constant temperature and allowed to equilibrate. The equilibrium solids content was determined by a mass balance on the screen after equilibrium had been established and the water vapor pressure was calculated from the known saturation pressure of the salt solution. The results are uncertain due to problems
of temperature control, reaction with the brass screen, and water adsorption by the screen, especially at very high solids. Robinson and Clay (1986) determined the boiling point rise of one black liquor above 85% solids at atmospheric pressure by suspending a pan of dry black liquor in a steam chamber (containing steam at atmospheric pressure) with a calibrated quartz spring. The chamber was placed in a temperature-controlled oven. At a fixed temperature, the water vapor was absorbed by black liquor, and at equilibrium conditions, the mass of the liquor in the pan was measured and the solids concentration of the liquor was determined by a mass balance. Although this method seems to be the best for determining vapor pressure of black liquors at high solids concentrations, it should be mentioned that the true equilibrium may not be achieved in reasonable times due to the slow diffusion of water vapor into the layer of black liquor in the pan. Also, the method of drying used to achieve dry black liquors may affect the nature of 100% black liquor solids. 3. Materials The black liquors used in this study are from a fourvariable, two-level central composite design for pulping slash pine. Kraft pulping of slash pine was conducted in a pilot scale digester of total volume of 0.1 m3 operated with liquor circulation. The four pulping variables were effective alkali (EA), sulfidity (S), cooking temperature (T), and time at temperature (t), respectively. The effective alkali is defined as NaOH + 1/2Na2S, expressed as Na2O, and the sulfidity is defined as Na2S/(NaOH + Na2S), expressed as Na2O (Casey, 1980). The range of the variables used in the design were 11.5-17.5% for EA, 12.5-42.5% for S, 160.0182.2°C for cooking temperature, and 20-60 min for cooking time. The liquor-to-wood ratio was fixed at 4/1, and the white liquor was adjusted with sodium carbonate and sodium sulfate to a causticizing efficiency of 85% and a reduction of 93%. After cooking, black liquor was quickly drained. The pulp was washed twice with water at 80 °F, and the two washes were combined with the black liquor. The combined liquor and washes were filtered to remove all fibers from the liquor, and samples of black liquor and washed pulp were taken for analysis. Black liquor solids were measured using TAPPI method T650 PM84. A sufficient amount of black liquor was loaded into the small-scale evaporator to determine the boiling point of the liquor as a function of pressure and solids mass fraction. 4. Brief Description of the Small-Scale Evaporator Apparatus and Experimental Procedure Details of the design of the apparatus and its schematic have been given in an earlier publication (Stoy et al., 1992). The basic apparatus is a 5 L still pot with three main parts: a glass sleeve, glass top, and stainless steel bottom. The top contains openings for the stirrer, condenser, pressure transducer, and a platinum resistance thermometer. The bottom clamps to the bottom of the glass sleeve and contains baffles and a drain port. The bottom can be removed for cleaning without disturbing the rest of the apparatus. Heating is provided by an electric jacket that slides over the stainless steel bottom, and the heating rate is controlled by controlling
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the voltage of the electric jacket. The heat input can be varied to produce up to 30 mL/min condensate but is usually operated to give 1-5 mL/min of condensate. The pressure control system consists of a PID controller, pressure transducer, bleed valve, and a vacuum pump. The controller operates the bleed valve, which is connected between the vacuum pump and still to control the pressure in the still. Turbine blade stirrers connected to a variable-speed D.C. motor are used for mixing. Stirrer speed can be varied from 0 to 300 rpm, and the direction of stirring is reversible. A stuffing box, filled with Teflon packing, seals the stirrer shaft to prevent air leakage into the system. The temperature of the liquor is measured with a platinum resistance thermometer that is connected to a Rosemount 414L (P) transducer bridge to produce a signal of 1 mV/ °C output that is read with a digital voltmeter. The evaporator is equipped with two condensers connected in series. Therefore, the system can be used for either concentrating the liquor or measuring the vapor-liquid equilibria under alternating cycles of total reflux and evaporation. The operation can be changed by turning the water on or off to the first condenser. Before using this system for black liquors, the boiling point of water was determined as a function of pressure using this apparatus. Boiling points of water determined at pressures from 120 to 760 mmHg agreed with the literature values to within (0.06 °C or better. Also, sucrose solutions (up to 65% by weight) were used to determine the effect of viscosity and solids concentrations. Experimental boiling point measurements agreed with literature values within (0.15 °C or better. With black liquors, the liquor was concentrated from the initial concentration at a low pressure (200 mmHg) and a moderately high evaporation rate (8-12 mL/min). When the desired concentration was reached, the evaporation rate was reduced to 1-1.5 mL/min and the still was operated at total reflux. While under constant reflux, a series of boiling point versus pressure data were taken by first setting the pressure and then recording the temperature when equilibrium was attained. Equilibrium at each pressure was reached normally in 15-30 min. When a complete data set was finished, the sample was concentrated to the next solids concentration and the procedure repeated. A load cell was used to record the amount of condensate produced during concentration, and the solids mass fraction in the liquor was calculated from the mass balance. Also, at the end of each run, small samples of the liquor were taken to determine the final solids mass fraction, and this was compared with the mass balance result. 5. Results 5.1. Effect of Pressure and Solids Concentrations. A large number of kraft black liquors from laboratory cooks of slash pine have been studied. The cooking conditions for these liquors are summarized in Table 1. Overall, the behavior observed for all liquors was similar. Figure 1 is a plot of vapor pressure as a function of temperature at different solids concentrations for one of the liquors used in this study that is a typical result. The vapor pressure decreases as the solids content increases, and its behavior with increasing temperature is similar at all solids concentrations. The liquors used in this work had been previously evaporated to 25-30% solids and the soap skimmed. Therefore, volatile organics had been removed. Chemi-
Table 1. Pulping Conditions for the Liquors Used in This Studya cook no.
t, min
T, K
EA, %
S, %
ABAFX011,12 ABAFX013,14 ABAFX015,16 ABAFX019,20 ABAFX021,22 ABAFX023,24 ABAFX025,26 ABAFX029,30 ABAFX031,32 ABAFX033,34 ABAFX037,38 ABAFX039,40 ABAFX043,44 ABAFX053,54 ABAFX055,56 ABAFX059,60 ABAFX067,68 ABAFX069,70 ABAFX071,72 ABAFX075,76 ABAFX077,78
40 80 80 80 40 40 80 40 40 80 80 80 60 60 60 60 60 60 60 20 40
438.75 449.85 438.75 438.75 449.85 438.75 449.85 449.85 438.75 449.85 449.85 438.75 444.25 444.25 444.25 444.25 444.25 433.15 455.35 444.25 449.85
13 13 16 13 13 16 16 13 16 16 13 16 14.5 11.5 17.5 14.5 14.5 14.5 14.5 14.5 16
20 20 20 35 35 35 35 20 20 20 35 35 27.5 27.5 27.5 42.5 12.5 27.5 27.5 27.5 20
a Key: t, cooking time; T, cooking temperature; EA, effective alkali; S, sulfidity.
Figure 1. Vapor pressure of experimental softwood kraft black liquor ABAFX025,26 at different solids concentrations.
cal analysis of the samples of condensate indicated no detectable salts and only traces of organic carbon; therefore, the condensate was essentially water. It is important to note that the log plots of vapor pressure as a function of 1/T are nearly linear over the whole range of solids concentrations for all liquors studied. If we treat black liquor as a binary solution of organic and inorganic components in water, at equilibrium conditions, where the liquor and the water vapor are at equilibrium with one another at vapor pressure, P, the following relationship applies (Lewis and Randall, 1961):
∆H dP ) dT T∆V
(1)
where P ) pressure (atm), T ) absolute temperature (K), ∆H ) heat of vaporization, and ∆V ) the increase in volume when an amount of the substance vaporizes. If we assume that the vapor phase is an ideal gas, integrating eq 1 will yield
ln(P) ) A -
B T
(2)
where A and B are constants. Equation 1 was applied
278 Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998 Table 2. Parameters A and B as in Eq 2 for Three Different Black Liquors black liquor
% solids
A
B
R2
ABAFX011,12
17.8 29.5 34.9 36.5 43.1 49.8 56.4 75.4 84.1 23.6 36.7 47.4 50.8 59.1 67.2 70.0 75.0 20.9 36.7 49.0 57.7 64.8 71.2 78.2 81.7
13.54 13.59 13.52 13.30 13.52 13.40 13.22 13.51 13.48 13.45 12.74 12.96 12.95 13.31 12.83 13.27 12.83 13.07 13.04 13.02 12.94 12.78 13.32 13.18 12.77
5124.5 5091.9 5132.2 5145.5 5113.4 5114.6 5083.8 5269.3 5327.3 5045.4 4814.2 4913.3 4931.1 5099.1 4932.8 5114.9 4976.1 4894.9 4915.6 4947.1 4957.7 4945.8 5201.5 5187.7 5075.2
0.99 0.99 0.99 0.99 0.98 0.99 0.99 0.97 0.97 0.97 0.97 0.98 0.99 0.99 0.97 0.98 0.98 0.97 0.98 0.99 0.99 0.97 0.98 0.99 0.99
ABAFX013,14
ABAFX025,26
to fit the data successfully for all the liquors listed in Table 1 over a wide range of solids concentrations. The results for some of the liquors at different solids concentrations are listed in Table 2. As can be observed, both A and B vary with the solids concentrations and type of the liquor. If the heat of vaporization were constant, then there should not be variation in the values of B for a particular liquor with changes in the solids concentrations. However, our earlier work indicates that black liquors are nonideal solutions (Stoy, 1992; Stoy and Fricke, 1994; Zaman et al., 1996; Zaman and Fricke, 1996) and their heat of dilution (or mixing) cannot be ignored in evaporation experiments. Therefore, variation in the values of B are due to the nonideal nature of black liquors. The thermodynamic consistency of treating black liquor as a binary solution was studied by Stoy (1992) and Stoy and Fricke (1994). They determined fugacities and vapor pressures for one liquor by using the enthalpy data and one vapor pressure equilibrium value obtained from direct measurements. They developed vapor pressure equilibrium curves at different concentrations and compared these to experimentally determined vapor pressure equilibrium curves at different solids contents. The agreement was very good. Thus, vapor pressures can be determined from enthalpy data that are as accurate as the measured values, and it is thermodynamically consistent to treat black liquor as a pseudobinary solution. 5.2. Boiling Point Elevation. The boiling point elevation for black liquors is defined as the difference between the boiling point of black liquor solution and that of pure water at the same pressure. Figure 2 shows the boiling point elevation for one of the liquors as a function of solids concentration at two pressures. It is evident that the boiling point rise pressure dependency is more significant at higher solids concentrations. This plot is typical of all the results. At 80% solids, the boiling point elevation for this liquor at atmospheric pressure is 22 °C. Clay and Grace (1984) have reported a value of 24.5 °C, and Szymonski and Grace (1985) have reported a value of 30 °C for boiling point elevation
Figure 2. Boiling point elevation of experimental softwood kraft black liquor ABAFX063,64 at two pressures.
Figure 3. Boiling point elevation of three different experimental softwood kraft black liquors at atmospheric pressure.
of black liquors. The range of boiling point elevation determined for a large number of experimental liquors as a function of solids concentration in this work are generally lower than the values that are reported in the literature by other investigators. Figure 3 is a plot of boiling point rise as a function of solids concentration at atmospheric pressure for three of the liquors used in this study. It can be observed that the boiling point rise is a function of the solids composition and that the variation in boiling point rise from liquor-to-liquor becomes larger as the solids concentration increases. At solids concentrations lower than 35% boiling point elevation does not change significantly with the solids composition in the liquor. Using a binary solution assumption, the boiling point elevation for black liquors can be described as (Wennberg, 1985)
BPR ) KBm
(3)
where BPR is the boiling point rise (°C), KB is a constant, and m is the molality of the solution. Since the molality of the liquor solution is proportional to the dissolved solids in water, eq 3 can be written as
S BPR ) K 1-S
(4)
where S is the solids mass fraction and S/(1 - S) is the ratio of solids to water. Equation 4 was applied by Frederick et al. (1980) to fit the boiling point elevation of black liquors to the solids mass fraction. Frederick et al. (1980) and Wennberg (1985) have shown that eq 4 is valid for concentrations up to about 50% solids. At
Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998 279
Figure 4. Boiling point elevation as a function of the dissolved solids for black liquor ABAFX019,20 at different pressures.
Figure 7. Parameters a′ and b′ as in eq 6 as a function of pressure for black liquor ABAFX015,16.
Figures 4 and 5 demonstrate the variation of the boiling point elevation with solids mass fraction to water ratio S/(1 - S) at different pressures for two of the liquors used in this study. Two distinguishable regions of boiling point rise can be observed from these plots. At high values of S/(1 - S), the slope of the curves decreased significantly at all pressures. For all slash pine kraft black liquors used in this study, the break point occurs at concentrations around 65% solids. As mentioned by Frederick et al. (1980), this is the solubility limit concentration for slash pine kraft liquors. The lower and upper portions of boiling point elevation data for all liquors at each pressure were described as a function of S/(1 - S) using the following equations: Figure 5. Boiling point elevation as a function of the dissolved solids for black liquor ABAFX051,52 at different pressures.
(1) the lower portion: S BPR ) a 1-S
(5)
(2) the upper portion: S BPR ) a′ + b′ 1-S
Figure 6. Parameter a as in eq 5 as a function of pressure for black liquor ABAFX059,60.
higher concentrations, a change in the slope of the line was observed. Frederick et al. (1980) concluded that the change in the slope is due to the solubility limit of Na2SO4 and Na2CO3 in the liquor. Above the solubility limit, Na2SO4 and Na2CO3 begin to precipitate, and as a result, the ratio of the dissolved solids to water increases more slowly above the solubility limit. Frederick et al. (1980) observed that, above the solubility limit, the boiling point of a particular black liquor did not change by adding Na2CO3 to the liquor. This indicates that the boiling point elevation depends only on the dissolved solids and increases more slowly with total solids content above the solubility limit. The solids content at which the slope of the boiling point elevation versus S/(1 - S) starts to change was taken to be the solubility limit of Na2CO3 and Na2SO4 in black liquor.
(6)
where a, a′, and b′ are constants that are pressure dependent and vary from liquor to liquor due to the differences in the composition of the liquors. It was found that the parameters a, a′, and b′ for every single liquor have a linear relation with pressure, as indicated in Figures 6 and 7, which are plots of these parameters as a function of pressure. Figure 6 is a plot of the constant a versus pressure for liquor ABAFX059,60, and Figure 7 shows plots of a′ and b′ versus pressure. This is a very unique and interesting result and can be used as a basis for data reduction and for obtaining universal correlations for boiling point elevation of black liquors as a function of pressure and solids concentration. These parameters were described as
a ) a1 + b1P
(7)
a′ ) a2 + b2P
(8)
b′ ) a3 + b3P
(9)
where a1, b1, a2, b2, a3, and b3 are constants that are composition dependent and P (mmHg) is pressure. By
280 Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998 Table 3. Parameters a1, b1, a2, b2, a3, and b3 as in Eqs 7-9 for the Liquors Used in This Study liquor
a1
b1
R2
a2
b2
R2
a3
b3
R2
ABAFX011,12 ABAFX013,14 ABAFX015,16 ABAFX019,20 ABAFX021,22 ABAFX023,24 ABAFX025,26 ABAFX029,30 ABAFX031,32 ABAFX033,34 ABAFX037,38 ABAFX039,40 ABAFX043,44 ABAFX053,54 ABAFX055,56 ABAFX059,60 ABAFX067,68 ABAFX069,70 ABAFX071,72 ABAFX075,76 ABAFX077,78
6.4779 4.5268 4.8986 4.0383 2.9704 5.4163 4.7320 3.3071 5.5057 3.9646 3.7780 3.9872 4.8785 4.0645 4.7771 3.7005 6.6487 7.5405 5.4164 6.1824 7.5048
0.0027 0.0036 0.0058 0.0039 0.0052 0.0048 0.0040 0.0046 0.0037 0.0050 0.0033 0.0050 0.0038 0.0033 0.0042 0.0050 0.0024 0.0039 0.0036 0.0022 0.0033
0.99 0.95 0.99 0.98 0.97 0.99 0.98 0.98 0.98 0.99 0.98 0.98 0.98 0.98 0.99 0.98 0.97 0.98 0.98 0.98 0.98
6.4100 1.9760 4.9615 4.9886 1.0695 3.3479 2.5867 1.9893 9.1854 1.9807 3.2908 2.0465 3.8232 4.4454 4.9451 1.6317 4.3682 8.2448 3.3423 0.3336 3.2589
0.0039 0.0037 0.0030 0.0020 0.0032 0.0027 0.0061 0.0023 0.0032 0.0027 0.0047 0.0037 0.0042 0.0017 0.0017 0.0047 0.0034 0.0027 0.0034 0.0032 0.0032
0.98 0.99 0.99 0.99 0.99 0.99 0.98 0.99 0.96 0.99 0.98 0.98 0.96 0.98 0.98 0.99 0.98 0.97 0.98 0.99 0.99
2.4198 2.9903 2.9424 2.5190 3.8406 4.4275 3.6788 3.0659 1.6429 4.6920 2.6992 3.4490 3.3749 2.9695 3.2131 3.8420 3.2226 3.4371 2.6084 5.7703 4.1303
0.0001 0.0005 0.0012 0.0011 0.0006 0.0010 0.0003 0.0016 0.0012 0.0003 0.0007 0.0013 0.0009 0.0007 0.0017 0.0006 0.0002 0.0004 0.0007 0.0006 0.0006
0.95 0.99 0.97 0.97 0.92 0.98 0.91 0.97 0.99 0.99 0.99 0.97 0.96 0.98 0.98 0.99 0.99 0.99 0.98 0.98 0.98
Table 4. Complete Linear Regression Model Parameters for a1, b1, a2, b2, a3, and b3 parameters variable
a1
b1
a2
b2
a3
b3
intercept t T EA S Tt tEA tS TEA TS EAS t2 T2 EA2 S2 tTEAS TEAS tEAS tTEA tTS R2
4540.587 -16.2538 -16.1168 -158.044 -23.1972 0.038256 1.552048 0.144971 0.366594 0.055112 2.723327 -0.00035 0.013012 -0.06044 0.001154 6.68 × 10-5 -0.00634 -0.0282 -0.0036 -0.00038 0.99
1.662954 -0.02459 -0.0055 -0.06679 -0.04131 5.46 × 10-5 0.001758 0.000853 0.000147 9.33 × 10-5 0.003052 -1.21 × 10-8 4.04 × 10-6 1.65 × 10-5 -2.3 × 10-7 1.34 × 10-7 -6.9 × 10-6 -6 × 10-5 -3.9 × 10-6 -1.9 × 10-6 0.99
2383.602 2.368055 -12.4561 92.30174 30.13235 -0.00497 -0.46844 0.010979 -0.21085 -0.06784 -3.04463 -0.00181 0.015983 0.096815 -0.00366 -1.6 × 10-5 0.006832 0.006587 0.001043 0 1.0
1.724884 -0.06696 7.03 × 10-5 -0.24535 -0.10507 0.000152 0.004644 0.001875 0.000576 0.000241 0.007878 -4.3 × 10-7 -9.3 × 10-6 -0.00028 -6.7 × 10-7 3.17 × 10-7 -1.8 × 10-5 -0.00014 -1.1 × 10-5 -4.3 × 10-6 1.0
-3927.78 80.05616 10.01004 196.7819 132.8823 -0.17875 -4.94237 -2.74947 -0.43624 -0.29599 -7.95667 0.001477 -0.00282 -0.03122 0.000711 -0.00038 0.017748 0.17061 0.011022 0.006124 0.98
-0.96978 -0.00879 0.003514 0.030668 0.004923 1.87 × 10-5 0.000504 0.000309 -7.5 × 10-5 -1.3 × 10-5 -0.00055 -3.3 × 10-7 -2.8 × 10-6 3.33 × 10-5 -2.2 × 10-6 3.35 × 10-8 1.38 × 10-6 -1.6 × 10-5 -1.1 × 10-6 -6.6 × 10-7 1.0
substituting eq 7 in eq 5 and eqs 8 and 9 in eq 6, the boiling point elevation of each slash pine kraft black liquor as a function of solids mass fraction and pressure can be defined as
S for S < 0.65 BPR ) (a1 + b1P) 1-S
(10)
S for S g 0.65 BPR ) (a2 + b2P) + (a3 + b3P) 1-S (11) The constants a1, b1, a2, b2, a3, and b3 were determined for all slash pine kraft black liquors used in this study, and these are listed in Table 3. As can be observed, the R2 of the fits are very high which indicates that the fits are very accurate. These constants vary from liquor to liquor due to the fact that the liquors are compositionally different. Since the effects of pressure and solids mass fraction can be combined together and the boiling point elevation can be defined with a single correlation of these variables, the effects of solids composition arising from differences in pulping conditions can be partitioned and the constants of the model
for each liquor can be used to study the effect of the pulping conditions on boiling point elevation of black liquors. As described earlier, the liquors used in this study are from a two-level, four-variable factorially designed experiment with center and star points for kraft pulping of slash pine. The resulting rotatable composite design supplies enough information for determining nonlinear responses and for obtaining proper predictive models for the constants of eqs 10 and 11 as a function of pulping conditions that can be applied to determine the boiling point rise and boiling point for other slash pine black liquors. One of the main objectives of the current study was to represent proper correlations that can be applied to estimate the boiling point rise and boiling point of black liquors at operating conditions from the knowledge of pulping variables. To date, no similar study has been reported in the literature. In order to obtain predictive models for boiling point elevation of slash pine kraft black liquors, different statistical approaches were used to establish proper models for a1, b1, a2, b2, a3, and b3 as a function of the cooking conditions. By replacing these models in eqs
Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998 281
10 and 11, one can determine the boiling point elevation for any slash pine kraft black liquor of known cooking conditions at a desired pressure and solids concentration. 5.3. Statistical Modeling. In order to investigate the influence of the cooking conditions on boiling point rise of slash pine black liquors and obtain proper predictive models, different statistical approaches were used to develop these models for a1, b1, a2, b2, a3, and b3 as a function of the pulping variables. As a basis for model selection, the following linear regression model that consists of the main variables, second-, third-, and fourth-order interactions, and quadratic terms was employed 4
Y ) β0 +
∑ i)1
4
βiXi +
∑ i
