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Acid hydrolysis of wood cellulose under low water condition
Biomass 6 (1984) 93-100
Acid Hydrolysis of Wood Cellulose under Low Water Condition S. K. Song* and Y. Y. Leet Department of Chemical Engineering, Auburn University, Alabama 36849, USA (Received: 27 May, 1984)
ABSTRACT Kinetic parameters in acid hydrolysis o f cellulose were determined covering the conditions 170-190°C, 4.4-12% sulfuric acM and 1 : 1.6 solid-to-liquid ratio. The findings were generally in accordance with dilute acM hydrolysis in that the glucose yieM and concentration increased with reaction temperature and acid concentration. Batch hydrolysis o f lignocellulose using low water content was found to be efficient in obtaining a concentrated glucose solution, thus potentially redudcing the heat input in the process. The concentrated glucose solution trapped within the biomass structure couM be extracted without significant dilution using a countercurrent leaching system. Key words: biomass, ligno-cellulosics, wood, acid hydrolysis, alcohol,
glucose, kinetics.
INTRODUCTION Acid-catalyzed hydrolysis of cellulose is one of the major process schemes currently being developed as part o f an integral biomass conversion process. The technology is now confronting a number of obstacles that must be overcome before it becomes an economicaUy * Present address: Department of Chemical Engineering, Busan National University, Busan, Korea. t To whom correspondence should be addressed. 93 Biomass 0144-4565/84/$03.30- © Elsevier Applied Science Publishers Ltd, England, 1984. Printed in Great Britain
94
S. K. Song, Y. Y. Lee
viable one. The specific technical problems include low yield, low concentration and poor fermentability of the sugar product. Substantial research effort has been aimed at improving the yield. 1-6 In terms of economics and energy efficiency in the process, however, the concentration level of the sugar product is also vitally important. This is realized by examining the steam requirement in the conventional cellulose hydrolysis process, which amounts to ¢4-6 kg-1 of glucose (for example, the percolation process). For a product with probable market value of ¢10-20 kg-1 the steam value is indeed a major cost item. The potential cost saving by reduction of heat input is therefore deemed quite substantial. The major heat input in the hydrolysis process is required in reactor heat up and post evaporation to remove the water content in the hydrolyzate. Reducing the amount of water in the reactor as much as possible provided that solid feed is uniformly in contact with liquid would be advantageous by improving heat efficiency. On the other hand, the recovery of liquid product from the solid biomass becomes more difficult as the ratio of solid to liquid increases. Therefore, to carry out the hydrolysis reaction under high solid-to-liquid ratio, an efficient solid-liquid separation becomes a necessity. This paper deals with two technical points in cellulose hydrolysis: (1) conducting the hydrolysis reaction using the least possible amount of water, and (2) recovering hydrolyzate at the highest possible concentration.
EXPERIMENTAL METHODS
Substrate Hardwood of southern red oak species supplied in the form of sawdust was screened and the 40-100 mesh fraction was used as feed material. Prehydrolysis was conducted in an autoclave (conditions: 160°C, 0.2% H2504, one solid to five liquid volumes ratio for 2 h) 7 to remove hemicellulose and amorphous cellulose in the wood structure. The residue remaining after prehydrolysis (primarily cellulose and lignin) was oven-dried and used in all hydrolysis studies. The composition of the feed material after pre-hydrolysis as determined by standard procedure s-l° was 57.0% cellulose, 6.5% hemicellulose and 36.5% lignin.
Acid hydrolysis of wood cellulose under low water condition
95
Hydrolysis reactor and operation The batch reactors (tubing bomb) used for hydrolysis reaction were constructed out of 316 stainless steel tubing with dimensions of 15-2 cm long, 1.27 cm OD and 0.89 mm wall thickness giving an effective reactor volume of 14.3 ml. The reactors were flanged and O-ring sealed at both ends. The center of the flange was threaded to fit a plug and thermocouple to be used for pressure release and monitoring reaction temperature. For preparation of solid feed, a known amount of pre-hydrolyzed sample was mixed with sulfuric acid solution in 1 : 2 solid-to-liquid ratio to ensure uniform wetting. The mixed sample was then placed in a vacuum oven at 50°C to reduce the water content until the desired solid-to-liquid ratio of 1 : 1.6 was achieved. About 10 g of the hardwood sample was packed into multiples of the tubing bomb reactors. A thermocouple was inserted into one of the reactors to monitor the reaction temperature. To initiate the reaction, the reactors were simultaneously placed into an oil bath (Haake, Model FS-2) which was preadjusted to be at a temperature 50°C higher than the desired reaction temperature. When the reactor temperature approached the set point, the reactors were transferred into another oil bath pre-set at the desired reaction temperature. This was done to minimize the pre-heating time to less than 30 s. Time zero was designated as the time when the set point was reached. Sugar analysis Samples taken from the reactor were diluted to give about 1% glucose solution and neutralized by barium hydroxide before analysis. Glucose analysis was done by HPLC (Waters Associate, RI detector) using an ion-exchange resin column prepared according to the procedure given by Ladisch and Tsao. la
Counter-current leaching The experimental setup for counter-current leaching was constructed following the concept of shanks system, ~2 which is a multi-state cascade extractor. In operation of this system, the solvent (water) was pumped from one end of the cascade in which six pre-reacted columns were
96
s. K. Song, Y. Y. Lee
connected end to end. After a pre-set period of time the exhausted column at the fluid inlet side was disconnected and removed. At the same time, a new pre-reacted column was added at the other end (product withdrawal side). This procedure was repeated until a steadystate was reached.
RESULTS AND DISCUSSION
Hydrolysis kinetics Microstructural analyses have shown that the pore space o f wood is equal to or slightly greater than the solid volume; thus it seems feasible to carry out the reaction under a solid-to-liquid ratio o f 1 to 1-2. This would mean that the liquid (acid solution) fills only the pores within the wood structure leaving the external reactor voidage essentially free of water. The reaction can then be carried out in a batch manner. Since the water content is at such a low level one can afford to use a much higher level of acid than is usual 1-3' s in dilute sulfuric acid processes. Our preliminary calculations suggested that acid concentration could be raised up to 10% before it became a prohibitive cost factor. The kinetic study therefore was designed to cover a range of sulfuric acid concentrations from 4 to 12%. This plan also reflects the expectation that higher kinetic yield would be obtained using higher acid level, as indicated by earlier studies on dilute acid hydrolysis. 1-3 In terms of reaction temperature, the range 170-190°C was chosen, the upper level being limited by experimental difficulties. The solid-to-liquid ratio was set at a constant level of 1 : 1.6. According to our experience, solid-to-liquid ratios higher than this gave rise to inconsistent results, perhaps due to non-uniform wetting in the biomass. Acid hydrolysis of cellulose can be represented by the following sequential reaction in which each reaction is to first-order: Cellulose i Glucose 2 Decomposed products
(1)
The kinetic investigation was carried out by repeated batch experimentation using hardwood particles with the hemicellulose extracted. Glucose variation data were then used to statistically determine the rate constants in the model. The batch reaction data are shown in Fig. 1. Several findings were noteworthy. Clearly, the glucose concentration
Acid hydrolysis o f wood cellulose under low water condition
97
a.
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,
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,
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Fig. 1.
TIME,
~ 2o.oo
, 25.00
.50.00
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Batch reaction data: o, 170°C; t~, 180°C; A, 190°C. H2SO4concentration: (a) 4-41%; (b) 7.87%; (c) 12-19%.
S. K. Song, Y. Y. Lee
98
in the product increased with reaction temperature, and high sulfuric acid concentrations increased glucose concentration and thus the yield. Particularly significant, however, is that glucose concentrations up to 17% corresponding to hydrolysis yields of 55% are achievable under the low water hydrolysis scheme, thus meeting our primary objective. The results also indicate that by employing reaction temperatures higher than 190°C the concentration and yield could be further improved, a point we could not prove at this time due to experimental difficulties arising from extremely short reaction time. The experimental results were further analyzed to verify the kinetic mode. To account for the effects of temperature and acid concentration, the rate constants were expressed by an Arrhenius type equation with the addition of an acid term such that: k i =- kio (S) ni exp
where
(2)
s = sulfuric acid concentration (wt %) kio = frequency factor E i - activation energy
n; = acid concentration exponent i = reaction index (1 for hydrolysis, 2 for composition) The kinetic parameters in this reaction model were simultaneously determined employing data from nine experimental runs. A computer program for non-linear regression analysis was set up for this purpose initially from SAS NLIN DUD. 13 A part of the program was later modified to enhance the confidence level following the methods suggested by Ralston and Jennrich. a4 The resulting kinetic parameters are shown in Table 1. The tendency of increasing yield with reaction temperature
TABLE 1 Kinetic Parameters in Hydrolysis of Hardwood Cellulosea
i
k lo (mitt- 1) {% acid) - ni
ni
E i (cal g-mole-1)
1 2
6.6 × 1016 6.4 × 101~
1.64 1-10
39 500 30 800
a Data coverage: 170-190°C, 4.41-12-19% H2SO4.
Acid hydrolysis o f wood cellulose under low water condition
99
and acid concentration is reconfirmed here by the fact that the activation energy and acid exponent for the hydrolysis reaction are substnatially higher than those for the decomposition reaction. The kinetic parameters are generally in accordance with those previously reported for dilute acid hydrolysis,1-3 indicating that there is no significant change in reaction mechanism under low water and high acid condition. As shown in Fig. 1, the agreement between the experimental data and the model prediction was satisfactory. The kinetic model and associated parameters provided in this study should thus be useful tools in studies involving acid hydrolysis of cellulose.
Counter-current leaching in the foregoing kinetic investigation it was proved that the low water hydrolysis scheme is quite satisfactory in providing high glucose concentration and yield. In order to make it a workable system, however, an efficient leaching mechanism must be incorporated to recover the sugar product without significant dilution. To this end, the use of a counter-current leaching process was investigated. As mentioned previously, the actual experimental design was based on a multi-stage cascade extractor known as shanks system. 12 Sample performance data for this leaching system are shown in Fig. 2 in which the outlet glucose concen-
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.......
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o
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I
NO. OF REACTOR REMOVED FROM EXTRACTION SYSTEM
Fig. 2.
Sampleperformance data for a counter-current leaching system.
1oo
s.K. Song, Y. Y. Lee
tration is plotted against the column number removed from the cascade. This analysis shows that the sugar product is recovered at a concentration of about 15%. Further analysis o f the data indicated that 90% of sugar recovery and 85% o f concentration retainment (or only 15% reduction of glucose concentration) could be achieved using this system.
ACKNOWLEDGMENT The authors gratefully acknowledge the support for this research by the National Science Foundation under Grant No. CPE-8211654 and by the Engineering Experiment Station of Auburn University.
REFERENCES 1. Thompson, D. R. & Grethlein, H. E. (1979).Ind. Eng. Chem.,Prod. Res. Dev., 18, 166. 2. Church, J. A. & Wooldridge, D. (1981). Ind. Eng. Chem., Prod. Res. Dev., 20, 371. 3. Seaman, J. F. (1945).Ind. Eng. Chem., 37, 43. 4. Song, S. K. & Lee, Y. Y. (1982). Chem. Eng. Commun., 17, 23. 5. Fagan, R. D., Grethlein, H. E., Converse, A. O. & Porteous, A. (1971).Environmental Science & Technology, 5, 545. 6. Rugg, B. (1981). Optimization of twin screw reactor, SERI Contractors Conference, Denver, Colorado, April. 7. Cahela, D. R., Lee, Y. Y. & Chambers, R. P. (1983). Biotechnology & Bioengineering, 25, 3. 8. Tappi Standard, T204 05-76 (1978). Alcohol-benzene and dichloromethane solubles in wood and pulp. 9. Wise, L. E. (1946). Tappi, 40, 568. 10. Timmel, T. E. (1957). Tappi, 40, 568. 11. Ladisch, M. R. & Tsao, G. T. (1978). J. Chromatogr., 166, 85. 12. King, C. J. (1980). Separation processes, McGraw-Hill, New York, p. 172. 13. Helwig, J. T. & Council, K. A. (Eds) (1979). SAS User's Guide, SAS Institute, Cary, p. 317. 14. Ralston, M. L. & Jennrich, R. I. (1978). Technometrics, 20, 7.
Acid Hydrolysis of Wood Cellulose under Low Water Condition S. K. Song* and Y. Y. Leet Department of Chemical Engineering, Auburn University, Alabama 36849, USA (Received: 27 May, 1984)
ABSTRACT Kinetic parameters in acid hydrolysis o f cellulose were determined covering the conditions 170-190°C, 4.4-12% sulfuric acM and 1 : 1.6 solid-to-liquid ratio. The findings were generally in accordance with dilute acM hydrolysis in that the glucose yieM and concentration increased with reaction temperature and acid concentration. Batch hydrolysis o f lignocellulose using low water content was found to be efficient in obtaining a concentrated glucose solution, thus potentially redudcing the heat input in the process. The concentrated glucose solution trapped within the biomass structure couM be extracted without significant dilution using a countercurrent leaching system. Key words: biomass, ligno-cellulosics, wood, acid hydrolysis, alcohol,
glucose, kinetics.
INTRODUCTION Acid-catalyzed hydrolysis of cellulose is one of the major process schemes currently being developed as part o f an integral biomass conversion process. The technology is now confronting a number of obstacles that must be overcome before it becomes an economicaUy * Present address: Department of Chemical Engineering, Busan National University, Busan, Korea. t To whom correspondence should be addressed. 93 Biomass 0144-4565/84/$03.30- © Elsevier Applied Science Publishers Ltd, England, 1984. Printed in Great Britain
94
S. K. Song, Y. Y. Lee
viable one. The specific technical problems include low yield, low concentration and poor fermentability of the sugar product. Substantial research effort has been aimed at improving the yield. 1-6 In terms of economics and energy efficiency in the process, however, the concentration level of the sugar product is also vitally important. This is realized by examining the steam requirement in the conventional cellulose hydrolysis process, which amounts to ¢4-6 kg-1 of glucose (for example, the percolation process). For a product with probable market value of ¢10-20 kg-1 the steam value is indeed a major cost item. The potential cost saving by reduction of heat input is therefore deemed quite substantial. The major heat input in the hydrolysis process is required in reactor heat up and post evaporation to remove the water content in the hydrolyzate. Reducing the amount of water in the reactor as much as possible provided that solid feed is uniformly in contact with liquid would be advantageous by improving heat efficiency. On the other hand, the recovery of liquid product from the solid biomass becomes more difficult as the ratio of solid to liquid increases. Therefore, to carry out the hydrolysis reaction under high solid-to-liquid ratio, an efficient solid-liquid separation becomes a necessity. This paper deals with two technical points in cellulose hydrolysis: (1) conducting the hydrolysis reaction using the least possible amount of water, and (2) recovering hydrolyzate at the highest possible concentration.
EXPERIMENTAL METHODS
Substrate Hardwood of southern red oak species supplied in the form of sawdust was screened and the 40-100 mesh fraction was used as feed material. Prehydrolysis was conducted in an autoclave (conditions: 160°C, 0.2% H2504, one solid to five liquid volumes ratio for 2 h) 7 to remove hemicellulose and amorphous cellulose in the wood structure. The residue remaining after prehydrolysis (primarily cellulose and lignin) was oven-dried and used in all hydrolysis studies. The composition of the feed material after pre-hydrolysis as determined by standard procedure s-l° was 57.0% cellulose, 6.5% hemicellulose and 36.5% lignin.
Acid hydrolysis of wood cellulose under low water condition
95
Hydrolysis reactor and operation The batch reactors (tubing bomb) used for hydrolysis reaction were constructed out of 316 stainless steel tubing with dimensions of 15-2 cm long, 1.27 cm OD and 0.89 mm wall thickness giving an effective reactor volume of 14.3 ml. The reactors were flanged and O-ring sealed at both ends. The center of the flange was threaded to fit a plug and thermocouple to be used for pressure release and monitoring reaction temperature. For preparation of solid feed, a known amount of pre-hydrolyzed sample was mixed with sulfuric acid solution in 1 : 2 solid-to-liquid ratio to ensure uniform wetting. The mixed sample was then placed in a vacuum oven at 50°C to reduce the water content until the desired solid-to-liquid ratio of 1 : 1.6 was achieved. About 10 g of the hardwood sample was packed into multiples of the tubing bomb reactors. A thermocouple was inserted into one of the reactors to monitor the reaction temperature. To initiate the reaction, the reactors were simultaneously placed into an oil bath (Haake, Model FS-2) which was preadjusted to be at a temperature 50°C higher than the desired reaction temperature. When the reactor temperature approached the set point, the reactors were transferred into another oil bath pre-set at the desired reaction temperature. This was done to minimize the pre-heating time to less than 30 s. Time zero was designated as the time when the set point was reached. Sugar analysis Samples taken from the reactor were diluted to give about 1% glucose solution and neutralized by barium hydroxide before analysis. Glucose analysis was done by HPLC (Waters Associate, RI detector) using an ion-exchange resin column prepared according to the procedure given by Ladisch and Tsao. la
Counter-current leaching The experimental setup for counter-current leaching was constructed following the concept of shanks system, ~2 which is a multi-state cascade extractor. In operation of this system, the solvent (water) was pumped from one end of the cascade in which six pre-reacted columns were
96
s. K. Song, Y. Y. Lee
connected end to end. After a pre-set period of time the exhausted column at the fluid inlet side was disconnected and removed. At the same time, a new pre-reacted column was added at the other end (product withdrawal side). This procedure was repeated until a steadystate was reached.
RESULTS AND DISCUSSION
Hydrolysis kinetics Microstructural analyses have shown that the pore space o f wood is equal to or slightly greater than the solid volume; thus it seems feasible to carry out the reaction under a solid-to-liquid ratio o f 1 to 1-2. This would mean that the liquid (acid solution) fills only the pores within the wood structure leaving the external reactor voidage essentially free of water. The reaction can then be carried out in a batch manner. Since the water content is at such a low level one can afford to use a much higher level of acid than is usual 1-3' s in dilute sulfuric acid processes. Our preliminary calculations suggested that acid concentration could be raised up to 10% before it became a prohibitive cost factor. The kinetic study therefore was designed to cover a range of sulfuric acid concentrations from 4 to 12%. This plan also reflects the expectation that higher kinetic yield would be obtained using higher acid level, as indicated by earlier studies on dilute acid hydrolysis. 1-3 In terms of reaction temperature, the range 170-190°C was chosen, the upper level being limited by experimental difficulties. The solid-to-liquid ratio was set at a constant level of 1 : 1.6. According to our experience, solid-to-liquid ratios higher than this gave rise to inconsistent results, perhaps due to non-uniform wetting in the biomass. Acid hydrolysis of cellulose can be represented by the following sequential reaction in which each reaction is to first-order: Cellulose i Glucose 2 Decomposed products
(1)
The kinetic investigation was carried out by repeated batch experimentation using hardwood particles with the hemicellulose extracted. Glucose variation data were then used to statistically determine the rate constants in the model. The batch reaction data are shown in Fig. 1. Several findings were noteworthy. Clearly, the glucose concentration
Acid hydrolysis o f wood cellulose under low water condition
97
a.
z o
1.72
I
V-
< D
F- ~)
1.29
x
g
';
0.86
¢j
o
(,.D 0.4'5
_a (3
O,OC
I
I
I
REACTION
b.
z
1
~b
1.2cp
o I<_
I
I
0.00 I 0.00 20.00 30.00 40.00 .50.00 60.00
"
7
2
~
TIME,
MINUTES
q
~
30.00
40.00
,,=,x 0 zT. o J
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\
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0 I0.00
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REACTION C.
Z
1.72
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TIME,
,
,
.50.00
60.00
MINUTES ,
,
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(3
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,
~"~......
z~,
5.00
i o.oo
i 5.00
REACTION
Fig. 1.
TIME,
~ 2o.oo
, 25.00
.50.00
MINUTES
Batch reaction data: o, 170°C; t~, 180°C; A, 190°C. H2SO4concentration: (a) 4-41%; (b) 7.87%; (c) 12-19%.
S. K. Song, Y. Y. Lee
98
in the product increased with reaction temperature, and high sulfuric acid concentrations increased glucose concentration and thus the yield. Particularly significant, however, is that glucose concentrations up to 17% corresponding to hydrolysis yields of 55% are achievable under the low water hydrolysis scheme, thus meeting our primary objective. The results also indicate that by employing reaction temperatures higher than 190°C the concentration and yield could be further improved, a point we could not prove at this time due to experimental difficulties arising from extremely short reaction time. The experimental results were further analyzed to verify the kinetic mode. To account for the effects of temperature and acid concentration, the rate constants were expressed by an Arrhenius type equation with the addition of an acid term such that: k i =- kio (S) ni exp
where
(2)
s = sulfuric acid concentration (wt %) kio = frequency factor E i - activation energy
n; = acid concentration exponent i = reaction index (1 for hydrolysis, 2 for composition) The kinetic parameters in this reaction model were simultaneously determined employing data from nine experimental runs. A computer program for non-linear regression analysis was set up for this purpose initially from SAS NLIN DUD. 13 A part of the program was later modified to enhance the confidence level following the methods suggested by Ralston and Jennrich. a4 The resulting kinetic parameters are shown in Table 1. The tendency of increasing yield with reaction temperature
TABLE 1 Kinetic Parameters in Hydrolysis of Hardwood Cellulosea
i
k lo (mitt- 1) {% acid) - ni
ni
E i (cal g-mole-1)
1 2
6.6 × 1016 6.4 × 101~
1.64 1-10
39 500 30 800
a Data coverage: 170-190°C, 4.41-12-19% H2SO4.
Acid hydrolysis o f wood cellulose under low water condition
99
and acid concentration is reconfirmed here by the fact that the activation energy and acid exponent for the hydrolysis reaction are substnatially higher than those for the decomposition reaction. The kinetic parameters are generally in accordance with those previously reported for dilute acid hydrolysis,1-3 indicating that there is no significant change in reaction mechanism under low water and high acid condition. As shown in Fig. 1, the agreement between the experimental data and the model prediction was satisfactory. The kinetic model and associated parameters provided in this study should thus be useful tools in studies involving acid hydrolysis of cellulose.
Counter-current leaching in the foregoing kinetic investigation it was proved that the low water hydrolysis scheme is quite satisfactory in providing high glucose concentration and yield. In order to make it a workable system, however, an efficient leaching mechanism must be incorporated to recover the sugar product without significant dilution. To this end, the use of a counter-current leaching process was investigated. As mentioned previously, the actual experimental design was based on a multi-stage cascade extractor known as shanks system. 12 Sample performance data for this leaching system are shown in Fig. 2 in which the outlet glucose concen-
z" 0 I-< Q:: I-Z hi (J Z O C.) bJ OD O O .J (.9
20.00
.......
I 5.OG
n
^lJ
g
o
o 0
0
O
I O.OC
5.00
0.00;
I
I
I
I
NO. OF REACTOR REMOVED FROM EXTRACTION SYSTEM
Fig. 2.
Sampleperformance data for a counter-current leaching system.
1oo
s.K. Song, Y. Y. Lee
tration is plotted against the column number removed from the cascade. This analysis shows that the sugar product is recovered at a concentration of about 15%. Further analysis o f the data indicated that 90% of sugar recovery and 85% o f concentration retainment (or only 15% reduction of glucose concentration) could be achieved using this system.
ACKNOWLEDGMENT The authors gratefully acknowledge the support for this research by the National Science Foundation under Grant No. CPE-8211654 and by the Engineering Experiment Station of Auburn University.
REFERENCES 1. Thompson, D. R. & Grethlein, H. E. (1979).Ind. Eng. Chem.,Prod. Res. Dev., 18, 166. 2. Church, J. A. & Wooldridge, D. (1981). Ind. Eng. Chem., Prod. Res. Dev., 20, 371. 3. Seaman, J. F. (1945).Ind. Eng. Chem., 37, 43. 4. Song, S. K. & Lee, Y. Y. (1982). Chem. Eng. Commun., 17, 23. 5. Fagan, R. D., Grethlein, H. E., Converse, A. O. & Porteous, A. (1971).Environmental Science & Technology, 5, 545. 6. Rugg, B. (1981). Optimization of twin screw reactor, SERI Contractors Conference, Denver, Colorado, April. 7. Cahela, D. R., Lee, Y. Y. & Chambers, R. P. (1983). Biotechnology & Bioengineering, 25, 3. 8. Tappi Standard, T204 05-76 (1978). Alcohol-benzene and dichloromethane solubles in wood and pulp. 9. Wise, L. E. (1946). Tappi, 40, 568. 10. Timmel, T. E. (1957). Tappi, 40, 568. 11. Ladisch, M. R. & Tsao, G. T. (1978). J. Chromatogr., 166, 85. 12. King, C. J. (1980). Separation processes, McGraw-Hill, New York, p. 172. 13. Helwig, J. T. & Council, K. A. (Eds) (1979). SAS User's Guide, SAS Institute, Cary, p. 317. 14. Ralston, M. L. & Jennrich, R. I. (1978). Technometrics, 20, 7.