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SW—Soil and Water
J. agric. Engng Res., (2001) 78 (1), 89 } 97 doi:10.1006/jaer.2000.0627, available online at http://www.idealibrary.com on SW*Soil and Water
Methodology to simultaneously optimize the Macrocation and the Macroanion Composition of Nutrient Solutions, using Mixture Theory G. De Rijck; E. Schrevens Department of Applied Plant Sciences, Faculty of Agricultural and Applied Biological Sciences, K.U. Leuven, Willem de Croylaan 42, B-3001 Heverlee (Belgium); e-mail of the corresponding author: [email protected] (Received 8 December 1998; accepted in revised form 5 August 2000; published online 22 September 2000)
To illustrate the use of mixture theory to simultaneously optimize the macrocation and the macroanion composition of a hydroponical nutrient solution, a +3,2;3,2, double lattice design was established in the double mixture factor space (K>, Ca>, Mg>, NO\, H PO\ and SO\). Only one response variable was considered: the calcium content of English ryegrass. The macronutrient composition signi"cantly a!ected the calcium content of the ryegrass. A reduced second-degree double mixture model with 20 instead of 36 parameters was selected to represent the response surface over the experimental region. The macrocation composition had a more pronounced e!ect on the calcium content of the ryegrass than the macroanion composition. 2001 Silsoe Research Institute
Optimizing the macrocation and the macroanion composition simultaneously, demands a more extensive experimental design and makes it possible to prove which combination of the macrocations and macroanions yields the best results. This research elaborates a justi"ed methodology to simultaneously optimize, for one total ionic strength, the concentration of the six essential macronutrients (K>, Ca>, Mg>, NO\, H PO\ and SO\) of hydroponical nutrient solutions. The experiment is carried out with English ryegrass (¸olium perenne) cultivated in hydroponics. To illustrate the elaborated methodology, only one dependent variable is considered: the calcium content of the ryegrass.
1. Introduction Nutrient solutions are aqueous solutions of inorganic ions, submitted to the ionic balance constraint. This constraint de"nes the cation and the anion composition of nutrient solutions as &mixture systems' (Schrevens, 1988; Schrevens & Cornell, 1993; De Rijck & Schrevens, 1994; De Rijck, 1996; De Rijck & Schrevens, 1998a) and makes it impossible to experiment with the mineral composition of nutrient solutions in a classical orthogonal or in a unifactorial way. A change in the concentration of one ion must be accompanied by either a corresponding change for an ion of the opposite charge, a complementary change for other ions of the same charge or both (Hewitt, 1966). Therefore, experimenting with the mineral composition of nutrient solutions is only possible in a multifactorial way. The previous research (De Rijck & Schrevens, 1998b) elaborated a methodology to optimize the macrocation and the macroanion composition, separately. This methodology demands only a limited amount of experimental treatments. The combination of the optimal macrocation and the optimal macroanion composition constitutes a good nutrient solution. However, optimizing the macrocation and the macroanion composition separately does not prove that their combination constitutes the best possible nutrient solution. 0021-8634/01/010089#09 $35.00/0
2. Theoretical considerations 2.1. Mixture systems The properties of the mixture systems are determined by the proportions of their components, rather than by their quantitative amount. The proportion of each component varies between zero and one. The sum of the proportions of all mixture components equals one (Cornell, 1981, 1990). For a mixture of q components, with x G 89
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G . D E R IJCK ; E . S CH R EV EN S
the proportion of the ith mixture component, this results in the following mixture constraints: 0)x )1 for i"1, 2, 3, 2, q G
(1)
O x "1 G G
(2)
The ionic balance constraint postulates that the sum of the cations equals the sum of the anions, expressed in eq l\ and de"nes the cation and the anion composition of nutrient solutions as mixture systems. Simplifying a nutrient solution to the six essential macronutrients, neglecting H> and OH\ ions and the possible dissociation forms of H PO\ renders: [K>]#[Ca>]#[Mg>] "[NO\]#[H PO\]#[SO\]"C
(3)
where: K>, Ca>, Mg>, NO\, H PO\ and SO\ are expressed in eq l\ and C the total equivalent concentration. Dividing each cation (anion) concentration by the total concentration of cations (anions) expressed in eq l\ yields the proportion of each cation (anion).
2.2. Double mixture systems When optimizing the macrocation and the macroanion compositions separately, they are both considered as a mixture system: cations: K>#Ca>#Mg>"1
(4)
respectively (Schrevens, 1988; De Rijck, 1996). M "1 G G M "M "0)5
(ionic balance constraint)
m "M or: K>#Ca>#Mg>"0)5 G G
(6) (7) (8)
where K>, Ca> and Mg> are expressed in proportions m "M or: NO\#H PO\#SO\"0)5 (9) G G where NO\, H PO\ and SO\ are expressed in pro portions. The dimensionality of this double mixture factor space equals 4 (sum of the independent components in both mixture systems).
2.3. Constraints Beside the ionic balance and the mixture constraints, other chemical and physiological constraints need to be taken into account (De Rijck & Schrevens, 1997a, 1997b, 1998c, 1998d). These constraints make it impossible to experiment over the whole factor space in hydroponic plant nutritional research (De Rijck & Schrevens, 1999a). If the experimental subregion is homomorphic with the whole factor space, the subregion can be transformed in the whole factor space, using a pseudocomponent transformation (Kurotory, 1966). The use of double mixture pseudocomponent proportions results in minimum correlations between the experimental factors.
where K>, Ca> and Mg> are expressed in proportions; anions: NO\#H PO\#SO\"1
(5)
where NO\, H PO\ and SO\ are expressed in propor tions. Each mixture system is optimized separately. When optimizing the macrocation and the macroanion composition simultaneously, the macronutrient composition is considered as a &double mixture system'. In a double mixture system, the mixture components are called &major components' M , which in turn are also G mixture systems, consisting of one or more &minor components' m . The macronutrient composition is a double GG mixture system consisting of the two major components &cations' and &anions', both present in a proportion 0)5 (ionic balance constraint). The minor components are +K>, Ca>, Mg>, and +NO\, H PO\, SO\,,
2.4. Model To represent the response as a function of the mineral composition of the nutrient solution a second-degree double mixture model was selected. This model consists of 36 terms: 9 linear and 27 interaction terms and is obtained by combining the +3,2, canonical polynomial of the cations with the +3,2, canonical polynomial of the anions (De Rijck & Schrevens, 1999b) f (x)"b P #b P #b P IL IN IQ #b
P #b P #b P ILN ILQ INQ
#b
P #b P #b P A?L A?N A?Q
#b
P #b P #b P A?LN A?LQ A?NQ
MA C R O CA TI O N AN D M A CR O AN I O N O F N U TRI EN T S O LU TI O NS
P #b P #b P KEL KEN KEQ #b P #b P #b P KELN KELQ KENQ #b P #b P #b P IA?L IA?N IA?Q #b P #b P #b P IA?LN IA?LQ IA?NQ #b P #b P #b P IKEL IKEN IKEQ #b P #b P #b P IKELN IKEL IKENQ #b P #b P #b P A?KEL A?KEN A?KEQ #b P #b P #b P A?KELN A?KELQ A?KENQ (10) #b
where b is the parameter estimate for the respective
variable; P , P , 2 are the pseudocomponents proporI A? tions and P , P , ... are the pseudocomponent inter-acIL IN tions
2.5. Design The second-degree double mixture model is associated with a +3,2;3,2, double lattice design (Lambrakis, 1968, 1969) Fig. 1. The &n' in &+n , m !1; n , m !1, double lattice design' indicates the number of factors in the "rst (macrocations) and the second (macroanions) mixture system, while &m' is the number of levels for each factor in the respective mixture system. Combining each of the six design points of a +3,2, simplex lattice design (De Rijck
91
& Schrevens, 1998a) in the macroanions (large simplex on the background of Fig. 1) with each of the six design points of a +3,2, simplex lattice design in the macrocations (small simplices on the foreground of Fig. 1) yields the 36 design points of the +3,2;3,2, double lattice design (Table 1).
3. Material and methods 3.1. Experimental region In the double mixture factor space, an experimental subregion was selected homomorphic with the whole factor space (De Rijck & Schreven, 1999b), de"ned by the following lower bounds: 0)22)K>
(11)
0)045)Ca>
(12)
0)06)Mg>
(13)
0)28)NO\ (14) 0)045)H PO\ (15) 0.045)SO\ (16) The following pseudocomponent transformations transform the selected experimental region in the whole double mixture factor space: K>!0)22 P" I 0)35
(17)
Ca>!0)045 P " A? 0)35
(18)
Mg>!0)06 P " KE 0)35
(19)
NO\!0)28 P" L 0)26
(20)
H PO\!0)045 P" N 0)26
(21)
SO\!0)045 P" Q 0)26
(22)
In the double mixture pseudocomponents (P , P , P , I A? KE P , P and P ) optimal whole simplex designs can be used, L N Q with optimal collinearity diagnostics.
3.2. Experimental set-up Fig. 1. (3,2;3,2) double lattice design in the pseudocomponents (Pk , Pca , Pmg , Pn , Pp and Ps )
A +3,2;3,2, double lattice design (Lambrakis, 1968, 1969) is set-up in the double mixture factor space consisting
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Table 1 Experimental units matrix in the double mixture pseudocomponents Pk, Pca, Pmg, Pn, Pp and Ps Design point
P I
P A?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
0)5 0)5 0)5 0)5 0)5 0)5 0 0 0 0 0 0 0 0 0 0 0 0 0)25 0)25 0)25 0)25 0)25 0)25 0 0 0 0 0 0 0)25 0)25 0)25 0)25 0)25 0)25
0 0 0 0 0 0 0)5 0)5 0)5 0)5 0)5 0)5 0 0 0 0 0 0 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0 0 0 0 0 0
of 36 design points (Fig. 1). The 36 treatments are represented in the experimental units matrix (Table 1). All the nutrient solutions were prepared with demineralized water, which had a total salt concentration of 6)25 meq l\ and contained the same amount of micronutrients (Table 2). The macrocation and the macroanion composition of each of the 36 nutrient solutions can be calculated by transforming the double mixture pseudocomponent proportions (Table 1) to the double mixture component proportions, with the pseudocomponent transformations [Eqns (17)}(22)]. The double mixture component proportion is multiplied by 2 [ionic balance constraint, Eqn (7)] and by the total equivalent concentration (6)25 meq l\). This yields the ionic concentration in meq l\. For each of the 36 nutrient solutions, elemental speciation was calculated with the computer speciation program Geochem PC version 2.0 (Parker et al., 1995). The
P KE 0 0 0 0 0 0 0 0 0 0 0 0 0)5 0)5 0)5 0)5 0)5 0)5 0 0 0 0 0 0 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25
P L
P N
P Q
0)5 0 0 0)25 0 0)25 0)5 0 0 0)25 0 0)25 0)5 0 0 0)25 0 0)25 0)5 0 0 0)25 0 0)25 0)5 0 0 0)25 0 0)25 0)5 0 0 0)25 0 0)25
0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0
0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25
pH of the nutrient solutions varied between 4)171 and 4)178. On 6 June 1996, 2)5 g English ryegrass (¸olium perenne) seed was sown on polyurethane slabs (9)8 cm by 14)8 cm and 6 cm high). Four of these slabs were placed Table 2 Micro-element concentration in lmol l!1 of the nutrient solutions Micro-element Fe Mn Cu Zn B Mo
Concentration, kmol l!1 31)8 11 0)155 0)47 7)815 0)15
Salt supplied FeHEDTA 4)5% MnSO ) H O CuSO ) 5H O ZnSO ) 7H O H BO (NH ) Mo O ) 4H O
MA C R O CA TI O N AN D M A CR O AN I O N O F N U TRI EN T S O LU TI O NS
in a plastic Libra tray (18 cm by 135 cm and 9 cm high). During the "rst week, each Libra tray was "lled with 13 l of a speci"c nutrient solution, resulting in a nutrient solution level of 5 cm in height. From the second week onwards, the Libra trays were "lled with 7 l of nutrient solutions. To prevent the depletion of some ions with more than 5% of their quantitative amount, the nutrient solutions were renewed 3 times a week. The three extreme vertices of the cation factor space combined with the three extreme vertices of the anion factor space were repeated, resulting in nine nutrient solutions (1, 2, 3, 7, 8, 9, 13, 14 and 15). This resulted in 45 Libra trays (experimental units). The Libra trays were randomized in a greenhouse with climate control. On 28 June 1996, 22 days after sowing, the ryegrass was harvested. After drying for 3 days in a ventilated oven at 703C, the dry plant material from two polyurethane slabs was mixed together, forming one sample. A quantity of 0)5 g of the ground dry plant material was used for a wet digestion with 20 ml HNO /HClO (17/3, v/v). The calcium content in the inorganic plant digest was determined with atomic absorption spectrophotometry (Varian SpectrAA plus).
3.3. Statistical analysis The experiment is arranged as a hierarchical classi"cation, with the e!ect of the polyurethane slabs nested within the repetitions of the nutrient solution, and the e!ect of the repetitions nested within the nutrient solutions. Since only nine out of 36 nutrient solutions are repeated, the data are &unbalanced'. Therefore, analysis of variance is carried out, using the general linear model procedure (SAS, 1993). The &Reg' procedure (SAS, 1993) was used to calculate the mixture model that represents the calcium content of the ryegrass as a function of the mineral composition of the nutrient solution. 4. Results and discussion 4.1. Analysis of variance The e!ect of the mineral composition of the nutrient solutions was tested with the repetitions of the nutrient solutions as an error e!ect. The e!ect of the composition of the nutrient solutions on the calcium content of the ryegrass, was highly signi"cant (probability P"0)0001). 4.2. Regression analysis The complete 36 terms double mixture model was reduced using the SAS &stepwise' procedure. Parameters
93
were selected, which were signi"cantly di!erent from zero (F-statistic) at a con"dence level of 95%. The reduced second-degree double mixture model [Eqn (23)] consisted of 20 variables, making interpretation less complex. The adjusted coe$cient of determination R of the model equals 0)99, while the variation coe$cient of the model equals 4)5. Ca"219)81 P #218)25 P #223)86 P #667)86 P IL IN IQ A?L #670)23 P #704)74 P #239)39 P A?N A?Q A?LQ #184)03 P #137)27 P #123)51 P KEL KEN KEQ #826)10 P #424)05 P #283)61 P IA?L IA?N IA?Q #1934)28 P !289)43 P !226)60 P IA?LN IKEN IKEQ !1922)86 P #584)98 P #657)45 P IKELQ A?KEL A?KEN #463)85 P (23) A?KEQ where calcium is measured in mmol kg\ dry weight (DW); P , P , P , P , P , P are the pseudocomponents I A? KE L N Q proportions and P , P , ... are the interactions between IL IN the respective pseudocomponents. The model consists of the nine variables of the "rstdegree double mixture model extended with 11 variables of the second-degree double mixture model.
4.3. Response surface Equation (23) can be used to calculate the calcium content of the ryegrass for each nutritional composition within the experimental region, as a function of the pseudocomponent proportions. For example, to calculate the calcium content of the ryegrass at the highest calcium and the highest nitrate concentration, the pseudocomponents P , P , P , P , P and P in Eqn (23) I A? KE L N Q have to be replaced with 0, 0)5, 0, 0)5, 0 and 0, respectively, resulting in the following equation: Ca"(667)86)(0)5)(0)5)"167 mmol kg\DW
(24)
The double mixture proportions can be calculated with the pseudocomponents transformation Eqns (17)}(22) resulting in 0)22, 0)045, 0)06, 0)28, 0)045 and 0)045 for K>, Ca>, Mg>, NO\, H PO\ and SO\, respectively. Multiplying these proportions by 2 (ionic balance constraint) and by the total equivalent concentration (6)25 meq l\) yields a K>, Ca>, Mg>, NO\, H PO\ and SO\ concentration of 2)75, 2)75, 0)75, 5)125, 0)5625 and 0)5625 meq l\, respectively. For each nutritional composition within the experimental region, the calcium content of the ryegrass can be predicted in the same way.
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The pseudocomponent proportions are linearly related with the nutrient concentrations. An increase in the pseudocomponent proportions corresponds to an increase in the respective nutrient concentrations. Therefore, for further discussion of the results the pseudocomponents P , P , P , P , P and P are treated as the I A? KE L N Q nutrients K>, Ca>, Mg>, NO\, H PO\ and SO\, respectively. The response surface over the experimental region is determined by six variables (mixture components). Therefore, the option is taken to represent the response in two "gures (Figs 2 and 3). In Fig. 2, the simplex on the foreground represents the anion factor space. For each of the six anion proportions of the experimental design, the response is plotted for the whole cation factor space, superimposed on the anion simplex.
In these small simplices, the vertical axis represents the calcium content of the ryegrass in mmol kg\ dry weight, while the two horizontal axes represent the proportions of the pseudocomponents P and P . I A? At the point where the pseudocomponents P and P I A? are both zero, the pseudocomponent P equals 0)5. KE At all the other points, the proportion of the pseudocomponent P can easily be calculated as 0)5 minus the KE proportion P and minus the proportion P . So in each I A? point of the six response surfaces the proportions of the pseudocomponents P , P , P , P , P and P can be read, I A? KE L N Q as well as the expected calcium content of the ryegrass grown on a nutrient solution with that particular composition. As a function of the mineral composition of the nutrient solution, the calcium content of the ryegrass varied between 30)8 and 179)1 mmol kg\ DW (Fig. 2).
Fig. 2. Calcium content mmol kg!1 DW of the ryegrass (cation factor space superimposed on the anion simplex): Pk , Pca , Pmg , Pn , Pp and Ps pseudocomponent
MA C R O CA TI O N AN D M A CR O AN I O N O F N U TRI EN T S O LU TI O NS
95
Fig. 3. Calcium content mmol kg!1 DW of the ryegrass (anion factor space superimposed on the cation simplex): Pk , Pca , Pmg , Pn , Pp and Ps pseudocomponent
The e!ect of the anion composition of the nutrient solution on the calcium content of the ryegrass, was only limited in comparison with the e!ect of the cation composition. The calcium concentration of the nutrient solution had a strong positive e!ect on the calcium content of the ryegrass. The highest calcium content in the ryegrass was obtained at a high calcium proportion, a low potassium and magnesium proportion, an intermediate nitrate and sulphate proportion and a low dihydrogenphosphate proportion. Reducing the calcium concentration of the nutrient solution reduced the calcium content of the ryegrass in both the magnesium and the potassium direction. The lowest calcium content was obtained with a nutrient solution with a low potassium, a low calcium, a high magnesium and a low nitrate concentration in the nutrient solution.
As indicated by the model Eqn (23) there existed a synergistic interaction between the calcium and potassium and between the calcium and the magnesium proportion of the nutrient solution for the calcium content of the ryegrass. The negative parameter estimates for the variables P , P and P indicate IKEN IKEQ IKELQ the antagonistic interaction between potassium and magnesium. In Fig. 3, the cation factor space is represented by the simplex on the foreground. For each of the six cation proportions of the experimental design, the response is plotted for the whole anion factor space, superimposed on the cation simplex. In these small simplices, the two horizontal axes represent the nitrate and the dihydrogenphosphate proportions. The point where the nitrate and the dihydrogenphosphate proportion are both zero, the sulphate proportion equals 0.5.
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The calcium content of the ryegrass is mainly determined by the cation composition of the nutrient solution, while the anion composition had only a limited e!ect on the calcium content. Figure 3 clearly indicated the strong positive e!ect of the calcium proportion of the nutrient solution on the calcium content in the ryegrass The parameter estimate of the variable P in the IKELQ model, indicates the antagonistic interaction between nitrate and sulphate at intermediate potassium and magnesium proportions. If more than one response variable is considered, the nutritional zone with an optimal response for all the response variables can be considered as an optimal nutritional zone. The zone with optimal response can be determined in a mathematical way (simultaneous equations, partial derivatives), in a statistical way (multivariate analysis, canonical correlation analysis) or in a graphical way (response surfaces).
5. Conclusions The macrocation and the macroanion composition of hydroponical nutrient solutions can be considered as &mixture systems', so they can be optimized separately in a multifactorial way. Combining the obtained optimal macrocation and macroanion composition constitutes a good nutrient solution, but does not prove that it is the best possible nutrient solution. When the macronutrients of hydroponical nutrient solutions are considered as &double mixture systems', the macrocation and the macroanion composition can be optimized simultaneously in a multifactorial way. This type of experimental design makes it possible to "nd the optimal combination of macrocations and macroanions. To illustrate the methodology, a +3,2;3,2, double lattice design is established in the cation and the anion factor space (K>, Ca>, Mg>, NO\, H PO\ and SO\). This model is used to investigate the e!ect of the mineral composition of the nutrient solution on the calcium content of English ryegrass (¸olium perenne). The mineral composition of the nutrient solution signi"cantly in#uenced the calcium content of the ryegrass. A reduced second-degree double mixture model with 20 terms was selected to represent the response over the experimental region. The ryegrass cultivated on a nutrient solution with a high calcium, a low potassium, magnesium and dihydrogenphosphate and an intermediate nitrate and sulphate proportion had the highest calcium content.
Acknowledgement This research was supported by the &Vlaams Instituut voor de bevordering van het Wetenschappelijk-Technologisch onderzoek in de industrie' (I.W.T., Brussels, Belgium)
References Cornell J A (1981). Experiments with Mixtures. Designs, Models and the Analysis of Mixture Data. John Wiley and Sons, New York Cornell J A (1990). Experiments with Mixtures. Designs, Models and the Analysis of Mixture Data. John Wiley and sons, New York De Rijck G; Schrevens E (1994). Application of mixture-theory for the optimization of the composition of the nutrient solution. &International Symposium on growing media and plant nutrition in horticulture'. I.S.H.S. 10}16 September 1994. Naaldwijk, The Netherlands. Acta Horticulturae, 401, 283}291 De Rijck G (1996). Design and analysis of constrained mixture systems and chemical constraints of nutrient solutions as prerequisites in hydroponic plant nutritional research. Ph.D. Thesis, Katholieke Universiteit Leuven De Rijck G; Schrevens E (1997a). Elemental bioavailibility in nutrient solutions in relation to dissociation reactions. Journal of Plant Nutrition, 20(7 and 8), 901}910 De Rijck G; Schrevens E (1997b). pH in#uenced by the mineral composition of nutrient solutions. Journal of Plant Nutrition, 20(7 and 8), 911}923 De Rijck G; Schrevens E (1998a). Multifactorial optimization of the nutritient solution for hydroponic grown chicory plants. Scientia Horticulturae, 76, 149}159 De Rijck G; Schrevens E (1998c). Elemental bioavailability in nutrient solutions in relation to complexation reactions. Journal of Plant Nutrition, 21(5), 849}859 De Rijck G; Schrevens E (1998d). Elemental bioavailability in nutrient solutions in relation to precipitation reactions. Journal of Plant Nutrition, 21(10), 2103}2113 De Rijck G; Schrevens E (1999a). Chemical feasibility region for nutrient solutions in hyroponic plant nutrition. Journal of Plant Nutrition, 22(2) De Rijck G; Schrevens E (1999b). Guidelines to optimise the macrocation and the macroanion composition of nutrient solutions, using mixture theory. Journal of Agricultural Engineering Research, 72(4), 355}362 Hewitt E J (1966). Sand and water culture methods used in the study of plant nutrition. Technical communication No 22 (revised). Commonwealth Bureau of Horticulture and Plantation Crops, East malling, Maidstone, Kent, England Kurotory J S (1966). Experiments with mixtures of components having lower bounds. Industrial Quality Control, 22, 592}596 Lambrakis D P (1968). Experiments with mixtures: a generalisation of the simplex lattice design. Journal of the Royal Statistical Society, B 30, 123}136 Lambrakis D P (1969). Experiments with mixtures: estimated regression function of the multiple lattice design. Journal of the Royal Statistical Society, B 31, 276}284
MA C R O CA TI O N AN D M A CR O AN I O N O F N U TRI EN T S O LU TI O NS
Parker D R; Norvell W A; Chaney R L (1995). GEOCHEMPC: a chemical speciation program for IBM and compatible personal computers. In: Chemical Equilibrium and Reaction Models. pp 253}269. (Loeppert RH et al. eds) SSSA Spec. Publ. 42, SSSA, ASA, Madison, WI SAS (1993). Statistical Analysis System for Windows 3.2, release 6.11
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Schrevens E (1988). Design and analysis of mixture systems: application in hydroponic plant nutritional research. Ph.D. Thesis. Katholieke Universiteit Leuven Schrevens E; Cornell J (1993). Design and analysis of mixture systems. Applications in hydroponic plant nutritional research. Plant and Soil, 154, 45}52
Methodology to simultaneously optimize the Macrocation and the Macroanion Composition of Nutrient Solutions, using Mixture Theory G. De Rijck; E. Schrevens Department of Applied Plant Sciences, Faculty of Agricultural and Applied Biological Sciences, K.U. Leuven, Willem de Croylaan 42, B-3001 Heverlee (Belgium); e-mail of the corresponding author: [email protected] (Received 8 December 1998; accepted in revised form 5 August 2000; published online 22 September 2000)
To illustrate the use of mixture theory to simultaneously optimize the macrocation and the macroanion composition of a hydroponical nutrient solution, a +3,2;3,2, double lattice design was established in the double mixture factor space (K>, Ca>, Mg>, NO\, H PO\ and SO\). Only one response variable was considered: the calcium content of English ryegrass. The macronutrient composition signi"cantly a!ected the calcium content of the ryegrass. A reduced second-degree double mixture model with 20 instead of 36 parameters was selected to represent the response surface over the experimental region. The macrocation composition had a more pronounced e!ect on the calcium content of the ryegrass than the macroanion composition. 2001 Silsoe Research Institute
Optimizing the macrocation and the macroanion composition simultaneously, demands a more extensive experimental design and makes it possible to prove which combination of the macrocations and macroanions yields the best results. This research elaborates a justi"ed methodology to simultaneously optimize, for one total ionic strength, the concentration of the six essential macronutrients (K>, Ca>, Mg>, NO\, H PO\ and SO\) of hydroponical nutrient solutions. The experiment is carried out with English ryegrass (¸olium perenne) cultivated in hydroponics. To illustrate the elaborated methodology, only one dependent variable is considered: the calcium content of the ryegrass.
1. Introduction Nutrient solutions are aqueous solutions of inorganic ions, submitted to the ionic balance constraint. This constraint de"nes the cation and the anion composition of nutrient solutions as &mixture systems' (Schrevens, 1988; Schrevens & Cornell, 1993; De Rijck & Schrevens, 1994; De Rijck, 1996; De Rijck & Schrevens, 1998a) and makes it impossible to experiment with the mineral composition of nutrient solutions in a classical orthogonal or in a unifactorial way. A change in the concentration of one ion must be accompanied by either a corresponding change for an ion of the opposite charge, a complementary change for other ions of the same charge or both (Hewitt, 1966). Therefore, experimenting with the mineral composition of nutrient solutions is only possible in a multifactorial way. The previous research (De Rijck & Schrevens, 1998b) elaborated a methodology to optimize the macrocation and the macroanion composition, separately. This methodology demands only a limited amount of experimental treatments. The combination of the optimal macrocation and the optimal macroanion composition constitutes a good nutrient solution. However, optimizing the macrocation and the macroanion composition separately does not prove that their combination constitutes the best possible nutrient solution. 0021-8634/01/010089#09 $35.00/0
2. Theoretical considerations 2.1. Mixture systems The properties of the mixture systems are determined by the proportions of their components, rather than by their quantitative amount. The proportion of each component varies between zero and one. The sum of the proportions of all mixture components equals one (Cornell, 1981, 1990). For a mixture of q components, with x G 89
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the proportion of the ith mixture component, this results in the following mixture constraints: 0)x )1 for i"1, 2, 3, 2, q G
(1)
O x "1 G G
(2)
The ionic balance constraint postulates that the sum of the cations equals the sum of the anions, expressed in eq l\ and de"nes the cation and the anion composition of nutrient solutions as mixture systems. Simplifying a nutrient solution to the six essential macronutrients, neglecting H> and OH\ ions and the possible dissociation forms of H PO\ renders: [K>]#[Ca>]#[Mg>] "[NO\]#[H PO\]#[SO\]"C
(3)
where: K>, Ca>, Mg>, NO\, H PO\ and SO\ are expressed in eq l\ and C the total equivalent concentration. Dividing each cation (anion) concentration by the total concentration of cations (anions) expressed in eq l\ yields the proportion of each cation (anion).
2.2. Double mixture systems When optimizing the macrocation and the macroanion compositions separately, they are both considered as a mixture system: cations: K>#Ca>#Mg>"1
(4)
respectively (Schrevens, 1988; De Rijck, 1996). M "1 G G M "M "0)5
(ionic balance constraint)
m "M or: K>#Ca>#Mg>"0)5 G G
(6) (7) (8)
where K>, Ca> and Mg> are expressed in proportions m "M or: NO\#H PO\#SO\"0)5 (9) G G where NO\, H PO\ and SO\ are expressed in pro portions. The dimensionality of this double mixture factor space equals 4 (sum of the independent components in both mixture systems).
2.3. Constraints Beside the ionic balance and the mixture constraints, other chemical and physiological constraints need to be taken into account (De Rijck & Schrevens, 1997a, 1997b, 1998c, 1998d). These constraints make it impossible to experiment over the whole factor space in hydroponic plant nutritional research (De Rijck & Schrevens, 1999a). If the experimental subregion is homomorphic with the whole factor space, the subregion can be transformed in the whole factor space, using a pseudocomponent transformation (Kurotory, 1966). The use of double mixture pseudocomponent proportions results in minimum correlations between the experimental factors.
where K>, Ca> and Mg> are expressed in proportions; anions: NO\#H PO\#SO\"1
(5)
where NO\, H PO\ and SO\ are expressed in propor tions. Each mixture system is optimized separately. When optimizing the macrocation and the macroanion composition simultaneously, the macronutrient composition is considered as a &double mixture system'. In a double mixture system, the mixture components are called &major components' M , which in turn are also G mixture systems, consisting of one or more &minor components' m . The macronutrient composition is a double GG mixture system consisting of the two major components &cations' and &anions', both present in a proportion 0)5 (ionic balance constraint). The minor components are +K>, Ca>, Mg>, and +NO\, H PO\, SO\,,
2.4. Model To represent the response as a function of the mineral composition of the nutrient solution a second-degree double mixture model was selected. This model consists of 36 terms: 9 linear and 27 interaction terms and is obtained by combining the +3,2, canonical polynomial of the cations with the +3,2, canonical polynomial of the anions (De Rijck & Schrevens, 1999b) f (x)"b P #b P #b P IL IN IQ #b
P #b P #b P ILN ILQ INQ
#b
P #b P #b P A?L A?N A?Q
#b
P #b P #b P A?LN A?LQ A?NQ
MA C R O CA TI O N AN D M A CR O AN I O N O F N U TRI EN T S O LU TI O NS
P #b P #b P KEL KEN KEQ #b P #b P #b P KELN KELQ KENQ #b P #b P #b P IA?L IA?N IA?Q #b P #b P #b P IA?LN IA?LQ IA?NQ #b P #b P #b P IKEL IKEN IKEQ #b P #b P #b P IKELN IKEL IKENQ #b P #b P #b P A?KEL A?KEN A?KEQ #b P #b P #b P A?KELN A?KELQ A?KENQ (10) #b
where b is the parameter estimate for the respective
variable; P , P , 2 are the pseudocomponents proporI A? tions and P , P , ... are the pseudocomponent inter-acIL IN tions
2.5. Design The second-degree double mixture model is associated with a +3,2;3,2, double lattice design (Lambrakis, 1968, 1969) Fig. 1. The &n' in &+n , m !1; n , m !1, double lattice design' indicates the number of factors in the "rst (macrocations) and the second (macroanions) mixture system, while &m' is the number of levels for each factor in the respective mixture system. Combining each of the six design points of a +3,2, simplex lattice design (De Rijck
91
& Schrevens, 1998a) in the macroanions (large simplex on the background of Fig. 1) with each of the six design points of a +3,2, simplex lattice design in the macrocations (small simplices on the foreground of Fig. 1) yields the 36 design points of the +3,2;3,2, double lattice design (Table 1).
3. Material and methods 3.1. Experimental region In the double mixture factor space, an experimental subregion was selected homomorphic with the whole factor space (De Rijck & Schreven, 1999b), de"ned by the following lower bounds: 0)22)K>
(11)
0)045)Ca>
(12)
0)06)Mg>
(13)
0)28)NO\ (14) 0)045)H PO\ (15) 0.045)SO\ (16) The following pseudocomponent transformations transform the selected experimental region in the whole double mixture factor space: K>!0)22 P" I 0)35
(17)
Ca>!0)045 P " A? 0)35
(18)
Mg>!0)06 P " KE 0)35
(19)
NO\!0)28 P" L 0)26
(20)
H PO\!0)045 P" N 0)26
(21)
SO\!0)045 P" Q 0)26
(22)
In the double mixture pseudocomponents (P , P , P , I A? KE P , P and P ) optimal whole simplex designs can be used, L N Q with optimal collinearity diagnostics.
3.2. Experimental set-up Fig. 1. (3,2;3,2) double lattice design in the pseudocomponents (Pk , Pca , Pmg , Pn , Pp and Ps )
A +3,2;3,2, double lattice design (Lambrakis, 1968, 1969) is set-up in the double mixture factor space consisting
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G . D E R IJCK ; E . S CH R EV EN S
Table 1 Experimental units matrix in the double mixture pseudocomponents Pk, Pca, Pmg, Pn, Pp and Ps Design point
P I
P A?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
0)5 0)5 0)5 0)5 0)5 0)5 0 0 0 0 0 0 0 0 0 0 0 0 0)25 0)25 0)25 0)25 0)25 0)25 0 0 0 0 0 0 0)25 0)25 0)25 0)25 0)25 0)25
0 0 0 0 0 0 0)5 0)5 0)5 0)5 0)5 0)5 0 0 0 0 0 0 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0 0 0 0 0 0
of 36 design points (Fig. 1). The 36 treatments are represented in the experimental units matrix (Table 1). All the nutrient solutions were prepared with demineralized water, which had a total salt concentration of 6)25 meq l\ and contained the same amount of micronutrients (Table 2). The macrocation and the macroanion composition of each of the 36 nutrient solutions can be calculated by transforming the double mixture pseudocomponent proportions (Table 1) to the double mixture component proportions, with the pseudocomponent transformations [Eqns (17)}(22)]. The double mixture component proportion is multiplied by 2 [ionic balance constraint, Eqn (7)] and by the total equivalent concentration (6)25 meq l\). This yields the ionic concentration in meq l\. For each of the 36 nutrient solutions, elemental speciation was calculated with the computer speciation program Geochem PC version 2.0 (Parker et al., 1995). The
P KE 0 0 0 0 0 0 0 0 0 0 0 0 0)5 0)5 0)5 0)5 0)5 0)5 0 0 0 0 0 0 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25 0)25
P L
P N
P Q
0)5 0 0 0)25 0 0)25 0)5 0 0 0)25 0 0)25 0)5 0 0 0)25 0 0)25 0)5 0 0 0)25 0 0)25 0)5 0 0 0)25 0 0)25 0)5 0 0 0)25 0 0)25
0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0
0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25 0 0 0)5 0 0)25 0)25
pH of the nutrient solutions varied between 4)171 and 4)178. On 6 June 1996, 2)5 g English ryegrass (¸olium perenne) seed was sown on polyurethane slabs (9)8 cm by 14)8 cm and 6 cm high). Four of these slabs were placed Table 2 Micro-element concentration in lmol l!1 of the nutrient solutions Micro-element Fe Mn Cu Zn B Mo
Concentration, kmol l!1 31)8 11 0)155 0)47 7)815 0)15
Salt supplied FeHEDTA 4)5% MnSO ) H O CuSO ) 5H O ZnSO ) 7H O H BO (NH ) Mo O ) 4H O
MA C R O CA TI O N AN D M A CR O AN I O N O F N U TRI EN T S O LU TI O NS
in a plastic Libra tray (18 cm by 135 cm and 9 cm high). During the "rst week, each Libra tray was "lled with 13 l of a speci"c nutrient solution, resulting in a nutrient solution level of 5 cm in height. From the second week onwards, the Libra trays were "lled with 7 l of nutrient solutions. To prevent the depletion of some ions with more than 5% of their quantitative amount, the nutrient solutions were renewed 3 times a week. The three extreme vertices of the cation factor space combined with the three extreme vertices of the anion factor space were repeated, resulting in nine nutrient solutions (1, 2, 3, 7, 8, 9, 13, 14 and 15). This resulted in 45 Libra trays (experimental units). The Libra trays were randomized in a greenhouse with climate control. On 28 June 1996, 22 days after sowing, the ryegrass was harvested. After drying for 3 days in a ventilated oven at 703C, the dry plant material from two polyurethane slabs was mixed together, forming one sample. A quantity of 0)5 g of the ground dry plant material was used for a wet digestion with 20 ml HNO /HClO (17/3, v/v). The calcium content in the inorganic plant digest was determined with atomic absorption spectrophotometry (Varian SpectrAA plus).
3.3. Statistical analysis The experiment is arranged as a hierarchical classi"cation, with the e!ect of the polyurethane slabs nested within the repetitions of the nutrient solution, and the e!ect of the repetitions nested within the nutrient solutions. Since only nine out of 36 nutrient solutions are repeated, the data are &unbalanced'. Therefore, analysis of variance is carried out, using the general linear model procedure (SAS, 1993). The &Reg' procedure (SAS, 1993) was used to calculate the mixture model that represents the calcium content of the ryegrass as a function of the mineral composition of the nutrient solution. 4. Results and discussion 4.1. Analysis of variance The e!ect of the mineral composition of the nutrient solutions was tested with the repetitions of the nutrient solutions as an error e!ect. The e!ect of the composition of the nutrient solutions on the calcium content of the ryegrass, was highly signi"cant (probability P"0)0001). 4.2. Regression analysis The complete 36 terms double mixture model was reduced using the SAS &stepwise' procedure. Parameters
93
were selected, which were signi"cantly di!erent from zero (F-statistic) at a con"dence level of 95%. The reduced second-degree double mixture model [Eqn (23)] consisted of 20 variables, making interpretation less complex. The adjusted coe$cient of determination R of the model equals 0)99, while the variation coe$cient of the model equals 4)5. Ca"219)81 P #218)25 P #223)86 P #667)86 P IL IN IQ A?L #670)23 P #704)74 P #239)39 P A?N A?Q A?LQ #184)03 P #137)27 P #123)51 P KEL KEN KEQ #826)10 P #424)05 P #283)61 P IA?L IA?N IA?Q #1934)28 P !289)43 P !226)60 P IA?LN IKEN IKEQ !1922)86 P #584)98 P #657)45 P IKELQ A?KEL A?KEN #463)85 P (23) A?KEQ where calcium is measured in mmol kg\ dry weight (DW); P , P , P , P , P , P are the pseudocomponents I A? KE L N Q proportions and P , P , ... are the interactions between IL IN the respective pseudocomponents. The model consists of the nine variables of the "rstdegree double mixture model extended with 11 variables of the second-degree double mixture model.
4.3. Response surface Equation (23) can be used to calculate the calcium content of the ryegrass for each nutritional composition within the experimental region, as a function of the pseudocomponent proportions. For example, to calculate the calcium content of the ryegrass at the highest calcium and the highest nitrate concentration, the pseudocomponents P , P , P , P , P and P in Eqn (23) I A? KE L N Q have to be replaced with 0, 0)5, 0, 0)5, 0 and 0, respectively, resulting in the following equation: Ca"(667)86)(0)5)(0)5)"167 mmol kg\DW
(24)
The double mixture proportions can be calculated with the pseudocomponents transformation Eqns (17)}(22) resulting in 0)22, 0)045, 0)06, 0)28, 0)045 and 0)045 for K>, Ca>, Mg>, NO\, H PO\ and SO\, respectively. Multiplying these proportions by 2 (ionic balance constraint) and by the total equivalent concentration (6)25 meq l\) yields a K>, Ca>, Mg>, NO\, H PO\ and SO\ concentration of 2)75, 2)75, 0)75, 5)125, 0)5625 and 0)5625 meq l\, respectively. For each nutritional composition within the experimental region, the calcium content of the ryegrass can be predicted in the same way.
94
G . D E R IJCK ; E . S CH R EV EN S
The pseudocomponent proportions are linearly related with the nutrient concentrations. An increase in the pseudocomponent proportions corresponds to an increase in the respective nutrient concentrations. Therefore, for further discussion of the results the pseudocomponents P , P , P , P , P and P are treated as the I A? KE L N Q nutrients K>, Ca>, Mg>, NO\, H PO\ and SO\, respectively. The response surface over the experimental region is determined by six variables (mixture components). Therefore, the option is taken to represent the response in two "gures (Figs 2 and 3). In Fig. 2, the simplex on the foreground represents the anion factor space. For each of the six anion proportions of the experimental design, the response is plotted for the whole cation factor space, superimposed on the anion simplex.
In these small simplices, the vertical axis represents the calcium content of the ryegrass in mmol kg\ dry weight, while the two horizontal axes represent the proportions of the pseudocomponents P and P . I A? At the point where the pseudocomponents P and P I A? are both zero, the pseudocomponent P equals 0)5. KE At all the other points, the proportion of the pseudocomponent P can easily be calculated as 0)5 minus the KE proportion P and minus the proportion P . So in each I A? point of the six response surfaces the proportions of the pseudocomponents P , P , P , P , P and P can be read, I A? KE L N Q as well as the expected calcium content of the ryegrass grown on a nutrient solution with that particular composition. As a function of the mineral composition of the nutrient solution, the calcium content of the ryegrass varied between 30)8 and 179)1 mmol kg\ DW (Fig. 2).
Fig. 2. Calcium content mmol kg!1 DW of the ryegrass (cation factor space superimposed on the anion simplex): Pk , Pca , Pmg , Pn , Pp and Ps pseudocomponent
MA C R O CA TI O N AN D M A CR O AN I O N O F N U TRI EN T S O LU TI O NS
95
Fig. 3. Calcium content mmol kg!1 DW of the ryegrass (anion factor space superimposed on the cation simplex): Pk , Pca , Pmg , Pn , Pp and Ps pseudocomponent
The e!ect of the anion composition of the nutrient solution on the calcium content of the ryegrass, was only limited in comparison with the e!ect of the cation composition. The calcium concentration of the nutrient solution had a strong positive e!ect on the calcium content of the ryegrass. The highest calcium content in the ryegrass was obtained at a high calcium proportion, a low potassium and magnesium proportion, an intermediate nitrate and sulphate proportion and a low dihydrogenphosphate proportion. Reducing the calcium concentration of the nutrient solution reduced the calcium content of the ryegrass in both the magnesium and the potassium direction. The lowest calcium content was obtained with a nutrient solution with a low potassium, a low calcium, a high magnesium and a low nitrate concentration in the nutrient solution.
As indicated by the model Eqn (23) there existed a synergistic interaction between the calcium and potassium and between the calcium and the magnesium proportion of the nutrient solution for the calcium content of the ryegrass. The negative parameter estimates for the variables P , P and P indicate IKEN IKEQ IKELQ the antagonistic interaction between potassium and magnesium. In Fig. 3, the cation factor space is represented by the simplex on the foreground. For each of the six cation proportions of the experimental design, the response is plotted for the whole anion factor space, superimposed on the cation simplex. In these small simplices, the two horizontal axes represent the nitrate and the dihydrogenphosphate proportions. The point where the nitrate and the dihydrogenphosphate proportion are both zero, the sulphate proportion equals 0.5.
96
G . D E R IJCK ; E . S CH R EV EN S
The calcium content of the ryegrass is mainly determined by the cation composition of the nutrient solution, while the anion composition had only a limited e!ect on the calcium content. Figure 3 clearly indicated the strong positive e!ect of the calcium proportion of the nutrient solution on the calcium content in the ryegrass The parameter estimate of the variable P in the IKELQ model, indicates the antagonistic interaction between nitrate and sulphate at intermediate potassium and magnesium proportions. If more than one response variable is considered, the nutritional zone with an optimal response for all the response variables can be considered as an optimal nutritional zone. The zone with optimal response can be determined in a mathematical way (simultaneous equations, partial derivatives), in a statistical way (multivariate analysis, canonical correlation analysis) or in a graphical way (response surfaces).
5. Conclusions The macrocation and the macroanion composition of hydroponical nutrient solutions can be considered as &mixture systems', so they can be optimized separately in a multifactorial way. Combining the obtained optimal macrocation and macroanion composition constitutes a good nutrient solution, but does not prove that it is the best possible nutrient solution. When the macronutrients of hydroponical nutrient solutions are considered as &double mixture systems', the macrocation and the macroanion composition can be optimized simultaneously in a multifactorial way. This type of experimental design makes it possible to "nd the optimal combination of macrocations and macroanions. To illustrate the methodology, a +3,2;3,2, double lattice design is established in the cation and the anion factor space (K>, Ca>, Mg>, NO\, H PO\ and SO\). This model is used to investigate the e!ect of the mineral composition of the nutrient solution on the calcium content of English ryegrass (¸olium perenne). The mineral composition of the nutrient solution signi"cantly in#uenced the calcium content of the ryegrass. A reduced second-degree double mixture model with 20 terms was selected to represent the response over the experimental region. The ryegrass cultivated on a nutrient solution with a high calcium, a low potassium, magnesium and dihydrogenphosphate and an intermediate nitrate and sulphate proportion had the highest calcium content.
Acknowledgement This research was supported by the &Vlaams Instituut voor de bevordering van het Wetenschappelijk-Technologisch onderzoek in de industrie' (I.W.T., Brussels, Belgium)
References Cornell J A (1981). Experiments with Mixtures. Designs, Models and the Analysis of Mixture Data. John Wiley and Sons, New York Cornell J A (1990). Experiments with Mixtures. Designs, Models and the Analysis of Mixture Data. John Wiley and sons, New York De Rijck G; Schrevens E (1994). Application of mixture-theory for the optimization of the composition of the nutrient solution. &International Symposium on growing media and plant nutrition in horticulture'. I.S.H.S. 10}16 September 1994. Naaldwijk, The Netherlands. Acta Horticulturae, 401, 283}291 De Rijck G (1996). Design and analysis of constrained mixture systems and chemical constraints of nutrient solutions as prerequisites in hydroponic plant nutritional research. Ph.D. Thesis, Katholieke Universiteit Leuven De Rijck G; Schrevens E (1997a). Elemental bioavailibility in nutrient solutions in relation to dissociation reactions. Journal of Plant Nutrition, 20(7 and 8), 901}910 De Rijck G; Schrevens E (1997b). pH in#uenced by the mineral composition of nutrient solutions. Journal of Plant Nutrition, 20(7 and 8), 911}923 De Rijck G; Schrevens E (1998a). Multifactorial optimization of the nutritient solution for hydroponic grown chicory plants. Scientia Horticulturae, 76, 149}159 De Rijck G; Schrevens E (1998c). Elemental bioavailability in nutrient solutions in relation to complexation reactions. Journal of Plant Nutrition, 21(5), 849}859 De Rijck G; Schrevens E (1998d). Elemental bioavailability in nutrient solutions in relation to precipitation reactions. Journal of Plant Nutrition, 21(10), 2103}2113 De Rijck G; Schrevens E (1999a). Chemical feasibility region for nutrient solutions in hyroponic plant nutrition. Journal of Plant Nutrition, 22(2) De Rijck G; Schrevens E (1999b). Guidelines to optimise the macrocation and the macroanion composition of nutrient solutions, using mixture theory. Journal of Agricultural Engineering Research, 72(4), 355}362 Hewitt E J (1966). Sand and water culture methods used in the study of plant nutrition. Technical communication No 22 (revised). Commonwealth Bureau of Horticulture and Plantation Crops, East malling, Maidstone, Kent, England Kurotory J S (1966). Experiments with mixtures of components having lower bounds. Industrial Quality Control, 22, 592}596 Lambrakis D P (1968). Experiments with mixtures: a generalisation of the simplex lattice design. Journal of the Royal Statistical Society, B 30, 123}136 Lambrakis D P (1969). Experiments with mixtures: estimated regression function of the multiple lattice design. Journal of the Royal Statistical Society, B 31, 276}284
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Parker D R; Norvell W A; Chaney R L (1995). GEOCHEMPC: a chemical speciation program for IBM and compatible personal computers. In: Chemical Equilibrium and Reaction Models. pp 253}269. (Loeppert RH et al. eds) SSSA Spec. Publ. 42, SSSA, ASA, Madison, WI SAS (1993). Statistical Analysis System for Windows 3.2, release 6.11
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Schrevens E (1988). Design and analysis of mixture systems: application in hydroponic plant nutritional research. Ph.D. Thesis. Katholieke Universiteit Leuven Schrevens E; Cornell J (1993). Design and analysis of mixture systems. Applications in hydroponic plant nutritional research. Plant and Soil, 154, 45}52