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J. Agric. Engng Res. (1999) 73, 29 } 33 Article No. jaer.1998.0389, available online at http://www.idealibrary.com on Automated Composition Control of Nutrient Solution in Closed Soilless Culture Systems D. Savvas; G. Manos Technological Education Institution (T.E.I.) of Epirus, Faculty of Agricultural Technology, P.O. Box 110, Arta 47100, Greece; e-mail: savvas@teiep.gr (Received 27 April 1998; accepted in revised form 28 October 1998) A computer algorithm was developed to perform replenishment and reuse of the drain nutrient solution in close hydroponic systems on the basis of dispensing as many nutrients and water in it as were needed to maintain a target electrical conductivity in the irrigation solution. To achieve this in each watering application, dilution ratios of the concentrated fertilizer solutions are automatically adjusted through a computer program based on this algorithm, in relation to the volume and the electrical conductivity of the drain solution. It was proved that the returning nutrient solution could be corrected and reused e$ciently through this method. However, the concentration of P, Mn and Zn in the irrigation nutrient solution was extremely low in samples taken after continual reuse of the drain solution for a fortnight. Since the analysed irrigation solution was prepared using a large quantity of drain solution, it seems likely that reduced P, Zn and Mn concentrations in the former were due to the high pH of the latter, and this may have caused immobilization of these elements. ( 1999 Silsoe Research Institute 1. Introduction Plants take up nutrients and water in ratios which #uctuate widely during the growing period. 1,2 Therefore, uptake ratios of nutrient to water by plants cannot cor- respond continually to those prevailing in the nutrient solution, even if the latter has a balanced composition. Consequently, in substrate culture, a di!erent amount of drain solution with variable ionic composition is pro- duced after each irrigation application. As a result, when drain solution is reused, it should be replenished with variable amounts of nutrients and water at eacdh irriga- tion application. 3 This fact complicates automation of nutrient solution preparation in soilless culture, when the runo! water from the substrates is collected and reused. Particularly, when nutrient solution is prepared by auto- matic equipment which dilutes concentrated stock solu- tions according to preset ratios, recycling of drain water results in a progressive increase of the electrical conduct- ivity (EC) in the irrigation solution, accompanied by a rapid appearance of imbalances in nutrient ratios. However, the EC can be controlled and the appearance of nutrient imbalances may be drastically retarded if dilution ratios of stock solutions change automatically in each irrigation application, in accordance with the amount and the EC of the collected and reused drain solution. In this paper, the possibility of using an algorithm in special computer programs is discussed to control the recycling of the drain solution in hydroponic installa- tions in which concentrated solutions are automatically dispensed through fertilizer injection systems. In each watering cycle, dilution ratios of concentrated solutions are automatically calculated via this algorithm, depend- ing on current EC and amount of the reused drain solution, in order to prepare an irrigation nutrient solu- tion of a certain, desired electrical conductivity. Further- more, some results concerning the application of the method, are presented and discussed. 2. Theoretical considerations In soilless culture systems, the nutrient solution is usually prepared by various fertilizer injection systems, which dilute concentrated solutions with the irrigation 0021-8634/99/050029#05 $30.00/0 29 ( 1999 Silsoe Research Institute

Automated Composition Control of Nutrient Solution in Closed Soilless Culture Systems

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Page 1: Automated Composition Control of Nutrient Solution in Closed Soilless Culture Systems

J. Agric. Engng Res. (1999) 73, 29}33Article No. jaer.1998.0389, available online at http://www.idealibrary.com on

002

Automated Composition Control of Nutrient Solution in Closed SoillessCulture Systems

D. Savvas; G. Manos

Technological Education Institution (T.E.I.) of Epirus, Faculty of Agricultural Technology, P.O. Box 110, Arta 47100, Greece;e-mail: [email protected]

(Received 27 April 1998; accepted in revised form 28 October 1998)

A computer algorithm was developed to perform replenishment and reuse of the drain nutrient solution in closehydroponic systems on the basis of dispensing as many nutrients and water in it as were needed to maintaina target electrical conductivity in the irrigation solution. To achieve this in each watering application, dilutionratios of the concentrated fertilizer solutions are automatically adjusted through a computer program based onthis algorithm, in relation to the volume and the electrical conductivity of the drain solution. It was proved thatthe returning nutrient solution could be corrected and reused e$ciently through this method. However, theconcentration of P, Mn and Zn in the irrigation nutrient solution was extremely low in samples taken aftercontinual reuse of the drain solution for a fortnight. Since the analysed irrigation solution was prepared usinga large quantity of drain solution, it seems likely that reduced P, Zn and Mn concentrations in the former weredue to the high pH of the latter, and this may have caused immobilization of these elements.

( 1999 Silsoe Research Institute

1. Introduction

Plants take up nutrients and water in ratios which#uctuate widely during the growing period.1,2 Therefore,uptake ratios of nutrient to water by plants cannot cor-respond continually to those prevailing in the nutrientsolution, even if the latter has a balanced composition.Consequently, in substrate culture, a di!erent amount ofdrain solution with variable ionic composition is pro-duced after each irrigation application. As a result, whendrain solution is reused, it should be replenished withvariable amounts of nutrients and water at eacdh irriga-tion application.3 This fact complicates automation ofnutrient solution preparation in soilless culture, when theruno! water from the substrates is collected and reused.Particularly, when nutrient solution is prepared by auto-matic equipment which dilutes concentrated stock solu-tions according to preset ratios, recycling of drain waterresults in a progressive increase of the electrical conduct-ivity (EC) in the irrigation solution, accompanied bya rapid appearance of imbalances in nutrient ratios.However, the EC can be controlled and the appearanceof nutrient imbalances may be drastically retarded if

1-8634/99/050029#05 $30.00/0 29

dilution ratios of stock solutions change automatically ineach irrigation application, in accordance with theamount and the EC of the collected and reused drainsolution.

In this paper, the possibility of using an algorithm inspecial computer programs is discussed to control therecycling of the drain solution in hydroponic installa-tions in which concentrated solutions are automaticallydispensed through fertilizer injection systems. In eachwatering cycle, dilution ratios of concentrated solutionsare automatically calculated via this algorithm, depend-ing on current EC and amount of the reused drainsolution, in order to prepare an irrigation nutrient solu-tion of a certain, desired electrical conductivity. Further-more, some results concerning the application of themethod, are presented and discussed.

2. Theoretical considerations

In soilless culture systems, the nutrient solution isusually prepared by various fertilizer injection systems,which dilute concentrated solutions with the irrigation

( 1999 Silsoe Research Institute

Page 2: Automated Composition Control of Nutrient Solution in Closed Soilless Culture Systems

30 D. SAVVAS; G. MANOS

water in proper ratios. The ratios, in which the con-centrated solutions are diluted, are regulated eitherby dispensing stock solutions according to real-timemeasurements of EC in order to achieve a preset ECvalue, or by setting dilution proportions directly to thecontrolling system. In closed hydroponic installationsoperating according to the latter principle, the dilutionratios of stock solutions should be related to the amountand the EC of the drain solution in each watering cycle,in order to maintain a target EC in the irrigation solu-tion. To achieve this, a computer algorithm can be used,based on the following equation:

Ri"

Ai<t(E

t!E

w)

Et<t!E

r<r!E

w<t#E

w<r), i"1,2 , n, (1)

where Riare the ratios in which the n stock solutions

should be diluted with the mixture of drain solution andirrigation water (litres of water that are mixed with 1 l ofstock solution) to produce an outgoing solution of givenelectrical conductivity E

t; A

iare the water to fertilizer

dilution ratios, indicating the strength of the n stocksolutions compared to the irrigation nutrient solution,when the latter is prepared without reusing of any drainsolution; <

tand E

tare the desired volume in l and the

target EC in dS/m of the irrigation nutrient solutionto be prepared, respectively; E

wis the electrical conduct-

ivity in dS/m of the tap water; and <r

and Er

are thevolume in 1 and the EC in dS/m of the reused drainsolution, respectively. Furthermore, in each irrigationapplication, A

i, E

w, <

tand E

tare preset values, whereas

<r

and Er

are measured by sensors in real-time condi-tions. The chain of reasoning leading to Eqn (1) is de-scribed in Appendix 1.

If fertilizer dispensers giving a constant injection rateare used (peristaltic pumps, venturi, etc.), the amount ofeach stock solution added, when a fresh solution is pre-pared, is a linear relationship of injection time. For eachstock solution, the injection time ¹

iin s, is related to

Riby the expression

¹i"

<t

JRi

(2)

where J is the injection rate of the stock solution dis-pensers in 1/s.

Thus, in each irrigation application, the computer canautomatically calculate and control the duration of stocksolution injection through Eqn (2).

3. Materials and Methods

In a glasshouse used for soilless cultivation of cut#owers, a computer-controlled system for automatic

nutrient solution preparation, supply and recycling wasinstalled. The glasshouse was divided into four sectorswhich were equipped with separate solution ducts forsupply and recycling and planted with roses, gerbera,chrysanthemum and carnation, respectively. Plants werecultivated in containers "lled with pumice (0}4 mm).

The computer program was created in Windows 95environment using a visual programming language.It contains its own libraries and database and is multi-tasking. Some of the parameters that can be adjustedby entering preset values to the computer program are(a) volume of nutrient solution to be supplied, (b) dilu-tion ratios of stock solutions, and (c) pH and EC of theirrigation solution. However, direct EC adjustment ispossible only when the drain solution is reused, by settinga desired value for E

t. Moreover, the computer program

is capable of registering volume, pH and EC of the reuseddrain solution, as well as of the supplied fresh solution ineach irrigation cycle.

The solution is prepared in a cylindrical mixing tankby injecting up to "ve stock solutions to the irrigationwater in adjustable ratios. Since the diameter of thecylindrical mixing tank is known, the volume of water ornutrient solution into the mixing tank can be monitoredcontinually using a water pressure gauge and, thus, auto-matically controlled. The fertilizer injectors are peristalticpumps with a #ow rate of 20 l/h. After automatic adjust-ment of pH, the nutrient solution is supplied to theplants. Reuse of the drain solution is optional. When thedrain solution is not reused, the stock solutions areinjected in preset ratios having been given as A

i. How-

ever, if the returning solution is recycled, the dilutionratios of stock solutions R

iare adjusted automatically in

each watering cycle, through a computer program basedon Eqn (1).

To control the e!ectiveness of the algorithm basedon Eqn (1) under real conditions, mean EC values fromall irrigation applications over a fortnight period (25August}9 September) were calculated in each crop. Thiswas performed using the data registered automatically bythe computer program. During the fortnight period,drain nutrient solution was always recycled automati-cally with the aid of the computer program based onEqn (1). Moreover, at the beginning of the period (25August) as well as at the end (9 September), three sampleswere taken from the irrigation nutrient solution suppliedto roses. The nutrient solution samples were analysed inorder to determine the di!erences in nutrient concentra-tions due to recycling the drain solution for a particularperiod of 15 days.

Analyses of the nutrient solution samples were per-formed according to Eaton et al.4 The concentration ofK and Na was measured by #ame photometry, whereasCa, Mg, Fe, Mn, Zn and Cu were determined by atomic

Page 3: Automated Composition Control of Nutrient Solution in Closed Soilless Culture Systems

CLOSED SOILLESS CULTURE SYSTEMS 31

absorption spectrophotometry. Mineral phosphate wasestimated colorimetrically at 880 nm by the ascorbic acidmethod. Nitrate was reduced to nitrite by cadmium anddetermined colorimetrically at 543 nm by the diazo-method; NH`

4was determined by distillation and titra-

tion with H2SO

4; SO2~

4was measured by gravimetric

analysis, HCO~3

by titration with H2SO

4; and Cl~ by

silver nitrate precipitation. Boron was measured by theazomethane method.

4. Results and Discussion

As can be seen in Table 1, in all crops, mean EC valuesof the irrigation solution from a fortnight approximatedclosely to the desired levels for E

t. Consequently, Eqn (1)

can be incorporated in particular computer programs toautomatically inject the proper fertilizer quantities in thedrain solution, when it is replenished with nutrients andwater up to a target EC and a desired volume in order tobe reused. In all crops, however the mean EC values inTable 1 tend to be slightly higher than the target ECvalues. This is due to the fact that, in a few exceptionalcases, when the values of E

rand <

rfor the drain solution

were very high, the EC of the resulting irrigation nutrientsolution could not be reduced to the desired level of E

t. It

is a common observation in soilless culture that the EC ofthe drain solution is, in most cases, higher than the EC ofthe supplied fresh solution. For given values of E

t, A

i,

<tand E

win Eqn (1), as the values of E

tand <

rincrease,

the value of Ri

becomes negative. In these cases, thecomputer program based on Eqn (1) indicates no addi-tion of stock solution to the drain solution. However,when negative values of R

iare calculated through

Eqn (1), the EC of the resulting irrigation solution re-mains above the value of E

t, although the drain solution

is mixed to the desired value <twith tap water only. As

can be concluded by data presesnted by Sonneveld,5 theelectrical conductivity of the drain solution E can be

r

Table 1Target electrical conductivity (EC) in dS/m of the irrigationnutrient solution for four di4erent cut 6ower crops grownhydroponically and means of EC over a fortnight, achieved byautomatically calculating the dilution ratio of stock solutions

to replenish and reuse the excessive drain solution

Crop Target EC, Mean of measureddS/m EC, dS/m

Roses 1)90 1)97Gerbera 2)10 2)22Chrysanthemum 1)80 1)82Carnation 2)30 2)41

reduced by setting a value of Etnear the lower limits

of the optimal range. When the amount of nutrientsolution supplied is related to water consumption byfollowing a proper irrigation schedule, the volumes<r

can also be reduced. However, when the value of<r

becomes very low, the degree of salt leaching(100<

r/<

t) from the root environment may be inadequate.

Therefore, manipulation of Etwhen de"ning the settings

in the computer program seems to be the main tool toavoid calculation of negative values of R

ithrough

Eqn (1).As can be seen in Table 2, the concentrations of P, Zn

and Mn in the irrigation nutrient solution were reduceddramatically, compared to the initial target values, afterreplenishing and reusing the drain solution according toEqn (1) for a fortnight. Table 3 indicates that the irriga-tion nutrient solution sampled and analysed after 15 daysof recycling was prepared by using a large quantity ofdrain solution (high <

r) with a high value of E

r(2)67 dS/m), compared to the target EC (E

t"1)90 dS/m).

As a result, extremely small quantities of nutrients wereadded in the form of stock solution to achieve the valueof E

t. It is therefore obvious that the low concentrations

of P, Zn and Mn in the irrigation solution were due toa sharp decrease of their concentrations in the drainsolution. Since the concentration of these mineral ele-ments tends to decrease with increasing pH,6+8 their lowconcentrations in the drain solution should be ascribedto the high pH of the latter (Table 3) rather than to anintensive uptake rate by the plants. Therefore, to avoidinadequate supply of plants with P, Zn and Mn in closedsoilless culture systems, it is essential to restrict theamount of surplus solution that runs o!, up to a neces-sary 20}30% of the total solution supply.9

Moreover, as Table 2 indicates, after a fortnightperiod of replenishment and reuse of the drain solu-tion, the concentration of K and Fe was approximatelyhalved and that of NO

3and B was reduced slightly,

whereas Ca, Mg and SO4

tended to accumulate in theirrigation solution. Accumulation of bivalent ions suchas Ca, Mg and SO

4and depletion of K are well-

known alterations occurring in the composition of nutri-ent solutions when they are recycled.5 However, thedecrease of Fe concentration seems to be an e!ect of highpH in the drain solution. The reduction of Fe concentra-tion was less markedly, compared to P, Zn and Mn,probably due to the use of Fe-EDDHA, which reducesFe immobilization in solutions with pH higher thanoptimal.10

The concentrations of the ballast ions Na and Cl in theirrigation nutrient solution were approximately doubled,after continual reuse of drain solution for a fortnight(Table 2). However, the absolute Na and Cl concen-trations measured (37 and 56 mg/l, respectively) hardly

Page 4: Automated Composition Control of Nutrient Solution in Closed Soilless Culture Systems

Table 2Ionic concentration in mg/l of the irrigation nutrient solution for roses: (a) at the beginning of recycling and (b) after a fortnight

of drain solution reuse

Macro- Initial Concentration Micro- Initial Concentrationelement concentration, after a element concentration, after a

mg/l fortnight, mg/l mg/l fortnight, mg/l

Ca2` 140 178 Fe2` 0)90 0)40Mg2` 58 70 Mn2` 0)26 (0)05K` 287 130 Zn2` 0)17 (0)02Na` 16 37 B 0)23 0)19NH`

4!N 8 6

NO~3!N 203 183

H2PO~

4!P 37 5

SO2~4

!S 54 77CI~ 29 56HCO~

328 27

32 D. SAVVAS; G. MANOS

exceed even the NaCl concentration level of 1)5 mmol/lwhich is accepted as the upper limit for irrigation waterof good quality.11 Therefore, in soilless cultivation ofroses, Na and Cl accumulation in the nutrient solutiondue to recycling of drain solution does not approxi-mate to detrimental levels in short periods of up to 2}3weeks, when tap water with relatively low Na and Clconcentrations is available. However, even if the concen-trations of Na and Cl in the tap water are higher thanthe optimal range, resulting in a more intensive accu-mulation of them in the nutrient solution, a correspond-ing increase of target solution concentration E

tat, say,

weekly intervals may delay the time of nutrient solutionremoval.

In the current form, as presented in this paper, Eqn (1)is used to control the EC of the irrigation solution whenthe drain solution is reused. However, using speci"csensors, it would be possible to monitor the concentra-tions of single nutrients in the drain solution.3,12,13 Whensuch facilities are available for commercial application,Eqn (1) could also be used to maintain target concentra-

Table 3Data referring to automatic preparation of the irrigation nutrientsolution whose composition is given in Table 2 for roses at the end

of the fortnight

Parameter Targetvalues

Irrigationsolution

Drainsolution

Tap water

Volume, 1 160 167 101 66EC, dS/m 1)90 1)92 2)67 0)53pH 5)50 5)53 7)05 7)38

EC, electrical conductivity.

tions of single nutrients in the irrigation solution, aftera slight modi"cation.

5. Conclusions

The equation used to provide computer-automatedcalculations and adjustments of concentrated fertilizerinjections when the drain solution is replenished andreused in hydroponics, proved to be e$cient at maintain-ing a target electrical conductivity (EC) in the resultingirrigation solution. This was accomplished by relatingthe dilution ratios of the stock solutions to the volumeand the EC of the drain solution. Due to its comprehen-sive possibilities to support further automation, themathematical concept developed can be adopted incommercial hydroponics to facilitate automatic recyclingof the drain solution.

When a large amount of drain solution is producedafter a watering application (expressed as a percent-age of the nutrient solution supplied), target EC mightnot be achieved in the irrigation solution, even if noaddition of stock solutions is indicated by the com-puter program. A similar situation might be observed ifthe EC of the drain solution is much higher than thetarget EC of the irrigation solution. However, throughproper irrigation and nutrition management, based onsupply according to crop consumption, these problemscan be eliminated.

Moreover, in soilless culture, when the drain solu-tion is replenished and reused, it is essential to ensurean adequate supply of P, Mn and Zn to the crop,through a proper nutrition and watering schedule thatreduces the volume, the EC and the pH of the returningsolution.

Page 5: Automated Composition Control of Nutrient Solution in Closed Soilless Culture Systems

CLOSED SOILLESS CULTURE SYSTEMS 33

Acknowledgements

We thank Dr K. Adamidis, Department of Mathemat-ics, University of Ioannina, Greece, for his contributionto the pressentation of the mathematical conception, onwhich this work is based.

References

1 Van Goor B J; De Jager A; Voogt W Nutrient uptake bysome horticultural crops during the growing period.ISOSC, Proceedings 7th International Congress on Soil-less Culture, Wageningen, Netherlands, 1988 : 163}176

2 Savvas D; Lenz F NaK hrsto!aufnahme von Aubergine(Solanum melongena L.) in Hydrokultur (Nutrient uptakeby eggplant (Solanum melongena L.) in soilless culture).Gartenbauwissenschaft, 1995, 60(1), 29}33

3 Gieling T H; Van Os E A; De Jager A The application ofchemo-sensors and bio-sensors for soilless cultures. ActaHorticulturae, 1988, 230, 357}361

4 Eaton A D; Clesceri L S; Greenberg A E (Eds) StandardMethods for the Examination of Water and Wastewater,19th edn, Washington: American Public Health Associ-ation, 1995

5 Sonneveld C Items for application of macro-elements in soil-less culture. Acta Horticulturae, 1981, 126, 187}195

6 Robson A D; Pitman M G Interactions between nutrients inhigher plants. In: Encyclopedia of Plant Physiology, NewSeries, Vol. 15A: Inorganic Plant Nutrition (LaK uchli A;Bieleski RL, eds), pp. 154}156. Berlin: Springer, 1983

7 Sonneveld C; Voogt S J The application of manganese innutrient solutions for tomatoes grown in a recirculatingsystem. Acta Horticulturae, 1980, 98, 171}178

8 Marschner H Mineral Nutrition of Higher Plants, 2nd edn,London pp. 664}665. Academic Press, 1995

9 Sonneveld C Fertigation in the greenhouse industry. In:Proceedings of the Dahlia Greidinger International Sym-posium on Fertigation, Technion } Israel Institute ofTechnology, Haifa, Israel, 1995 : 121}140

10 Wreesman C T J Chelated micro-nutrients for soilless cul-ture. ISOSC, Proceedings 9th International Congresson Soilless Culture, Wageningen, Netherlands, 1996 :559}572

11 Van Os E A Dutch developments in soilless culture. Outlookon Agriculture, 1982, 11(4), 165}171

12 Albery W J; Haggett B G D; Svanberg R The development ofsensors for hydroponics. Biosensors, 1985, 1, 369}397

13 Morard P Possible use of ion selective electrodes for nutrientsolutions in recirculated systems. ISOSC, Proceedings 9thInternational Congress on Soilless Culture, Wageningen,Netherlands, 1996 : 291}298

14 De Kreij C; Voogt W; Van Den Bos A L; Baas R Voedin-gsoplossingen gesloten teeltsystemen (Nutrient solutionsfor closed cultivation systems). Brochures 1}16. ResearchStation for Floriculture and Glasshouse Vegetables (PBG),Naaldwijk, Netherlands, 1997

15 US Salinity Laboratory Sta4 Diagnosis and improvement ofsaline and alkali soils. United States Department of Agri-culture, Agriculture Handbook No 60, 1954

Appendix 1

Currently, for most vegetable and ornamental crops, thetarget EC of the nutrient solution supplied to the plants rangesfrom 1)1 dS/m for anthurium to 2)8 dS/m for tomatoes, whenthey are grown in closed hydroponic systems.14 Thus, whenstock solutions are injected to the mixture of the drain solutionand water, the EC change occurring in the latter is usually nothigher than 2)8 dS/m. However, from the graphical representa-tion of data obtained by US Salinity Laborator Sta!,15 it can beconcluded that EC di!erences of this magnitude and evenlarger, are almost linearly related to the corresponding saltconcentration changes in meq/l. Any discrepancy from linearitywithin this range of EC is negligible, especially with respect tothe fact that in applied hydroponics EC is used as a rathercoarse indicator for total salt concentration in the nutrientsolution. Thus, when various amounts of a certain concentratedsolution are added to the mixture of water and drain solution,the changes occurring in the total salt concentration C in meq/land in the electrical conductivity E in dS/m may be approxim-ately related to each other through the equation

*C"a*E (A1)

where a is a factor depending on the composition of the particu-lar concentrated solution.

Furthermore, the solution resulting from the mixture of drainsolution with tap water to a volume of<

thas electrical conduct-

ivity Em

given by

Em"

<r<t

Er#

(<t!<

r)

<t

Ew

(A2)

During nutrient solution preparation, addition of stock solu-tions increases the electrical conductivity by

*E1"E

t!E

w(A3)

when no recycling is applied and by

*E2"E

t!E

m(A4)

when the drain solution is mixed with tap water and reused.The changes in total salt concentration in meq/l occurring inthe irrigation solution due to the addition of the stock solutionsare de"ned as *C

f1in the former case and *C

f2in the latter.

Since the fertilizer concentration in each of the stock solutionsis identical, regardless of recycling application or not, the re-spective dilution ratios of the stock solutions A

iand R

iare

related with *Cf1

and *Cf2

through the equation

*Cf1

*Cf2

"

Ri

Ai

(A5)

Moreover, since *Cf1

and *Cf2

are concentration di!erencesbrought about by injecting di!erent amounts of the same saltconcentrate, the factor a is identical when substituting each ofEqns (A3) and (A4) in Eqn (A1):

*Cf1"a (E

t!E

w) (A6)

*Cf2"a (E

t!E

m) (A7)

Thus, the dilution ratios Rifor the n stock solutions in Eqn (1)

are derived by substituting Eqns (A6), (A7) and (A2) intoEqn (A5) and manipulating the resulting expression.