Measurement and data interpretation of the freezing point depression of milks

Printklfn

Great Britain. All rights rcwrvcd 02hO-X77JiY6 $lS.OO +0.(N)

Measurement and Data Interpretation of the Freezing Point Depression of Milks Ping Chen, Xiao Dong Chen* & Kevin W. Free Food Science and Process Engineering Group, Department of Chemical and Materials Engineering, The University of Auckland. Private Bag 920 IO. Auckland. New Zealand (Received

14 November

1994: accepted

12 April 19%;

ARSTR4 CT Freezing point depression of milks of various concentrations (i.e. 0-3.5 wt%) were measured by the thermistor cryoscope method. The method was initiall!, validated using aqueous NaCl and sucrose solutions at high concentrations as their data are readily available in literature. The effect of fat content on freezing point depression of milk was found to be minimal. Ejjective molecular weights were calculated based on existing semi-empirical thermodynamic relationships. The effect of aging of skim milk at high concentrations on frcezing point depression was also observed and the quantitative results are reported here. Copyright 0 1996 Elsevier Science Limited

NOTATION

C l? F K 1211

L,,

Water activity (g mol water p mol ’ solution) Bound water (kg water kg solids) Effective percentage composition of milk powder An empirical constant The ratio of the molecular weight of water and the molecular weight of solute Fat content in dry milk powder (wt%) A constant, K = RT$L,, Fusion latent heat of water at freezing point of pure water by weight (kcal kg ~ ‘) Fusion heat of water at freezing point of pure water by molar weight (kcal kmol- ‘)

‘“To whom correspondence

should be addressed. 239

240

Ping Chen et al.

A constant in eqn (1) Effective molecular weight (g/mole’) The molecular weight of dilute solution in eqn (7) (g mall’) A constant in eqn (1) The universal gas constant (1.987 kcal kg-’ mol-’ K- ‘) Reading from the strip chart recorder (mV) Temperature (“C) Freezing point temperature of pure water (“C) Solids content by weight (%) Effective concentration of dissolved solids (%) At

Freezing point depression

(“C)

Subscripts

: i

Experimental data set 1 Experimental data set 2 Composition i

INTRODUCTION Freezing point depression (FPD) is one of the most important physical properties of an aqueous solution for its freezing process design (Schwartzberg, 1976; van Pelt & Swinkels, 1984). FPD data of many general chemical agents have already been published in the Handbook of Chemistry and Physics (Wolf et al., 1982); however, there are fewer accurate FPD data for liquids food, such as milk and fruit juice. The effective molecular weight of dissolved matter in a liquid food can also be calculated using the FPD data, which gives better understanding of the dissolution behaviour of the solute at the freezing temperature (Chen, 1986, 1987). The effect of chemical composition of a mixture solution of more than three compounds has not been studied in detail. Commonly cited in literature, there is one set of FPD data for a whole milk that can be derived from an enthalpy-concentration diagram (Riedel, 1977). This diagram was obtained using a micro-calorimeter (Riedel, 1977). Another set of FPD data available in literature on a whole milk was reported by Swinkels (1985). There are a number of measurement methods which have been used to obtain FPD data of aqueous solutions. Firstly, a method reported by Lerici et al. (1983) was used to determine the FPD data by cooling the sample (100 ml) in a container to at least -30°C. The sample was constantly agitated with a magnetic stirrer, and the container and the solution temperatures were recorded with thermoresistors connected to a digital thermometer and a graphic recorder. The FPD was obtained from the constant temperature plateau in the temperature-time curve. The maximum error for NaCl solution was +0*05”C and for sucrose solution _tO*25”C. Lerici et al. (1983) also measured the FPD of a freeze-dried skim milk using this method. The Hortvet method and the thermistor cryoscope method (IDF, 1986; BSI, 1988; AOAC, 1990) are also recognized as suitable procedures for determining FPDs of milks. Highly accurate FPD data can be obtained for herd milk or whole and skim milk at very low concentrations using the thermistor cryoscope method (i.e. the

Freezitlg point depression

241

in milks

measurement resolution is +O.OOl”C). Large errors, however, can occur at high concentrations. In this study, a thermistor cryoscopic unit (H. Knauer Wissenschaftl Gerate GmbH & Co., Germany) was used to measure the FPD of reconstituted concentrated milks. The Handbook FPD values of NaCl and sucrose solutions were compared with the results from the current instrument. Accurate FPDs of milks with different fat contents at solid concentrations ranging from 0 to 35 wt% were then obtained. An interpretation of the effect of fat content on FPD is made. It is well known that, for a milk of more than 30 wt% total solids, so-called age thickening occurs over a long period of storage (Snoeren et al., 1984), i.e. the apparent viscosity of the milk increases with time. Similarly, an interesting observation on the aging of FPD has also been made and the results are reported here.

EXPERIMENTAL Sample preparation The instrument was initially calibrated using distilled water and Knauer 400 mosmol kg-’ standard NaCl solution. In order to further validate the measuring procedure in the range of high concentrations, the aqueous solutions were made up using both NaCl and sucrose crystals which are analytical reagents. The FPDs of these solutions are available in literature so a comparison between the current results and the literature data can be made. Firstly, the NaCl and sucrose crystals were laid aside in an oven in which the temperature was kept constant at 105°C for more than 6 h to remove the free moisture contained in the particles. Secondly, the quantity of the particles was weighed out using an electronic analytical balance ( kO.05 mg accuracy), and the sample was then put into a clean and dry container. An appropriate amount of distilled water at 20°C was then added. The sample was ready after stirring fully. A sample solution of about 200 ml was prepared in each case. Samples of reconstituted milks were made up using the whole and skim milk powder which were obtained from local manufacturers. The compositions of the whole and skim milk powder are shown in Table 1. A similar procedure (without pre-drying) as that for NaCl and sucrose was used to prepare the milk samples in accordance with the standard procedures (Richardson, 1990).

TABLE 1 Composition of the Whole and Skim Milk Powder (Analytical Zealand Composition

Dairy Research

Institute,

Whole milk powder

Results Pers. Comm.)

Provided

hy New

Skim milk powder

Et

26.5

0.8

Total protein Lactose Minerals Moisture

28.0 36.8 5.0 2.8

37.x 40.8 7.X 38

242

Ping Chen et al.

FPD measurement and analysis When an aqueous solution is cooled without stirring to below the freezing point (i.e. supercooling) the temperature of the solution initially reduces to induce nucleation and then reaches a point when a sufficient number (or size) of nuclei become available to trigger an auto-crystallization. From this point onwards, the temperature rises rapidly from the supercooling temperature to a plateau of relatively constant temperature level that corresponds to the freezing point of the sample (when the heat of crystallization is balanced by the heat removed from the sample solution by the coolant) (Knauer, 1990). Mechanical vibration at some point during nucleation can also induce an auto-crystallization (perhaps by dispersing uniformly the nuclei). The highest temperature which occurs at the plateau of this freezing curve is known as the solution’s freezing point (refer to Fig. 1). The thermistor cryoscope method used in this study is based on this principle. During the cryoscopic measurement, a sample of only 0.15 ml was cooled continuously in a cooling batch with constant cooling power. The supercooling was terminated by activating a vibration wire. The temperature-time curve of freezing was recorded. The Knauer cryoscopic unit and the Knauer strip chart recorder (H. Knauer Wissenschaftl Gerate GmbH & Co., Germany) were used for the measurement. The instrument was initially calibrated by following the standard procedures for calibration (i.e. using distilled water for zeroing and a Knauer 400 mosmol kg- ’ (11.259% solids) standard NaCl solution for reference temperature at the constant supercooling temperature). The repeatability of the FPD measurements for both liquids were found to be within fO.OOl”C. The instrument was then examined against two

Temp. rc> 0

__...

At

Time (min) Fig. 1. Freezing curve of an aqueous solution.

Freezing point depression

A Comparison Solution

between

Concentration (%)

NaCl

0

1WI3 2.258 306X 3.75 1 4.443 5.119 5.383 6.75 1 Sucrose

0 16.787 19.831 23.096 27662 30.387 35wl6 40.1 18

233

in milks

TABLE 2 the Measured FPD Data and the Handbook and Sucrose Solution FPD (Handbook) 0

(“C)

FPD (Measured) 0

I.104 1.341 I .832 2.254 2.689 3.124 3.29ti 4.211

1~100 I .333 I ~805 2.214 2.62 I 3.044 3.202 4G-WI

0 I.166 1.449

I.177

1.809 2.329 2.702 3.458 4.480

0

(“C)

Data for NaCl Solution Absolute error (“C)

Rdatiw (wror (%J

0

0

-WOOS - om8 - 0.027 - 0.040 - 0468 - 0~079 ~ 0.094 ~ 0.125

fO.41 * 0.58 * 146 i I.76 f 2.53 5 2.53 & 2.85 _t 2% -_ 0 5 0+2

0 0.0 I I O~OOh

& 040

1.455 I .803

~ 0406

_+0.32

2.306 2%62 3.390 3.369

~ 0~022 - 0.039 - 0~069 -0.1 11

3 0.95 3. I.45 3: I~%3

$-. 2.30 -.-

for which detailed FPD data are available in the Handbook of Chemistry and Physics (Wolf et al., 1982). Triplicates were obtained for each sample. The average values were obtained as the FPDs of the samples. The measured results for NaCl and sucrose solutions are listed in Table 2, and the differences between the measured data and the published data are shown in Fig. 2 (top), (bottom). It can be seen that the differences between the current data and the previous data are small although there is a trend of increasing as concentration goes up. To this end, the current method may be more accurate, in fact, as distinct supercooling can be found in all the tests at the small sample size used. As such, the method used in this work was believed to be suitable for the purpose of measuring accurately the FPDs of milks. Typically, the repeatability (i.e. the difference between the results of any two single determinations, carried out in succession on the same sample) was found to be + 0.003”C for a concentration of 2.26 wt% NaCl solution and +O-006°C for a concentration of 27.66 wt% sucrose solution, according to the readout of the numeric values from the special chart record of the system. solutions

RESULTS

AND DISCUSSION

FPD of milk

The FPDs of reconstituted spray-dried milks measured are listed in Table 3 for the above method. The FPD curve for the whole milk is plotted in Fig. 3 together with

244

Ping Chen et al.

Riedel’s (1976) and Swinkels’ (1985) curve. The FPD curve for the skim milk is plotted in Fig. 4 against Lerici’s (1983) curves. It can be seen that in Fig. 3, although Swinkels’ curve is reasonably smooth, an extrapolation of the curve to the FPD axis would not go through the origin, which is the freezing point for pure water. The

4.5 4 3.5 c

3

p z E

2.5

k

1.5

2

1 0.5 0 0

2

1

4

3

CONCENTRATION -B- handbook data

+

6

5

7

~3 (Oh)

measured data

4.5 4 3.5 G

3

tf 2.5 z 2 2 IJi 1.5 1 0.5 0 0

20

IO

+-handbook Fig. 2. (top) Measured

data

FPD of the NaCl solution; solution.

40

30

CONCENTRATION

50

XI (%)

-I- measured data (bottom)

measured

FPD of the sucrose

Freezing point depression

245

in milks

TABLE 3 FPD Data and Effective Concentrations for Whole Milk, Skim milk and Mixed milk Milk

Whole milk

Skim milk

x, (wt96)

9.146 1I.994 28.673 34,816 39654

6.889 13+%8 22.808 28.191 32.568

4.318 12.455 23054 28.383 33.416

4.285 12.368 17.833 22.92 1 18.220 33.238

(I.230 0.700 I.1 IO I .48Y I YY? :!.s34

7%0 17.641 24.591 32.171 37.139

6.940 IS.609 21.972 29.055 33.782

0.364 092x I .3c)‘) 2G39 2513

Il+xi I

50% Whole milk+SO% skim milk

FPD (“C) - -__ 0.365 (NO I

x, (wt%)

I .47s I975 2.461

4 3.5 3 2.5 2 1.5 1 0.5 0 0

10

20

30

CONCENTRATION

40 xs(%)

Fig. 3. FPD curves for whole milk.

50

60

246

Ping Chen et al.

4.5

Lerici (1983) g P

2.5

z

2

2

1.5

1 0.5

0

5

10

15

20

25

30

35

40

CONCENTRATION xs(%) Fig. 4. FPD curves for skim milk.

shape of Riedel’s curve in Fig. 4 also appears to be less consistent because of the ‘kinks’ that occur at the concentration of 20 wt% solids. For the skim milk, the only data available in literature are those of Lerici (1983). The difference between Lerici’s data and the current data may be caused by compositional differences of the milks used in the two cases. It should be noted that Lerici’s curve is not as smooth as the current result. Effect of fat content on FPD The FPDs of milk with different fat content were measured (see Table 3) and the FPD curves are plotted in Fig. 5. It is expected that only the dissolved solids in milk affect FPD values. The fat content which is insoluble during the freezing process is usually included in the solids content of milk. This apparent concentration does not directly correspond to the FPD of the dissolved matter. The concentration of the dissolved solids, i.e. the effective solids content, should be used and can be calculated by the following equation:

x, =

a(1 -F) 1 -x, F

(1)

where X, and X, are the solids content and effective solids content, respectively, and F is the fat content in dry milk powder. The data of the effective concentration JY, are shown in Table 3 and the curves of FPDs versus .Y, are plotted in Fig. 6. Comparing the curves in Fig. 6 with the curve in Fig. 5, it can be seen that all the FPD curves of different milks with different fat contents have coalesced into one

Freezing point depression

247

in milks

3.5 3 2.5 G ?f -J

2

z

1.5

50% skim +50% whole

E 1 0.5 0 10

5

0

20

15

25

CONCENTRATION

+

0.8% fat

+

26.5% fat

35

30

40

xl(%)

+

13.65% fat

Fig. 5. FPD curves of milk with various fat content.

2.5

s 2 e

p, 1.5 8

.a n

1

l n

A

0.5 0 5

0

IO

EFFECTIVE

1

n

skim milk

15

20

25

CONCENTRATION

A whole milk

Fig. 6. FPDs versus effective concentration

30

35

X, (%)

4 mixed milk

1

of the milks in Fig. 5.

248

Ping Chen et al. TABLE 4

Results of Effective Molecular X,f%,)

Weight for Milks Calculated

Model I

4.285 6.889 12.368 13.888 15.609 17.833 21.972 22.808 28.191 29.055 32.598 33.416 average

from Various Models

Model 2

Model 3

M,

h

M,

c

MY

361.57 376.53 375.18 374.74 370.71 363.74 374.25 372.70 369.80 373.54 365.50 368.36 370.42

- 1.512 0.053 0.058 0.423 0.528 - 0.495 0.276 0.078 - 0.778 0.259 - 0.542 -0.164

338.64 378.03 378.26 402.20 410.86 328.43 405.75 381.51 283.29 417.94 289.55 340.35 362.90

- 1.433 0.066 0.078 0.693 0.994 - 0.605 0.560 0.150 - 0.873 0.757 - 0.725 - 0.308

339.37 378.25 378.80 410.79 428.23 324.50 420.26 385.49 278.79 455.67 279.08 330.46 367.48

Note: X, (%) evaluated from eqn (2); Model 1 evaluated from eqn (4); Model 2 evaluated from eqns (6) and (7); Model 3 evaluated from eqns (9) and (10).

4 5

0

10

EFFECTIVE .

model

Fig. 7. Effective molecular

x mode2

15

20

25

CONCENTRATION .

mode3

30

35

XI (%) -

weight versus effective concentration

average

value

I

for various milks.

Freezingpoint depression in tnilks curve. This confirms that there is little effect of fat content on the FPDs of milks.

240 (which is not dissolvable)

Calculation of the effective molecular weight There have been a number of theoretical or semi-empirical models of FPD published to describe the effect of increasing solute concentration. For an ideal solution (i.e. a dilute solution), the van’t Hoff law for liquid-solid transition (Pitzer & Brewer, 1961) defines the relationship between FPD and water activity a,,. as follows: At=

_-

RTT,,

ha,

L,,

(2)

where R is the universal gas constant (1.987 kcal kg--’ mol ~’ K- ‘); T is the temperature; T,, is the freezing point temperature of pure water; and L,,, is the fusion heat of water at the freezing point of pure water by molar weight. The water activity is given by Raoult’s law, i.e. a,. =

1--x.5 I -x, +EY,

where E is the ratio of the molecular weight of water and the molecular weight of the solute(s). A simple model for calculating effective molecular weight from the measured FPD data has been developed by Schwartzberg (1976) - M&l 1:

MS=

Kx’

(1 -x,)At

(4)

where K = (RTEIL,) in which Lo is the fusion heat of water at the freezing point of pure water by weight. Based on the concept of ‘bound water’ (immobilised water molecules), Schwartzberg (1976) modified eqn (3) to the following semi-empirical expression: Uw=

1 -x,, - bx, 1 --xv-bx,+Ex,

(5)

where b is the amount of water which is ‘bound’ to the solid components. This bound water is unavailable for freezing at any temperature. Chen (1986) assumed that the water activity can also be increased due to the complex interactions between solutes and water and that it can be expressed by an equivalent increase or decrease in free water. Thus, the value of b can either be positive or negative depending on the nature of freezing point data. Equation (6) can then be obtained for calculation of effective molecular weight (Chen, 1986) -Model 2:

M., =

fi,

(1 -x,-bx,,)At

(6)

Ping Chen et al.

250

and b=

X~I AtZ-XS2At,

-1

Xs,X,,(At,-At,)

I

where b can be calculated using any two sets of FPDs (At, and At2) and concentration data (X,, and XA2which correspond to At, and At, respectively) using eqn (7). The average value of b obtained from all the experimental data points can then be used to calculate the average value of MS defined in eqn (6). A further deviation of the molecular weight model, also proposed by Chen (1987) is a linear equation:

M., = 1+cX-,

(8)

M”6

where Mi denotes M, which is determined using eqn (4) and C is an empirical constant which is a linear coefficient of concentration. The value of C can be either positive or negative depending on the nature of various food systems. Finally, there is a thfrd model - Model 3: M = Kx,(1+cx) s

where C is called as the ‘concentration the following equation:

c=

(9)

(1-XJAt coefficient’

which can be determined

from

X,,(l-X,,)At,-X,,(l-X,,)Atl X,:,(1 -X,l)At,

-X:,(1

-Xs&&

I

The effective molecular weights calculated using the FPD data of milks measured in this study at effective concentrations using various models are listed in Table 4. As experienced in previous studies, the semi-empirical models tend to give better predictions of the FPD than the ideal model. These semi-empirical models, however, do not give any drastic differences in their predictions of the molecular weight of the same solute. The results of the effective molecular weight are also plotted in Fig. 7. The estimated effective molecular weights of milks were found to be between 362 and 370 g molt ’ according to the calculated results in Table 4. The average value was 366.9 g mol-’ which was calculated from the data at low concentrations using Model 1, Model 2 and Model 3. This value is close to the molecular weight of lactose, i.e. 342.3 g mol- ‘, indicating there are only small amounts of the other soluble components in the milks. Effect of aging at high concentrations The ‘aging effect’ on the FPD of the concentrated skim milk (i.e. the FPD of milk changes with storage time at a constant solids content) was found to occur at the concentrations above 34 wt%. The aging curves of FPD at different concentrations are plotted in Fig. 8 and the curves of FPD at different storage times are shown in Fig. 9. The latter figure shows that the FPD at the concentration of 29.02 wt% kept constant as the storage time increased. The FPDs at the concentrations of 44.95 and

Freezing point depression

in milks

10 9 /

8

1 skim milk ]

7 6 5 4 3 2

1 0 10

0

30

20

40

CONCENTRATION I

~

time=Omin +u- time=300min

-

time=1 OOmin time=480min

Fig. 8. FPD curves at different

60

50

x3 (%) time=200min

storage times.

9 8 7 G6 t?5 u av4

.

A

.

I

E3 2 1 0 500

0

1000

1600

TIME (min) m

C=29.02%

A

Fig. 9. Aging behaviour

C=44.95% of FPD for the skim milk.

2000

252

Ping Chen et al.

54.12 wt% decreased rather rapidly with time at the beginning, then the FPDs levelled off. The age thickening in terms of becoming more viscous at these high concentrations (Snoeren et al., 1984) was also evident (the time taken for the same volume of milk draining from a capillary tube was measured, which became much longer). The reasons for the effect of aging on FPD were probably Maillard type reactions between protein and lactose (Andrews, 1975) which generate insoluble materials and/or less mobile water molecules as their interactions with protein micelles become stronger with increasing hydration time [being shifted in the ‘continuum of hindered domains between the extremes of water molecules in ice and the bulk liquid water’ according to Slade et al. (1988)]. Although this aging effect appears to correspond to the age thickening [which has been linked largely to the effect of cross-linking of protein micelles as summarised by Chen (1994)], the linkage between the two cannot be established without a detailed chemical analysis. As the current work only gives the results of FPD (i.e. only a single measure of the milk freezing problem), in-depth interpretation of the aging behaviour observed here is not possible.

CONCLUSIONS The thermistor cryoscope method has been used to measure the FPDs of several aqueous solutions for a large range of solute concentrations. The FPD data of the reconstituted whole and skim milks ranging in solids content from 0 to 35 wt% have been accurately measured by the method. Comparison with the published data showed that the measured FPD data are reliable. It has been demonstrated that there is little effect of fat content on FPD. An effective concentration can be derived to correlate whole milk FPDs of various fat contents. The effect of aging on FPD occurs at concentrations over about 34 wt%, i.e. the freezing point temperature of milk decreases with increasing storage time at room temperature and at these high concentrations. The FPD aging of skim milk is apparently similar to that of the viscosity change of the skim milk in terms of the concentration at which the aging starts to happen. Detailed chemical analysis has to be adopted to reveal the mechanism of the aging of FPD. This is, however, beyond the capacity of the current study.

ACKNOWLEDGEMENTS Thanks are given to the Chemistry Department of Auckland University for lending us the Knauer cryoscopic unit. Financial support for the first author (a PhD research fellow) from the New Zealand Dairy Board and Electricity Corporation New Zealand is very much appreciated.

REFERENCES Andrews, A. T. (1975). Properties of aseptically packed ultra-high-temperature milk: III. Formation of polymerized protein during storage at various temperatures. J. Dairy Rex, 42, 89-99.

253

Freezirzg point depressiorz in nzilks

AOAC Official

(1990). Freezing Metlzods

point of milk-thermistor

qf Analysis qf the

Association

cryoscope of Ofjicial

method, Analytical

IDF-ISO-AOAC. In Clzemists, 15th win.

AOAC, Arlington, VA. BSI (1988). British standard method for determination of the freezing-point depression of milk, BS 3095. British Standard Institution. Chen, C. S. (1986). Effective molecular weight of aqueous solutions and liquid foods calculated from the freezing point depression, 1. Food Sci., 51(h), 1537-1539. 1553. Chcn C. S. (1987). Sorption isotherm and freezing point depression equations of glycerol solutions. Trans. ASAE, 30(l), 279-280. Chcn, X. D. (1994). Towards a comprehensive model based control of milk drying procesxes. Dryirzg Technol., 12(5). I 105 1 130. IDF (1986). Determination of freezing point (thermistor cryoscope method), IDF IOKA, IDF Bulletin 207, 203. Knauer (1990). Cryoscopic unit-operating manual. No.V7186, Version 0990, H. Knauer Wissenschaftl Gerate GmbH & Co., Germany. Lerici, C. R., Piva, M. & Dalla-Rosa, M. (1983). Water activity and freezing point depression of aqueous solutions and liquid foods. J. Food Sci., 48, 1667-1669. Pitzer, K. S. & Brewer, L. (1961). Thermodynamics, 2nd edn. McGraw-Hill, New York. Richardson, G. H. (1990). Dairy Products. In Official Methods qf Analysis of the Associaziorz qf Qficial Analytical Chemists, I4- 15 edn. AOAC. Arlington, pp. 802-803. Riedel, L. (1977). Enthalpy determinations of food. Chem. Mikrohiol. Technol. Leherzsm.. 5(4), 118-127. Schwartzberg, H. G. (1976). Effective heat capacities for the freezing and thawing of food. ./. Food Sci., 41. 152-156. Sladc, L., Levine. H. CycFinley, J. W. (1988). Protein-water interactions: water as a plasticizer of Glutinand and other protein polymers. In Proteirz Quality arzd tlze Effects of Proc.essing, eds R. D. Phillips & J. W. Finley. Marcel Dekker. New York, pp. 9-124. Snoeren, T. H. M., Brinkhuis, J. A., Damman, A. J. & Klok, H. J. (lY84). Viscosity and agethickening of skim-milk concentrate. Netherlands Milk Da@ Journal, 38, 43-53. Swinkcls, W. J. (1985). Recent developments in freeze concentration. In New Dab? Products \ia New Technology Proceedings of IDF seminars. Atlanta, GA, pp. 173- 187. van Pelt, W. H. J. M. & Swinkels, W. J. M. (1984). Freeze concentration: an alternative for evaporation in the citrus industry. Trans. 1984 Citrus Engiineering Corz,ference. ASME. Lakeland, IL. Wolf, A. V., Brown, M. G. & Prentiss, P. G. (1982). Concentrative properties of aqueous solutions: conversion tables. In CRC Handbook o,f C’hemistqt and P/zysics, 63rd edn. CRC Press, Boca Raton, FL.