An isostere equation for some common seeds

J. agric. Engng Res. (1987) 37, 93-105

An Isostere Equation for some C o m m o n Seeds ALECK J. HUNTER* Hygroscopy of food seeds affects storage, handling and processing. The thermodynamic relationship upon which the isostere equation is based was originally presented by Othmer. 1 The ratio of latent heats (bound water to free water) is part of the isostere equation and is fitted to data for nine common food seeds using a transition function having logarithmic asymptotes. The isostere equation is completed by the evaluation of a further constant termed the intercept pressure. A reverted form of the isostere equation is also presented to assist with computation. 1. Introduction The word isostere is used to specify lines of constant adsorped material. An isostere equation therefore gives the variation of relative humidity with temperature for a fixed seed moisture content. The purpose of this paper is to present an isostere equation which is accurate and simple, is based on thermodynamic principles and is applicable to as wide a range of biological materials as possible. Othmer's 1 equation is based on thermodynamic principles and gives an isostere equation with two unknown functions of moisture content. The data indicate that one of these functions may be taken to be a constant for a given seed and the other (the ratio of latent heats as a function of moisture content) must be fitted empirically. To fit the ratio of latent heats various authors have used an exponential decay (Gallaher, 2 Chung and Pfost3), a power law (Agrawal et al. 4) and a number of other transcendental functions incorporating two parameters. This limits the applicability of the resulting isostere equation to a particular type of biological material. For example, Pixton and Howe s state that the Chung and Pfost 3 equation is ideal for starchy grains and flour but is inadequate for grains richer in protein. The transition function used in this paper, having five independent parameters overcomes this limitation and has been found to fit a wide range of seeds. Hygroscopy of seeds affects storage, handling and processing. The water vapour pressure in equilibrium with various common seeds has been the subject of much experimentation by different methods but it is still a difficult quantity to measure accurately (Haynesr). Fortunately, and for no apparent reason, the ratio of latent heats (bound water to free water) seems to be reasonably independent of temperature for a given moisture content, over the range of ordinary ambient temperatures. As a result, Othmer 1 showed that the equilibrium vapour pressure is related to saturation vapour pressure by a simple mathematical expression. By fitting this expression through experimental data by a least squares error method, it should be possible to obtain a more accurate estimation of the true equilibrium vapour pressure. It was found that if it is assumed that isosteres on the Othmer plot for the seeds considered intersect at a single point on the saturation line, the fit to the data lies within the range of the probable experimental error. Strohman and Yoeger 7 used this same assumption, but they claim (erroneously in the author's opinion) that it can be justified in terms of units. It is incorrect to have any transcendental function such as a logarithm of a dimensioned quantity such as pressure. The reason is that a transcendental * Division of Chemical and Wood Technology, CSIRO, P.O. Box 26, Graham Road, Highett, Victoria, Australia 3190 Received 28 October 1985; accepted in revised form 23 May 1986 93

94

ISOTERE

EQUATION

FOR

SEEDS

Notation a, c, b, d e

f(w), 9(w) h ho ha

constants Euler's constant, 2.71828 arbitrary functions

Pb r t T w wo w1

base unit of pressure, Pb = 1 Pa relative humidity temperature, ~ temperature, K moisture content, dry basis transitional moisture content data curve value of w corresponding to (h~/hv)- 1 = h o ~t,/3 constants

(h~/hv)- 1 transitional value of (hs/hv)- 1 data curve value of (h~/h,)- 1 corresponding to w = wo h s latent heat o f b o u n d water, J/kg hv latent heat of free water, J/kg m,/I constant exponents Po intercept pressure, Pa

The superscript ' refers to data

function represents a power series and if the argument has units then every term in the expansion will have different units. The argument of a transcendental function can be a ratio of dimensioned quantities.

2. The Clapeyron relation Since the isostere equation is expressed in terms of the pressure p~ of water vapour at saturation, and not temperature, a modified Clapeyron equation (Hunter a) is given for reference. Hence Oe e_O/T,

=

(1)

where ct and/3 are constants equal to 6 x 1025 Pa K s and 6800 K respectively. The accuracy of this expression is better than 0.33/0 in the range 0 to 100~

3. The isostere equation Othmer 1 derived the thermodynamic relation

1 dp_

1 dps

h~ p

hv p~

(2)

p is the vapour pressure, p~ the saturated vapour pressure, h~ the latent heat of bound water and hv the latent heat of free water. Relative humidity r is defined by

p = rps

(3)

and p can be eliminated from (2) and (3) so that d(rh- ~- - l ) d P S ' r

(4)

If hs/hv is assumed to be independent of temperature, (4) can be integrated to

r = ( p s ~ 'hs/hv'-I

(hs/hv)- I is a function

C5)

of w and so is Po. Both these functions must be evaluated when (5) is fitted to the data (Appendix A).

A. J.

HUNTER

95

Having examined the experimental data, it seems that it is a reasonable simplification to take P0 independent of seed moisture content and so to have a single value for each type of seed. To complete the isostere equation it is necessary to express h~/hv as a function of moisture content w (dry-basis). It was found that if h~/h~- 1 is plotted against In w, that for low w and for high w a straight line asymptote results, there being a transition region between. A suitable function for the entire range of w is therefore hs

1 = a In

h~

bw- (W/Wo)"cIn dw 1 -(W/Wo) n

(6)

a and b are evaluated by a least squares fit to the data for small w, and c and d similarly for large w. wo indicates the transition region and is the value of w at which the logarithms are equal. Therefore

Wo

(7)

\de/

On the (h~/h,)- 1 versus In w graph, a smooth curve is drawn which has straight asymptotes and passes through the transition data. h~ is read corresponding to w = Wo and n is calculated from c--a

n-

hl - a In

(8)

bwo"

which is derived in Appendix B. Eqns (5) and (6) constitute the isostere equation.

4. Reversion of the isostere equation

There are occasions when one wishes to express w as a function of r and t instead of r as a function of w and t. To this end (5) and (6) may be reverted as follows. Equation (5) becomes In r

(hJhv)- 1 - In (PJPo)"

(9)

Since a logarithm reverted is an exponential, (6) may be written approximately as (h)~l

lh'c

h

ho 1-

where h has been written for exponentials and so

(10)

(hs/h,,)- 1. Similar to (7) above, h o is the intercept of the ac

ho = - -

c--a

In

(b/d).

(11)

Setting w = w0 and h = h~ in (10) and solving for m,

l Vb(eh'/C-dwo)l n L ~ j /,t/~--

In

(hl/ho)

(12)

96

ISOTERE E Q U A T I O N FOR SEEDS

Table 1 Parameters for use in Eqns (5), (6), (7), (9), (10) and (11) Seed

Po, Pa

n

m

a

b

c

d

~arley s Corn TM Peanuts 4 Rapeseed 11 RiceTM Sorghum s Soybeans TM Sunflower TM Wheat 11

6"9176 x 105 1'1584 x 105 1-2119 x 105 6-5175 x 106 9'7481 x 105 7.4031 x 107 8"5377 X 106 5'0304 X 106 4"4571X 106

8-4783 11'4758 11.2951 7"3356 12"0556 56"3811 10-3580 6-0718 17.6706

1'7788 3"7880 4.4464 2-6469 4'9670 9.7154 5.6651 3.0445 4.9682

--0.26249 -0"50160 --0.30395 --0'22976 --0.29907 --0'17662 --0.22297 --0.27797 --0'21133

4.6097 5.6323 7"4899 10-5894 5'1427 5'1872 8'1659 12.6731 4-7403

--0'00910 --0'09832 -0.07120 -0'01706 --0.14485 --0-06118 --0.04524 -0.04931 -0.03483

1"2485 3"1041 3.9970 2.4461 3.8106 4"2819 2'8957 4'4794 2-6210

5. Parameter values for nine seeds The parameter values calculated for nine seeds are given in Table 1. The references from which the data are taken are indicated in the Table. Additional information concerning the different seeds is given below. Barley: Corn: Peanuts: Rapeseed: Rice: Sorghum: Soya beans: Sunflower: Wheat:

Sultan, desorption, smoothed data. shelled, desorption, actual data. pods, interpolated. Candle, desorption, smoothed data. Calrose, desorption, smoothed data. smoothed data. desorption, smoothed data. desorption, smoothed data. Glen lea, desorption, smoothed data.

The plot originally proposed by Othmer used co-ordinates In (P/Pb) and In (P,/Pb), Pb being the base unit of pressure. The co-ordinates used here are In r and In (Ps/Pb); the saturation 1.0

.w=0.351 .w=0"282 w=0'220 "w=O.191

--fi

~

~ ~

~

-.,=o.176 ~

~

0"5

~J

-""

~ ~

w=0.163 w=0.149 -w=0.136 w=0.124

>=

"w=0.111

o

0"2 0%

IO~

20~

30~

I 0.1

i 3xlO 2

,

,

,,,,

I

I ,

,

,i

,

103

,

, , ,

104

Saturation pressure of woler vopour, Po

Fig. 1. M o d i f i e d Othmer plot for barley showing the f i t o f the isostere equation. Data: Pixton and

Warburton s

A. J. H U N T E R

97 1.0

w = 0-26 w =0,22 w=O. 18 -

o o

~

-

~

w=O'16 ~ - - ~ - ~ - - ~ : ' - ~ - - - ' ~ ' - ~ 0"5

w =0"14 /

o.._.....~~

~

-g

~'0-2

4-4C ~ " ~

15'50C

30'0"C

I

I

w=O.O8 / o.,

,

,

, , ,I,,

I,

3xlO z

,

37"70C

I

,I

,

,I,

10 3

,,

104

Saturalion pressure of water vapour, Pa

Fig. 2. Modified Othmer plot for corn showing the'fit of the isostere equation. Data: Rodriguez-Arias et al.lO

line is thus made horizontal and the plot fits more conveniently onto rectangular paper. The plot has been termed the modified Othmer plot because of this difference. The modified Othmer plots are presented in Figs I to 9 and the ratio of latent heats in Figs 10 to 18. A modified Othmer plot showing the convergence of isosteres at the saturation line is shown in Fig. 19. 6. Discussion An indication of the accuracy of experimental measurements is given by the maximum deviation of data points from a smooth curve drawn by eye through each set of points. The 1.0

w =0.=91;

~

~

n

w =0-163 ~

w=O'lll >. 0.5

~

w =0.087 - - - ~ - ~ ~ -

E w =0 - 0 6 8 . - -

~" o.2 IO~

o, 3 x l O z,,,,,

10 3

I

21.1~

,[,

52.2oc

,I

....

104

Soturolion pressure of woter vopour, Po

Fig. 3. Modified Othmer plot for peanuts showing the fit of the isostere equation. Data: Agrawal

et al. 4

98

ISOTERE E Q U A T I O N FOR SEEDS I'0

w: 0 . 2 2 0 ~ w = O" 190"~ - . ~ - ' - - ~

.o

=o.,s37/

=,0.5

w= 0 " 1 5 6 7 w=O. iii//-----'r w =0'087

,,

~

v

__.___.o____.-

.o----------'-'w= 0 . 0 6 4

E

.g >= s 0'2

w :0-042

/

5~

15~

250C

55"C

I O'l

I

I

I

I

I

I

I

3xlO 2

i

I

I

I

I

I

I

I

I

103

104

Saturotion pressure of water vopour,

Pa

Fig. 4. Modified Othrner plot for rapeseed showing the fit of the isostere equation. Data: Pixton and Henderson11

accuracy so determined is given by Pixton and Warburton is for their data as 1.5% r.h. or 0.2% moisture content. These figures account for nearly all of the differences between the experimental data and the function fits presented in this paper. The ratio of latent heats hs/hv as well as being a necessary part of the isostere equation is needed when analysing heat and mass transfer processes in seedbulks. Although hJhv will be given to a reasonable accuracy by (6), it should be noted that for some seeds (for example corn, Fig. 2) the fractional error in (hJhv)-1 may be unacceptably high, particularly for high moisture contents. The error results from the assumption that the isosteres intersect on the saturation line. Previous authors have tested several transcendental functions to fit the hs/hv data including the exponential (Gallaher 2) and a power-law (Agrawal et al.4). Fig. 12

I'0 w : 0"220

c,

o

w=O'191 w :0"163

.-(3

~'-

Q

n

_--.----o---------------"-6"--'_..__.....~..._o..---~ - - - - - ' - ' ~

>,0"5

/

w = 0 . 1 3 6 __------o----

E w=O'lll

>= 0'2

~

w =0.087

20"C

IO*C

O'

I

~xlO 2

J

I

I

I

J

I

i

30"C

I

I

103

38"C

=

i

J

i

1

104

Saturation pressure of water vapour, Pa

Fig. 5. Modified Othmer plot for rice showing the fit of the isostere equation. Data: Putranon et

al. 12

A. J. HUNTER

99 1.0

w=0'20 w=O.15

o

w : O . lO --'-'-"-

o

0.5

E=

L

"6

0-2 22'2~

400C w=O.05

.0"1

i

I

i

i

i

i

5xlO z

i

i

~

L

,...~.---~'~"~,I, .

103 Saturalion

I ,

104 pressure

of

water

vapour, Pa

Fig. 6. Modified Othmer plot for sorghum showing-the ffit of t[ze isostere equation. Data." Haynes s

gives the estimated ratio of latent heats for peanuts. Shown also is the transition function, a least squares exponential fit and a least squares power law fit. The transition function fits best as is to be expected since there are five fitted constants. Both the exponential and the power law are asymptotic to zero when w approaches infinity. This implies that the seeds could be completely immersed in water and still exert an influence on the bonding energy at the free surface. In fact there will be a certain finite moisture content (somewhat less than when each seed is completely immersed in water) at which the latent heat of bound water must equal the latent heat of free water. This phenomenon is accommodated by the transition function used here. The moisture content at which this occurs is given by the reciprocal of the parameter d given in Table 1.

1.0 w = 0-203 w=O'160 ~ w=0'133

~

o .o _~

n

,~

.__o._....-.--~ >, 0-5

w = 0-092

== w = 0"062

o~ 0.2 15~

o

5 x I0

25*C

I

103 Soluration

35"C

pressure

of

weter

vopour,

I

104

Po

Fig. 7. Modified Othmer plot for soybeans showing the fit of the isostere equation. Data." Pixton and Warburton13

100

ISOTERE EQUATION

FOR SEEDS

1.0 w=O.164 w=O'l19

n n

w=0"098 --

~'

0

n

(3

n

_

>, 0 ' 5 E .go ~

w= 0 - 0 4 6 ~ 55%

'~ o.2

25 ~ 15~

I 0"I

i

I

I

I

I

I

I

5xlO 2

I

l

I

i

,

i

i

IO s

10 4

Saluration pressure of water vapour, Pa

Fig. 8. Modified Othmer plat for sunflower seed showing the fit of the isostere equation. Data." Pixton and Warburton 13 1.0

w =0-282-

2~

e

e

w = 0 . 2 5 0 ~

~

"

w = 0.220~/~

"~

=--

":"

15"C

25"C

w=O. 191-w = O' 165 " ~ ' ~ ' ' - ~

--

~

0.5 w=0.136 E

w=O. III

0.2

~

w : 0.087 5"C

0.1

I 3xlO

I

I

I

I

2

i

I

I

I

I

I

I

I

1

10 4

10 3

Saturation pressure of water vapour, Pa

Fig. 9. Modified Othmer plot for wheat showing the fit of the isostere equation. Data: Pixton and Henderson 11 0.2,

T

.~. 04 s

-2"5

-2"0

- 1'5 Ln w

-I'0

-0"5

Fig. 10. Ratio of latent heats f.or barley showing the fit of the transition function

A.

J.

101

HUNTER 0.5 0.4 0-3

~S o.z 0.1

-3'0

-2"5

-2'0 Ln w

-1"5

-I'0

Fig. 11. Ratio of latent heats for corn showing the fit of the transition function \ 0.2 T

0 -3'0

-2"5

-2-0 In w

-1"5

- I'0

Fig. 12. Ratio of latent heats for peanuts showing the fit of the transition function [Eqn (6)] together with the exponential of Gallaher2 ( - - - ), and the power law of Agrawal et al.4 ( - - - ) 0,2

7

o.;

0 -4.0

-3-0

-2.0 Lnw

-I-0

0

Fig. 13. Ratio of latent heats for rapeseed showing the fit of the transition function

Six parameter values are involved in the complete isostere equation used here for each type of seed. The transition function fits the hs/h, data very well as seen in Figs 10 to 18. This is necessary in order that the isostere equation fit the equilibrium data adequately. Thus the number of parameters used is justified.

102

ISOTERE EQUATION

FOR S E E D S

0.2 T '~

0-1

-2"5

- 2"0

- 1'5 Ln w

-ll.O

-0"5

Fig. 14. Ratio of latent heats for rice showing the fit of the transition function

0.2 T ~0.1

0 -5,0

-2.5

-2.0

-I.5

- I'0

9

Ln w

Fig. 15. Ratio of latent heats for sorghum showing the fit of the transition function 0-2

~.o.t

~

-,'.s

-,.,

Ln w

Fig. 16. Ratio of latent heats for soybeans showing the fit of the transition function

7. Conclusions The isostere equation presented together with the parameter values given for the nine common seeds should be a useful reference for designers involved in processing, handling and storage of food seeds. The isostere equation is based on thermodynamic principles and is therefore preferable to purely empirical formulations.

A. J. H U N T E R

103 0.5 0.4 T 0.3

~'~-~0.2 0.1

-25.5

- 5.0

- 2-5

- 2"0

- I '5

In w

Fig. 17. Ratio of latent heats for sunflower showing the fit of the transition function 0"2

-7 -.~0.1

0 -2"5

-2"0

- 1.5

- I'0

- 0"5

Ln w

Fig. 18. Ratio of latent heats for wheat showing the fit of the transition function X=Xo= In 0

5-

xl

x2

X3

Po/Pb

I

X = Ln

(Ps/Pb)

Fig. 19. Modified Othmer plot showing convergence of isostere "fan" at Ps = Po

The ratio of latent heats is presented in functional form as part of the isostere equation and is in itself necessary information when analysing heat and mass transfer processes in seedbulks. References 10thmer, D. F. Correlating vapour pressure and latent heat data. Journal of Industrial and Engineering Chemistry 1940, 32(6): 841

104

ISOTERE E Q U A T I O N FOR SEEDS

2 Gallaher, G. L. A method of determining the latent heat of agricultural crops. Agricultural Engineering 1951, 32(I): 34 s Chung, D. S.; Pfost, H. B. Adsorption and desorption of water vapour by cereal grains and their products. Transactions of the American Society of Agricultural Engineers 1967, 10(4): 520 4 Agrawal, K. K.; Clary, B. L.; Nelson, G. L. Investigation into the theories of desorption isotherms for rough rice and peanuts. Journal of Food Science 1971, 36:919 s Pixton, S. W.; Howe, R. W. The suitability of various linear transformations to present the sigmoid relationship of humidity and moisture content. Journal of Stored Products Research 1983, 190): 1-18 6 Haynes, B. C. Jr. Vapour pressure determination of seed hygroscopicity. Technical Bulletin No. 1229, Agricultural Research Science, United States Department of Agriculture, January 1961 7 Strohman, R. D.; Yoerger, R. R. A new equilibrium moisture content equation. Transactions of the American Society of Agricultural Engineers 1967, 10(5): 675 a Hunter, A. J. Thermodynamic criteria for optimum cooling performance of aeration systems for stored seed. Journal of Agricultural Engineering Research 1986, 33(2): 83-99 9 Pixton, S. W.; Warburton, S. Moisture content/relative humidity equilibrium of some cereal grains at different temperatures. Journal of Stored Products Research 1970, 6(4): 283-293 lo Rodriguez-Arias, J. H.; Hall, C. W.; Bakker-Arkema, F. W. Heat of vaporization of corn. Cereal Chemistry 1963, 40:676-683 11 Pixton, S. W., Henderson, S. The moisture content-equilibrium relative humidity relationships of five varieties of Canadian wheat and of Candle rapeseed at different temperatures. Journal of Stored Products Research 1981, 17(4): 187-190 12 Putranon, R.; Bowrey, R. G.; Eccleston, J. Sorbtion isotherms for two cultivars of paddy rice grown in Australia. Food Technology Australia 1979, 31(2): 510-515 13 Pixton, S. W.; Warburton, S. Moisture content relative humidity equilibrium, at different temperatures, of some oilseeds of economic importance. Journal of Stored Products Research 1971, 7(4): 261-269 14 Pixton, S. W.; Warburton, S. The moisture content/equilibrium relative humidity relationship and oil composition of rapeseed. Journal of Stored Products Research 1977, 13:77-81

Appendix A. Least squares fit of Eqn (5) to data Eqn (5) may also be written In r = ((hs/hv)- 1) In (PdPo)

= ((hs/hv)- 1)[In ( P J P b ) - In (Po/Pb)]" The above equation may be written where

y = ax + b, y = In r,

a = (hs/hv)- 1, and

x = In (Ps/Pb),

b = -((hJhv)-1)

In (Po/Pb).

An example of the geometry concerned is shown in Fig. 19. In terms of x and y, the equations to the lines are

Yi~ -- aj(x~- Xo).

(A1)

It is required to evaluate Xo and the aj to give a least squares fit of (A1) to the data y'~j. The sum of the squares of the deviations is e. = ~.. (Yij -- y~j)2. U

(A2)

A. J.

HUNTER

105

Using (A1)

e = ~ (ai(x i - Xo) - y'ij)2.

(A3)

ij

Differentiating (A3), equating to zero

z'

xiYij

-- XO

Yij

aj = E x ~ - 2 X o ~ xi+ x~ • i" i

"

(A4)

i

For any Xo the aj can be calculated from (A4) and e evaluated from (A3). Xo is taken as the value which minimizes e. The aj are given by (A4), and are equal to (hs/h,)-1, and x o is equal to In Po/Pb.

Appendix B. Derivation of the exponent in the function for (h,lh,)- 1 [-Eqn (8)] The general form of a transition function for asymptotes which intercept is

h = f ( w ) - (W/Wo)"g(w)

(B 1)

1 -(W/Wo)" The obvious choice for the value of wo is the value of w at which f ( w ) = O ( w ) . As w approaches w o the limiting value of h is given by L'Hospital's rule as

:i wo)+ wo

I .i ] L~ I.o- ~ woj

Iff(w) = a In bw and g(w) = c In dw, then (B2) gives r

hi = a In bw o + - - ,

(B3)

n

or r

n = hl - a In bw o"

(B4)