Beek, G., van

Verbeek, W.

English

Name not available

Calculation of thermophysical properties of horticultural produce from their composition between - 40C and +20C

van Beek, G.

NARCIS 

Wageningen Yield 

S P R E N G E R I N S T I T U U T 
Haagsteeg 6, 6708 PM Wageningen 
t e l . : 08370-19013 
(Publikatie uitsluitend met 
toestemming van da directeur) 
RAPPORT NO. 1959 
I r . G. van Beek 
and i n g . W. Verbeek 
CALCULATION OF THERMOPHYSI CAL 
PROPERTIES OF HORTICULTURAL 
PRODUCE FROM THEIR COMPOSITION 
BETWEEN -*tO°C AND +20°C 
Uitgebracht aan de directeur van het Sprenger Inst i tuut 
Project no. 49 
C O N T E N T S biz. 
1.0 INTRODUCTION 3 
2.0 CALCULATION OF THE DENSITY 
2.1 Density of product 4 
2.2 Density of bulk 4 
3.0 SPECIFIC HEAT OF PRODUCT AND BULK 5 
3.1 Temperature above the initial freezing point 5 
3.2 Temperature below the initial freezing point 5 
4.0 THERMAL CONDUCTIVITY OF PRODUCT 6 
5.0 THERMAL DIFFUSIVITY 8 
6.0 ENTHALPY 8 
7.0 EXAMPLE 9 
8.0 LITERATURE 9 
9.0 SYMBOLS AND DIMENSIONS 10 
(5) 
follows from: 
pmeasured, bulk 
e
 •
 k , i = 1 (3) 
a -\ Y> hi i I v \ i 
ppr 
'air, bulk 
3.0 SPECIFIC HEAT OF PRODUCT AND BULK 
3.1 Temperature above the initial freezing point 
The specific heat of product and bulk are almost equal because of the ne-
glect of the mass of enclosed air. The specific heat is: 
c , -,, = E(c. • q. ) (4) 
pr - Dulk v i a v y ' 
3.2 Temperature below the initial freezing point 
Now the water and ice fractions vary with temperature. The relation 
between ice fraction and temperature is given by the law of Raoult (3)(4) 
9\ = 1 - — (5) 
t 
The specific heat is : 
c ,-,, = c -,., (1 - a ) + c • q (1 - q'. ) + e.« q'. • q -pr = bulk solids v "wy w Jw ^ b i ; n ai Jw 
dgi (6) 
r • q • o yw dt 
(6) 
Differentiation of equation 5 to t gives: 
dgi tf 
_ = . _ (7) 
dt t2 
while the specific heat of the solid components (not ice) is : 
Z(9i * c.) 
Solids = <8> 
Zgi 
4.0 THERMAL CONDUCTIVITY OF PRODUCT 
We use the equation of Eugen-Maxwell (5): 
2 • X1 + X2 - 2 • E 2 (Xj - X2) 
\
 + 2 = * À l ^ 
2 • X, + X2 + £o (^ 1 " *?) 
The index 1 is related to the continuous phase and 2 to the disperse phase. 
Table 2 points out which components have index 1 or 2 in the different 
calculations, if a, > 0,5. 
(7) 
Table 2 
' l + 2 'Al n2 equa- t empor;!L uro ^1 ,0 , A„ ( o 
L1 on 
9 t > tj- flesh water protein protein + fat 
a f f f 
9, t > tr product flesh air air 
9 t ^ tr - AT flesh ice water water + carbohydrates 
9 1 t < tr. - AT product flesh air air 
9 all t bulk product air air 
The value of AT is approximately 0,5 K. The calculation of A, -,, (t -< tr) 
follows 3 steps: 1st 9,-, 2nd 9,, 3rd 9 . 
c d e 
The volume fraction of protein, fat and carbohydrates are: 
p 
pr 
Ei = 9i (10) 
pi 
The volume fraction of water below the initial freezing point is : 
v 
£
w = — • 3W ' (1 - 9 i } ( U ) 
pw 
(8) 
.5.0 THERMAL DIFFUSIVITY 
The thermal diffusivity is for all temperatures equal to: 
X 
a = . (12) 
p • c 
6.0 ENTHALPY 
At t = 0 C the zero point for the enthalpy of foods is chosen. Above the 
initial freezing point t^ the enthalpy is: 
h = t - Z( C i - 9 i) (13) 
Below the freezing point: 
t 
h = ƒ Cpr • dt + tf • Z(Ci • g.) (14) 
Substitution of equation 6 and 7 in equation 14 gives 
h =
 Solids0"8^ ^ " ^ + VV1*'lntJ + ci-gw-(t-tf) - c r g w . l n ^ 
- r 0 -g w - ( l^) . + t f-E(C1-gi) (15) 
(9) 
7.0 EXAMPLE 
The example is given for apple. Figure 1 to 5 give the results of the 
calculations. Report 1919 (6) discusses the difference between calculation 
and experimental values above the free&ing point, and report.2025 (7) shows that 
the calculated enthalpy is in good agreement with the measured enthalpy 
below the freezing point. 
8.0 LITERATURE 
1) Har tog, C. den, Nederlandse Voedingsmiddelentabel. 
2) Produktgegevens groente en fruit, Sprenger Instituut, mededeling 30. 
3) Fikiin, A.G., Sur les paramètres thermophysiques des produits 
alimentaires congelés. IIF, Annexe 1974-3, Bressanone. 
4) Heldman, D.R. and D.P. Gorby, Computer simulation of ice cream 
freezing, IIF3 Annexe 1974-3, Bressanone. 
5) Tye, R.P., Thermal conductivity, Academic Press, London, 1969, 
Volume I + II. 
6) Beek, G. van en W. Verbeek, Berekening van thermofysische eigenschap-
pen van tuinbouwprodukten uit hun samenstelling, Sprenger Instituut, 
rapport no. 1919 (1975) 
7) Verbeek, W., Calorimetris che bepaling van het vriespunt, de enthalpie 
en de warmteproduktie van leliebollen cv. Enchantment na opslag bis 
temperaturen beneden het vriespunt, Sprenger Instituut, rapport no. 
2025 (1978). 
r ?-:^ï%. 
(10) 
9.0 SYMBOLS AND DIMENSIONS 
a 
c 
g 
h 
ro 
t 
*f 
AT 
e 
X 
P 
m2/s 
J/(kg 
kg/kg 
J/kg 
J/kg 
°C 
°C 
K 
m3/m3 
W/(m • 
kg/m3 
• K) 
K) 
thermal diffusivity 
specific heat 
mass fraction 
enthalpy 
latent heat of fusion 
temperature 
initial freezing point 
temperature difference 
volume fraction 
thermal conductivity 
density 
Wageni ngen, 20-09-1978 — GvB/WV/j1 
• 1 1 -
Composition of the apple 
component 
p r o t e i n 
f a t 
carbohydrate 
Ca 
Na 
K 
rest 
water 
in 
in 
' i n 
i n 
in 
in 
in 
in 
g) 
q) 
g) 
mg) 
mq) 
mg) 
g) 
g) 
f r a c -
t i o n 
0.000 
0.000 
0.100 
0.000 
0.000 
0.001 
0.028 
0.870 
densi ty 
kg/m^ 
1380 
930 
1550 
2400 
2400 
2400 
1550 
1000 
s p e c i f i c heat 
kJ / (kg-K) 
1.900 
1.900 
1.220 
0.840 
0.840 
0.840 
1.220 
4.182 
Initial freezing point: 
Density and porosity 
density (in kg/m^) 
porosity (in % ) 
-1°C 
product 
800 
23.7 
bulk 
500 
37.5 
- 1 2 -
Thermophysical p roper t ies o f the apple 
10 
M 
14 
18 
20 
24 
26 
28 
30 
32 
34 
36 
38 
40 
42 
46 
t emp, 
'C 
- 1 . 6 
- 1 . 7 
- 1 . 8 
- 1 . 9 
- 2 . 0 
- 2 . 1 
- 2 . 2 
—2. 
- 2 . 
..^..r). 
48 
50 
- 3 . 8 
- 3 . 9 
- 4 . 0 
- 5 . 0 
- 1 0 . 0 
- 1 5 . 0 
- 2 0 . 0 
- 2 S . 0 
- 3 0 . 0 
~3S. 0 
- 4 0 . 0 
k J 
kg 
k J 
k g . K 
l ( p ) 
W 
a ( p > 
m2 
K b ) 
w 
a ( b > 
tn2 
m.K m.K 
2 0 . 0 7 6 
1 5 . 0 57 
1 0 . 0 38 
5 . 0 19 
0 . 0 0 
- 1 . 0 - 4 
- 1 . 5 - 1 0 3 
• I IS 
•126 
•136 
-144 
•152 
•160 
•166 
3 - 1 7 2 
4 - 1 7 8 
5 - 1 8 3 
6 -187 
— 7 - 1 9 2 
- 2 . 8 - 1 9 6 
- 2 . 9 - 2 0 0 
- 3 . 0 - 2 0 3 
• - 3 . 1 - 2 0 7 
- 3 . 2 -210 
- 3 . 3 - 2 1 3 
- 3 . 4 -216 
• - 3 . S - 2 1 8 
- 3 . 6 - 2 2 1 
- 3 . 7 - 2 2 3 
- 2 2 5 
- 2 2 8 
-230 
-246 
-286 
-305 
-320 
- 3 3 3 
-345 
-356 
-367 
3 .80 
3 .80 
3 .80 
3 .80 
3 .80 
3 .80 
132 .75 
116 .99 
103 .92 
92 .96 
83 .69 
75 .77 
68 .95 
63 .04 
57 .88 
53 .35 
49 .35 
4 5 . 8 1 
4 2 . 6 4 
3 9 . 8 1 
3 7 . 2 7 
34 .98 
32 .90 
3 1 . 0 2 
29 .30 
27 .73 
2 6 . 3 0 
24 .98 
2 3 . 7 6 
2 2 . 6 4 
2 1 . 6 1 
2 0 . 6 5 
1 4 . 0 0 
5 .04 
3 . 3 2 
2 . 38 
2 . 1 9 
2 . 0 6 
1.96 
0 . 4 2 
0 . 4 2 
0 . 4 1 
0 .40 
0 . 4 0 
0 .40 
0 . 4 0 
0.89 
0 .92 
0 .95 
0 .97 
0 .99 
1.01 
1.03 
1 .05 
1.06 
1.08 
1.09 
1.10 
1.12 
l v l 3 
1.14 
1.15 
1.16 
1.17 
1.18 
1.18 
1.19 
1.20 
1.21 
1.21 
1.22 
1.27 
1.40 
1.46 
1.51 
1.55 
1.59 
1.63 
1.66 
1.38E-07 
1.3ÓE-07 
1.34E-07 
1.32E-07 
1.31E-07 
1.30E-07 
3 .73E-09 
9 .S0E-09 
1 .10E-08 
1.27E--08 
1 .44E-03 
1.63E-08 
1.33E-08 
2 . 0 4 E - 0 8 
2 . 2 6 E - 0 8 
2 . 4 8 E - 0 8 
2 . 7 2 E - 0 3 
2 . 9 7 E - 0 8 
3 .23E-08 
3 .S0E-08 
3 . 7 8 E - 0 8 
4 .06E-08 
4 . 3 6 E - 0 8 
4 . 6 6 E - 0 8 
4.97E--08 
S .29E-08 
5 .62E-08 
5 .9SE-08 
Ó.30E-08 
6 .6SE-08 
7 . 0 1 E - 0 3 
7 .37E-08 
1 .13E-07 
3 .46E-07 
S .50E-07 
7 .01E-07 
8 . 17E-07 
9 .09E-07 
9 .89E-07 
1.06E-06 
0 .24 
0 . 2 3 
0 . 2 3 
0 . 2 3 
0 . 2 2 
0 . 2 2 
0 . 2 2 
0 . 4 8 
0 . 5 0 
0 . 5 1 
0 . 5 2 
0 .54 
0 . 55 
0 . 5 6 
0 . 5 7 
0*57 
0 . 5 8 
0 . 5 9 
0 . 6 0 
0 . 6 0 
0 . 6 1 
0 . 6 1 
0 . 6 2 
0 . 6 2 
0 . 6 3 
0 . 6 3 
0 . 6 4 
0 .64 
0 .64 
0 . 6 5 
0 . 6 5 
0 . 6 6 
0 . 6 3 
0 . 7 5 
0 . 7 8 
0 . 8 1 
0 . 8 3 
0 . 8 5 
0 . 8 7 
0 . 8 9 
1. 
1. 
3 . 
1 .24E-07 
1.23E-07 
1.21E-07 
1.19E--07 
17E-07 
17E-07 
35E-09 
8 .24E-09 
9 .57E-09 
1.10E-08 
1.25E-08 
1.41E-08 
1.S8E-08 
1.76E-03 
1.9SE-08 
2.15E--03 
2 .3SE-08 
2 .56E-08 
2 . 7 9 E - 0 8 
3 . 0 2 E - 0 8 
3 .26E-03 
3 .50E-08 
3 .7SE-08 
4 .01E-08 
4 .23E-08 
4 .56E-08 
4 .84E-08 
S .13E-08 
S .42E-08 
5 .72E-03 
6 .03E-08 
6 .34E-03 
9 . 7 5 E - 0 3 
2 .97E-07 
4 .71E-07 
6 .01E-07 
6 .99E-07 
7 .78E-07 
8 .4SE-07 
9 .05E-07 
<%> 
0 . 5 
0 . 5 
0 . 5 
3 3 . 8 
3 8 . 0 
4 1 . 6 
4 4 . 9 
4 7 . 8 
5 0 . 5 
5 2 . 3 
5 5 . 0 
5 7 . 0 
5 3 . 8 
6 0 . 5 
6 2 . 0 
6 3 . 4 
6 4 . 7 
6 6 . 0 
6 7 . 1 
6 8 . 2 
6 9 . 2 
7 0 . 1 
7 1 . 0 
7 1 . 9 
7 2 . 7 
7 3 . 4 
7 4 . 1 
7 4 . 3 
7 5 . 5 
8 0 . 5 
9 0 . 5 
9 3 . 8 
9 5 . 5 
9 6 . 5 
9 7 . 1 
9 7 . 6 
9 8 . 0 
Explanation of symbols and indices: 
h 
c 
1 
a 
(P) 
(b) 
g 
= 
= 
= 
= 
= 
= 
= 
enthalpy 
speci f ic heat 
thermal conduct iv i ty 
thermal d i f f u s i v i 
of product 
of bulk 
ice f ract ion 
ty 
- 1 3 -
specific heat 
10 r- kJ 
kg-K 
0 J_ _L j 
4 0 - 2 0 0 20 
t e m p e r a t u r e °C 
therma l conductivity 
2.0 
1.5 
1.0 
0.5 
W 
m- k 
product 
bulk 
0 
\ 
L 
- 4 0 - 2 0 0 2 0 
tempera tu re °C 
F i g . I F i g . 2 
t he rma l diffusivity m 2 
1 2 r x 1 0 ~ 7 
product 
•40 - 2 0 0 20 
t empera tu re °C 
enthalpy 
2 0 0 
- 2 0 0 
-40Q l L 
4 0 - 2 0 O 20 
tempera tu re °C 
F i g . 3 F i g . 4 
- 1 4 -
ice fraction 
100r°/o 
7 5 -
50 -
25 
40 -20 0 20 
temperature °C 
Fig . 5 


Calculation of thermophysical properties of horticultural produce from their composition between - 40C and +20C

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