Theoretical Half-Cooling Time of Mass-Average Temperature of Fruits and Vegetables

In the case of spherical products under the forced air cooling

Abstract

The half-cooling time of the mass-average temperature of the spherical fruits and vegetables are caluculated as the solution of the following equation by electronic computer in the range of cooling-air velocity between 2.0m/sec and 5.0m/sec, and in the range of diameter of fruits and vegetables between 1.0cm and 30.0cm (Table 1).
0.5=∑
^∞_n=16 (ah) e^{-κα_n2THALF}/(aα_n)²((aα_n)²+(ah) (ah-1)) (1)
where
Thalf: half-cooling time, κ: thermal diffusivity of fruits and vegetables, a: radius of spherical fruits and vegetables, h=H/K_p, H: surface heat transfer coefficient, K_p: thermal conductivity of fruits and vegetables, α_n: the n th roots of
(aα)cot(aα)+ah-1=0 (2)
The surface heat transfer coefficients H are caluculated with the following empirical formula by McAdams:
HD/K_f=0.33(wD/ν)^0.6 (3)
where
w: velocity of cooling air, D=diameter of spherical fruits and vegetables, K_f: thermal conductivity of cooling air,
ν: kinematic viscosity of cooling air.
The thermal property of the air and the fruits and vegetables are assumed that K=5.0×10^-4m²/hr, K_p=0.36Kcal/mh°C, K_f=0.0207Kcal/mh°C (at 0°C), ν=0.138×10^-4m²/sec (at 0°C).
The following approximate formula are derived on the assumption that the terms of the series of the equation (1) after the first may be neglected, and the half-cooling time are also caluculated with the formula (Table 2).
The cooling time can be estimated accurately by the following formula with the half-cooling time in Table (1) in the range that the value of Table (1) is equal to that of Table (2):
t=3.322 T_HALF log10((V_po-V_f)/(v-V_f)) (4)
where
t: cooling time (hr), V_po: initial temperature of fruits and vegetables, V_f: temperature of air, v: final mass average temperature of fruits and vegetables.