COOLING AND FREEZING CALCULATIONS
Q.T. Pham, School of Chemical Engineering and Industrial Chemistry, University of New South Wales
REFERENCES
Freezing of slab, infinite cylinder and sphere
Freezing/thawing of rods and bricks
Freezing/thawing of ellipses and ellipsoids
Freezing/thawing of ellipses and ellipsoids
COOLING AND FREEZING CALCULATIONS
Q.T. Pham, School of Chemical Engineering and Industrial Chemistry, University of New South Wales
REFERENCES
Freezing of slab, infinite cylinder and sphere
Pham, Q.T.; Freezing of foodstuffs with variations in environmental conditions. 1986. International Journal of
Refrigeration; 9; 290-295.
Freezing/thawing of rods and bricks
Cleland D.J.; Cleland A.C.; Earle R.L. Prediction of freezing & thawing times for multidimensional shapes by simple
formulae. Part : Regular shapes. 1987. International Journal of Refrigeration; 10; 157-164.
Freezing/thawing of ellipses and ellipsoids
Hossain, M.M.; Cleland, D.J.; Cleland, A.C. Prediction of freezing and thawing times for foods of two-dimensional
irregular shape by using a semi-analytical geometric factor. International Journal of Refrigeration; 1992; 15(4); 235-
240.
Freezing/thawing of ellipses and ellipsoids
Hossain, M.M.; Cleland, D.J.; Cleland, A.C. Prediction of freezing and thawing times for foods of three-dimensional
irregular shape by using a semi-analytical geometric factor. International Journal of Refrigeration; 1992; 15(4); 241-
246.
FREEZING TIME CALCULATION
INPUTS:
Dimensions (diameter, height or thickness):
Smallest dimension: 0.12 m
Second smallest dimension: 0.24 m
Largest dimension: 0.50 m
Thermal properties of the product:
Specific heat of unfrozen product (cu): 3800 J/kgK RESULTS:
Thermal conductivity of unfrozen product (ku): 0.47 W/mK
Density of unfrozen product: 1050 kg/m3
Specific heat of frozen product (cf): 1900 J/kgK Slab
Thermal conductivity of frozen product (kf): 1.35 W/mK Cylinder
Density of frozen product: 970 kg/m3 Sphere
Latent heat 209000 J/kg Rod
Freezing conditions: Ellipse
Initial product temperature: 35 oC Cyl D>H
Air temperature: -40 oC Cyl D
Heat transfer coefficient: 11 W/m2K Brick
Desired final centre temperature: -18 oC Ellipsoid
SOLUTION
Freezing time of simple shapes (Pham 1986 method)
Slab Cylinder Sphere
Half-thickness R = 0.06 m
Biot number, Bi (based on frozen product) = 0.49
Tfm = -7.13 oC
DH1 = rho_u.cu (Ti-Tfm) = 1.68E+08 J/m3
DT1 = (Ti+Tfm)/2 – Ta = 53.9 oC
DH2 = rho_f [Lf + cf(Tfm –Tc)] 2.23E+08 J/m3
DT2 = Tfm-Ta = 32.9 oC
E_freeze = 1 2 3
tf = 67165 33582 22388 s
Freezing time in hours = 18.7 9.33 6.22 h
Freezing time of Rod Brick Cylinder Cylinder
H
G1 1 1 1 2
G2 1 1 2 0
G3 0 1 0 1
b1 2.00 2.00 2.00 1.00
2.32 b1^ -1.77 0.680 0.680 0.680 2.320
X(2.32 b1^ -1.77) 0.412 0.412 0.412 0.705
E1 0.282 0.282 0.282 0.920
b2 1.0E+30 4.17 2.00 2.00
2.32 b2^ -1.77 0.000 0.186 0.680 0.680
X(2.32 b2^ -1.77) 0.000 0.161 0.412 0.412
E2 0.000 0.041 0.229 0.229
E_freeze = 1.282 1.323 1.564 2.229
tf 52394 50782 42948 30134
Freezing time in hours = 14.6 14.1 11.9 8.4
Freezing time of Ellipse Ellipsoid
b1 2.00 2.00
b2 4.17
2nd term in eqn for E_freeze 0.418 0.418
3rd term in eqn for E_freeze 0 0.148
E_freeze = 1.418 1.566
tf 47369 42893 s
Freezing time in hours = 13.2 11.9 h
time, h
18.7
9.3
6.2
14.6
13.2
11.9
8.4
14.1
11.9
s
h