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Comparison of quasi-steady-state and unsteady-state formulations in a freeze dryer modeling

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An unsteady-state model is developed for primary and secondary stages of freeze drying process of skim milk. The results are compared with those obtained from a quasi-steady-state (QSS) formulation. The QSS formulation is not valid where the applied heat load is high. The applied heat load affects on the drying time the most compared to other parameters like chamber pressure and the radiation surface temperature.

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C p :

Heat capacities

C :

Concentration (kg/m3)

c :

Weight fraction of internal water

h :

Sample thickness (m)

H :

Position of interface (m)

k 2, k 4 :

Self diffusivity constant (m2/s)

k 1, k 3 :

Bulk diffusivity constant (m2/s)

k :

Thermal conductivity (kW/m K)

f (T z ):

Water vapor pressure–temperature functional form (N/m2)

ΔH sub :

Heat of sublimation of ice (kJ/kg)

ρ :

Density (kg/m3)

K g :

Internal mass transfer coefficient (s)

M :

Molecular weight

m s :

Weight of dry solid

N :

Mass flux (kg/m2 s)

N t :

Total flux (Nt = Nw + Nin)

P :

Partial pressure (N/m2)


Partial pressure of water vapor at z = 0 (N/m2)

q :

Heat flux at the bottom of the vial (kW/m2)

R :

Universal gas constant

S :

Vial surface area (m2)

t :

Time (s)

T :

Temperature (K)

V :

Velocity of interface (m/s)

V tot :

Volume of the sample

X :

Weight fraction of total moisture (dry basis)

z :

Sample length coordinate

α :

Thermal diffusivity

ε :


σ :

Stefan-Boltzmann constant

ρ :

Density (kg/m3)

e :

Effective value

eq :

Equilibrium value

f :

Frozen layer

d :

Dried layer

in :

Inert gas

up :

Irradiation surface

w :

Water vapor

wh :


o :

Initial value at time zero

t :

Value at time t






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Correspondence to Amir Rahimi.

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Adhami, S., Rahimi, A. & Hatamipour, M.S. Comparison of quasi-steady-state and unsteady-state formulations in a freeze dryer modeling. Heat Mass Transfer 50, 1291–1300 (2014).

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