Chapter-4 Mathematical Model CHAPTER MATHEMATICAL MODEL A moving boundary layer model is used to carry out drying kinetics analysis of the proposed system and illustrated in this chapter. Figure 4.1 shows a schematic of the one-dimensional physical model of the AFD process. The mathematical model consists of the applicable conservation equations of energy and mass. It is based on the work of (Jaakko and Impola, 1995) who considered drying of wood slab. y Q Pva pvwa Ta ……………………………………………… Cold dry air (s) ps ……………………………………………………………………… m Dry ……………………………………………………………………… Layer ……………………………………………………………………… ……………………………………………………………………… (f) Ice Layer X Figure 4.1 Physical model of atmospheric freeze drying (f- interface; s-surface; Q-heat transfer; m-mass transfer; Ta-temperature gradient; Pva-partial pressure gradient; pspartial pressure of water vapor around the product surface; pvwa- partial pressure of water in the drying chamber) With references to Figure 4.1, the ice interface (f) recedes to the centre line as heat of sublimation (Q) flows from the surface (s) to the interface to a temperature gradient (Ta) represented by the dotted curve. Simultaneously, water vapor flows through the 51 Chapter-4 Mathematical Model dry layer in response to the water vapor pressure (Pva) gradient represented by the firm line curve. Following mechanism is considered in the model: convective heat transfer from the carrier gas to the surface of the solid mass; radiant heat transfer from the IR radiation heater to the material surface and conductive heat transfer within the solid mass. 4.1 Assumptions • One dimensional heat and mass transfer (thin slab). • There is equilibrium between ice and water vapor at the interface. • Supplied energy is used to remove only ice at the sublimation front. • The frozen region is considered to have homogeneous and uniform thermal conductivity, density and specific heat. • The shape of the product remains constant during the drying period considered. Shrinkage and deformation are neglected. 4.2 Math Model The conservation equation of energy for dry layer is: ρp CP ∂ 2T ∂T ∂T = Kp - C& ′′ ∂x ∂t ∂x (1) The conservation equation of water vapor inside the dry layer is: ρg ∂Y ∂t = ρg D p ∂Y ∂ 2Y ′ ′ & m ∂x ∂x (2) Here, Lv . () () p g v satvt satgt gg p g v v g gg s D ShD Zg pp pp R ShD D ShD Z Y Y R SHD m 2 ,11 ln 2 2 11 1 1 ln 2 , , Γ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − = −+ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − = ′′ ϕ ρ ρ & ( 24) The thermophysical and transport properties of the carrier gas, potato and carrot used in the simulations are summarized in Table 4. 1 (a) and Table 4. 1(b). Chapter -4 Mathematical. the model: convective heat transfer from the carrier gas to the surface of the solid mass; radiant heat transfer from the IR radiation heater to the material surface and conductive heat transfer. Chapter -4 Mathematical Model 51 CHAPTER 4 MATHEMATICAL MODEL A moving boundary layer model is used to carry out drying kinetics analysis of the proposed system and illustrated