A food product with the moisture content of $m_c$ and dimensions $A(m)$ length, $B$

$(m)$ width, and $(m)$ thickness $[$or length $A(m)$ and diameter $D(m)]$ is frozen

when in contact with a refrigerant with a temperature of $T_R.$ The convective heat$-$

transfer coefficient of the refrigerant $h_c,$ the initial product temperature $T_i,$ the

initial freezing temperature of $T_F,$ the final product temperature $($or the final centre

temperature$)T_C,$ the thermal conductivity of frozen product $k_F,$ the specific heat

capacity of frozen product of $C_{pF},$ the specific heat capacity of unfrozen product

$C_{pu},$ the density of unfrozen product ${}_u,$ and the density of frozen product ${}_F$ are

given in Table $1.$

Using the Pham's model and a spreadsheet numerical solution:

a$)$ Calculate the pre$-$freezing, phase change, post$-$freezing and total freezing

times. $($Explain and show your detailed sample calculations$).$

b$)$ Plot the final product temperature $($or the final centre temperature$)T_c$ as a

function of the freezing time $($Explain and show your detailed sample

calculations$).$