A food product containing 65 percent moisture is to be freeze-dried in a vacuum dryer operating at 0.5 mm Hg and a condensing temperature of -25°C. The material is 1.0 cm thick, and drying is to take place from both sides with a 1-mm gap between the sample and the heating surfaces, which are at -5°C. The thermal conductivities of the dry solid and the vapor film are about 1.2 x 10^-4 and 4.8 x 10^-5 cal/s-cm-K respectively. 

(a) Allowing for radiation, what is the coefficient for heat transfer from the hot surface to the exterior of the solid? 

(b) When one-half the moisture has bee removed, what is the overall heat-transfer coefficient?

(c) Using this value as an average coefficient, estimate the required time for drying. 

(d) If the material was it layers 0.5 cm thick, by what factor would the drying time be reduced?