It is desired to develop a simple model for predicting the temperature–time history of a plate during the drying cycle in a dishwasher. Following the wash cycle the plate is at T_{p}(t)=T_{p}(0)=65^{\circ} \mathrm{C}T p
(t)=T p
(0)=65 ∘
C and the air in the dishwasher is completely saturated \left(\phi_{\infty}=1.0\right)(ϕ [infinity]
=1.0) at T_{\infty}=55^{\circ} \mathrm{C}T [infinity]
=55 ∘
C. The values of the plate surface area A_{s}A s
, mass M, and specific heat c are such that M c / A_{s}=1600 \mathrm{J} / \mathrm{m}^{2} \cdot \mathrm{K}Mc/A s
=1600J/m 2
⋅K. (a) Assuming the plate is completely covered by a thin film of water and neglecting the thermal resistances of the film and plate, derive a differential equation for predicting the plate temperature as a function of time. (b) For the initial conditions (t=0) estimate the change in plate temperature with time, d T / d t\left(^{\circ} \mathrm{C} / \mathrm{s}\right)dT/dt( ∘
C/s), assuming that the average heat transfer coefficient on the plate is 3.5 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}3.5W/m 2
⋅K.
