A 2 cm-thick slab of cloth-filled thermoplastic resin,...

A $2 \mathrm{~cm}$-thick slab of cloth-filled thermoplastic resin, initially at $20^{\circ} \mathrm{C}$, is placed in a press with walls maintained at $100^{\circ} \mathrm{C}$ by condensing steam. As soon as the center temperature reaches $85^{\circ} \mathrm{C}$, the slab is removed and allowed to cool in $20^{\circ} \mathrm{C}$ air until the center temperature falls to $40^{\circ} \mathrm{C}$. How long does each stage of the process take? Use $k=0.6 \mathrm{~W} / \mathrm{m} \mathrm{K}, \rho=1700 \mathrm{~kg} / \mathrm{m}^{3}$, and $c=1200 \mathrm{~J} / \mathrm{kg} \mathrm{K}$. The heat transfer coefficient for the cooling process can be taken as $7 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$.

A $2 \mathrm{~cm}$-thick slab of cloth-filled thermoplastic resin, initially at $20^{\circ} \mathrm{C}$, is placed in
a press with walls maintained at $100^{\circ} \mathrm{C}$ by condensing steam. As soon as the center temperature reaches $85^{\circ} \mathrm{C}$, the slab is removed and allowed to cool in $20^{\circ} \mathrm{C}$ air until the center temperature falls to $40^{\circ} \mathrm{C}$. How long does each stage of the process take? Use $k=0.6 \mathrm{~W} / \mathrm{m} \mathrm{K}, \rho=1700 \mathrm{~kg} / \mathrm{m}^{3}$, and $c=1200 \mathrm{~J} / \mathrm{kg} \mathrm{K}$. The heat transfer coefficient for the cooling process can be taken as $7 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$.

Key Concepts

Convective Heat Transfer

Convective heat transfer refers to the mechanism by which heat is transferred between a surface and a moving fluid, characterized by a convective heat transfer coefficient. It is particularly relevant during the cooling process where the slab loses heat to ambient air. Newton’s law of cooling, which relates the heat flux at the surface to the temperature difference between the surface and the surrounding air, is a critical component for setting up the boundary conditions in the cooling phase of this problem.

Biot Number

The Biot number is a dimensionless parameter defined as hL/k, where h is the convective heat transfer coefficient, L is the characteristic length, and k is the thermal conductivity. It indicates the ratio of internal resistance to heat conduction within the object to the external resistance to heat transfer. Assessing the Biot number helps decide whether a lumped capacitance model may be applied or if a distributed parameter model (i.e., solving the transient conduction equation) is required for accurate prediction of the temperature evolution in the slab.

Boundary Conditions

Boundary conditions specify how the material interacts with its environment at its surfaces. In this problem, the slab undergoes two sets of boundary conditions: constant surface temperature during the heating stage (due to condensing steam at 100°C) and a convective boundary condition during the cooling stage (where heat is lost to air at 20°C). Proper formulation of these boundary conditions is essential in solving the heat equation and determining the time required for the center temperature to reach the target values.

Transient Heat Conduction

This concept involves the study of how temperature within a material changes with time as heat is conducted. It is crucial here because both the heating and cooling stages involve the evolution of the temperature profile within a finite slab, governed by the heat equation. Understanding transient conduction helps in setting up the problem with the appropriate initial conditions and internal energy balances to compute the time taken for a specified change in the center temperature.

Thermal Diffusivity

Thermal diffusivity, defined as k/(?c), measures the rate at which heat diffuses through a material and plays a central role in transient heat conduction problems. In the context of this question, it helps determine how quickly the temperature profile will adjust when the slab is exposed to new boundary conditions, whether during the rapid heating in a steam press or the slower convective cooling process.

Please give Ace some feedback