A large slab of polyvinylchloride 1 cm thick is in contact with fluid on either side. On one si

step 1 In this question we have given a uniform slab which is kept between two heat reservoirs whose te

A large slab of polyvinylchloride 1 cm thick is in contact...

A large slab of polyvinylchloride $1 \mathrm{~cm}$ thick is in contact with fluid on either side. On one side the heat transfer coefficient is $20 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$, and on the other it is 40 $\mathrm{W} / \mathrm{m}^{2} \mathrm{~K}$. Initially both fluids are at $20^{\circ} \mathrm{C}$, and the system is in thermal equilibrium. Suddenly the fluid temperatures are raised to $100^{\circ} \mathrm{C}$. Determine the time required for the center of the plate to reach $80^{\circ} \mathrm{C}$.

Key Concepts

Transient Heat Conduction

This concept deals with the time-dependent process of heat transfer in a material, where the temperature distribution changes over time. It involves solving the heat equation with appropriate initial and boundary conditions to predict how a system evolves from one thermal state to another after a sudden change, such as a sudden increase in the temperature of the surrounding fluid.

Thermal Diffusivity

Thermal diffusivity quantifies how quickly heat spreads through a material and is defined as the ratio of thermal conductivity to the product of density and specific heat capacity. It plays a critical role in transient conduction problems by influencing the rate at which a material responds to changes in its thermal environment.

Convection Heat Transfer

Convection heat transfer is the mechanism by which heat is transferred between a solid surface and a fluid moving past it. The efficiency of this process is expressed by the convective heat transfer coefficient, which serves as a boundary condition in conduction problems and directly affects the rate at which the solid's surface temperature changes.

Biot Number

The Biot number is a dimensionless parameter defined as the ratio of the internal thermal resistance of a body to the external convective thermal resistance. It indicates whether the temperature within a solid can be considered uniform (lumped system analysis for low Biot numbers) or if significant internal temperature gradients exist (high Biot numbers), thereby influencing the analytical approach for solving heat conduction problems.

Boundary Conditions in Heat Transfer

Boundary conditions, such as convective heat transfer at surfaces, are essential for solving heat conduction problems. They define how the system interacts with its environment and determine the rate of heat transfer into or out of the material. Correctly applying these conditions is crucial for accurately modeling the thermal response of a system subjected to transient changes.

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