A 4 cm-thick slab has an initial temperature of 100^∘ C. At time t=0 the temperature of the one

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A 4 cm-thick slab has an initial temperature of 100^∘ C. At...

A $4 \mathrm{~cm}$-thick slab has an initial temperature of $100^{\circ} \mathrm{C}$. At time $t=0$ the temperature of the one surface is lowered to $50^{\circ} \mathrm{C}$ and the other to $0^{\circ} \mathrm{C}$. Determine the heat flux out the colder side at $t=2000 \mathrm{~s}$. Take $k=0.35 \mathrm{~W} / \mathrm{m} \mathrm{K}$, $\alpha=0.15 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ for the slab material.

Key Concepts

Boundary and Initial Conditions in Heat Transfer

Specifying boundary and initial conditions is essential in transient heat conduction problems as they uniquely determine the solution to the heat diffusion equation. The initial temperature distribution sets the starting point, while the boundary conditions, such as fixed temperatures at the surfaces, drive the subsequent evolution of the temperature profile.

Thermal Diffusivity

Thermal diffusivity is a key material property that characterizes the rate at which heat spreads through a material. It is defined as the ratio of thermal conductivity to the product of density and specific heat capacity, and it influences the speed at which a material responds to changes in its thermal environment.

Analytical Solution Methods for the Heat Equation

Analytical methods, such as separation of variables and eigenfunction expansion, are commonly employed to solve the heat equation in transient conduction problems. These methods decompose the problem into simpler parts, enabling the derivation of series solutions that describe the temperature distribution as a function of both space and time.

Transient Heat Conduction

This concept involves analyzing how heat is distributed and changes within a material over time. It deals with solving the heat diffusion equation under time-dependent conditions, where both the initial temperature of the material and the temperature at its boundaries influence the evolution of the temperature field.

Fourier’s Law of Conduction

Fourier’s law is the foundational principle that relates the heat flux through a material to the temperature gradient within it. It states that the heat flux is proportional to the negative of the temperature gradient, thereby providing a direct method to calculate the rate of heat transfer based on how temperature changes spatially within the body.

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