During vulcanization a tire carcass is heated by exposure to...

During vulcanization a tire carcass is heated by exposure to saturated steam at $150^{\circ} \mathrm{C}$ on both sides. If the carcass is $2 \mathrm{~cm}$ thick and has an initial temperature of $20^{\circ} \mathrm{C}$, how long will it take for the centerplane to reach $130^{\circ} \mathrm{C}$ ? Take $\alpha=0.06 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ for the rubber.

Key Concepts

Transient Heat Conduction

This concept addresses how temperature fields develop and evolve over time within a body, rather than assuming a steady, equilibrium state. It involves solving the heat diffusion equation, which provides the temperature distribution as a function of both time and space. Understanding transient conduction is key to determining how long it takes for a material’s interior to reach a desired temperature when subject to changing surface conditions.

Thermal Diffusivity

Thermal diffusivity is a material property that quantifies how quickly heat spreads through a substance. It is a ratio of the thermal conductivity to the product of density and specific heat capacity, and it plays a central role in the heat diffusion equation by determining the rate of internal temperature changes. This property is crucial for predicting transient temperature responses in materials.

Boundary Conditions in Heat Transfer

Boundary conditions specify the thermal interactions between a material and its surroundings and are essential for solving heat conduction problems. They can include conditions like prescribed surface temperatures or convective heat transfer, which significantly influence the internal temperature evolution. Clear definition of these conditions, such as symmetric heating from both sides, is critical for accurately modeling the temperature rise in the material.

One-Dimensional Conduction Analysis

One-dimensional conduction analysis simplifies the heat transfer process by considering temperature variations only along one spatial dimension, assuming that heat flow in the other directions is negligible. This approach is often applicable to thin slabs or materials with large lateral dimensions compared to thickness, and it allows for simpler mathematical treatment of the thermal response over time.

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