Consider the manufacture of photovoltaic silicon, as...

Consider the manufacture of photovoltaic silicon, as described in Problem 1.42. The thin sheet of silicon is pulled from the pool of molten material very slowly and is subjected to an ambient temperature of $T_{\infty}=527^{\circ} \mathrm{C}$ within the growth chamber. A convection coefficient of $h=7.5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ is associated with the exposed surfaces of the silicon sheet when it is inside the growth chamber. Calculate the maximum allowable velocity of the silicon sheet $V_{\mathrm{si}}$. The latent heat of fusion for silicon is $h_{x f}=1.8 \times 10^{6}$ $\mathrm{J} / \mathrm{kg}$. It can be assumed that the thermal energy released due to solidification is removed by conduction along the sheet.

Key Concepts

Process Velocity Determination via Energy Removal

This concept links the rate at which a process occurs—such as a sheet being pulled during manufacturing—with the ability to remove the heat generated by the process (for example, through latent heat release during solidification). It involves calculating a maximum allowable process speed that ensures heat removal keeps pace with the thermal energy generated, thus maintaining process integrity and material quality.

Heat Conduction in Thin Sheets

Heat conduction in thin sheets involves the transfer of thermal energy through the material. In contexts where a material is being processed, such as in manufacturing, conduction along the sheet's plane is critical for removing latent heat during phase changes. Understanding this concept is vital for analyzing how temperature gradients develop and how efficiently heat can be removed from a localized region.

Thermal Energy Balance at the Phase Change Interface

This key concept is based on the principle of energy conservation at the interface where phase change occurs. It requires that the energy released (or absorbed) during phase transition must be balanced by the energy transfer mechanisms, such as conduction or convection, that remove or supply the excess energy. This balance is fundamental for determining process limits, like the rate at which a material can solidify.

Convection Heat Transfer

This concept involves the transfer of heat between a solid surface and a fluid (either a liquid or gas) in motion. The convection heat transfer coefficient quantifies the effectiveness of this process. It plays an essential role in applications where cooling (or heating) by a fluid is used to control the temperature of a material or process, particularly when dealing with uniform ambient conditions.

Latent Heat of Fusion

Latent heat of fusion is the amount of energy released or absorbed during the phase change between solid and liquid states at a constant temperature. It is an intrinsic material property and is critical when analyzing processes such as manufacturing or solidification, as it dictates how much thermal energy must be removed or supplied to complete the phase transition without changing the temperature.

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