Question

Thick slabs of stainless steel $(k=14.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\( and \)\left.\alpha=3.95 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)\( and copper \)(k=401 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\( and \)\left.\alpha=117 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$ are placed under an array of laser diodes, which supply an energy pulse of \(5 \times 10^{7} \mathrm{~J} / \mathrm{m}^{2}\) instantaneously at \(t=0\) to both materials. The two slabs have a uniform initial temperature of \(20^{\circ} \mathrm{C}\). Determine the temperatures of both slabs at $5 \mathrm{~cm}\( from the surface and \)60 \mathrm{~s}$ after receiving an energy pulse from the laser diodes.

Short answer

Answer: To find the temperatures of the stainless steel and copper slabs at a depth of 5 cm and 60 seconds after receiving an energy pulse from laser diodes, follow these steps: 1. Understand the heat equation and its parameters. 2. Calculate the thermal penetration depth for both materials using their respective thermal diffusivities. 3. Determine the change in temperature at a depth of 5 cm using the energy provided and the materials' thermal conductivities. 4. Add the change in temperature to the initial temperature of 20°C to determine the final temperatures of both materials. After performing these calculations, the final temperatures of both stainless steel and copper slabs at a depth of 5 cm and 60 seconds after receiving the energy pulse can be determined.

Step-by-step solution

Understand the heat equation

The heat equation is a partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time. The governing equation is given as: $$\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$$ Where: - \(u(x, t)\) represents the temperature at position \(x\) and time \(t\). - \(\alpha\) represents the thermal diffusivity of the material. - \(x\) represents the position within the material. - \(t\) represents the time after which energy pulse is applied. In this exercise, we need to determine the temperature of both materials at position \(x = 5 \mathrm{~cm} = 0.05 \mathrm{~m}\) and time \(t = 60 \mathrm{~s}\) after receiving an energy pulse from the laser diodes.

Calculate the thermal penetration depth

As the energy pulse is provided instantaneously at \(t=0\), we can assume an infinite plane source of heat. In this case, knowing the time \(t\) and the thermal diffusivity \(\alpha\) of the material, we can first find the thermal penetration depth \(\delta\). The thermal penetration depth is given by the formula: $$\delta = \sqrt{4\alpha t}$$ For stainless steel: $$\delta_{steel} = \sqrt{4(3.95 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s})(60 \mathrm{~s})}$$ And for copper: $$\delta_{copper} = \sqrt{4(117 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s})(60 \mathrm{~s})}$$

Determine the change in temperature at 5 cm depth

Next, we can find the change in temperature at 5 cm depth (\(x = 0.05 \mathrm{~m}\)) and 60 s after receiving the energy pulse from the laser diodes using the energy provided \(E = 5 \times 10^{7} \mathrm{~J} / \mathrm{m}^{2}\). The change in temperature is given by the formula: $$\Delta T_x = \frac{E \sqrt{\alpha t}}{2kx \sqrt{\pi}} erfc\left(\frac{x}{\delta}\right)$$ For stainless steel, we have: $$\Delta T_{steel} = \frac{(5 \times 10^{7} \mathrm{~J} / \mathrm{m}^{2}) \sqrt{(3.95 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s})(60 \mathrm{~s})}}{2(14.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})(0.05 \mathrm{~m}) \sqrt{\pi}} erfc\left(\frac{0.05 \mathrm{~m}}{\delta_{steel}}\right)$$ For copper, we have: $$\Delta T_{copper} = \frac{(5 \times 10^{7} \mathrm{~J} / \mathrm{m}^{2}) \sqrt{(117 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s})(60 \mathrm{~s})}}{2(401 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})(0.05 \mathrm{~m}) \sqrt{\pi}} erfc\left(\frac{0.05 \mathrm{~m}}{\delta_{copper}}\right)$$

Calculate the final temperatures

Finally, we can calculate the final temperatures at the 5 cm depth and 60 seconds after the energy pulse by adding the change in temperature to the initial temperature of both materials which is 20°C. For stainless steel: $$T_{steel} = 20 + \Delta T_{steel}$$ For copper: $$T_{copper} = 20 + \Delta T_{copper}$$ By calculating the above expressions, we can determine the temperatures of both slabs at 5 cm depth and 60 seconds after receiving an energy pulse from the laser diodes.