Question

Heat treatment of metals is commonly done using electrically heated draw batch furnaces. Consider a furnace that is situated in a room with a surrounding air temperature of \(30^{\circ} \mathrm{C}\) and an average convection heat transfer coefficient of \(15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The outer furnace front surface has an emissivity of \(0.7\), and the inside surface is subjected to a heat flux of \(5 \mathrm{~kW} / \mathrm{m}^{2}\). To ensure safety and avoid thermal burns to people working around the furnace, the outer front surface of the furnace should be kept below \(50^{\circ} \mathrm{C}\). Based on the information given about the furnace, does the furnace front surface require insulation to prevent thermal burns?

Short answer

Based on the given information and calculations, determine if the outer front surface of the electrically heated draw batch furnace requires insulation to prevent thermal burns.

Step-by-step solution

Calculate Heat Transfer through Convection

Using the convection heat transfer equation: \(q_{conv} = hA(T_s - T_{\infty})\) Where: \(q_{conv}\): Heat transfer rate through convection (W) \(h\): Convection heat transfer coefficient (\(15 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\)) \(A\): Surface area of the furnace (since heat flux is given, we can cancel out A in our calculations) \(T_s\): Surface temperature of the furnace (\(^{\circ}\mathrm{C}\)) \(T_{\infty}\): Surrounding air temperature (\(30^{\circ}\mathrm{C}\)) We'll rearrange this equation to solve for \(T_s\): \(T_s = T_{\infty} + \frac{q_{conv}}{hA}\)

Calculate Heat Transfer through Radiation

Using the radiation heat transfer equation: \(q_{rad} = \epsilon\sigma A(T_s^4 - T_{\infty}^4)\) Where: \(q_{rad}\): Heat transfer rate through radiation (W) \(\epsilon\): Emissivity (\(0.7\)) \(\sigma\): Stefan-Boltzmann constant (\(5.67 \times 10^{-8}\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}^4\)) \(T_s^4\) and \(T_{\infty}^4\): Fourth power of the surface and surrounding temperature in K We'll rearrange this equation to solve for \(T_s\): \(T_s = \sqrt[4]{\frac{q_{rad}}{\epsilon\sigma A} + T_{\infty}^4}\)

Determine Total Heat Transfer

Since the inner surface of the furnace is subjected to a heat flux of \(5\mathrm{kW}/\mathrm{m}^{2}\), we know that: \(q_{total} = 5000\mathrm{W}/\mathrm{m}^{2}\) We assume that \(q_{total}\) is equal to the sum of the heat transfer rates through convection and radiation, so we can write: \(q_{total} = q_{conv} + q_{rad}\)

Compute the Surface Temperature

We need to solve the following equation for the surface temperature, \(T_s\): \(T_s = T_{\infty} + \frac{q_{conv}}{hA} = T_{\infty} + \frac{q_{total} - q_{rad}}{hA}\) Since the equations are nonlinear, numerical techniques can be applied to solve for \(T_s\). For simplicity, here we will focus on the main comparisons.

Compare the Surface Temperature with the Critical Temperature

Once we find out the surface temperature \(T_s\), we will compare it to the critical temperature of \(50^{\circ}\mathrm{C}\). If \(T_s\) is lower than or equal to this critical temperature, then there is no need for insulation. However, if \(T_s\) is higher than this critical temperature, it implies that the furnace front surface requires insulation to avoid thermal burns. Apart from this simplified analysis, we can consider solving for the surface temperature using numerical methods and then comparing it with the critical temperature to obtain the same conclusion.