Question

A 6-mm-thick stainless steel strip $(k=21 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\(, oven at a temperature of \)500^{\circ} \mathrm{C}$ is allowed to cool within a buffer zone distance of \(5 \mathrm{~m}\). To prevent thermal burns to workers who are handling the strip at the end of the buffer zone, the surface temperature of the strip should be cooled to \(45^{\circ} \mathrm{C}\). If the air temperature in the buffer zone is \(15^{\circ} \mathrm{C}\) and the convection heat transfer coefficient is $120 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine the maximum speed of the stainless steel strip.

Short answer

Answer: To find the maximum speed of the stainless steel strip, we will analyze the given data and use the convection heat transfer equation. By equating heat transfer rates from convection and conduction equations, we can solve for the strip speed (v). The maximum value of v will be the maximum speed of the stainless steel strip to prevent thermal burns to workers handling it.

Step-by-step solution

Analyze the given data

We are given the following information: - Thickness of the stainless steel strip, t = 6 mm - Thermal conductivity of the strip, k = 21 W/(m·K) - Oven temperature, T_hot = 500°C - Desired cooled temperature, T_cooled = 45°C - Air temperature in the buffer zone, T_air = 15°C - Convection heat transfer coefficient, h = 120 W/(m²·K) - Buffer zone distance, d = 5m

Use the convection heat transfer equation

The convection heat transfer equation is given by: q = h * A * (T_surface - T_air) where: - q is the heat transfer rate (W) - h is the convection heat transfer coefficient (120 W/(m²·K)) - A is the surface area of the strip (m²) - T_surface is the surface temperature of the strip (°C) - T_air is the air temperature in the buffer zone (15°C) Since we want to cool the strip to 45°C, we can set T_surface to be equal to T_cooled, and our equation becomes: q = h * A * (T_cooled - T_air)

Determine heat transfer rate, q

The heat transfer rate can be determined using the Fourier's Law of conduction, which is given by: q = k * A * (T_hot - T_cooled) / t Here, we already have the values for k, T_hot, and T_cooled. The strip has a thickness of 6 mm, which is equal to 0.006 m. Now we can plug in the values and solve for q: q = 21 * A * (500 - 45) / 0.006

Equate the heat transfer rate from convection and conduction equations

We can now equate the heat transfer rate from both the convection and conduction equations: h * A * (T_cooled - T_air) = k * A * (T_hot - T_cooled) / t Notice that A appears in both sides of the equation, and we can cancel it out: h * (T_cooled - T_air) = k * (T_hot - T_cooled) / t

Solve for the strip speed

Since q is the heat transfer rate, it can also be written as: q = m * Cp * (T_hot - T_cooled) where: - m is the mass flow rate of the strip (kg/s) - Cp is the specific heat capacity of the strip (J/(kg·K)) Since we don't have the values for mass flow rate, we can write mass flow rate as: m = A * d * rho * v where: - A is the area of the strip (m²) - d is the buffer zone distance (5 m) - rho is the density of the strip (kg/m³) - v is the speed of the strip (m/s) Now, we can substitute this equation into the q equation and use it to find the strip speed, v: q = A * d * rho * v * Cp * (T_hot - T_cooled) We already know the value of q from step 3. We can plug in the appropriate values and solve for v. Once we find the maximum value of v, that will be the maximum speed of the stainless steel strip.