A long steel strip is being conveyed through a 3 -m long furnace to be heat treated at a speed of \(0.01 \mathrm{~m} / \mathrm{s}\). The steel strip \(\left(k=21 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=8000 \mathrm{~kg} / \mathrm{m}^{3}\right.\), and \(c_{p}=570 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) ) has a thickness of \(5 \mathrm{~mm}\), and it enters the furnace at an initial temperature of \(20^{\circ} \mathrm{C}\). Inside the furnace, the air temperature is maintained at \(900^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of \(80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Using EES (or other) software, determine the surface temperature gradient of the steel strip as a function of location inside the furnace. By varying the location in the furnace for \(0 \leq x \leq 3 \mathrm{~m}\) with increments of \(0.2 \mathrm{~m}\), plot the surface temperature gradient of the strip as a function of furnace location. Hint: Use the lumped system analysis to calculate the plate surface temperature. Make sure to verify the application of this method to this problem.

Step by step solution

01

Calculate the Biot number for verification

The Biot number (Bi) is defined as: \(\mathrm{Bi} = \frac{h\delta}{k}\) Bi = h * delta / k Calculate the Biot number with the given values: h = 80 \(\mathrm{~W/m^{2}\cdot K}\) delta = 5mm = \(0.005 \mathrm{~m}\) k = 21 \(\mathrm{~W/m \cdot K}\) Bi = 80 * 0.005 / 21 Bi ≈ 0.019 When Bi is below 0.1, it is considered as valid to use the lumped system analysis. In this case, the Biot number is approximately 0.019, so the lumped system analysis can be applied.

02

Calculate the time required to move through the furnace

First, we need to find the time it takes for the strip to move through the 3-meter furnace with a speed of \(0.01 \mathrm{~m} / \mathrm{s}\). t = L / v t = 3 / 0.01 t = 300s The time required for the steel strip to move through the furnace is 300 seconds.

03

Calculate the dimensionless time (Fourier number)

Next, we will calculate the dimensionless time (Fourier number): Fo = \(\frac{\alpha t}{\delta^{2}}\) where: α (alpha) is the thermal diffusivity of the material and is calculated by: \(\alpha=\frac{k}{\rho c_{p}}\) α = k / (ρ * cp) First, calculate alpha for the steel strip: α = 21 / (8000 * 570) α ≈ 4.91 x 10⁻⁶ m²/s Now we can find the Fourier number (Fo): Fo = (4.91 x 10⁻⁶ * 300) / 0.005² Fo ≈ 5.88 x 10⁻²

04

Calculate the surface temperature

Apply the lumped system analysis to find the surface temperature of the steel strip at different positions in the furnace. The temperature distribution is given by: \(T(x,t) = T_{\infty} + (T_{i} - T_{\infty})\cdot \exp{(-\mathrm{Bi} \cdot \mathrm{Fo})}\) For each position \(x\) from 0 to 3 meters with increments of \(0.2 \mathrm{~m}\) , calculate the surface temperature using the above formula.

05

Find the surface temperature gradient

The surface temperature gradient can be found by calculating the derivative of the surface temperature with respect to the distance x (location inside the furnace). Calculate the gradient for each position and store the values.

06

Plot the surface temperature gradient

Use the calculated gradient values and plot a graph with the surface temperature gradient on the vertical axis and location inside the furnace on the horizontal axis. The resulting graph will show how the surface temperature gradient varies as the steel strip moves through the furnace.

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