Question

In a meat processing plant, 2-cm-thick steaks $\left(k=0.45 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.\( and \)\left.\alpha=0.91 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right)\( that are initially at \)25^{\circ} \mathrm{C}$ are to be cooled by passing them through a refrigeration room at \(-11^{\circ} \mathrm{C}\). The heat transfer coefficient on both sides of the steaks is \(9 \mathrm{~W} / \mathrm{m}^{2}\). K. If both surfaces of the steaks are to be cooled to \(2^{\circ} \mathrm{C}\), determine how long the steaks should be kept in the refrigeration room. Solve this problem using the analytical one-term approximation method.

Short answer

Answer: The steaks should be kept in the refrigeration room for approximately 45.79 minutes.

Step-by-step solution

Calculate the Biot number (Bi)

The Biot number (Bi) is given by the formula: $$ Bi = \frac{hL_c}{k} $$ Where: - \(h\) is the heat transfer coefficient (9 W/m² K), - \(L_c\) is the characteristic length, which is half the thickness of the steak (0.01 m), and - \(k\) is the thermal conductivity of the steak (0.45 W/m K). We can now calculate the Biot number: $$ Bi = \frac{9 \times 0.01}{0.45} = 0.2 $$

Calculate the Fourier number (Fo)

The analytical one-term approximation method is based on the solution of the transient heat conduction equation with the assumption of negligible internal temperature gradients. The Fourier number (Fo) can be expressed as: $$ Fo = \frac{\alpha t}{L_c^2} $$ Where: - \(\alpha\) is the thermal diffusivity of the steak (0.91 x 10⁻⁷ m²/s), - \(t\) is the time in seconds, and - \(L_c\) is the characteristic length (0.01 m). For the one-term approximation method, we use the following relationship between the centerline temperature (\(T_c\)) and the surface temperature (\(T_s\)): $$ \frac{T_c - T_s}{T_i - T_s} = 1 - 2 \left( \frac{1 - e^{-Bi^2 Fo}}{Bi^2} \right) $$ Where \(T_i\) is the initial temperature of the steak (25°C), and \(T_s\) is the desired surface temperature (2°C). We can now solve for Fo: $$ Fo = \frac{1}{Bi^2} \left( 1 - \frac{T_c - T_s}{T_i - T_s} \right) $$ We need to find \(T_c\) to calculate Fo. Since there are no internal temperature gradients, we can assume that \(T_c = T_s\) and substitute the given values to find Fo: $$ Fo = \frac{1}{0.2^2} \left( 1 - \frac{2 - 2}{25 - 2} \right) = 25 $$

Calculate the cooling time t

Now that we have Fo, we can calculate the cooling time: $$ t = \frac{Fo \cdot L_c^2}{\alpha} $$ Substitute the given values: $$ t = \frac{25 \times (0.01)^2}{0.91 \times 10^{-7}} = 2747.25 \ seconds $$

Convert the cooling time to minutes

To convert the cooling time (\(t\)) from seconds to minutes, simply divide by 60. $$ t_{minutes} = \frac{2747.25}{60} = 45.79 \ minutes $$ The steaks should be kept in the refrigeration room for approximately 45.79 minutes to cool both surfaces to 2°C using the analytical one-term approximation method.