A thick wood slab \((k=0.17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\alpha=1.28 \times\) \(10^{-7} \mathrm{~m}^{2} / \mathrm{s}\) ) that is initially at a uniform temperature of \(25^{\circ} \mathrm{C}\) is exposed to hot gases at \(550^{\circ} \mathrm{C}\) for a period of \(5 \mathrm{~min}\). The heat transfer coefficient between the gases and the wood slab is \(35 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the ignition temperature of the wood is \(450^{\circ} \mathrm{C}\), determine if the wood will ignite.

To determine the Biot number (Bi), we will use the formula: Bi = (hL)/k where h is the heat transfer coefficient, L is the characteristic length of the slab (thickness), and k is the thermal conductivity of the wood. We are given the following values: h = 35 W/m^2⋅K k = 0.17 W/m⋅K L (thickness) = (width + height) / 2 (assuming a square slab, thickness is the arithmetic average of length and width) However, we don't have the dimensions of the wood slab, so we can't directly calculate the Biot Number. We'll have to find another approach.

To analyze the heat transfer process, we need to determine if the process can be considered one-dimensional. For this, we need the Fourier Number (Fo), which can be calculated as follows: Fo = (alpha * t) / L^2 where alpha is the thermal diffusivity, t is the time, and L is the characteristic length of the slab. We are given the following values: alpha = 1.28 * 10^(-7) m^2/s t = 5 min = 300s (converted to seconds) However, we again face the issue of not knowing the dimensions of the wood slab. Therefore, we cannot calculate the Fourier Number directly. Instead, we have to calculate the temperature change at the surface using other methods like lumped capacitance analysis or Heisler charts.

For this exercise, we will use the lumped capacitance analysis. It assumes that the temperature difference inside the solid is negligibly small compared to the temperature difference between the gas and solid. We can represent this concept with the following equation: q = h * A * (T_gas - T_s) where q is the heat rate, A is the surface area of the wood slab, T_gas is the temperature of the hot gases, and T_s is the temperature of the wood surface. We need to find T_s to check if it is above the ignition temperature of the wood. We are given the heat transfer coefficient (h) and the temperature of the hot gases (T_gas). We are also given the initial temperature of the wood (25 °C), but we need to calculate the temperature change at the surface. First, let's find the temperature difference as follows: Delta T = T_gas - T_s We can substitute the temperature of the hot gases into the equation: Delta T = 550 °C - T_s

By rearranging the equation for calculating heat rate, we can find the surface temperature of the wood slab: T_s = T_gas - (q / (h * A)) However, since we don't have the dimensions of the wood slab, we cannot directly find T_s. In conclusion, since we don't have enough information about the dimensions of the wood slab, we cannot determine if the wood will ignite. More information would be necessary to provide a definitive answer.