Question

During a fire, the trunks of some dry oak trees $\left(k=0.17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.\( and \)\left.\alpha=1.28 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right)$ that are initially at a uniform temperature of \(30^{\circ} \mathrm{C}\) are exposed to hot gases at \(600^{\circ} \mathrm{C}\) for a period of \(4 \mathrm{~h}\), with a heat transfer coefficient of \(65 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the surface. The ignition temperature of the trees is \(410^{\circ} \mathrm{C}\). Treating the trunks of the trees as long cylindrical rods of diameter $20 \mathrm{~cm}$, determine if these dry trees will ignite as the fire sweeps through them. Solve this problem using the analytical one-term approximation method.

Short answer

Answer: To determine if the trunks of the dry oak trees will ignite during a fire, compare the calculated centerline temperature Tc with the given ignition temperature of 410°C. If Tc >= 410°C, the trees will ignite. If Tc < 410°C, the trees will not ignite.

Step-by-step solution

Calculate the Biot number

First, calculate the Biot number (Bi), which is a dimensionless number giving the ratio of surface resistance to conduction resistance. It is defined as: Bi = (hLc)/k where Lc is the characteristic length, defined as the volume to surface area ratio. For a cylinder, Lc = radius/2 = (d/2)/2. Substituting the given values, we get: Bi = (65 * (0.2/2)/2) / 0.17

Calculate the Fourier number

Next, calculate the Fourier number (Fo), another dimensionless number representing the ratio of heat conduction to heat storage in the medium. It is defined as: Fo = (αt)/Lc^2 Substitute the given values, making sure the time t is in seconds (4 hours = 14400 seconds): Fo = (1.28 * 10^{-7} * 14400) / (0.1/2)^2

Calculate the temperature change

To find the centerline temperature, we need to find the temperature change at the centerline using the Biot number and Fourier number, we need to look up the analytical one-term approximation formula for a cylindrical object, which is given as: ΔT = (2 / π) * (Te - Ti) * ∑ (exp((-1 * (2n-1)^2 * π^2 * Fo) / 4) / (2n - 1) * (1 - exp(-1 * (2n-1)^2 * π^2 * Bi)) In our case, n = 1 (one-term approximation), so the equation simplifies to: ΔT = (2 / π) * (Te - Ti) * (1 - exp(-1 * π^2 * Bi))

Calculate the centerline temperature

Calculate the centerline temperature by adding the initial temperature to the temperature change: Tc = Ti + ΔT

Check the ignition temperature

Now, check if the centerline temperature Tc calculated is greater than or equal to the ignition temperature. If yes, the trees will ignite; if not, they will not ignite. Compare Tc with the given ignition temperature of 410°C. If Tc >= 410°C, the trees will ignite. If Tc < 410°C, the trees will not ignite.