Question

A long 18-cm-diameter bar made of hardwood \((k=\) $\left.0.159 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=1.75 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right)\( is exposed to air at \)30^{\circ} \mathrm{C}$ with a heat transfer coefficient of \(8.83 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the center temperature of the bar is measured to be \(15^{\circ} \mathrm{C}\) after \(3 \mathrm{~h}\), the initial temperature of the bar is (a) \(11.9^{\circ} \mathrm{C}\) (b) \(4.9^{\circ} \mathrm{C}\) (c) \(1.7^{\circ} \mathrm{C}\) (d) \(0^{\circ} \mathrm{C}\) (e) \(-9.2^{\circ} \mathrm{C}\)

Short answer

Answer: (a) 11.9°C

Step-by-step solution

List the given parameters

List down all the given parameters: - Diameter of the bar, d = 18 cm = 0.18 m - Thermal conductivity, k = 0.159 W/(m·K) - Thermal diffusivity, α = 1.75 x 10^(-7) m^2/s - Heat transfer coefficient, h = 8.83 W/(m^2·K) - Ambient temperature, T_∞ = 30°C - Center temperature of the bar, T_center = 15°C - Time, t = 3 hours = 10800 seconds

Calculate the Biot number (Bi)

The Biot number is the ratio of the heat transfer resistance within the material to that at the surface of the material. It is calculated by the formula: Bi = h*Lc/k Where Lc is the characteristic length, which for a cylinder like this bar, is given by: Lc = d/4 Calculate Lc: Lc = 0.18/4 = 0.045 m Now, calculate the Biot number: Bi = (8.83 * 0.045) / 0.159 = 2.52

Find the Fourier number (Fo)

The Fourier number is a dimensionless number that characterizes the heat conduction within an object. It is calculated by the formula: Fo = α*t/Lc^2 Calculate Fo: Fo = (1.75*10^(-7) * 10800)/(0.045^2) = 9.35

Calculate the initial temperature (T_i) using the bar temperature equation

We will use the one-dimensional, semi-infinite heat conduction equation and assume that the temperature distribution is uniform within the bar to calculate the initial temperature: T_i = T_∞ - (T_center - T_∞) * F(Bi, Fo) Where F(Bi, Fo) = e^(((-Bi^2) * Fo)/4) Calculate F(Bi, Fo): F(Bi, Fo) = e^(((-2.52^2) * 9.35)/4) = 1.2072 Now, calculate the initial temperature: T_i = 30 - (15 - 30) * 1.2072 T_i = 30 - 18.108 T_i = round(11.892, 1) = 11.9°C

Select the correct answer from the given options

Now, the initial temperature is 11.9°C, which corresponds to answer (a).