Question

A long cylindrical wood \(\log (k=0.17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\left.\alpha=1.28 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right)\) is \(10 \mathrm{~cm}\) in diameter and is initially at a uniform temperature of \(15^{\circ} \mathrm{C}\). It is exposed to hot gases at \(550^{\circ} \mathrm{C}\) in a fireplace with a heat transfer coefficient of \(13.6 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the surface. If the ignition temperature of the wood is \(420^{\circ} \mathrm{C}\), determine how long it will be before the log ignites. Solve this problem using analytical one-term approximation method (not the Heisler charts).

Short answer

Answer: First, calculate the Biot number and then use the one-term approximation method formula to find the time needed for ignition, as specified in the provided step-by-step solution.

Step-by-step solution

Calculate the Biot Number

Calculate the Biot number using the formula: Bi = (h * L)/k where h is the heat transfer coefficient, L is the log radius(diameter/2), and k is the thermal conductivity. We can assume the one-term approximation method is appropriate for this exercise if the Biot number is less than or equal to 0.1. Bi = (13.6 (W/m²⋅K) * 0.05 (m)) / 0.17 (W/m⋅K)

Calculate the time needed for ignition

We can use the following formula to find the time needed for ignition using the one-term approximation method: t = [ (h * L * (T_s - T_i) * sqrt(pi))/(3 * k * (T_i - T_o)) ]² / alpha where T_i= Initial temperature of the wood T_o= Temperature of the hot gases T_s= Ignition temperature of the wood L= Radius of the log k= Thermal conductivity of the wood h= Heat transfer coefficient alpha= Thermal diffusivity of the wood t = [ (13.6 (W/m²⋅K) * 0.05 (m) * (420 (°C) - 15 (°C)) * sqrt(pi))/(3 * 0.17 (W/m⋅K) * (15 (°C) - 550 (°C))) ]² / (1.28 * 10^(-7) (m^2/s)) Calculate the value of t to find the time it takes for the log to ignite.

Key concepts

Thermal Conductivity

Thermal conductivity is a material property that indicates how well a material conducts heat. It's measured in watts per meter per degree Kelvin (W/m·K). In the context of our exercise, the thermal conductivity of the wood log is given as 0.17 W/m·K. This relatively low value tells us that wood is not a great conductor of heat, which is why it doesn't transfer heat quickly and is often used as an insulating material.

In the problem, thermal conductivity (\(k\)) plays a crucial role because it directly affects the rate at which heat can be transferred through the log. Higher thermal conductivity would result in the inner part of the log reaching the ignition temperature more quickly. Understanding this concept helps us comprehend why and how materials react differently to heat.

• It's critical in determining the Biot Number (covered in the next section), which helps decide how uniform the temperature distribution is within a material.
• It influences cooling and heating times, which is vital for predicting scenarios like when the log will ignite.

Biot Number

The Biot Number (\(Bi\)) is a dimensionless number that encapsulates the ratio of heat transfer resistance within a body to the heat transfer resistance across the boundary of the body. It is calculated using the formula:
\[Bi = \frac{h \cdot L}{k}\]

Where:
\(h\) is the heat transfer coefficient,
\(L\) is the characteristic length (in this case, the radius of the log), and
\(k\) is the thermal conductivity of the material.

The Biot Number is important in determining the suitability of the one-term approximation method. If the Biot Number is less than or equal to 0.1, as it is in this exercise, it implies that the temperature distribution inside the object is relatively uniform. This information is crucial because:

• A small Biot Number means the log's center reaches a temperature close to the surface temperature more quickly.
• It allows for simpler models to be used in calculations, like the one-term approximation method, which can provide accurate results without complex computations.

One-term Approximation Method

The one-term approximation method is an analytical approach used to simplify the estimation of temperature changes in a solid over time. It's particularly useful when the Biot Number indicates a relatively uniform internal temperature. This method enables us to avoid using complex charts or lengthy calculations by expressing the temperature change as a simple function.

In our exercise, the one-term approximation method helps us calculate the time (\(t\)) it takes for the log to reach its ignition temperature using the equation:
\[t = \left(\frac{h \cdot L \cdot (T_s - T_i) \cdot \sqrt{\pi}}{3 \cdot k \cdot (T_i - T_o)} \right)^2 / \alpha\]

Where:

• \(T_s\) is the ignition temperature.
• \(T_i\) is the initial temperature of the log.
• \(T_o\) is the temperature of the hot gases.
• \(\alpha\) is the thermal diffusivity of the wood.

By substituting these values into the equation, we can efficiently calculate the time required for ignition. This method is crucial as it leverages the assumption of uniform temperature to reduce computational complexity while maintaining accuracy.