Answer :
Final answer:
It would take approximately 6 minutes for the thermally thin piece of paper to ignite (c).
Explanation:
To calculate the time it takes for the paper to ignite, we can use the one-dimensional transient heat conduction equation:
[tex]\[ \frac{{\partial^2 T}}{{\partial x^2}} = \frac{{\alpha}}{{\alpha t}} \][/tex]
Where [tex]\( \alpha \)[/tex] is the thermal diffusivity, given by [tex]\( \alpha = \frac{{k}}{{\rho \cdot c}} \)[/tex], where k is the thermal conductivity, [tex]\( \rho \)[/tex] is the density, and c is the specific heat.
First, we calculate the thermal diffusivity:
[tex]\[ \alpha = \frac{{0.05 \, \text{W/mK}}}{{700 \, \text{kg/m}^3 \cdot 1.34 \, \text{kJ/kgK}}} = 5.970 \times 10^{-5} \, \text{m}^2/\text{s} \][/tex]
Now, we can use the heat conduction equation to solve for the time it takes to reach the ignition temperature from the ambient temperature:
[tex]\[ t = \frac{{\Delta x^2}}{{2 \alpha}} \][/tex]
Given that the initial temperature difference [tex]\( \Delta T = 220^\circ \text{C} - 18^\circ \text{C} = 202^\circ \text{C} \)[/tex], and the paper is 1 mm thick [tex](\( \Delta x = 0.001 \, \text{m} \))[/tex], we can substitute these values into the equation:
[tex]\[ t = \frac{{(0.001 \, \text{m})^2}}{{2 \cdot 5.970 \times 10^{-5} \, \text{m}^2/\text{s}}} \][/tex]
[tex]\[ t = \frac{{0.000001 \, \text{m}^2}}{{2 \cdot 5.970 \times 10^{-5} \, \text{m}^2/\text{s}}} \][/tex]
[tex]\[ t = \frac{{0.000001}}{{2 \times 5.970}} \, \text{s} \][/tex]
t ≈ 8.4 s
Converting seconds to minutes, we get approximately 6 minutes, indicating option (c) as the correct answer.
