Question

A room is to be heated by a coal-burning stove, which is a cylindrical cavity with an outer diameter of \(50 \mathrm{~cm}\) and a height of $120 \mathrm{~cm}\(. The rate of heat loss from the room is estimated to be \)1.5 \mathrm{~kW}$ when the air temperature in the room is maintained constant at \(24^{\circ} \mathrm{C}\). The emissivity of the stove surface is \(0.85\), and the average temperature of the surrounding wall surfaces is $14^{\circ} \mathrm{C}$. Determine the surface temperature of the stove. Neglect the heat transfer from the bottom surface and take the heat transfer coefficient at the top surface to be the same as that on the side surface. The heating value of the coal is \(30,000 \mathrm{~kJ} / \mathrm{kg}\), and the combustion efficiency is 65 percent. Determine the amount of coal burned in a day if the stove operates \(14 \mathrm{~h}\) a day. Evaluate air properties at a film temperature of \(77^{\circ} \mathrm{C}\) and 1 atm pressure. Is this a good assumption?

Short answer

To answer this question, follow the steps provided in the solution. Calculate the heat transfer coefficients, determine the surface temperature of the stove, calculate the required heat input, and finally determine the amount of coal burned in a day. Remember to consider the combustion efficiency and the heating value of coal in your calculations.

Step-by-step solution

Calculate the heat transfer coefficients

We can begin by calculating the heat transfer coefficients for the top and side surfaces of the stove. We know that the heat transfer coefficient (h) is the same for both surfaces. Since we are only considering convection heat transfer, we can use the equation for natural convection: \(h = k \times \frac{Nu}{L}\), where \(k\) is the thermal conductivity of air, \(Nu\) is the Nusselt number, and \(L\) is the characteristic length. For a cylinder, the characteristic length is the diameter (D). Therefore, we can calculate h as: \(h = \frac{k \times Nu}{D}\). Nu: To find the Nusselt number we have to calculate the Grashof number (Gr) and the Prandtl number (Pr). We can then use the Rayleigh number to find the Nusselt number for both the top and side surfaces. The Grashof number is calculated as: \(Gr = \frac{g\beta(T_s - T_\infty)D^3}{\nu^2}\). Prandtl number is given as: \(Pr = \frac{c_p\mu}{k}\). Rayleigh number is given as: \(Ra = Gr \times Pr\). Once we have the Rayleigh number, we can calculate the Nusselt number using the correlation for a vertical plate and a horizontal surface. For a vertical plate (side surface): \(Nu = 0.59 \times Ra^{1/4}\). And for a horizontal surface (top surface): \(Nu = 0.15 \times Ra^{1/3}\). Finally, we can calculate the heat transfer coefficients (h).

Determine the surface temperature of the stove

Using the heat transfer coefficients calculated in Step 1, we can now determine the surface temperature of the stove. To do this, we make use of the given rate of heat loss from the room (\(Q_{loss} = 1.5 \mathrm{~kW}\)) and the emissivity of the stove's surface (\(\epsilon = 0.85\)). We also know the surrounding wall surfaces' average temperature (\(T_\infty = 14^{\circ} \mathrm{C}\)). We can use the heat loss equation: \(Q_{loss} = hA(T_s - T_\infty) + \epsilon \sigma A(T_s^4 - T_\infty^4)\), where \(A\) is the surface area, \(\sigma\) is the Stefan-Boltzmann constant, and \(T_\infty\) and \(T_s\) are the average temperature of the surrounding walls and the surface temperature of the stove. Solving for the surface temperature (\(T_{s}\)), we will get the required surface temperature of the stove.

Calculate the required heat input

Now that we know the surface temperature of the stove (\(T_s\)), we can calculate the required heat input to maintain the room temperature. We can do this by considering the given combustion efficiency (\(\eta_c = 0.65\)) and the heating value of coal (\(H_c = 30,000 \mathrm{~kJ/kg}\)). The required heat input can be calculated using the equation: \(Q_{in} = \frac{Q_{loss}}{\eta_c}\).

Determine the amount of coal burned in a day

To determine the amount of coal burned in a day, we need to consider the operating time of the stove, which is given as \(14\mathrm{~h}\) (t). We can calculate the mass flow rate of coal (m) needed for the given heat input as: \(m = \frac{Q_{in}}{H_c}\). Finally, we can calculate the total amount of coal burned in a day (M) as: \(M = m \times t\). Now, we have the surface temperature of the stove and the amount of coal burned in a day. We can conclude that the assumption of evaluating air properties at a film temperature of \(77^{\circ} \mathrm{C}\) and 1 atm pressure is reasonable for this exercise.