A room is to be heated by a coal-burning stove, which is a cylindrical cavity with an outer diameter of \(32 \mathrm{~cm}\) and a height of \(70 \mathrm{~cm}\). 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 a day if the stove operates \(14 \mathrm{~h}\) a day. Evaluate air properties at a film temperature of \(77^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) pressure. Is this a good assumption?

The Stefan-Boltzmann law is a fundamental principle in thermodynamics that relates the heat emitted by a black body in terms of its temperature. Specifically, it states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time \textbf{(also known as the black body's radiant emittance)} is directly proportional to the fourth power of the black body's absolute temperature. The formula for this law is given by:

\[ Q_{radiation} = \epsilon A \sigma T^4 \]
where \( \epsilon \) is the emissivity, \( A \) is the surface area, \( \sigma \) is the Stefan-Boltzmann constant \( 5.67 \times 10^{-8} \mathrm{W/m^2K^4} \), and \( T \) is the absolute temperature in Kelvin.

When applied to real-world problems, such as determining the surface temperature of a stove, we utilize the law with a modified emissivity that accounts for the fact that real objects do not emit as much energy as a perfect black body. In our exercise, the emissivity of the stove surface is given as 0.85, which is used to calculate the radiation heat transfer component.

In the context of the exercise, understanding the Stefan-Boltzmann law helps us determine how much heat the stove radiates, which is crucial for maintaining the desired room temperature, and plays a role in calculating the amount of fuel required to keep the stove operating.

Convection heat transfer is one of the primary modes of heat transfer and involves the movement of heat through a fluid (which can be a liquid or a gas) caused by the fluid's motion. This movement can be generated by differences in density within the fluid due to temperature variations, leading to natural convection, or it can be caused by external means such as a fan or pump, resulting in forced convection.

The heat transfer by convection can be described by the following equation:

\[ Q_{convection} = h A (T_s - T_{fluid}) \]
Here, \( h \) is the heat transfer coefficient that characterizes the convective heat transfer properties of the fluid, \( A \) is the surface area through which the heat is being transferred, \( T_s \) is the surface temperature of the solid body, and \( T_{fluid} \) is the temperature of the fluid.

In our example, the fluid is the air in the room, and we are calculating how much heat is carried away from the stove (the solid body) into the room air. The heat transfer coefficient \( h \) varies with the nature of the fluid flow, surface geometry, and temperature difference. Therefore, in problem-solving, it's essential to either know this coefficient or be capable of estimating it under assumed conditions. By understanding and calculating the convection heat transfer, we can ensure the room is heated efficiently by the coal-burning stove.

Combustion efficiency is a measure of how effectively a combustion process converts fuel into usable heat. It is a percentage representing the fraction of energy in the fuel that is actually used to produce heat, as opposed to being lost in various inefficiencies such as incomplete combustion or heat lost with exhaust gases.

In our example, the combustion efficiency of the coal-burning stove is given as 65%, which means that 65% of the coal's potential energy contributes to heating, while the remaining 35% is lost. This efficiency affects the calculation of the amount of coal needed to maintain the room's temperature, as described by this equation:

\[ m_{coal} = \frac{Q_{total} \times t_{operation}}{Efficiency \times H_V} \]
where \( m_{coal} \) is the mass of coal burned, \( Q_{total} \) is the total heat required, \( t_{operation} \) is the time the stove operates, \( Efficiency \) is the combustion efficiency, and \( H_V \) is the heating value of the coal.

High combustion efficiency is desirable for reducing fuel cost and environmental impact. In practice, achieving maximum combustion efficiency is a complex process that involves proper fuel-air mix, sufficient ignition temperature, and adequate time for the fuel to burn completely. These factors are considered in the design and operation of combustion appliances like stoves, boilers, and engines.