Question

13.62 Coal with a mass analysis of \(88 \% \mathrm{C}, 6 \% \mathrm{H}, 4 \% \mathrm{O}, 1 \% \mathrm{~N}\), 1\% S burns completely with the theoretical amount of air. Determine (a) the amount of \(\mathrm{SO}_{2}\) produced, in \(\mathrm{kg}\) per \(\mathrm{kg}\) of coal. (b) the air-fuel ratio on a mass basis. (c) For environmental reasons, it is desired to separate the \(\mathrm{SO}_{2}\) from the combustion products by supplying the products at \(340 \mathrm{~K}, 1\) atm to a device operating isothermally at \(340 \mathrm{~K}\). At steady state, a stream of \(\mathrm{SO}_{2}\) and a stream of the remaining gases exit, each at \(340 \mathrm{~K}, 1 \mathrm{~atm}\). If the coal is burned at a rate of \(10 \mathrm{~kg} / \mathrm{s}\), determine the minimum theoretical power input required by the device, in \(\mathrm{kW}\). Heat transfer with the surroundings occurs, but kinetic and potential energy effects can be ignored.

Short answer

The \(\mathrm{SO_2}\) produced is 0.02 kg per kg of coal. The air-fuel ratio is 11.22 kg-air/kg-coal. The theoretical minimum power required is zero.

Step-by-step solution

Analyze composition of coal

The coal composition is given as: \(88 \, \text{wt}\text{\text{%}} \text{C}, 6 \, \text{wt}\text{\text{%}} \text{H}, 4 \, \text{wt}\text{\text{%}} \text{O}, 1 \, \text{wt}\text{\text{%}} \text{N}, 1 \, \text{wt}\text{\text{%}} \text{S}\).

Determine the amount of \(\mathrm{SO_2}\) produced

Since 1% of the coal's mass is sulfur, for 1 kg of coal burned, we have 0.01 kg of sulfur. The reaction of sulfur during combustion is: \( \mathrm{S} + \mathrm{O}_{2} \rightarrow \mathrm{SO}_{2} \). Thus, the mass of \(\mathrm{SO_2}\) produced per kg of coal is calculated as: \( \text{mass of } \mathrm{SO_2} = 0.01 \, \text{kg of S} \times \dfrac{64 \, \text{g/mol}}{32 \, \text{g/mol}}= 0.02 \, \text{kg} \).

Calculate the air-fuel ratio

For complete combustion of carbon, hydrogen and sulfur in 1 kg of coal, the required air is determined as: \( \text{C:} \, 0.88 \text{ kg of C} \times \dfrac{32 \, \text{g/mol}}{12 \, \text{g/mol}} = 2.33 \text{ kg of } \mathrm{O_2} \) \( \text{H:} \, 0.06 \text{ kg of H} \times \dfrac{8 \, \text{g/mol}}{2 \, \text{g/mol}} = 0.24 \text{ kg of } \mathrm{O_2} \) \( \text{S:} \, 0.01 \text{ kg of S} \times \dfrac{32 \text{ g}}{32 \text{ g}} = 0.01 \text{ kg of } \mathrm{O_2} \) Total \( \mathrm{O_2} = 2.33 + 0.24 + 0.01 = 2.58 \text{ kg} \) of \( \mathrm{O_2} \). Since air contains 23% oxygen by mass: \( \mathrm{Air-fuel \, ratio} = \dfrac{2.58 \, \text{kg}}{\dfrac{23}{100}} = 11.22 \, \text{kg/kg fuel} \)

Determine minimum theoretical power for separating \(\mathrm{SO_2}\)

The power required for isothermal separation involves calculating the Gibbs free energy change. The change \( \Delta G\) is given by the chemical potential \( \mu \)of \( \mathrm{SO_2}\) at given conditions, calculated as: \( \Delta G_{m} = RT \, \ln \dfrac{p}{p_{0}} \) where \( T = 340 \, \text{K}, \ p = 1 \, \text{atm} \) Plugging in the values: \( \, \Delta G_{m} = (8.314 \, \text{J/mol K})(340 \, \text{K}) \, \ln (1) = 0 \, \text{J/mol} \). As there is no change in free energy, the theoretical minimum power is zero.

Key concepts

Combustion

Combustion, commonly known as burning, is a high-temperature exothermic reaction between a fuel and an oxidant. In this context, coal undergoes combustion in the presence of oxygen from the air. The complete combustion of coal typically yields carbon dioxide, water, nitrogen, and sulfur dioxide as products. The chemical reaction governing combustion is essential for understanding how different elements in the fuel, such as carbon, hydrogen, sulfur, and nitrogen, react with oxygen. Here, sulfur in the coal burns to form sulfur dioxide (SO₂): \( \mathrm{S} + \mathrm{O}_{2} \rightarrow \mathrm{SO}_{2} \). Knowing the precise composition of the coal helps predict the quantities of each combustion product accurately, crucial for engineering and environmental studies.

Air-Fuel Ratio

The air-fuel ratio is an important concept in thermodynamics and combustion engineering. It defines the amount of air required to completely combust a given amount of fuel. This ratio is imperative for achieving efficient burning with minimal pollutants. For coal with known percentages by mass of carbon, hydrogen, sulfur, and other elements, determining the air-fuel ratio allows engineers to design combustion systems that ensure complete oxidation of all elements in the fuel, minimizing unburnt components and possible pollutants. The calculation involves summing the required oxygen masses for each element and then using the fact that air is about 23% oxygen by mass. For example, the calculation for 1 kg of coal with the given composition results in an air-fuel ratio of 11.22 kg of air per kg of coal.

Thermodynamics in Engineering

Thermodynamics plays a pivotal role in engineering, especially in processes involving energy transfer, like combustion. The exercise involves principles such as energy balance, Gibbs free energy, and isothermal processes. When dealing with isothermal separation devices, the concept of Gibbs free energy (\(Delta G\)) is crucial. It gives an understanding of how spontaneous a reaction is under constant temperature and pressure conditions. In an isothermal separation of SO₂, the power input can be theoretically zero if there is no change in Gibbs free energy, as shown by the calculation for this system operating at 340K. Understanding thermodynamics helps engineers design systems that maximize efficiency while minimizing environmental impacts.

Sulfur Dioxide Emissions

Sulfur dioxide (SO₂) emissions are a major environmental concern associated with the combustion of sulfur-containing fuels like coal. SO₂ can lead to acid rain and respiratory problems in living beings. Hence, it is important to quantify and manage SO₂ emissions. In this problem, we'd determine the amount of SO₂ produced by burning coal, which was calculated to be 0.02 kg per kg of coal. This quantification is essential for complying with environmental regulations and designing emission control systems. One such control system is an isothermal device that separates SO₂ from other gases at high efficiency. The accurate design and power input calculations for these devices are critical to minimize pollution and protect the environment.