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Chapter 9: Problem 58

An ultimate analysis of a coal is a series of operations that yields the percentages by mass of carbon, hydrogen, nitrogen, oxygen, and sulfur in the coal. The heating value of a coal is best determined in a calorimeter, but it may be estimated with reasonable accuracy from the ultimate analysis using the Dulong formula: $$H H V(\mathrm{k} J / \mathrm{kg})=33,801(\mathrm{C})+144,158[(\mathrm{H})-0.125(\mathrm{O})]+9413(\mathrm{S})$$ where (C), (H), (O), and (S) are the mass fractions of the corresponding elements. The 0.125(O) term accounts for the hydrogen bound in the water contained in the coal. (a) Derive an expression for the higher heating value ( \(H H V\) ) of a coal in terms of \(\mathrm{C}, \mathrm{H}, \mathrm{O},\) and \(\mathrm{S},\) and compare your result with the Dulong formula. Suggest a reason for the difference. (b) A coal with an ultimate analysis of \(75.8 \mathrm{wt} \% \mathrm{C}, 5.1 \% \mathrm{H}, 8.2 \% \mathrm{O}, 1.5 \% \mathrm{N}, 1.6 \% \mathrm{S},\) and \(7.8 \%\) ash (noncombustible) is burned in a power-plant boiler fumace. All of the sulfur in the coal forms \(\mathrm{SO}_{2}\) The gas leaving the furnace is fed through a tall stack and discharged to the atmosphere. The ratio \(\phi\) (\(\mathrm{kg} \mathrm{SO}_{2}\) in the stack gas/kJ heating value of the fuel) must be below a specified value for the power plant to be in compliance with Environmental Protection Agency regulations regarding sulfur emissions. Estimate \(\phi\), using the Dulong formula for the heating value of the coal. (c) An earlier version of the EPA regulation specified that the mole fraction of \(\mathrm{SO}_{2}\) in the stack gas must be less than a specified amount to avoid a costly fine and the required installation of an expensive stack gas scrubbing unit. When this regulation was in force, a few unethical plant operators blew clear air into the base of the stack while the furnace was operating. Briefly explain why they did so and why they stopped this practice when the new regulation was introduced.

Short Answer

Expert verified
The Dulong formula does not need further derivation. The mass ratio of \(SO_2\) emitted per kJ of energy produced can be calculated from the mass fractions of the elements and the Dulong formula. The plant operators stopped blowing fresh air into the stack after the regulations changed because it no longer helped lower the mass ratio of \( SO_2\) that is emitted.

Step by step solution

01

Converting mass percentages into mass fractions

Before proceeding with the task, first, the mass percentages need to be converted into mass fraction. Mass fraction is the ratio of the mass of a component to the total mass of the mixture. Hence, for each element in the coal, its mass fraction would be its mass percentage divided by 100.
02

Deriving the higher heating value (HHV)

With the corresponding mass fractions of the elements, we can derive the Higher Heating Value (HHV) from the Dulong formula. The Dulong formula already states the heating value as: \[HHV (kJ/kg) = 33,801(C) + 144,158[(H)-0.125(O)] + 9413(S)\]. Since this is already in terms of C, H, O, and S, no further derivation is required.
03

Calculating the heating value using the Dulong formula

To calculate \(\phi\), first, determine the heating value of the coal using the Dulong formula, substituting the mass fractions of the elements obtained in step 1.
04

Computing the amount of \(SO_2\)

Since all Sulphur turns into \(SO_2\), for every kg of S, we can obtain 1 mole of \(SO_2\). By using the molar mass of \(SO_2\) (approximately 64 g/mol), we can calculate how many kg of \(SO_2\) a kg of coal produces.
05

Estimating \(\phi\)

The ratio \(\phi\) is the ratio of \(\frac{kg SO_2}{kJ of heating value of fuel}\). Divide the amount of \(SO_2\) produced (found in step 4) by the heating value of the fuel (found in step 3), to estimate \(\phi\).
06

Understanding environmental regulations' impact on plant operators' practice

For the last part of the exercise, we need to understand why some plant operators blow clear air into the stack while the furnace is operating. This helped dilute the \(SO_2\) in the stack gas, thereby decreasing set the mole fraction of \(SO_2\). When regulations changed and focused on the mass ratio instead of the molar ratio of \(SO_2\), this practice became ineffective, because just diluting the \(SO_2\) gas with air does not change the mass of \(SO_2\) emitted.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dulong Formula
The Dulong Formula is a classic way to estimate the higher heating value (HHV) of coal from its elemental composition. Dulong's formula provides a mathematical expression to calculate the energy produced when coal is burned, which is crucial for optimizing industrial energy usage. In the formula- \(HHV (kJ/kg) = 33,801(C) + 144,158[(H)-0.125(O)] + 9413(S)\),- \(C\) is the mass fraction of carbon,\(H\) is the mass fraction of hydrogen,\(O\) is the mass fraction of oxygen,\(S\) is the mass fraction of sulfur.The term \(0.125(O)\) adjusts for the oxygen forming water with hydrogen in the coal. It's crucial to realize this formula isn't an exact calorimetric measure, but rather an estimate based on coal's ultimate analysis. It's especially useful in situations where direct measurement isn't feasible.
Higher Heating Value
The Higher Heating Value (HHV) quantifies the total energy released in combustion, considering the latent heat of vaporization of water. HHV differs from Lower Heating Value (LHV), which excludes the energy from condensing water vapor. In the context of coal, understanding HHV is vital for assessing fuel efficiency. HHV provides insights into how much usable energy is obtained from burning a specific fuel mass. In Dulong's formula: - HHV incorporates corrections for water vapor, - Meaning the latent heat from water turning to vapor is included in energy output. This comprehensive understanding aids engineers and scientists in evaluating and comparing fuel efficiency, which is integral for power plants and thermal energy systems.
Environmental Regulations
Environmental regulations, set by authorities like the Environmental Protection Agency (EPA), aim to minimize pollution and its impacts on the environment. For coal-burning power plants, specific standards limit sulfur dioxide (SO鈧�) emissions. Historically, regulations have evolved: - Earlier, mole fraction limits for SO鈧� in stack gases were enforced. - Now, the focus is on the mass of SO鈧� per unit of energy produced. By changing the focus to the mass ratio rather than the mole ratio, the regulation pushed for actual reductions in sulfur emissions, rather than simply diluting them with additional air. This is crucial for ensuring that environmental standards increase air quality and protect ecosystems from acid rain and other harmful consequences of sulfur pollution.
Sulfur Emissions
Sulfur emissions from burning coal can result in the production of sulfur dioxide (SO鈧�), a significant environmental pollutant. When coal containing sulfur is burned, it reacts with oxygen to form SO鈧� gas, which can contribute to acid rain.Key points about sulfur emissions include:- For each mass of sulfur combusted, a specific mass of SO鈧� is produced.- Understanding the sulfur content in coal helps estimate potential SO鈧� emissions.- Managing sulfur emissions is essential for meeting environmental standards.By calculating the ratio \(\phi = \frac{kg \, of \, SO_2}{ kJ \, of \, heating \, value}\), power plants can assess compliance with emission limits, contributing to reduced environmental impact.
Ultimate Analysis
Ultimate analysis in coal analysis refers to the detailed breakdown of a fuel's elemental composition. It identifies the percentages of carbon, hydrogen, sulfur, nitrogen, and oxygen present in the coal. Understanding ultimate analysis helps in: - Estimating energy content, like through the Dulong Formula, - Predicting emissions from combustion, - Deciding which coals are most suitable for specific applications. Ultimate analysis is foundational for industries in selecting the right type of coal for energy production. It provides essential data to optimize combustion processes and adherence to environmental regulations.

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Most popular questions from this chapter

Methane is burned completely with 40\% excess air. The methane enters the combustion chamber at \(25^{\circ} \mathrm{C},\) the combustion air enters at \(150^{\circ} \mathrm{C},\) and the stack gas \(\left[\mathrm{CO}_{2}, \mathrm{H}_{2} \mathrm{O}(\mathrm{v}), \mathrm{O}_{2}, \mathrm{N}_{2}\right]\) exits at \(450^{\circ} \mathrm{C} .\) The chamber functions as a preheater for an air stream flowing in a pipe through the chamber to a spray dryer. The air enters the chamber at \(25^{\circ} \mathrm{C}\) at a rate of \(1.57 \times 10^{4} \mathrm{m}^{3}(\mathrm{STP}) / \mathrm{h}\) and is heated to \(181^{\circ} \mathrm{C}\). All of the heat generated by combustion is used to heat the combustion products and the air going to the spray dryer (i.e., the combustion chamber may be considered adiabatic). (a) Draw and completely label the process flow diagram and perform a degree- of-freedom analysis. (b) Calculate the required molar flow rates of methane and combustion air (kmol/h) and the volumetric flow rates \(\left(\mathrm{m}^{3} / \mathrm{h}\right)\) of the two effluent streams. State all assumptions you make. (c) When the system goes on line for the first time, environmental monitoring of the stack gas reveals a considerable quantity of CO, suggesting a problem with either the design or the operation of the combustion chamber. What changes from your calculated values would you expect to see in the temperatures and volumetric flow rates of the effluent streams [increase, decrease, cannot tell without doing the calculations]?

Calcium chloride is a salt used in a number of food and medicinal applications and in brine for refrigeration systems. Its most distinctive property is its affinity for water. in its anhydrous form it efficiently absorbs water vapor from gases, and from aqueous liquid solutions it can form (at different conditions) calcium chloride hydrate \(\left(\mathrm{CaCl}_{2} \cdot \mathrm{H}_{2} \mathrm{O}\right)\) dihydrate \(\left(\mathrm{CaCl}_{2} \cdot 2 \mathrm{H}_{2} \mathrm{O}\right)\) tetrahydrate \(\left(\mathrm{CaCl}_{2} \cdot 4 \mathrm{H}_{2} \mathrm{O}\right),\) and hexahydrate \(\left(\mathrm{CaCl}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O}\right)\) You have been given the task of determining the standard heat of the reaction in which calcium chloride hexahydrate is formed from anhydrous calcium chloride: $$\mathrm{CaCl}_{2}(\mathrm{s})+6 \mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \rightarrow \mathrm{CaCl}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O}(\mathrm{s}): \quad \Delta H_{\mathrm{r}}^{\circ}(\mathrm{k} \mathrm{J})=?$$ By definition, the desired quantity is the heat of hydration of calcium chloride hexahydrate. You cannot carry out the hydration reaction directly, so you resort to an indirect method. You first dissolve 1.00 mol of anhydrous \(\mathrm{CaCl}_{2}\) in \(10.0 \mathrm{mol}\) of water in a calorimeter and determine that \(64.85 \mathrm{kJ}\) of heat must be transferred away from the calorimeter to keep the solution temperature at \(25^{\circ} \mathrm{C}\). You next dissolve 1.00 mol of the hexahydrate salt in 4.00 mol of water and find that 32.1 kJ of heat must be transferred to the calorimeter to keep the temperature at \(25^{\circ} \mathrm{C}\). (a) Use these results to calculate the desired heat of reaction. (Suggestion: Begin by writing out the stoichiometric equations for the two dissolution processes.) (b) Calculate the standard heat of reaction in \(\mathrm{kJ}\) for \(\mathrm{Ca}(\mathrm{s}), \mathrm{Cl}_{2}(\mathrm{g})\) and \(\mathrm{H}_{2} \mathrm{O}\) reacting to form \(\mathrm{CaCl}_{2}\) (aq, \(r=10\) ). (c) Speculate about why the standard heat of reaction in forming calcium chloride hexahydrate cannot be measured directly by reacting the anhydrous salt with water in a calorimeter.

The standard heat of the combustion reaction of liquid \(n\) -hexane to form \(\mathrm{CO}_{2}(\mathrm{g})\) and \(\mathrm{H}_{2} \mathrm{O}(\mathrm{l}),\) with all reactants and products at \(77^{\circ} \mathrm{F}\) and 1 atm, is \(\Delta H_{\mathrm{r}}^{\prime}=-1.791 \times 10^{6} \mathrm{Btu} .\) The heat of vaporization of hexane at \(77^{\circ} \mathrm{F}\) is \(13,550 \mathrm{Btu} / \mathrm{b}\) -mole and that of water is \(18.934 \mathrm{Btu} / \mathrm{h}\) -mole. (a) Is the reaction exothermic or endothermic at \(77^{\circ} \mathrm{F}\) ? Would you have to heat or cool the reactor to keep the temperature constant? What would the temperature do if the reactor ran adiabatically? What can you infer about the energy required to break the molecular bonds of the reactants and that released when the product bonds form? (b) Use the given data to calculate \(\Delta H_{\mathrm{r}}^{\mathrm{r}}\) (Btu) for the combustion of \(n\) -hexane vapor to form \(\mathrm{CO}_{2}(\mathrm{g})\) and \(\overline{\mathrm{H}}_{2} \mathrm{O}(\mathrm{g})\) (c) If \(\dot{Q}=\Delta \dot{H},\) at what rate in \(\mathrm{B}_{\text {tu } / \mathrm{s}}\) is heat absorbed or released (state which) if \(120 \mathrm{lb}_{\mathrm{n}} / \mathrm{s}\) of \(\mathrm{O}_{2}\) is consumed in the combustion of hexane vapor, water vapor is the product, and the reactants and products are all at \(77^{\circ} \mathrm{F} ?\) (d) If the reaction were carried out in a real reactor, the actual value of \(\dot{Q}\) would be greater (less negative) than the value calculated in Part (c). Explain why.

A gaseous fuel containing methane and ethane is burned with excess air. The fuel enters the furnace at \(25^{\circ} \mathrm{C}\) and 1 atm, and the air enters at \(200^{\circ} \mathrm{C}\) and 1 atm. The stack gas leaves the furnace at \(800^{\circ} \mathrm{C}\) and 1 atm and contains 5.32 mole\% \(\mathrm{CO}_{2}, 1.60 \%\) CO, \(7.32 \%\) O \(_{2}, 12.24 \% \mathrm{H}_{2} \mathrm{O}\), and the balance \(\mathrm{N}_{2}\). (a) Calculate the molar percentages of methane and ethane in the fuel gas and the percentage excess air fed to the reactor. (b) Calculate the heat (kJ) transferred from the reactor per cubic meter of fuel gas fed. (c) A proposal has been made to lower the feed rate of air to the furnace. State advantages and a drawback of doing so.

Synthetically produced ethanol is an important industrial commodity used for various purposes, including as a solvent (especially for substances intended for human contact or consumption); in coatings, inks, and personal-care products; for sterilization; and as a fuel. Industrial cthanol is a petrochemical synthesized by the hydrolysis of ethylene: $$\mathrm{C}_{2} \mathrm{H}_{4}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{v}) \rightleftharpoons \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(\mathrm{v})$$ Some of the product is converted to diethyl ether in the undesired side reaction $$2 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(\mathrm{v}) \rightleftharpoons\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2} \mathrm{O}(\mathrm{v})+\mathrm{H}_{2} \mathrm{O}(\mathrm{v}) $$The combined feed to the reactor contains 53.7 mole \(\% \mathrm{C}_{2} \mathrm{H}_{4}, 36.7 \% \mathrm{H}_{2} \mathrm{O}\) and the balance nitrogen, and enters the reactor at \(310^{\circ} \mathrm{C}\). The reactor operates isothermally at \(310^{\circ} \mathrm{C}\). An ethylene conversion of \(5 \%\) is achieved, and the yield of ethanol (moles cthanol produced/mole cthylene consumed) is 0.900 . Data for Diethyl Ether $$\begin{aligned}&\Delta \hat{H}_{f}^{\circ}=-271.2 \mathrm{kJ} / \mathrm{mol} \text { for the liquid }\\\ &\left.\Delta \hat{H}_{v}=26.05 \mathrm{kJ} / \mathrm{mol} \quad \text { (assume independent of } T\right)\end{aligned}$$ $$C_{p}\left[\mathrm{kJ} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\right]=0.08945+40.33 \times 10^{-5} T\left(^{\circ} \mathrm{C}\right)-2.244 \times 10^{-7} T^{2}$$ (a) Calculate the reactor heating or cooling requirement in \(\mathrm{kJ} / \mathrm{mol}\) feed. (b) Why would the reactor be designed to yield such a low conversion of ethylene? What processing step (or steps) would probably follow the reactor in a commercial implementation of this process?
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