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Chapter 9: Problem 69

A gas stream consisting of \(n\) -hexane in methane is fed to a condenser at \(60^{\circ} \mathrm{C}\) and 1.2 atm. The dew point of the gas (considering hexane as the only condensable component) is \(55^{\circ} \mathrm{C}\). The gas is cooled to \(5^{\circ} \mathrm{C}\) in the condenser, recovering pure hexane as a liquid. The effluent gas leaves the condenser saturated with hexane at \(5^{\circ} \mathrm{C}\) and 1.1 atm and is fed to a boiler furnace at a rate of \(207.4 \mathrm{L} / \mathrm{s}\), where it is burned with \(100 \%\) excess air that enters the furnace at \(200^{\circ} \mathrm{C}\). The stack gas emerges at \(400^{\circ} \mathrm{C}\) and 1 atm and contains no carbon monoxide or unburned hydrocarbons. The heat transferred from the furnace is used to generate saturated steam at 10 bar from liquid water at \(25^{\circ} \mathrm{C}\). (a) Calculate the mole fractions of hexane in the condenser feed and product gas streams and the rate of hexane condensation (liters condensate/s). (b) Calculate the rate at which heat must be transferred from the condenser (kW) and the rate of generation of steam in the boiler ( \(\mathrm{kg} / \mathrm{s}\) ).

Short Answer

Expert verified
The operation of the condenser and furnace involves principles of thermodynamics and physical chemistry. Calculating mole fractions involves knowledge of the dew point. The rate of hexane condensation and the heat transferred from the condenser can be calculated using energy balance equations, while the rate of steam formation involves both energy balance and the properties of steam.

Step by step solution

01

Mole fractions of hexane

Subtract the dew point of the gas (55掳C) from the temperature of the condenser (60掳C) to get the temperature difference. Use this information along with the equation for the dew point, which relates the partial pressure of the hexane to its mole fraction in the gas. Solve for the mole fractions of hexane in the condenser feed and product gas.
02

Rate of hexane condensation

Calculate the total moles of gas entering the condenser using the Ideal Gas Law. Use this with the mole fraction of hexane in the feed gas to get the total moles of hexane entering the condenser. Similarly, get the total moles of hexane leaving the condenser using the product gas mole fraction and calculate the difference between these two quantities to get the rate of hexane condensation.
03

Heat transfer from condenser

Use principles of energy balance, assuming no work is done, the heat transferred from the condenser will be energy change of the hexane gas as it gets cooled in the condenser. This can be calculated using the specific heat capacity of the gas and the temperature difference.
04

Generation of steam in the boiler

Again applying energy balance, the heat generated in the furnace is used to convert water into steam. Using the specific heat capacity of water and the latent heat of steam formation, calculate the rate of steam formation.

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Key Concepts

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

Mole Fractions
Understanding mole fractions can be incredibly useful in chemical processes. In essence, a mole fraction is a way to express the concentration of a component in a mixture. It's defined as the moles of a compound divided by the total moles of all components in the mixture. For a gas mixture, it's particularly important when considering processes like condensation.

To put it into context, let's imagine you have a bag filled with red and blue marbles. If you want the fraction of red marbles, you'd count up all the red marbles and divide that by the total number of marbles in the bag. In chemical terms, the moles of hexane in the gas stream (the 'red marbles') divided by the total moles of hexane and methane (all 'marbles') gives us the mole fraction of hexane. This is a critical step when calculating the quantity of hexane that will condense out of a gas stream in a condenser.
Hexane Condensation
Hexane condensation from a gas mixture is a practical application of thermodynamics and phase equilibrium. Condensation occurs when a gas is cooled below its dew point at a given pressure, transitioning from the gas phase to the liquid phase.

In chemical engineering, this is an essential separation process, especially when the gas stream contains valuable components such as hexane. To optimize the condensation process, engineers must understand both the conditions under which hexane will condense (like temperature and pressure) and how to calculate the rate at which it condenses. Calculating the rate of hexane condensation involves using the ideal gas law to determine the moles of gas entering the condenser and then applying mole fractions to find out how much hexane is in the gas stream.
Energy Balance
An energy balance is a fundamental concept in chemical engineering and thermodynamics that equates the energy entering a system to the energy leaving the system, plus any changes in energy stored within the system. It's akin to balancing your checkbook, making sure that what you spend matches what you earn and save.

In the context of our condenser problem, the heat required to condense hexane from the gas mixture can be found by applying an energy balance. The energy removed from the gas as it cools is equal to the condensation heat of hexane, accounting for the heat lost due to the change in temperature and phase. By understanding this concept, engineers can determine the required heat transfer rates to ensure efficient condensation and process operation.
Ideal Gas Law
The ideal gas law is a cornerstone of gas phase chemistry and is expressed as PV=nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is the temperature in Kelvin. It's a simple yet powerful tool for understanding how gases behave under various conditions of temperature and pressure.

The law assumes that gases consist of a large number of tiny particles that are far apart, with no interactions between them, and that all collisions are elastic. This model is quite helpful when calculating conditions in processes like the condensation of hexane. By using the ideal gas law, you can compute the total moles of gas entering the condenser, which is a necessary step when solving for mole fractions and calculating the rate of hexane condensation in the given exercise.

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Most popular questions from this chapter

The heating value of a fuel oil is to be measured in a constant-volume bomb calorimeter. The bomb is charged with oxygen and \(0.00215 \mathrm{lb}_{\mathrm{m}}\) of the fuel and is then sealed and immersed in an insulated container of water. The initial temperature of the system is \(77.00^{\circ} \mathrm{F}\). The fuel-oxygen mixture is ignited, and the fuel is completely consumed. The combustion products are \(\mathrm{CO}_{2}(\mathrm{g})\) and \(\mathrm{H}_{2} \mathrm{O}(\mathrm{v}) .\) The final calorimeter temperature is \(89.06^{\circ} \mathrm{F}\). The mass of the calorimeter, including the bomb and its contents, is 4.62 \(\mathrm{Ib}_{\mathrm{m}},\) and the average heat capacity of the system \(\left(C_{v}\right)\) is \(0.900 \mathrm{Btu} /\left(\mathrm{b}_{\mathrm{m}} \cdot^{\circ} \mathrm{F}\right)\). (a) Calculate \(\Delta \hat{U}_{\mathrm{S}}^{\circ}\) \(\left(\mathrm{B} \mathrm{tu} / \mathrm{lb}_{\mathrm{m}} \text { oil }\right)\) for the combustion of the fuel oil at \(77^{\circ} \mathrm{F}\). Briefly explain your calculation. (b) What more would you need to know to determine the higher heating value of the oil?

Sulfur dioxide is oxidized to sulfur trioxide in a small pilot-plant reactor. SO \(_{2}\) and \(100 \%\) excess air are fed to the reactor at \(450^{\circ} \mathrm{C}\). The reaction proceeds to a \(65 \% \mathrm{SO}_{2}\) conversion, and the products emerge from the reactor at \(550^{\circ} \mathrm{C}\). The production rate of \(\mathrm{SO}_{3}\) is \(1.00 \times 10^{2} \mathrm{kg} / \mathrm{min}\). The reactor is surrounded by a water jacket into which water at \(25^{\circ} \mathrm{C}\) is fed. (a) Calculate the feed rates (standard cubic meters per second) of the \(\mathrm{SO}_{2}\) and air feed streams and the extent of reaction, \(\xi\) (b) Calculate the standard heat of the SO_ oxidation reaction, \(\Delta H_{\mathrm{t}}^{\mathrm{r}}(\mathrm{kJ}) .\) Then, taking molecular species at \(25^{\circ} \mathrm{C}\) as references, prepare and fill in an inlet-outlet enthalpy table and write an energy balance to calculate the necessary rate of heat transfer ( \(\mathrm{kW}\) ) from the reactor to the cooling water. (c) Calculate the minimum flow rate of the cooling water if its temperature rise is to be kept below \(15^{\circ} \mathrm{C}\) (d) Briefly state what would have been different in your calculations and results if you had taken elemental species as references in Part (b).

In a coal gasification process, carbon (the primary constituent of coal) reacts with steam to produce carbon monoxide and hydrogen (synthesis gas). The gas may either be burned or subjected to further processing to produce any of a variety of chemicals. A coal contains 10.5 wt\% moisture (water) and 22.6 wt\% noncombustible ash. The remaining fraction of the coal contains 81.2 wife \(\mathrm{C}, 13.4 \%\) O, and \(5.4 \%\) H. A coal slurry containing \(2.00 \mathrm{kg}\) coal/kg water is fed at \(25^{\circ} \mathrm{C}\) to an adiabatic gasification reactor along with a stream of pure oxygen at the same temperature. The following reactions take place in the reactor: $$\begin{array}{l}\mathrm{C}(\mathrm{s})+\mathrm{H}_{2} \mathrm{O}(\mathrm{v}) \rightarrow \mathrm{CO}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g}): \quad \Delta H_{\mathrm{r}}^{\circ}=+131.3 \mathrm{kJ} \\\\\mathrm{C}(\mathrm{s})+\mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{CO}_{2}(\mathrm{g}): \quad \Delta H_{\mathrm{r}}^{\circ}=-393.5 \mathrm{kJ} \\ 2 \mathrm{H}(\mathrm{in} \mathrm{coal})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{v}): \quad \Delta H_{\mathrm{r}}^{\circ} \approx-242 \mathrm{kJ}\end{array}$$ Gas and slag (molten ash) leave the reactor at \(2500^{\circ} \mathrm{C}\). The gas contains \(\mathrm{CO}, \mathrm{H}_{2}, \mathrm{CO}_{2},\) and \(\mathrm{H}_{2} \mathrm{O}^{14}\) (a) Feeding oxygen to the reactor lowers the yield of synthesis gas, but no gasifier ever operates without supplementary oxygen. Why does the oxygen lower the yield? Why it is nevertheless always supplied. (Hint: All the necessary information is contained in the first two stoichiometric equations and associated heats of reaction shown above.) (b) Suppose the oxygen gas fed to the reactor and the oxygen in the coal combine with all the hydrogen in the coal (Reaction 3) and with some of the carbon (Reaction 2), and the remainder of the carbon is consumed in Reaction 1. Taking a basis of 1.00 kg coal fed to the reactor and letting \(n_{0}\) equal the moles of \(\mathrm{O}_{2}\) fed, draw and label a flowchart. Then derive expressions for the molar flow rates of the four outlet gas species in terms of \(n_{0}\). [Partial solution: \(n_{\mathrm{H}_{2}}=\left(51.3-n_{0}\right)\) mol \(\mathrm{H}_{2} . \mathrm{J}\) (c) The standard heat of combustion of the coal has been determined to be -21,400 kJ/kg, taking \(\mathrm{CO}_{2}(\mathrm{g})\) and \(\mathrm{H}_{2} \mathrm{O}(\mathrm{l})\) to be the combustion products. Use this value and the given clemental composition of the coal to prove that the standard heat of formation of the coal is \(-1510 \mathrm{kJ} / \mathrm{kg}\). Then use an energy balance to calculate \(n_{0},\) using the following approximate heat capacities in your calculation: Take the heat of fusion of ash (the heat required to convert ash to slag) to be \(710 \mathrm{kJ} / \mathrm{kg}\).

Biodiesel fuel - a sustainable alternative to petroleum diesel as a transportation fuel- -is produced via the transesterification of triglyceride molecules derived from vegetable oils or animal fats. For every \(9 \mathrm{kg}\) of biodiesel produced in this process, \(1 \mathrm{kg}\) of glycerol, \(\mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}_{3},\) is produced as a byproduct. Finding a market for the glycerol is important for biodiesel manufacturing to be economically viable. A process for converting glycerol to the industrially important specialty chemical intermediates acrolein, \(C_{3} \mathrm{H}_{4} \mathrm{O},\) and hydroxyacetone (acetol), \(\mathrm{C}_{3} \mathrm{H}_{6} \mathrm{O}_{2},\) has been proposed. $$\begin{array}{l}\mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}_{3} \rightarrow \mathrm{C}_{3} \mathrm{H}_{4} \mathrm{O}+2 \mathrm{H}_{2} \mathrm{O} \\ \mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}_{3} \rightarrow \mathrm{C}_{3} \mathrm{H}_{6} \mathrm{O}_{2}+\mathrm{H}_{2} \mathrm{O} \end{array}$$ The reactions take place in the vapor phase at \(325^{\circ} \mathrm{C}\) in a fixed bed reactor over an acid catalyst. The feed to the reactor is a vapor stream at \(325^{\circ} \mathrm{C}\) containing 25 mol\% glycerol, \(25 \%\) water, and the balance nitrogen. All of the glycerol is consumed in the reactor, and the product stream contains acrolein and hydroxyacctone in a 9: 1 mole ratio. Data for the process species are shown below. $$\begin{array}{|l|c|c|}\hline \text { Species } & \Delta \hat{H}_{\mathrm{f}}(\mathrm{kJ} / \mathrm{mol}) & C_{p}\left[\mathrm{kJ} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\right] \\ \hline \text { glycerol(v) } & -620 & 0.1745 \\ \hline \text { acrolein(v) } & -65 & 0.0762 \\\\\hline \text { hydroxyacetone(v) } & -372 & 0.1096 \\ \hline \text { water(v) } & -242 & 0.0340 \\\\\hline \text { nitrogen(g) } & 0 & 0.0291 \\ \hline\end{array}$$ (a) Assume a basis of 100 mol fed to the reactor, and draw and completely label a flowchart. Carry out a degree-of-freedom analysis assuming that you will use extents of reaction for the material balances. Then calculate the molar amounts of all product species. (b) Calculate the total heat added or removed from the reactor (state which it is), using the constant heat capacities given in the above table. (c) Assuming this process is implemented along with biodiesel production, how would you determine whether the biodiesel is an cconomically viable alternative to petroleum diesel? (d) If you do a degree-of-freedom analysis based on atomic species balances, you are likely to count one more equation than you have unknowns, and yet you know the system has zero degrees of freedom. Guess what the problem is, and then prove it.

The standard heat of combustion \(\left(\Delta \hat{H}_{c}\right)\) of liquid 2,3,3 -trimethylpentane \(\left[\mathrm{C}_{8} \mathrm{H}_{18}\right]\) is reported in a table of physical properties to be \(-4850 \mathrm{kJ} / \mathrm{mol} .\) A footnote indicates that the reference temperature for the reported value is \(25^{\circ} \mathrm{C}\) and the presumed combustion products are \(\mathrm{CO}_{2}(\mathrm{g})\) and \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\). (a) In your own words, briefly explain what all that means. (b) There is some question about the accuracy of the reported value, and you have been asked to determine the heat of combustion experimentally. You burn 2.010 grams of the hydrocarbon with pure oxygen in a constant-volume calorimeter and find that the net heat released when the reactants and products \(\left[\mathrm{CO}_{2}(\mathrm{g}) \text { and } \mathrm{H}_{2} \mathrm{O}(\mathrm{g})\right]\) are all at \(25^{\circ} \mathrm{C}\) is sufficient to raise the temperature of \(1.00 \mathrm{kg}\) of liquid water by \(21.34^{\circ} \mathrm{C}\). Write an energy balance to show that the heat released in the calorimeter equals \(n_{\mathrm{C}_{3} \mathrm{H}_{18}} \Delta \hat{U}_{\mathrm{c}}^{\mathrm{S}},\) and calculate \(\Delta \tilde{U}_{\mathrm{c}}^{\mathrm{o}}(\mathrm{kJ} / \mathrm{mol}) .\) Then calculate \(\Delta \hat{H}_{c}^{c}\) (See Example 9.1-2.) By what percentage of the measured value does the tabulated value differ from the measured one? (c) Use the result of Part (b) to estimate \(\Delta \hat{H}_{f}\) for 2,3,3 -trimethylpentane. Why would the heat of formation of 2,3,3 -trimethylpentane probably be determined this way rather than directly from the formation reaction?
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