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Chapter 10: Problem 18

A \(40.0-\mathrm{ft}^{3}\) oxygen tent initially contains air at \(68^{\circ} \mathrm{F}\) and 14.7 psia. At a time \(t=0\) an enriched air mixture containing \(35.0 \%\) v/v \(\mathrm{O}_{2}\) and the balance \(\mathrm{N}_{2}\) is fed to the tent at \(68^{\circ} \mathrm{F}\) and 1.3 psig at a rate of \(60.0 \mathrm{ft}^{3} / \mathrm{min},\) and gas is withdrawn from the tent at \(68^{\circ} \mathrm{F}\) and 14.7 psia at a molar flow rate equal to that of the feed gas. (a) Calculate the total Ib-moles of gas \(\left(\mathrm{O}_{2}+\mathrm{N}_{2}\right)\) in the tent at any time. (b) Let \(x(t)\) equal the mole fraction of oxygen in the outlet stream. Write a differential mole balance on oxygen, assuming that the tent contents are perfectly mixed (so that the temperature, pressure, and composition of the contents are the same as those propertics of the exit stream). Convert the balance into an equation for \(d x / d t\) and provide an initial condition. (c) Integrate the equation to obtain an expression for \(x(t)\). How long will it take for the mole fraction of oxygen in the tent to reach 0.33 ? Sketch a plot of \(x\) versus \(t,\) labeling the value of \(x\) at \(t=0\) and the asymptotic value at \(t \rightarrow \infty\)

Short Answer

Expert verified
The lb-moles of gas in the tent at any time will be calculated using Ideal Gas Law, which relates the pressure, volume, number of moles and temperature of a gas. A differential mole balance on oxygen will be derived assuming the tent contents are perfectly mixed. The initial mole fraction of oxygen in the outlet stream, consequently in the tent, will be considered as 0.21, or that of the natural atmosphere. Upon further analysis and solving the linear first order differential equation, an expression for the mole fraction of oxygen with time will be obtained. In order to find when the mole fraction of oxygen reaches 0.33, \(x(t)\) will be set to 0.33 and solving for t will result in the desired time. A graph of mole fraction versus time will be a decaying exponential graph, with an asymptote at the equilibrium value of mole fraction as t goes to infinity.

Step by step solution

01

Calculating the total lb-moles of gas in the tent at any time

Given the volume of the tent as 40.0 ft鲁, the temperature as 68掳F (convert to Rankine by adding 460, so T = 528R), and the pressure as 14.7 psia, we can use the Ideal Gas Law Pv = nRT to find the total moles. Here P is the pressure, v is the volume, n is the number of moles, R is the Ideal Gas Constant and T is the temperature. The Ideal Gas Law can be rearranged to solve for n (n = Pv/RT). Substitution of values (P = 14.7 psia, v = 40.0 ft鲁, R = 10.73 psia.ft鲁/lb-mole.R and T = 528R) gives a value for n, which represents total lb-moles of the gas in the tent at any time.
02

Creating a differential mole balance

Given that the tent is perfectly mixed, the mole fraction of oxygen in the feed mixture (y) is equal to the mole fraction of the oxygen in the exit stream x(t). The feed rate determines how much oxygen enters and leaves the tent, so we have the equation \(d(n \cdot x(t)) / dt = Q \cdot (y - x(t))\), where n is the number of moles in the tent, \(x(t)\) is the mole fraction of oxygen at time t, Q is the feed rate, and y is the mole fraction of the feed gas. Since we have found n as a constant in the previous step, we can simplify the equation to the form of a first order linear differential equation: \(dx(t) / dt = (Q / n) \cdot (y - x(t))\).
03

Providing an initial condition

At the beginning, at t = 0, the mole fraction of oxygen in the tent and hence in the exit stream is equal to the mole fraction of oxygen in the ambient air, which is approximately 0.21. Therefore, the initial condition can be written as \(x(0) = 0.21\).
04

Solving the differential equation

The given equation is a first order linear non-homogeneous differential equation of the form dx/dt + P*x = Q where P = - Q/n and Q = y*Q/n. Solving equations of this type should result in \(x(t) = Q / P + (C - Q / P) * e^{(- P * t)}\), where C is the constant of integration. Substituting values will give the function of the mole fraction in terms of time.
05

Determining the time required

To find the time required for mole fraction to reach 0.33, we can equate \(x(t)\) to 0.33 and solve for t.
06

Sketching the plot

The curve of \(x(t)\) versus t is expected to be an exponential decay curve starting from the initial condition 0.21 and reaching an asymptote at \(Q / P\). The shape of the curve should indicate that as t approaches infinity, \(x(t)\) approaches an equilibrium value of \(Q / P\).

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Key Concepts

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

Ideal Gas Law
The Ideal Gas Law is a fundamental equation that models the behavior of gases under various conditions. It is given by the formula: \(PV = nRT\), where \(P\) represents the pressure of the gas, \(V\) is the volume it occupies, \(n\) is the amount of substance of gas (measured in moles), \(R\) is the ideal gas constant, and \(T\) is the temperature of the gas in an absolute scale (such as Kelvin or Rankine).

For chemical engineers, the Ideal Gas Law is crucial in calculating the moles of gas, as it relates the physical parameters of a gaseous system that can be easily measured. For instance in the exercise, the total lb-moles of gas in the oxygen tent at any time is determined using this law. This involves conversion of temperature to Rankine and then substituting the given values into the equation to solve for \(n\), the number of moles. It serves as a cornerstone concept in understanding gas systems in engineering applications. The universality of the Ideal Gas Law makes it highly applicable across various scenarios involving ideal gases.
Differential Mole Balance
A differential mole balance is an application of the conservation of mass to the mole quantities within a chemical process. It takes into account the rates at which species enter and leave a system, as well as the rate of reaction within the system. The general form of a differential mole balance is \( \frac{d(n_i)}{dt} = \text{rate in} - \text{rate out} + \text{rate of generation} - \text{rate of consumption} \), where \(n_i\) represents the number of moles of species \(i\).

In the exercise provided, a perfectly mixed system is assumed, simplifying the differential mole balance. There is no chemical reaction, so the balance focuses on the rates of inflow and outflow of the mole fraction of oxygen. The provided solution employs this mole balance to relate the change in the mole fraction of oxygen, \( dx(t)/dt \), to the flow rate and the difference in mole fractions between the inlet and the outlet stream.
First Order Linear Differential Equation
A first order linear differential equation is a type of differential equation characterized by derivatives of only the first degree, and the variables and their derivatives appear in a linear fashion. The standard form is \( \frac{dx}{dt} + P(x) = Q(t) \), where \(P\) and \(Q\) are functions of \(x\) and \(t\), respectively. To solve for \(x(t)\), various methods such as separation of variables, integrating factors, or simply recognizing patterns that match standard solutions can be used.

In the context of the exercise, after setting up a differential mole balance, we are led to a first order linear differential equation of the form \( \frac{dx(t)}{dt} = (Q/n) \times (y - x(t)) \). This equation describes the rate of change in the mole fraction of oxygen over time, and by integrating with proper initial conditions, one can obtain an expression for the oxygen mole fraction as a function of time. Correctly solving these types of equations is essential for predicting the behavior of gaseous systems over time in the field of chemical engineering.

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Most popular questions from this chapter

Water is added at varying rates to a 300 -liter holding tank. When a valve in a discharge line is opened, water flows out at a rate proportional to the height and hence to the volume \(V\) of water in the tank. The flow of water into the tank is slowly increased and the level rises in consequence, until at a steady input rate of \(60.0 \mathrm{L} / \mathrm{min}\) the level just reaches the top but does not spill over. The input rate is then abruptly decreased to \(40.0 \mathrm{L} / \mathrm{min}\). (a) Write the equation that relates the discharge rate, \(\dot{V}_{\text {out }}(\mathrm{L} / \mathrm{min}),\) to the volume of water in the tank, \(V(\mathrm{L}),\) and use it to calculate the steady-state volume when the input rate is \(40 \mathrm{L} / \mathrm{min}\). (b) Write a differential balance on the water in the tank for the period from the moment the input rate is decreased \((t=0)\) to the attainment of steady state \((t \rightarrow \infty),\) expressing it in the form \(d V / d t=\cdots \cdot\) Provide an initial condition. (c) Without integrating the equation, use it to confirm the steady-state value of \(V\) calculated in Part (a) and then to predict the shape you would anticipate for a plot of \(V\) versus \(t\). Explain your reasoning. (d) Separate variables and integrate the balance equation to derive an expression for \(V(t)\). Calculate the time in minutes required for the volume to decrease to within \(1 \%\) of its steady-state value.

An electrical coil is used to heat \(20.0 \mathrm{kg}\) of water in a closed well-insulated vessel. The water is initially at \(25^{\circ} \mathrm{C}\) and 1 atm. The coil delivers a steady \(3.50 \mathrm{kW}\) of power to the vessel and its contents. (a) Write a differential energy balance on the water, assuming that \(97 \%\) of the energy delivered by the coil goes into heating the water. What happens to the other \(3 \% ?\) (b) Integrate the equation of Part (a) to derive an expression for the water temperature as a function of time. (c) How long will it take for the water to reach the normal boiling point? Will it boil at this temperature? Explain your answer.

Methane is generated via the anaerobic decomposition (biological degradation in the absence of oxygen) of solid waste in landfills. Collecting the methane for use as a fuel rather than allowing it to disperse into the atmosphere provides a useful supplement to natural gas as an energy source. If a batch of waste with mass \(M\) (tonnes) is deposited in a landfill at \(t=0,\) the rate of methane generation at time \(t\) is given by $$\dot{V}_{\mathrm{CH}_{4}}(t)=k L_{0} M_{\text {waste }} e^{-k t}$$ where \(\dot{V}_{\mathrm{CH}_{4}}\) is the rate at which methane is generated in standard cubic meters per year, \(k\) is a rate constant, \(L_{0}\) is the total potential yield of landfill gas in standard cubic meters per tonne of waste, and \(M_{\text {watte is the tonnes of waste in the landfill at } t=0}\). (a) Starting with Equation 1, derive an expression for the mass generation rate of methane, \(\dot{M}_{\mathrm{CH}_{4}}(t)\) Without doing any calculations, sketch the shape of a plot of \(M_{\mathrm{CH}, \text { versus } t \text { from } t=0 \text { to } t=3 \mathrm{y},}\) and graphically show on the plot the total masses of methane generated in Years \(1,2,\) and \(3 .\) Then derive an expression for \(M_{\mathrm{CH}_{4}}(t),\) the total mass of methane (tonnes) generated from \(t=0\) to a time \(t\) (b) A new landfill has a yield potential \(L_{0}=100\) SCM CH \(_{4}\) /tonne waste and a rate constant \(k=0.04 \mathrm{y}^{-1} .\) At the beginning of the first year, 48,000 tonnes of waste are deposited in the landfill. Calculate the tonnes of methane generated from this deposit over a three-year period. (c) A colleague solving the problem of Part (b) calculates the methane produced in three years from the \(4.8 \times 10^{4}\) tonnes of waste as $$M_{\mathrm{CH}_{4}}(t=3)=\dot{M}_{\mathrm{CH}_{4}}(t=0) \times 1 \mathrm{y}+\dot{M}_{\mathrm{CH}_{4}}(t=1) \times 1 \mathrm{y}+\dot{M}_{\mathrm{CH}_{4}}(t=2) \times 1 \mathrm{y}$$ where \(\dot{M}_{\mathrm{CH}_{4}}\) is the first expression derived in Part (a). Briefly state what has been assumed about the rate of methane generation. Calculate the value determined with this method and the percentage error in the calculation. Show graphically what the calculated value corresponds to on another sketch of \(M_{\mathrm{CH}_{4}}\) versus \(t\) (d) The following amounts of waste are deposited in the landfill on January 1 in each of three consecutive years. Exploratory Exercises - Research and Discover (e) Explain in your own words the benefits of reducing the release of methane from landfills and of using the methane as a fuel instead of natural gas. (f) One way to avoid the environmental hazard of methane generation is to incinerate the waste before it has a chance to decompose. What problems might this alternative process introduce?

One hundred fifty kmol of an aqueous phosphoric acid solution contains 5.00 mole\% \(\mathrm{H}_{3} \mathrm{PO}_{4}\). The solution is concentrated by adding pure phosphoric acid at a rate of \(20.0 \mathrm{L} / \mathrm{min}\). (a) Write a differential mole balance on phosphoric acid and provide an initial condition. [Start by defining \(n_{\mathrm{p}}(\mathrm{kmol})\) to be the total quantity of phosphoric acid in the tank at any time.] Without solving the equation, sketch a plot of \(n_{\mathrm{p}}\) versus \(t\) and explain your reasoning. (b) Solve the balance to obtain an expression for \(n_{\mathrm{p}}(t) .\) Use the result to derive an expression for \(x_{\mathrm{p}}(t)\) the mole fraction of phosphoric acid in the solution. Without doing any numerical calculations, sketch a plot of \(x_{\mathrm{p}}\) versus \(t\) from \(t=0\) to \(t \rightarrow \infty,\) labeling the initial and asymptotic values of \(x_{\mathrm{p}}\) on the plot. Explain your reasoning. (c) How long will it take to concentrate the solution to \(15 \% \mathrm{H}_{3} \mathrm{PO}_{4} ?\)

A kettle containing 3.00 liters of water at a temperature of \(18^{\circ} \mathrm{C}\) is placed on an electric stove and begins to boil in three minutes. (a) Write an energy balance on the water and determine an expression for \(d T / d t,\) neglecting evaporation of water before the boiling point is reached, and provide an initial condition. Sketch a plot of \(T\) versus \(t\) from \(t=0\) to \(t=4\) minutes. (b) Calculate the average rate (W) at which heat is being added to the water. Then calculate the rate (g/s) at which water vaporizes once boiling begins. (c) The rate of heat output from the stove element differs significantly from the heating rate calculated in Part (b). In which direction, and why?
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