/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 A ventilation system has been de... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 10

A ventilation system has been designed for a large laboratory with a volume of \(1100 \mathrm{m}^{3}\). The volumetric flow rate of ventilation air is \(700 \mathrm{m}^{3} / \mathrm{min}\) at \(22^{\circ} \mathrm{C}\) and 1 atm. (The latter two values may also be taken as the temperature and pressure of the room air.) A reactor in the laboratory is capable of emitting as much as 1.50 mol of sulfur dioxide into the room if a seal ruptures. An \(\mathrm{SO}_{2}\) mole fraction in the room air greater than \(1.0 \times 10^{-6}(1 \mathrm{ppm})\) constitutes a health hazard. (a) Suppose the reactor seal ruptures at a time \(t=0,\) and the maximum amount of \(\mathrm{SO}_{2}\) is emitted and spreads uniformly throughout the room almost instantaneously. Assuming that the air flow is sufficient to make the room air composition spatially uniform, write a differential SO_ balance, letting \(N\) be the total moles of gas in the room (assume constant) and \(x(t)\) the mole fraction of \(\mathrm{SO}_{2}\) in the laboratory air. Convert the balance into an equation for \(d x / d t\) and provide an initial condition. (Assume that all of the \(\left.\mathrm{SO}_{2} \text { emitted is in the room at } t=0 .\right)\) (b) Predict the shape of a plot of \(x\) versus \(t\). Explain your reasoning, using the equation of Part (a) in your explanation. (c) Separate variables and integrate the balance to obtain an expression for \(x(t)\). Check your solution. (d) Convert the expression for \(x(t)\) into an expression for the concentration of \(\mathrm{SO}_{2}\) in the room, \(C_{\mathrm{SO}_{2}}\) (mol \(\mathrm{SO}_{2} / \mathrm{L}\) ). Calculate (i) the concentration of \(\mathrm{SO}_{2}\) in the room two minutes after the rupture occurs, and (ii) the time required for the \(S O_{2}\) concentration to reach the "safe" level. (e) Why would it probably not yet be safe to enter the room after the time calculated in Part (d)? (Hint:One of the assumptions made in the problem is probably not a good one.)

Short Answer

Expert verified
The mole fraction of \(\mathrm{SO}_{2}\) decreases with time due to ventilation and approaches constant steady state. The \(\mathrm{SO}_{2}\) concentration reaches safe level after a certain period of time. However, the room might not be safe to enter due to non-uniform distribution of \(\mathrm{SO}_{2}\).

Step by step solution

01

Differential SO_2 balance

From the problem, \(N\) is the total moles of gas in the room, \(V\) is the volume of the room, \(Q\) is the volumetric flow rate, \(\mathrm{SO}_{2}\) is being added at the rate of \(N_{0}\), and the rate of change of the \(\mathrm{SO}_{2}\), \(dx/dt\) = \(Q/N)\) * \((x_{0} - x)\), where \(x_0\) is the incoming \(\mathrm{SO}_{2}\) concentration. Here, the initial condition \(x(0) = N_{0}/N\). Let's put the values into the above equation.
02

Shape of the plot x vs t

The plot of mole fraction of \(\mathrm{SO}_{2}\) (x) versus time (t) will be an exponential decay one. The reason being as time passes, the mole fraction of SO2 will decrease due to the ventilation system.
03

Equation for \(x(t)\)

By separating variables and integrating the balance, we get \(\int_{N_{0}/N}^{x}\) dx / \((x_{0}-x)\) = \(-Q/N)*\int_{0}^{t} dt\). Solving this equation gives \(x(t)\) = \(x_{0}\) + \((N_{0}/N - x_{0})*(e^{-Qt/N})\). This is the expression needed.
04

Concentration of \(\mathrm{SO}_{2}\)

To convert the expression for \(x(t)\) into the concentration of \(\mathrm{SO}_{2}\), use \(C_{\mathrm{SO}_{2}}=x(N/V)\). The \(\mathrm{SO}_{2}\) concentration in the room at any time \(t\) will be given by \(C_{\mathrm{SO}_{2}}=x_{0}(N/V) + (N_{0}/V - x_{0}(N/V)*e^{-Qt/N}\). Now, insert the given values and calculate \(C_{\mathrm{SO}_{2}}\) two minutes after rupture and the time when \(C_{\mathrm{SO}_{2}}\) reaches safe level.
05

Safety considerations

It will probably not be safe to enter the room even after the \(\mathrm{SO}_{2}\) concentration has reached safe levels because we have assumed the ventilation system perfectly mixed the room air, but in reality, there might still be some areas with high \(\mathrm{SO}_{2}\) concentration in the room.

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Key Concepts

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

Sulfur Dioxide Emission
Sulfur dioxide (SO\(_2\)) is a chemical compound that can pose significant health hazards. It is often released during industrial processes involving the burning of fossil fuels or smelting of mineral ores. In laboratory settings, such as the one described, there is a risk of sudden emissions in the case of equipment failure, like a seal rupture.
The emissions of SO\(_2\) in a closed room can lead to dangerous concentrations if not addressed swiftly. For instance, even a momentary spike as low as 1 ppm can create health risks. Prolonged exposure may lead to respiratory issues and other health complications. Understanding SO\(_2\) emissions and mitigating their effects through engineered safety measures, such as a well-designed ventilation system, is crucial in chemical process safety.
  • Immediate identification and response to emission events are vital for safety.
  • Understanding emission dynamics helps in designing appropriate control measures.
  • The goal is to ensure safe air composition and minimize health risks.
Ventilation System Design
A ventilation system is crucial in maintaining safe air quality by diluting and removing hazardous emissions. The design of such a system in a laboratory setting must consider the maximum potential release of harmful substances and ensure the rapid dispersion of these substances to safe levels.
In our scenario, we have a laboratory with a precise volumetric flow rate of 700 \ \( \text{m}^3/\ \text{min} \.\) For effective safety management, this system must be capable of reducing the concentration of SO\(_2\) from an initial high to an acceptable low.
A sound design takes into account:
  • The rate of air exchange, ensuring that air is replaced frequently.
  • The uniform distribution of air flow throughout the room.
  • Consideration of temperature and pressure conditions, as seen with the specified 22掳C and 1 atm, to optimize system performance.
Ultimately, the ventilation system must be reliable enough to maintain a uniform room atmosphere quickly, diluting potential emissions like SO\(_2\) to prevent any health hazards.
Differential Balance Equations
Differential balance equations are essential in modeling how the concentration of a substance changes in a system over time. In this specific case, a differential balance is used to describe how the concentration of SO\(_2\) in a laboratory changes with time due to ventilation.
The core principle involves setting up a differential equation that represents the rate of change of SO\(_2\) concentration, taking into account both the inflow of air and the natural decay of SO\(_2\) due to the ventilatory outflow.
The key balance equation here is:\[ \frac{dx}{dt} = \frac{Q}{N} \cdot (x_0 - x) \]
  • Where \(x\) is the mole fraction of SO\(_2\), \(Q\) is the volumetric flow rate, and \(N\) is the total moles of gas.
  • The initial condition, \(x(0) = \frac{N_0}{N}\), reflects the condition right after the maximum possible SO\(_2\) has been emitted.
This setup allows for the calculation of the concentration over time and aids in predicting when concentrations will return to safe levels. Understanding and solving differential balance equations are crucial for designing systems to manage and mitigate the risks of hazardous emissions.
Exponential Decay in Chemical Processes
Chemical processes often involve concepts like exponential decay, where the quantity of a substance decreases at a rate proportional to its current value. In the context of the study exercise, this means that the mole fraction of sulfur dioxide (SO\(_2\)) in the laboratory decreases exponentially over time due to effective ventilation.
The equation derived shows the exponential nature of decay in this particular scenario:
\[ x(t) = x_0 + \left(\frac{N_0}{N} - x_0\right)\cdot e^{-\frac{Qt}{N}} \]
This equation indicates that:
  • The SO\(_2\) concentration decreases more quickly at higher flow rates \((Q)\).
  • The decrease is slower if the initial SO\(_2\) concentration \((N_0/N)\) is high, highlighting the need for rapid ventilation under high emission scenarios.
Exponential decay reflects how contaminants diminish as fresh air continuously dilutes them. Understanding this pattern helps in predicting how long it takes for a system to return to a safe state, thereby enhancing the safety measures in chemical process management.

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Most popular questions from this chapter

A 7.35 million gallon tank used for storing liquefied natural gas (LNG, which may be taken to be pure methane) must be taken out of service and inspected. All the liquid that can be pumped from the tank is first removed, and the tank is allowed to warm from its service temperature of about \(-260^{\circ} \mathrm{F}\) to \(80^{\circ} \mathrm{F}\) at 1 atm. The gas remaining in the tank is then purged in two steps: (1) Liquid nitrogen is sprayed gently onto the tank floor, where it vaporizes. As the cold nitrogen vapor is formed, it displaces the methane in a piston-like flow until the tank is completely filled with nitrogen. Once all the methane has been displaced, the nitrogen is allowed to warm to ambient temperature. (2) Air is blown into the tank where it rapidly and completely mixes with the nitrogen until the composition of the gas leaving the tank is very close to that of air. (a) Use the ideal-gas equation of state to estimate the densities of methane at \(80^{\circ} \mathrm{F}\) and 1 atm and of nitrogen at \(-260^{\circ} \mathrm{F}\) and 1 atm. How confident are you about the accuracy of each estimate? Explain. (b) If the density of liquid nitrogen is \(50 \mathrm{lb}_{\mathrm{m}} / \mathrm{ft}^{3}\), how many gallons will be required to displace all the methane from the tank? (c) How many cubic feet of air will be required to increase the oxygen concentration to \(20 \%\) by volume? (d) Explain the logic behind vaporizing nitrogen in the manner described. Why purge with nitrogen first as opposed to purging with air?

A gas-phase decomposition reaction with stoichiometry \(2 \mathrm{A} \rightarrow 2 \mathrm{B}+\mathrm{C}\) follows a second-order rate law (see Problem 10.19): $$r_{\mathrm{d}}\left[\operatorname{mol} /\left(\mathrm{m}^{3} \cdot \mathrm{s}\right)\right]=k C_{\mathrm{A}}^{2}$$ where \(C_{\mathrm{A}}\) is the reactant concentration in \(\mathrm{mol} / \mathrm{m}^{3}\). The rate constant \(k\) varies with the reaction temperature according to the Arrhenius law $$k\left[\mathrm{m}^{3} /(\mathrm{mol} \cdot \mathrm{s})\right]=k_{0} \exp (-E / R T)$$ where \(k_{0}\left[\mathrm{m}^{3} /(\mathrm{mol} \cdot \mathrm{s}]\right)=\) the preexponential factor \(E(\mathrm{J} / \mathrm{mol})=\) the reaction activation energy \(R=\) the gas constant \(T(\mathrm{K})=\) the reaction temperature (a) Suppose the reaction is carried out in a batch reactor of constant volume \(V\left(\mathrm{m}^{3}\right)\) at a constant temperature \(T(\mathrm{K}),\) beginning with pure \(\mathrm{A}\) at a concentration \(C_{\mathrm{A} 0} .\) Write a differential balance on A and integrate it to obtain an expression for \(C_{\mathrm{A}}(t)\) in terms of \(C_{\mathrm{A} 0}\) and \(k\) (b) Let \(P_{0}(\text { atm })\) be the initial reactor pressure. Prove that \(t_{1 / 2}\), the time required to achieve a \(50 \%\) conversion of \(\mathrm{A}\) in the reactor, equals \(R T / k P_{0},\) and derive an expression for \(P_{1 / 2},\) the reactor pressure at this point, in terms of \(P_{0} .\) Assume ideal- gas behavior. (c) The decomposition of nitrous oxide \(\left(\mathrm{N}_{2} \mathrm{O}\right)\) to nitrogen and oxygen is carried out in a 5.00 -liter batch reactor at a constant temperature of \(1015 \mathrm{K},\) beginning with pure \(\mathrm{N}_{2} \mathrm{O}\) at several initial pressures. The reactor pressure \(P(t)\) is monitored, and the times \(\left(t_{1 / 2}\right)\) required to achieve \(50 \%\) conversion of \(\mathrm{N}_{2} \mathrm{O}\) are noted. $$\begin{array}{|c|c|c|c|c|} \hline P_{0}(\mathrm{atm}) & 0.135 & 0.286 & 0.416 & 0.683 \\ \hline t_{1 / 2}(\mathrm{s}) & 1060 & 500 & 344 & 209 \\ \hline \end{array}$$ Use these results to verify that the \(\mathrm{N}_{2} \mathrm{O}\) decomposition reaction is second-order and determine the value of \(k\) at \(T=1015 \mathrm{K}\) (d) The same experiment is performed at several other temperatures at a single initial pressure of 1.00 atm, with the following results: $$\begin{array}{|c|c|c|c|c|} \hline T(\mathrm{K}) & 900 & 950 & 1000 & 1050 \\ \hline t_{1 / 2}(\mathrm{s}) & 5464 & 1004 & 219 & 55 \\ \hline \end{array}$$ Use a graphical method to determine the Arrhenius law parameters ( \(k_{0}\) and \(E\) ) for the reaction. (e) Suppose the reaction is carried out in a batch reactor at \(T=980 \mathrm{K},\) beginning with a mixture at 1.20 atm containing 70 mole \(\%\) N \(_{2}\) O and the balance a chemically inert gas. How long (minutes) will it take to achieve a \(90 \%\) conversion of \(\mathrm{N}_{2} \mathrm{O} ?\)

A 10.0 -ft compressed-air tank is being filled. Before the filling begins, the tank is open to the atmosphere. The reading on a Bourdon gauge mounted on the tank increases linearly from an initial value of 0.0 to 100 psi after 15 seconds. The temperature is constant at \(72^{\circ} \mathrm{F}\), and atmospheric pressure is 1 atm. (a) Calculate the rate \(\dot{n}\) (lb-mole/s) at which air is being added to the tank, assuming ideal-gas behavior. (Suggestion: Start by calculating how much is in the tank at \(t=0 .)\) (b) Let \(N(t)\) equal the number of Ib-moles of air in the tank at any time. Write a differential balance on the air in the tank in terms of \(N\) and provide an initial condition. (c) Integrate the balance to obtain an expression for \(N(t)\). Check your solution two ways. (d) Estimate the Ib-moles of oxygen in the tank after two minutes. List reasons your answer might be inaccurate, assuming there are no mistakes in your calculation.

A steam coil is immersed in a stirred tank. Saturated steam at 7.50 bar condenses within the coil, and the condensate emerges at its saturation temperature. A solvent with a heat capacity of \(2.30 \mathrm{kJ} /\left(\mathrm{kg} \cdot^{\cdot} \mathrm{C}\right)\) is fed to the tank at a steady rate of \(12.0 \mathrm{kg} / \mathrm{min}\) and a temperature of \(25^{\circ} \mathrm{C},\) and the heated solvent is discharged at the same flow rate. The tank is initially filled with \(760 \mathrm{kg}\) of solvent at \(25^{\circ} \mathrm{C},\) at which point the flows of both steam and solvent are commenced. The rate at which heat is transferred from the steam coil to the solvent is given by the expression $$\dot{Q}=U A\left(T_{\mathrm{steam}}-T\right)$$ where \(U A\) (the product of a heat transfer coefficient and the coil surface area through which the heat is transferred) equals \(11.5 \mathrm{kJ} /\left(\min \cdot^{\circ} \mathrm{C}\right) .\) The tank is well stirred, so that the temperature of the contents is spatially uniform and equals the outlet temperature. (a) Prove that an energy balance on the tank contents reduces to the equation given below and supply an initial condition. \frac{d T}{d t}=1.50^{\circ} \mathrm{C} / \mathrm{min}-0.0224 T (b) Without integrating the equation, calculate the steady-state value of \(T\) and sketch the expected plot of \(T\) versus \(t,\) labeling the values of \(T_{\mathrm{b}}\) at \(t=0\) and \(t \rightarrow \infty\) (c) Integrate the balance equation to obtain an expression for \(T(t)\) and calculate the solvent temperature after 40 minutes. (d) The tank is shut down for routine maintenance, and a technician notices that a thin mineral scale has formed on the outside of the steam coil. The coil is treated with a mild acid that removes the scale and reinstalled in the tank. The process described above is run again with the same steam conditions, solvent flow rate, and mass of solvent charged to the tank, and the temperature after 40 minutes is \(55^{\circ} \mathrm{C}\) instead of the value calculated in Part (c). One of the system variables listed in the problem statement must have changed as a result of the change in the stirrer. Which variable would you guess it to be, and by what percentage of its initial value did it change?

The flow rate of a process stream has tended to fluctuate considerably, creating problems in the process unit to which the stream is flowing. A horizontal surge drum has been inserted in the line to maintain a constant downstream flow rate even when the upstream flow rate varies. A cross-section of the drum, which has length \(L\) and radius \(r,\) is shown below. The level of liquid in the drum is \(h\), and the expression for liquid volume in the drum is $$V=L\left[r^{2} \cos ^{-1}\left(\frac{r-h}{r}\right)-(r-h) \sqrt{r^{2}-(r-h)^{2}}\right]$$ Here is how the drum works. The rate of drainage of a liquid from a container varies with the height of the liquid in the container: the greater the height, the faster the drainage rate. The drum is initially charged with enough liquid so that when the input rate has its desired value, the liquid level is such that the drainage rate from the drum has the same value. A sensor in the drum sends a signal proportional to the liquid level to a control valve in the downstream line. If the input flow rate increases, the liquid level starts to rise; the control valve detects the rise from the transmitted signal and opens to increase the drainage rate, stopping when the level comes back down to its set-point value. Similarly, if the input flow rate drops, the control valve closes enough to bring the level back up to its set point. (a) The drum is to be charged initially with benzene (density \(=0.879 \mathrm{g} / \mathrm{cm}^{3}\) ) at a constant rate \(\dot{m}(\mathrm{kg} / \mathrm{min})\) until the tank is half full. If \(L=5 \mathrm{m}, r=1 \mathrm{m},\) and \(\dot{m}=10 \mathrm{kg} / \mathrm{min},\) how long should it take to reach that point? (b) Now suppose the flow rate into the tank is unknown. A sight gauge on the tank allows determination of the liquid level, and instructions are to stop the flow when the tank contains 3000 kg. At what value of \(h\) should this be done? (c) After the tank has been charged, the flow rate into the drum, \(\dot{m}_{1}\), varies with upstream operations, and the flow rate out is \(10 \mathrm{kg} / \mathrm{min}\). Write a mass balance around the drum so that you obtain a relationship between \(\dot{m}_{1}\) and the rate of change in the height of liquid in the tank \((d h / d t)\) as a function of \(h .\) Estimate the flow rate into the tank when \(h\) has an approximate value of \(50 \mathrm{cm}\), and \(d h / d t=1 \mathrm{cm} / \mathrm{min} .\) (Hint: Although an analytical solution is feasible, you may find it easier to create plots of \(V\) and \(d V / d h\) at \(0.1 \mathrm{m}\) increments in \(h,\) which can be used in obtaining an approximate solution to the problem.) (d) Speculate on why the drum would provide better performance than feeding a signal proportional to the flow rate directly to the control valve that would cause the valve to close if the flow rate drops below the set point and to open if the flow rate rises above that point.
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