Elementary Principles of Chemical Processes

Richard M. Felder, Ronald W. Rousseau, Lisa G. Bullard

Chapter 10

Balances on Transient Processes - all with Video Answers

Educators

Chapter Questions

Problem 1

A solution containing hydrogen peroxide with a mass fraction $x_{\mathrm{p} 0}$ $\left(\mathrm{kg} \mathrm{H}_{2} \mathrm{O}_{2} / \mathrm{kg} \text { solution }\right)$ is added to a storage tank at a steady rate $\dot{m}_{0}(\mathrm{kg} / \mathrm{h})$. During this process, the liquid level reaches a corroded spot in the tank wall and a leak develops. As the filling continues, the leak rate $\dot{m}_{1}(\mathrm{kg} / \mathrm{h})$ becomes progressively worse. Moreover, once it is in the tank the peroxide begins to decompose at a rate
$$r_{\mathrm{d}}(\mathrm{kg} / \mathrm{h})=k M_{\mathrm{p}}$$
where $M_{\mathrm{p}}(\mathrm{kg})$ is the mass of peroxide in the tank. The tank contents are well mixed, so that the peroxide concentration is the same at all positions. At a time $t=0$ the liquid level reaches the corroded spot. Let $M_{0}$ and $M_{\mathrm{p} 0}$ be the total liquid mass and mass of peroxide, respectively, in the tank
at $t=0,$ and let $M(t)$ be the total mass of liquid in the tank at any time thereafter.
(a) Show that the leakage rate of hydrogen peroxide at any time is $\dot{m}_{1} M_{\mathrm{p}} / M$
(b) Write differential balances on the total tank contents and on the peroxide in the tank, and provide initial conditions. Your solution should involve only the quantities $\dot{m}_{0}, \dot{m}_{1}, x_{\mathrm{p} 0}, k, M, M_{0}, M_{\mathrm{p}}$
$M_{\mathrm{p} 0},$ and $t$

Arun Bana

Arun Bana

Numerade Educator

Problem 2

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 $15 \% \mathrm{H}_{3} \mathrm{PO}_{4} ?$

Lottie Adams

Numerade Educator

Problem 3

Methanol is added to a storage tank at a rate of $1200 \mathrm{kg} / \mathrm{h}$ and is simultaneously withdrawn at a rate $\dot{m}_{w}(t)(\mathrm{kg} / \mathrm{h})$ that increases linearly with time. At $t=0$ the tank contains $750 \mathrm{kg}$ of the liquid and $\dot{m}_{w}=750 \mathrm{kg} / \mathrm{h} .$ Five hours later $\dot{m}_{\mathrm{w}}$ equals $1000 \mathrm{kg} / \mathrm{h}$
(a) Calculate an expression for $\dot{m}_{w}(t),$ letting $t=0$ signify the time at which $\dot{m}_{w}=750 \mathrm{kg} / \mathrm{h},$ and incorporate it into a differential methanol balance, letting $M(\mathrm{kg})$ be the mass of methanol in the tank at any time.
(b) Integrate the balance equation to obtain an expression for $M(t)$ and check the solution two ways. (See Example 10.2-1.) For now, assume that the tank has an infinite capacity.
(c) Calculate how long it will take for the mass of methanol in the tank to reach its maximum value, and calculate that value. Then calculate the time it will take to empty the tank.
(d) Now suppose the tank volume is $3.40 \mathrm{m}^{3}$. Draw a plot of $M$ versus $t$, covering the period from $t=0$ to an hour after the tank is empty. Write expressions for $M(t)$ in each time range when the function changes.

Arun Bana

Numerade Educator

Problem 4

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.

Lottie Adams

Numerade Educator

Problem 5

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?

Lottie Adams

Numerade Educator

Problem 6

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.

Lottie Adams

Numerade Educator

Problem 7

A gas storage tank with a floating roof receives a steady input of $540 \mathrm{m}^{3} / \mathrm{h}$ of a natural gas. The rate of withdrawal of gas from the tank, $\dot{V}_{w}$, varies more or less randomly during the day and is recorded at 10-min intervals. At 8: 00 a. $m$. one morning the volume of stored gas is $3.00 \times 10^{3} \mathrm{m}^{3}$. The withdrawal rate data for the next 4 hours are as follows: The temperature and pressure of the inlet, stored, and outlet gases are equal and nearly constant throughout the given time period.
(a) Write a differential balance on the moles of gas in the tank, and prove that when integrated it yields the following equation for the gas volume:
$$V(t)=3.00 \times 10^{3} \mathrm{m}^{3}+\left(9.00 \frac{\mathrm{m}^{3}}{\mathrm{min}}\right) t-\int_{0}^{t} \dot{V}_{w} d t$$
where $t(\min )$ is the time elapsed since 8: 00 a.m.
(b) Calculate the stored gas volume at noon, using Simpson's rule (Appendix A.3) to evaluate the integral.
(c) Althougharunning estimate of the tank volume is important to have, in practice it would probably not be obtained in the manner indicated. Speculate on how it would more likely be obtained. What might you infer if the value estimated in Part (b) is greater than that obtained by the more accurate method?

Lottie Adams

Numerade Educator

Problem 8

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.

Lottie Adams

Numerade Educator

Problem 9

The production supervisor of a small pharmaceutical firm has observed a decreasing demand for potassium regurgitol (PRG) over a two-month period, and since the plant manager has been throwing up the low sales of this product at the weekly staff meetings, the supervisor decides to discontinue its production immediately. On the day of this decision, the inventory of PRG is 28,000 kg. Based on the orders on hand, the manager projects the following weekly demand for the next six weeks:
(a) Use a semilog plot of the projected demand figures to derive an equation for $\dot{D}$ as a function of $t$ (weeks) from the present time.
(b) Write a differential balance on the inventory $I(\mathrm{kg})$ of $\mathrm{PRG},$ and integrate it to determine $I$ as a function of $t$
(c) If the demand continues to follow the projected trend of the next six weeks, how much PRG will eventually have to be discarded?

Lottie Adams

Numerade Educator

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.)

Lottie Adams

Numerade Educator

Problem 11

Purification of proteins for use as biopharmaceuticals is often accomplished by ion exchange chromatography, in which a process fluid passes through a column packed with small resin beads whose ionic surface charge causes them to adsorb some stream components more strongly than others. An ion-exchange run takes place in two steps: (1) the load step, in which the process stream flows through the column and the target protein (the product) and some undesired impurities are adsorbed onto the resin; and (2) the elution step, during which another fluid passes through the column and desorbs the impurities and the protein from the resin. The elution fluid consists of an aqueous solution of a solute known as Tris diluted with an $\mathrm{NaCl}$ solution, with the NaCl-to-Tris ratio starting at 0 and steadily increasing with time. The impurities desorb into the fluid when the NaCl concentration is low, and the effluent is collected in a waste vessel. As the NaCl concentration increases, the target protein desorbs. When analysis of the effluent reveals the presence of the target protein, the flow is switched to the product collection vessel, and the effluent is collected until no more product is detected in the effluent. The collected product is then subjected to additional process steps to further isolate the protein, and the column is cleaned for reuse.
Consider an elution step in which solutions of 1 M $\mathrm{NaCl}$ (solution A) and 50 mM Tris (solution B) are mixed and fed to a loaded ion-exchange column. The system is programmed to keep the total volumetric flow rate $\left(\dot{V}_{\Lambda}+\dot{V}_{B}\right)$ into the column constant at 120 Lh while linearly increasing the volume fraction of solution A in the feed from $0 \%$ to $20 \%$ over a period of 33.6 minutes, at which point the elution is declared to be complete. The flowchart is shown below:
(a) Calculate $V_{\mathrm{t}}(\mathrm{L}),$ the total amount of solution fed to the column.
(b) Derive an equation for the volumetric flow rate of solution $A, \dot{V}_{A}(t),$ assuming that the densities of both fluids are the same. Use the calculated value to determine calculate $V_{\mathrm{A}}(\mathrm{L}),$ the total volume of that solution fed to the column, and $m_{\mathrm{A}}(\mathrm{g} \mathrm{NaCl})$, the total mass of $\mathrm{NaCl}$ fed. Then determine $\dot{V}_{B}(t)$
and $V_{\mathrm{B}}(\mathrm{L})$ and $n_{\mathrm{B}}(\text { mol } \text { Tris) }, \text { the total volume of solution } \mathrm{B}$ and total moles of Tris fed, respectively. (Hint: Once you've done the calculations for solution A, those for B should be trivial.)
(c) Suppose in one run product is detected in the effluent at the same time impurities are detected $-$
that is, product protein starts desorbing earlier than in previous runs. List up to five possible causes of the problem.

Lottie Adams

Numerade Educator

Problem 12

A gas leak has led to the presence of 1.00 mole $\%$ carbon monoxide in a $350-\mathrm{m}^{3}$ laboratory. $^{4}$ The leak was discovered and sealed, and the laboratory is to be purged with clean air to a point at which the air contains less than the OSHA (Occupational Safety and Health Administration) specified Permissible Exposure Level (PEL) of 35 ppm (molar basis). Assume that the clean air and the air in the laboratory are atthe same temperature and pressure and that the laboratory air is perfectly mixed throughout the purging process.
(a) Let $t_{\mathrm{r}}(\mathrm{h})$ be the time required for the specified reduction in the carbon monoxide concentration. Write a differential CO mole balance, letting $N$ equal the total moles of gas in the room (assume constant), the mole fraction of CO in the room air, and $\dot{V}_{\mathrm{p}}\left(\mathrm{m}^{3} / \mathrm{h}\right)$ the flow rate of purge air entering the room (and also the flow rate of laboratory air leaving the room). Convert the balance into an equation for $d x / d t$ and provide an initial condition. Sketch a plot of $x$ versus $t,$ labeling the value
of $x$ at $t=0$ and the asymptotic value at $t \rightarrow \infty$
(b) Integrate the balance to derive an equation for $t_{r}$ in terms of $\dot{V}_{\mathrm{p}}$
(c) If the volumetric flow rate is $700 \mathrm{m}^{3} / \mathrm{h}$ (representing a tumover of two room volumes per hour), how long will the purge take? What would the volumetric flow rate have to be to cut the purge time in half?
(d) Give several reasons why it might not be safe to resume work in the laboratory after the calculated purge time has elapsed. What precautionary steps would you advise taking at this point?

Lottie Adams

Numerade Educator

Problem 13

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?

Lottie Adams

Numerade Educator

Problem 14

Ninety kilograms of sodium nitrate is dissolved in $110 \mathrm{kg}$ of water. When the dissolution is complete (at time $t=0$ ), pure water is fed to the tank at a constant rate $\dot{m}(\mathrm{kg} / \mathrm{min}),$ and solution is withdrawn from the tank at the same rate. The tank may be considered perfectly mixed.
(a) Write a total mass balance on the tank and use it to prove that the total mass of liquid in the tank remains constant at its initial value. (b) Write a balance on sodium nitrate, letting $x(t, \dot{m})$ equal the mass fraction of $\mathrm{NaNO}_{3}$ in the tank and outlet stream. Convert the balance into an equation for $d x / d t$ and provide an initial condition.
(c) On a single graph of $x$ versus $t,$ sketch the shapes of the plots you would expect to obtain for $\dot{m}=50 \mathrm{kg} / \mathrm{min}, 100 \mathrm{kg} / \mathrm{min},$ and $200 \mathrm{kg} / \mathrm{min} .$ (Don't do any calculations.) Explain your reason-
ing, using the equation of Part (b) in your explanation.
(d) Separate variables and integrate the balance to obtain an expression for $x(t, \dot{m})$. Check your solution. Then generate the plots of $x$ versus $t$ for $\dot{m}=50 \mathrm{kg} / \mathrm{min}, 100 \mathrm{kg} / \mathrm{min},$ and $200 \mathrm{kg} / \mathrm{min}$ and show them on a single graph. (A spreadsheet is a convenient tool for carrying out this step.)
(e) If $\dot{m}=100 \mathrm{kg} / \mathrm{min}$, how long will it take to flush out $90 \%$ of the sodium nitrate originally in the tank? How long to flush out 99\%? 99.9\%?
(f) The stream of water enters the tank at a point near the top, and the exit pipe from the tank is located on the opposite side toward the bottom. One day the plant technician forgot to turn on the mixing impeller in the tank. On the same chart, sketch the shapes of the plots of $x$ versus $t$ you would expect to see with the impeller on and off, clearly showing the differences between the two curves at small values and large values of $t .$ Explain your reasoning.

Rashmi Sinha

Numerade Educator

Problem 15

A stirred tank contains $1500 \mathrm{lb}_{\mathrm{m}}$ of pure water at $70^{\circ} \mathrm{F}$. At time $t=0,$ two streams begin to flow into the tank and one is withdrawn. One input stream is a $20.0 \mathrm{wt} \%$ aqueous solution of $\mathrm{NaCl}$ at $85^{\circ} \mathrm{F}$ flowing at a rate of $15 \mathrm{lb}_{\mathrm{m}} / \mathrm{min},$ and the other is pure water at $70^{\circ} \mathrm{F}$ flowing at $10 \mathrm{lb}_{\mathrm{m}} / \mathrm{min} .$ The mass of liquid in the tank is held constant at $1500 \mathrm{lb}_{\mathrm{m}}$. Perfect mixing in the tank may be assumed, so that the outlet stream has the same $\mathrm{NaCl}$ mass fraction $(x)$ and temperature $(T)$ as the tank contents. Also assume that the heat of mixing is zero and the heat capacity of all fluids is $C_{p}=1 \mathrm{Btu} /\left(\mathrm{lb}_{\mathrm{m}} \cdot^{\circ} \mathrm{F}\right)$
(a) Write differential material and energy balances and use them to derive expressions for $d x / d t$ and $d T / d t$
(b) Without solving the equations derived in Part (a), sketch plots of $T$ and $x$ as a function of time $(t)$ Clearly identify values at time zero and as $t \rightarrow \infty$

Lottie Adams

Numerade Educator

Problem 16

A radioactive isotope decays at a rate proportional to its concentration. If the concentration of an isotope is $C(\mathrm{mg} / \mathrm{L}),$ then its rate of decay may be expressed as $$r_{\mathrm{d}}[\mathrm{mg} /(\mathrm{L} \cdot \mathrm{s})]=k C$$ where $k$ is a constant.
(a) A volume $V(\mathrm{L})$ of a solution of a radioisotope whose concentration is $C_{0}(\mathrm{mg} / \mathrm{L})$ is placed in a closed vessel. Write a balance on the isotope in the vessel and integrate it to prove that the half-life
$t_{1 / 2}$ of the isotope $-$ by definition, the time required for the isotope concentration to decrease to half of its initial value- equals ( $\ln 2$ )/ $k$.
(b) The half-life of $^{56} \mathrm{Mn}$ is $2.6 \mathrm{h}$. A batch of this isotope that was used in a radiotracing experiment has been collected in a holding tank. The radiation safety officer declares that the activity (which
is proportional to the isotope concentration) must decay to $1 \%$ of its present value before the solution can be discarded. How long will this take?

Lottie Adams

Numerade Educator

Problem 17

A tracer is used to characterize the degree of mixing in a continuous stirred tank. Water enters and leaves the mixer at a rate of $\dot{V}\left(\mathrm{m}^{3} / \mathrm{min}\right) .$ Scale has built up on the inside walls of the tank, so that the effective volume $V\left(\mathrm{m}^{3}\right)$ of the tank is unknown. At time $t=0,$ a mass $m_{0}(\mathrm{kg})$ of the tracer is injected into the tank and the tracer concentration in the outlet stream, $C\left(\mathrm{kg} / \mathrm{m}^{3}\right),$ is monitored.
(a) Write a differential balance on the tracer in the tank in terms of $V, C,$ and $\dot{V},$ assuming that the tank contents are perfectly mixed, and convert the balance into an equation for $d C / d t$. Provide an initial condition, assuming that the injection is rapid enough so that all of the tracer may be considered to be in the tank at $t=0 .$ Without doing any calculations, sketch a plot of $C$ versus $t$ labeling the value of $C$ at $t=0$ and the asymptotic value at $t \rightarrow \infty$
(b) Integrate the balance to prove that $$C(t)=\left(m_{0} / V\right) \exp (-\dot{V} t / V)$$ (c) Suppose the flow rate through the mixer is $\dot{V}=30.0 \mathrm{m}^{3} / \mathrm{min}$ and that the following data are taken: (For example, at $t=1$ min, $C=0.223 \times 10^{-3} \mathrm{kg} / \mathrm{m}^{3}$.) Verify graphically that the tank is functioning as a perfect mixer- -that is, that the expression of Part (b) fits the data- -and determine the effective volume $V\left(\mathrm{m}^{3}\right)$ from the slope of your plot.
(d) A solution of a radioactive element with a fairly short half-life (see Problem 10.16 ) is often used as a tracer for applications like the one in this problem. The advantage of doing so is that the concentration of the tracer at the outlet can be measured with a sensitive radiation detector mounted outside the exit pipe rather than having to draw fluid samples from the pipe and analyze them. What is a potential drawback of radiotracers? Why is it important that the half-life of the tracer be neither too short nor too long?

Lottie Adams

Numerade Educator

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 $t \rightarrow \infty$

Lottie Adams

Numerade Educator

Problem 19

- A chemical reaction with stoichiometry $\mathrm{A} \rightarrow$ products is said to follow an $n^{\text {th }}$ -order rate law if $\mathrm{A}$ is consumed at a rate proportional to the $n$ th power of its concentration in the reaction mixture. If $r_{\mathrm{A}}$ is the rate of consumption of A per unit reactor volume, then
$$r_{\mathrm{A}}[\operatorname{mol} /(\mathbf{L} \cdot \mathbf{s})]=k C_{\mathrm{A}}^{n}$$ where $C_{A}(m o l / L)$ is the reactant concentration, and the constant of proportionality $k$ is the reaction rate constant. A reaction that follows this law is referred to as an $n^{\text {th }}$ order reaction. The rate constant
is a strong function of temperature but is independent of the reactant concentration.
(a) Suppose a first-order reaction ( $n=1$ ) is carried out in an isothermal batch reactor of constant volume $V$. Write a material balance on $A$ and integrate it to derive the expression
$$C_{\mathrm{A}}=C_{\mathrm{A} 0} \exp (-k t)$$
where $C_{\mathrm{A} 0}$ is the concentration of $\mathrm{A}$ in the reactor at $t=0$
(b) The gas-phase decomposition of sulfuryl chloride
$$\mathrm{SO}_{2} \mathrm{Cl}_{2} \rightarrow \mathrm{SO}_{2}+\mathrm{Cl}_{2}$$ is thought to follow a first-order rate law. The reaction is carried out in a constant-volume isothermal batch reactor and the concentration of $\mathrm{SO}_{2} \mathrm{Cl}_{2}$ is measured at several reaction times, with the following results: Verify the proposed rate law graphically [i.e., demonstrate that the expression given in Part (a) fits the data for $\left.C_{\mathrm{A}}(t)\right]$ and determine the rate constant $k,$ giving both its value and its units.

Arun Bana

Numerade Educator

Problem 20

The demand for biopharmaceutical products in the form of complex proteins is growing. These proteins are most often produced by cells genetically engineered to produce the protein of interest, known as a recombinant protein. The cells are grown in a liquid culture, and the protein is harvested and purified to generate the final product. Sf9 cells obtained from the fall armyworm can be used to produce protein therapeutics. Consider the growth of Sf9 cells in a bench-top bioreactor operating at $22^{\circ} \mathrm{C}$, with a liquid volume of 4.0 liters that may be assumed constant. Oxygen required for cell growth and protein production is supplied in air fed at $22^{\circ} \mathrm{C}$ and 1.1 atm. During the process, the gas leaving the bioreactor at $22^{\circ} \mathrm{C}$ and 1 atm is analyzed continuously. The data can be used to calculate the rate at which oxygen is taken up in the culture, which in turn can be used to determine the Sf9 cell growth rate (a quantity difficult to measure directly) and consistency of the operation from batch to batch.
(a) Analysis of the exhaust gas at a time 25 hours after the process is started shows a composition of
$15.5 \mathrm{mol} \% \mathrm{O}_{2}, 78.7 \% \mathrm{N}_{2},$ and the balance $\mathrm{CO}_{2}$ and small amounts of other gases. Determine the value of the oxygen use rate (OUR) in mmol $\mathrm{O}_{2}$ consumed $(\mathrm{L} \cdot \mathrm{h})$ at that point in time. Assume that nitrogen is not absorbed by the culture.
(b) OUR is related to cell concentration, $X(\mathrm{g} \text { cells } / \mathrm{L}),$ by $\mathrm{OUR}=q_{0_{2}} X,$ where $q_{0_{2}}$ is the specific rate of oxygen consumption. Analysis of a sample of the culture taken at $t=25 \mathrm{h}$ finds that the concentration of cells is $5.0 \mathrm{g}$ cells/L. What is the value of $q_{\mathrm{O}_{2}} ?$ (Do not forget to include its units.)
(c) Six hours after this measurement, the exhaust gas contains 14.5 mol\% $\mathrm{O}_{2}$ and the percentage of $\mathrm{N}_{2}$ is unchanged. What is the concentration of cells, $X,$ at that point? Assume that the specific rate of oxygen consumption does not change as long as the process temperature is constant.
(d) The growth rate of cells can be expressed as: $$\frac{d X}{d t}=\mu_{\mathrm{g}} X$$ where $\mu_{g}$ is the specific growth rate, with units of $\mathrm{h}^{-1}$. Beginning with this equation and treating $\mu_{\mathrm{g}}$ as a constant, derive an expression for $t(X) .$ Use the data from the previous parts of the problem to determine $\mu_{\mathrm{g}}$ (include units). Then calculate the cell-doubling time $\left(t_{\mathrm{d}}\right),$ defined as the time for the cell concentration to double.

Lottie Adams

Numerade Educator

Problem 21

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 $\mathrm{N}_{2} \mathrm{O} ?$

Lottie Adams

Numerade Educator

Problem 22

In an enzyme-catalyzed reaction with stoichiometry $\mathrm{A} \rightarrow \mathrm{B}, \mathrm{A}$ is consumed at a rate given by an expression of the Michaelis-Menten form:
$$r_{\mathrm{A}}[\operatorname{mol} /(\mathrm{L} \cdot \mathrm{s})]=\frac{k_{1} C_{\mathrm{A}}}{1+k_{2} C_{\mathrm{A}}}$$ where $C_{\mathrm{A}}(\operatorname{mol} / \mathrm{L})$ is the reactant concentration, and $k_{1}$ and $k_{2}$ depend only on temperature.
(a) The reaction is carried out in an isothermal batch reactor with constant reaction mixture volume $V$ (liters), beginning with pure $A$ at a concentration $C_{\mathrm{A} 0}$. Derive an expression for $d C_{\mathrm{A}} / d t$, and provide an initial condition. Sketch a plot of $C_{\mathrm{A}}$ versus $t,$ labeling the value of $C_{\mathrm{A}}$ at $t=0$ and the asymptotic value as $t \rightarrow \infty$
(b) Solve the differential equation of Part (a) to obtain an expression for the time required to achieve a specified concentration $C_{\mathrm{A}}$
(c) Use the expression of Part (b) to devise a graphical method of determining $k_{1}$ and $k_{2}$ from data for In versus the pour plot should involve fitting a straight line and determining the two parameters $C_{\mathrm{A}}$ (int the parting of the partating and and the conting are a contation a conting from the slope and intercept of the line. (There are several possible solutions.) Then apply your method to determine $k_{1}$ and $k_{2}$ for the following data taken in a 2.00 -liter reactor, beginning with
A at a concentration $C_{\mathrm{A} 0}=5.00 \mathrm{mol} / \mathrm{L}$
$$\begin{array}{|l|l|l|l|l|l|}
\hline t(\mathrm{s}) & 60.0 & 120.0 & 180.0 & 240.0 & 480.0 \\
\hline C_{\mathrm{A}}(\mathrm{mol} / \mathrm{L}) & 4.484 & 4.005 & 3.561 & 3.154 & 1.866 \\
\hline
\end{array}$$

Arun Bana

Numerade Educator

Problem 23

Phosgene (COCl_) is formed by CO and Cl_ reacting in the presence of activated charcoal: $$\mathrm{CO}+\mathrm{Cl}_{2} \rightarrow \mathrm{COCl}_{2}$$ At $T=303.8 \mathrm{K}$ the rate of formation of phosgene in the presence of 1 gram of charcoal is $$R_{\mathrm{f}}(\mathrm{mol} / \mathrm{min})=\frac{8.75 C_{\mathrm{CO}} C_{\mathrm{C}_{2}}}{\left(1+58.6 C_{\mathrm{C}_{2}}+34.3 C_{\mathrm{COC}_{2}}\right)^{2}}$$ where C denotes concentration in mollL.
(a) Suppose the charge to a 3.00 -liter batch reactor is $1.00 \mathrm{g}$ of charcoal and a gas initially containing 60.0 mole\% CO and the balance $\mathrm{Cl}_{2}$ at $303.8 \mathrm{K}$ and 1 atm. Calculate the initial concentrations (mol/L) of both reactants, neglecting the volume occupied by the charcoal. Then, letting $C_{\mathrm{P}}(t)$ be the concentration of phosgene at an arbitrary time $t,$ derive relations for
$C_{\mathrm{P}}$ $C_{\mathrm{CO}}$ and $C_{\mathrm{C}_{2}}$ in terms of (b) Write a differential balance on phosgene and show that it simplifies to $$\frac{d C_{\mathrm{P}}}{d t}=\frac{2.92\left(0.02407-C_{\mathrm{P}}\right)\left(0.01605-C_{\mathrm{P}}\right)}{\left(1.941-24.3 C_{\mathrm{P}}\right)^{2}}$$ Provide an initial condition for this equation.
(c) A plot of $C_{\mathrm{P}}$ versus $t$ starts at $C_{\mathrm{P}}=0$ and asymptotically approaches a maximum value. Explain how you could predict that behavior from the form of the equation of Part (b). Without attempting to solve the differential equation, determine the maximum value of $C_{\mathrm{P}}$
(d) Starting with the equation of Part (b), derive an expression for the time required to achieve a $75 \%$ conversion of the limiting reactant. Your solution should have the form $t=a$ definite integral.
(e) The integral you derived in Part (d) can be evaluated analytically; however, more complex rate laws than the one given for the phosgene formation reaction would yield an integral that must be evaluated numerically. One procedure is to evaluate the integrand at a number of points between the limits of integration and to use a quadrature formula such as the trapezoidal rule or Simpson's
rule (Appendix A.3) to estimate the value of the integral. Usea spreadsheet to evaluate the integrand of the integral of Part (c) at $n_{p}$ equally spaced points between and including the limits of integration, where $n_{p}$ is an odd number, and then to evaluate the integral using Simpson's rule. Perform the calculation for $n_{p}=5,21,$ and $51 .$ What can you conclude about the number of points needed to obtain a result accurate to three significant figures?

Lottie Adams

Numerade Educator

Problem 24

A gas that contains $\mathrm{CO}_{2}$ is contacted with liquid water in an agitated batch absorber. The equilibrium solubility of $\mathrm{CO}_{2}$ in water is given by Henry's law (Section $6.4 \mathrm{b}$ ) $$C_{\mathrm{A}}=p_{\mathrm{A}} / H_{\mathrm{A}}$$ where $C_{\mathrm{A}}\left(\mathrm{mol} / \mathrm{cm}^{3}\right)=$ concentration of $\mathrm{CO}_{2}$ in solution, $p_{\mathrm{A}}(\mathrm{atm})=$ partial pressure of $\mathrm{CO}_{2}$ in the gas phase, and $H_{\mathrm{A}}\left[\mathrm{atm} /\left(\mathrm{mol} / \mathrm{cm}^{3}\right)\right]=$ Henry's law constant. The rate of absorption of $\mathrm{CO}_{2}$ (i.e., the rate of transfer of $\mathrm{CO}_{2}$ from the gas to the liquid per unit area of gas-liquid interface) is given by the expression $$r_{\mathrm{A}}\left[\operatorname{mol} /\left(\mathrm{cm}^{2} \cdot \mathrm{s}\right)\right]=k\left(C_{\mathrm{A}}^{*}-C_{\mathrm{A}}\right)$$ where $C_{A}=$ actual concentration of $\mathrm{CO}_{2}$ in the liquid, $C_{\mathrm{A}}^{*}=$ concentration of $\mathrm{CO}_{2}$ in the liquid that would be in equilibrium with the $\mathrm{CO}_{2}$ in the gas phase, and $k(\mathrm{cm} / \mathrm{s})=$ a mass transfer coefficient. The gas phase is at a total pressure $\mathrm{P}\left(\text { atm) and contains } y_{\mathrm{A}}\left(\mathrm{mol} \mathrm{CO}_{2} / \mathrm{mol}\text { gas), and the liquid }\right.\right.$ phase initially consists of $V\left(\mathrm{cm}^{3}\right)$ of pure water. The agitation of the liquid phase is sufficient for the composition to be considered spatially uniform, and the amount of $\mathrm{CO}_{2}$ absorbed is low enough
for $P, V,$ and $y_{\mathrm{A}}$ to be considered constant throughout the process.
(a) Derive an expression for $d C_{\mathrm{A}} / d t$ and provide an initial condition. Without doing any calculations, sketch a plot of $C_{\mathrm{A}}$ versus $t,$ labeling the value of $C_{\mathrm{A}}$ at $t=0$ and the asymptotic value at $t \rightarrow \infty$ Give a physical explanation for the asymptotic value of the concentration.
(b) Prove that $$C_{\mathrm{A}}(t)=\frac{p_{\mathrm{A}}}{H_{\mathrm{A}}}[1-\exp (-k S t / V)]$$ where $S\left(\mathrm{cm}^{2}\right)$ is the effective contact area between the gas and liquid phases.
(c) Suppose the system pressure is 20.0 atm, the liquid volume is 5.00 liters, the tank diameter is $10.0 \mathrm{cm},$ the gas contains 30.0 mole $\% \mathrm{CO}_{2},$ the Henry's law constant is $9230 \mathrm{atm} / \mathrm{mole} / \mathrm{cm}^{3}$ ), and the mass transfer coefficient is $0.020 \mathrm{cm} / \mathrm{s}$. Calculate the time required for $C_{\mathrm{A}}$ to reach $0.620 \mathrm{mol} / \mathrm{L}$ if the gas-phase properties remain essentially constant.
(d) If A were not $\mathrm{CO}_{2}$ but instead a gas with a moderately high solubility in water, the expression for $C_{\mathrm{A}}$ given in Part (b) would be incorrect. Explain where the derivation that led to it would break down.

Lottie Adams

Numerade Educator

Problem 25

A liquid-phase chemical reaction with stoichiometry $\mathrm{A} \rightarrow \mathrm{B}$ takes place in a semibatch reactor. The rate of consumption of A per unit volume of the reactor contents is given by the first-order rate expression (see Problem 10.19) $$r_{\mathrm{A}}[\operatorname{mol} /(\mathrm{L} \cdot \mathrm{s})]=k C_{\mathrm{A}}$$ where $C_{\Lambda}(\text { mol } A / L)$ is the reactant concentration. The tank is initially empty. Beginning at a time $t=0$ a solution containing $\mathrm{A}$ at a concentration $\mathrm{C}_{\mathrm{A} 0}(\mathrm{mol} \mathrm{A} / \mathrm{L})$ is fed to the tank at a constant rate $\dot{V}(\mathrm{L} / \mathrm{s})$
(a) Write a differential balance on the total mass of the reactor contents. Assuming that the density of the contents always equals that of the feed stream, convert the balance into an equation for $d V / d t$ where $V$ is the total volume of the contents, and provide an initial condition. Then write a differential mole balance on the reactant, A, letting $N_{\mathrm{A}}(t)$ equal the total moles of A in the vessel, and provide an initial condition. Your equations should contain only the variables $N_{\mathrm{A}}, V,$ and $t$ and the constants $\dot{V}$ and $C_{\mathrm{A} 0}$. (You should be able to eliminate $C_{\mathrm{A}}$ as a variable.)
(b) Without attempting to integrate the equations, derive a formula for the steady-state value of $N_{\mathrm{A}}$.
(c) Integrate the two equations to derive expressions for $V(t)$ and $N_{\mathrm{A}}(t),$ and then derive an expression for $C_{\mathrm{A}}(t)$. Determine the asymptotic value of $N_{\mathrm{A}}$ as $t \rightarrow \infty$ and verify that the steady-state value obtained in $\operatorname{Part}(\mathbf{b})$ is correct. Briefly explain how it is possible for $N_{\mathrm{A}}$ to reach a steady value when you keep adding A to the reactor and then give two reasons why this value would never be reached in a real reactor.
(d) Determine the limiting value of $C_{\mathrm{A}}$ as $t \rightarrow \infty$ from your expressions for $N_{\mathrm{A}}(t)$ and $V(t) .$ Then explain why your result makes sense in light of the results of Part (c).

Lottie Adams

Numerade Educator

Problem 26

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?

Mahnoor Amin

Numerade Educator

Problem 27

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.

Dading Chen

Numerade Educator

Problem 28

An iron bar $2.00 \mathrm{cm} \times 3.00 \mathrm{cm} \times 10.0 \mathrm{cm}$ at a temperature of $95^{\circ} \mathrm{C}$ is dropped into a barrel of water at $25^{\circ} \mathrm{C} .$ The barrel is large enough so that the water temperature rises negligibly as the bar cools. The rate at which heat is transferred from the bar to the water is given by the expression $$\dot{Q}(\mathrm{J} / \mathrm{min})=U A\left(T_{\mathrm{b}}-T_{\mathrm{w}}\right)$$ where $U\left[=0.050 \mathrm{J} /\left(\mathrm{min} \cdot \mathrm{cm}^{2} \cdot^{\circ} \mathrm{C}\right)\right]$ is a heat transfer coefficient, $A\left(\mathrm{cm}^{2}\right)$ is the exposed surface area of the bar, and $T_{\mathrm{b}}\left(^{\circ} \mathrm{C}\right)$ and $T_{\mathrm{w}}\left(^{\circ} \mathrm{C}\right)$ are the surface temperature of the bar and the water temperature, respectively. The heat capacity of the bar is $0.460 \mathrm{J} /\left(\mathrm{g} \cdot^{\circ} \mathrm{C}\right) .$ Heat conduction in iron is rapid enough for the temperature $T_{\mathrm{b}}(t)$ to be considered uniform throughout the bar.
(a) Write an energy balance on the bar, assuming that all six sides are exposed. Your result should be
an expression for $d T_{\mathrm{b}} / d t$ and an initial condition.
(b) Without integrating the equation, sketch the expected plot of $T_{\mathrm{b}}$ versus $t$, labeling the values of $T_{\mathrm{b}}$ at $t=0$ and $t \rightarrow \infty$
(c) Derive an expression for $T_{\mathrm{b}}(t)$ and check it three ways. How long will it take for the bar to cool to $30^{\circ} \mathrm{C} ?$

Arun Bana

Numerade Educator

Problem 29

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?

Aadit Sharma

Numerade Educator

Problem 30

At 9: 30 one morning, a graduate student measures 350 grams of liquid benzene at $20^{\circ} \mathrm{C}$ into a glass flask dirty enough that its contents cannot be seen, puts the open flask on a bunsen burner, turns the burner on, and goes for a coffee break. The conversation at the break is lively, and he doesn't get back until 10:10 a.m. He looks down into the flask, sees the liquid is boiling, turns the burner off, feels a little irritation in his eye and rubs the eye with his hand, picks up the flask, says "Ouch" (or something roughly equivalent), puts the flask down on his laboratory partner's thermodynamics homework, and starts to prepare the next step of the experiment.
(a) Suppose the heat input rate to the flask contents is 40.2 W. Calculate the time at which the benzene temperature reaches $40^{\circ} \mathrm{C}$. Neglect evaporation of benzene during the heating and take the heat capacity of liquid benzene to be constant at $1.77 \mathrm{J} /\left(\mathrm{g} \cdot^{\circ} \mathrm{C}\right)$
(b) Calculate the quantity of benzene left in the flask at 10: 10 a.m., assuming that once the benzene starts boiling, the rate of heat input to the flask (40.2 W) equals the rate of vaporization (g/s) times the heat of vaporization (J/g).
(c) The graduate student was lucky. First, neither his faculty advisor nor the departmental safety officer came into the laboratory during this episode. More importantly, he was still alive and well at the end of the day. Identify as many of his safety violations as you can, explaining the danger and suggesting for each violation what he should have done instead.

Crystal Wang

Numerade Educator

Problem 31

A steam radiator is used to heat a $60-\mathrm{m}^{3}$ room. Saturated steam at 3.0 bar condenses in the radiator and emerges as a liquid at the saturation temperature. Heat is lost from the room to the outside at a rate $$\dot{Q}(\mathrm{kJ} / \mathrm{h})=30.0\left(T-T_{0}\right)$$ where $T\left(^{\circ} \mathrm{C}\right)$ is the room temperature and $T_{0}=0^{\circ} \mathrm{C}$ is the outside temperature. At the moment the radiator is turned on, the temperature in the room is $10^{\circ} \mathrm{C}$.
(a) Let $\dot{m}_{\mathrm{s}}(\mathrm{kg} / \mathrm{h})$ denote the rate at which steam condenses in the radiator and $n(\mathrm{kmol})$ the quantity of
air in the room. Write a differential energy balance on the room air, assuming that $n$ remains constant at its initial value, and evaluate all numerical coefficients. Take the heat capacity of air $\left(C_{v}\right)$ to be constant at $20.8 \mathrm{J} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)$
(b) Write the steady-state energy balance on the room air and use it to calculate the steam condensation rate required to maintain a constant room temperature of $24^{\circ} \mathrm{C}$. Without integrating the transient balance, sketch a plot of $T$ versus $t,$ labeling both the initial and maximum values of $T$
(c) Integrate the transient balance to calculate the time required for the room temperature to rise by $99 \%$ of the interval from its initial value to its steady-state value, assuming that the steam condensation rate is that calculated in Part (b).

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 32

An immersed electrical heater is used to raise the temperature of a liquid from $20^{\circ} \mathrm{C}$ to $60^{\circ} \mathrm{C}$ in 20.0 min. The combined mass of the liquid and the container is $250 \mathrm{kg}$, and the mean heat capacity of the system is 4.00 kJ/(kg.'C). The liquid decomposes explosively at 85"C. At 10: 00 a.m. a batch of liquid is poured into the vessel, and the operator turns on the heater and answers a call on his cell phone. Ten minutes later, his supervisor walks by and looks at the computer display of the power input. This what she sees. The supervisor immediately shuts off the heater and charges off to pass on to the operator several brief observations that come to her mind.
(a) Calculate the required constant power input $\dot{Q}(\mathrm{k} \mathrm{W})$, neglecting energy losses from the container.
(b) Write and integrate using Simpson's rule (Appendix A.3) an energy balance on the system to estimate the system temperature at the moment the heater is shut off. Use the following data from the recorder chart:
$$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|}
\hline t(\mathrm{s}) & 0 & 30 & 60 & 90 & 120 & 150 & 180 & 210 & 240 & 270 & 300 \\
\hline \dot{Q}(\mathrm{kW}) & 33 & 33 & 34 & 35 & 37 & 39 & 41 & 44 & 47 & 50 & 54 \\
\hline
\end{array}$$
$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline t(\mathrm{s}) & 330 & 360 & 390 & 420 & 450 & 480 & 510 & 540 & 570 & 600 \\
\hline \dot{Q}(\mathrm{kW}) & 58 & 62 & 66 & 70 & 75 & 80 & 85 & 90 & 95 & 100 \\
\hline
\end{array}$$
(c) Suppose that if the heat had not been shut off, $\dot{Q}$ would have continued to increase linearly at a rate of $10 \mathrm{kW} / \mathrm{min}$. At what time would everyone in the plant realize that something was wrong?

Arun Bana

Numerade Educator

Problem 33

A 2000 -liter tank initially contains 400 liters of pure water. Beginning at $t=0$, an aqueous solution containing $1.00 \mathrm{g} / \mathrm{L}$ of potassium chloride flows into the tank at a rate of $8.00 \mathrm{L} / \mathrm{s}$ and an outlet stream simultaneously starts flowing at a rate of $4.00 \mathrm{L} / \mathrm{s}$. The contents of the tank are perfectly mixed, and the densities of the feed stream and of the tank solution, $\rho(g / L),$ may be considered equal and constant.
Let $V(t)(\mathrm{L})$ denote the volume of the tank contents and $C(t)(\mathrm{g} / \mathrm{L})$ the concentration of potassium chloride in the tank contents and outlet stream.
(a) Write a balance on total mass of the tank contents, convert it to an equation for $d V / d t$, and provide an initial condition. Then write a potassium chloride balance, show that it reduces to $$\frac{d C}{d t}=\frac{8-8 C}{V}$$ and provide an initial condition. (Hint: You will need to use the mass balance expression in your derivation.)
(b) Without solving either equation, sketch the plots you expect to obtain for $V$ versus $t$ and $C$ versus $t$
If the plot of $C$ versus $t$ has an asymptotic limit as $t \rightarrow \infty,$ determine what it is and explain why it makes sense.
(c) Solve the mass balance to obtain an expression for $V(t)$. Then substitute for $V$ in the potassium chloride balance and solve for $C(t)$ up to the point when the tank overflows. Calculate the $\mathrm{KCl}$ concentration in the tank at that point.

Rashmi Sinha

Numerade Educator

Problem 34

The diagram below shows three continuous stirred tanks connected in series and initially filled with water. The flow and mixing patterns in this system are studied by dissolving $1500 \mathrm{g}$ of a salt $(\mathrm{S})$ in the first tank, and then starting the 40 L/s flow through the system. Each tank outlet stream is monitored with an on-line thermal conductivity detector calibrated to provide instantaneous readings of salt concentration. The data are plotted versus time, and the results are compared with the plots that would be expected if the tanks are all perfectly mixed. Your job is to generate the latter plots.
(a) Assuming that pure water is fed to the first tank and that each tank is perfectly mixed (so that the salt concentration in a tank is uniform and equal to the concentration in the outlet stream from that tank), write salt balances on each of the three tanks, convert them to expressions for $d C_{\mathrm{S} 1} / d t$ $d C_{\mathrm{S} 2} / d t,$ and $d C_{\mathrm{S} 3} / d t,$ and provide appropriate initial conditions.
(b) Without doing any calculations, on a single graph sketch the forms of the plots of $C_{\mathrm{S} 1}$ versus $t$ Con versus to to to the reaction of cout pout partiefly explain your reasoning. (s) expect to obtain. $\mathrm{s}$ versus $t$ you would expecting
(c) Use a differential equation-solving program to solve the three equations, proceeding to a time at which $C_{\mathrm{S} 3}$ has fallen below $0.01 \mathrm{g} / \mathrm{L},$ and plot the results.

Linda Hand

Numerade Educator

Problem 35

The following chemical reactions take place in a liquid-phase batch reactor of constant volume $V$. $$\begin{aligned}
&\mathrm{A} \rightarrow 2 \mathrm{B} \quad r_{1}[\mathrm{mol} \mathrm{A} \text { consumed } /(\mathrm{L} \cdot \mathrm{s})]=0.100 C_{\mathrm{A}}\\
&\mathbf{B} \rightarrow \mathbf{C} \quad r_{2}[\mathrm{mol} \mathbf{C} \text { generated } /(\mathbf{L} \cdot \mathbf{s})]=0.200 C_{\mathrm{B}}^{2}
\end{aligned}$$ where the concentrations $C_{\mathrm{A}}$ and $C_{\mathrm{B}}$ are in mol/L. The reactor is initially charged with pure $\mathrm{A}$ at a concentration of 1.00 mol/L.
(a) Write expressions for ( $i$ ) the rate of generation of $\mathrm{B}$ in the first reaction and (ii) the rate of consumption of $\mathrm{B}$ in the second reaction. (If this takes you more than about 10 seconds, you're missing the point.)
(b) Write mole balances on A, B, and C, convert them into expressions for $d C_{\mathrm{A}} / d t, d C_{\mathrm{B}} / d t$, and $d C_{\mathrm{C}} / d t,$ and provide boundary conditions.
(c) Without doing any calculations, sketch on a single graph the plots you would expect to obtain of $C_{\mathrm{A}}$ versus $t, C_{\mathrm{B}}$ versus $t,$ and $C_{\mathrm{C}}$ versus $t .$ Clearly show the function values at $t=0$ and $t \rightarrow \infty$ and the curvature (concave up, concave down, or linear) in the vicinity of $t=0 .$ Briefly explain your reasoning.
(d) Solve the equations derived in Part (b) using a differential equation-solving program. On a single graph, show plots of $C_{\mathrm{A} \text { versust }}, C_{\mathrm{B}}$ versus $t,$ and $C_{\mathrm{C}}$ versus $t$ from $t=0$ to $t=50$ s. Verify that your predictions in Part (c) were correct. If they were not, change them and revise your explanation.

Lottie Adams

Numerade Educator

Problem 36

A liquid mixture containing 70.0 mol of $n$ -pentane and 30.0 mol of $n$ -hexane initially at $46^{\circ} \mathrm{C}$ is partially vaporized at $P=1$ atm in a single-stage distillation apparatus (Rayleigh still). The heat added to the system, $\dot{Q}$, vaporizes liquid at the rate $\dot{n}_{\mathrm{V}}(\mathrm{mol} / \mathrm{s})$. The vapor product and remaining liquid at a given moment are always in equilibrium with each other. The relationship between the mole fraction of pentane in the liquid ( $x$ ) and that in the vapor ( $y$ ) is of the form $$y=\frac{a x}{x+b}$$ so that the system involves four time-dependent variables $-N_{\mathrm{L}}, \dot{n}_{\mathrm{V}}, x,$ and $y,$ where $N_{L}$ is the total moles of liquid in the still at any time. (We will suppose that the rate of heat transfer to the evaporator,
Q. is constant and known.) Four equations relating the unknowns will be required to determine these variables. The equations are two material balances, an energy balance, and the vapor-liquid equilibrium relationship just given.
(a) When $x=1,$ what must $y$ equal? (Think of the definitions of these quantities.) Use your answer and the vapor-liquid equilibrium expression to derive an equation relating the parameters $a$ and $b$
(b) Use Raoult's law (Equation 6.4-1) and the Antoine equation to calculate the mole fraction of pentane in the vapor phase in equilibrium with the $70 \%$ pentane- $30 \%$ hexane feed mixture at the initial system temperature of $46^{\circ} \mathrm{C}$ and a pressure of 1 atm. Then use this result and that of Part (a)
to estimate $a$ and $b$. (Assume that these values remain the same over the range of compositions and temperatures to be undergone by the system.)
(c) Taking the residual liquid in the still as your system, write a differential balance on total moles to obtain an expression for $d N_{\mathrm{L}} / d t .$ Then write a balance on pentane, recognizing that both $N_{\mathrm{L}}$ and $x$ are functions of time. (Hint: Remember the product rule for differentiation.) Prove that the pentane balance can be converted into the following equation:
$$\frac{d x}{d t}=\frac{\dot{n}_{\mathrm{V}}}{N_{\mathrm{L}}}\left(\frac{a x}{x+b}-x\right)$$ Supply initial conditions for your two differential equations.
(d) In Part (c), you derived two equations in three unknown dependent variables $-\dot{n}_{\mathrm{V}}(t), N_{L}(t),$ and
$x(t) .$ To determine these variables, we need a third relationship. An energy balance provides it.
A rigorous energy balance would take into account the changing composition of the liquid,
the slightly different heats of vaporization of pentane and hexane, and the enthalpy changes associated with temperature changes, and would make the problem relatively hard to solve. A reasonable approximation is to assume that (i) the liquid has a constant heat of vaporization of 27.0 $\mathrm{kJ} / \mathrm{mol}$, independent of composition and temperature; and (ii) all heat supplied to the still $\underline{I} \dot{Q}(\mathrm{kJ} / \mathrm{s})]$ goes to vaporize liquid (i.e., we neglect energy that goes into raising the temperature of the liquid or the vapor). Make these assumptions, consider $\dot{Q}$ to be constant and known, and derive a simple expression for $\dot{n}_{\mathrm{V}}$ that can be used to eliminate this variable in the differential equations of Part (c). From there, derive the following expression: $$\frac{d x}{d t}=-\frac{\dot{Q} / 27.0}{100.0 \mathrm{mol}-\dot{Q} t / 27.0}\left(\frac{a x}{x+b}-x\right)$$ (e) Use a differential equation-solving program to calculate $x, y, N_{\mathrm{L}},$ and $\dot{n}_{\mathrm{V}}$ from $t=0$ until the time
at which the liquid completely evaporates. Do the calculation for (i) $\dot{Q}=1.5 \mathrm{kJ} / \mathrm{s}$ and
(ii) $\dot{Q}=3.0 \mathrm{kJ} / \mathrm{s} .$ On a single graph, plot $x$ and $y$ versus $t,$ showing curves for both values of $\dot{Q}$.
(f) In a short paragraph, describe what happens to the compositions of the vapor product and residual liquid over the course of a run. Include a statement of what the initial and final vapor compositions are and how the heating rate affects the system behavior.

Lottie Adams

Numerade Educator