# 7.6: Problems

## Section 7.2

1. In the production of formaldehyde ($$\ce{CH2O}$$) by catalytic oxidation of methanol ($$\ce{CH3OH}$$) an equimolar mixture of methanol and air (21% oxygen and 79% nitrogen) is sent to a catalytic reactor. The reaction is catalyzed by finely divided silver supported on alumina as suggested in Figure $$\PageIndex{1}$$ where we have indicated that carbon dioxide ($$\ce{CO2}$$) is produced as an undesirable product.

The conversion for methanol ($$\ce{CH3OH}$$) is given by

${ C}= \text{ Conversion of }\ce{CH3OH}=\frac{- \mathscr{R}_{ \ce{CH3OH}} }{ \left(\dot{M}_{\ce{CH3OH} }\right)_{1} } =0.20\nonumber$

and the selectivity for methanol/carbon dioxide is given by

${ S}= \text{ Selectivity of } \ce{CH2O}/\ce{CO2} =\frac{\mathscr{R}_{ \ce{CH2O}} }{\mathscr{R}_{ \ce{CO2} } } =8.5\nonumber$

In this problem you are asked to determine the mole fraction of all components in the Stream #2 leaving the reactor.

2. Use the pivot theorem (Sec. 6.4) with Eq. 6 of Example $$7.2.1$$ to develop a solution for $$\mathscr{R}_{\ce{H2}}$$ and $$\mathscr{R}_{\ce{C2H6}}$$ using ethylene ($$\ce{C2H4}$$), methane ($$\ce{CH4}$$) and propylene ($$\ce{C3H6}$$) as the pivot species. Compare your solution with Eqs. 7 and 8 of Example $$7.2.1$$. In order to use ethylene ($$\ce{C2H4}$$), ethane ($$\ce{C2H6}$$), and propylene ($$\ce{C3H6}$$) as the pivot species, one needs to use a column/row interchange (see Sec. 6.2.5) with Eq. 6 of Example $$7.2.1$$. Carry out the appropriate column/row interchange and the necessary elementary row operations and use the pivot theorem to develop a solution for $$\mathscr{R}_{\ce{H2}}$$ and $$\mathscr{R}_{\ce{CH4}}$$.

3. Acetic anhydride can be made by direct reaction of ketene ($$\ce{CH2CO}$$) with acetic acid. Ketene ($$\ce{CH2CO}$$) is an important intermediary chemical used to produce acetic anhydride. The pyrolysis of acetone ($$\ce{CH3} \ce{COCH3}$$) in an externally heated empty tube, illustrated in Figure $$\PageIndex{2}$$, produces ketene ($$\ce{CH2CO}$$) and methane ($$\ce{CH4}$$); however, some of the ketene reacts during the pyrolysis to form ethylene ($$\ce{C2H4}$$) and carbon monoxide ($$\ce{CO}$$). In turn, some of the ethylene is cracked to make coke ($$\ce{C}$$) and hydrogen ($$\ce{H2}$$).

At industrial reactor conditions, the yields for this set of reactions are given by

${ Y}_{\ce{CH2CO}} = \text{ Yield of } \ce{CH2CO} { /}\ce{CH3} \ce{COCH3} =\frac{\mathscr{R}_{ \ce{CH2CO} } }{- \mathscr{R}_{ \ce{CH3} \ce{COCH3} } } =0.95\nonumber$

${ Y}_{\ce{C2H4}} = \text{ Yield of } \ce{C2H4} { /}\ce{CH3} \ce{COCH3} =\frac{\mathscr{R}_{ \ce{C2H4} } }{- \mathscr{R}_{ \ce{CH3} \ce{COCH3} } } =0.015\nonumber$

${ Y}_{\ce{H2}} = \text{ Yield of } \ce{H2} { /}\ce{CH3} \ce{COCH3} =\frac{\mathscr{R}_{ \ce{H2}} }{- \mathscr{R}_{ \ce{CH3} \ce{COCH3} } } =0.02\nonumber$

Given the conversion of acetone

${ C}= \text{ Conversion of } \ce{CH3} \ce{COCH3} =\frac{- \mathscr{R}_{ \ce{CH3} \ce{COCH3} } }{ (\dot{M}_{ \ce{CH3} \ce{COCH3} } )_{feed}} =0.98\nonumber$

determine the mole fraction of all components in the reactor product stream.

4 $$\ddagger$$. Ethylene oxide can be produced by catalytic oxidation of ethane using pure oxygen. The stream leaving the reactor illustrated in Figure $$\PageIndex{3}$$ contains non-reacted ethane and oxygen as well as ethylene oxide, carbon monoxide, carbon dioxide, and water.

A gas stream of 10 kmol/min of ethane and oxygen is fed to the catalytic reactor with the mole fraction specified by

Stream #1: $y_{\ce{C2H6}} =y_{\ce{O2}} =0.50 \label{1}$

The reaction occurs over a platinum catalyst at a pressure of one atmosphere and a temperature of 250 C. Some of the mole fractions in the exit stream have been determined experimentally and the values are given by

Stream #2: $y_{\ce{C2H4O}} = 0.287 , \quad y_{\ce{C2H6}} = 0.151 , \quad y_{\ce{O2}} = 0.076 \label{2}$

In this problem you are asked to determine the global rates of production of all the species participating in the catalytic oxidation reaction.

5 $$\ddagger$$. In a typical experimental study, such as that described in Problem 4, one would normally determine the complete composition of the entrance and exit streams. If these compositions were given by

Inlet Stream: \begin{aligned} (y_{\ce{C2} \ce{H6} } )_{1} = 0.50 , && (y_{\ce{O2}} )_{1} = 0.50 , && (y_{ \ce{H2O}} )_{1} = 0.0 \\ (y_{\ce{ CO}} )_{1} = 0.0 , && (y_{\ce{CO2} } )_{1} = 0.0 , && (y_{ \ce{C2H4O}} )_{1} = 0.0 \end{aligned}

Outlet Stream: \begin{aligned} (y_{\ce{C2} \ce{H6} } )_{2} = 0.14 , && (y_{\ce{O2}} )_{2} = 0.08 , && (y_{ \ce{H2O}} )_{2} = 0.43 \\ (y_{\ce{ CO}} )_{2} = 0.05 , && (y_{\ce{CO2} } )_{2} = 0.03 , && (y_{ \ce{C2H4O}} )_{2} = 0.27 \end{aligned}

how would you determine the six global rates of production associated with the partial oxidation of ethane? One should keep in mind that the experimental values of the mole fractions have been conditioned so that they sum to one.

6. Aniline is an important intermediate in the manufacture of dyes and rubber. A traditional process for the production of aniline consists of reducing nitrobenzene in the presence of iron and water at low pH. Aniline and water are fed in vapor form, at 250 C and atmospheric pressure, to a fixed-bed reactor containing the iron particles as illustrated in Figure $$\PageIndex{4}$$. The solid iron oxide produced in the reaction remains in the reactor and it is later regenerated with hydrogen. The conversion is given by

${ C}= \text{ Conversion of } \ce{C6H5} \ce{NO2} =\frac{- \mathscr{R}_{ \ce{ C6H5} \ce{NO2} } }{ (\dot{M}_{ \ce{C6H5} \ce{NO2} } )_{feed} } =0.80\nonumber$

and the feed consists of 100 kg/hr of an equimolar gaseous mixture of aniline and water. Determine the mole fraction of all components leaving the reactor. If the reactor is initially charged with 2000 kg of iron (Fe), use Eq. ($$7.1.5$$) to determine the time required to consume all the iron. Assume that the reactor operates at a steady state until the iron is depleted.

7. Vinyl chloride ($$\ce{CH2CHCl}$$) is produced in a fixed-bed catalytic reactor where a mixture of acetylene ($$\ce{C2} \ce{H2}$$) and hydrogen chloride ($$\ce{HCl}$$) react over in the presence of mercuric chloride supported on activated carbon. Assume that an equilibrium mixture leaves the reactor with the equilibrium relation given by

$K_{eq} =\frac{y_{A} }{y_{B} y_{C} } =300\nonumber$

Here $$y_{A}$$, $$y_{B}$$ and $$y_{C}$$ are the mole fractions in the exit stream of vinyl chloride ($$\ce{CH2CHCl}$$), acetylene ($$\ce{C2} \ce{H2}$$) and hydrogen chloride ($$\ce{HCl}$$) respectively. Determine the excess of hydrogen chloride over the stoichiometric amount in order to achieve a conversion given by

${ C}= \text{ Conversion of } \ce{ C2} \ce{H2} =\frac{- \mathscr{R}_{ \ce{C2} \ce{H2} } }{ (\dot{M}_{\ce{ C2} \ce{H2} } )_{feed} } =0.99\nonumber$

8. Carbon dioxide ($$\ce{CO2}$$) gas reacts over solid charcoal ($$\ce{C}$$) to form carbon monoxide in the so-called Boudouard reaction illustrated in Figure $$\PageIndex{5}$$.

At 940 K the equilibrium constant for the reaction of carbon dioxide with carbon to produce carbon monoxide is given by

$K_{eq} =\frac{p_{ \ce{CO}}^{2} }{p_{ \ce{CO2} }} =1.2 \ atm$

A mixture of carbon dioxide and nitrogen is fed to a fixed bed reactor filled with charcoal at 940 K. The mole fraction of carbon dioxide entering the reactor is 0.60. Assuming that the exit stream is in equilibrium with the solid charcoal, compute the mole fraction of all components in the exit stream.

9. Hydrogen cyanide is made by reacting methane with anhydrous ammonia in the so-called Andrussov process illustrated in Figure $$\PageIndex{6}$$. The equilibrium constant for this reaction at atmospheric pressure and 1300 K is given by

$K_{eq} =\frac{p_{ \ce{HCN}} p_{\ce{H2} }^{3} }{p_{\ce{CH4} } p_{\ce{ NH3} } } =380 \ atm^{2}\nonumber$

A mixture of 50% by volume of methane and 50% anhydride ammonia is sent to a chemical reactor at atmospheric pressure and 1300 K. Assuming the output of the reactor is in equilibrium, what would be the composition of the mixture of gases leaving the reactor, in mole fractions?

10. Teflon, tetrafluoroethylene ($$\ce{C2F4}$$) is made by pyrolysis of gaseous monocholorodifluoromethane ($$\ce{ CHClF2}$$) in a reactor such as the one shown in Figure $$7.2.1$$. The decomposition of $$\ce{ CHClF2}$$ produces $$\ce{C2F4}$$ and $$\ce{HCl}$$ in addition to the undesirable homologous polymer, $$\ce{ H}\left(\ce{CF2} \right)_{ 3} \ce{ Cl}$$. The conversion of gaseous monocholorodifluoromethane ($$\ce{ CHClF2}$$) is given by

${ C}=\text{ Conversion of }\left(\ce{ CHClF2} \right)=\frac{- \mathscr{R}_{ \ce{ CHClF2} } }{ (\dot{M}_{ \ce{CHClF2} } )_{feed} } =0.98\nonumber$

and the yield for Teflon takes the form

${ Y}=\text{ Yield of }\left(\ce{C2F4} { /} \ce{ CHClF2} \right)=\frac{\mathscr{R}_{ \ce{C2F4} } }{- \mathscr{R}_{ \ce{ CHClF2} } } =0.475\nonumber$

For the first part of this problem, find a relation between the molar flow rate of the exit stream and the molar flow rate of the entrance stream in terms of the number of monomer units in the polymer species. For the second part, assume that $$\alpha \ce{CHClF_2} \Rightarrow \left[ \ce{H}(\ce{CF2})_3\ce{Cl}\right]_{\alpha}$$ where $$\alpha = 10$$ and assume that the input flow rate is $$\dot{M}_1 = 100$$ mol/s in order to determine the mole fractions of the four components in the product stream leaving the reactor.

11. In Example $$7.2.1$$ numerical values for the conversion, selectivity and yield were given, in addition to the conditions for the inlet stream, i.e., $$\dot{M}=100$$ mol/s and $$x_{\ce{C2H6}} =1.0$$. Indicate what other quantities had to be measured in order to determine the conversion, selectivity and yield.

12. In Example $$7.2.1$$ the reaction rates for methane and hydrogen were determined to be $$\mathscr{R}_{ \ce{CH4} } ={ 1.0}$$ mol/s and $$\mathscr{R}_{ \ce{H2} } =19.0$$ mol/s. Determine the rates of production represented by $$\mathscr{R}_{ \ce{C2} \ce{H6} }$$, $$\mathscr{R}_{ \ce{C2H4} }$$, and $$\mathscr{R}_{ \ce{C3} \ce{H6} }$$.

## Section 7.3

13. Show how to obtain the row reduced echelon form of $$\left[N_{JA} \right]$$ given in Equation ($$7.3.3$$) from the form given in Equation ($$7.3.1$$).

14. A stream of pure methane, $$\ce{CH4}$$, is partially burned with air in a furnace at a rate of 100 moles of methane per minute. The air is dry, the methane is in excess, and the nitrogen is inert in this particular process. The products of the reaction are illustrated in Figure $$\PageIndex{7}$$.

The exit gas contains a 1:1 ratio of $$\ce{H2O} : \ce{H2}$$ and a 10:1 ratio of $$\ce{ CO} : \ce{CO2}$$. Assuming that all of the oxygen and 94% of the methane are consumed by the reactions, determine the flow rate and composition of the exit gas stream.

15. Consider the special case in which two molecular species represented by $$\ce{ C}_{m} \ce{ H}_{n} \ce{ O}_{p}$$ and $$\ce{ C}_{q} \ce{ H}_{r} \ce{ O}_{s}$$ provide the fuel for complete combustion as illustrated in Figure $$\PageIndex{8}$$. Assume that the molar flow rates of the two molecular species in the fuel are given in order to develop an expression for the theoretical air.

16. A flue gas (Stream #1) composed of carbon monoxide, carbon dioxide, and nitrogen can undergo reaction with “water gas” (Stream #2) and steam to produce the synthesis gas (Stream #3) for an ammonia converter. The carbon dioxide in the synthesis gas must be removed before the gas is used as feed for an ammonia converter. This process is illustrated in Figure $$\PageIndex{9}$$, and the product gas in Stream #3 is required to contain hydrogen and nitrogen in a 3 to 1 molar ratio. In this problem you are asked to determine the ratio of the molar flow rate of the flue gas to the molar flow rate of the water gas, i.e., $$\dot{M}_{1} /\dot{M}_{2}$$, that is required in order to meet the specification that $$(y_{ \ce{H2} } )_{3} =3 (y_{ \ce{N2} } )_{3}$$.

Answer: $$\dot{M}_{1} /\dot{M}_{2} =0.467$$

17. A process yields 10,000 ft$$^{3}$$ per day of a mixture of hydrogen chloride ($$\ce{HCl}$$) and air. The volume fraction of hydrogen chloride ($$\ce{HCl}$$) is 0.062, the temperature of the mixture is 550$$^{\circ}$$F, and the total pressure is represented by 29.2 inches of mercury. Calculate the mass of limestone per day required to neutralize the hydrogen chloride ($$\ce{HCl}$$) if the mass fraction of calcium carbonate ($$\ce{ CaCO3}$$) in the limestone is 0.92. Determine the cubic feet of gas liberated per day at 70$$^{\circ}$$F if the partial pressure of carbon dioxide ($$\ce{ CO2}$$) is 1.8 inches of mercury. Assume that the reaction between $$\ce{HCl}$$ and $$\ce{CaCO3}$$ to form $$\ce{ CaCl2}$$, $$\ce{CO2}$$, $$\ce{H2O}$$ goes to completion.

18. Carbon is burned with air with all the carbon being oxidized to $$\ce{CO2}$$. Calculate the flue gas composition when the percent of excess air is 0, 50, and 100. The percent of excess air is defined as:

$\left\{\begin{array}{c} \text{ percent of} \\ \text{ excess air} \end{array}\right\}=\frac{\left(\begin{array}{c} \text{ molar flow} \\ \text{ rate of oxygen} \\ \text{ entering} \end{array}\right) - \left(\begin{array}{c} \text{ molar rate of} \\ \text{ consumption of oxygen} \\ \text{ owing to reaction} \end{array}\right)}{\left(\begin{array}{c} \text{ molar rate of} \\ \text{ consumption of oxygen} \\ \text{ owing to reaction} \end{array}\right)} \times 100\nonumber$

Take the composition of air to be 79% nitrogen and 21% oxygen. Assume that no NOX is formed.

19. A waste gas from a petrochemical plant contains 5% HCN with the remainder being nitrogen. The waste gas is burned in a furnace with excess air to make sure the HCN is completely removed. The combustion process is illustrated in Figure $$\PageIndex{10}$$. Assume that the percent excess air is 100% and determine the composition of the stream leaving the furnace.

The percent of excess air is defined as:

$\left\{\begin{array}{c} \text{ percent of} \\ \text{ excess air} \end{array}\right\}=\frac{\left(\begin{array}{c} \text{ molar flow} \\ \text{ rate of oxygen} \\ \text{ entering} \end{array}\right) - \left(\begin{array}{c} \text{ molar rate of} \\ \text{ consumption of oxygen} \\ \text{ owing to reaction} \end{array}\right)}{\left(\begin{array}{c} \text{ molar rate of} \\ \text{ consumption of oxygen} \\ \text{ owing to reaction} \end{array}\right)} \times 100\nonumber$

20. A fuel composed entirely of methane and nitrogen is burned with excess air. The dry flue gas composition in volume percent is: $$\ce{CO2}$$, 7.5%, $$\ce{ O2}$$, 7%, and the remainder nitrogen. Determine the composition of the fuel gas and the percentage of excess air as defined in Problem 19.

21. In this problem we consider the production of sulfuric acid illustrated in Figure $$\PageIndex{11}$$. The mass flow rate of the dilute sulfuric acid stream is specified as $$\dot{m}_{1} =100 \ { lb}_{ m} /{ hr}$$ and we are asked to determine the mass flow rate of the pure sulfur trioxide stream, $$\dot{m}_{2}$$. As is often the custom with liquid systems the percentages given in Figure $$\PageIndex{11}$$ refer to mass fractions, thus we desire to produce a final product in which the mass fraction of sulfuric acid is 0.98.

22. In Example $$7.3.2$$ use elementary row operations to obtain Eq. 10 from Eq. 9, and apply the pivot theorem to Eq. 10 to verify that Eqs. 11 are correct.

## Section 7.4

23. Given $$(x_{A} )_{1}$$, $$(x_{A} )_{2}$$ and any three molar flow rates for the splitter illustrated in Figure $$\PageIndex{3}$$, demonstrate that the compositions and total molar flow rates of all the streams are determined. For the three specified molar flow rates, use either species molar flow rates, or total molar flow rates, or a combination of both.

24. Given any three total molar flow rates and any two species molar flow rates for the splitter illustrated in Figure $$\PageIndex{3}$$, demonstrate that the compositions and total molar flow rates of all the streams are determined. If the directions of the streams in Figure $$\PageIndex{3}$$ are reversed we obtain a mixer as illustrated in Figure $$7.4.3$$. Show that six specifications are needed to completely determine a mixer with three input streams, i.e., $$S = 3$$.

25. Given $$(x_{A} )_{1}$$, $$(x_{B} )_{1}$$, $${\dot{M}_{3} / \dot{M}_{2} }$$, $${\dot{M}_{4} / \dot{M}_{2} }$$, and any species molar flow rate for the splitter illustrated in Figure $$\PageIndex{3}$$, demonstrate that the compositions and total molar flow rates of all the streams are determined.

26. Show how Equation 6 is obtained from Equation 5 in Example $$7.4.2$$ using the concepts discussed in Sec. 6.2.5.

27. In the catalytic converter shown in Figure $$\PageIndex{12}$$, a reaction of the form $$A\to$$products takes place. The product is completely separated from the stream leaving the reactor, and pure $$A$$ is recycled via stream #5. A certain fraction, $$\varphi$$, of species $$A$$ that enters the reactor is converted to product, and we express this idea as

$(x_{A} )_{3} \dot{M}_{3} = (1-\varphi ) \dot{M}_{2}\nonumber$

In this problem you are asked to derive an expression for the ratio of molar flow rates, $$\dot{M}_{5} /\dot{M}_{1}$$, in terms of $$\varphi$$ given that pure $$A$$ enters the system in stream #1.

28. A simple chemical reactor in which a reaction, $$A\to$$products, is shown in Figure $$\PageIndex{13}$$. The reaction occurs in the liquid phase and the feed stream is pure species $$A$$. The overall extent of reaction is designated by $$\xi$$ where $$\xi$$ is defined by the relation

$(\omega _{A} )_{4} =\left(1-\xi \right) (\omega _{A} )_{2}\nonumber$

Here we see that $$\xi =0$$ when no reaction takes place, and when $$\xi =1$$ the reaction is complete and no species $$A$$ leaves the reactor. We require that the mass fraction of species $$A$$ entering the reactor be constrained by

$(\omega _{A} )_{2} =\varepsilon (\omega _{A} )_{1}\nonumber$

in which $$\varepsilon$$ is some number less than one and greater than zero. Since the products of the reaction are not specified, assume that they can be lumped into a single “species” $$B$$. Under these conditions the reaction can be expressed as $$A\to B$$ and the reaction rates for the two species system must conform to $$r_{A} = -r_{B}$$. The objective in this problem is to derive an expression for the ratio of mass flow rates $$\dot{m}_{ 5} /\dot{m}_{ 4}$$ in terms of $$\xi$$ and $$\varepsilon$$.

29. Solve problem 28 using an iterative procedure (see Appendix B) for $$\xi = 0.5$$ and $$\varepsilon = 0.3$$ for a feed stream flow rate of $$\dot{m}_{1} = 100,000$$ mol/h. Use a convergence criteria of 0.1 kmol/h for Stream #5.

30. Assume that the system described in Problem 28 contains $$N$$ molecular species, thus species $$A$$ represents the single reactant and there are $$N-1$$ product species. The reaction rates for the products can be expressed as

$r_{B} = - r_{A}^{ I} , \quad r_{C} = - r_{A}^{ II} , \quad r_{D} = - r_{A}^{ III} , \quad ......., \quad r_{N} = - r_{A}^{ N - 1}\nonumber$

where the overall mass rate of production for species $$A$$ is given by

$r_{A} =r_{A}^{ I} + r_{A}^{ II} + r_{A}^{ III} + ...... + r_{A}^{ N - 1}\nonumber$

Begin your analysis with the axioms for the mass of an $$N$$-component system and identify the conditions required in order that $$N-1$$ product species can be represented as a single species.

31. In the air drier illustrated in Figure $$\PageIndex{14}$$, part of the effluent air stream is to be recycled in an effort to control the inlet humidity. The solids entering the drier (Stream #3) contain 20 % water on a mass basis and the mass flow rate of the wet solids entering the drier is 1000 lb$$_m$$/hr. The dried solids (stream #4) are to contain a maximum of 5 % water on a mass basis. The partial pressure of water vapor in the fresh air entering the system (Stream #1) is equivalent to 10 mm of mercury and the partial pressure in the air leaving the drier (Stream #5) must not exceed 200 mm of mercury. In this particular problem the flow rate of the recycle stream (stream #6) is to be regulated so that the partial pressure of water vapor in the air entering the drier is equivalent to 50 mm Hg. For this condition, calculate the total molar flow rate of fresh air entering the system (Stream #1) and the total molar flow rate of the recycle stream (Stream #6).

32. Solve problem 31 using one of the iterative procedures described in Appendix B. Assume that the partial pressure of water in the fresh air stream (Stream #1) is 20 mm of Hg and that the maximum partial pressure in the stream leaving the unit (Stream #5), does not exceed 180 mm Hg. If you use a spreadsheet or a Matlab program to solve this problem, make sure all variables are conveniently labeled. Use a convergence criteria of 1 mm Hg for Stream #6.

33. By manipulating the operating conditions (temperature, pressure and catalyst) in the reactor described in Example $$7.4.2$$, the conversion can be increased from 0.30 to 0.47, i.e.,

${ C}= \text{ Conversion of } \ce{C2H4Cl2} =\frac{- \mathscr{R}_{ \ce{C2H4Cl2} } }{ (\dot{M}_{ \ce{C2H4Cl2} } )_{2} } =0.47\nonumber$

For this conversion, what are the changes in the molar flow rate of vinyl chloride in Stream #4 and the molar flow rate of dichloroethane in Stream #5?

34. For the conditions given in Example $$7.4.2$$, determine the total molar flow rate and composition in Stream #3.

35. Metallic silver can be obtained from sulfide ores by first roasting the sulfides ($$\ce{Ag2S}$$) to produce silver sulfates ($$\ce{Ag2SO4}$$) which are leached from the ore using boiling water11. Next the silver sulfate ($$\ce{Ag2SO4}$$) is reacted with copper ($$\ce{Cu}$$) to produce copper sulfate ($$\ce{CuSO4}$$) and silver ($$\ce{Ag}$$) as illustrated in Figure $$\PageIndex{15}$$. The product leaving the second separator contains 90% (by mass) silver and 10% copper. The percent of excess copper in Stream #1 is defined by

$\left\{\begin{array}{c} \text{ percentage of} \\ \text{ excess copper} \end{array}\right\}=\frac{\left(\begin{array}{c} \text{ molar flow rate of copper } \\ \text{entering in the feed stream} \end{array}\right) - \left(\begin{array}{c} \text{ molar rate of consumption } \\ \text{ of copper in the reactor} \end{array}\right)}{\left(\begin{array}{c} \text{ molar rate of consumption } \\ \text{ of copper in the reactor} \end{array}\right)} \times 100\nonumber$

and the conversion of silver sulfate is defined by

${ C}=\frac{\left\{\begin{array}{c} \text{ molar rate of consumption} \\ \text{ of } \ce{Ag2SO4} \text{ in reactor} \end{array}\right\}}{\left\{\begin{array}{c} \text{ molar flow rate of } \ce{Ag2SO4} \\ \text{ entering the reactor} \end{array}\right\}} = \frac{- \mathscr{R}_{ \ce{Ag2SO4} } }{(\dot{M}_{ \ce{Ag2SO4} } )_{3} }\nonumber$

For the conditions given, what is the percentage excess of copper? If the conversion is given by $$C = 0.75$$, what is $$\dot{m}_6/\dot{m}_5$$?

36. In Figure $$\PageIndex{16}$$ we have illustrated a process in which $$\ce{ NaHCO3}$$ is fed to a combined drying and calcining unit in which $$\ce{ Na2} \ce{ CO3}$$, $$\ce{H2O}$$ and $$\ce{ CO2}$$ are produced. The partial pressure of water vapor in the entering air stream is equivalent to 12.7 mm Hg and the system operates at one atmosphere (760 mm Hg). The exit gas leaves at 300$$^{\circ}$$F and a relative humidity of 5%. The $$\ce{NaHCO3}$$ is 70% solids and 30% water (mass basis) when fed to the system. The $$\ce{Na2} \ce{ CO3}$$ leaves with a water content of 3% (mass basis). To improve the character of the solids entering the system, 50% (mass basis) of the dry material is moistened to produce $$\ce{ Na2} \ce{ CO3} \cdot \ce{H2O}$$ and then recycled.

Calculate the following quantities per ton of $$\ce{ Na2} \ce{ CO3}$$ product.

1. mass of wet $$\ce{ NaHCO3}$$ fed at 30% water.
2. mass of water fed to moisten recycle material.
3. cubic feet of dry air fed at 1 atm and 273 K.
4. total volume of exit gas at 1 atm and 300 F.

37. In Figure $$7.4.2$$ we have illustrated an ammonia “converter” in which the unconverted gas is recycled to the reactor. In this problem we consider a process in which the feed stream is a stoichiometric mixture of nitrogen and hydrogen containing 0.2% argon. In the reactor, 10% of the entering reactants (nitrogen and hydrogen) are converted to ammonia which is removed in a condenser. To be explicit, the conversion is given by

${ C}= \text{ Conversion of } \ce{N2} =\frac{- \mathscr{R}_{ \ce{ N2} } }{(\dot{M}_{ \ce{N2} } )_{1} } =0.10\nonumber$

The unconverted gas is recycled to the converter, and in order to avoid the buildup of argon in the system, a purge stream is incorporated in the recycle stream. In this problem we want to determine the fraction of recycle gas that must be purged if the argon entering the reactor is limited to 0.5% on a molar basis.

## Section 7.5

38. Solve Problem 37 using one of the following methods described in Appendix B.

1. The bi-section method
2. The false position method
3. Newton’s method
4. Picard’s method
5. Wegstein’s method

1. The process shown in Figure $$7.2.1$$ should be considered in terms of the principle of stoichiometric skepticism described in Sec. 6.1.1.↩
2. See Figure $$1.3.3$$.↩
3. Here we note that the concentrations of $$\ce{HCl}$$ and $$\ce{C2H2Cl}$$ are zero in Stream #2 and this information has been ignored in the degree-of-freedom analysis. The explanation for this requires a degree-of-freedom analysis of the mixer that is left as an exercise for the student.↩
4. This result could also be obtained directly by enclosing the entire system in a control volume and noting that the rate at which nitrogen enters the system should be equal to the rate at which nitrogen leaves the system.↩
5. See Figure $$1.3.1$$ in Chapter 1.↩
6. In the last century, this problem was resolved by assuming a “basis” of 100 mol/hr for $$(\dot{M}_{\ce{C2H4Cl2}}$$.↩
7. Bradie, B. 2006, A Friendly Introduction to Numerical Analysis, Prentice Hall, Upper Saddle River, New Jersey.↩
8. See Appendix B4 for details.↩
9. Wegstein, J.H. 1958, Accelerating convergences of iterative processes, Comm. ACM 1, 9-13.↩
10. See Appendix B5 for details.↩
11. R. H. Bradford, 1902, The Reactions of the Ziervogel Process and Their Temperature Limits, PhD these, Columbia University↩