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Chapter 4: Problem 5

Draw and label the given streams and derive expressions for the indicated quantities in terms of labeled variables. The solution of Part (a) is given as an illustration. (a) A continuous stream contains 40.0 mole\% benzene and the balance toluene. Write expressions for the molar and mass flow rates of benzene, \(\dot{n}_{\mathrm{B}}\left(\operatorname{mol} \mathrm{C}_{6} \mathrm{H}_{6} / \mathrm{s}\right)\) and \(\dot{m}_{\mathrm{B}}\left(\mathrm{kg} \mathrm{C}_{6} \mathrm{H}_{6} / \mathrm{s}\right),\) in terms of the total molar flow rate of the stream, \(\dot{n}(\mathrm{mol} / \mathrm{s})\) (b) The feed to a batch process contains equimolar quantities of nitrogen and methane. Write an expression for the kilograms of nitrogen in terms of the total moles \(n(\) mol) of this mixture. (c) A stream containing ethane, propane, and butane has a mass flow rate of \(100.0 \mathrm{g} / \mathrm{s}\). Write an expression for the molar flow rate of ethane, \(\dot{n}_{\mathrm{E}}\left(\text { Ib-mole } \mathrm{C}_{2} \mathrm{H}_{6} / \mathrm{h}\right)\), in terms of the mass fraction of this species, \(x_{\mathrm{E}}\). (d) A continuous stream of humid air contains water vapor and dry air, the latter containing approximately 21 mole \(\% \mathrm{O}_{2}\) and \(79 \% \mathrm{N}_{2}\). Write expressions for the molar flow rate of \(\mathrm{O}_{2}\) and for the mole fractions of \(\mathrm{H}_{2} \mathrm{O}\) and \(\mathrm{O}_{2}\) in the gas in terms of \(\dot{n}_{1}\left(\mathrm{lb}-\mathrm{mole} \mathrm{H}_{2} \mathrm{O} / \mathrm{s}\right)\) and \(\dot{n}_{2}(\text { lb- mole dry air/s })\) (e) The product from a batch reactor contains \(\mathrm{NO}, \mathrm{NO}_{2},\) and \(\mathrm{N}_{2} \mathrm{O}_{4} .\) The mole fraction of \(\mathrm{NO}\) is 0.400. Write an expression for the gram-moles of \(\mathrm{N}_{2} \mathrm{O}_{4}\) in terms of \(n(\mathrm{mol}\) mixture) and \(y_{\mathrm{NO}_{2}}\left(\operatorname{mol} \mathrm{NO}_{2} / \mathrm{mol}\right)\)

Short Answer

Expert verified
(b) \(m_{N} = 0.5 * n * 28\), (c) \(\dot{n}_{E} = x_{E} * \dot{m}/30\), (d) \(\dot{n}_{O2} = 0.21 * \dot{n}_{2}\), \(y_{O2} = \dot{n}_{O2}/(\dot{n}_{1} + \dot{n}_{2})\), \(y_{H2O} = \dot{n}_{1}/(\dot{n}_{1} + \dot{n}_{2})\), (e) \(n_{N2O4} = n*0.60 - n*y_{NO2}\)

Step by step solution

01

Part (b) Solution

The mixture contains equal molar quantities of Nitrogen and Methane. This means half of the total moles (\(n\)) would be Nitrogen. Then, the mass of Nitrogen can be calculated by multiplying moles of Nitrogen with its molecular weight. Nitrogen (\(N_2\)) has a molecular weight of approximately 28. Therefore, Mass \(m_{N}\) of Nitrogen is given by: \(m_{N} = 0.5 * n * 28\).
02

Part (c) Solution

Given the total mass flow rate \(\dot{m}\) and mass fraction \(x_{E}\) of ethane, the mass flow rate \(\dot{m}_{E}\) of ethane is \(x_{E} * \dot{m}\). To convert from mass flow rate to molar flow rate, we divide by the molecular weight of ethane (approximately 30). Therefore, the molar flow rate \(\dot{n}_{E}\) of ethane is given by: \(\dot{n}_{E} = x_{E} * \dot{m}/30\).
03

Part (d) Solution

The mole flow rate of \(O_2\) (\(\dot{n}_{O2}\)) is composed of the mole fraction of \(O_2\) in dry air, which is approximately \(0.21 * \dot{n}_{2}\). The mole fractions of \(H_2O\) and \(O_2\) are calculated by dividing their flow rates by the total flow rate. So, for \(O_2\), \(y_{O2} = \dot{n}_{O2}/(\dot{n}_{1} + \dot{n}_{2})\) and for \(H_2O\), \(y_{H2O} = \dot{n}_{1}/(\dot{n}_{1} + \dot{n}_{2})\).
04

Part (e) Solution

The mole fraction of \(NO\) is 0.40, so the combined mole fractions of \(NO_2\) and \(N_2O_4\) is \(1 - y_{NO} = 0.60\). The gram-moles of \(N_2O_4\) can be expressed in terms of the total moles \(n\) of the mixture and the mole fraction \(y_{NO2}\) of \(NO_2\). After subtraction of the moles of \(NO\) and \(NO_2\), we find: \(n_{N2O4} = n*0.60 - n*y_{NO2}\).

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Key Concepts

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

Mole Fraction
Mole fraction, often denoted by the symbol \( y \) or \( x \), represents the proportion of moles of a particular substance to the total moles in a mixture. It is a dimensionless quantity and is a way to express concentration in chemistry. To calculate a mole fraction, you divide the number of moles of the component of interest by the total moles in the mixture.

For example, if a solution has 4 moles of sodium chloride (NaCl) and 1 mole of water (H2O), the mole fraction of NaCl would be \( \frac{4}{4+1} = 0.8 \) and for water, it would be \( \frac{1}{4+1} = 0.2 \). Mole fractions are particularly useful for understanding vapor-liquid equilibrium, calculating partial pressures in gas mixtures, and in the stoichiometry of chemical reactions.
Molar Flow Rate
The molar flow rate refers to the amount of substance that passes through a given surface per unit time, measured in moles per second \( (\frac{mol}{s}) \) or moles per hour \( (\frac{mol}{h}) \). It is an essential concept in chemical engineering because it allows engineers to quantify and analyze the transformation of substances in chemical processes.

To calculate the molar flow rate \( (\dot{n}) \), you may need information like the total number of moles passing through a point and the time taken. In a process where different chemical species are present, the molar flow rate can be used to find the rate at which each individual species is traveling through the system. For instance, in an equation such as \( \dot{n}_{\mathrm{B}} = 0.40 \times \dot{n} \), if the total molar flow rate \( \dot{n} \) is given, one can easily find the molar flow rate of benzene \( (\dot{n}_{\mathrm{B}}) \) by multiplying by the mole fraction of benzene (0.40 in this case).
Mass Flow Rate
Mass flow rate, symbolized by \( \dot{m} \), is the mass of a substance that passes through a given surface per unit time. It is indicated in units such as kilograms per second \( (\frac{kg}{s}) \) or grams per hour \( (\frac{g}{h}) \). This concept is pivotal for examining the conservation of mass in a chemical process and for designing equipment that can handle the required mass throughput.

The relation between mass flow rate and molar flow rate is characterized by the molecular weight (or molar mass) of the substance. Since mass is equivalent to the number of moles multiplied by the molecular weight, you can find the mass flow rate of a particular component in a mixture by multiplying its molar flow rate by its molecular weight. For example, if ethane\( (C_{2}H_{6}) \) has a molar flow rate \( \dot{n}_{\mathrm{E}} \), the mass flow rate \( \dot{m}_{\mathrm{E}} \) would be \( \dot{m}_{\mathrm{E}} = \dot{n}_{\mathrm{E}} \times \text{molecular weight of ethane} \).

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Most popular questions from this chapter

A Claus plant converts gaseous sulfur compounds to elemental sulfur, thereby eliminating emission of sulfur into the atmosphere. The process can be especially important in the gasification of coal, which contains significant amounts of sulfur that is converted to \(\mathrm{H}_{2}\) S during gasification. In the Claus process, the \(\mathrm{H}_{2}\) S-rich product gas recovered from an acid-gas removal system following the gasifier is split, with one-third going to a furnace where the hydrogen sulfide is burned at 1 atm with a stoichiometric amount of air to form SO \(_{2}\). $$\mathrm{H}_{2} \mathrm{S}+\frac{3}{2} \mathrm{O}_{2} \rightarrow \mathrm{SO}_{2}+\mathrm{H}_{2} \mathrm{O}$$ The hot gases leave the furnace and are cooled prior to being mixed with the remainder of the \(\mathrm{H}_{2}\) S-rich gases. The mixed gas is then fed to a catalytic reactor where hydrogen sulfide and \(\mathrm{SO}_{2}\) react to form elemental sulfur. $$2 \mathrm{H}_{2} \mathrm{S}+\mathrm{SO}_{2} \rightarrow 2 \mathrm{H}_{2} \mathrm{O}+3 \mathrm{S}$$ The coal available to the gasification process is 0.6 wt\% sulfur, and you may assume that all of the sulfur is converted to \(\mathrm{H}_{2} \mathrm{S}\), which is then fed to the Claus plant. (a) Estimate the feed rate of air to the Claus plant in \(\mathrm{kg} / \mathrm{kg}\) coal. (b) While the removal of sulfur emissions to the atmosphere is environmentally beneficial, identify an environmental concern that still must be addressed with the products from the Claus plant.

The fresh feed to an ammonia production process contains nitrogen and hydrogen in stoichiometric proportion, along with an inert gas (I). The feed is combined with a recycle stream containing the same three species, and the combined stream is fed to a reactor in which a low single-pass conversion of nitrogen is achieved. The reactor effluent flows to a condenser. A liquid stream containing essentially all of the ammonia formed in the reactor and a gas stream containing all the inerts and the unreacted nitrogen and hydrogen leave the condenser. The gas stream is split into two fractions with the same composition: one is removed from the process as a purge stream, and the other is the recycle stream combined with the fresh feed. In every stream containing nitrogen and hydrogen, the two species are in stoichiometric proportion. (a) Let \(x_{10}\) be the mole fraction of inerts in the fresh feed, \(f_{\mathrm{sp}}\) the single-pass conversion of nitrogen (and of hydrogen) in the reactor, and \(y_{p}\) the fraction of the gas leaving the condenser that is purged (mol purged/mol total). Taking a basis of 1 mol fresh feed, draw and fully label a process flowchart, incorporating \(x_{10}, f_{\mathrm{sp}},\) and \(y_{\mathrm{p}}\) in the labeling to the greatest possible extent. Then, assuming that the values of these three variables are given, write a set of equations for the total moles fed to the reactor \(\left(n_{\mathrm{r}}\right),\) moles of ammonia produced \(\left(n_{\mathrm{p}}\right),\) and overall nitrogen conversion \(\left(f_{\mathrm{ov}}\right) .\) Each equation should involve only one unknown variable, which should be circled. (b) Solve the equations of Part (a) for \(x_{10}=0.01, f_{\mathrm{sp}}=0.20,\) and \(y_{\mathrm{p}}=0.10\) (c) Briefly explain in your own words the reasons for including (i) the recycle stream and (ii) the purge stream in the process design. (d) Prepare a spreadsheet to perform the calculations of Part (a) for given values of \(x_{10}, f_{\mathrm{sp}},\) and \(y_{\mathrm{p}} .\) Test it with the values in Part (b). Then in successive rows of the spreadsheet, vary each of the three input variables two or three times, holding the other two constant. The first six columns and first five rows of the spreadsheet should appear as follows:Summarize the effects on ammonia production \(\left(n_{\mathrm{P}}\right)\) and reactor throughput \(\left(n_{\mathrm{r}}\right)\) of changing each of the three input variables.

Methane reacts with chlorine to produce methyl chloride and hydrogen chloride. Once formed, the methyl chloride may undergo further chlorination to form methylene chloride ( \(\mathrm{CH}_{2} \mathrm{Cl}_{2}\) ), chloroform, and carbon tetrachloride. A methyl chloride production process consists of a reactor, a condenser, a distillation column, and an absorption column. A gas stream containing 80.0 mole \(\%\) methane and the balance chlorine is fed to the reactor. In the reactor a single-pass chlorine conversion of essentially \(100 \%\) is attained, the mole ratio of methyl chloride to methylene chloride in the product is \(5: 1,\) and negligible amounts of chloroform and carbon tetrachloride are formed. The product stream flows to the condenser. Two streams emerge from the condenser: the liquid condensate, which contains essentially all of the methyl chloride and methylene chloride in the reactor effluent, and a gas containing the methane and hydrogen chloride. The condensate goes to the distillation column in which the two component species are separated. The gas leaving the condenser flows to the absorption column where it contacts an aqueous solution. The solution absorbs essentially all of the HCl and none of the \(\mathrm{CH}_{4}\) in the feed. The liquid leaving the absorber is pumped elsewhere in the plant for further processing, and the methane is recycled to join the fresh feed to the process (a mixture of methane and chlorine). The combined stream is the feed to the reactor. (a) Choose a quantity of the reactor feed as a basis of calculation, draw and label a flowchart, and determine the degrees of freedom for the overall process and each single unit and stream mixing point. Then write in order the equations you would use to calculate the molar flow rate and molar composition of the fresh feed, the rate at which HCI must be removed in the absorber, the methyl chloride production rate, and the molar flow rate of the recycle stream. Do no calculations. (b) Calculate the quantities specified in Part (a), either manually or with an equation-solving program. (c) What molar flow rates and compositions of the fresh feed and the recycle stream are required to achieve a methyl chloride production rate of \(1000 \mathrm{kg} / \mathrm{h} ?\)

A mixture of propane and butane is burned with pure oxygen. The combustion products contain 47.4 mole \(\% \mathrm{H}_{2} \mathrm{O}\). After all the water is removed from the products, the residual gas contains 69.4 mole \(\% \mathrm{CO}_{2}\) and the balance \(\mathrm{O}_{2}\) (a) What is the mole percent of propane in the fuel? (b) It now turns out that the fuel mixture may contain not only propane and butane but also other hydrocarbons. All that is certain is that there is no oxygen in the fuel. Use atomic balances to calculate the elemental molar composition of the fuel from the given combustion product analysis (i.e., what mole percent is \(C\) and what percent is \(\mathrm{H}\) ). Prove that your solution is consistent with the result of Part (a).

The reaction between ethylene and hydrogen bromide to form ethyl bromide is carried out in a continuous reactor. The product stream is analyzed and found to contain 51.7 mole \(\% \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br}\) and 17.3\% HBr. The feed to the reactor contains only ethylene and hydrogen bromide. Calculate the fractional conversion of the limiting reactant and the percentage by which the other reactant is in excess. If the molar flow rate of the feed stream is \(165 \mathrm{mol} / \mathrm{s}\), what is the extent of reaction?
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