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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

An evaporation-crystallization process of the type described in Example \(4.5-2\) is used to obtain solid potassium sulfate from an aqueous solution of this salt. The fresh feed to the process contains 19.6 wt\% \(\mathrm{K}_{2} \mathrm{SO}_{4}\). The wet filter cake consists of solid \(\mathrm{K}_{2} \mathrm{SO}_{4}\) crystals and a \(40.0 \mathrm{wt} \% \mathrm{K}_{2} \mathrm{SO}_{4}\) solution, in a ratio \(10 \mathrm{kg}\) crystals/kg solution. The filtrate, also a \(40.0 \%\) solution, is recycled to join the fresh feed. Of the water fed to the evaporator, 45.0\% is evaporated. The evaporator has a maximum capacity of 175 kg water evaporated/s. (a) Assume the process is operating at maximum capacity. Draw and label a flowchart and do the degree-of-freedom analysis for the overall system, the recycle-fresh feed mixing point, the evaporator, and the crystallizer. Then write in an efficient order (minimizing simultaneous equations) the equations you would solve to determine all unknown stream variables. In each equation, circle the variable for which you would solve, but don't do the calculations. (b) Calculate the maximum production rate of solid \(\mathrm{K}_{2} \mathrm{SO}_{4}\), the rate at which fresh feed must be supplied to achieve this production rate, and the ratio kg recycle/kg fresh feed. (c) Calculate the composition and feed rate of the stream entering the crystallizer if the process is scaled to 75\% of its maximum capacity. (d) The wet filter cake is subjected to another operation after leaving the filter. Suggest what it might be. Also, list what you think the principal operating costs for this process might be. (e) Use an equation-solving computer program to solve the equations derived in Part (a). Verify that you get the same solutions determined in Part (b).

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?

A liquid-phase chemical reaction \(\mathrm{A} \rightarrow \mathrm{B}\) takes place in a well-stirred tank. The concentration of \(\mathrm{A}\) in the feed is \(C_{\mathrm{A} 0}\left(\operatorname{mol} / \mathrm{m}^{3}\right),\) and that in the tank and outlet stream is \(C_{\mathrm{A}}\left(\mathrm{mol} / \mathrm{m}^{3}\right) .\) Neither concentration varies with time. The volume of the tank contents is \(V\left(\mathrm{m}^{3}\right)\) and the volumetric flow rate of the inlet and outlet streams is \(\dot{V}\left(\mathrm{m}^{3} / \mathrm{s}\right)\). The reaction rate (the rate at which \(\mathrm{A}\) is consumed by reaction in the tank) is given by the expression $$r(\text { mol } A \text { consumed } / \mathrm{s})=k V C_{\mathrm{A}}$$ (a) Is this process continuous, batch, or semibatch? Is it transient or steady-state? (b) What would you expect the reactant concentration \(C_{\mathrm{A}}\) to equal if \(k=0\) (no reaction)? What should it approach if \(k \rightarrow \infty\) (infinitely rapid reaction)? (c) Write a differential balance on \(A,\) stating which terms in the general balance equation (accumulation = input + generation - output - consumption) you discarded and why you discarded them. Use the balance to derive the following relation between the inlet and outlet reactant concentrations: $$C_{\mathrm{A}}=\frac{C_{\mathrm{A} 0}}{1+k V / \dot{V}}$$ Verify that this relation predicts the results in Part (b).

L-Serine is an amino acid that often is provided when intravenous feeding solutions are used to maintain the health of a patient. It has a molecular weight of \(105,\) is produced by fermentation and recovered and purified by crystallization at \(10^{\circ} \mathrm{C}\). Yield is enhanced by adding methanol to the system, thereby reducing serine solubility in aqueous solutions. An aqueous serine solution containing 30 wt\% serine and \(70 \%\) water is added along with methanol to a batch crystallizer that is allowed to equilibrate at \(10^{\circ} \mathrm{C}\). The resulting crystals are recovered by filtration; liquid passing through the filter is known as filtrate, and the recovered crystals may be assumed in this problem to be free of adhering filtrate. The crystals contain a mole of water for every mole of serine and are known as a monohydrate. The crystal mass recovered in a particular laboratory run is \(500 \mathrm{g},\) and the filtrate is determined to be \(2.4 \mathrm{wt} \%\) serine, \(48.8 \%\) water, and \(48.8 \%\) methanol. (a) Draw and label a flowchart for the operation and carry out a degree-of- freedom analysis. Determine the ratio of mass of methanol added per unit mass of feed. (b) The laboratory process is to be scaled to produce \(750 \mathrm{kg} / \mathrm{h}\) of product crystals. Determine the required aqueous serine solution rates of aqueous serine solution and methanol.

A liquid mixture contains \(60.0 \mathrm{wt} \%\) ethanol \((\mathrm{E}), 5.0 \mathrm{wt} \%\) of a dissolved solute \((\mathrm{S}),\) and the balance water. A stream of this mixture is fed to a continuous distillation column operating at steady state. Product streams emerge at the top and bottom of the column. The column design calls for the product streams to have equal mass flow rates and for the top stream to contain 90.0 wt\% ethanol and no S. (a) Assume a basis of calculation, draw and fully label a process flowchart, do the degree-of-freedom analysis, and verify that all unknown stream flows and compositions can be calculated. (Don't do any calculations yet.) (b) Calculate (i) the mass fraction of \(S\) in the bottom stream and (ii) the fraction of the ethanol in the feed that leaves in the bottom product stream (i.e., \(\mathrm{kg} \mathrm{E}\) in bottom stream/kg \(\mathrm{E}\) in feed) if the process operates as designed. (c) An analyzer is available to determine the composition of ethanol-water mixtures. The calibration curve for the analyzer is a straight line on a plot on logarithmic axes of mass fraction of ethanol, \(x\) (kg E/kg mixture), versus analyzer reading, \(R\). The line passes through the points \((R=15, x=\) 0.100) and \((R=38, x=0.400)\). Derive an expression for \(x\) as a function of \(R(x=\cdots\) ) based on the calibration, and use it to determine the value of \(R\) that should be obtained if the top product stream from the distillation column is analyzed. (d) Suppose a sample of the top stream is taken and analyzed and the reading obtained is not the one calculated in Part (c). Assume that the calculation in Part (c) is correct and that the plant operator followed the correct procedure in doing the analysis. Give five significantly different possible causes for the deviation between \(R_{\text {measured and }} R_{\text {prediced }}\), including several assumptions made when writing the balances of Part (c). For each one, suggest something that the operator could do to check whether it is in fact the problem.
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