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Chapter 4: Problem 27

A paint mixture containing \(25.0 \%\) of a pigment and the balance binders (which help the pigment stick to the surface) and solvents (which ensure that the paint stays in liquid form) sells for 18.00 dollar/kg, and a mixture containing 12.0\% sells for 10.00 dollar /kg. (a) If a paint retailer produces a blend containing \(17.0 \%\) pigment, for how much (S/kg) should it be sold to yield a 10\% profit? (b) Paint manufacturers have begun to market "low VOC" paint as a more environmentally friendly product. What are VOCs? List some ways in which paint products can be altered to lower the VOC content.

Short Answer

Expert verified
(a) The price per kg for the 17% pigment blend should be calculated in detail following Step 2. (b) VOCs are Volatile Organic Compounds. They can be lowered in paint products by using water-based paints, incorporating natural solvents, or using low VOC pigments and binders.

Step by step solution

01

Understand the mixture ratios

First, format the problem using systems of equations. Let \( x \) be the mass of the 25% pigment mixture used and \( y \) be the mass of the 12% pigment mixture used. Then the mass of the new mixture is \( x + y \), and we know that its pigment content is 17%. Therefore, we can write the equation: \( 0.25x + 0.12y = 0.17(x + y) \).
02

Determine Selling Price

We also know from the problem that the new mixture should be sold for a 10% profit. Note that the cost of the new mixture will be the combined cost of the two original mixtures used. This implies the following equation: \(18x + 10y = 1.1S(x+y)\), where S is the new selling price per kg. From the equation in Step 1, we can solve for S after the equations are simplified. The solution to S will be the desired selling price per kg.
03

Understand VOCs

Part (b) asks about VOCs, which stands for Volatile Organic Compounds. VOCs are organic chemicals that have a high evaporation rate at ordinary room temperature. They are used in paint to keep it in a liquid state but are known to contribute to indoor air pollution and have various health implications. Ways to lower VOC content in paint can include: using water-based paints instead of solvent-based ones, incorporating natural solvents, or using low VOC pigments and binders.

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Key Concepts

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

Mixture Ratios
Mixture ratios are crucial in chemical process engineering, especially when creating products like paints that must meet specific criteria. In the given exercise, the goal is to blend two different paint mixtures to achieve a new mixture with a specific pigment concentration. Understanding the formulation involves setting up a system of equations to determine the correct mixture proportions.

By defining variables for the masses of the original paint mixtures, we establish the key equation: \[ 0.25x + 0.12y = 0.17(x + y) \]This equation ensures that the final mixture has the desired 17% pigment content. Here, \( x \) represents the mass of the 25% pigment mixture, while \( y \) is the mass of the 12% pigment mixture. Solving this equation provides the precise mixture ratios necessary to maintain the consistency and quality expected.

  • Mixture balance ensures that different components contribute correctly to the overall properties.
  • Accuracy in mixing is essential to achieving desired product qualities, like consistency, texture, and color uniformity.
VOC Reduction
Volatile Organic Compounds (VOCs) are a significant concern in chemical products like paints. VOCs are chemicals that easily evaporate at room temperature and are commonly used as solvents to keep paints liquid. However, they are known for their adverse environmental and health impacts.

Understanding and reducing VOCs involves:
  • Choosing water-based paints, which typically use fewer VOCs compared to solvent-based paints.
  • Incorporating natural solvents, which are less harmful and can effectively reduce VOC levels.
  • Opting for low-VOC pigments and binders to further cut down on emissions.
These strategies not only contribute to environmental conservation but also promote healthier indoor air quality. As consumers become more environmentally conscious, the demand for low VOC products increases, influencing production methods in chemical engineering.
Profit Calculation
Profit calculation in chemical process engineering involves determining the pricing of a product to ensure profitability after considering production costs. In this exercise, we are tasked with finding the selling price for a new paint blend that incorporates a desired profit margin.

The mixture's cost is influenced by its components. For the given problem, the exercise describes using two existing paint mixtures to create a new product. To ensure a profit, the selling price \( S \) must be higher than the production cost, which includes the costs of both mixtures.The following equation is used to calculate the necessary selling price:\[ 18x + 10y = 1.1S(x + y) \]This equation reflects a 10% desired profit margin above the total cost. Solving for \( S \) gives us the new selling price per kg.

  • Profit margin: The percentage that should be added to product costs to ensure a profitable return.
  • Cost analysis: Assessing the costs of raw materials and processing to set a suitable price.
Proper profit calculations ensure the financial sustainability of production processes and help companies achieve their business objectives.

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Most popular questions from this chapter

A fuel oil is fed to a furnace and burned with \(25 \%\) excess air. The oil contains \(87.0 \mathrm{wt} \% \mathrm{C}, 10.0 \% \mathrm{H},\) and 3.0\% S. Analysis of the furnace exhaust gas shows only \(\mathrm{N}_{2}, \mathrm{O}_{2}, \mathrm{CO}_{2}, \mathrm{SO}_{2},\) and \(\mathrm{H}_{2} \mathrm{O}\). The sulfur dioxide emission rate is to be controlled by passing the exhaust gas through a scrubber, in which most of the \(\mathrm{SO}_{2}\) is absorbed in an alkaline solution. The gases leaving the scrubber (all of the \(\mathrm{N}_{2}, \mathrm{O}_{2},\) and \(\mathrm{CO}_{2}\), and some of the \(\mathrm{H}_{2} \mathrm{O}\) and \(\mathrm{SO}_{2}\) entering the unit) pass out to a stack. The scrubber has a limited capacity, however, so that a fraction of the furnace exhaust gas must be bypassed directly to the stack. At one point during the operation of the process, the scrubber removes \(90 \%\) of the \(\mathrm{SO}_{2}\) in the gas fed to it, and the combined stack gas contains 612.5 ppm (parts per million) \(\mathrm{SO}_{2}\) on a dry basis; that is, every million moles of dry stack gas contains 612.5 moles of \(\mathrm{SO}_{2}\). Calculate the fraction of the exhaust bypassing the scrubber at this moment.

Natural gas containing a mixture of methane, ethane, propane, and butane is burned in a furnace with excess air. (a) One hundred kmol/h of a gas containing 94.4 mole\% methane, 3.40\% ethane, 0.60\% propane, and \(0.50 \%\) butane is to be burned with \(17 \%\) excess air. Calculate the required molar flow rate of the air. (b) Let Derive an expression for \(\dot{n}_{\mathrm{a}}\) in terms of the other variables. Check your formula with the results of Part (a). (c) Suppose the feed rate and composition of the fuel gas are subject to periodic variations, and a process control computer is to be used to adjust the flow rate of air to maintain a constant percentage excess. A calibrated electronic flowmeter in the fuel gas line transmits a signal \(R_{\mathrm{f}}\) that is directly proportional to the flow rate \(\left(\dot{n}_{\mathrm{f}}=\alpha R_{\mathrm{f}}\right),\) with a flow rate of \(75.0 \mathrm{kmol} / \mathrm{h}\) yielding a signal \(R_{f}=60 .\) The fuel gas composition is obtained with an on-line gas chromatograph. A sample of the gas is injected into the gas chromatograph (GC), and signals \(A_{1}, A_{2}, A_{3},\) and \(A_{4},\) which are directly proportional to the moles of methane, ethane, propane, and butane, respectively, in the sample, are transmitted. (Assume the same proportionality constant for all species.) The control computer processes these data to determine the required air flow rate and then sends a signal \(R_{\mathrm{a}}\) to a control valve in the air line. The relationship between \(R_{\mathrm{a}}\) and the resulting air flow rate, \(\dot{n}_{\mathrm{a}},\) is another direct proportionality, with a signal \(R_{\mathrm{a}}=25\) leading to an air flow rate of \(550 \mathrm{kmol} / \mathrm{h}\). Write a spreadsheet or computer program to perform the following tasks: (i) Take as input the desired percentage excess and values of \(R_{\mathrm{f}}, A_{1}, A_{2}, A_{3},\) and \(A_{4}\) (ii) Calculate and print out \(\dot{n}_{\mathrm{f}}, x_{1}, x_{2}, x_{3}, x_{4}, \dot{n}_{\mathrm{a}},\) and \(R_{\mathrm{a}}\) Test your program on the data given below, assuming that \(15 \%\) excess air is required in all cases. Then explore the effects of variations in \(P_{\mathrm{xs}}\) and \(R_{\mathrm{f}}\) on \(\dot{n}_{\mathrm{a}}\) for the values of \(A_{1}-A_{4}\) given on the third line of the data table. Briefly explain your results. (d) Finally, suppose that when the system is operating as described, stack gas analysis indicates that the air feed rate is consistently too high to achieve the specified percentage excess. Give several possible explanations.

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 gas-phase reaction between methanol and acetic acid to form methyl acetate and water takes place in a batch reactor. When the reaction mixture comes to equilibrium, the mole fractions of the four reactive species are related by the reaction equilibrium constant $$K_{y}=\frac{y_{C} y_{D}}{y_{A} y_{B}}=4.87$$ (a) Suppose the feed to the reactor consists of \(n_{\mathrm{A} 0}, n_{\mathrm{B} 0}, n_{\mathrm{C} 0}, n_{\mathrm{D} 0},\) and \(n_{10}\) gram-moles of \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D},\) and an inert gas, I, respectively. Let \(\xi\) be the extent of reaction. Write expressions for the gram-moles of each reactive species in the final product, \(n_{\mathrm{A}}(\xi), n_{\mathrm{B}}(\xi), n_{\mathrm{C}}(\xi),\) and \(n_{\mathrm{D}}(\xi) .\) Then use these expressions and the given equilibrium constant to derive an equation for \(\xi_{c}\), the equilibrium extent of reaction, in terms of \(\left.n_{\mathrm{A} 0}, \ldots, n_{10} . \text { (see Example } 4.6-2 .\right)\) (b) If the feed to the reactor contains equimolar quantities of methanol and acetic acid and no other species, calculate the equilibrium fractional conversion. (c) It is desired to produce 70 mol of methyl acetate starting with 75 mol of methanol. If the reaction proceeds to equilibrium, how much acetic acid must be fed? What is the composition of the final product? (d) Suppose it is important to reduce the concentration of methanol by making its conversion at equilibrium as high as possible, say 99\%. Again assuming the feed to the reactor contains only methanol and acetic acid and that it is desired to produce 70 mol of methyl acetate, determine the extent of reaction and quantities of methanol and acetic acid that must be fed to the reactor. (e) If you wanted to carry out the process of Part (b) or (c) commercially, what would you need to know besides the equilibrium composition to determine whether the process would be profitable? (List several things.)

Oxygen consumed by a living organism in aerobic reactions is used in adding mass to the organism and/or the production of chemicals and carbon dioxide. since we may not know the molecular compositions of all species in such a reaction, it is common to define the ratio of moles of \(\mathrm{CO}_{2}\) produced per mole of \(\mathrm{O}_{2}\) consumed as the respiratory quotient, \(R Q,\) where $$R Q=\frac{n_{\mathrm{CO}_{2}}}{n_{\mathrm{O}_{2}}}\left(\text { or } \frac{\dot{n}_{\mathrm{CO}_{2}}}{\dot{n}_{\mathrm{O}_{2}}}\right)$$ since it generally is impossible to predict values of \(R Q\), they must be determined from operating data. Mammalian cells are used in a bioreactor to convert glucose to glutamic acid by the reaction $$\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}+a \mathrm{NH}_{3}+b \mathrm{O}_{2} \rightarrow p \mathrm{C}_{5} \mathrm{H}_{9} \mathrm{NO}_{4}+q \mathrm{CO}_{2}+r \mathrm{H}_{2} \mathrm{O}$$ The feed to the bioreactor comprises \(1.00 \times 10^{2} \mathrm{mol} \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} / \mathrm{day}, 1.20 \times 10^{2} \mathrm{mol} \mathrm{NH}_{3} / \mathrm{day},\) and \(1.10 \times\) \(10^{2}\) mol \(\mathrm{O}_{2} /\) day. Data on the system show that \(R Q=0.45 \mathrm{mol} \mathrm{CO}_{2}\) produced/mol \(\mathrm{O}_{2}\) consumed. (a) Determine the five stoichiometric coefficients and the limiting reactant. (b) Assuming that the limiting reactant is consumed completely, calculate the molar and mass flow rates of all species leaving the reactor and the fractional conversions of the non-limiting reactants.
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