91Ó°ÊÓ

A stream of hot dry nitrogen flows through a process unit that contains liquid acetone. A substantial portion of the acetone vaporizes and is carried off by the nitrogen. The combined gases leave the recovery unit at \(205^{\circ} \mathrm{C}\) and 1.1 bar and enter a condenser in which a portion of the acetone is liquefied. The remaining gas leaves the condenser at \(10^{\circ} \mathrm{C}\) and 40 bar. The partial pressure of acetone in the feed to the condenser is 0.100 bar, and that in the effluent gas from the condenser is 0.379 bar. Assume ideal-gas behavior. (a) Calculate for a basis of \(1 \mathrm{m}^{3}\) of gas fed to the condenser the mass of acetone condensed ( \(\mathrm{kg}\) ) and the volume of gas leaving the condenser \(\left(\mathrm{m}^{3}\right)\) (b) Suppose the volumetric flow rate of the gas leaving the condenser is \(20.0 \mathrm{m}^{3} / \mathrm{h}\). Calculate the rate (kg/h) at which acetone is vaporized in the solvent recovery unit.

Step by step solution

01

Calculate mass of acetone condensed

The number of moles of acetone corresponds to its partial pressure. \ Hence, the moles of acetone in feed \(n_{feed}\) = \(\frac{P_{feed}}{RT}\) where R = 8.314 \(\frac{J}{mol.K}\), T = 205+273 (converted to K), and P = 0.100 (in bar converted to Pa is \(0.100*10^5\)). \ The moles in the effluent \(n_{eff}\) = \(\frac{P_{eff}}{RT}\), where T = 10+273 (as the gas is cooled to this temperature), and P = 0.379 (in bar converted to Pa is \(0.379*10^5\)). \ The moles condensed \(n_{condensed}\) = \(n_{feed} - n_{eff}\). \ The mass of acetone condensed is then given by \(m_{condensed} = n_{condensed} * M_{acetone}\) where \(M_{acetone}=58.08 \, g/mol\) is the molar mass of acetone (convert to kg).

02

Calculate volume of gas leaving the condenser

The volume of gas can be calculated using the ideal gas equation \(V = \frac{nRT}{P}\). The pressure and temperature are the conditions for the gas leaving the condenser (40 bar and \(10^{\circ}\mathrm{C}\)). The number of moles is the sum of nitrogen and acetone moles i.e \(n_{nitrogen} + n_{eff}\) where \(n_{nitrogen} = \frac{P_{nitrogen}}{RT}\). List the partial pressure of nitrogen as \(P_{nitrogen} = 1-P_{feed}-P_{eff}\) (in bar, convert it into Pa by multiplying by \(10^5\)).

03

Calculate the rate at which acetone is vaporized in the solvent recovery unit

Given the volumetric flow rate of the gas leaving the condenser is \(20 \, m^3/h\), the rate of acetone vaporized \(m_{vaporized/h}\) on basis of \(1 \, m^3\) of gas fed is given by \(V_{out} = 20 \, m^3/h\) and \(V_{in} = 1 \, m^3\). Then, the rate of acetone vaporized is \(m_{vaporized} = m_{condensed} * (V_{out}/V_{in})\) per hour, with \(m_{condensed}\) obtained from Step 1.

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Key Concepts

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

Vapor-Liquid Equilibrium

Understanding vapor-liquid equilibrium (VLE) is essential in chemical engineering, particularly when dealing with separation processes like distillation. VLE describes the distribution of a chemical species between the vapor and liquid phases at a particular temperature and pressure. It's a state where the rate of evaporation of a liquid equals the rate of condensation of its vapor phase.

In the given exercise, the system reaches equilibrium in the condenser, where acetone vapor is partially condensed when cooled. The amount of acetone that remains in the vapor phase is related to its vapor pressure, which depends on the temperature and the presence of other components, such as nitrogen. The equilibrium can be shifted by changing the temperature, as seen when the gas is cooled from 205°C to 10°C, leading to more acetone being condensed.

Ideal Gas Law

The ideal gas law is a fundamental equation in chemical engineering that provides a close approximation to the behavior of many gases under certain conditions. It is represented as PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the universal gas constant, and T is the absolute temperature in Kelvin.

In our exercise, the ideal gas law is used twice: first, to find the moles of acetone before and after entering the condenser, and second, to find the volume of the gas leaving the condenser. This law assumes the gas molecules do not interact and occupy no volume, which simplifies calculations for systems such as the one in this exercise. It's important to ensure that pressures are consistently used in SI units (Pascals) and temperatures in Kelvin for accurate calculations.

Partial Pressure

Partial pressure is a concept that describes the pressure that a single gas component in a mixture would exert if it alone occupied the entire volume of the mixture. The sum of the partial pressures of all the gases in the mixture equals the total pressure exerted by the gas mixture. This conforms to Dalton's Law of Partial Pressures.

In this example, the partial pressure of acetone is provided for both the feed to the condenser and the effluent gas. By knowing these pressures, and assuming ideal gas behavior, we can calculate the moles of acetone in both stages using the ideal gas law. It is crucial that when we look for the number of moles of nitrogen, we consider the total pressure and subtract the partial pressures of acetone from it to avoid any overestimation.

Mass and Energy Balance

Mass and energy balance principles are used to track and quantify the input and output streams of mass and energy in any process system. For a steady-state process, the mass entering a system must equal the mass leaving it. The same concept applies to energy. These balances are vital in designing and scaling up chemical processes.

In the exercise at hand, the mass balance is focused on the amount of acetone that is carried by nitrogen before and after the condenser. By calculating the mass of acetone in the input and output streams and the volume of gas leaving the condenser, students learn to apply these principles practically. Additionally, the rate at which acetone is vaporized is determined by comparing the volumetric flow rates pre- and post-condenser, which ties back into maintaining a mass balance on the acetone.