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A stream of liquid \(n\) -pentane flows at a rate of \(50.4 \mathrm{L} / \mathrm{min}\) into a heating chamber, where it evaporates into a stream of air \(15 \%\) in excess of the amount needed to burn the pentane completely. The temperature and gauge pressure of the entering air are \(336 \mathrm{K}\) and \(208.6 \mathrm{kPa}\). The pentane-laden heated gas flows into a combustion furnace in which a fraction of the pentane is burned. The product gas, which contains all of the unreacted pentane and no \(\mathrm{CO},\) goes to a condenser in which both the water formed in the furnace and the unreacted pentane are liquefied. The uncondensed gas leaves the condenser at \(275 \mathrm{K}\) and 1 atm absolute. The liquid condensate is separated into its components, and the flow rate of the pentane is measured and found to be \(3.175 \mathrm{kg} / \mathrm{min}\). (a) Calculate the fractional conversion of pentane achieved in the furnace and the volumetric flow rates ( \(\mathrm{L} / \mathrm{min}\) ) of the feed air, the gas leaving the condenser, and the liquid condensate before its components are separated. (b) Sketch the apparatus that could have been used to separate the pentane and water in the condensate. Hint: Remember that pentane is a hydrocarbon and recall what is said about oil (hydrocarbons) and water.

Step by step solution

01

Calculation of the mass flow rate of pentane

The flow rate of pentane entering the heating chamber is given as 50.4 L/min. The density of \(n\)-pentane at room temperature is about 0.626 g/mL. Use this and convert the volume flow rate to mass flow rate: \( \text{mass flow rate of pentane} = 50.4\,L/min \times 1000\,mL/L \times 0.626\,g/mL \times 1\,kg/1000\,g = 31.55\,kg/min \)

02

Calculation of the fractional conversion of pentane

In this step, calculate the fraction of the initial pentane that gets converted in the furnace. It is found by comparing the mass flow rate of pentane before and after the combustion furnace. Fractional conversion \( X \) can be calculated as \( X = \frac{{\text{Initial flow rate} - \text{Final flow rate}}}{\text{Initial flow rate}} \). Plugging the known values: \( X = \frac{{31.55\,kg/min - 3.175\,kg/min}}{31.55\,kg/min} = 0.899 = 89.9 \% \)

03

Calculation of the flow rates

First calculate the flow rate of air. Using the ideal gas law \(PV = nRT\), where \(P\) is the pressure, \(V\) is volume, \(n\) the number of moles, \(R\) the gas constant and \(T\) the temperature in Kelvin, the volume flow rate can be calculated as \( V_{air} = \frac{{nRT}}{P} \). From the combustion reaction of \(n\)-Pentane (C5H12 + 15/2 O2 -> 5CO2 + 6H2O), the molar ratio of O2 to \(n\)-Pentane is 15/2 : 1. Thus, the flow rate of the feed air is 15% more the required amount, i.e., \(n_{O2} = 1.15 \times \frac{15}{2} \times \text{flow rate of \(n\)-pentane} \). The flow rate of the gas leaving the condenser can be calculated too using the ideal gas law under the new conditions. The flow rate of the liquid condensate is equivalent to the unreacted pentane and the water formed, i.e., 3.175 kg/min + 6 moles of water per mole of \(n\)-pentane reacted \( \times (1 - X) \times \text{flow rate of \(n\)-pentane} \times \text{Molar mass of water} \).

04

Sketch of the separation apparatus

To separate pentane and water in the condensate, a separating funnel can be used given the fact that pentane is insoluble in water. The mixture of pentane and water is added to the separating funnel which is then shaken gently to ensure complete mixing. Once it's left to settle, two layers will form with pentane being the upper layer due to its lesser density. The tap of the funnel can be opened to allow water (bottom layer) to run out, discharging the liquid until there's only pentane left in the funnel. After ensuring water has completely drained out, the remaining pentane can now be collected.

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Key Concepts

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

Fractional Conversion Calculation

Understanding the fractional conversion calculation in a chemical process is essential for evaluating the efficiency of a reaction. It measures the fraction of a reactant that has been converted into a product during a chemical process. In the context of the exercise provided, the fractional conversion, denoted as \(X\), of pentane in the heating chamber is determined by the decrease in its mass flow rate after passing through the combustion furnace.

To calculate the fractional conversion of pentane, we apply the formula: \(X = \frac{{\text{Initial flow rate} - \text{Final flow rate}}}{{\text{Initial flow rate}}}\). In the given problem, the initial mass flow rate of pentane was 31.55 kg/min and the final mass flow rate after the process was 3.175 kg/min. Substituting these values gives a fractional conversion of 89.9%, indicating a significant portion of pentane was converted during the process.

The concept of fractional conversion is a cornerstone in chemical engineering as it provides insight into the progress of reactions and the efficiency of the process, guiding the optimization of operational conditions for better performance.

Volumetric Flow Rate

The volumetric flow rate is a fundamental parameter in chemical engineering, defining the volume of fluid that passes through a given point per unit time. It's often measured in liters per minute (L/min) or cubic meters per second (m^3/s). This value becomes crucial when designing systems for chemical processing, where maintaining proper flow rates is necessary to ensure desired reaction times and overall process efficiency.

In the exercise, we encounter calculations of volumetric flow rates for different stages in the process. For instance, after finding the mass flow rate, the volume flow rate of the air fed into the reaction can be deduced from the ideal gas law, demonstrating how temperature, pressure, and the stoichiometric requirements of the reaction dictate the volume required for complete combustion. Subsequent calculations based on this parameter facilitate a deeper comprehension of the gas dynamics involved in the separation process.

Mass Flow Rate

Mass flow rate is the mass of a substance passing through a given surface per unit time. It's a key concept in chemical engineering, particularly when assessing the material balance around a process. In the context of this problem, the mass flow rate is critical for determining the fractional conversion of pentane. The given mass flow rate of the pentane inlet is 31.55 kg/min, found by converting the volumetric flow rate and considering the density of the pentane.

To enhance comprehension of mass flow rate, it's important to recognize it as a measure of the 'amount' of mass being transferred, distinct from the volumetric flow, which considers only volume. Grasping both concepts allows students to seamlessly switch between different units based on the phase of the compound (liquid or gas), and to understand the processes of combustion and separation more deeply.

Separation Apparatus Design

Separation apparatus design is vital in chemical engineering for the purification and isolation of components from a mixture. The design must consider the properties of the components, such as density, solubility, and boiling points. In the problem, the separation of pentane from water is required, which is achieved using a simple yet effective device: a separating funnel.

This apparatus leverages the difference in density between pentane and water. Pentane, being a hydrocarbon, is less dense and insoluble in water, forming a distinct upper layer when mixed. By gently mixing and allowing the mixture to settle, the two liquids separate out into their respective layers. The separating funnel is then used to drain the denser water from the bottom, leaving behind the lighter pentane. This represents a practical application of separation principles — a fundamental skill in the development and operation of chemical processes.