91Ó°ÊÓ

A stream consisting of 44.6 mole \(\%\) benzene and \(55.4 \%\) toluene is fed at a constant rate to a process unit that produces two product streams, one a vapor and the other a liquid. The vapor flow rate is initially zero and asymptotically approaches half of the molar flow rate of the feed stream. Throughout this entire period, no material accumulates in the unit. When the vapor flow rate has become constant, the liquid is analyzed and found to be 28.0 mole\% benzene. (a) Sketch a plot of liquid and vapor flow rates versus time from startup to when the flow rates become constant. (b) Is this process batch or continuous? Is it transient or steady-state before the vapor flow rate reaches its asymptotic limit? What about after it becomes constant? (c) For a feed rate of 100 mol/min, draw and fully label a flowchart for the process after the vapor flow rate has reached its limiting value, and then use balances to calculate the molar flow rate of the liquid and the composition of the vapor in mole fractions.

Step by step solution

01

Identify the nature of the process

The process starts with a single input feed and divides it into two product streams. One stream is vapor and the other is liquid. There are no losses, conversion or accumulation which meets the criteria of a splitting process.

02

Understanding of batch & continuous process

In a batch process, material is processed in definite batches and at the end of processing, the entire batch is removed from the equipment. Also, batch processing typically involves time variation. On the other hand, in a continuous process, the feed and products are continuously added and taken from the unit respectively. Here, since the operation is performed at a constant feed rate, the process is continuous.

03

Understanding of transient & steady-state process

In a transient state, the process variables change with time until the steady state is reached. Once steady state is achieved, variables like flow rate, concentration, temperature do not change with time. Here, the vapor flow rate initially varies and then eventually becomes constant indicating that the process moves from transient to steady state.

04

Draw a plot of liquid and vapor flow rates versus time

At startup, the vapor flow rate is zero and it increases over time eventually becoming a constant value, which is half of the molar feed rate. On the other hand, since the total matter does not accumulate in the system, any increase in the vapor flow rate must be counteracted by a reduction in the liquid flow rate. Therefore, at startup, the liquid flow rate is equal to the feed rate and it decreases until it reaches its steady state value, which is also half of the molar feed rate.

05

Mass balance calculation

Perform a mass balance calculation for both benzene and toluene to determine the molar flow rate and composition of the vapor product. For benzene, it is given by \( 0.446 * Feed Rate = Vapor Flow * Vapor mole fraction + Liquid Flow * Liquid mole fraction \). Similarly, a balance for toluene can be calculated. Substituting feed rate as 100 mol/min, vapor flow rate as 50 mol/min and liquid mole fraction of benzene as 0.280 in the above mentioned equation enables us to solve for the vapor mole fraction of benzene.

06

Calculation of toluene in vapor and liquid streams

As the process involves only benzene and toluene, the sum of mole fractions in every stream should be equal to 1. Therefore, the mole fraction of toluene in vapor and liquid streams can be calculated by subtracting the respective mole fractions of benzene from 1.

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Key Concepts

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

Mass Balance

In chemical process engineering, performing a mass balance is critical for analyzing how stream compositions and flow rates interact in a system. Mass balance involves accounting for the intake, distribution, and exit of materials. In the exercise, a mass balance focuses on a feed stream being split into vapor and liquid streams. Applying this concept ensures that the sum of the input equals the sum of the outputs since no accumulation occurs within the unit. For example, the mass flow rate of benzene and toluene entering the system must match the combined rates leaving in vapor and liquid forms. A mass balance equation might look like this for benzene:

• \( 0.446 \, \text{mole fraction} \times \text{Feed Rate} = \text{Vapor Flow Rate} \times \text{Vapor mole fraction} + \text{Liquid Flow Rate} \times \text{Liquid mole fraction} \).

This equation ensures all molecules in the process are accounted for and transitions the understanding toward the equilibrium and split behavior of these substances.

Transient State

The transient state in a chemical process refers to the period during which variables—such as flow rate or concentration—change with time. This phase occurs before reaching steady state, highlighting the system's gradual change towards equilibrium. In our exercise, the process starts with a newly introduced feed. Initially, no vapor is present, marking a transient state. Consequently, the vapor flow rate will rise from zero until it stabilizes. During this time, you observe shifting compositions and flow rates that create dynamic conditions. Understanding transient conditions is essential for predicting how a system reaches its final output or equilibrium state, accommodating any operational variability and ensuring system stability.

Steady State

Thriving at steady state is the goal for many continuous processes in chemical engineering. Once reached, steady state means that key variables remain constant over time. In the context of the exercise, once the vapor flow rate becomes constant, so do other variables such as composition and flow rates of the liquid and vapor outlets. At this point, one can consistently predict outputs, optimizing the process for efficiency and desired product quality. Steady state greatly simplifies calculations and control since introductions and removals balance out, leading to stable operation irrespective of the complex dynamics that may have preceded it.

Continuous Process

Continuous processes are characterized by the uninterrupted feeding of materials and extraction of products. In contrast to batch processes, which require stopping and starting, continuous processes maintain a state of equilibrium. In this exercise, the feed is introduced to the system at a constant rate, indicating a continuous process. By maintaining consistent input and output levels, downtime is minimized, allowing for efficient material use and consistent product quality. Continuous processes are ideal for large-scale production due to their ability to manage sustained operations with less intervention.

Vapor-Liquid Equilibrium

Vapor-liquid equilibrium (VLE) is a crucial concept for understanding the distribution of compound components between vapor and liquid phases at given conditions. In the context of the exercise, VLE principles help define how benzene and toluene distribute themselves between the vapor and liquid streams.

• In a state of equilibrium, at a given temperature and pressure, the chemical potential of each component is equal in both phases.
• This means that the concentrations in the liquid and vapor phases become stable.

Understanding VLE allows engineers to predict component behavior, facilitating the design of efficient separation and purification processes. The composition in each phase is important for calculating mole fractions of the liquid and vapor streams, making VLE analysis integral to process optimization.