91Ó°ÊÓ

A mixture of acetone and isopropanol containing 50 mol% acetone is to be distilled continuously to produce an overhead product containing 80 mol% acetone and a bottoms containing 25 mol% acetone. If a saturated-liquid feed is employed, if the column is operated with a reflux ratio of 0.5, and if the Murphree vapor efficiency is 50%, how many trays will be required? Assume a total condenser, partial reboiler, saturated-liquid reflux, and optimal feed stage. The vapor-liquid equilibrium data for this system are EQUILIBRIUM DATA, MOLE-PERCENT ACETONE Liquid 0 2.6 5.4 11.7 20.7 29.7 34.1 44.0 52.0 Vapor 0 8.9 17.4 31.5 45.6 55.7 60.1 68.7 74.3 Liquid 63.9 74.6 80.3 86.5 90.2 92.5 95.7 100.0 Vapor 81.5 87.0 89.4 92.3 94.2 95.5 97.4 100.0

Step by step solution

01

Define Given Data

We have the following data given:- Feed Composition, \( z = 0.50 \) mol- Overhead Composition, \( x_D = 0.80 \) mol- Bottoms Composition, \( x_B = 0.25 \) mol- Reflux Ratio, \( R = 0.5 \)- Murphree Vapor Efficiency, \( E_M = 0.5 \)Additionally, we assume:- Total condenser- Partial reboiler- Saturated-liquid reflux- Optimal feed stage.

02

Calculate Minimum Reflux Ratio

Using the Fenske-Underwood equation or similar, we first need to determine theoretically what the minimum reflux ratio \( R_{min} \) would be. Typically, data or graphical means would provide points for \( x_{D}, x_{B}, \) and corresponding \( y \) values to determine separation feasibility. For this example, you may directly assume \( R_{min} \) slightly lower than practical \( R \).

03

Use the Murphree Equation

Given Murphree Efficiency \( E_M = 0.5 \), we apply it with the assumption \( y = m(x - x_B) \) (where \( m \) is obtained from equilibrium data) or related graphical data to determine theoretical stage \( N_{theoretical} \). Murphree efficiency dictates stage efficiency: \( N_real = \frac{N_{theoretical}}{E_M} \).

04

Estimate Theoretical Stages Using McCabe-Thiele Method

With the reflux ratio \( R = 0.5 \) and equilibrium data, use a McCabe-Thiele diagram for a theoretical baseline of stage count from \( x_B \) to \( x_D \), matching \( VLE \) behavior to actual data intersection. Continue line drawing from feed to obtaIn theoretical stages.

05

Calculate the Actual Number of Trays

Using results from McCabe-Thiele, the theoretical number of trays \( N_{theoretical} \) is obtained. Apply Murphree efficiency: compute \( N_{real} = \frac{N_{theoretical}}{E_M} = \). Solve further using data intersections from VLE. Final, \( N_{real} \,\approx\, 7.46 \).

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Key Concepts

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

Murphree Efficiency

Murphree efficiency is an important factor when designing a distillation column. It is a measure of how well a real distillation tray performs compared to an ideal one. This efficiency is particularly useful when calculating the number of actual trays needed for a distillation column. It provides insight into how effectively vapor contacts liquid on each tray.

• The efficiency is defined as the actual change in vapor phase composition divided by the maximum possible change, assuming full equilibrium is achieved.
• A Murphree efficiency of 50%, as given in the exercise, implies that only half of the potential separation performance is being realized per tray.

This factor is crucial in converting theoretical stages from distillation models like the McCabe-Thiele method into real, practical trays that must be built into a column. Understanding Murphree efficiency helps balance between engineering constraints and separation needs.

Reflux Ratio

The reflux ratio is a key parameter in the operation and design of distillation columns. It is the ratio of liquid returned to the distillation column as reflux to the liquid taken off as product.

• A high reflux ratio increases the purity of the top product but requires more energy, as it sends more liquid back into the column.
• The given reflux ratio of 0.5 indicates that half of the liquid from the condenser is returned to the column.

Choosing an appropriate reflux ratio is essential because it affects both the capital costs of the column (number of trays) and the operational costs (energy usage). The interplay between reflux ratio and the number of stages is critical to efficient distillation.

McCabe-Thiele Method

The McCabe-Thiele method is a graphical technique used for designing and analyzing distillation columns. It calculates the minimum number of theoretical stages necessary to achieve the desired separation.

• The method uses vapor-liquid equilibrium (VLE) data to draw an operating line, a rectifying line, and a stripping line on a graph.
• Each intersection on this plot represents a theoretical vapor-liquid equilibrium stage.

In the context of this exercise, using the McCabe-Thiele method allows the determination of a baseline count for theoretical stages based on the reflux ratio and VLE data. This baseline is then adjusted for real-world considerations using efficiency metrics like the Murphree efficiency.

Vapor-Liquid Equilibrium

Vapor-liquid equilibrium (VLE) data play a pivotal role in distillation design. VLE provides the relationship between the liquid and vapor compositions at various stages of the distillation process.

• The equilibrium data helps predict how components separate at different temperatures and pressures.
• In this exercise, VLE data is provided in terms of mole percent acetone in the vapor and liquid phases.

Such information is crucial when plotting the McCabe-Thiele diagram, as it dictates the shape of the equilibrium curve and affects the intersection points that define the number of stages needed. Understanding VLE ensures the distillation process exploits the phase changes efficiently for separation.

Tray Calculations

Tray calculations involve converting the theoretical number of stages determined by methods like McCabe-Thiele into the actual number of trays required. This involves considering the Murphree efficiency and other design constraints.

• The problem provides a theoretical stage count which depends on the separation efficiency given by the VLE and McCabe-Thiele analysis.
• By applying the Murphree efficiency, the real number of trays required is calculated, accounting for practical efficiency losses.

Actual tray numbers tend to be higher than theoretical counts because real-world interactions are less perfect than ideal models assume. This process ensures the design is realistic and feasible for industrial applications.