91Ó°ÊÓ

A liquid mixture contains \(60.0 \mathrm{wt} \%\) ethanol \((\mathrm{E}), 5.0 \mathrm{wt} \%\) of a dissolved solute \((\mathrm{S}),\) and the balance water. A stream of this mixture is fed to a continuous distillation column operating at steady state. Product streams emerge at the top and bottom of the column. The column design calls for the product streams to have equal mass flow rates and for the top stream to contain 90.0 wt\% ethanol and no S. (a) Assume a basis of calculation, draw and fully label a process flowchart, do the degree-of-freedom analysis, and verify that all unknown stream flows and compositions can be calculated. (Don't do any calculations yet.) (b) Calculate (i) the mass fraction of \(S\) in the bottom stream and (ii) the fraction of the ethanol in the feed that leaves in the bottom product stream (i.e., \(\mathrm{kg} \mathrm{E}\) in bottom stream/kg \(\mathrm{E}\) in feed) if the process operates as designed. (c) An analyzer is available to determine the composition of ethanol-water mixtures. The calibration curve for the analyzer is a straight line on a plot on logarithmic axes of mass fraction of ethanol, \(x\) (kg E/kg mixture), versus analyzer reading, \(R\). The line passes through the points \((R=15, x=\) 0.100) and \((R=38, x=0.400)\). Derive an expression for \(x\) as a function of \(R(x=\cdots\) ) based on the calibration, and use it to determine the value of \(R\) that should be obtained if the top product stream from the distillation column is analyzed. (d) Suppose a sample of the top stream is taken and analyzed and the reading obtained is not the one calculated in Part (c). Assume that the calculation in Part (c) is correct and that the plant operator followed the correct procedure in doing the analysis. Give five significantly different possible causes for the deviation between \(R_{\text {measured and }} R_{\text {prediced }}\), including several assumptions made when writing the balances of Part (c). For each one, suggest something that the operator could do to check whether it is in fact the problem.

Step by step solution

01

Calculate mass fraction of S in the bottom stream from distillation column

From the problem statement, the column is designed such that the product streams have equal mass flow rates. The top stream contains 90% of ethanol E and no solute S. Therefore, since input equals output in a steady-state process, the bottom stream should also contain equal masses of S and water compared to the feed stream. For a mixture containing 60% E, 5% S and 35% water, the bottom stream will be 5% S and 95% water since no S goes to the top stream.

02

Calculate fraction of ethanol in feed that leaves in the bottom stream

Although the distillation column operates such that the two output streams are of equal mass, evenly split between the two outlet streams since equal mass flow rates was a design constraint. Then, half the ethanol in the feed will be in top steam and the other half in the bottom stream. Hence, the mass fraction of the ethanol that leaves the bottom stream compared to the feed will be 50%.

03

Derive an expression for x as a function of R

Using the two given points, we can form two simultaneous equations in loge form. Let the equation of the straight line on the calibration graph be y = mx + c, where y = log(R), x = log(X), m is the slope and c is the intercept. Substituting the two given points into this equation, we can solve for m and c. With those values, we can then rewrite the equation in a suitable form to find x = f(R), where 'f' designates a function which involves exponential because of the logarithms used.

04

Use the formula derived in Step 3 to determine R for the top-stream

Having the x as a function of R, we can calculate the value of R for the distillate from the column. Given that the top stream is 90 wt% ethanol, therefore, x in this analysis is 0.9. Substituting x = 0.9 into the equation obtained in Step 3, we can find the corresponding R value which is the predicted optical density.

05

Analyze possible causes of deviation in measured and predicted R

Possible causes for the difference between the measured and predicted R could include: inaccuracies in the calibration curve, the presence of trace components in the sample which might interfere with the analyzer response, variability in the distillation process leading to inconsistencies in mixture composition, errors in sample collection or handling, and inaccuracies in your calculations. In order to check these assumptions, the operator could discard the first few outputs from the analyzer to remove possible contaminants, rerun the calibration curve, recheck the calculations, or collect and handle samples more carefully according to the standard procedure.

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Key Concepts

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

Mass Balance Calculations

Understanding the concept of mass balance calculations is crucial when analyzing a distillation column process. These calculations are based on the principle of conservation of mass, which states that mass cannot be created or destroyed in a process. In a distillation column, this means that the mass of each component entering the column must equal the mass exiting, either in the distillate (top stream) or the bottoms (bottom stream).

For instance, if a feed contains a certain percentage of ethanol (E), water, and a solute (S), we can use mass balance to calculate the composition of the product streams. Assuming a steady state and no accumulation within the column, the mass fraction of the solute in the bottom stream would be equal to that in the feed if the solute is non-volatile and does not distill with ethanol. Similarly, understanding the distribution of ethanol between the top and bottom streams hinges on the mass balance - with equal mass flow rates for both streams, half of the ethanol in the feed will be found in each product stream.

When there are deviations in the process or unexpected results, mass balance calculations can be revisited to identify potential errors or changes in the process conditions.

Steady-State Process

A steady-state process is a condition where the variables (mass flow, temperature, composition, etc.) that define the operation of a process do not change with time. In the context of a distillation column, this means that the feed rate, composition, and mass flow rates of the top and bottom streams remain constant with time.

This assumption simplifies mass balance calculations because it eliminates the need to consider changes over time. In practice, achieving and maintaining steady-state requires precise control of the process conditions and is crucial for consistent product quality. If the steady state is not maintained, variables such as the mass fraction of a component in the output streams can fluctuate, leading to deviations from the expected performance of the column.

Calibration Curve Analysis

Calibration curve analysis is an essential tool in analytical chemistry, used to understand the relationship between an instrument's reading and the actual concentration of a substance in a sample. A calibration curve generally plots the instrument response against known concentrations of a substance.

In the exercise, the calibration curve is used to correlate the analyzer reading (R) to the mass fraction of ethanol (x). By knowing two points, a line can be drawn on a log-log plot, and an equation can be derived linking R and x. This equation allows for the prediction of unknown concentrations from their respective analyzer readings. When the measured R deviates from the predicted R, the calibration curve itself may need to be reassessed, or it might indicate a problem with the sample or the analyzer. Regular calibration and verification using standard solutions are essential for maintaining accuracy in such analytical instruments.

Process Flowchart

A process flowchart provides a visual representation of the steps, materials, and streams involved in a process. For the distillation column exercise, a well-labeled flowchart would illustrate the feed entering the column, the two product streams (top and bottom), and the components present in each stream.

Creating a flowchart is a key step in process analysis, as it helps in visually organizing the information and identifying what is known and what needs to be calculated. It serves as a reference point for mass balance calculations and for doing a degree-of-freedom analysis. Ensuring that each stream is labeled with the correct mass flow rate and composition, as mentioned in the exercise's instructions, supports accuracy in calculations and process design.

Degree-of-Freedom Analysis

Degree-of-freedom analysis is a method used to determine whether a system of equations is solvable as it relates to the number of unknowns versus the number of independent equations. In the exercise, before performing any calculations, it's crucial to establish whether there are enough equations to solve for all unknowns associated with the distillation process.

For the provided exercise, the variables include stream flow rates, compositions, and the mass fractions of components within each stream. By setting up a degree-of-freedom analysis, we identify if the system is under-specified, over-specified, or just right. If the degrees of freedom are zero, it indicates that we have exactly as many independent equations as we have unknowns, and the system can be solved with a high degree of confidence.

Considering the degree-of-freedom at an early stage in the problem-solving process helps in understanding the solvability of the system, and it can be a great indicator of whether additional data or assumptions are needed to proceed with the calculations.