91Ó°ÊÓ

An important parameter in the design of gas absorbers is the ratio of the flow rate of the feed liquid to that of the feed gas. The lower the value of this ratio, the lower the cost of the solvent required to process a given quantity of gas but the taller the absorber must be to achieve a specified separation.Propane is recovered from a 7 mole \(\%\) propane \(-93 \%\) nitrogen mixture by contacting the mixture with liquid \(n\) -decane. An insignificant amount of decane is vaporized in the process, and \(98.5 \%\) of the propane entering the unit is absorbed.(a) The highest possible propane mole fraction in the exiting liquid is that in equilibrium with the propane mole fraction in the feed gas (a condition requiring an infinitely tall column). Using Raoult's law to relate the mole fractions of propane in the feed gas and liquid, calculate the ratio \(\left(\dot{n}_{L_{1}} / \dot{n}_{G_{2}}\right)\) corresponding to this limiting condition.(b) Suppose the actual feed ratio \(\left(\dot{n}_{L_{1}} / \dot{n}_{G_{2}}\right)\) is 1.2 times the value calculated in Part (a) and the percentage of the entering propane absorbed is the same (98.5\%). Calculate the mole fraction of propane in the exiting liquid.(c) What are the costs and benefits associated with increasing \(\left(\dot{n}_{L_{1}} / \dot{n}_{G_{2}}\right)\) from its minimum value [the value calculated in Part (a)]? What would you have to know to determine the most cost-effective value of this ratio?

Step by step solution

01

Apply Raoult's Law for Limiting Condition

Raoult's law is given by \(y_{i}P = x_{i}P_{i}^{sat}\), where \(y_{i}\) is the mole fraction of propane in the gas phase, \(P\) is the total pressure, \(x_{i}\) is the mole fraction in the liquid phase, and \(P_{i}^{sat}\) is the saturation pressure of propane. Given that the mole fraction in the feed gas is 7%, apply the law to solve for \(x_{i}\), the mole fraction in the liquid.

02

Calculate the Ratio

Given that all the propane in the absorbed gas (7% mole fraction) goes to the liquid phase and becomes \(x_{i}\) mole fraction after absorption, the actual feed ratio \(\left(\dot{n}_{L_{1}} / \dot{n}_{G_{2}}\right)\) can be calculated using the quantities of propane in the gas \(y_{i}\) (7%) and in the liquid \(x_{i}\).

03

Calculate the Mole Fraction for Part b

For Part (b), it is stated that the actual feed ratio is 1.2 times the value calculated in Part (a), with the same percentage of propane absorbed. To find the new mole fraction of propane in the exiting liquid, multiply the propane absorbed by 1.2.

04

Evaluate Costs and Benefits

In Part (c), we need to evaluate how changing the ratio from its minimum value affects the overall cost and efficiency. It is known that as the ratio increases, the height of the tank (and thus the cost) decreases. However, the cost of solvent also increases. To determine the most cost-effective value, we need information such as the cost of the solvent, the cost of tank construction and operation, and the market value of propane.

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Key Concepts

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

Raoult's Law

Raoult's Law is a fundamental principle in thermodynamics that relates the partial pressure of a component in a gas phase to its mole fraction in a liquid phase. This law is crucial in processes such as gas absorption, where we separate components through contact with a liquid.
In essence, Raoult's Law expresses that for an ideal solution, the partial pressure of a component (\(y_i P\)) is equal to the mole fraction of that component in the liquid phase (\(x_i\)) multiplied by the component's saturation pressure (\(P_i^{sat}\)). This relationship forms the basis of calculating the distribution of propane between gaseous and liquid phases in our gas absorber exercise.
To apply Raoult's Law effectively, we first need to understand the composition of the starting gas mixture. In this exercise, we are dealing with a 7% propane mixture, which helps us determine \(x_i\) once we have \(y_i\) and \(P_i^{sat}\). This relationship helps calculate the limiting condition, where the column must be infinitely tall for completion.
A practical application: the higher the propane mole fraction we achieve in the liquid by applying Raoult's Law, the more efficient the absorber becomes.

Solvent Cost Evaluation

When considering gas absorbers, solvent cost evaluation becomes a key factor. Simply put, solvent cost evaluation is assessing the expenses associated with using a specific liquid to capture and separate gases.
This involves not just the direct purchase price of the solvent, but also indirect costs like transportation, storage, and disposal. Additionally, the choice of solvent impacts the overall operational costs, such as energy consumption for heating or cooling to achieve optimal absorption.
Incorporating solvent costs into our decision-making is essential for determining the most economical approach. While a more efficient solvent might improve absorber performance, it might also come at a high cost, affecting the project's overall budget.

• Initial solvent purchase and replenishment
• Energy costs for regulating absorber temperature
• Maintenance and handling

Evaluating these costs against the rates of absorption and column efficiency gives a clearer picture of the best practice. The exercise hints at this by suggesting adjustments to the feed ratio, which could save costs but require a critical balance between solvent expense and column efficiency.
Choosing the right solvent and optimizing its use is a continuous puzzle of cost-effectiveness and performance.

Absorption Efficiency

Absorption efficiency in a gas absorber refers to the percentage of a target gas component that is successfully captured and removed from a gas mixture during the absorption process.
This efficiency is an indicator of how effective a particular absorber is in separating gases. In the provided exercise, 98.5% of the entering propane is absorbed, indicating high absorption efficiency.
Several factors directly influence absorption efficiency:

• Solvent selection: The properties of the solvent, such as its affinity for the target gas, are critical.
• Operating conditions: Temperature and pressure settings must be optimized for maximum efficiency.
• Absorber design: The physical characteristics, such as height and surface area for contact, play a role in effectiveness.

Achieving high absorption efficiency often necessitates a trade-off between operational cost and the physical footprint of the absorber (i.e., height). A taller column may achieve better separation but at a higher cost due to materials and construction expenses.
Thus, balancing these aspects ensures that the overall performance of the gas absorbing system is optimized, with efficiency tallied against economic constraints.