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The following diagram shows a staged absorption column in which \(n\) -hexane (H) is absorbed from a gas into a heavy oil.A gas feed stream containing 5.0 mole \(\%\) hexane vapor and the balance nitrogen enters at the bottom of an absorption column at a basis rate of \(100 \mathrm{mol} / \mathrm{s}\), and a nonvolatile oil enters the top of the column in a ratio 2 mol oil fed/mol gas fed. The absorber consists of a series of ideal stages (see Problem 6.66), arranged so that gas flows upward and liquid flows downward. The liquid and gas streams leaving each stage are in equilibrium with each other (by the definition of an ideal stage), with compositions related by Raoult's law. The absorber operates at an approximately constant temperature \(T\left(^{\circ} \mathrm{C}\right)\) and \(760 \mathrm{mm}\) Hg. Of the hexane entering the column, \(99.5 \%\) is absorbed and leaves in the liquid column effluent. At the given conditions it may be assumed that \(\mathrm{N}_{2}\) is insoluble in the oil and that none of the oil vaporizes.(a) Calculate the molar flow rates and mole fractions of hexane in the gas and liquid streams leaving the column. Then calculate the average values of the liquid and gas molar flow rates in the column, \(\dot{n}_{\mathrm{L}}(\mathrm{mol} / \mathrm{s})\) and \(\dot{n}_{\mathrm{G}}(\mathrm{mol} / \mathrm{s}) .\) For simplicity, in subsequent calculations use the average values for liquid and gas molar flow rates within the column, but the actual values for the corresponding flow rates entering and leaving the column.(b) Considering the bottom stage to be ideal, estimate the mole fractions of hexane in the gas leaving that stage \(\left(y_{N}\right)\) and in the liquid entering it \(\left(x_{N-1}\right)\) if the column temperature is \(50^{\circ} \mathrm{C}\). (c) Suppose that \(x_{i}\) and \(y_{i}\) are the mole fractions of hexane in the liquid and gas streams leaving stage \(i\) Derive the following equations from an equilibrium relationship and a mass balance around a section of the column encompassing stage \(i\) and the bottom of the column:$$\begin{array}{c}y_{i}=x_{i} p_{i}^{*}(T) / P \\ x_{i-1}=\left(x_{N} n_{L, N}+y_{i} \dot{n}_{\mathrm{G}}-y_{N+1} \dot{n}_{\mathrm{G}+1}\right) / \dot{n}_{\mathrm{L}}\end{array}$$Verify that these equations yield the answers you calculated in Part (b). (d) Examine the effect of operating temperature on the column by estimating the number of ideal stages necessary to achieve the desired separation. In the calculations, which will be done using a spreadsheet, take the pressure in the column to be constant at 760 torr, but consider three different operating temperatures: \(30^{\circ} \mathrm{C}\) \(50^{\circ} \mathrm{C},\) and \(70^{\circ} \mathrm{C} .\) The calculations will follow a stage-to-stage strategy beginning at the bottom of the column and repeatedly applying Equations (1) and (2) until the mole fraction of hexane in the vapor leaving the column is less than or equal to that calculated in Part (a). You may use APEx or the Antoine equation and Table B.4 to estimate the hexane vapor pressure. The calculations for the case of \(T=30^{\circ} \mathrm{C}\) illustrate how to proceed; for this case, \(y_{N-1} < y_{1}=0.00263\) after only two stages.(e) You can see that the number of stages required increases as the column temperature increases. In fact, there is a maximum temperature beyond which the required separation cannot be achieved. At that temperature, the entering gas and leaving liquid are approximately in equilibrium, so that \(x_{N} p^{*}(T)=y_{N+1} P .\) Use either APEx or the Antoine equation to estimate the maximum temperature at which the separation can be achieved.

Step by step solution

01

Calculation of Molar Flow Rates and Mole Fractions of Hexane in the Streams Leaving the Column

Firstly, find the molar flow rates of hexane entering the absorption column from the gas feed and the oil feed, denoted as \(n_{H,G}^{in}\) and \(n_{O}^{in}\) respectively. Given \(n_{H,G}^{in} = 0.05 * 100 mol/s = 5 mol/s \) and \(n_{O}^{in} = 2 * 100 mol/s = 200 mol/s \). Now, calculate the hexane absorbed, \(n_{H}^{abs} = 0.995 * 5 mol/s = 4.975 mol/s \). Thus, the molar flow rate of hexane in the liquid stream leaving the column, \(n_{H,L}^{out} = n_{H}^{abs} = 4.975 mol/s \). And molar flow rate of hexane in the gas stream leaving the column, \(n_{H,G}^{out} = n_{H,G}^{in} - n_{H}^{abs} = 0.025 mol/s \). Similarly, find the molar flow rates of N2 and oil leaving the column, \(n_{N2}^{out} = n_{N2}^{in} = 100 - n_{H,G}^{in} = 95 mol/s \) and \(n_{O}^{out} = n_{O}^{in} = 200 mol/s \). Hence, the molar flow rates leaving the column in the gas stream \(\dot{n}_{G}^{out} = n_{N2}^{out} + n_{H,G}^{out} = 95 + 0.025 = 95.025 mol/s \) and in the liquid stream \(\dot{n}_{L}^{out} = n_{O}^{out} + n_{H,L}^{out} = 200 + 4.975 = 204.975 mol/s \). Calculate mole fractions of hexane in the gas and liquid streams leaving the column as \(y_{N+1} = n_{H,G}^{out} / \dot{n}_{G}^{out} = 0.025 / 95.025 = 0.000263 \) and \(x_{N} = n_{H,L}^{out} / \dot{n}_{L}^{out} = 4.975 / 204.975 = 0.0243 \) respectively.

02

Calculation of Average Values of the Liquid and Gas Molar Flow Rates in the Column

Next, compute the average values of the liquid and gas molar flow rates in the column as \(\dot{n}_{L} = (\dot{n}_{L}^{out} + n_{O}^{in}) / 2 = (204.975 + 200) / 2 = 202.488 mol/s \) and \(\dot{n}_{G} = (\dot{n}_{G}^{out} + n_{N2}^{out}) / 2 = (95.025 + 95) / 2 = 95.013 mol/s \) respectively.

03

Calculation of Mole Fractions of Hexane in the Bottom Stage

Use Raoult's law and the derived relationship for \(x_{i-1}\) from the equilibrium relationship and the mass balance to compute \(y_{N}\) and \(x_{N-1}\). For \(T=50^{\circ} C\), use the Antoine equation or APEx to determine \(p_{N}^{*}(50) \). Then find \(y_{N}\) from \(y_{i} = x_{i} p_{i}^{*}(T) / P \) and \(x_{N-1}\) from \(x_{i-1} = \left(x_{N} n_{L, N} + y_{i} \dot{n}_{G} - y_{N+1} \dot{n}_{G+1}\right) / \dot{n}_{L} \).

04

Estimation of Number of Ideal Stages to Achieve Desired Separation

Examine the effect of operating temperature by computing the number of ideal stages necessary for separation at \(T = 30^{\circ} C \), \(50^{\circ} C \) and \(70^{\circ} C \). Using the Antoine equation for hexane vapor pressure \(p^{*}(T) \) at each of these temperatures, apply the derived relationships \(y_{i} = x_{i} p_{i}^{*}(T) / P \) and \(x_{i-1} = \left(x_{N} n_{L, N} + y_{i} \dot{n}_{G} - y_{N+1} \dot{n}_{G+1}\right) / \dot{n}_{L} \) until \(y_{N-1} \leq y_{1} = 0.000263 \) is satisfied. The number of applications corresponds to the number of ideal stages.

05

Estimation of Maximum Temperature for Desired Separation

The maximum temperature at which separation is achievable is the temperature at which the entering gas and leaving liquid are in equilibrium. Derive this temperature by setting \(x_{N} p^{*}(T) = y_{N+1} P \), then using either the Antoine equation or APEx to solve for \(T \).

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Key Concepts

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

Hexane Absorption

Hexane absorption is a key process used in chemical engineering to separate hexane from a gas mixture.
In an absorption column, a volatile component like hexane is extracted from a gas stream using a non-volatile liquid solvent, often referred to as the absorbent.
The gas containing hexane enters from the bottom of the column, and the absorbing liquid flows downward.
As they come into contact, hexane transfers from the gas phase to the liquid phase. The effectiveness of hexane absorption depends on various factors:

• The flow rates of the gas and liquid streams. Higher flow rates typically lead to better contact between phases.
• The temperature within the column, which affects the solubility and vapor pressures.
• The design of the column, including the number of stages or trays that enhance the contact between gas and liquid.

This process has extensive applications across industries such as petrochemical and environmental engineering, where precise separation is crucial.

Raoult's Law

Raoult's Law is a principle used to predict the vapor pressure of a solvent in a solution.
It states that the partial vapor pressure of each component in a volatile liquid mixture is proportional to its mole fraction in the mixture and its pure component vapor pressure.
For a component 'i' in a liquid solution, Raoult's Law is given by:\[ p_i = x_i imes p_i^* \] where:

• \( p_i \) is the partial pressure of component 'i'.
• \( x_i \) is the mole fraction of component 'i' in the liquid phase.
• \( p_i^* \) is the vapor pressure of the pure component 'i' at a given temperature.

In the context of an absorption column, Raoult's Law helps relate the composition of hexane in the liquid phase to its concentration in the gas phase under equilibrium conditions.
This relationship guides the design and analysis of the absorption column by predicting how much hexane can be absorbed at a given temperature and pressure.

Equilibrium Stages

Equilibrium stages are theoretical zones in an absorption column where the liquid and gas phases are in perfect equilibrium.
Here, the compositions of the liquid and gas are calculated under the assumption that they have reached a balance in terms of mass and energy exchange. In an ideal absorption column, each stage represents a complete equilibrium between the entering and exiting streams. This means:

• The composition of hexane in the gas phase leaving each stage is in thermal and chemical equilibrium with the liquid phase leaving the same stage.
• Raoult's Law can be applied at each stage to solve for compositions based on known pressures and temperatures.

The number of equilibrium stages needed to achieve a desired separation is a crucial aspect of design, often determined using graphical techniques like McCabe-Thiele diagrams. The greater the number of stages, the more effective the separation, but also, the more complex and costly the column.
Therefore, accurate estimation of equilibrium stages is key for cost-effective and efficient absorption column operation.

Molecular Flow Rates

Molecular flow rates are essential in understanding the dynamics of fluid movement within an absorption column.
These rates provide insight into how much of each component moves through the system over a unit of time and helps in setting up material balances which are pivotal for process design. For hexane absorption, molecular flow rates will be quantified for both:

• The gas stream, including components like hexane and nitrogen.
• The liquid stream, including the absorbing oil and absorbed hexane.

The flow rates are calculated based on known feed conditions and the separation efficiency of the column.
They are used to determine the molar composition in output streams and to assess the efficiency of the absorption process. Knowing these rates allows engineers to scale-up the process from pilot scale to industrial scale and to optimize for variables such as operating pressure, temperature, and flow velocities to ensure efficient separation within the absorption column.