EQUATIONS OF STATE Objective 1. To determine the specific volume of pure p ure substances and mixtures using MS Excel® and Aspen Plus® 2. To compare the calculated specific volume of pure substances and mixtures using different Equations of State.

Theoretical Discussion

Solving equations of state allows the determination of the specific volume of a gaseous mixture of chemicals ch emicals at a specified temperature and pressure. By determining the specific volume, the size size – – and and therefore, the cost — — of of the plant, including the diameter of pipes, the horsepower of compressors and pumps, and the diameter of distillation towers and chemical reactors. The specific volume is also the first step in calculating the enthalpy and vapor-liquid properties of mixtures. Enthalpy is important when making energy balances. In order to solve equations of state, algebraic equations must be solved. Functions in MS Excel® are are available for such calculations. calculations. Aspen Plus® used modified equations of state state to easily and accurately solve problems involving gaseous mixtures. The ideal gas equation of state relates the pressure, temperature and specific volume.

or ̂ where ̂

3.1

where p where p is is the absolute pressure, V is is the volume, n is the number of moles, R moles, R is is the gas constant and T is the absolute temperature. temperature. This equation is quite adequate when the pressure is is low. However, many chemical processes take place at very high pressure. Other equations of states have been developed, usually in conjunction with process simulators, to address the chemical processes at high pressure. One generalization of the ideal gas law is the van der Waals equation of state:

̂− −̂

3.2

In this equation, a accounts for the interaction force between two molecules and b accounts for the excluded volume.

̂− −̂ ̂+ 0.42748

The Redlich-Kwong Redlich-Kwong Equation of state is a modification mod ification of van der Waal’s Equation of State:

where

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0.08664 1.

3.5 3.6 3.7

In these equations, T c is the critical temperature, pc is the critical pressure and T r is the reduced temperature. particular to the Redlich-Kwong Equation of State.

is

The Redlich-Kwong Equation of State was further modified by Soave to give the Redlich-Kwong-Soave Equation of State which is common in process simulators.

̂− −̂ ̂+ 1+ 1−. 0.480+1.574+0.176

The parameter is given by a different formula.

3.3 3.8 3.9

is the acentric factor, which is a tabulated quantity for many substances.

̂− −̂ ̂+ + ̂−

The Peng-Robinson equation is another variation:

3.10

These equations can be rearranged into a cubic function of specific volume. the form of the Redlich-Kwong and Redlich-Kwong-Soave Equation of State is:

̂ − ̂ + ̂ − −−0

3.11

For a pure component, the parameters a and b are determined from the critical temperature, the critical pressure, and the acentric factor. For mixtures, it is necessary to combine the values of a and b for each component according to the composition of the gaseous mixture according to the following mixing rules:

0.42748 ; ∑= . 0.08664 ; ∑= Computer Applications in ChE

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1. for Redℎ−Kwong [1+ (1−.)] for Redlich−Kwong−Soave

where

3.14 3.15

Illustrative Example Determining the specific volume of a substance using MS Excel® G oal Seek

Find the specific volume of n- butane at 500 K and 18 atm using the Redlich-Kwong Equation of State. 1. Find the critical temperature and pressure. Perry’s Chemical Engineers’ Handbook gives Tc = 425.2 K and pc = 37.5 atm. 2. Calculate the values of a and b using Equation 3.8 and 3.9. 3. Prepare a spreadsheet for the parameters and results. Supply the cells with the needed formula. Refer to the equations given in the Theoretical Discussion. Figure 3.1 shows a sample spreadsheet.

Figure 3.1. Sample Spreadsheet

4. Use the Goal Seek command to make f(v) [cell I7] equal to zero by changing cell v [I6]. The Goal Seek function may be found under Data – Data Tools – What If Analysis. Figure 3.2 shows the Goal Seek dialog box.

Figure 3.2. Goal Seek Dialog Box

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5. Figure 3.3 shows the Goal Seek Results, including the value of v.

Figure 3.3. Goal Seek Results

Determining the specific volume of a substance using using Aspen Plus®

Find the specific volume of n- butane at 500 K and 18 atm using the Redlich-Kwong-Soave Equation of State. 1. Start Aspen Plus® and choose Template. When the New window appears, choose General with Metric Units. Choose Property Analysis in the Run Type (lower right-hand corner). Figure 3.4 shows the New window.

Figure 3.4. New Window

2.

The Data Browser opens next. On the left pane, the following folders are shown: Setup, Components, Properties, Reactions, Flowsheeting Options, Summary. Note that in Aspen Plus®, folders marked in red represent incomplete data inputs. Figure 3.5 shows the Data Browser window.

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Figure 3.5. Data Browser Window

3. Choose Component – Specifications and enter the name or formula of the chemical (n-butane). If Aspen Plus® does not recognize your chemical, a window appears that allows you to search again, and it will suggest a number o f possibilities. Note that it is important that there is an entry for the chemical in the column Component Name. The first column is what you are naming the chemicals but the third column is what Aspen Plus® is using when it gets the physical properties. Note, too, that you no longer need to look for the properties of your component in the Perry’s Chemical Engineers’ Handbook, since Aspen Plus® has a database for such. Figure 3.6 shows the Components Specification window.

Figure 3.6. Components Specification Window

4. Once the component specifications are complete, choose Property – Specifications. Use the following data: Process Type: All Base Method: RK – Soave Property Method: RK – Soave Use True Components:

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EQUATIONS OF STATE 5. On the menus at the top, choose Tools – Analysis – Property – Pure. On the Pure Component Properties Analysis Window that appears, choose the following: Process Type: Thermodynamic Property: V Phase: Vapor Selected Components: n-butane Temperature Units: K List of Values 500, 510 Pressure 18 atm Property Method: RK – Soave 6. A graph appears with the plotted results. A table is also generated (behind the graph).

Determining the specific volume of a mixture using Aspen Plus®

Find the specific volume of a mixture consisting of 630 kmol/h of carbon monoxide, 1130 km/h, 189 kmol/h or carbon dioxide, and 63 kmol/h of hydrogen at 1 atm and 500 K. 1. Start Aspen Plus® and choose Template. Choose General with Metric Units. Choose Flowsheet in the Run Type. 2. The Process Flowsheet Window appears, as shown in Figure 3.7. Note that the bottom part shows the Model Library.

Figure 3.7. Process Flowsheet Window

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EQUATIONS OF STATE 3. To place a model or unit operation (called block) from the Model Library to the flowsheet, click on the desired model and click on the screen where you intend to place the block. Blocks may be renamed or deleted by choosing the right option upon right click. On the Model Library, choose Pressure Changers, and place a Compressor. 4. To add material streams to and from blocks, click on Material Streams. Click on the flowsheet where you want the stream to originate and drag the cursor to where you want it to end. When done, click the arrow on the upper left of Material Streams box. Note that if any red arrows show in the flowsheet, it means that the block is not properly connected. Figure 3.8 shows a single compressor with material streams as placed in the flowsheet.

Figure 3.8. Compressor with Material Streams

5. Click on the Data Browser (glasses icon). This brings up the menu much like in the previous example, only with more folders. Specify the components in the same manner as the previous example. 6. Choose RK-Soave on the Property Specification. 7. On the Streams folder, note that there are two subfolders, 1 and 2. Be careful in specifying the properties of each stream. Choose Stream 1 – Input . Specify the following: Substream Name Mixed State Variables Temperature and Pressure Total Flow Mole Composition Mole-Flow (kmol/hr) – specify flow rates 8. Choose Blocks – Specifications: Type Discharge Pressure

Isentropic 7.09275 bar

9. Once all data are complete, click on Next (N→ icon). Once the calculations are complete, click the Results box to return to the regular menu. Look at Results – Streams. The stream data will appear in tabular form. 10. The specific volume can be obtained by dividing the volumetric flow rate by the molar flow rate.

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11. The flowsheet and mass and energy balance may be transferred to a word processing program by select, copy, paste. The stream table may also be selected and pasted into MS Excel® (Results – Summary – Stream). 12. Detailed information can be obtained using View – Report .

Reference Finlayson, Bruce. Introduction to Chemical Engineering Computing . New Jersey: John Wiley & Sons, Inc., 2006.

Preliminary Data Answer the following problems using the specified tool. Save the file on the mapped network drive using the specified filename format. 1. Find the molar volume of ammonia gas at 56 atm an d450 K using the Redlich-Kwong equation of state using MS Excel®. Save as MP3A-

Reminder

Your final report should explain what problem you have solved and how you solved it. Focus on the chemical engineering information rather than the d etailed step-by-step process of solution.

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