Calculating Power of a CO 2 Compressor
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1 Undergraduate Journal of Mathematical Modeling: One + Two Volume Fall Issue 1 Article 9 Calculating Power of a CO 2 Compressor Jennifer Wedebrock University of South Florida Advisors: Mohamed Elhamdadi, Mathematics and Statistics Scott Campbell, Chemical & Biomedical Engineering Problem Suggested By: Scott Campbell Abstract. In this project the goal is to find the power of a CO 2 compressor used in a process of storing CO 2 underground. Although the compressor is not 100% efficient and does not exist under ideal conditions, the power can first be calculated as if it were under ideal conditions by calculating its enthalpy and entropy. Residual terms are then added to both the enthalpy and the entropy to account for the behavior of CO 2 under non-ideal conditions. Since the sum of the change in entropy under ideal conditions and the residual terms for entropy is zero, a system of equations with the Redlich-Kwong equation of state can then be set up and solved for the unknown temperature and volume at the desired pressure. Once these are determined, they can be used to find the total change in enthalpy when the compressor is not under ideal conditions. Then, since the molar flow rate of CO 2 is known, the power of the compressor, if it were 100% efficient, can be calculated and divided by its actual efficiency. The power needed to run the CO 2 compressor was found to be watts. Keywords. CO2 Compressor, Redlich-Kwong equation of state, Power Follow this and additional works at: Part of the Mathematics Commons UJMM is an open access journal, free to authors and readers, and relies on your support: Donate Now Recommended Citation Wedebrock, Jennifer (2009) "Calculating Power of a CO 2 Compressor," Undergraduate Journal of Mathematical Modeling: One + Two: Vol. 2: Iss. 1, Article 9. DOI: Available at:
2 Wedebrock: Calculating Power of a CO<sub>2</sub> Compressor 2 JENNIFER WEDEBROCK TABLE OF CONTENTS Problem Statement... 3 Motivation... 6 Mathematical Description and Solution Approach... 6 Discussion Conclusion and Recommendations Nomenclature Appendix Produced by The Berkeley Electronic Press, 2009
3 Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 2, Iss. 1 [2009], Art. 9 CALCULATING POWER OF A CO 2 COMPRESSOR 3 PROBLEM STATEMENT To prevent its release to the atmosphere, it is proposed to store recovered carbon dioxide in underground reservoirs. During one step of the process, it is required to adiabatically compress the CO 2 from 1 bar and 300 K to 10 bars at steady state. The efficiency of the compressor is 80% and the demand is 0.1 mol/s of CO 2. The goal of this project is to determine the power (in J/s or W) required for the compression. The power W required by the actual compressor is related to the power required by an ideal (isentropic) compressor by where is the efficiency of the actual compressor. From thermodynamics, the power required by the ideal compressor is: where is the molar flow rate of the gas and is the molar enthalpy change of the gas between the entrance and exit of the compressor. Since the ideal compressor is isentropic: where is the molar entropy of the gas. One commonly expresses the enthalpy and entropy changes by calculating what they would be for an ideal gas and then adding residual terms to account for non-ideal gas behavior. Specifically, one writes: DOI:
4 Wedebrock: Calculating Power of a CO<sub>2</sub> Compressor 4 JENNIFER WEDEBROCK and where the superscript refers to the ideal gas change and superscript refers to the residual property of the gas at either the entrance (subscript 1) or exit (subscript 2) of the compressor. Thermodynamics provides relations for the ideal gas changes. They are: and where and are the inlet temperature and pressure and and are the outlet temperature and pressure. The quantity is the ideal gas heat capacity for the gas. Expressions for the residual properties are also provided by thermodynamics. They are: and where is the pressure, is the molar volume, is the universal gas constant, is temperature and is the compressibility factor. To calculate the ideal gas changes and residual properties, one needs two things: A formula for the ideal gas heat capacity and an equation of state that connects, and. For carbon dioxide, the ideal gas heat capacity is: where is the temperature in K. The pressures used here are too high to assume that CO 2 follows the ideal gas law. It is suggested that you use the Redlich-Kwong equation of state: Produced by The Berkeley Electronic Press, 2009
5 Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 2, Iss. 1 [2009], Art. 9 CALCULATING POWER OF A CO 2 COMPRESSOR 5 where is in K and where and are constants. By using properties at the critical point, it can be shown that: and where and are the critical temperature and pressure of the gas. For CO 2, these are: and. The universal gas constant can be expressed different ways, depending on what it is being used for. Two convenient values are: or. DOI:
6 Wedebrock: Calculating Power of a CO<sub>2</sub> Compressor 6 JENNIFER WEDEBROCK MOTIVATION CO 2 compressors are a necessary part of the process of storing CO 2 in underground reservoirs. This is important because when CO 2 is stored above ground there is a risk of it escaping into the atmosphere. For each desired pressure after compression, given the input pressure and temperature of the CO 2, there will be a certain amount of power needed to run the compressor. To find this we will start by assuming that all conditions are ideal when calculating the change in enthalpy and entropy, and then we will add in residual terms to account for non-ideal behavior. We will also need to evaluate and at the critical temperature and pressure for CO 2, as these are further adjustments for non-ideal behavior. In the process of determining the power needed to run a CO 2 compressor under the conditions specified, we will also have determined the output temperature, output volume, and input volume of CO 2. MATHEMATICAL DESCRIPTION AND SOLUTION APPROACH I. Calculating the change in entropy for an ideal gas: Using and we have = (1) Produced by The Berkeley Electronic Press, 2009
7 Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 2, Iss. 1 [2009], Art. 9 CALCULATING POWER OF A CO 2 COMPRESSOR 7 II. Calculate the change in enthalpy for an ideal gas: Using and we have ] (2) III. Evaluate,, and for the Redlich-Kwong equation of state: Differentiating gives. Combining these we find DOI:
8 Wedebrock: Calculating Power of a CO<sub>2</sub> Compressor 8 JENNIFER WEDEBROCK (3) Again using we have (4) Recall, so (5) IV. Integrate using partial fraction decomposition: (6) so and at we have, i.e.. At we have, i.e.. Produced by The Berkeley Electronic Press, 2009
9 Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 2, Iss. 1 [2009], Art. 9 CALCULATING POWER OF A CO 2 COMPRESSOR 9 It follows that (7) V. Integrate to find : Using that and, becomes (8) Using the partial fraction decomposition from (6) above, (9) Recall, and. Using the above formulas for and (10) Recall,. Using the above formulas for and DOI:
10 Wedebrock: Calculating Power of a CO<sub>2</sub> Compressor 10 JENNIFER WEDEBROCK (11) VI. Find the values of and for CO 2 using critical temperature and pressures: Using,, and, (12) and (13) VII. Use the Redlich-Kwong equation of state to find unknown : Knowing that, use goal seek on Excel to change the value of until. A reasonable guess for is needed, so assume everything is ideal and use the ideal gas law to make a first guess. Produced by The Berkeley Electronic Press, 2009
11 Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 2, Iss. 1 [2009], Art. 9 CALCULATING POWER OF A CO 2 COMPRESSOR 11 Using and we have (14) Excel s goal-seek found that when,. (See Table 2 in the Appendix) VIII. Set up a system of equations (Redlich-Kwong equation of state and the expression for ) and solve them simultaneously to find the remaining two unknowns: (15) IX. Manually change and use goal seek, changing to make the Redlich-Kwong equation of state equal 0, until the expression for also equals 0: Goal-seek found the Redlich-Kwong equation at and when and. (16) DOI:
12 Wedebrock: Calculating Power of a CO<sub>2</sub> Compressor 12 JENNIFER WEDEBROCK X. Now that all unknowns have been found, solve for : Recall from (10) that, Using the appropriate substitutions we find (17) Produced by The Berkeley Electronic Press, 2009
13 Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 2, Iss. 1 [2009], Art. 9 CALCULATING POWER OF A CO 2 COMPRESSOR 13 Using and can be found: (18) (19) DISCUSSION In this project, the power needed to use the CO 2 compressor under the given conditions, starting at 300 K and 1 bar and increasing the pressure to 10 bar, would be only watts if the compressor were 100% efficient as calculated. However, since the compressor is only 80% efficient, the compressor actually requires watts to run under these conditions. This result meets the objective of the project, which was to explicitly find the power needed to run a CO 2 compressor at given temperature and pressure. For the industrial application of storing CO 2 underground, the result of watts seems like a reasonable result as well. CONCLUSION AND RECOMMENDATIONS This project shows the relationship between the Redlich-Kwong equation, entropy, and enthalpy. By using the laws of thermodynamics you can extrapolate information expediently. This can be done by starting off assuming the system is under ideal conditions and then adding residual terms until we have expressions that are rigorous and make no assumptions at all. In this case, and are the residual terms added to the enthalpy of an ideal gas to find the change in enthalpy of any gas; and are the residual terms added to the entropy of an ideal DOI:
14 Wedebrock: Calculating Power of a CO<sub>2</sub> Compressor 14 JENNIFER WEDEBROCK gas in order to find the change in entropy of a non-ideal gas; the constants a and b vary based on the critical pressure and temperature of the non-ideal gas being used, and each account for different non-ideal behaviors. Others doing similar projects could examine how the power required to run the CO 2 compressor changes when the given values for temperature, input pressure and output pressure change. One could also check if multiple values of temperature and volume work when using goal seek on the Redlich-Kwong equation of state and the expression for. Alternatively, if one knew a maximum power that could be supplied to the CO 2 compressor, they could perhaps reverse the process used in this project, and given at least some other piece of information, determine everything else using the expressions for, and the Redlich-Kwong equation of state. One could also investigate at how a change in any one of the variables (input temperature, output temperature, input pressure, output pressure, input volume, output volume, molar flow rate) affects the power needed to run the CO 2 compressor. Produced by The Berkeley Electronic Press, 2009
15 Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 2, Iss. 1 [2009], Art. 9 CALCULATING POWER OF A CO 2 COMPRESSOR 15 NOMENCLATURE Symbol Term Units universal gas constant universal gas constant work work for ideal gas efficiency molar flow rate molar enthalpy change molar enthalpy change - ideal gas residual enthalpy term molar entropy change molar entropy change - ideal gas residual entropy term compressibility factor ideal gas heat capacity input temperature output temperature critical temperature input pressure output pressure critical pressure input volume output volume DOI:
16 Wedebrock: Calculating Power of a CO<sub>2</sub> Compressor 16 JENNIFER WEDEBROCK APPENDIX Symbol Reference Value Given Given (16) Given (14) (16) (12) (13) Given Given Given Figure 1: Values needed to find, Guess Goal-Seek Goal-Seek 24,942 24, Figure 2: Guessing and using goal-seek to find the value of when Symbol Value Redlich-Kwong Goal-Seek Figure 3: Trial-and-error to find that allows goal-seek with and Redlich-Kwong to equal Produced by The Berkeley Electronic Press, 2009
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