Critical Role of Control Volume in Thermodynamics Education for the Analysis of Energy and Entropy Balance.

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1 Academia Journal of Scientific Research 4(4): , April 2016 DOI: /ajsr ISSN: Academia Publishing Research Paper Critical Role of Control Volume in Thermodynamics Education for the Analysis of Energy and Entropy Balance. Accepted 1 st April, 2016 ABSTRACT Dun-Yen Kang* and Kai-Hsin Liou Department of Chemical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan. *Corresponding author. dunyen@ntu.edu.tw. Control volume must be specified explicitly in order to apply balance equations to the analysis of thermodynamic systems. A failure to do so would render any analysis meaningless. Unfortunately, the role of control volume with regard to the energy and entropy balance is generally overlooked in the thermodynamics education for chemical engineers. We addressed this issue by investigating the classic thermodynamic problem involving the expansion of confined gas within a piston-and-cylinder device. Three sets of control volumes were used in the analysis of a given process, the results of which demonstrate the fact that many of the key questions, such as the origin of entropy generation can be unambiguously answered through the judicious selection of control volume. We then compared the results from a variety of approaches in order to elucidate the degree of consistency among these methods. The proposed method of varying control volumes to facilitate the analysis of thermodynamic systems enables the diagnosis of local energy gain and loss and could be used to optimize the design of thermodynamic devices. Key words: Undergraduate thermodynamics, control volume, energy balance, entropy generation. INTRODUCTION Core courses in the second and third years of an undergraduate chemical engineering education typically deal with transport phenomena (Bird 2002, Deen 2012, Plawsky 2010), reaction engineering (Davis 2003, Fogler 2006, Levenspiel 1999), and thermodynamics (Atkins and De Paula 2010, Mortimer 2008, Sandler 2006, Silbey, et al. 2005, Smith, et al. 2008, Sonntag, et al. 1998). These three courses provide the fundamentals required of a competent chemical engineer. Balance equations are commonly included in the curriculum for extensive physical quantities: mass, energy, momentum and entropy. Employing balance equations in any system of interest requires that the control volume be clearly specified upfront; otherwise, any further analysis would be irrelevant. Most textbooks on transport phenomena (Bird, 2002; Deen, 2012; Plawsky, 2010) and reaction engineering (Fogler, 2006; Levenspiel, 1999; Davis, 2003) specified control volume prior to performing balance equations to solve illustrative examples. Unfortunately, most existing textbooks on thermodynamics fail to address the importance of defining the control volume (Atkins and De Paula, 2010; Smith et al., 2008; Mortimer, 2008; Sandler, 2006; Silbey et al., 2005; Sonntag et al., 1998). Specifically, the concept of control volume is introduced in the derivation of equations for the balance of mass, energy, and entropy; however, this concept is not even touched upon in the application of these balance equations in solving illustrative problems. Take for example, the expansion of gas in a piston-and-cylinder device, which appears in many textbooks on thermodynamics as an illustration of energy and entropy balance. In the existing chemical engineering thermodynamics education, the energy and entropy balance equations are used for solving the piston-and-cylinder problems without explicitly specifying the control volume. This approach is unacceptable because thermodynamic quantities, such as

2 Academia Journal of Scientific Research; Kang and Liou. 086 work, heat and entropy, vary with the control volume. A failure to specifically define the control volume prior to performing balance equations could lead to ambiguous and/or confusing results. For example, control volume can be specified with or without the piston and the magnitude of work would vary accordingly. Furthermore, varying the selection of the control volume can help to disclose detailed information related to the process of interest (Sahin, 2012). In the case of confined gas expanding in a piston-and-cylinder device, no entropy is generated during expansion, such that analysis using an appropriate set of control volumes would make it possible to identify the main source of the entropy generation. In this work, we addressed the issue of specifying the control to facilitate the analysis of energy and entropy balance in thermodynamic systems or devices. For the sake of illustration, we adopted the classic example involving the expansion of gas in a piston-and-cylinder device. Energy and entropy balance equations were applied using three different sets of control volumes. The thermodynamic quantities of internal energy, work, heat, entropy and entropy generation were computed for different control volumes according to the balance of energy and entropy. We discussed the origin of entropy generation based upon our obtained results and also discussed the consistency of analysis results obtained using various control volumes. THEORY Balance equations Investigating changes in energy and entropy during the expansion of confined gas in a piston-and-cylinder device requires balance equations for these two thermodynamic quantities. The energy balance equation for a given control volume during the span of time from t 1 to t 2 is as follows (Sandler, 2006): q(1) Where U is the total internal energy in a given control volume, k is the k th mass stream flowing across the control surface, l is the molar flow rate of the k th stream, l is the molar enthalpy of the k th stream, Q represents heat conduction across the control surface, and W PV refers to the PV work due to the expansion or contraction of the control volume. It should be noted that the energy balance equation in Equation (1) does not include work related to the shaft, because it is not a part of the device being discussed. The entropy balance for a control volume is written as follows (Sandler, 2006): (2) Where S is the total entropy in the control volume,. is the molar entropy of the k th stream, T indicates the temperature of the control surface across q which the heat isconducted, and Sgen represents entropy generation in the control volume due during the span of time from t 1 to t 2. Constitutive equations Balance equations as well as, several constitutive equations are required to express PV work, internal energy, enthalpy and entropy. PV work can be expressed as follows (Sandler, 2006): (3) Where V is the volume of the control volume, V 1 and V 2 represent the volumes at t 1 and t 2 of the control volume, respectively, and P MB indicates the pressure at the boundary that moves during the expansion process. Using a control volume for a piston-and-cylinder device in which the moving surface is identical to the cross section of the cylinder, Equation (3) can be rewritten as follows: (4) Where F MB is the normal force exerted on the control surface due to pressure at the surface, l is the distance traveled by the moving surface, and l 1 and l 2 indicate the positions of the moving surface at times t 1 and t 2, respectively. The internal molar energy for an ideal gas with a fixed heat capacity at constant volume can be written as follows (Sandler, 2006): (5) Where C v is the heat capacity at constant volume, T is the temperature in the control volume, and T R is the reference temperature, which is an arbitrary value. The molar enthalpy of an ideal gas with a fixed heat capacity under constant pressure can be written as follows (Sandler, 2006):

Academia Journal of Scientific Research; Kang and Liou. 087 Figure 1. Illustration of the expansion of confined gas in piston-and-cylinder device.

3 Academia Journal of Scientific Research; Kang and Liou. 087 Figure 1. Illustration of the expansion of confined gas in piston-and-cylinder device. (6) Where C p is the heat capacity under constant pressure and Ris the gas constant. The molar entropy of an ideal gas is written as follows (Sandler, 2006): (7) Where V is the molar volume of the gas in its present state, V R is the molar volume of the gas in the reference state and indicates the molar entropy in the reference state. In this work, we considered a confined ideal gas, which follows the ideal gas law as follows: (8) Where P represents the pressure of the gas. RESULTS AND DISCUSSION Figure 1 illustrates the expansion of a confined gas in a piston-and-cylinder device. In this case, we considered 1 mole of an ideal gas confined within a piston-and-cylinder device with a 100 kg weight on the top. The weight is removed at t 1 to initiate the expansion of gas, which occurs under isothermal conditions at K. The external pressure outside the cylinder is Pa. The piston is assumed to be massless with a cross-sectional area of 0.01 m 2. The initial pressure of the confined gas is Pa and the initial volume is m 3. Friction present between the piston and the interior wall of the cylinder means that the piston will eventually be damped, at which time the expansion of the gas will cease. At the end of the gas expansion (t 2), the external pressure and the mass of the piston produced a final confined gas pressure of Pa and final volume of m 3. During the expansion process, the piston travels m. In this study, we calculated the changes in internal energy and entropy, the PV work, and the entropy generation in the system between t 1 and t 2. Analysis is performed using three sets of control volumes, each of which may contain more than one control volume. Finally, we compared the results obtained from the analysis using the three sets of control volumes. Approach A: Analysis using a single control volume We began our investigation of the energy and entropy balance in the expansion of confined gas using a single control volume. This control volume referred to as CV-A, encloses the entirety of the confined gas and piston and the top surface of the control volume is identical to that of the piston (Figure 2). The top surface of the control volume is a moving boundary traveling with the top surface of the piston during the expansion of the gas. This situation is identical to the gas-expansion problem found in most thermodynamics textbooks; however, the control volume is generally not specified. In the case of this control volume, no mass is transferred across the control surface and the entire process occurs isothermally, such that the energy balance equation, Equation (1) can be simplified as follows: Q +W PV = 0 (9) For this specified control volume, the only moving boundary of the control surface is its top surface and the pressure at the top surface is Pa, which remains constant throughout the entire expansion process. The

4 Academia Journal of Scientific Research; Kang and Liou. 088 Figure 2. Illustration of CV-A. Figure 3. Illustration of CV-B-I and CV-B-II. work performed on the control volume can be computed using Equation (3), which yields W PV= J. The heat transferred across the control surface can be calculated using Equation (9), which results in Q= J. As mentioned earlier, no mass is transferred across this control volume; therefore, the entropy balance in Equation (2) can be expressed as follows:. (10) with the gas expansion process has been derived using the conventional approach to the specifying of the control volume. This approach to the derivation of control volume is valid; however, it leaves one unanswered critical question: What is the origin of entropy generation? Or more specifically, which part of the piston-and-cylinder device is responsible for entropy generation? This question answered by investigating the same process using a different set of control volumes. Changes in the entropy of the control volume (the lefthand side of Equation (10) can be estimated using Equation (7), which yields. We can then obtain S gen = J/ using Equation (10). Thus far, every thermodynamic property of interest associated Approach B: Analysis using two control volumes In the following, we analyzed the same expansion of a confined gas using two control volumes, as illustrated in Figure 3. The first control volume, referred to as CV-B-I, encloses only the confined gas. The top surface of CV-B-I is identical to the bottom surface of the piston. The second

5 Academia Journal of Scientific Research; Kang and Liou. 089 control volume, referred to as CV-B-II, encloses the entire piston, such that the control volume moves with the piston. The energy balance for CV-B-I can be expressed by in Equation (9), in which W PV can be estimated using Equation (3). The only moving boundary of CV-B-I is the top control surface, such that P MB should be identical to the pressure of the confined gas. Thus, the W PV of CV-B-I can be computed as follows: Table 1. Thermodynamic properties in CV-B-I and CV-B-II. Thermodynamic properties CV-B-I CV-B-II (J) 0 0 (J) (J) (J/K) (J/K) (11) Where Q for CV-B-I is obtained using Equation (9). Applying Equations (7) and (9) to CV-II makes it possible to deduce the changes in entropy and entropy generation in CV-I: and. The aforementioned derivation leads to one important finding: no entropy is generated in CV-B-I. At first, this may sound surprising; however, it is quite reasonable because none of the potential sources of entropy generation (friction, concentration gradient, temperature gradient, or viscous force) are present in CV-B-I. By taking into account our analysis, the entire piston-and-cylinder device (CV-A) has entropy generation of J/K during the gas expansion process, which implies that entropy generation must be due to the piston rather than the confined gas. We assessed this speculation by performing energy and entropy balance on CV-B-II. The top and bottom control surfaces provided two moving boundaries against pressure in CV-II. The top control surface moves against external pressure, whereas the bottom control surface moves against the pressure of the confined gas. Thus, the work caused by the top and bottom control surfaces should be estimated separately. Summation of these two work values gives the net work performed on CV-B-II. Using Equation (4), the work induced by the movement of the top and bottom control surfaces is J and J, respectively. Thus, the net work performed on CV-B-II is J. Next, the Q of CV- II calculated using Equation (9) is J. CV-B-II does not encloses any fluid; therefore, there should be no change in entropy; that is,. According to Equation (10), the S gen in CV-B-II during the expansion of gas is J/K, which is identical to the total entropy generatedin CV-A. Several things concerning the aforementioned derivations are worth mentioning. First, we have demonstrated that the generation of entropy during the expansion of gas can be attributed to the piston. Second, the origin of the entropy generated by the piston is the heat conducted across the control surface, which is produced by friction. This supposition is in agreement (J/K) with the findings in our previous work (Kang et al., 2015). Third, our analysis in this section demonstrated that decomposing a device or a system into several control volumes makes it possible to identify the parts of a device or system responsible for the generation of entropy. The thermodynamic properties of CV-B-I and CV-B-II are summarized in Table 1. Approach C: Analysis using two other control volumes Here, we illustrated how the control volume of a given device can be specified in an entirely arbitrary manner. In this case, we used two control volumes to enclose the entire piston-and-cylinder device, as illustrated in Figure 4. The first control volume, referred to as CV-C-I, is identical to the initial volume of the confined gas. CV-C-I has a fixed volume: that is, no moving boundary is present. The initial volume of the second control volume, referred to as CV-C-II, is identical to the volume of the piston. The bottom control surface of CV-C-II is fixed, whereas the top control surface moves with the top surface of the piston. Unlike CV-A, CV-B-I and CV-B-II, the control volumes CV-C- I and CV-C-II present the transfer of mass across their control surfaces. Specifically, during the expansion of gas, there is a departure of mass from the top control surface of CV-C-I, as well as, the entry of mass into CV-C-II from its bottom control surface. As a consequence, the terms and in Equations (1) and (2) are no longer absent when balancing energy and entropy incv-c-i and CV-C-II. The estimation of each term in Equations (1) and (2) can be realized using Equations (3 to 8). The details involved in these calculations are beyond the scope of this study; however, the thermodynamic properties in CV-C-I and CV-C-II are outlined in Table 2. There are several observations worth pointing out. First, no work is performed on or by CV-C-I, because its volume is fixed. Second, there is zero entropy generation in CV-C-I. The reason is very similar to the

6 Academia Journal of Scientific Research; Kang and Liou. 090 Figure 4. Illustration of CV-C-I and CV-C-II. Table 2.Thermodynamic properties of CV-C-I and CV-C-II. Thermodynamic properties CV-C-I CV-C-II (J) (J) (J) (J) (J/K) (J/K) (J/K) (J/K) same observation in CV-B-I; that is, friction, temperature and concentration gradient are absent from these two control volumes. Third, entropy generation is observed only in CV-C-II, because this control volume encloses the piston, which generates friction (the origin of entropy generation) through its interaction with the interior wall of the cylinder. Comparison of various approaches We have outlined three different approaches using different sets of control volumes for the analysis of the energy and entropy balance in the expansion of gas in a piston-and-cylinder device. Here, we compared the results obtained using these methods and discussed the consistency among them. Geometrically, the union of CV-B- I and CV-B-II and the union of CV-C-I and CV-C-II are identical. Furthermore, both of these unions are equivalent to CV-A (Figure 5). The thermodynamic quantities associated with CV-B can be estimated by summing up the quantities of CV-B-I and CV-B-II. The thermodynamic quantities of CV-C are estimated by analogy. The computed thermodynamic properties for CV-A, CV-B (CV-B-IUCV-B- II), and CV-C (CV-C-IUCV-C-II) are summarized in Table 3. Our results demonstrated that all physical quantities obtained using CV-A, CV-B and CV-C is identical, thereby demonstrating the consistency among these approaches to analysis using various sets of control volumes. This degree of consistency demonstrated the validity of determining the balance of energy and entropy using various control volumes for a given device or system, as long as the

7 Academia Journal of Scientific Research; Kang and Liou. 091 Figure 5. Illustration of the control volume equality among CV-A, CV-B, and CV-C. Table 3. Thermodynamic properties of various control volumes discussed in this work. Thermodynamic properties CV-A CV-B = CV-B-I CV-B-II CV-C = CV-C-I CV-C-II (J) 0 0* 0* (J) * * (J) * * (J/K) * 5.625* (J/K) * 4.087* (J/K) * 1.537* *These values are computed by summing the values for CV-B-I and CV-B-II; **these values are computed as by summing the values for CV-C-I and CV-C-II

8 Academia Journal of Scientific Research; Kang and Liou. 092 balance and constitutive equations are applied correctly. Conclusions The role of the control volume has largely been overlooked in the education of thermodynamics for chemical engineers. Students will have ambiguous and confusing results when they failed to specify the control volume prior to performing the energy and entropy balance on thermodynamic systems. The most important point in this work is the need to specify the control volume prior to the application of energy and entropy balance to thermodynamic devices or systems. We strongly suggest that the role of control volume be more strongly emphasized in the instruction of thermodynamics at the undergraduate as well as graduate levels. Concluding remarks are listed in the following: (1) The application of energy and entropy balance to a thermodynamic device/system without explicitly specifying the control volume is meaningless. (2) For any given device or system, the specifications of control volume are arbitrary; i.e., they remain entirely independent from the device or system. As illustrated by the piston-and-cylinder model in this study, there should theoretically be an infinite number of ways to specify the control volumes for a device or system of interest. (3) The judicious selection of control volume(s) makes it possible to identify the part of a device or system responsible for the generation of entropy. REFERENCES Atkins (PW), De Paula J (2010). Atkins' Physical chemistry, Oxford University Press, New York. Bird RB (2002). Transport phenomena, 2nd ed. ed., J. Wiley, New York. Davis ME (2003). Fundamentals of chemical reaction engineering, International ed. ed., Boston, McGraw-Hill. Deen WM (2012). Analysis of transport phenomena, Oxford University Press, New York. Fogler HS (2006). Elements of chemical reaction engineering, Prentice Hall PTR, Upper Saddle River, NJ. Kang DY, Liou KH, Chang WL (2015). Investigating Friction as a Main Source of Entropy Generation in the Expansion of Confined Gas in a Piston-and-Cylinder Device. J. Chem. Educ. 92(10): Levenspiel O (1999). Chemical reaction engineering, Wiley, New York. Mortimer RG (2008), Physical chemistry, Academic Press/Elsevier, Boston. Plawsky JL (2010). Transport phenomena fundamentals, CRC Press, Boca Raton. Sahin AZ (2012). The thermodynamic quantity minimized in steady heat and fluid flow processes: A control volume approach. Energy Conv. Manag. 59: Sandler SI (2006), Chemical, biochemical, and engineering thermodynamics, 4th ed. ed., Hoboken, NJ, John Wiley & Sons. Silbey RJ, Alberty RA, Bawendi MG (2005). Physical chemistry, Wiley, Hoboken, NJ. Smith JM, Abbott MM, Van Ness HC, Yang Z (2008). Introduction to chemical engineering thermodynamics = Hua gong re li xue, Mei shang mai ge luo.xi er guo ji gong si, Tai bei shi. Sonntag RE, Borgnakke C, Van Wylen GJ (1998). Fundamentals of thermodynamics, Wiley, New York. The discussion in the present work is limited to an ideal gas as the working fluid. Nonetheless, this analysis is applicable to real fluids as well in the education of thermodynamics for chemical engineers. We expect that in the future, chemical engineers will be able to use the strategy of specifying control volume(s) proposed in this work for a given system or a device in order to facilitate investigations into local work production and entropy generation. This methodology could lead to a novel approach to performance optimization in existing and new chemical engineering processes and devices. ACKNOWLEDGMENTS This work was jointly supported by the Ministry of Science and Technology (MOST) of Taiwan (MOST E MY3 and MOST E MY2) and National Taiwan University (NTU-CDP-105R7814). The authors would like to thank Prof. Shiang-Tai Lin for his valuable insights. Cite this article as: Kang DY, Liou KH (2016). Critical Role of Control Volume in Thermodynamics Education for the Analysis of Energy and Entropy Balance. Acad. J. Sci. Res. 4(4): Submit your manuscript at

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