Piston Surface Pressure of Piston- Cylinder System with Finite Piston Speed

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1 44, Issue (08) 55-5 Journal of Adanced Research in Fluid Mechanics and Thermal Sciences Journal homeage: ISSN: Piston Surface Pressure of Piston- Cylinder System with Finite Piston Seed Oen Access Siti Nurul Akmal Yusof, Yutaka Asako,, Tan Lit Ken, Nor Azwadi Che Sidik Deartment of Mechanical Precision Engineering, Malaysia-Jaan International Institute of Technology, Uniersity Technology Malaysia, Jalan Sultan Yahya Petra, 5400 Kuala Lumur, Malaysia ARTICLE INFO Article history: Receied 9 February 08 Receied in reised form Aril 08 Acceted 9 Aril 08 Aailable online Aril 08 Keywords: irreersible comression rocess, ressure, ideal gas, kinetic molecular theory ABSTRACT Understanding the thermodynamics fundamental are imortant for the adance of energy and enironmental technologies. The study aims to derie an exression of the rage ressure on the iston surface, in an adiabatic iston-cylinder system during an irreersible rocess with finite iston seed using the kinetic molecular theory. Exressions for the rage ressure on the iston surface, are deried from the change of momentum and change of energy. The Maxwell- Boltzmann distribution is used to estimate the gas molecule elocities. The iston surface ressure obtained from the change of momentum is 37.8% lower than that obtained by the reious study. The iston surface ressure obtained from the change of energy is.94% lower than that obtained by the reious study. Coyright 08 PENERBIT AKADEMIA BARU - All rights resered. Introduction Design of heat engines, energy deices in a ower lant and thermo-fluid deices h increased the need for fundamental understanding of thermodynamics for the adance of energy and enironmental technologies [-]. A rocess of a comression or an exansion of a gas in a iston-cylinder system is common in many alications, howeer an irreersible rocess is not well understood [3-9]. In thermodynamics, state quantities at a final state in a reersible rocess can be determined. A reersible rocess may occur when a system is maintained continuously and thermally at an equilibrium state [9-]. Therefore, the reersible rocess is also called a quasistatic or a quasi-equilibrium rocess [0]. The rocess is reersible when a iston moes with zero elocity in a iston-cylinder system. On the other hand, the thermal equilibrium state breaks in the system and the rocess becomes irreersible when the iston moes with infinite elocity. In general, we cannot determine the state quantities at a final state in an irreersible rocess. The only excetion is a throttling rocess when a gas or a steam asses through a caillary tube or a Corresonding author. address: y.asako@utm.my (Yutaka Asako) 55

2 orous material. For examle, in an adiabatic throttling rocess, the secific enthaly at the final state is identical to the secific enthaly at the initial state. Then, the state quantities at the final state can be determined. This strictly highlighted in a thermodynamic textbook [] that we can make calculations only for a reersible rocess. Exressions for the rage ressure on the iston surface in an adiabatic iston-cylinder system during an irreersible rocess with finite iston elocity using the kinetic molecular theory has been deried by Petrescu et al., [4]. Howeer, the deriing rocess and the molecular elocity distribution they used are not clearly described. The study aims to derie an exression for the rage ressure on the iston surface, in an adiabatic iston-cylinder system during an irreersible comression rocess with finite iston elocity using the kinetic molecular theory with Maxwell-Boltzmann distribution.. An Ideal Gas Laws Various studies h inestigated of a reersible rocess in closed system because it can be used as a medium to analyze the irreersible rocess occur in the system and as an imortant source in understanding the fundamental of thermodynamics [3-8,5]. A reersible rocess is the rocess in which the system and surroundings can be restored to the initial state from the final state without roducing any changes in their equilibrium states. The iston moes with ery slow moement will result in a uniform gas ressure. This henomenon can be exressed in first laws of thermodynamics for a closed system [-3, 0-, ]. d q = di + d w () where dwis the secific work done by the fluid to the iston. The secific work done by the fluid to the iston is exressed as d w = d () where is the rage ressure on the iston surface. Note that Eqs. () and () are alid for both reersible and irreersible rocesses. Substituting Eq. () into Eq. () for the case of an adiabatic rocess ( d q = 0), the change of internal energy is gien as di = d (3) For an ideal gas di = C dt (4) If the rocess is a reersible rocess d = = (5) dw d RT Substituting Eqs. (4) and (5) into Eq. (3), the following equation can be obtained 5

3 d CdT = RT () Note that for ideal gas, R = C C, γ is the ratio between the secific heats, C C. For a diatomic molecule, γ =.4. Then, by integrating Eq. () between ( T, ) and (, ) equation can be obtained i i T, the following γ i T Ti = (7) Eq. (7) is the general equation that can be used to obtain the initial and final temeratures and olumes for an adiabatic reersible rocess in a iston-cylinder system. The ressure in a system is exressed as γ i = i (8) Note that in case of a reersible rocess, the ressure in the system is uniform and is identical to the rage ressure on the iston surface =. In a case of an irreersible rocess, the ressure in the system is not uniform. The following correlation for the work done by the fluid is widely known when the rage ressure in the system is denoted by d w < d (9) Howeer, it is not known how the work done by the fluid is less than d. Note that Eq. () is aailable for both cases of reersible and irreersible rocesses. Substituting Eq. (9) into Eq. (), we obtain d d (0) Therefore, the following correlation for a comression rocess is obtained (comression rocess) () The rage ressure on the iston surface, using the kinetic molecular theory will be deried in the next section. 3. Kinetic Molecular Theory The kinetic molecular theory for an ideal gas simly exressed that all articles are moing in a random, constant and straight-line motion. The ressure of a gas then results from the collisions of the molecules of the gas with the walls of the container. 57

4 3. Maxwell (-Boltzmann) distribution Figure shows a stationary container in which single-atomic gas such as Ar, Ne or He is filled. Since the container is stationary, the gas molecule trls with the thermal elocity denoted by C r. The thermal elocity is not a unique alue and obeys the Maxwell distribution [7-8]. The Maxwell distribution is exressed as m mc f ( C) = 4π C ex π k T k T 3 () where m, k and T are mass of a molecule in kg, the Boltzmann constant and temerature in K, resectiely. Fig.. Molecules in a container The Maxwell distributions of Ar, Ne and He gases at room temerature (300 K) are lotted in Figure as a form of the robability distribution. The effect of temerature on the molecular elocity of Ar gas is shown in Figure 3. Note that the y-axis in Figures and 3 is in s/m so that the area under any section of the cure is dimensionless. Fig.. Maxwell distribution of Ar, Ne and He gases at 300 K Fig. 3. Effect of temerature on molecular elocity of Ar gas 58

5 3. Boltzmann constant Table listed the arameters inoled in obtaining Boltzmann constant. Table The arameters inoled in obtaining Boltzmann constant The number density, N Volume, V Pressure, Temerature, T ( N = N / V ) A m The Boltzmann constant, ( k = / NT ) Pa 73.5 K where N A is the Aogadro constant (N A =.04 0 /kmol) and V reresents the olume of a gas of kmol at arbitrary ressure and temerature. 3.3 Tyical elocities There are three tyical elocities, they are the mean elocity, most robable elocity and rootmean-square elocity can be obtained from roerties of the Maxwell distribution. The most robable elocity, C m, is the elocity most likely to be ossessed by any molecule (of the same mass m) in the system and corresonds to the maximum alue. To find it, we calculate df/dc, set it to zero and sole for C. ( ) df C 0 dc = (3) which yields C kt m R T M 0 m = = = RT (4) where R 0 is the uniersal gas constant ( J/(kmol K)), M is the molar weight of a substance and R is the gas constant, J/(kg K). For diatomic nitrogen, N, at room temerature (300 K), this gies C m = 4 m/s. The mean elocity is the exected alue of the elocity distribution 8kT C = C f C dc = = C π m π ( ) m (5) 0 The root mean square elocity is 3kT 3R0T 3 Crms = C = C f C dc = = = C m M 0 ( ) m () These tyical elocities are related as follows 59

6 ( 9 ) ( 5 ) C < C =. C < C =. C (7) m m rms m 3.4 Number of Molecules which Collide with a Wall Although the hard shere model redicts more accurate one but using the molecular-meanfree-ath model the ressure will be deried. The molecule is treated as a lumed mass. Besides, the following assumtions will be made. - The molecules are in random motion. - The collisions with the walls and other molecules are erfectly elastic. This means, the kinetic energy is consered during the collision. - Excet during collisions, the interaction among molecules is negligible. Note that intermolecular forces (e.g. Van der Waals force) are considered in the molecular dynamics (MD). In the kinetic theory, the interaction among molecules is not considered. Based on these assumtions, we can agree that mechanical energy is consered in collisions between molecules. For the sake of simlicity, we should consider a rectangular container (unit area length C rms, see Figure 4) and assume that all gas molecules trl with the root-mean-square elocity C rms. Since the olume of this container is C rms, the number of the molecules in the container is N C where N is the number density which reresents number of the molecules in rms unit olume. Since each NCrms molecules trl in ±x, ±y and ±z directions, the number of molecules which collide with one wall er unit area and er unit time is NC rms (8) 3.5 Pressure When a gas molecule collides with the wall, the molecule bounces off in the oosite direction with the same elocity (an elastic collision). The change in momentum during the collision is ( ) mc mc = mc (9) rms rms rms The change in momentum er unit area and er unit time is N Crms mcrms N m Crms N m C 3 3 ( ) = = (0) The change in momentum er unit area and er unit time reresents the force acting on the wall er unit area. Thus, the ressure = N m Crms = N m C = ρ Crms = ρ C ()

7 C rms [m] Fig. 4. Collision of molecules in a rectangular container 3. Collision with moing wall (uniform temerature case) Consider a iston cylinder system in which single-atomic gas is filled and the iston is moing with constant elocity, r, as shown in Figure 5. Here, we assume that the temerature in the iston-cylinder system including the iston surface is uniform. Each molecule in the iston-cylinder system trls with the thermal elocity, C r i based on the uniform temerature. Attention will now be turned to the kinetic energy that the molecule brings to the iston when the molecule collides with the iston. The sum of the kinetic energy that the molecules bring to the iston er unit area and unit time is exressed as NC rms Earrie = mc r i i= () We further assume that the molecule which arries on the iston surface is thermally fully accommodated with the surface. If the reflection on the iston surface is fully diffuse, molecules reflect with the thermal elocity lus the iston elocity as in Figure 5. Therefore, the total kinetic energy les from the wall is exressed as NC rms r Ele = m C + i= r ( i ) (3) mr m C r i r m r r C + i Fig. 5. Energy transort by molecules colliding with the wall

8 Consider the mean-square of the thermal elocity denoted by () as C r. C r can be obtained from Eq. NC rms r = i = rms NC rms i= r C C C (4) Substituting Eq. (4) into Eqs. () and (3), then E arrie NCrms m = Crms (5) ( rms ) NCrms m Ele = C + () The difference of Eq. (5) and () is increase of the energy er unit time and it is equal to the work done by the iston as ( rms ) rms ( ) NCrms m NCrms m NCrms m C + C = = (7) Eq. (7) can be rewritten as NCrms m = + (8) Substituting Eq. () into Eq. (8), we obtain = + (9) 4C rms Substituting Eq. () into Eq. (9), Eq. (9) can be rewritten as γ γ = + = + = Ma + (30) 4 3RT 4 3 γ RT 4 3 Howeer, Petrescu et al., [4] obtained = ± 3γ = ± γ Ma (3) 3RT When a gas molecule collides with the iston, the molecule bounces off in the oosite direction with the same elocity (an elastic collision). The change in momentum during the collision is

9 ( ) mc mc = mc + (3) rms rms rms The change in momentum er unit area and er unit time is nc rms ( mcrms + m ) = NmC + NmC rms rms 3 (33) The change in momentum er unit area and er unit time reresents the force acting on the iston surface er unit area. Thus, the rage ressure on the iston surface = + (34) 3 NmC NmC rms rms Then, γ = + = + = + Ma (35) C 3RT 3 rms To comare the ratio of rage ressure on the iston surface to the rage ressure, / obtained using energy equation from Eq. (30) and the change of momentum from Eq. (35) with the results obtained by Petrescu et al., [4] from Eq. (3), all the results are lotted as a function of Mach number, Ma in Fig.. Both the ratios of rage ressure on the iston surface to the rage ressure, / obtained from Eqs. (30) and (35) are lower than the results obtained by Petrescu et al., [4]. In the case of Mach number, 0.5, the differences ratio of rage ressure on the iston surface to the rage ressure, / between Petrescu et al., [4] with the Eqs. (35) and (30) are 37.8% and.94%, resectiely. Fig.. Comarison of / with reious results The differences are because the reious studies [4] did not considered the energy conersion from the kinetic energy to thermal energy after the iston sto. And they also derie the Eq. (3) based on the following assumtions 3

10 - Clausius kinetic-molecular model to determine the analytical exressions of temerature, internal energy, and ressure for a system of refect gases in equilibrium, deending on the total number and the seed of the inside molecules. - All molecules can h on a continuous scale, any seed alue within this range (from a minimum seed threshold C and a maximum C N ). - After collision a certain number of molecules changes its trajectory and les the cluster, they are immediately relaced by an equal number of molecules migrated from other clusters. 4. Conclusions In this aer, a single comression rocess in an adiabatic iston-cylinder system has been analysed using kinetic molecular theory. The following conclusions are obtained.. The ratio of rage ressure on the iston surface to the rage ressure, / obtained from the change in momentum, Eq. (35) and energy equation, Eq. (30) are lower than the results obtained by Petrescu el al., [4].. The differences ratio of rage ressure on the iston surface to the rage ressure, / between Petrescu el al., with the Eqs. (35) and (30) are 37.8% and.94%, resectiely. 3. The results show differences between reious studies because they did not consider the energy conersion from the kinetic energy to thermal energy after the iston sto. Acknowledgement The authors would like to exress their areciation to Uniersity Technology Malaysia and Takasago Thermal Engineering Co, Ltd., Jaan, for roiding financial suort for this work through Takasago education and research grant (Vote No: R. K B34). References [] Abdel-Aziz Farouk Abdel-Aziz Mohamed, Hybrid Nanocrystal Photooltaic/Wind Turbine Power Generation System in Buildings, J. Ad. Res. Mater. Sci. 40, no. (08): 8-9. [] Ny, G., N. Barom, S. Noraziman, and S. Yeow. "Numerical study on turbulent-forced conectie heat transfer of Ag/Heg water nanofluid in ie." J. Ad. Res. Mater. Sci., no. (0): -7. [3] Anacleto, Joaquim, and J. M. Ferreira. "On the reresentation of thermodynamic rocesses." Euroean Journal of Physics3, no. 3 (05): [4] Anacleto, Joaquim, J. M. Ferreira, and A. A. Soares. "When an adiabatic irreersible exansion or comression becomes reersible." Euroean Journal of Physics 30, no. 3 (009): 487. [5] Gislason, Eric A., and Norman C. Craig. "First law of thermodynamics; Irreersible and reersible rocesses." Journal of chemical education 79, no. (00): 93. [] Plotniko, L. V., and B. P. Zhilkin. "The gas-dynamic unsteadiness effects on heat transfer in the intake and exhaust systems of iston internal combustion engines." International Journal of Heat and Mass Transfer 5 (07): 8-9. [7] Sieniutycz, Stanislaw. "Variational setting for reersible and irreersible fluids with heat flow." International Journal of Heat and Mass Transfer 5, no. - (008): [8] Zhao, Yingru, and Jincan Chen. "An irreersible heat engine model including three tyical thermodynamic cycles and their otimum erformance analysis." International Journal of Thermal Sciences 4, no. (007): [9] Norton, John D. "The imossible rocess: Thermodynamic reersibility." Studies in History and Philosohy of Science Part B: Studies in History and Philosohy of Modern Physics55 (0): 43-. [0] Paul, D. R., and J. W. Barlow. "Basic thermodynamics." J Macromol Sol (C) 8 (980): 09-. [] Cengel, Yunus A., and Michael A. Boles. "Thermodynamics: an engineering aroach." Sea 000 (00): 88. [] Klein, Martin J. "Thermodynamics in Einstein's thought." Science 57, no (97):

11 [3] Christians, Joseh. "Aroach for teaching olytroic rocesses based on the energy transfer ratio." International Journal of Mechanical Engineering Education 40, no. (0): [4] Petrescu, S., B. Borcila, M. Costea, E. Banches, G. Poescu, N. Boriaru, C. Stanciu, and C. Dobre. "Concets and fundamental equations in Thermodynamics with Finite Seed." In IOP Conference Series: Materials Science and Engineering, ol. 47, no., IOP Publishing, 0. [5] Sommerfeld, Arnold. Lectures on Theoretical Physics: Thermodynamics and statistical mechanics. Vol. 5. Academic ress, 0. [] Sobel, Michael I. "Kinetic theory deriation of the adiabatic law for ideal gases." American Journal of Physics 48 (980): [7] Semat, Henry, and Robert Katz. "Physics, Chater : Kinetic Theory of Gases." (958). [8] Gruber, Christian, and Gary P. Morriss. "A Boltzmann equation aroach to the dynamics of the simle iston." Journal of statistical hysics 3, no. - (003):