Mathematical Method for Evaluating Heat Capacity Ratio and Joule-Thompson Coefficient for Real Gases Wordu, A. A, Ojong, O. E Department of Chemical/Petrochemical Engineering University of Science and Technology Nkpolu, Port Harcourt, Nigeria. ABSTRACT Mathematical procedure for estimating Joule- Thompson coefficient η, heat capacity ratio γ and Shaft work done W S using C V, C P, z, R as precursor parameters for calculation is presented. The research significance lies on its applications as a quick check for these parameters by design engineers during design of some process plant units i.e compressors, reactors, refrigeration system, power plant and utilities, piping systems etc. The method gave acceptable deviation that can be applied in process designs. They are shaft work done 10.6%, heat capacity ratio 1.42% and Joule- Thompson coefficient 2.05%. Keywords: Mathematics, Joule-Thompson coefficient, heat capacity ratio, Shaft work done, derivative-compressibility, real gases. 1. INTRODUCTION A gas is a collection of particles in a constant, rapid and random motion. It is one of the four states of matter which includes; solid, liquid, gas and plasma. Gases can be subdivided into ideal gas and real gas. Real gases are made up of atoms or molecules that actually take up some space (i.e. they have volume) no matter how small, there exist forces of attraction between their particles, there collisions are inelastic and they do not obey the ideal gas law. Real gases approximate ideal gases at low temperature (298K) and pressure (1bar). However, at higher pressures, there exists a wide variation in behavior of real gases with respect to ideal gas. Consequently, accurate determination of real gases physical properties can be achieved by accounting for the deviation from ideality. Some physical properties of interest for real gases includes; temperature, pressure, viscosity, heat-capacity ratio, joule-thomson coefficient. Physical properties of real gases enables engineers to know the state of a system, monitor operational performance of process equipment and streams, determine operational efficiency etc. Physical properties of real gases are an essential tool for proper design of equipment such as compressors, piping systems, reactors, refrigeration systems [Reid, Prausnitz, and Poling, 1987]. Many engineers prefer to look up physical property data in tables. However, some commonly needed properties of real gases, such as heat capacity ratio, r, and the joule-thomson coefficient, n, are frequently not available in the tables. Consequently engineers require alternative procedures of determining these properties. Hence, research focuses on mathematical procedure for evaluating two real gases properties and further replicates the research for engineering practice. The significance of this research is to provide a theoretical and/or analytical pathway for estimating physical properties of real gases. 2. MATERIALS The necessary materials for evaluating real gases physical properties comprises of various thermodynamic and hydraulic equations obtained from literatures [Perry and Green, 2008], [Reid and Sherwood, 1977]. Compressibility factor z, Mollier chart, Pitzer Accentric factor, Critical point, Critical pressure, Critical temperature, Reduced temperature, and Heat capacity are utilized [Perry and Green, 2008], [Reid and Sherwood, 1977]. 3. METHOD The derivational process of the heat capacity ratio, Joule-Thompson coefficient, shaft work done and compressibility are elucidated from known thermodynamic principles or laws. 3.1 Heat capacity ratio Typically, when working with flows of compressible fluids, design engineers need a value for γ [Smith and Van Ness, 1992] ISSN: 2231-5381 http://www.ijettjournal.org Page 406
(1) The real gas heat capacity at constant pressure C p is obtained from equation (2) below as given by [Lee and Kessler 1999]. (2) (3) (4) The real gases heat capacity at constant volumes,, are not as readily available. To derive, use is made of the concept of derivative compressibility factors [Reid and Sherwood, 1977] gave the definitions for these quantities and provide tables as calculated by [Reid and Valbert, 1962]. (5) (6) To get an expression for 1962]., we start with the Bridgman table of differentials as given by [Reid and Valbert, (7) (8) By definition: Hence is obtained explicitly by dividing (7) by (8). Hence (9) From ideal gas, (10) By comparing the expression for in (9) to the ideal gas expression for in equation (10), may be said to provide a correction from non ideality. ISSN: 2231-5381 http://www.ijettjournal.org Page 406
Can be expressed from non ideality of and the Lee kesler departure function, by substituting equation (3) into equation (9): (11) Is usually calculated from an empirical polynomial in terms of absolute Temperature: (12) Where A, B, C and D are constants that are specific for different substances. Hence, an explicit expression for real gas heat capacity is: (13) This is obtained when equations (12) and (4) are substituted into equation (3). Combing equations (13) and (9) results in an explicit expression for : (14) By substituting equations (14) and (13) into equation (1), an explicit expression for γ is obtained. (15) can now be calculated in terms of intensive variables and known or tabulated constants and functions. Like the departure functions for heat capacity and other important thermodynamic functions, the derivativecompressibility have been tabulated in a two term format, where the first term gives the simple (ideal) fluid function and the second term gives the departure from the simple fluid: (16) (17) For reference: (18) Equations (16) and (17) are obtained from equation(18) compressibility z converting ideal to real gases. (19) Equation (19) is Peng-Robinson cubic equation for evaluating compressibility z T [Peng and Robinson, 1976]. ISSN: 2231-5381 http://www.ijettjournal.org Page 407
3.2 Joule Thompson Coefficient ( ) as:, reflects the change in temperature of a gas as pressure changes, while enthalpy remains constant. It is defined (19) To calculate this quantity, we refer to the Bridgman tables: (20) (21) To get an explicit expression for divide equation (20) by equation (21): (22) By combining equation (22) with the relations for C p o in equation (12), and heat capacity departure from (3) and (4), η is in terms of tabulated values and extensive variables: (23) Summarily, Basic steps for evaluating γ and η are as outlined below: 1. Having values for w, T c and P c, get, (0),Z P and from the attached tables. 2. Calculate Z T using the equation (16) above. 3. Calculate Z P using the equation (17) above. 4. Look up values for and from the appropriate texts (e.g Reid, Prausnitz, and Poling). 5. Look up values for A, B, C and D for use in the heat capacity equation (12) above. 6. Calculate γ from the equation (15) or η from the equation (23) above. Specific calculations showing how to derive a value for γ and η, as well as their uses in process calculations is as demonstrated below: Problem Statement: Assuming adiabatic operation and a constant value of γ, estimate the power required to compress 50 lb/min of nitrous oxide (N 2 O) from 80 F (288.667k) and 250Psia (17.237bars) to 800Psia (55.158bars). Also, calculate the outlet temperature using the mollier diagram in Perry s Handbook, the enthalpy change is approximately 2.97 Btu/lb, yielding a requirement of 35.0hp. Solution stepwise Substance: Nitrous Oxide (N 2 O): ISSN: 2231-5381 http://www.ijettjournal.org Page 408
Molecular Weight: 44.013 Critical Temperature, T c = 310K Critical Pressure, P c = 12.40bars Accentric Factor, w = 0.165 Gas Constant, R = 8.314 J/gmolK Inlet conditions: Outlet conditions: Temperature = 80 F = 299.667K Pressure = 800Psia = 55.158bars Pressure = 250Psia = 17.237bars Reduced pressure, P r = 0.762 Reduced Temperature, T r = 0.968 Reduced pressure, P r = 0.238 Initially, we evaluate the physical properties, and hence γ, at the inlet conditions. Then, once a first value of T 2 is available, calculate a mean temperature and pressure and re-evaluate γ and continue, until successive values of T 2 converge. The ideal gas heat capacity is usually expressed as a polynomial of the absolute temperature: (24) Constants shown are from Reid, Prausnitz and Poling, using T in Kelvin. A = 21.620 e+00, B = 72.810e-03, C = -57.780e-06, D = 1.830e-09, J/gmolk Using the Reid and Valbert tables, interpolate to get the derivative compressibility functions at the desired (reduced) temperature and pressure Using these values, interpolate to find And Therefore For ; Using these values, interpolation gives ISSN: 2231-5381 http://www.ijettjournal.org Page 409
For Z T : Using these values, interpolate to find Using these values, interpolating gives From, For Using these values, interpolation gives For Given Using these values, interpolation gives ISSN: 2231-5381 http://www.ijettjournal.org Page 410
By using equation (4), γ = 1.419 Calculate the Exit Temperature We use equation [25] thermodynamic function (25) From which, T 2 = 410.919k Shaft work, ws; we use the equation; (26) From equation [26] shaft work done can be calculated as, 3361.81 J/gmol N 2 O =76.4J/g N 2 O =76.4 KJ/Kg N 2 O W, mass flow = 22.7 Kg/min Therefore P, power required = 1734.28 KJ/min = 38.72hp Power required from Mollier chart = 35.0hp approx. Deviation = 10.62% or 0.106 3.3 Determining Joule Thomson Coefficient η Estimate the final temperature when refrigerant 12 (dichlorodifluoromethane: CCl 2 F 2 ) expands through a nozzle from 300 psia (20.684 bars) and 180 F (355.2k) to 125 psia (8.618 bars). The enthalpy pressure chart in Perry s handbook shows a final temperature of -144 F (175k) Solutions Techniques Declaration of industrial and literature data for estimating γ and η Substance: Refrigerant 12(CCl 2 F 2 ) Molecular Weight = 120.913, Critical Temperature T c = 385.0k, Critical Pressure P c = 41.40 bars, Accentric Factor, w = 0.304, Gas Constant, R = 8.314 J/gmol K Inlet conditions: Outlet conditions: Temperature = 355.2k Pressure = 8.618 bars Pressure = 20.684 bars Reduced Pressure, Pr = 0.208 Reduced Temperature, Tr = 0.923 Reduced Pressure Pr = 0.500 The ideal gas heat capacity is usually expressed as a polynomial in terms of the absolute temperature: Constants shown here are from Reids, Pravsnitz and Poling, with temperature in Kelvin. A = 31.60 e+00, B = 178.20e-03, C = -150.90e-06, D = 43.420e-09, = 77.806 J/gmol K Using the Reid Valbert table and interpolate to get the derivative- compressibility functions at the desired (reduced) temperature and pressure. Since the inlet and outlet pressure are known, using the geometric mean of these two values as a reference point for interpolation. ISSN: 2231-5381 http://www.ijettjournal.org Page 411
Using these values and interpolating, we get Using these values, interpolating gives From For Z T : Using these values, interpolating gives From tables after interpolation, at P r = 0.354 and T r =0.932 = 1.521 From For Using these values, interpolation gives For Given ISSN: 2231-5381 http://www.ijettjournal.org Page 412
Using these values, interpolation gives The compressibility z is evaluated by solving the Peng Robinson cubic equation of state given by; Where a = 11.93e+06, b = 60.155, A* = 0.183, B* = 0.027, Z = 0.829 With a value of Z at hand as well as Z T, the ideal gas heat capacity and the heat capacity deviation, η can be calculated using equation (23) to get η = 1.616 k/bars The outlet temperature may be estimated assuming this value of η is an average value over the entire process. Then: From which Outlet Temperature = 335.876K From Pressure enthalpy diagram, the outlet temperature for this change is 334.26K. Thus, deviation = 2.05% 4. RESULTS AND DISCUSSION The scarce nature of these physical properties for gases data has propelled the essence of embarking on alternative method of estimation. Hence, the mathematical method has been devised. The method actually gave good deviations for engineering design applications. 5. CONCLUSION This work demonstrates the mathematical procedures for obtaining Joule-Thomson coefficient η, heat capacity ratio γ and shaft work done W s using appropriate thermodynamic relations in situations where they are not readily available for design process. Nomenclature a = Parameter in cubic equation of state. A = Empirical constant in ideal gas heat capacity polynomial. A* = Parameter in cubic equation of state. B = Parameter in cubic equation of state. B = Empirical constant in ideal gas heat capacity polynomial. B* = Parameter in cubic equation of state. C = Empirical constant in ideal gas heat capacity polynomial. C P = Real gas heat capacity (at constant pressure), J/gmol. = Ideal gas heat capacity (at constant pressure), J/gmol. C v = Real gas heat capacity (at constant volume), J/gmol. = Residual heat capacity (at constant pressure), J/gmol. = Simple fluid contribution to residual heat capacity (at constant pressure), J/gmol. = Departure contribution to residual heat capacity (at constant pressure), J/gmol. ISSN: 2231-5381 http://www.ijettjournal.org Page 413
D = Empirical constant in ideal gas heat capacity polynomial. γ = Heat capacity ratio, C p /C v η = Joule Thomson coefficient, change in temperature with respect to pressure of constant enthalpy, K/bar. P = Pressure, bars (atm). P c = Critical Pressure, bars P r = Reduced Pressure, P/P c. T = Temperature K. T c = Critical Temperature K. 7) Reid, R. C, Prausnitz, J. M, and Sherwood, T. K [1977] The Properties of Gases and Liquids,3 rd edition. McGraw- Hill, New York. 8) Reid, R. C, Prausnitz, J. M and Poling, B. E. [1987] The Properties of Gases and Liquids, 4 rd edition. McGraw- Hill, New York. 9) Perry, R. H and Green, D. W [2008] Chemical Engineers, 8 th edition. McGraw-Hill, New York. 10) Peng, D. Y and Robinson, D. B [1976] A new two constant equation of state. Journal of Industrial and Engineering Chemistry: Fundamentals 15: 59-64. T r = Reduced Temperature T/T c ω = Pitzer acentric factor. Z = Compressibility. Z (0) = Simple fluid contribution to compressibility. Z (1) = Departure contribution to compressibility. Z P = Derivative-compressibility with respect to pressure. = Simple fluid contribution to derivativecompressibility with respect to pressure. = Departure contribution to derivativecompressibility with respect to pressure. Z T = Derivative-compressibility with respect to temperature. = Simple fluid contribution to derivativecompressibility with respect to Temperature. = Departure contribution to derivative compressibility with respect to Temperature. REFERENCES 1) Smith J. M. and Van Ness, H. C. [1992] Introduction to Chemical Engineering Thermodynamics 2 nd edition P 70 2) Lee, B. I. and Kessler, M. G. [1999] AIChE Journal. 21, 510, 3) Reid R. C. and Sherwood, T. K. [1966] The Properties of Gases and Liquids, 2 nd edition P 272. 4) Reid R. C. and Valbert J. R.[ 1962]. Industrial Engineering Chemistry Fundamentals. P 292. 5) Ikoku C. U. [1984]. Natural Gas Production Engineering 2 nd edition p.19-37. 6) Reid R. C. and others [2012] Properties of gases and Liquids, 6th edition. pp, 36-39 ISSN: 2231-5381 http://www.ijettjournal.org Page 414