Equations and Computer Program for Calculating the Properties of Gases and Combustion Products

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1 Thermal Engineering, Vol. 5, o., 5, pp. 9. Original ussian Text Copyright 5 by Teploenergetika, Aleksandrov, Ochkov, Orlov. English Translation Copyright 5 by MAIK auka /Interperiodica (ussia). Equations and Computer Program for Calculating the Properties of Gases and Combustion Products A. A. Aleksandrov, V. F. Ochkov, and K. A. Orlov Moscow Power Institute, ul. Krasnokazarmennaya 4, Moscow, 5 ussia Abstract Equations are presented for calculating the isobaric heat capacity, enthalpy and entropy of gases resulting from the combustion of fuels (taking dissociation into account). A computer program is described that was developed for determining the thermodynamic properties of these gases and their tures as a function of temperature, enthalpy, or entropy. The properties of gases in the ideal gas state are calculated for substances most frequently encountered in calculations of thermal power processes. Among these are nitrogen, oxygen O, carbon oxide CO, carbon dioxide CO, water steam H O, sulfur dioxide SO, air, atmospheric nitrogen, nitrogen oxide O, nitrogen dioxide O, argon Ar, neon e, and hydrogen H. Air consists (by volume) of 78.% nitrogen,.99% oxygen,.94% argon,.% hydrogen, and.% carbon dioxide []. Atmospheric nitrogen consists (by volume) of 98.7% nitrogen,.9% argon,.% hydrogen, and.4% carbon dioxide []. The following constants are used in these calculations: the absolute gas constant = 8.45 J/(mol K), the molar masses of individual gases (Table ), and the standard pressure p =. MPa. THE POCEDUE FO CALCULATIG THE THEMODYAMIC POPETIES OF IDIVIDUAL GASES The thermodynamic properties of substances and their tures in the ideal gas state are described by the system of equations from [], which employs a unified form of the equation for isobaric heat capacity c ---- p a i τ i a i -- τ i = i = i = 7 () where τ = T/T* is the relative temperature (T is the gas temperature, K, and T* = K); a i is the array of coefficients that are specific for each gas; is the absolute gas constant, J/(mol K); and c p is the isobaric heat capacity, J/(mol K). The coefficients a i for all substances except water steam and single-atom gases have been determined using the least square method; they are based on the data from [] for the temperature range from to 5 K and are given in Table. These coefficients, allow the values from [] to be calculated with the following standard error:.7 5 for oxygen,. 4 for nitrogen,. 4 for carbon dioxide,.9 4 for carbon oxide,.8 5 for nitrogen oxide,.54 5 for nitrogen dioxide,.44 5 for sulfur dioxide, and.5 5 for hydrogen. When water steam is in the ideal gas state, its isobaric heat capacity has previously been described in the International equation (IF-95) [4] in terms of an equation that was taken from [5], where it was derived using Table. List of unique gas identifiers Gas Molar mass, kg/mol Unique identifier itrogen.84 Oxygen O.9988 Carbon oxide CO.84 Carbon dioxide CO.4498 Water steam H O.85 4 Sulfur dioxide SO Air as a single gas Air Atmospheric nitrogen as air a single gas itrogen oxide O. 8 itrogen dioxide O.455 [] 9 Argon Ar.9948 [] eon e.79 [] Hydrogen H.58 [] Air as a ture of gases Air Atmospheric nitrogen as air a ture of gases

2 ALEKSADOV et al. Table. Values of the coefficients a i in Eq. () i O CO CO H O SO i Air O O H Ar, e air THEMAL EGIEEIG Vol. 5 o. 5

3 EQUATIOS AD COMPUTE POGAM FO CALCULATIG THE POPETIES data from []. In subsequent years, however, new calculations of the properties of water steam in this state were carried out [7 9], whose results at high temperatures differ from data in []. A new equation for water steam is therefore presented in this paper, which is based on the data from [7] and describes these properties with a root-mean-square error not exceeding.. The coefficients of Eq. () for air and atmospheric nitrogen were determined by summing the coefficients for components (using the procedure for calculating tures, which is described below) taking into account their concentrations given above. The discrepancies between the isobaric heat capacity values calculated from Eq. () and similar values obtained from some other studies are presented in Fig.. The greatest discrepancies are observed for water steam. When temperatures are below K, all data except those from [] are in strong agreement with one another (the error does not exceed ±.%). As the temperature increases, however, discrepancies grow. It is interesting to note that at T = 5 K, the values calculated from [5] are.8% greater and those calculated from [7] are.8% lower than those obtained using the proposed method. Deviations in the values calculated using the equation from [] within its applicability limits from to 77 K do not exceed.%. The results calculated by Eq. () for carbon dioxide coincide with the values from [,, ] over the entire temperature range with an error not exceeding.4%. Discrepancies with the data from [] remain within.% only at temperatures not exceeding K; the discrepancies with data from [] reach.4% at 8 K. The results obtained for nitrogen and oxygen were found to be in very strong agreement with the data from []: the discrepancy for nitrogen was within.% and for oxygen.%. At temperatures below 5 K a similarly strong agreement is observed with the values calculated from the data in [] (within.%), and from [4] (within.%). The discrepancy from the data in [] is somewhat greater: for nitrogen it is.8%, and for oxygen it is more than.% (at the boundaries of the temperature range). Accordingly, for air at temperatures to K the difference from the data in [5] is not more than.%, and from the data in [] it is not more than.%. Strong agreement is observed between the figures calculated by () and the data from [] for sulfur dioxide (within.%), and for carbon oxide (within.4%). The discrepancy between the results for carbon dioxide and the data in [] does not exceed.9%. Expressions for calculating other properties of gases can be obtained from Eq. () using well-known thermodynamic relationships. For example, the following expression was obtained for calculating enthalpy h h --- = T* i = 8 a i τ i a i 7 lnτ i = a i i -- i 7 τ h int, () where h int is an integration constant, which is determined from []. The following relation holds for calculating gas entropy s at pressure p: p s = s ln ----, () where s is the entropy at standard pressure p = kpa, which is calculated by the formula s --- = a lnτ i = 7 a i i -- τ i = (4) where s int is an integration constant (which was determined from the data in [] taking into account the changeover from the old standard pressure p =.5 kpa to its modern value p = kpa), and p is the gas pressure kpa. The integration constants in () and (4) are determined from the condition of equality of the specific enthalpy and specific entropy calculated from these equations to their values at the check points taken from [] (Table ). THE COMPUTE POGAM FO CALCULATIG THE THEMODYAMIC POPETIES OF IDIVIDUAL GASES elations () (4) were used to develop computer programs for calculating the properties of gases in accordance with a procedure similar to that used for developing the WaterSteamPro, a computer program for calculating the properties of water and water steam []. These two programs have now been incorporated into a single unit. This enables users to employ the functions for calculating properties of gases in all computer programs where the WaterSteamPro can be used. The following basic functions have been determined in the package for calculating the properties of gases, which correspond to relationships () (4): wspgcpidt(id, T); (5) wspghidt(id, T); () wspgsidt(id, T); (7) i p a ---τ i i i s , int THEMAL EGIEEIG Vol. 5 o. 5

4 4 ALEKSADOV et al.... O.4..8 Air..... CO 4. 4 SO H O CO H 4 8 Temperature, K 4 8 Temperature, K Fig.. Discrepancies between the isobaric heat capacity values calculated by () and data of other researchers. Source of data: () [ 5]; () []; () []; (4) []; (5) [8, ]; () [9]; (7) [7]; (8) [], and (9) [4, 5]. wspgsidpt(id, p, T), (8) where id is a unique gas identifier, T is the temperature, K, and p is the pressure, Pa. When the user wishes to calculate the properties of a specific gas, he or she must specify its unique identifier (Table ). This identifier specified, the program itself then finds the coefficients for relations () (4). It is important to note that the same identifier will be assigned to each gas in subsequent versions of the program. otice that air and atmospheric nitrogen appear twice in Table. In the first case they appear as individual gases (the program regards them not as a ture of gases but as a unique gas, whose properties are equivalent to those of the ture of the gases constituting it). THEMAL EGIEEIG Vol. 5 o. 5

5 EQUATIOS AD COMPUTE POGAM FO CALCULATIG THE POPETIES 5 Dissociation cannot be taken into account for them because the program does not store information on their components. In the second case these are tures of gases, and for them it is possible to calculate dissociation and to determine their components. This was done to facilitate a comparison and generate calculations using the proposed procedure. The user himself can select the best suitable option. Apart from initial functions (5) (8), there are additional functions, which allow the following parameters to be calculated: molar mass, kg/mol wspgmmid(id); (9) specific gas constant, J/(kg K) wspggcid(id); () specific volume determined from the Clapeyron- Mendeleyev equation wspgvidpt(id, p, T), () wspgvidt(id, T). () The last function is calculated at standard pressure. The program can also be used to determine inverse relations, namely, to calculate temperatures from specific enthalpy or entropy. To this end, we developed the following functions (using the ewton method), which allow the roots of functions () (8) to be determined more exactly: wspgtidh(id, h); () wspgtids(id, s ); (4) wspgtidps(id, p, s). (5) These functions make possible the quick and easy calculation of adiabatic expansion and compression processes. THE POCEDUE FO CALCULATIG THE THEMODYAMIC POPETIES OF GAS MIXTUES The properties of gas tures can be determined using the same functional relations () (4) that were used for individual gases c p, = a i, τ i a i i = i = 7, -- ; τ i () Table. Values of enthalpy and entropy at check points (T ch = 98.5 K and p ch =.5 kpa) Gas k ch, kj/mol s ch, J/(mol K) O CO CO H O SO Air air O O Zr e H However, the relation for standard entropy, which is given by s , = a a i, i = 7, lnτ i i -- τ i = a i, τ i i (8) now contains a term that expresses the change in the entropy when the gases are ed (9) The coefficients for the ture are calculated by the following formulas: s, int, s , s , = x j ln( x j ). a i j =, = x j a i j h int j =, ;, = x j h int j j =, ; () () h = T* i = 8 a i, i = a i, i -- τ i τ i i 7 a 7, lnτ h int, (7) s int, = x j s int j j = () where x j is the molar (or volume) share of the jth gas appearing in a ture consisting of gases, and a i, j is the ith coefficient of the jth gas.,, THEMAL EGIEEIG Vol. 5 o. 5

6 ALEKSADOV et al. The following volumetric fractions of the air components are assumed nitrogen: x = 78.% oxygen: argon: hydrogen: carbon dioxide: x O =.99% x O =.94% x H =.% x CO =.% The identifier of a new ture is obtained: Gases are added to the ture: wspgmmid(wspg ) = 8.4 g/mol wspgmmid(wspgo ) =.9988 g/mol wspgmmid(wspgar) = g/mol wspgmmid(wspgh ) =.58 g/mol wspgmmid(wspgco ) = g/mol air_ = wspgewmix() air_ = 8 wspgaddgasmomix(air_, wspg, x ) = wspgaddgasmomix(air_, wspgo, x O ) = wspgaddgasmomix(air_, wspgar, x Ar ) = wspgaddgasmomix(air_, wspgh, x H ) = wspgaddgasmomix(air_, wspgco, x CO ) = 4 The volumetric or mass fractions of the initial gases in the ture are determined: by volume: wspggasmfimix(air_, wspgo ) =.99% by mass: wspggasmfimix(air_, wspgo ) =.89% The parameters of the ture are calculated and the unique identifier for the turebased gas is formed: air_id = wspgewgasfommix(air_) air_id = It is now possible to calculate, for example, the molar mass of the ture or specific enthalpy: wspgmmid(air_id) = g/mol wspghidt(air_id, 5 K) = 5 kj/kg Fig.. Example of calculating a ture (air consisting of nitrogen, oxygen, argon, hydrogen, and carbon dioxide). THE POGAM FO CALCULATIG THE THEMODYAMIC POPETIES OF GAS MIXTUES A concept of creating tures was proposed for calculating tures of initial gases. This concept consists in the following. The user notifies the program that a new ture has to be calculated. This is done by calling the function wspgewmix(). () Function () assigns a unique identifier to the new ture, which will then be used for determining the parameters of this ture. Every time this function is called it gives a new identifier to the ture. Their total number is limited by the resources of the system; its maximal value cannot be greater than The identifier thus obtained is then used to call a function that takes into account the addition of gases to the ture: wspgaddgasmmix(_id, gas_id, mass), (4) where _id is the ture identifier obtained earlier when function () was called; gas_id is the identifier of the gas that is added (see Table ); and mass is the mass of the gas that is added. It should be noted that the user can specify not only absolute, but also relative values of the mass of a gas in the ture (for example, mass content d). Another function for taking into account the addition of a gas into a ture appears as follows: wspgaddgasmomix(_id, gas_id, moles), (5) where moles is the quantity of moles of the gas that is added. The approach implemented in these functions is somewhat different from the commonly adopted one, in which the fractions (mass and volume) of the substances composing the ture are usually specified. Everything is much easier in the new approach. Suppose, for example, that the user wants to calculate the properties of a ture consisting of kg of air and g of water steam. There is no need to find the relative fractions of these substances in the ture; the only thing that has to be done is to call function (4) with the following parameters: gas_id = 7, mass =, gas_id = 4, and mass =.. The program itself will determine the necessary fractions of substances in the ture. Function (5) is used in a similar way. The parameter moles can be specified as a quantity of moles or as a volume fraction of the substance in the ture. THEMAL EGIEEIG Vol. 5 o. 5

7 EQUATIOS AD COMPUTE POGAM FO CALCULATIG THE POPETIES 7 Figure shows a fragment of the calculation of the properties of air as a ture of gases. After all of the components are added to the ture, the coefficients for Eqs. () (9) must be determined. This is done by calling the following function: wspgewgasfommix(_id). () Once all required parameters are calculated, function () will return the gas identifier gas_id for the ture specified by the parameter _id. The gas identifier gas_id thus obtained can be used when calling functions (5) (5) for calculating the properties of gases. The program can identify for the user the fraction of each component in a ture at any time in the course of calculating its properties. This is done by introducing the following functions: for the mass fraction of the gas with the identifier gas_id in the ture with the identifier _id wspggaswfimix(_id, gas_id), for the molar (volume) fraction of the gas with the identifier gas_id in the ture with the identifier _id wspggasmfimix(_id, gas_id). The program that has been developed can be used to calculate tures of gases that are themselves tures of other gases; that is, a ture composed of tures of primary gases can be calculated. To this end, Eq. (9) is written in the form s , = x ln x j j j s j = j s, , where, is the ing entropy of the ture of which gas j is composed. The function for calculating the fraction of a gas that is a component of a ture is augmented with a special mechanism, which allows the total fraction of the gas to be determined even though it is part of tures of which the ture being studied is composed. This possibility can be used to calculate, for example, the fraction of oxygen in flue gases from a gas-turbine unit; it may also be used when a consecutive calculation of tures has to be carried out: first for dry air, then for humid air (a ture of dry air and water steam), then for a ture of humid air and the products that result from burning fuel in the combustion chamber, and then for the ing of the ture thus obtained with the cooling air, and so on. The program will itself scan all components of the ture and determine the fraction of the gas in it. When calling functions () and (), the program allocates a memory space for saving data on the tures and gases composed of them. If the user wishes to allocate this memory, this can be done using the following special functions: to allocate the memory occupied by data on a gas composed using a ture wspgdeletegas(gas_id); to allocate the memory occupied by data on a ture wspgdeletemix(_id). THE POCEDUE FO TAKIG ITO ACCOUT DISSOCIATIO I GAS MIXTUES AT HIGH TEMPEATUES The equations described above were set up without considering the dissociation of gases at high temperatures. If we wish this effect to be taken into account, the chemical equilibrium of the components in the given ture must be calculated. Quite a number of methods and computer programs are now available for calculating this equilibrium [7 9]. Most of these methods and programs require a considerable amount of iterative calculation for the precise determination of the chemical equilibrium of a ture, because this involves the necessity of solving systems of nonlinear equations. However, the calorific effect of dissociation within the required temperature range can be approximated accurately enough without knowing the precise equilibrium composition of the ture. An approximate method was therefore used to calculate this effect, which was proposed in []. This method is based on the assumption that under conditions typically encountered in power engineering applications and at temperatures below K, the change in the properties of interest depends mostly on the formation of products of the following reactions: CO CO --O ; H O H --O ; --H H; --O O; --H O 4 --O OH; -- --O O. The equilibrium constants for these reactions were represented in [] (using the data from [8, 9]) by the empirical formula K j = A j exp( B j /T). (7) The values of A j and B j are given in Table 4. The variables U j are then introduced, which characterize the concentration of the reaction products U x CO p A B = exp x O p (8) THEMAL EGIEEIG Vol. 5 o. 5

8 8 ALEKSADOV et al. Table 4. Coefficients of Eqs. (7) (4) Gas A j B j, K U x H O p A B = exp x O p U A x 4 H O x p O B = exp (9) () () () () The contribution that the thermal effect of each reaction introduces into the isobaric heat capacity at temperatures from to K is given by the formula D V j C j E = j j. (4) T T The coefficients of (4) are also given in Table 4; changes in the thermodynamic properties of the ture due to dissociation are calculated as follows: p U 4 A 4 U p B = exp p U 5 A 5 x p O B = exp p B U = A x x exp O and the total value U tot = U j. j = dis c p = U j V j ; U tot j = dis h T = dis s dis h = = T C j, J/(mol K) U tot j = T U j V ; j B j U tot j = D j, J/mol U j V j B j E j, GJ K/mol CO H OH H O O After that, the properties of the ture of dissociating gases are determined by summing the values thus obtained with the values calculated by () (9) c p, dis h dis s dis = = = c p, h s dis c p ; dis h ; dis s. When implemented in practice, this method will allow the user to determine, without iterative calculations, the thermodynamic properties of a ture of dissociating gases with high accuracy at temperatures to K. The user can specify settings for the mode when the dissociation effect is taken into account, using the following function: wspgsetcalcdissmode(mode), where mode is the number of the mode connected with taking the dissociation effect into account (when mode =, the thermal effect of dissociation is not calculated; when mode =, the effect of dissociation is calculated at all temperatures; and when mode =, the effect of dissociation is calculated only for temperatures above K). This study was supported by a grant from the ussian Fund for Fundamental esearch. EFEECES. S. L. ivkin, Thermodynamic Properties of Gases. A Handbook, 4th ev. Edition (Energoatomizdat, Moscow, 987) [in ussian].. V. V. Sychev, A. A. Aleksandrov, A. V. Matveev, et al., A Set of Application and Educational Interactive Computer Programs for Calculating the Thermodynamic Properties of Working Media and Coolants, Izv. Vyssh. Uchebn. Zaved. Energetika 9, 8 (99).. L. V. Gurvich, I. V. Veits, V. A. Medvedev, et al., Thermodynamic Properties of Individual Substances. A Handbook, rd evised and Extended Edition (auka, Moscow, 978), Vol., Book [in ussian]. 4. elease on the IAPWS Formulation-995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use International Association for the Properties of Water and Steam. Executive Secretary. B. Dooley (Electric Power esearch Institute, Palo Alto, CA USA). 5. J.. Cooper, epresentation of the Ideal-Gas Thermodynamic Properties of Water, Int. J. Thermophys., 5 4 (98).. A. S. Friedman and L. Haar, High Speed Machine Computing of Ideal Gas Thermodynamic Functions for Isotopic Water Molecules, J. Chem. Phys., 5 58 (954). 7. M. Vidler and J. Tennyson, Accurate Partition Function and Thermodynamic Data for Water, J. Chem. Phys. (), 5 (). THEMAL EGIEEIG Vol. 5 o. 5

9 EQUATIOS AD COMPUTE POGAM FO CALCULATIG THE POPETIES 9 8. H. W. Woolley, Thermodynamic Properties for H O in the Ideal Gas State. Water and Steam: Their Properties and Current Industrial Application, in Proceedings of the 9th International Conference on the Properties of Water and Steam (Pergamon Press, ew York, 98), pp H. W. Woolley, Ideal Gas Thermodynamic Functions for Water, J. es. atl. Bur. Stand. 9, 5 5 (987).. D. Bucker,. Span, and W. Wagner, Thermodynamic Property Models for Moist Air and Combustion Gases, ASME Journal of Engineering for Gas Turbines and Power 5, ().. H. W. Woolley, Thermodynamic Functions for Carbon Dioxide in the Ideal Gas State, Journal of esearch of the ational Bureau of Standards 5 (), 89 (954).. Liquid and Gaseous Carbon Dioxide. Density, Compressibility Factor, Enthalpy, Entropy, Isobaric Heat Capacity, Sound Velocity, and Volumetric Expansion Coefficient at Temperatures of K and Pressures of. MPa. Standard eference Tables (Izd. Standartov, Moscow, 98) [in ussian].. V. V. Sychev, A. A. Vasserman, A. D. Kozlov, et al., Thermodynamic Properties of Oxygen, in GSSSD, Ser. Monografii (Izd. Standartov, Moscow, 98) [in ussian]. 4. V. V. Sychev, A. A. Vasserman, A. D. Kozlov, et al., Thermodynamic Properties of itrogen (Izd. Standartov, Moscow, 977) [in ussian]. 5. V. V. Sychev, A. A. Vasserman, A. D. Kozlov, et al., Thermodynamic Properties of Air, in GSSSD, Ser. Monografii (Izd. Standartov, Moscow, 978) [in ussian].. A. A. Aleksandrov, A. V. Ochkov, V. F. Ochkov, and K. A. Orlov, A Set of Programs for Calculating the Thermophysical Properties of Water and Steam (WaterSteamPro) (OSPATET, Moscow, Certificate o. 8, August 5, ). 7. V. S. Yungman et al., Ivtanthermo for Windows. Database on Thermodynamic Properties of Individual Substances and Thermodynamic Modeling Software. Version. (Glushko ThermoCenter of uss. Acad. of Sci., 99 ). 8. B. J. McBride and S. Gordon, Computer Program for Calculation of Complex Chemical Equilibrium Composition and Applications (ASA P-, ational Aeronautics and Space Administration, Washington, DC, 99). 9. B. J. McBride and S. Gordon, Computer Program for Calculation of Complex Chemical Equilibrium Composition and Applications. Version (CEA) (ASA P-, ational Aeronautics and Space Administration, Washington, DC, ). THEMAL EGIEEIG Vol. 5 o. 5

STP-TS THERMOPHYSICAL PROPERTIES OF WORKING GASES USED IN WORKING GAS TURBINE APPLICATIONS

STP-TS THERMOPHYSICAL PROPERTIES OF WORKING GASES USED IN WORKING GAS TURBINE APPLICATIONS THERMOPHYSICAL PROPERTIES OF WORKING GASES USED IN WORKING GAS TURBINE APPLICATIONS THERMOPHYSICAL PROPERTIES OF WORKING GASES USED IN GAS TURBINE APPLICATIONS Prepared by: ASME Standards Technology, LLC