Thermofluids Data Book for Part I of the Engineering Tripos

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1 Thermofluids Data Book for Part I of the Engineering Tripos 004 Edition Cambridge University Engineering Department

2 CONTENTS Thermodynamic definitions & relationships... Ideal gas relationships... 3 Perfect gas relationships... 3 Mixtures of perfect gases... 3 Non-dimensional groups... 4 Heat transfer... 5 Equations for systems... 6 Equations for control volumes... 7 Equations for streamlines... 9 Incompressible viscous pipe flow... 9 Equations of motion in differential form for viscous steady flow of a fluid... 9 Thermodynamic efficiencies... 0 Combustion... Properties of perfect gases... Molar enthalpies of common gases at low pressures... 3 Thermochemical data for equilibrium reactions... 4 Data for steam... 6 Triple point data for steam... 6 Critical point data for steam... 6 Properties of saturated water & steam: Temperatures from the triple point to the critical point. 7 Properties of saturated water & steam: Pressures from the triple point to the critical point... 9 Specific enthalpy of water and steam...3 Specific Entropy of water and steam... 4 Density of water and steam... 5 Specific internal energy of water and steam... 6 Transport properties of saturated water & steam... 7 Transport properties of steam... 8 Transport properties of air... 8 Transport properties of carbon dioxide... 9 Transport properties of hydrogen... 9 Perfect gas relations for compressible flow for γ= Properties of gases at sea level conditions... 3 Properties of liquids at sea level conditions... 3 The International Standard Atmosphere... 3 Properties of the International Standard Atmosphere at altitude... 3 Physical constants Conversion of Non-SI to SI Units Conversion of Non-SI to SI Units cont Sources of Information Critical point data for Refrigerant R-34a (CH FCF 3 ) Properties Table for Refrigerant R-34a (CH FCF 3 ) Properties Chart for Refrigerant R-34a (CH FCF 3 ) Properties of steam /09/04

3 THERMODYNAMIC DEFINITIONS & RELATIONSHIPS Specific enthalpy Specific heat capacity at constant volume Specific heat capacity at constant pressure Ratio of specific heat capacities Coefficient of volume expansion Coefficient of compressibility For a simple compressible substance, in the absence of capillarity, electric and magnetic fields h u + c c v p γ pv u T v h T c c p v p v β v T p v κ v p T Tds = du + pdv = dh vdp 06/09/04

4 IDEAL GAS RELATIONSHIPS Equation of state Relationship between c p, c v and R pv = nrt pv = mrt pv = RT p = ρrt c p cv = R Speed of sound a = γrt PERFECT GAS RELATIONSHIPS Change in specific internal energy u u = cv ( T ) T Change in specific enthalpy h h = c p ( T ) T Change in specific entropy For Isentropic changes s T v c v ln +Rln T v T p s = c p ln Rln T p p ln v c +c ln v p p v γ pv = const. γ Tv = const. ( γ T / p ) γ = const. MIXTURES OF PERFECT GASES For a mixture of N perfect gases where, for component -i, m i = mass, p i = partial pressure, h i = h i (T) = partial specific enthalpy, s i = s i (T, p i ) = partial specific entropy, n i = number of mols and the overbar signifies a partial molar quantity: Pressure of the mixture = p mixture = p i i= i N Enthalpy of the mixture Entropy of the mixture H S mixture mixture = = i = N i= i= N i= m h i m s i i i = = i= N n h i i= i = N n s i i= i i 06/09/04 3

5 Reynolds Number Mach Number Froude Number Prandtl Number Drag Coefficient Lift Coefficient NON-DIMENSIONAL GROUPS ρvd Vd Re = = µ ν V M = a Fr = V gz µ c p Pr = = λ = D C D ρ V = L C L ρ V τ Skin Friction Coefficient c w f = V ρ Friction Factor Discharge Coefficient Nusselt Number Grashof Number Stanton Number f = 4c f C d m = m hd Nu = λ actual ideal 3 ν α A A gd β T Gr = ν St = Nu RePr h = ρvc p 06/09/04 4

6 HEAT TRANSFER Conduction, Convection and Radiation Rate of heat transfer Q by convection from a body of surface area A ( ) Q = ha T body T surroundings Rate of heat transfer Q by conduction along a straight bar of cross-sectional area A dt Q = λa dx Rate of heat transfer Q by conduction radially in a straight circular bar of length L dt Q = λπrl dr Rate of heat transfer Q by radiation from a grey body of surface area A and emissivity ε 4 Q = ε Q black = εσat where σ = 5.67x0-8 Wm - K -4 is the Stefan-Boltzmann constant. Logarithmic-Mean Temperature Difference (LMTD) T m T = ln T ( T T ) Convective heat transfer for fully developed flow in circular pipes of diameter d Overall heat transfer for laminar flow with constant wall temperature Nu = 3.66 Re < 300 d Overall heat transfer for turbulent flow with constant wall temperature Nu d = 0.8 d 0.03Re Pr 0.4 d 7 0. < Pr 300 Re <0 0.5 < Pr < 0 < d Heat transfer due to natural convection from a vertical isothermal plate of height L Overall heat transfer ( ) 0. 5 Nu L = 0.5 GrPr Pr Reynolds Analogy c f St = Note that the net rate of heat transfer will be less due to incident radiation 06/09/04 5

7 Definition of a system A system is a fixed quantity of matter. Conservation of mass applied to a system The mass of a system is constant, i.e. EQUATIONS FOR SYSTEMS m = const. st Law of Thermodynamics applied to a system When heat transfer to, and work done by, a system are defined as positive and the system is undergoing a cyclic process dq = dw When heat transfer to, and work done by, a system of mass m undergoing a process between state and state are defined as positive and in the absence of capillarity, electric and magnetic fields Q W = E E = ( U + mv + mgz ) ( U + mv + mgz ) nd Law of Thermodynamics applied to a system When heat transfer to a system undergoing a cyclic process is defined as positive, the Clausius inequality is dq 0 T When heat transfer to a system of mass m undergoing a process is defined as positive dq mds = + mds irrev where dsirrev T where ds is the entropy created per unit mass by irreversibilities. irrev pdv work for a system When the surface of a system undergoes displacement between state and state and the pressure causing the displacement is known over the entire surface that is displaced, the displacement work done by the system during this process is W = pdv 0 06/09/04 6

8 EQUATIONS FOR CONTROL VOLUMES Definition of a control volume A control volume is that region of space that is enclosed by a rigid control surface. The requirements for steady flow within a control volume For steady-flow, conditions within the control volume are not, on average, changing so that the mass, momentum, energy and entropy within the control volume remain constant. Conservation of mass applied to a control volume - the Continuity Equation In general, this may be written in vector form as d dt cv ρdv + cs ρv da = 0 where da is positive out of the control volume. It may also be written as dmcv + m out m in = 0 dt For steady flow, the above becomes m out = m in Momentum equations applied to a control volume, including the Steady Flow Momentum Equation (SFME) In general, this may be written in vector form as d dt cv ρvdv + cs ρvv da = F where da is positive out of the control volume and F is the sum of all the non-pressure forces on the flow. cs pda For steady flow, the above becomes the Steady Flow Momentum Equation (SFME) and in x and y coordinates, the SFME is m m out out ρvv da = F cs V V cs x, out m invx, in y, out m inv y, in = = pda F F x y ( pa) x ( pa) y where care is needed with respect to the sign of the V x, V y and A terms. 06/09/04 7

9 st Law of Thermodynamics applied to a control volume, including the Steady Flow Energy Equation (SFEE) In the absence of capillarity, electric and magnetic fields, the st Law becomes ( h + V + gz ) m ( h + V + gz ) = Q W x decv + m out out dt out out in in in in where E cv is the total energy within the control volume, Q is the rate of heat transfer to the control volume and W x is the rate of shaft work transferred from the control volume. For steady flow, the above becomes the Steady Flow Energy Equation (SFEE) m out ( hout + Vout + gzout ) m in ( hin + Vin + gzin ) = Q W x nd Law of Thermodynamics applied to a control volume When heat transfer to a control volume is defined as positive ds dt cv dq + mout sout minsin S = + irrev where S irrev 0 T A where dq is the rate of heat transfer to an area da of the control surface at temperature T and the integration is over the whole area A of the control surface, S cv is the total entropy within the control volume and S irrev is the rate of entropy creation within the control volume due to irreversibilities. 06/09/04 8

10 EQUATIONS FOR STREAMLINES Bernoulli's Equation for incompressible inviscid steady flow along a streamline p + ρv + ρgz = const. The pressure gradient normal to a streamline with a radius of curvature R V dp dn ρv = R n,e n e s s R INCOMPRESSIBLE VISCOUS PIPE FLOW The pressure drop along a pipe of constant diameter d and length L, with viscous flow L L p = 4c f ρ V = f ρv d d EQUATIONS OF MOTION IN DIFFERENTIAL FORM FOR VISCOUS STEADY FLOW OF A FLUID Mass conservation (continuity) n Intrinsic coordinates. ( ) = 0 e s + e s Vector Notation. ( ρv ) = 0 n V e s Momentum Intrinsic coordinates Vector Notation V V p p ρ V es e n = es en + ρ g + viscous s R s n ρv V = p + ρ g + viscous 06/09/04 9

11 THERMODYNAMIC EFFICIENCIES Efficiency of a cycle W ηcycle Q but so W η net cycle = Q B net B Q Q = Q A B A Isentropic efficiencies of compressors and turbines. Q B. Q A. W Net η c Ideal Work Input Actual Work Input = h s h h h η t Actual Work Output Ideal Work Output h = h 3 3 h h 4 4s 06/09/04 0

12 COMBUSTION The SFEE applied to stoichiometric combustion at constant T and p When heat transfer to a control volume is defined as positive and in the absence of shaft work, changes in KE and PE, capillarity, electric and magnetic fields, the rate of heat transfer and the calorific value (CV ) are related by where ( CV ) n h Q = m = fuel m = M fuel n fuel Calorific values of common fuels (Enthalpies of Reaction) In the following table, the calorific value is equal and opposite to the enthalpy of reaction when the reactants and products are at 5 C and.034 bar. In the evaluation of the lower calorific value and lower enthalpy of reaction, the steam is assumed to be dry saturated. Stoichiometric Equation C Molar Mass M of Fuel kg kmol fuel Phase Calorific Value MJ/kg O CO solid C + O CO solid CO + O CO 8 gas 0.00 Higher: H O Lower: H O to water to steam + O H O gas H CH4 + O CO + HO 6 gas H + 3.5O CO + H O 30 gas C 6 3 C3H8 + 5 O 3CO + 4H O 44 gas H + 6.5O 4CO + H O 58 gas C4 0 5 C8H8 +.5O 8CO + 9H O 4 gas H +.5O 8CO + H O 4 liquid C8 8 9 A more exact value is.06 06/09/04

13 PROPERTIES OF PERFECT GASES Values of M, R, c p, c v and γ At normal atmospheric conditions, and over a limited range of temperature and pressure, the gases listed below may be assumed to behave as perfect gases. That is, they may be assumed to have the equation of state pv = RT, and to have constant specific heat capacities. Gas Molar mass M kg/kmol Gas constant R kj/kg K c p kj/kg K c v kj/kg K Air # Atmospheric nitrogen N O A H * He CO CO SO CH C H C 3 H Real gases are not perfect gases, and the rounded values for R, c p c v and c p cv listed above do not exactly satisfy the relationships between these quantities that would be obtained for perfect gases. γ c c p v Molar (universal) gas constant R = MR = kj/kmol K Molar volume of a perfect gas kmol of any perfect gas occupies a volume of approximately.4 m 3 at s.t.p. (0 C and bar) and contains 6.0x0 6 particles. # Air contains.0% O and 79.0% atmospheric nitrogen by volume (Volumetric and Molar Analyses); 3.% O and 76.8% atmospheric nitrogen by weight (Gravimetric Analysis). Air contains 0.93 % of argon (A) and traces of other gases; these and the nitrogen together are called atmospheric nitrogen. * A more exact value is /09/04

14 MOLAR ENTHALPIES OF COMMON GASES AT LOW PRESSURES At low pressures, and over the temperature range quoted, the gases listed in this Table behave as semi-perfect gases. That is, while having the molar equation of state pv = RT, their specific heat capacities c p and c v are not constant but are functions only of temperature. Gas Air N O H CO CO H O Gas Molar Mass kg/kmol Molar Mass kg/kmol Temperature K Molar enthalpy MJ/kmol Temperature K =5 C C= Notes: () The molar enthalpies listed are those in the ideal gas state at zero pressure, but the values given are also valid at and around atmospheric pressure. () In this table, the arbitrary datum state for zero enthalpy is that of the substance in the ideal gas state at zero pressure and zero absolute temperature. (3) The values for atmospheric nitrogen, * N, may be taken to be the same as those for N. A more exact value is.06 06/09/04 3

15 THERMOCHEMICAL DATA FOR EQUILIBRIUM REACTIONS The tables of equilibrium constants and standard enthalpy change on the next page relate to the reactions listed below Stoichiometric equations vi A i = 0 i where v i is the stoichiometric coefficient of the substance whose chemical symbol is A i. () H + H = 0 (4) NO + N + O = 0 (7) CO ½ O + CO = 0 () N + N = 0 (5) H ½ O + H O = 0 (8) CO H O + CO + H = 0 (3) O + O = 0 (6) ½ H OH + H O = 0 (9) ½ N 3 H + NH 3 = 0 Equilibrium constants The equilibrium constant * i K p is given by ln ( K p ) = where p pi p0 pi partial pressure of species Ai in bars p standard pressure bar 0 i i * i v ln p Thus * p i is numerically equal to p i but is dimensionless. Standard free enthalpy of reaction At a given temperature, the standard free enthalpy of reaction (or the standard Gibbs function change) ln by the following equation: 0 G T may be calculated from the listed value of ( ) K p G 0 T = RT ln K p = T ln( K ) kj p Standard enthalpy of reaction The Standard enthalpy of reaction is given by where 0 ( f T ) i where [ ] H 0 0 [ h ] = ν ([ H ] ) ν i i T i 0 ~ T = i ~ [ hi ] T = ([ H f ] 98 ) + ([ h ] T [ h ] 98 ) i Η is the standard enthalpy of formation of species i at temperature T and a pressure p 0 = bar. i i f T i 06/09/04 4

16 Equilibrium constants & standard enthalpies of reaction Reaction Number v i = 0 ½ ½ ½ 0 i Temperature Equilibrium Constant ln( ) K Temperature K K p Standard Enthalpy of Reaction MJ 0 H T Warning: These tables list absolute temperatures 06/09/04 5

17 DATA FOR STEAM Source of data The follow ing tables have been produced using equations from the IAPW S Form ulation 995 for the Therm odynam ic Properties ofordinary W ater Substance for G eneraland Scientific Use and the Supplem entary R elease: Saturation Properties of Ordinary W ater Substance. These docum ents can be found on the InternationalA ssociation for the Properties of W ater and Steam w ebsite w w.iapw s.org. TRIPLE POINT DATA FOR STEAM Temperature = 73.6 K (0.0 o C) Pressure = bar Phase Specific volume Specific enthalpy Specific entropy m 3 /kg kj/kg kj/kg K Ice Water Steam CRITICAL POINT DATA FOR STEAM Temperature = K ( C) Pressure = 0.64 bar Density = 3 kg/m 3 06/09/04 6

18 PROPERTIES OF SATURATED WATER & STEAM: Temperatures from the triple point to the critical point Temp. Pressure Specific volume Spec. int. energy Specific enthalpy Specific entropy Temp. o C bar m 3/ kg kj/kg kj/kg kj/kg K o C T p v f v g u f u g h f h fg h g s f s g T T p v f v g u f u g h f h fg h g s f s g T 06/09/04 7

19 Properties of Saturated Water & Steam continued: Temperatures from the triple point to the critical point Temp. Pressure Specific volume Spec. int. energy Specific enthalpy Specific entropy Temp. o C bar m 3/ kg kj/kg kj/kg kj/kg K o C T p v f v g u f u g h f h fg h g s f s g T T p v f v g u f u g h f h fg h g s f s g T 06/09/04 8

20 PROPERTIES OF SATURATED WATER & STEAM: Pressures from the triple point to the critical point Pressure Temp. Specific volume Spec. int. energy Specific enthalpy Specific entropy Pressure bar o C m 3/ kg kj/kg kj/kg kj/kg K bar p T v f v g u f u g h f h fg h g s f s g p p T v f v g u f u g h f h fg h g s f s g p 06/09/04 9

21 Properties of Saturated Water & Steam continued: Pressures from the triple point to the critical point Pressure Temp. Specific volume Spec. int. energy Specific enthalpy Specific entropy Pressure bar o C m 3/ kg kj/kg kj/kg kj/kg K bar p T v f v g u f u g h f h fg h g s f s g p p T v f v g u f u g h f h fg h g s f s g p 06/09/04 0

22 Properties of Saturated Water & Steam continued: Pressures from the triple point to the critical point Pressure Temp. Specific volume Spec. int. energy Specific enthalpy Specific entropy Pressure bar o C m 3/ kg kj/kg kj/kg kj/kg K bar p T v f v g u f u g h f h fg h g s f s g p p T v f v g u f u g h f h fg h g s f s g p 06/09/04

23 Properties of Saturated Water & Steam continued: Pressures from the triple point to the critical point Pressure Temp. Specific volume Spec. int. energy Specific enthalpy Specific entropy Pressure bar o C m 3/ kg kj/kg kj/kg kj/kg K bar p T v f v g u f u g h f h fg h g s f s g p p T v f v g u f u g h f h fg h g s f s g p 06/09/04

24 Pressure (bar) SPECIFIC ENTHALPY OF WATER AND STEAM kj/kg Critical Isobar Temp ( o C) /09/04 3

25 Pressure (bar) SPECIFIC ENTROPY OF WATER AND STEAM kj/kg K Critical Isobar Temp ( o C) /09/04 4

26 Pressure (bar) DENSITY OF WATER AND STEAM kg/m Critical Isobar Temp ( o C) /09/04 5

27 Pressure (bar) SPECIFIC INTERNAL ENERGY OF WATER AND STEAM kj/kg Critical Isobar Temp ( o C) /09/04 6

28 Temp. C TRANSPORT PROPERTIES OF SATURATED WATER & STEAM Specific volume m 3 /kg v f vg Isobaric specific heat capacity kj/kg K c p f c pg Thermal conductivity W/m K λ f 3 λg µ f /0 Dynamic viscosity kg/s m 6 µ /0 g f Prandtl number = µ c p λ Pr Prg Temp. C 06/09/04 7

29 Temp. C TRANSPORT PROPERTIES OF STEAM Isobaric sp. heat capacity kj/kg K Thermal conductivity W/m K Dynamic viscosity kg/s m Prandtl number c λ 6 µ /0 Pr = µ c p λ p Values for water at atmospheric pressure between 0 C and 00 C are given with sufficient accuracy by the saturated values in the previous table. The above values are correct for a pressure of atm =.035 bar but may be used with sufficient accuracy at other pressures. Temp. C TRANSPORT PROPERTIES OF AIR Isobaric sp. heat capacity kj/kg K Thermal conductivity W/m K Dynamic viscosity kg/s m Prandtl number c λ 6 µ /0 Pr = µ c p λ p This table may be used with reasonable accuracy for values of c p, γ, µ and Pr of N, O and CO. The above values are correct for a pressure of atm =.035 bar but may be used with sufficient accuracy at other pressures. 06/09/04 8