THERMODYNAMICS PROPERTIES OF SINGLE-COMPONENT SYSTEMS For an ideal gas, Pv = RT or PV = mRT, and Nomenclature P1v1/T1 = P2v2/T2, where 1. Intensive properties are independent of mass. P = pressure, 2. Extensive properties are proportional to mass. v = speci c volume, 3. Speci c properties are lowercase (extensive/mass). m = mass of gas, R = gas constant, and State Functions (properties) 2 Absolute Pressure, P (lbf/in or Pa) T = absolute temperature. Absolute Temperature, T (°R or K) V = volume Volume, V (ft3 or m3) R is speci c to each gas but can be found from 3 3 Speci c Volume, vVm= (ft /lbm or m /kg) = R R ^h, where Internal Energy, U (Btu or kJ) mol. wt Speci c Internal Energy, R = the universal gas constant uUm= (usually in Btu/lbm or kJ/kg) = 1,545 ft-lbf/(lbmol-°R) = 8,314 J/(kmol⋅K). Enthalpy, H (Btu or KJ) For ideal gases, c – c = R Speci c Enthalpy, p v h = u + Pv = H/m (usually in Btu/lbm or kJ/kg) Also, for ideal gases: bb2h ll2u Entropy, S (Btu/°R or kJ/K) 2 ==002 P TTv Speci c Entropy, s = S/m [Btu/(lbm-°R) or kJ/(kg•K)] Gibbs Free Energy, g = h – Ts (usually in Btu/lbm or kJ/kg) For cold air standard, heat capacities are assumed to be constant at their room temperature values. In that case, the Helmholz Free Energy, following are true: a = u – Ts (usually in Btu/lbm or kJ/kg) Δu = c ΔT; Δh = c ΔT = bl2h v p Heat Capacity at Constant Pressure, cp 2 Δ T P s = cp ln (T2 /T1) – R ln (P2 /P1); and 2 = blu Δs = c ln (T /T ) + R ln (v /v ). Heat Capacity at Constant Volume, cv 2 v 2 1 2 1 T v For heat capacities that are temperature dependent, the value Quality x (applies to liquid-vapor systems at saturation) is to be used in the above equations for Δh is known as the mean de ned as the mass fraction of the vapor phase: `j heat capacity cp and is given by x = mg /(mg + mf), where #T2 mg = mass of vapor, and cdTp = T1 cp mf = mass of liquid. TT21- Also, for constant entropy processes: Speci c volume of a two-phase system can be written: - k k 1 v = xv + (1 – x)v or v = v + xv , where P v T P k g f f fg 2 ==d 1 n ; 2 d 2 n P v T P vf = speci c volume of saturated liquid, 1 2 1 1 - vg = speci c volume of saturated vapor, and k 1 T2 d v1 n ==, where kccpv vfg = speci c volume change upon vaporization. T1 v2 = vg – vf For real gases, several equations of state are available; one Similar expressions exist for u, h, and s: such equation is the van der Waals equation with constants u = xug + (1 – x) uf or u = uf + xufg based on the critical point: h = xh + (1 – x) h or h = h + xh g f f fg cP +-=a m ^vbh RT v2 s = xsg + (1 – x) sf or s = sf + xsfg 2 RT2 RT For a simple substance, speci cation of any two intensive, where a ==c 27 m f c p, b c independent properties is suf cient to [ x all the rest. 64 Pc 8Pc where Pc and Tc are the pressure and temperature at the critical point, respectively, and v is the molar speci c volume. THERMODYNAMICS 73 FIRST LAW OF THERMODYNAMICS us = speci c internal energy of system, and The First Law of Thermodynamics is a statement of Qo = rate of heat transfer (neglecting kinetic and potential conservation of energy in a thermodynamic system. The net energy of the system). energy crossing the system boundary is equal to the change in energy inside the system. Special Cases of Open Systems Constant Volume: Heat Q is energy transferred due to temperature difference w = – v (P – P ) and is considered positive if it is inward or added to the rev 2 1 system. Constant Pressure: w = 0 Closed Thermodynamic System rev No mass crosses system boundary Constant Temperature: ( ideal gas) Pv = constant Q – W = ΔU + ΔKE + ΔPE w = RT ln (v /v ) = RT ln (P /P ) where rev 2 1 1 2 ΔKE = change in kinetic energy, and Isentropic (ideal gas): k ΔPE = change in potential energy. Pv = constant w = k (P v – P v )/(1 – k) Energy can cross the boundary only in the form of heat or rev 2 2 1 1 = kR (T2 – T1)/(1 – k) work. Work can be boundary work, wb, or other work forms ^kk- 1h/ (electrical work, etc.) = k > - d P2 n H wrev - RT1 1 bl= W k 1 P1 Work W w m is considered positive if it is outward or work done by the system. Polytropic: ∫ Pvn = constant Reversible boundary work is given by wb = P dv. wrev = n (P2v2 – P1v1)/(1 – n) Special Cases of Closed Systems Steady-State Systems Constant Pressure (Charles’ Law): The system does not change state with time. This assumption Δ wb = P v is valid for steady operation of turbines, pumps, compressors, (ideal gas) T/v = constant throttling valves, nozzles, and heat exchangers, including Constant Volume: boilers and condensers. RRoo``++-22jj +++-=o o wb = 0 mhi Vi //220 gZiee m h Ve gZe Q in W out (ideal gas) T/P = constant and Isentropic (ideal gas): RRmmoo= k ie Pv = constant where w = (P v – P v )/(1 – k) 2 2 1 1 o = R(T – T )/(1 – k) m = mass \ ow rate (subscripts i and e refer to inlet and exit 2 1 states of system), Constant Temperature (Boyle’s Law): g = acceleration of gravity, (ideal gas) Pv = constant Z = elevation, w = RTln (v / v ) = RTln (P /P ) b 2 1 1 2 V = velocity, and Polytropic (ideal gas): o n W = rate of work. Pv = constant w = (P2v2 – P1v1)/(1 – n) Special Cases of Steady-Flow Energy Equation Open Thermodynamic System Nozzles, Diffusers: Velocity terms are signi cant. No Mass crosses the system boundary elevation change, no heat transfer, and no work. Single mass There is ow work (Pv) done by mass entering the system. stream. +=+22 The reversible ow work is given by: hVi i //22 hVe e ∫ Δ Δ wrev = – v dP + ke + pe VV22- Isentropic Ef ciency (nozzle) = ei, where _i First Law applies whether or not processes are reversible. 2 hhies- FIRST LAW (energy balance) hes = enthalpy at isentropic exit state. RRoo88++-22BB ++ mhii Vi //22 gZiee mh Ve gZe Turbines, Pumps, Compressors: Often considered adiabatic +-o o =_i QWin net dmudt s s /,where (no heat transfer). Velocity terms usually can be ignored. o There are signi cant work terms and a single mass stream. Wnet = rate of net or shaft work transfer, ms = mass of \ uid within the system, hi = he + w 74 THERMODYNAMICS hh- IDEAL GAS MIXTURES Isentropic Ef ciency (turbine) = ie hhies- i = 1, 2, …, n constituents. Each constituent is an ideal gas. Mole Fraction: hhes- i Isentropic Ef ciency (compressor, pump) = Σ Σ hhei- xi = Ni /N; N = Ni; xi = 1 Throttling Valves and Throttling Processes: No work, no where Ni = number of moles of component i. heat transfer, and single-mass stream. Velocity terms are often Σ Σ Mass Fraction: yi = mi /m; m = mi; yi = 1 insigni cant. Σ Molecular Weight: M = m/N = xiMi hi = he Gas Constant: RRM= / Boilers , Condensers, Evaporators, One Side in a Heat To convert mole fractions x to mass fractions y : Exchanger: Heat transfer terms are signi cant. For a single- i i xM mass stream, the following applies: y = ii i R_i xMii hi + q = he To convert mass fractions to mole fractions: Heat Exchangers: No heat or work. Two separate ow rates o o yMii m1 and m2: x = i R _iyMii oo__ii mh11ie-= h 1 mh 2 2 ei - h 2 mRT Partial Pressures: PPP==R ; ii See MECHANICAL ENGINEERING section. ii V Mixers, Separators, Open or Closed Feedwater Heaters: mRT RRoo= Partial Volumes: VVV==! ; ii , where mhii mh ee and ii P RRmmoo= ie P, V, T = the pressure, volume, and temperature of the mixture. BASIC CYCLES xi = Pi /P = Vi /V Heat engines take in heat QH at a high temperature TH, produce a net amount of work W, and reject heat QL at a low Other Properties: temperature T . The ef ciency η of a heat engine is given by: Σ Σ Σ L u = (yiui); h = (yihi); s = (yisi) η u and h are evaluated at T, and = W/QH = (QH – QL)/QH i i s is evaluated at T and P . The most ef cient engine possible is the Carnot Cycle. Its i i ef ciency is given by: PSYCHROMETRICS η c = (TH – TL)/TH, where We deal here with a mixture of dry air (subscript a) and water vapor (subscript v): TH and TL = absolute temperatures (Kelvin or Rankine). P = P + P The following heat-engine cycles are plotted on P-v and T-s a v diagrams (see later in this chapter): Speci c Humidity (absolute humidity, humidity ratio) ω: ω Carnot, Otto, Rankine = mv /ma, where mv = mass of water vapor and Refrigeration cycles are the reverse of heat-engine cycles. ma = mass of dry air. Heat is moved from low to high temperature requiring work, ω = 0.622P /P = 0.622P /(P – P ) W. Cycles can be used either for refrigeration or as heat v a v v pumps.