Equation Compendium

COMPENDIUM OF EQUATIONS Unified Engineering Thermodynamics Note: It is with some reservation that I supply this compendium of equations. One of the common pitfalls for engineering students is that they solve problems through pattern matching rather than through applying the correct equation based upon a foundation of conceptual understanding. This type of training does not serve the students well when they are asked to perform higher-level cognitive tasks such as analysis, synthesis, and evaluation. Thus, while you are welcome to use this list as a guide and a study aid, I expect you to be able to derive each of these equations from their most general forms (e.g. work, the First Law, etc.). Do not settle for a shallow understanding of this material. I. Equation of State: pv = RT or p = ρRT for a thermally perfect gas II. Expressions for Work: A. Work for a simple compressible substance V2 W= ∫p ext dV V1 B. Work for a simple compressible substance undergoing a quasi-static process V2 W= ∫ pdV V1 C. Work for an isothermal, quasi-static process of a simple compressible substance ⎛v ⎞ W = mRT ⋅ ln ⎜ 2 ⎟ ⎝ v1 ⎠ ⎛p ⎞ = mRT ⋅ ln ⎜ 1 ⎟ ⎝ p2 ⎠ D. Work for an isobaric quasi-static process of a simple compressible substance W = p(V2 -V1 ) E. Work for a quasi-static adiabatic process W = - (U2 -U1 ) F. Work for quasi-static adiabatic process of an ideal gas W = -mcv (T2 -T1 ) -1- III. Forms of the First Law of Thermodynamics A. Most general forms ΔE = Q – W, Δe = q – w, dE = δQ – δW, and de = δq - δw B. Neglecting changes in kinetic and potential energy ΔU = Q - W Δu = q – w, dU = δQ - δW, and du = δq - δw C. Neglecting changes in kinetic and potential energy, in terms of enthalpy H = U + pV therefore dH = dU + pdV + Vdp so dH = δQ - δW + pdV + Vdp or dh = δq - δw + pdv + vdp D. For quasi-static processes where changes in kinetic and potential energy are not important. dU = δQ – pdV or du = δq – pdv dH = δQ + Vdp or dh = δq + vdp E. For quasi-static processes of an ideal gas where changes in kinetic and potential energy are not important. mcv dT = δQ – pdV or cv dT = δq – pdv mcp dT = δQ + Vdp or cp dT = δq + vdp IV. The First Law of Thermodynamics as a Rate Equation A. Most general form dE c.v. ˙ ˙ ˙ ine in − m ˙ oute out = Q c.v. − W c.v. + m dt ⎛ rate of change ⎞ =⎛ rate of heat ⎞ −⎛ rateof work⎞ +⎛ rate of energy ⎞ − ⎛ rate of energy ⎞ ⎝ of energyin c.v.⎠ ⎝ added to c.v.⎠ ⎝ done ⎠ ⎝ flowin to c.v.⎠ ⎝ flow outof c.v.⎠ -2- B. For a steady flow process d ˙ in = m ˙ ˙ out = m = 0 and m dt ˙ ˙ (eout − e in ) Q˙ c.v. − W c.v. = m or ˙ =m ˙ [(IE + KE + PE)out − (IE + KE + PE)in ] Q˙ c.v. − W c.v. C. For a steady flow process neglecting changes in potential energy ⎡⎛ ⎛ c2 ⎞ c2 ⎞ ⎤ ˙ ⎜ ⎜ ⎟ ˙ ⎟ ⎥ Q˙ c.v. − W = m u + − u + c.v. ⎢ 2 2 ⎝ ⎠ ⎝ ⎠ in ⎦ ⎣ out or c 2 c2 q1− 2 − w1− 2 = u2 − u1 + 2 − 1 2 2 written in terms of external or shaft work c 22 c12 q1− 2 − ws1− 2 = ( u2 + p 2v 2 ) − (u1 + p1v1 ) + − 2 2 or in terms of shaft work and enthalpy q1− 2 − ws1− 2 = h 2 − h1 + c2 2 c12 − 2 2 D. Steady flow energy equation for an ideal gas ⎛ c 2⎞ ⎛ c 2⎞ q1− 2 − ws1− 2 = ⎜ c p T2 + 2 ⎟ − ⎜ cp T1 + 1 ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ E. Steady flow energy equation for an ideal gas for an adiabatic process with no shaft work c2 2 c1 2 c pT 2 + = c p T1 + 2 2 The quantity that is conserved is called the stagnation temperature. -3- TT = T + ⎛ ⎝ or c2 2c p TT γ −1 2 =1+ M T 2 using a = γRT ⎞⎠ It is also convenient to define the stagnation enthalpy, h T c2 2 h T = c pT + so we can rewrite the Steady Flow Energy Equation in a convenient form as q1− 2 − ws1− 2 = h T2 − h T1 F. Steady flow energy equation for an ideal gas for a quasi-static adiabatic process with no shaft work pT ⎛ γ −1 2 ⎞ = ⎝1 + M ⎠ p 2 γ γ− 1 V. Other relationships A. Relationship between properties for quasi-static, adiabatic processes for thermally perfect gases pvγ = constant p2 ⎛⎜ T2 ⎞⎟ = p1 ⎝ T1 ⎠ γ γ−1 and T2 ⎛⎜ v1 ⎟⎞ = T1 ⎝ v2 ⎠ B. Thermal efficiency of a cycle η= net work Δw = heat input q comb. C. Entropy ds = cv dT dv +R T v For the case of a thermally perfect gas then T s − s0 = ∫ cv T0 ⎛ v⎞ dT + Rln ⎜ ⎟ T ⎝ v0 ⎠ -4- γ−1 and P2 ⎛⎜ v1 ⎞⎟ = P1 ⎝ v2 ⎠ γ or in situations with cv = constant ⎛ T⎞ ⎛ v⎞ s − s0 = c v ln ⎜ ⎟ + Rln⎜ ⎟ ⎝ T0 ⎠ ⎝ v0 ⎠ So for the case of a thermally perfect gas then T s − s0 = ∫ cp T0 ⎛ p⎞ dT − Rln ⎜ ⎟ T ⎝ p0 ⎠ or in situations with cp = constant ⎛ T⎞ ⎛ p⎞ s − s0 = c p ln ⎜ ⎟ − Rln⎜ ⎟ ⎝ T0 ⎠ ⎝ p0 ⎠ VI. Nomenclature a c cp cv e E h hT H m p pT q Q R s S t T TT u U v V w ws W γ η ρ speed of sound (m/s) velocity (m/s) specific heat at constant pressure (J/kg-K) specific heat at constant volume (J/kg-K) energy (J/kg) energy (J) enthalpy (J/kg) total or stagnation enthalpy (J/kg) enthalpy (J) mass (kg) pressure (kPa) total or stagnation pressure (kPa) heat (J/kg) heat (J) gas constant (J/kg-K) entropy (J/K) entropy (J/kg-K) time (s) temperature (K) total or stagnation temperature (K) internal energy (J/kg) internal energy (J) specific volume (m3 /kg) volume (m3 ) work (J/kg) shaft or external work (J/kg) work (J) ratio of specific heats, cp /cv thermal efficiency density (kg/m3 ) -5-