COMPENDIUM OF EQUAIONS Unified Engineering hermodynamics Note: It is with some reseration that I suly this comendium of equations. One of the common itfalls for engineering students is that they sole roblems through attern matching rather than through alying the correct equation based uon a foundation of concetual understanding. his tye of training does not sere the students well when they are asked to erform higher-leel cognitie tasks such as analysis, synthesis, and ealuation. hus, while you are welcome to use this list as a guide and a study aid, I exect you to be able to derie 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: = R or = ρr for a thermally erfect gas II. Exressions for Work: A. Work for a simle comressible substance V W = ext dv V 1 B. Work for a simle comressible substance undergoing a quasi-static rocess V W = dv V 1 C. Work for an isothermal, quasi-static rocess of a simle comressible substance W = mr ln 1 = mr ln 1 D. Work for an isobaric quasi-static rocess of a simle comressible substance W = (V -V 1 E. Work for a quasi-static adiabatic rocess W = - (U -U 1 F. Work for quasi-static adiabatic rocess of an ideal gas W = -mc ( - 1-1 -
III. Forms of the First Law of hermodynamics A. Most general forms ΔE = Q W, Δe = q w, de = δq δw, and de = δq - δw B. Neglecting changes in kinetic and otential energy ΔU = Q - W Δu = q w, du = δq - δw, and du = δq - δw C. Neglecting changes in kinetic and otential energy, in terms of enthaly H = U + V therefore dh = du + dv + Vd so dh = δq - δw + dv + Vd or dh = δq - δw + d + d D. For quasi-static rocesses where changes in kinetic and otential energy are not imortant. du = δq dv dh = δq + Vd or du = δq d or dh = δq + d E. For quasi-static rocesses of an ideal gas where changes in kinetic and otential energy are not imortant. mcd = δq dv or cd = δq d mc d = δq + Vd or c d = δq + d IV. he First Law of hermodynamics as a Rate Equation A. Most general form de c.. = Q c.. W c.. + m in e in m e dt rate of change rate of heat rateof work + rate of energy rate of energy = of energyin c.. added to c.. done flowin to c.. flow of c.. - -
B. For a steady flow rocess d = 0 and min = m = m dt Q c.. W c.. = m (e e in or Q = m c.. W c.. [(IE + KE + PE (IE + KE + PE in ] C. For a steady flow rocess neglecting changes in otential energy Q W c.. = m u + u + c c.. c in or q 1 w 1 = u u 1 + c written in terms of external or shaft work q 1 w s 1 = (u + (u 1 + 1 1 + c or in terms of shaft work and enthaly q1 w s1 = h h 1 + c D. Steady flow energy equation for an ideal gas c 1 q 1 w s 1 = c + c c 1 + E. Steady flow energy equation for an ideal gas for an adiabatic rocess with no shaft work c 1 c + c = c 1 + he quantity that is consered is called the stagnation temerature. - 3 -
c 1 M = + or = 1 + using a = R c It is also conenient to define the stagnation enthaly, h c h = c + so we can rewrite the Steady Flow Energy Equation in a conenient form as q 1 w s 1 = h h 1 F. Steady flow energy equation for an ideal gas for a quasi-static adiabatic rocess with no shaft work = 1 + 1 M 1 V. Other relationshis A. Relationshi between roerties for quasi-static, adiabatic rocesses for thermally erfect gases = constant 1 1 1 = and = P and = 1 1 1 1 P 1 B. hermal efficiency of a cycle net work η= = heat inut Δw q comb. C. Entroy d ds = c d + R For the case of a thermally erfect gas then d s s 0 = c + Rln 0 0-4 -
or in situations with c = constant s s 0 = c ln + Rln 0 0 So for the case of a thermally erfect gas then d s s 0 = c Rln 0 0 or in situations with c = constant VI. Nomenclature s s 0 = c ln Rln 0 0 a seed of sound (m/s c elocity (m/s c secific heat at constant ressure (J/kg-K c secific heat at constant olume (J/kg-K e energy (J/kg E energy (J h enthaly (J/kg h H total or stagnation enthaly (J/kg enthaly (J m mass (kg ressure (kpa q total or stagnation ressure (kpa heat (J/kg Q heat (J R gas constant (J/kg-K s entroy (J/K S entroy (J/kg-K t time (s temerature (K u total or stagnation temerature (K internal energy (J/kg U internal energy (J secific olume (m 3 /kg V olume (m 3 w work (J/kg ws W shaft or external work (J/kg work (J ratio of secific heats, c /c η thermal efficiency ρ density (kg/m 3-5 -