COMPENDIUM OF EQUATIONS Unified Engineering Thermodynamics
I. Equation of State:
pv = RT or p = RT for a thermally perfect gas
II. Expressions for Work: A. Work for a simple compressible substance V2 W p dV = ext 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 p W = mRT ln 2 = mRT ln 1 v1 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) III. Forms of the First Law of Thermodynamics A. Most general forms
E = Q – W, e = q – w, dE = Q – W, and de = q - w
B. Neglecting changes in kinetic and potential energy
U = Q - W u = q – w, dU = Q - W, and du = q - w
C. Neglecting changes in kinetic and potential energy, in terms of enthalpy
- 1 - H = U + pV therefore dH = dU + pdV + Vdp
so dH = Q - W + pdV + Vdp
or dh = q - w + pdv + vdp
D. For quasi-static processes where changes in kinetic and potential energy are not important.
dU = Q – pdV or du = q – pdv
dH = Q + Vdp or dh = q + vdp
E. For quasi-static processes of an ideal gas where changes in kinetic and potential energy are not important.
mcvdT = Q – pdV or cvdT = q – pdv
mcpdT = Q + Vdp or cpdT = q + vdp
IV. The First Law of Thermodynamics as a Rate Equation A. Most general form
dE c.v. = Q˙ W˙ + m˙ e m˙ e dt c.v. c.v. in in out out
rate of change rate of heat rate of work rate of energy rate of energy = + of energy in c.v. added to c.v. done flow in to c.v. flow out of c.v.
B. For a steady flow process
d = 0 and m˙ = m˙ = m˙ dt in out ˙ ˙ Qc .v. W c.v. = m˙ (eout ein ) or
Q˙ W˙ m˙ IE KE PE IE KE PE c.v. c.v. = []()+ + out ()+ + in
- 2 - C. For a steady flow process neglecting changes in potential energy
2 2 ˙ ˙ c c Qc .v. W c.v. = m˙ u + u + 2 out 2 in
or c 2 c 2 q w = u u + 2 1 12 12 2 1 2 2 written in terms of external or shaft work
2 2 c2 c1 q12 ws1 2 = ()u2 + p2v2 ()u1 + p1v1 + 2 2 or in terms of shaft work and enthalpy
c 2 c 2 q w = h h + 2 1 12 s12 2 1 2 2
D. Steady flow energy equation for an ideal gas
2 2 c2 c1 q12 ws12 = c pT2 + c pT1 + 2 2
E. Steady flow energy equation for an ideal gas for an adiabatic process with no shaft work
c 2 c 2 c T + 2 = c T + 1 p 2 2 p 1 2
The quantity that is conserved is called the stagnation temperature.
2 c TT 1 2 TT = T + or =1+ M usinga= RT 2c p T 2
It is also convenient to define the stagnation enthalpy, hT
c 2 h = c T + T p 2 so we can rewrite the Steady Flow Energy Equation in a convenient form as
q12 ws12 = hT 2 hT1
- 3 - F. Steady flow energy equation for an ideal gas for a quasi-static adiabatic process with no shaft work p 1 1 T = 1 + M 2 p 2
G. For a uniform state, uniform flow process from time t1 to time t2
c 2 c 2 c 2 c 2 m2 u + + gZ m1 u + + gZ + mout h + + gZ min h + + gZ = Qc.v. Ws,c.v. 2 2 2 2 t2 t1 c .v. out in
H. For a steady process in terms of molar flow rates and enthalpy per mole (neglecting changes in kinetic and potential energy)
˙ ˙ n˙ out hout n˙ in h in = Q W s
V. Other relationships A. Relationship between properties for quasi-static, adiabatic processes for thermally perfect gases pv = constant
1 p T 1 T v P v 2 = 2 and 2 = 1 and 2 = 1 p1 T1 T1 v2 P1 v2
B. Thermal efficiency of a cycle
net work w = = heat input qcomb. C. Quality, x, for two-phase systems
mass vapor v v f x = = or v = (1 x)v f + xvg total mass v fg
D. Entropy
dT dv ds = c + R v T v
- 4 - For the case of a thermally perfect gas then
T dT v s s0 = cv + Rln T v T0 0
or in situations with c v = constant
T v s s0 = cv ln + Rln T0 v0
So for the case of a thermally perfect gas then
T dT p s s0 = c p Rln T p T0 0
or in situations with c p = constant
T p s s0 = c p ln Rln T0 p0
VI. Nomenclature a speed of sound (m/s) c velocity (m/s) cp specific heat at constant pressure (J/kg-K) cv specific heat at constant volume (J/kg-K) e energy (J/kg) E energy (J) h enthalpy (J/kg) hT total or stagnation enthalpy (J/kg) H enthalpy (J) m mass (kg) n number of moles (moles) p pressure (kPa) pT total or stagnation pressure (kPa) q heat (J/kg) Q heat (J) R gas constant (J/kg-K) s entropy (J/K) S entropy (J/kg-K) t time (s) T temperature (K) TT total or stagnation temperature (K) u internal energy (J/kg) U internal energy (J) v specific volume (m3/kg) V volume (m3)
- 5 - w work (J/kg) ws shaft or external work (J/kg) W work (J) X quality ratio of specific heats, cp/cv thermal efficiency density (kg/m3)
- 6 -