Chapter 4 Formula Sheet

Formula Units
Displacement Work

    \[W=\int_{1}^{2}{Fdx}=\int_{1}^{2}{Pdv}\]

    \[J\]

Integration

    \[W=\int_{1}^{2}{Pdv}=P(V_2-V_1)\]

    \[J\]

Specific Work

    \[w=\frac{w}{m}(work \ per \ unit \ mass)\]

    \[J/kg\]

Power (rate of work)

    \[W=F\bar{V}=PV=T\omega\]

    \[W\]

  • Velocity

    \[\bar{V}=r\omega\]

    \[Rad/s\]

  • Torque

    \[T=Fr\]

    \[Nm\]

Isobaric Process

    \[W=P_o(V_2-V_1)\]

    \[P_0=P_1=P_2\]

    \[W\]

Polytropic Process ( n \neq 1 )

    \[PV^n = Const =P_1V_1^n=P_2V_2^n\]

    \[Pv^n=C\]

  • Polytropic Exponent

    \[n=\frac{ln (\frac{P_2}{P_1})}{ln (\frac{V_1}{V_2})}\]

  • n=1

    \[PV=Const=P_1V_1=P_2V_2\]

Polytropic Process Work

    \[W_{1-2}=\frac{(P_2V_2-P_1V_1)}{1-n}, n \neq 1\]

    \[J\]

Isothermal Process

    \[W_{1-2}=P_2V_2ln(\frac{V_2}{V_1})\]

    \[J\]

Adiabatic Process

    \[Q=0\]

Conduction Heat Transfer

    \[Q=-kA\frac{dT}{dx},k=thermal \ conductivity\]

    \[W\]

Convection Heat Transfer

    \[Q=hA\Delta T, h=heat \ transfer \ coefficient\]

    \[W\]

Radiation Heat Transfer

    \[Q=\epsilon \sigma A(T_s^4-T^4_{amb})\]

    \[W\]

Terminology
Q = heat
Q = heat transferred during the process between state 1 and state 2
Q = rate of heat transfer
W = work
W_{1-2} = work done during the change from state 1 to state 2
W = rate of work = power. 1w = 1 J/s
K = specific hear ratio
c_p = constant pressure
c_v = constant volume