5.5 Key equations

Claire Yu Yan

5. The First Law of Thermodynamics for a Control Volume

5.5 Key equations

Constant-pressure and constant-volume specific heats

Constant-pressure specific heat

C_{p} = {(\frac{\partial h}{\partial T})}_{p}

$C_p=\left(\displaystyle\frac{\partial\ h}{\partial\ T}\right)_p$

Constant-volume specific heat

C_{v} = {(\frac{\partial u}{\partial T})}_{v}

$C_v=\left(\displaystyle\frac{\partial\ u}{\partial\ T}\right)_v$

Relations between

C_{p}

$C_p$ and

C_{v}

$C_v$ for ideal gases

k = \frac{C_{p}}{C_{v}} C_{p} = C_{v} + R

$k=\displaystyle\frac{C_p}{C_v} \qquad \qquad C_p=C_v+R$

$C_v=\displaystyle\frac{R}{k-1} \qquad C_p=\displaystyle\frac{kR}{k-1}$

Specific enthalpy

Change in specific enthalpy

Δ h = h_{2} - h_{1}

$\Delta h = h_2-h_1$

Change in specific enthalpy for ideal gases

Δ h = h_{2} - h_{1} = C_{p} (T_{2} - T_{1})

$\Delta h = h_2-h_1 = C_p(T_2-T_1)$
(assuming constant

C_{p}

$C_p$ in the temperature range)

Relation between

Δ h

$\Delta h$ and

Δ u

$\Delta u$ for solids and liquids

Δ h \approx Δ u \approx C_{p} (T_{2} - T_{1})

$\Delta h \approx\Delta u\approx C_p(T_2-T_1)$

Mass conservation equations in a control volume

Volume flow rate

$\dot{\mathbb{V}}=\displaystyle\frac{d\mathbb{V}}{dt}=V_{avg,\ n}A=\dot{m}v$

Mass flow rate

$\dot{m}=\displaystyle\frac{dm}{dt}=\rho\ V_{avg,\ n}A=\rho\dot{\mathbb{V}}$

Transient flow

$\displaystyle\frac{dm_{CV}}{dt}=\sum{\dot{m}}_i-\sum{\dot{m}}_e\neq0$

Steady flow

$\displaystyle\frac{dm_{CV}}{dt}=\sum{\dot{m}}_i-\sum{\dot{m}}_e=0$

Energy conservation equations in a control volume

Transient flow

$\displaystyle\frac{dE_{CV}}{dt}\neq0$

$\begin{align*} \displaystyle\frac{dE_{CV}}{dt}={\dot{Q}}_{cv}-{\dot{W}}_{cv} &+\sum{{\dot{m}}_i(h_i+\frac{1}{2}V_i^2+gz_i)} \\&-\sum{{\dot{m}}_e(h_e+\frac{1}{2}V_e^2+gz_e)} \end{align*}$

Steady flow

$\displaystyle\frac{dE_{CV}}{dt}=0$

${\dot{Q}}_{cv} +\sum{{\dot{m}}_i\left(h_i+\displaystyle\frac{1}{2}V_i^2+gz_i\right) \\={\dot{W}}_{cv} +\sum{{\dot{m}}_e\left(h_e+\displaystyle\frac{1}{2}V_e^2+gz_e\right)}}$

Mass and energy conservation equations for steady-state, steady-flow (SSSF) devices

SSSF device

Assumptions

Mass conservation

Energy conservation

Expansion device

Adiabatic flow; Negligible work transfer with the surroundings; Negligible changes in kinetic and potential energies

${\dot{m}}_i={\dot{m}}_e$

$h_i=h_e$

Nozzle and diffuser

Adiabatic flow; Negligible work transfer with the surroundings; Negligible change in potential energy

${\dot{m}}_i={\dot{m}}_e$

$h_i+\displaystyle\frac{1}{2}V_i^2=h_e+\displaystyle\frac{1}{2}V_e^2$

Mixing chamber

Negligible work transfer with the surroundings; Negligible changes in kinetic and potential energies

$\sum{{\dot{m}}_i=\sum{\dot{m}}_e}$

${\dot{Q}}_{cv}+\sum{{\dot{m}}_ih_i=\sum{{\dot{m}}_eh_e}}$

Heat exchanger

Negligible work transfer with the surroundings; Negligible changes in kinetic and potential energies

${\dot{m}}_i={\dot{m}}_e$
(for each of the hot and cold streams, separately)

${\dot{Q}}_{cv}+\sum{{\dot{m}}_ih_i=\sum{{\dot{m}}_eh_e}}$

Turbine

Adiabatic flow; Negligible changes in kinetic and potential energies

${\dot{m}}_i={\dot{m}}_e=\dot{m}$

${\dot{W}}_{shaft}=\dot{m}(h_i-h_e)$

Compressor

Adiabatic flow; Negligible changes in kinetic and potential energies

${\dot{m}}_i={\dot{m}}_e=\dot{m}$

${\dot{W}}_{shaft}=\dot{m}(h_e-h_i)$

License

Icon for the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

Introduction to Engineering Thermodynamics Copyright © 2022 by Claire Yu Yan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

License

Share This Book