Subsections

5 Processes

5.1 Control Mass Processes

Specific heat and work:

\begin{displaymath}
\vphantom{q}_1q_2 = \frac{\vphantom{Q}_1Q_2}{m}
\quad
...
...m{q}_1q_2
\quad
\vphantom{W}_1W_2 = m\; \vphantom{w}_1w_2
\end{displaymath}

Continuity (mass conservation):

\begin{displaymath}
m_1 \left(+ m_{\mbox{\scriptsize added}} \right) = m_2
\end{displaymath}

The first law of thermo (energy conservation):

\begin{displaymath}
E_2 - E_1 = \vphantom{Q}_1Q_2 - \vphantom{W}_1W_2
\qquad (E = U + m {\textstyle \frac12}{\bf V}^2? + m g Z?)
\end{displaymath}

Work:

\begin{displaymath}
\vphantom{W}_1W_2 = \int_1^2 p \; {\rm d} V =
\left\{
...
...1}} pV\ln\left(\frac{V_2}{V_1}\right)
\end{array}
\right.
\end{displaymath}

In case of an ideal gas, note that $pV$ can be replaced by $mRT$.

Heat added:

\begin{displaymath}
\vphantom{Q}_1Q_2 =
\left\{
\begin{array}{rl}
\mbox{...
...tyle\vphantom{\int_1^2}
T (S_2-S_1)
\end{array}
\right.
\end{displaymath}

5.2 Rate Equations

The first law as a rate equation:

\begin{displaymath}
\frac{{\rm d}E}{{\rm d}t} = \dot Q - \dot W \qquad
\left...
...bf V}^2}{{\rm d}t}? +
m g \frac{{\rm d}Z}{{\rm d}t}?\right)
\end{displaymath}

Work as a rate equation:

\begin{displaymath}
\dot W = p \dot V
\end{displaymath}

For an ideal gas:

\begin{displaymath}
\frac{{\rm d}u}{{\rm d}t} = C_v \frac{{\rm d}T}{{\rm d}t}
\end{displaymath}

For liquids and solids:

\begin{displaymath}
\dot Q = m C_{(p)} \frac{{\rm d}T}{{\rm d}t}
\end{displaymath}

5.3 Steady State Control Volume Processes

The control volume is assumed to be steady state in all formulae below.

Specific work output and heat added (i.e., per unit mass flowing through):

\begin{displaymath}
w = \frac{\dot W}{\dot m} \quad q = \frac{\dot Q}{\dot m}
\qquad
\dot W = \dot m w \quad \dot Q = \dot m q
\end{displaymath}

In- and outflow velocities and pipe cross-sectional areas:

\begin{displaymath}
\dot m = \dot V/v = A {\bf V}/v \qquad A=\frac\pi 4 D^2
\end{displaymath}

Continuity (mass conservation):

\begin{displaymath}
\sum \dot m_i = \sum \dot m_e
\end{displaymath}

where $\sum$ means sum over all inflow/outflow points, if there is more than one.

The first law of thermo (energy conservation):

\begin{displaymath}
\dot Q +
\sum \dot m_i \left(h_i+ \frac12 {\bf V}_i^2 + ...
... m_e \left(h_e+ \frac12 {\bf V}_e^2 + g Z_e\right) +
\dot W
\end{displaymath}

The kinetic energy and potential energy terms are often ignored. Devices without moving parts do not do work, $\dot W = 0$. Adiabatic devices have no heat transfer, $\dot Q = 0$.