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

can be replaced by

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, $\dot Q = 0$ .