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\def\eo{{\strut\dotfill\it\HTML{U}{Exams 3 and final only}\dotfill}}
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\begin{document}

\title{Formulae Collection}
\author{}
\date{Revised 2/22/07 for Cengel}
\maketitle

\tableofcontents

\setlength{\parskip}{\baselineskip}\setlength{\parindent}{0pt}


\section{Introduction}

These formulae are not guaranteed to be complete and/or correct,
though we tried to make them so.  The {\em student} has the final
reponsibility for knowing all formulae covered in both class lectures
and homework correctly.  So, if you see an error, or miss a formulae
that you think was covered, (if you are mistaken about that, it could
get you in trouble!),
\begin{rawhtml}
  <a href="mailto:dommelen@eng.fsu.edu">
\end{rawhtml}
let us know immediately.
\begin{rawhtml}
  </a>
\end{rawhtml}

This document can {\em not} be taken to the exams.


\section{Units}

Basic SI units are kilogram (kg), meter (m), and second (s).  Many
derived units exist.  The book uses kPa, kJ, kW instead of
the ``normal'' units Pa, J, W.  Conversions may be needed
(inside back-cover.)

Definitions: 1 bar = $10^5$ Pa.  1 atm = 101,325 Pa.

Absolute temperature (required in most formulae and some tables): add
273.15 to convert Centigrade into Kelvin, substract 273.15 to convert
Kelvin into Centigrade.  However, temperature {\em differences} are
the same in Kelvin as in Centigrade; the two temperatures in the
difference do not need to be converted.


\section{Properties}

The intensive specific properties are lowercase, the corresponding
extensive properties are uppercase:
\begin{displaymath}
  v=\frac Vm \quad u= \frac Um \quad h=\frac Hm
  \qquad V=mv \quad U=mu \quad H=mh
\end{displaymath}

\eo
\begin{displaymath}
  s= \frac Sm
  \qquad S=ms
\end{displaymath}
\eeo

Definition of enthalpy:
\begin{displaymath}
  h \equiv u + p v
\end{displaymath}

\eo\\
Definition of entropy:
\begin{displaymath}
  s \equiv \int_{\mbox{\scriptsize ref}}
  \left(\frac{\delta q}{T}\right)_{\mbox{\scriptsize rev}}
\end{displaymath}
\eeo

Specific volume and density:
\begin{displaymath}
  v = \frac{1}{\rho}
\end{displaymath}

Pressure and force:
\begin{displaymath}
  F = p A
\end{displaymath}

Static pressure differences versus height $z$ differences:
\begin{displaymath}
  p_A - p_B = - \rho g \left(z_A - z_B\right)
\end{displaymath}


\section{Formulae for substances}

How you treat a substance depends on the type of substance.

\subsection{Substances with saturated and superheated tables}

Tables of specific properties such as tables A-4 through A-7 for water
are available.  Use requires normally drawing a $pv$-diagram, and/or a
$Tv$-diagram,
\\\eo\\
and/or a $Ts$-diagram,\\
\eeo\\
which students must master, and which is covered
\htlink{elsewhere}
{http://www.eng.fsu.edu/~dommelen/courses/eml3100/aids/tables}.

In the two phase region:
\begin{displaymath}
  v=v_f + x \left(v_g -v_f\right) \quad
  u=u_f + x \left(u_g -u_f\right) \quad
  h=h_f + x \left(h_g -h_f\right) \quad
\end{displaymath}
\eo
\begin{displaymath}
  s=s_f + x \left(s_g -s_f\right)
\end{displaymath}
\eeo\\
where $f$ stands for the saturated liquid ($x=0$) value at the same
pressure and same temperature, and $g$ for the saturated vapor ($x=1$)
value.  These can be found in the saturated tables.  The quality $x$
is the ratio of the vapor mass to the total mass of vapor and liquid.
The quality $x$ is undefined if the substance is not two-phase or
saturated.

For compressed liquids, often there are no suitable tables available.
Fortunately, good approximate values for $v$, $u$, $h$,\\
\eo\\
and $s$,\\
\eeo\\
can usually be taken from the saturated tables at the correct given
temperature.  The fact that the given pressure is not the same as the
saturated pressure must then be ignored.  (For $h$, it is more
accurate to write $h=u+pv$ and then take $u$ and $v$ from the
saturated tables, instead of $h$ itself.)


\subsection{Ideal Gases}

Number of kilomoles:
\begin{displaymath}
  N = \frac mM
\end{displaymath}
where $M$ is the molecular mass (table A-1.)

Specific gas constant $R$ in terms of the universal gas constant $R_u$
(back cover):
\begin{displaymath}
  R = \frac {R_u}{M}
\end{displaymath}
where $M$ is the molecular mass (table A-1, but this table already
includes $R$).

Forms of the ideal gas law:
\begin{displaymath}
  \begin{array}{cc}
    pv = RT \quad & \quad p\bar v = R_u T \\
    \vspace{-3pt} \\
    pV = m R T \quad & \quad p V = N R_u T
  \end{array}
\end{displaymath}

To correct for real gas effects, replace $R$ (or $R_u$) by $Z R$
($ZR_u$), where $Z$ is the compressibility factor (figure A-15, with
$P_R=p/p_{cr}$, $T_R=T/T_{cr}$ and $p_{cr}$ and $T_{cr}$ from A-1.)

For an ideal gas, $u$, $h$, $C_p$, $C_v$, and $k$ only depend on
temperature.  Also,
\begin{displaymath}
  u_2 - u_1 = \int_1^2 C_v \;{\rm d}T \qquad
  h_2 - h_1 = \int_1^2 C_p \;{\rm d}T
\end{displaymath}
(Using table A-2(c), these can actually be integrated.)  Note that for
ideal gasses we often do not have $u$, $h$, or $s$ themselves, and we
have to make do with differences.  (For a mixing chamber you will need
to replace $\dot m_3$ with $\dot m_1 + \dot m_2$ before you can use
differences.)

Specific heat relation:
\begin{displaymath}
  C_p - C_v = R
\end{displaymath}
Hence if we know $C_p$ (table A-2(c)) we can compute $C_v$ and
vice-versa.  Also, if $C_p$ is constant, then so is $C_v$ and vice-versa,

Definition of specific heat ratio:
\begin{displaymath}
  k = \frac{C_p}{C_v}
\end{displaymath}
If $k$ is constant, then so are $C_p$ and $C_v$, and vice-versa.

Note that {\em isothermal} ideal gasses are also polytropic with $n=1$.

\subsubsection{Ideal gases in A-17 through A-25}

The internal energy $u$ and enthalpy $h$ can be read off in the table
as a function of temperature.  Or vice-versa, for that matter.  Watch
the units.

\eo\\
The formula that gives you the entropy $s$ is, (in terms of differences),
\begin{displaymath}
      s_2 - s_1 =
      s_{2}^0-s_{1}^0
      - R \ln\left(\frac{p_2}{p_1}\right)
\end{displaymath}
where $s_{1}^0$, also written as $s^0(T_1)$, is whatever you find
in the table's $s^0$ column at the temperature $T_1$.  (It is what
the entropy would be if the pressure would have been one atmosphere.)
\\\eeo

\subsubsection{Ideal gasses with constant specific heats}

If no A-17 through A-25 table is available, you will have to assume
that the specific heats $C_p$ and $C_v$, as well as their ratio $k$
are constant.  Specific heats that can be assumed to be approximately
constant can be computed from the average process temperature using
table A-2(b) or (c).  Less accurately, near room temperature they can
be taken from A-2(a).  Use A-2(b) or (c) wherever possible unless you
are very close to room temperature.

Internal energy:
\begin{displaymath}
  u_2 - u_1 \approx
  {C_v}\strut_{\mbox{\scriptsize ave}} \left(T_2 - T_1\right)
\end{displaymath}

Enthalpy:
\begin{displaymath}
  h_2 - h_1 \approx
  {C_p}\strut_{\mbox{\scriptsize ave}} \left(T_2 - T_1\right)
\end{displaymath}

\eo\\
Entropy:
\begin{displaymath}
  \begin{array}{rcl}
    s_2 - s_1 & = &
    \displaystyle C_p\strut_{\mbox{\scriptsize ave}}
    \ln\left(\frac{T_2}{T_1}\right)
    - R \ln\left(\frac{p_2}{p_1}\right) \\
    \vspace{-3pt} \\
    & \approx &
    \displaystyle C_v\strut_{\mbox{\scriptsize ave}}
    \ln\left(\frac{T_2}{T_1}\right)
    + R \ln\left(\frac{v_2}{v_1}\right) \\
    \vspace{-3pt} \\
    & \approx &
    \displaystyle C_v\strut_{\mbox{\scriptsize ave}}
    \ln\left(\frac{p_2}{p_1}\right)
    + C_p\strut_{\mbox{\scriptsize ave}} \ln\left(\frac{v_2}{v_1}\right)
  \end{array}
\end{displaymath}
\eeo

If the process is isothermal,\\
\eo\\
or reversible adiabatic (isentropic),\\
\eeo\\
the following {\lq\lq}polytropic{\rq\rq} relations also apply (they are
equivalent to the equations above, but often much more convenient):
\begin{displaymath}
  \begin{array}
    {l@{\hspace{6pt}}c@{\hspace{6pt}}c@{\hspace{6pt}}c@{\hspace{6pt}}ll}
    \displaystyle \frac{p_2}{p_1}
    &=& \displaystyle
    \left(\frac{v_1}{v_2}\right)^n
    & = & \displaystyle
    \left(\frac{T_2}{T_1}\right)^{\textstyle\frac{n}{n-1}} \\
    \vspace{-3pt} \\
    \displaystyle \frac{T_2}{T_1}
    & = & \displaystyle
    \left(\frac{v_1}{v_2}\right)^{n-1}
    & = & \displaystyle
    \left(\frac{p_2}{p_1}\right)^{\textstyle\frac{n-1}{n}}
    & \displaystyle\qquad
    \left(
      \begin{array}{rl}
        \mbox{isothermal:} & n = 1 \\
        \mbox{isentropic and $k$ constant:} & n = k
      \end{array}
    \right) \\
    \vspace{-3pt} \\
    \displaystyle \frac{v_2}{v_1}
    & = & \displaystyle
    \left(\frac{p_1}{p_2}\right)^{\textstyle\frac{1}{n}}
    & = & \displaystyle
    \left(\frac{T_1}{T_2}\right)^{\textstyle\frac{1}{n-1}}
  \end{array}
\end{displaymath}


\subsection{Solids and Liquids without saturated tables}

Approximate formulae if no better tabulated values are available,
or to simplify things.

Heat added:
\begin{displaymath}
  Q_{12} = m \int_1^2 C_{(p)} \;{\rm d}T \approx
  m {C_{(p)}}\strut_{\mbox{\scriptsize ave}} \left(T_2 - T_1\right)
\end{displaymath}

Enthalpy:
\begin{displaymath}
  h_2-h_1 \approx {C_{(p)}}\strut_{\mbox{\scriptsize ave}}
  \left(T_2 - T_1\right)\quad \left[ + v\left(p_2-p_1\right) \right]
\end{displaymath}
The term in the square brackets can often be neglected.

\eo\\
Entropy:
\begin{displaymath}
  s_2 - s_1 \approx {C_{(p)}}\strut_{\mbox{\scriptsize ave}}
  \ln\left(\frac{T_2}{T_1}\right)
\end{displaymath}
\eeo

\section{Processes}

\subsection{Control Mass Processes (Closed Systems)}

Specific heat and work:
\begin{displaymath}
  q_{12} = \frac{Q_{12}}{m}
  \quad
  w_{12} = \frac{W_{12}}{m}
  \qquad
  Q_{12} = m\; q_{12}
  \quad
  W_{12} = m\; w_{12}
\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 = Q_{12} - W_{12}
  \qquad (E = U + m {\textstyle \frac12}{\bf V}^2? + m g z?)
\end{displaymath}

Work:
\begin{displaymath}
  W_{12} = \int_1^2 p \; {\rm d} V =
  \left\{
    \begin{array}{rl}
      \mbox{incompressible/constant volume (isochoric):} &
      \displaystyle\vphantom{\frac{V^2}{V_1}} 0 \\
      \mbox{constant pressure (isobaric):} &
      \displaystyle\vphantom{\frac{V^2}{V_1}} p(V_2-V_1) \\
      \mbox{pressure linear in V:} &
      \displaystyle\vphantom{\frac{V^2}{V_1}} \frac{p_1+p_2}{2}(V_2-V_1) \\
      \mbox{polytropic for $n\ne 1$ ($pV^n =$ constant):} &
      \displaystyle\vphantom{\frac{V^2}{V_1}} \frac{p_2 V_2 - p_1 V_1}{1-n}\\
      \mbox{polytropic for $n = 1$ ($pV =$ constant):} &
      \displaystyle\vphantom{\frac{V^2}{V_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 $mRT$.

No heat is added or removed if the process is adiabatic.\\
\eo\\
More generally:
\begin{displaymath}
  Q_{12} =
  \left\{
    \begin{array}{rl}
      \mbox{Adiabatic:} & \displaystyle\vphantom{\int_1^2} 0 \\
      \mbox{reversible:} & \displaystyle\vphantom{\int_1^2}
      \int_1^2 T\;{\rm d}S \\
      \mbox{reversible isothermal:} & \displaystyle\vphantom{\int_1^2}
      T (S_2-S_1)
    \end{array}
  \right.
\end{displaymath}

The second law for reversible adiabatic (isentropic) processes:
\begin{displaymath}
  S_2 = S_1
\end{displaymath}

The second law of thermo for general processes:
\begin{displaymath}
  S_{12,\mbox{\scriptsize gen}} + \sum \frac{Q_{k}}{T_{k}}
  = S_2 - S_1 \qquad S_{12,\mbox{\scriptsize gen}} \ge 0
\end{displaymath}
\eeo


\subsection{Rate Equations}

The first law as a rate equation:
\begin{displaymath}
  \frac{{\rm d}E}{{\rm d}t} = \dot Q - \dot W \qquad
  \left(
    \frac{{\rm d}E}{{\rm d}t}
    = m \frac{{\rm d}u}{{\rm d}t} +
    m \frac{{\rm d}{\textstyle \frac12}{\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}


\subsection{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}
(where ${\bf V}$ is velocity.)

Continuity (mass conservation):
\begin{displaymath}
  \sum \dot m_i = \sum \dot m_e
\end{displaymath}
where $\sum$ means sum over all inflow, respectively outflow, points,
if there is more than one.

The first law of thermo (energy conservation):
\begin{displaymath}
  \sum \dot m_e \left(h_e+ \frac12 {\bf V}_e^2 + g z_e\right) -
  \sum \dot m_i \left(h_i+ \frac12 {\bf V}_i^2 + g z_i\right) =
  \dot Q - \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, $\dot Q = 0$.

\eo

The second law of thermo for a reversible adiabatic (isentropic)
process inside a single entrance and exit CV:
\begin{displaymath}
  s_e = s_i
\end{displaymath}

The second law of thermo for a reversible isothermal process inside a
single entrance and exit CV:
\begin{displaymath}
  q = T \left(s_e - s_i\right)
\end{displaymath}

The second law for a general control volume:
\begin{displaymath}
  \sum \dot m_e s_e - \sum \dot m_i s_i
  = \sum \frac{\dot Q_k}{T_k} + \dot S_{\mbox{\scriptsize gen}}
  \qquad \dot S_{\mbox{\scriptsize gen}} \ge 0
\end{displaymath}

Specific work done during a reversible process inside a single
entrance and exit CV:
\begin{displaymath}
  w + {\textstyle \frac12} {\bf V}_e^2? - {\textstyle \frac12} {\bf V}_i^2?
  + g Z_e? - g Z_i? = - \int_i^e v \; {\rm d} p
\end{displaymath}
where
\begin{displaymath}
  - \int_i^e v \; {\rm d} p =
  \left\{
    \begin{array}{rl}
      \mbox{constant pressure (isobaric):} &
      \displaystyle\vphantom{\frac{V^e}{V_i}} 0 \\
      \mbox{incompressible (constant volume/isochoric):} &
      \displaystyle\vphantom{\frac{V^e}{V_i}} -v \left(p_e-p_i\right) \\
      \mbox{polytropic for $n\ne 1$ ($pv^n =$ constant):} &
      \displaystyle\vphantom{\frac{V^e}{V_i}}
      \frac{n\left(p_e v_e - p_i v_i\right)}{1-n}\\
      \mbox{polytropic for $n = 1$ ($pv =$ constant):} &
      \displaystyle\vphantom{\frac{V^e}{V_i}}
      - p v \ln\left(\frac{p_e}{p_i}\right)
    \end{array}
  \right.
\end{displaymath}
In case of an ideal gas, note that $pv$ can be replaced by $RT$.

Special case of (a) ideal gas, (2) polytropic, $n\ne 1$, (including
isentropic constant specific heats, then $n=k$,) and (3) reversible:
\begin{displaymath}
  - \int_i^e v \; {\rm d} p =
  \frac{nRT_1}{1-n}\left[\left(\frac{P_2}{P_1}\right)^{(n-1)/n}-1\right]
\end{displaymath}

Special case of (a) ideal gas, (2) constant specific heats, (3) polytropic,
$n=1$ (isothermal). and (4) reversible:
\begin{displaymath}
  - \int_i^e v \; {\rm d} p =
  -RT\ln\frac{P_2}{P_1}
\end{displaymath}

Adiabatic turbine and compressor/pump efficiencies:
\begin{displaymath}
  \eta_{\mbox{\scriptsize T}} = \frac{w_a}{w_s}
  \qquad
  \eta_{\mbox{\scriptsize C}} = \frac{w_s}{w_a}
\end{displaymath}
where subscript $s$ indicates an ideal isentropic device with the same
entrance conditions and exit pressure and $a$ the actual device.

\eeo

\section{Cycle Devices}

\eo

For cycle devices, the normal sign conventions (heat in and work out
are positive) are ignored.  Instead, the signs of heat and work are
defined such that $\dot Q_H$, $\dot Q_L$ and net work $\dot W$ are
normally positive.  The pictures below show the directions of positive
heat fluxes and work rates for a heat engine as compared to a heat
pump (or refrigeration cycle).

\setlength{\unitlength}{.00467truein}
\begin{picture}(1392,460)(0,0)

\put(196,0){
\begin{picture}(400,460)(0,0)
  \put(200,440){\makebox(0,0){{\it Heat Engine}}}
  \put(200,400){\makebox(0,0){High temperature $T_H$}}
  \put(0,400){\line(1,-1){30}}
  \put(400,400){\line(-1,-1){30}}
  \put(30,370){\line(1,0){340}}

  \put(200,370){\line(0,-1){100}}
  \put(201,323){\makebox(0,0){$\vee$}}
  \put(220,320){$\dot Q_H$}

  \put(270,270){\line(0,-1){140}}
  \put(270,270){\line(-1,0){140}}
  \put(130,130){\line(0,1){140}}
  \put(130,130){\line(1,0){140}}
  \put(200,200){\circle{60}}
  \put(270,200){$\Longrightarrow \dot W_{\mbox{\scriptsize (net)}}$}

  \put(200,130){\line(0,-1){100}}
  \put(201,83){\makebox(0,0){$\vee$}}
  \put(220,80){$\dot Q_L$}

  \put(30,30){\line(1,0){340}}
  \put(0,0){\line(1,1){30}}
  \put(400,0){\line(-1,1){30}}
  \put(200,0){\makebox(0,0){Low temperature $T_L$}}

\end{picture}
}

\put(796,0){
\begin{picture}(400,460)(0,0)
  \put(200,440){\makebox(0,0){{\it Heat Pump}}}
  \put(200,400){\makebox(0,0){High temperature $T_H$}}
  \put(0,400){\line(1,-1){30}}
  \put(400,400){\line(-1,-1){30}}
  \put(30,370){\line(1,0){340}}

  \put(200,370){\line(0,-1){100}}
  \put(201,323){\makebox(0,0){$\wedge$}}
  \put(220,320){$\dot Q_H$}

  \put(270,270){\line(0,-1){140}}
  \put(270,270){\line(-1,0){140}}
  \put(130,130){\line(0,1){140}}
  \put(130,130){\line(1,0){140}}
  \put(200,200){\circle{60}}
  \put(270,200){$\Longleftarrow \dot W_{\mbox{\scriptsize (net)}}$}

  \put(200,130){\line(0,-1){100}}
  \put(201,83){\makebox(0,0){$\wedge$}}
  \put(220,80){$\dot Q_L$}

  \put(30,30){\line(1,0){340}}
  \put(0,0){\line(1,1){30}}
  \put(400,0){\line(-1,1){30}}
  \put(200,0){\makebox(0,0){Low temperature $T_L$}}

\end{picture}
}
\end{picture}

The first law for a cycle:
\begin{displaymath}
  \mbox{Net work } {\mbox{out} \atop \mbox{in}} =
  \mbox{Net heat } {\mbox{in} \atop \mbox{out}}\mbox{:}
  \qquad W_{\mbox{\scriptsize (net)}} = Q_H - Q_L
\end{displaymath}

{\em The Kelvin-Planck Statement:} Any working heat engine must dump
some waste heat to a lower temperature than the input heat is provided
at.

{\em The Clausius Statement:} Any working heat pump must have positive
work going in.

Thermal efficiency:
\begin{displaymath}
  \eta_{\mbox{\scriptsize th}} \equiv
  \frac{\dot W_{\mbox{\scriptsize net}}}{\dot Q_H} =
  \frac{\dot Q_H- \dot Q_L}{\dot Q_H} \quad\le\quad
  \eta_{\mbox{\scriptsize th, rev}} =
  \frac{T_H-T_L}{T_H}
\end{displaymath}

Refrigeration coefficient of performance:
\begin{displaymath}
  {\rm COP}_{\rm R} \equiv
  \frac{\dot Q_L}{\dot W_{\mbox{\scriptsize net}}} =
  \frac{\dot Q_L}{\dot Q_H-\dot Q_L} \quad\le\quad
  {\rm COP}_{\rm R,rev} =
  \frac{T_L}{T_H-T_L}
\end{displaymath}

Heating heat pump coefficient of performance $\beta' = 1 + \beta$:
\begin{displaymath}
  {\rm COP}_{\rm HP} \equiv
  \frac{\dot Q_H}{\dot W_{\mbox{\scriptsize net}}} =
  \frac{\dot Q_H}{\dot Q_H-\dot Q_L} \quad\le\quad
  {\rm COP}_{\rm HP,rev} =
  \frac{T_H}{T_H-T_L}
\end{displaymath}

Ideal gas Carnot cycle:
\begin{displaymath}
  q_H = q_{12} = R T_H \ln\left(\frac{v_2}{v_1}\right)
  \qquad
  q_L = -q_{34} = R T_L \ln\left(\frac{v_3}{v_4}\right)
  \qquad
  \frac{v_2}{v_1} = \frac{v_3}{v_4}
\end{displaymath}

\eeo

\end{document}
