Commit ddf2bd69 authored by Olivier's avatar Olivier

Chapter 9: fix signs in derivation of speed of sound equation

In chapter 9, the derivation of equation 9/5 (starting from 9/2) was
incoherent with the sign convention used in figure 9.1. Thanks to
Afrinbanu Mehboob Merchant for finding and reporting the error.
parent 89e08df8
\renewcommand{\lastedityear}{2018}
\renewcommand{\lasteditmonth}{04}
\renewcommand{\lasteditday}{02}
\renewcommand{\lasteditmonth}{07}
\renewcommand{\lasteditday}{05}
\renewcommand{\numberofthischapter}{9}
\renewcommand{\titleofthischapter}{Introduction to compressible flow}
......@@ -53,21 +53,21 @@
Control volume fluid flow analysis does not frighten us anymore. From mass conservation (eq.~\ref{eq_rtt_mass} p.\pageref{eq_rtt_mass}) we write:
\begin{IEEEeqnarray}{rCl}
\rho c A &=& (\rho + \diff \rho) (c - \diff V) A\label{eq_continuity_pressurewave}
\rho c A &=& (\rho + \diff \rho) (c + \diff V) A\label{eq_continuity_pressurewave}
\end{IEEEeqnarray}
This equation~\ref{eq_continuity_pressurewave} relates the density change across the wave to the velocity change between inlet and outlet. Re-arranging, and focusing only on the case where the pressure wave is extremely weak (as are sound waves in practice), we see the product $\diff \rho \diff V$ vanish and obtain:
\begin{IEEEeqnarray}{rCl}
\rho c &=& \rho c - \rho \diff V + c \diff \rho - \diff \rho \diff V \nonumber\\
\rho \diff V &=& c \diff \rho \label{eq_tmpsoundone}
\rho c &=& \rho c + \rho \diff V + c \diff \rho + \diff \rho \diff V \nonumber\\
\rho \diff V &=& - c \diff \rho \label{eq_tmpsoundone}
\end{IEEEeqnarray}
In equation~\ref{eq_tmpsoundone}, which relates the speed of sound~$c$ to the density~$\rho$, we would now like to eliminate the~$\diff V$ term. For this, we turn to the control volume momentum equation (eq:~\ref{eq_rtt_linearmom} p.\pageref{eq_rtt_linearmom}):
\begin{IEEEeqnarray}{rCl}
F_\net &=& -\dot m_\inn V_\inn + \dot m_\out V_\out \nonumber\\
p A - (p + \diff p) A &=& - \rho c A c + (\rho - \diff \rho) (c - \diff V) A (c - \diff V) \nonumber\\
-\diff p A &=& \rho c A \left[-c + (c - \diff V)\right] \nonumber\\
\frac{\diff p}{c} &=& \rho \diff V \label{eq_tmpsoundtwo}
F_\net &=& -\dot m_\inn V_\inn + \dot m_\out V_\out = \dot m (V_\out - V_\inn) \nonumber\\
p A - (p + \diff p) A &=& \rho c A \left[(c + \diff V) - c\right] \nonumber\\
-\diff p &=& \rho c \diff V \nonumber\\
\frac{\diff p}{c} &=& -\rho \diff V \label{eq_tmpsoundtwo}
\end{IEEEeqnarray}
We can now combine eqs. (\ref{eq_tmpsoundone}) and (\ref{eq_tmpsoundtwo}) to obtain:
......
......@@ -132,7 +132,7 @@
In control volume, mass conservation means: \pause
\begin{IEEEeqnarray}{rCl}
\rho A c &=& (\rho + \diff \rho) A (c - \diff V)
\rho A c &=& (\rho + \diff \rho) A (c + \diff V)
\end{IEEEeqnarray} \pause
\footnotesize $\to$ the density change is caused by the velocity change from one side of the wave to the other.
......@@ -142,12 +142,12 @@
re-arrange:
\begin{IEEEeqnarray*}{rCl}
\rho c &=& \rho c - \rho \diff V + c \diff \rho - \diff \rho \diff V \pause
\rho c &=& \rho c + \rho \diff V + c \diff \rho + \diff \rho \diff V \pause
\end{IEEEeqnarray*}
from which we drop $\diff \rho \diff V$: \pause
\begin{IEEEeqnarray}{rCl}
\rho \diff V &=& c \diff \rho \label{eq_tmpsoundone} \pause
\rho \diff V &=& - c \diff \rho \label{eq_tmpsoundone} \pause
\end{IEEEeqnarray}
\footnotesize let’s eliminate this $\diff V$!
......@@ -158,10 +158,10 @@
Now turn to Newton: \pause
\small
\begin{IEEEeqnarray}{rCl}
F_\net &=& -\dot m_\inn V_\inn + \dot m_\out V_\out \nonumber\\ \pause
p A - (p + \diff p) A &=& - \rho c A c + (\rho - \diff \rho) (c - \diff V) A (c - \diff V) \nonumber\\\pause
-\diff p A &=& \rho c A \left[-c + (c - \diff V)\right] \nonumber\\\pause
\frac{\diff p}{c} &=& \rho \diff V \label{eq_tmpsoundtwo}
F_\net &=& \dot (V_\out - V_\inn) \nonumber\\ \pause
p A - (p + \diff p) A &=& \rho c A \left[(c + \diff V) - c\right] \nonumber\\\pause
-\diff p &=& \rho c \diff V \nonumber\\\\\pause
\frac{\diff p}{c} &=& -\rho \diff V \label{eq_tmpsoundtwo}
\end{IEEEeqnarray}
\end{frame}
......@@ -170,11 +170,11 @@ p A - (p + \diff p) A &=& - \rho c A c + (\rho - \diff \rho) (c - \diff V) A (c
\small
\begin{IEEEeqnarray*}{rCl}
\rho \diff V = c \diff \rho
\rho \diff V = -c \diff \rho
\end{IEEEeqnarray*}
and
\begin{IEEEeqnarray*}{rCl}
\frac{\diff p}{c} = \rho \diff V
\frac{\diff p}{c} = -\rho \diff V
\end{IEEEeqnarray*}
\normalsize
......
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