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Commit ddf2bd69 by 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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