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Next: Speed of Sound in Up: Speed of Sound Previous: Introduction   Index

Speed of sound in ideal and perfect gases

The speed of sound can be obtained easily for the equation of state for an ideal gas (also perfect gas as a sub set) because of a simple mathematical expression. The pressure for an ideal gas can be expressed as a simple function of density, $ \rho $ , and a function ``molecular structure'' or ratio of specific heats, $ k$ namely
and hence
$\displaystyle c = \sqrt{\Dxy{P}{\rho}} = k \times constant \times \rho^{k-1}$ $\displaystyle =k \times {\overbrace{ constant \times \rho^k}^{P} \over \rho} \hfill$    
  $\displaystyle = k \times {P \over \rho }$ (3.12)

Remember that $ P / \rho $ is defined for an ideal gas as $ RT$ , and equation (3.12) can be written as

Calculate the speed of sound in water vapor at $20 [bar]$ and $350
\celsius$, (a) utilizes the steam table
(b) assuming ideal gas.

The solution can be estimated by using the data from steam table3.3
$\displaystyle c = \sqrt{\Delta P \over \Delta \rho}_{s=constant}$ (3.14)

At $ 20[bar]$ and $ 350 \celsius$ : s = 6.9563 $ \left[ kJ \over K\;
kg\right]$ $ \rho $ = 6.61376 $ \left[ kg \over m^3 \right]$
At $ 18[bar]$ and $ 350 \celsius$ : s = 7.0100 $ \left[ kJ \over K\;
kg\right]$ $ \rho $ = 6.46956 $ \left[ kg \over m^3 \right]$
At $ 18[bar]$ and $ 300 \celsius$ : s = 6.8226 $ \left[ kJ \over K\;
kg\right]$ $ \rho $ = 7.13216 $ \left[ kg \over m^3 \right]$

After interpretation of the temperature:
At $ 18[bar]$ and $ 335.7 \celsius$ : s $ \sim$ 6.9563 $ \left[ kJ \over K\;
kg\right]$ $ \rho \sim$ 6.94199 $ \left[ kg \over m^3 \right]$
and substituting into the equation yields

$\displaystyle c = \sqrt{ 200000 \over 0.32823} = 780.5 \left[ m \over sec \right]$ (3.15)

for ideal gas assumption (data taken from Van Wylen and Sontag, Classical Thermodynamics, table A 8.)

$\displaystyle c = \sqrt{kRT} \sim \sqrt{ 1.327 \times 461 \times
(350 + 273)} \sim
771.5 \left[ m \over sec \right]

Note that a better approximation can be done with a steam table, and it will be part of the future program (Potto-GDC).

\index{speed of sound!linear temperature}
The temperature in the a...
...d to travel from point \lq\lq A'' to
point \lq\lq B'' under this assumption.?

The temperature is denoted at ``A'' as $ T_A$ and temperature in ``B'' is $ T_B$ . The distance between ``A'' and ``B'' is denoted as $ h$ .
$\displaystyle T = (T_B - T_A) {x \over h} + T_A$    

Where the distance $ x$ is the variable distance. It should be noted that velocity is provided as a function of the distance and not the time (another reverse problem). For an infinitesimal time $ dt$ is equal to
$\displaystyle d t = {d x \over \sqrt{kRT(x)} } = {dx \over \sqrt{ kRT_A \left( {(T_B - T_A) x \over T_A h} +1 \right) } }$    

integration of the above equation yields
For assumption of constant temperature the time is
Hence the correction factor
$\displaystyle {t_{corrected} \over t } = \sqrt{T_A \over \bar{T}} {2 \over 3} { T_A \over ( T_B - T_A )} \left( \left(T_B \over T_A\right)^{3\over 2}- 1 \right)$ (3.18)

This correction factor approaches one when $ T_B \longrightarrow T_A$ .

next up previous index
Next: Speed of Sound in Up: Speed of Sound Previous: Introduction   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21