Isentropic Flow

**Figure 4.1:** Flow thorough a converging diverging nozzle
$\begin{figure}\centerline{\includegraphics {cont/variableArea/nozzle}} \end{figure}$

In this chapter a discussion on a steady state flow through a smooth and continuous area flow rate is presented. A discussion about the flow through a converging-diverging nozzle is also part of this chapter. The isentropic flow models are important because of two main reasons: One, it provides the information about the trends and important parameters. Two, the correction factors can be introduced later to account for deviations from the ideal state.

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Speed of Sound in Solids

The situation with solids is considerably more complicated, with different speeds in different directions, in different kinds of geometries, and differences between transverse and longitudinal waves. Nevertheless, the speed of sound in solids is larger than in liquids and definitely larger than in gases.

Young's Modulus for a representative value for the bulk modulus for steel is 160

N /

Table 3.3: Solids speed of sound, after Aldred, John, Manual of Sound Recording, London:Fountain Press, 1972

material	reference	Value [m/sec]
Diamond		12000
Pyrex glass		5640
Steel	longitudinal wave	5790
Steel	transverse shear	3100
Steel	longitudinal wave (extensional wave)	5000
Iron		5130
Aluminum		5100
Brass		4700
Copper		3560
Gold		3240
Lucite		2680
Lead		1322
Rubber		1600

Speed of sound in solid of steel, using a general tabulated value for the bulk modulus, gives a sound speed for structural steel of

$\displaystyle c = \sqrt{ E \over \rho} = \sqrt{160 \times 10^{9} N/m^{2} \over 7860 Kg /m^3} = 4512 m/s$

(3.38)

Compared to one tabulated value the example values for stainless steel lays between the speed for longitudinal and transverse waves.

Speed of Sound in Almost Incompressible Liquid

Even liquid normally is assumed to be incompressible in reality has a small and important compressible aspect. The ratio of the change in the fractional volume to pressure or compression is referred to as the bulk modulus of the material. For example, the average bulk modulus for water is $2.2 \times 10^9$

. At a depth of about 4,000 meters, the pressure is about $4 \times 10^7$

. The fractional volume change is only about 1.8% even under this pressure nevertheless it is a change.

The compressibility of the substance is the reciprocal of the bulk modulus. The amount of compression of almost all liquids is seen to be very small as given in Table (3.5). The mathematical definition of bulk modulus as following

$\displaystyle B = \rho {\Dxy{P}{\rho}}$

(3.35)

In physical terms can be written as

$\displaystyle c = \sqrt{elastic\; property \over inertial\; property} = \sqrt{B \over \rho}$

(3.36)

For example for water

$\displaystyle c = \sqrt{2.2 \times 10^9 N /m^2 \over 1000 kg /m^3} = 1493 m/s %\label{sound:eq:waterSpeed}$

This agrees well with the measured speed of sound in water, 1482 m/s at $20^{\circ}C$ . Many researchers have looked at this velocity, and for purposes of comparison it is given in Table (3.5)

Table 3.1: Water speed of sound from different sources

Remark	reference	Value [m/sec]
Fresh Water ( $20^{\circ}C$ )	Cutnell, John D. & Kenneth W. Johnson. Physics. New York: Wiley, 1997: 468.	1492
Distilled Water at ( $25^{\circ}C$ )	The World Book Encyclopedia. Chicago: World Book, 1999. 601	1496
Water distilled	Handbook of Chemistry and Physics. Ohio: Chemical Rubber Co., 1967-1968:E37	1494

The effect of impurity and temperature is relatively large, as can be observed from the equation (3.37). For example, with an increase of 34 degrees from $0 ^{\circ}C$ there is an increase in the velocity from about 1430 m/sec to about 1546 [m/sec]. According to Wilson^3.5, the speed of sound in sea water depends on temperature, salinity, and hydrostatic pressure.

Wilson's empirical formula appears as follows:

$\displaystyle c(S,T,P) = c_0 + c_T + c_S + c_P + c_{STP},$

(3.37)

where

is about clean/pure water, $c_{T}$ is a function temperature, and

is a function salinity,

is a function pressure, and $c_{STP}$ is a correction factor between coupling of the different parameters.

Table 3.2: Liquids speed of sound, after Aldred, John, Manual of Sound Recording, London: Fountain Press, 1972

material	reference	Value [m/sec]
Glycerol		1904
Sea water	$25^{\circ}C$	1533
Mercury		1450
Kerosene		1324
Methyl alcohol		1143
Carbon tetrachloride		926

In summary, the speed of sound in liquids is about 3 to 5 relative to the speed of sound in gases

Speed of Sound in Real Gas

The ideal gas model can be improved by introducing the compressibility factor. The compressibility factor represents the deviation from the ideal gas.

$\begin{figure}\centerline{\includegraphics {cont/sound/CompressiblityChart}} \end{figure}$

**Figure:** `The Compressibility Chart`
$\begin{figure}\centerline{\includegraphics {cont/sound/CompressiblityChart}} \end{figure}$

Thus, a real gas equation can be expressed in many cases as

$\displaystyle P = z\rho R T$

(3.19)

The speed of sound of any gas is provided by equation (3.7). To obtain the expression for a gas that obeys the law expressed by (3.19) some mathematical expressions are needed. Recalling from thermodynamics, the Gibbs function (3.20) is used to obtain

$\displaystyle Tds = dh - {dP \over \rho}$

(3.20)

The definition of pressure specific heat for a pure substance is

$\displaystyle C_p = \left( \partial h \over \partial T \right)_P = T \left( \partial s \over \partial T \right)_P$

(3.21)

The definition of volumetric specific heat for a pure substance is

$\displaystyle C_v = \left( \partial u \over \partial T \right)_{\rho} = T \left( \partial s \over \partial T \right)_{\rho}$

(3.22)

From thermodynamics, it can be shown ^3.4

$\displaystyle dh = C_p dT + \left[ v -T \left( \partial v \over \partial T \right)_P \right]$

(3.23)

The specific volumetric is the inverse of the density as

and thus

$\displaystyle \left( \partial v \over \partial T \right)_P = \left( \partial \l... ..._P + {zR \over P} \cancelto {1} { \left( \partial T \over \partial T \right)_P}$

(3.24)

Substituting the equation (3.24) into equation (3.23) results

$\displaystyle dh = C_p dT + \left[ v - T \left( \overbrace{RT \over P}^{v \over... ...er \partial T \right)_P + \overbrace{zR \over P}^{v \over T} \right) \right] dP$

(3.25)

Simplifying equation (3.25) to became

$\displaystyle dh = C_p dT - \left[ {T v \over z} \left( \partial z \over \parti... ...ver z}\left(\partial z \over \partial T \right)_P {dP \over \rho} %\label{eq:}$

(3.26)

Utilizing Gibbs equation (3.20)

$\displaystyle Tds = \overbrace{C_p dT - {T \over z} \left(\partial z \over \par... ...{zRT} \left[ {T \over z}\left(\partial z \over \partial T \right)_P + 1 \right]$

(3.27)

Letting

for isentropic process results in

$\displaystyle {dT \over T} = {dP \over P } {R \over C_p} \left[ z + {T }\left(\partial z \over \partial T \right)_P \right]$

(3.28)

Equation (3.28) can be integrated by parts. However, it is more convenient to express

in terms of

and $d\rho / \rho$ as follows

$\displaystyle {dT \over T} = {d\rho \over \rho} {R \over C_v} \left[ z + { T }\left( \partial z \over \partial T \right)_{\rho} \right]$

(3.29)

Equating the right hand side of equations (3.28) and (3.29) results in

$\displaystyle {d\rho \over \rho} {R \over C_v} \left[ z + { T }\left( \partial ... ...R \over C_p} \left[ z + {T }\left(\partial z \over \partial T \right)_P \right]$

(3.30)

Rearranging equation (3.30) yields

$\displaystyle {d\rho \over \rho} = {dP \over P } {C_v \over C_p } \left[ z + {T... ...ght)_P \over z + { T }\left( \partial z \over \partial T \right)_{\rho} \right]$

(3.31)

If the terms in the braces are constant in the range under interest in this study, equation (3.31) can be integrated. For short hand writing convenience,

is defined as

$\displaystyle n = \overbrace{C_p \over C_v}^{k} \left( z + T \left( \partial z ... ...ight)_{\rho} \over z + T \left( \partial z \over \partial T \right)_{P} \right)$

(3.32)

Note that

approaches

when $z\rightarrow1$ and when

is constant. The integration of equation (3.31) yields

$\displaystyle \left(\rho_1 \over \rho_2 \right)^n = {P_1 \over P_2}$

(3.33)

Equation (3.33) is similar to equation (3.11). What is different in these derivations is that a relationship between coefficient n and

was established. This relationship (3.33) isn't new, and in-fact any thermodynamics book shows this relationship. But the definition of n in equation (3.32) provides a tool to estimate n . Now, the speed of sound for a real gas can be obtained in the same manner as for an ideal gas.

$\displaystyle {dP \over d\rho} = nz R T$

(3.34)

$\begin{examl} Calculate the speed of sound of air at $30\celsius$ and atmospher... ...as model (compressibility factor). Assume that $R = 287 [j /kg /K]$. \end{examl}$

SOLUTION $\;$
According to the ideal gas model the speed of sound should be

$\displaystyle c = \sqrt{kRT} = \sqrt{1.407 \times 287 \times 300} \sim 348.1[m/sec]$

For the real gas first coefficient

has

$\displaystyle c = \sqrt{znRT} = \sqrt{1.403 \times 0.995 times 287 \times 300} = 346.7 [m/sec]$

Solution

According to the ideal gas model the speed of sound should be

$\displaystyle c = \sqrt{kRT} = \sqrt{1.407 \times 287 \times 300} \sim 348.1[m/sec]$

For the real gas first coefficient

has

$\displaystyle c = \sqrt{znRT} = \sqrt{1.403 \times 0.995 times 287 \times 300} = 346.7 [m/sec]$

The correction factor for air under normal conditions (atmospheric conditions or even increased pressure) is minimal on the speed of sound. However, a change in temperature can have a dramatical change in the speed of sound. For example, at relative moderate pressure but low temperature common in atmosphere, the compressibility factor,

and $n\sim 1$ which means that speed of sound is only $\sqrt{0.3 \over 1.4}$ about factor of (0.5) to calculated by ideal gas model

Compressible Flow and Gas Dynamic

الجمعة، 10 أغسطس 2012

Isentropic Flow

Isentropic Flow

Speed of Sound in Solids

Speed of Sound in Almost Incompressible Liquid

Speed of Sound in Real Gas