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Sound Speed in Two Phase Medium

The gas flow in many industrial situations contains other particles. In actuality, there could be more than one speed of sound for two phase flow. Indeed there is double chocking phenomenon in two phase flow. However, for homogeneous and under certain condition a single velocity can be considered. There can be several models that approached this problem. For simplicity, it assumed that two materials are homogeneously mixed. Topic for none homogeneous mixing are beyond the scope of this book. It further assumed that no heat and mass transfer occurs between the particles. In that case, three extreme cases suggest themselves: the flow is mostly gas with drops of the other phase (liquid or solid), about equal parts of gas and the liquid phase, and liquid with some bubbles. The first case is analyzed.

The equation of state for the gas can be written as

$\displaystyle P_a = \rho_a R T_a$

(3.39)

The average density can be expressed as

$\displaystyle {1 \over \rho_m} = { \xi \over \rho_a} + { 1 - \xi \over \rho_b}$

(3.40)

where $\xi = {\dot{m}_b \over \dot{m}}$ is the mass ratio of the materials.

For small value of $\xi$ equation (3.40) can be approximated as

$\displaystyle {\rho \over \rho_a} = 1 + m$

(3.41)

where $m = {\dot{m}_b \over \dot{m}_a}$ is mass flow rate per gas flow rate.

The gas density can be replaced by equation (3.39) and substituted into equation (3.41)

$\displaystyle {P \over \rho} = { R \over 1 + m} T$

(3.42)

A approximation of addition droplets of liquid or dust (solid) results in reduction of

and yet approximate equation similar to ideal gas was obtained. It must noticed that

. If the droplets (or the solid particles) can be assumed to have the same velocity as the gas with no heat transfer or fiction between the particles isentropic relation can be assumed as

$\displaystyle {P \over {\rho_a}^{k} } = constant$

(3.43)

Assuming that partial pressure of the particles is constant and applying the second law for the mixture yields

$\displaystyle 0 = \overbrace{mC{dT \over T}}^{droplets} + \overbrace{C_p {dT \over T} -R {dP \over P}}^{gas} = {(C_p + mC) dT \over T} -R {dP \over P}$

(3.44)

Therefore, the mixture isentropic relationship can be expressed as

$\displaystyle {P^{\gamma - 1 \over \gamma} \over T} = constant$

(3.45)

where

$\displaystyle {\gamma -1 \over \gamma} = {R \over C_p + mC}$

(3.46)

Recalling that

reduces equation (3.46) into

$\displaystyle \gamma = {C_p + mC \over C_v + mC}$

(3.47)

In a way the definition of $\gamma$ was so chosen that effective specific pressure heat and effective specific volumetric heat are $C_p + mC \over 1 + m$ and $C_v + mC \over 1 + m$ respectively. The correction factors for the specific heat is not linear.

Since the equations are the same as before hence the familiar equation for speed of sound can be applied as

$\displaystyle c = \sqrt{\gamma R_{mix} T}$

(3.48)

It can be noticed that $R_{mix}$ and $\gamma$ are smaller than similar variables in a pure gas. Hence, this analysis results in lower speed of sound compared to pure gas. Generally, the velocity of mixtures with large gas component is smaller of the pure gas. For example, the velocity of sound in slightly wed steam can be about one third of the pure steam speed of sound.