Question 15.3: Velocity of Sound Steam flows through a duct with a velocity...
Velocity of Sound
Steam flows through a duct with a velocity of 300 m/s at 100 kPa and 200°C. Determine the velocity of sound using (a) the PG model and (b) PC model for steam.
Use Eq. (15.19) for the PG model and Eq. (15.18) for the PC model to evaluate the flow Mach number.
c=\sqrt{(1000 N / kN ) k R T} ; \quad\left[\frac{ m }{ s }\right] (15.19)
Assumptions
One-dimensional flow.
Analysis
From Table C-1, we obtain the material properties of H _{2} O \text { as } k=1.327 \text { and } R=0.462 kJ / kg \cdot K .
Equation (15.19) produces:
c=\sqrt{(1000 N / kN )(1.327)(0.462)(473)}=538.5 m / s
For the PC model, the partial derivative of Eq. (15.18) must be evaluated first. Representing the given conditions by State-1 and its differential neighbor—an isentropic state with, say, a one percent variation in pressure—by State-2 (Fig. 15.11), use the PC flow state TESTcalc, or the manual procedure described in Chapter 3, to obtain:
c=\sqrt{(1000 N / kN )\left(\frac{\partial p}{\partial \rho}\right)_{s}} ;\left[\frac{ m }{ s }\right] (15.18)
\text { State-1 }\left(\text { given } p_{1}, T_{1}\right): \quad \rho_{1}=0.46036 \frac{ kg }{ m ^{3}} ; \quad s_{1}=7.8342 \frac{ kJ }{ kg \cdot K }
\text { State-2 }\left(\text { given } s_{2}=s_{1}, p_{2}=1.01 p_{1}\right): \quad \rho_{2}=0.46381 \frac{ kg }{ m ^{3}}
Therefore,
\left(\frac{\partial p}{\partial \rho}\right)_{s} \cong\left(\frac{\Delta p}{\Delta \rho}\right)_{s}=\frac{101-100}{0.46381-0.46036}=290 \frac{ kPa \cdot m ^{3}}{ kg }
and
c=\sqrt{\left(1000 \frac{ N }{ kN }\right)\left(\frac{\partial p}{\partial \rho}\right)_{s}}=\sqrt{(1000)(290) \frac{ Pa }{ kPa } \frac{ kPa \cdot m ^{3}}{ kg }}=538.5 m / s
TEST Analysis
Launch the gas dynamics TESTcalc. Select H2O as the working fluid and evaluate State-1 with T1 = 200 deg-C. The velocity of sound c1 is calculated as 538.6 m/s. To use the PC model, launch the PC flow state TESTcalc. After evaluating State-1 with the given pressure and temperature, evaluate State-2 with p2 = 1.01*p1, and s2 = s1. In the I/O panel, evaluate the expression ‘=sqrt(1000*(p2 – p1)/(rho2 – rho1))’ as 538.4 m/s.
Discussion
Instead of varying pressure by 1%, any other property could be used to select a neighboring isentropic state. Under the given conditions, superheated steam behaves like a perfect gas, thereby producing almost identical results for the PG and PC models. As a state gets closer to saturation, the PG model results will not be as accurate. This can be verified by calculating the speed of sound at State-3 at 500 kPa (Fig. 15.11). Eq. (15.19) should be used only when the PG model assumptions (Sec. 3.5.2) are appropriate.
