Question

Steam flows through a device with a pressure of 120 psia, a temperature of \(700^{\circ} \mathrm{F}\), and a velocity of \(900 \mathrm{ft} / \mathrm{s}\). Determine the Mach number of the steam at this state by assuming ideal-gas behavior with \(k=1.3\).

Short answer

Answer: The Mach number of the steam flowing through the device is approximately 0.574.

Step-by-step solution

Determine the speed of sound

The formula for the speed of sound, \(a\), in an ideal gas is given by: $$a=\sqrt{k \cdot R \cdot T}$$ where \(k\) is the specific heat ratio, \(R\) is the specific gas constant, and \(T\) is the absolute temperature. For steam, we can use the specific gas constant, \(R = 85.34 \, ft \cdot lb/(lb_m\cdot R)\). However, our given temperature is in degrees Fahrenheit, and we need it in absolute temperature (Rankine). The conversion is: $$T_{Rankine} = T_{Fahrenheit} + 459.67 $$ Hence, $$T_{Rankine} = 700^\circ F + 459.67 = 1159.67\, R $$ Now, we can calculate the speed of sound in the steam: $$a = \sqrt{k \cdot R \cdot T} = \sqrt{(1.3)(85.34)(1159.67)}\, ft/s \approx 1567.08\, ft/s$$

Calculate the Mach number

The Mach number, \(M\), is the ratio of the fluid speed, \(V\), to the speed of sound, \(a\). It is given by: $$M = \frac{V}{a}$$ The problem statement provides the steam's velocity as \(V = 900 \, ft/s\). Now we can calculate the Mach number: $$M = \frac{900 \, ft/s}{1567.08 \, ft/s} \approx 0.574$$ Thus, the Mach number of the steam at this state is approximately 0.574.