Helium at 68^∘ F and 14.7 psia in a large tank flows steadily and isentropically through a conve

Helium at 68^∘ F and 14.7 psia in a large tank flows...

Key Concepts

Receiver Pressure Effect

The receiver pressure, or the back pressure, plays a significant role in determining whether the flow through the nozzle becomes choked. A higher receiver pressure can maintain subsonic conditions, while a lower receiver pressure may cause the flow to choke at the throat, limiting the mass flow rate. Understanding this effect is essential for predicting the operational characteristics of the nozzle and ensuring that the system performs as expected in different pressure scenarios.

Mass Flow Rate Calculation in Compressible Flow

The mass flow rate in compressible nozzle flows is calculated using relations derived from the conservation of mass, momentum, and energy, combined with the ideal gas law and isentropic flow equations. These calculations show that the rate depends on factors such as the upstream stagnation conditions, the throat area, the gas properties (such as the specific heat ratio), and the pressure ratio between the reservoir and receiver. This underpins the design and analysis of systems where precise control of mass flow is critical.

Temperature-Entropy (T-s) Diagrams

Temperature-entropy diagrams are graphical representations that show how the temperature and entropy of a substance change during a thermodynamic process. In the context of isentropic nozzle flows, a T-s diagram illustrates a vertical line (constant entropy) during ideal expansion or compression. Such diagrams help visualize the energy transformations involved and assess deviations when non-ideal effects are present.

Converging Nozzle Flow

A converging nozzle is designed to accelerate a compressible fluid by forcing it through a decreasing cross-sectional area. As the fluid passes through the nozzle, its velocity increases while its pressure and temperature decrease. The throat of the nozzle, being the smallest cross-sectional area, plays a critical role in determining the maximum achievable mass flow rate, especially under conditions where the flow becomes choked.

Isentropic Flow

Isentropic flow is a process in which the fluid undergoes expansion or compression without any change in entropy, meaning there is no irreversibility due to friction or dissipative effects. In such processes, the relationships between pressure, temperature, and density can be predicted using isentropic relations, which are fundamental in the analysis of high-speed gas flows in nozzles and other devices.

Choked Flow

Choked flow occurs when the velocity of the gas at the narrowest section (throat) of the nozzle reaches the speed of sound. Under choked conditions, the mass flow rate becomes independent of the downstream (receiver) pressure, as further reductions in receiver pressure do not increase the velocity at the throat beyond sonic speed. This concept is crucial in predicting the behavior of compressible flows in nozzles and ensuring that systems are designed to handle the maximum flow rates.

Transcript

00:01 A problem were given some information about helium at 68 degrees fahrenheit and 14 .7 psi.

00:06 It's a large tank flow steadily through converging nozzle to receiver pipe.

00:11 The cross -section area at the throat to the converging converging passage is 0 .05 feet cubed.

00:17 We're asked to determine the mass flow rate through the duct if the receiver pressure is first 10 psis and second 5 psis.

00:28 Okay, let's start with a.

00:30 Okay, that will be the following equation.

00:44 And we're going to use textbook table for helium, for helium info, and we'll get the following.

01:10 This will be 1 .66 over 1 .66 minus 1 .6.

01:15 And this will equal 7 .17 psi .8.

01:22 Now we're going to compare the receiver pressure is greater than my critical pressure.

01:37 That means we can use the receiver pressure as the throat pressure, and then we can evaluate the mock number.

02:04 And i'll have p over pt, and that'll be k minus 1 over k.

02:15 And this will equal, after we plug our values again, take that times 14 .7 over 10 to the 1 .66 minus 1 over 1 .66 minus 1 .6 minus 1.

02:45 And this will equal 7 .0 .78.

02:53 And then the temperature in the throat equal t over 1 plus k minus 1 over 2 times mok squared.

03:09 And this will equal 528 divided by 1 plus 1 .66 minus 1 ,000, and we'll take that over 2 and then we'll multiply that by 0 .78 squared.

03:42 This will be degrees rankin and then this will equal 453.

03:58 Okay and we can collect get the velocity at the throat and this will equal 0 .708 times the square root of 1 .6 times 1 .242 times 10 to the fourth times 453.

04:33 And this will be in, oopsie, feet per second.

04:41 And this will equal to 164 feet per second.

04:47 And i'm going to clean this one up a little, sort of squash it together.

04:52 1 .66 times 1 .242 times 102 .10.

05:07 N to the fourth times 453 feet squared.

05:18 Okay.

05:20 And then the density will be equal to 1 plus times m a squared to the 1k minus 1.

05:50 And let's do this here.

05:53 So pt will be equal to 3.

06:02 This will equal p over rt.

06:09 And that'll be times 1 over 1 plus k minus 1 over 2, ma squared 1 times k minus 1...