The nozzle is attached onto the end of the pipe. The air in...

The nozzle is attached onto the end of the pipe. The air in the pipe is at a stagnation temperature of $120^{\circ} \mathrm{C}$ and an absolute stagnation pressure of $800 \mathrm{kPa}$. Determine the two values of the backpressure that will choke the nozzle yet produce isentropic flow. Also, what is the maximum velocity of the isentropic flow?

Key Concepts

Maximum Velocity in Isentropic Flow

The maximum velocity in an isentropic flow corresponds to the condition where the entire stagnation enthalpy of the fluid is converted into kinetic energy. This concept is commonly quantified using formulas that relate the stagnation temperature and the specific heat capacity to the maximum achievable flow speed. It serves as a theoretical upper limit for the flow velocity that can be obtained under ideal isentropic expansion conditions.

Backpressure

Backpressure is the pressure imposed at the outlet of a nozzle or flow device, which affects the flow regime within the nozzle. Adjusting the backpressure can lead to different flow regimes, including subsonic, sonic (choked), or even supersonic conditions in different nozzle designs. Determining the specific backpressure limits is fundamental in achieving the desired flow characteristics, such as choked flow in an isentropic process.

Isentropic Flow

Isentropic flow refers to a process in which the fluid flow is both adiabatic (no heat transfer) and reversible, meaning that the entropy remains constant throughout the process. This assumption simplifies the analysis by allowing the use of conservation relationships that connect stagnation properties with static properties, which is crucial in determining flow behavior in nozzles and other devices under varying pressure conditions.

Stagnation Properties

Stagnation properties, specifically stagnation temperature and stagnation pressure, represent the conditions of the fluid when it is brought to a complete stop isentropically. These values serve as the reference from which changes in flow variables such as temperature, pressure, and velocity are determined as the fluid accelerates through the nozzle, making them essential for analyzing compressible flows.

Choked Flow

Choked flow occurs when the mass flow rate through a nozzle reaches a maximum limit due to the flow velocity at the throat reaching the speed of sound (Mach 1). At this condition, further decreases in downstream (back) pressure do not increase the mass flow rate. The analysis of choked flow helps identify the critical backpressure values that lead to sonic conditions at the nozzle throat.

Transcript

00:01 In this problem.

00:03 And i've got to go draw this.

00:05 I forgot to draw this.

00:08 I'm assuming i have one here.

00:10 Yes.

00:11 So here is my flow.

00:18 Retratcher, nozzle.

00:27 And this is from here to here, 60 millimeters.

00:33 And this is 20 millimeters.

00:38 There's my nozzle.

00:39 Okay.

00:41 Now we can solve.

00:44 So we are asked to find our, two values of back pressures that will choke the nozzle yet get a flow that is is entropic and the velocity max of the isotropic flow.

01:07 Okay.

01:09 So the first thing we're going to do is get our data, our data of our throat is 20 millimeters, excuse me, our diameter and our exit is 60 millimeters.

01:27 Meters.

01:30 So our area ratio will equal, since everything else crosses out, will equal 60 over 20 squared and that'll equal 9 .0.

01:55 Then we're going to get our exit mock number.

02:01 So let's use this.

02:38 This whole thing.

02:45 Okay.

02:47 And then we're going to have nine...