Air flows isentropically into the nozzle at MA=0.2 and out at MB=2. If the diameter of the nozzl

step 1 Okay, in this problem, we're going to be doing the following. And we're given air flows isotropi

Air flows isentropically into the nozzle at MA=0.2 and out...

Air flows isentropically into the nozzle at $\mathrm{M}_{A}=0.2$ and out at $\mathrm{M}_{B}=2$. If the diameter of the nozzle at $A$ is 30 $\mathrm{mm},$ determine the diameter of the throat and the diameter at $B$. Also, if the temperature at $A$ is $300 \mathrm{~K},$ determine the stagnation temperature and the temperature at $B$.

Key Concepts

Area-Mach Number Relationship

The area-Mach number relationship is a key concept in nozzle design that links the cross-sectional area of the flow passage to the local Mach number under isentropic conditions. This relationship is used to determine how the geometry of the nozzle (for example, the throat and exit areas) influences the acceleration or deceleration of the flow, enabling the transition from subsonic to supersonic speeds in converging-diverging nozzles.

Stagnation Temperature

Stagnation temperature represents the temperature a fluid parcel would reach if it were brought to rest isentropically. It is an important property in compressible flows because it remains constant along a streamline in the absence of energy losses. This concept is used to relate static temperature measurements to the total energy of the flow and to analyze the effects of velocity changes on temperature in the nozzle.

Isentropic Flow

Isentropic flow refers to a thermodynamic process in which the fluid moves without any change in entropy, meaning it is both adiabatic (no heat transfer) and reversible. This concept is fundamental in compressible fluid dynamics because it allows the use of simplified relations between pressure, temperature, and density, which are crucial for calculating flow properties within nozzles and diffusers.

Mach Number

The Mach number is defined as the ratio of the flow velocity to the local speed of sound. It is a critical parameter in compressible flow as it distinguishes between subsonic, transonic, and supersonic regimes. The behavior of the flow, including the response of pressure, temperature, and density to changes in geometry, depends largely on the Mach number.

Transcript

00:01 Okay, in this problem, we're going to be doing the following.

00:05 And we're given air flows isotropically into a nozzle at these conditions, two marks.

00:11 If the diameter is 300 millimeters, we're asked to determine the diameter of the throat and the diameter of b.

00:18 And we're told if the temperature at a is 300k, we're asked to determine the stagnation temperature and temperature at b.

00:32 Okay, let's see here.

00:49 And we're going to do something that's the same for the first two parts of this.

00:54 I've already solved that.

00:57 So this is good for this question two.

00:59 And that's to find dtp.

01:01 We're going to have to use our throat.

01:05 Then we're going to use this to find this value.

01:15 Plug this into this 2 .387.

01:22 And we'll, whoopsie, we're also going to get, i'm just highlighting each one of these so i know where i am.

01:33 This 174 we're using, and then we go a little bit differently.

01:39 Okay.

01:42 So for the next part of this that we're going to work on, we're going to find for mb2.

01:53 We know that my tb over tc is 0 .5566, and my ab over my ac will be 1 .688.

02:08 So then we can find some ratios here...

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