Nitrogen enters a converging-diverging nozzle at 700 kPa and 450 K with a negligible velocity.

Nitrogen enters a converging-diverging nozzle at 700 kPa and...

Key Concepts

Ideal Gas Behavior

Many calculations involving nozzle flows are based on the assumption that the working fluid behaves as an ideal gas. This allows the use of the ideal gas law to relate pressure, temperature, and density. The ideal gas assumption simplifies the analysis and enables the derivation of explicit expressions for the static and stagnation properties of the fluid.

Speed of Sound

The speed of sound in a fluid is a critical parameter in compressible flow, defined by the square root of the product of the specific heat ratio and the gas constant times the temperature. It represents the maximum speed at which small pressure disturbances travel through the fluid and determines the sonic conditions in the nozzle where the Mach number equals one.

Converging-Diverging Nozzle

A converging-diverging nozzle is a device used to accelerate compressible fluids from subsonic to supersonic speeds. In these nozzles, the cross-sectional area first decreases (converges) to a minimum at the throat and then increases (diverges), which allows for the proper acceleration of the flow and the possibility of reaching sonic conditions at the throat – a phenomenon known as choked flow.

Choked Flow and Critical Conditions

Choked flow occurs when the fluid velocity at the narrowest section (the throat) of the nozzle reaches the speed of sound (Mach 1). Under these conditions, the mass flow rate through the nozzle is maximized and becomes independent of downstream pressure. The conditions at the throat are often referred to as critical conditions and are essential for designing and analyzing high-speed flow systems.

Isentropic Flow Relations

In many nozzle flow problems, the flow is assumed to be isentropic, meaning that the process is both adiabatic and reversible. This assumption allows the use of specific isentropic relations which relate the static properties (pressure, temperature, density) at different points in the flow to the stagnation (total) conditions. These relations are crucial for determining the changes in thermodynamic properties as the fluid accelerates through the nozzle.

Transcript

00:01 Okay, so we know the critical properties occurs when mark number equal 1.

00:04 So now we need to use the table a32 at the back of the book to find the critical properties for nitrogen gas.

00:15 So at mark number equal 1, if you take a look at the table, we know p over p0 is equal 0 .5283, which means that critical pressure is equal to 0 .583 p0.

00:27 And row over row 0 is given as 0 .6339.

00:31 So therefore the critical density is equal to 0 .6339, row 0.

00:39 We have t over t0 is equal to 0 .833.

00:42 So we have a critical temperature is equal to 0 .8333 t0.

00:47 So now that's trying to determine the critical pressure first.

00:50 So when we know the initial pressure, or in other words, stagnation pressure is given at 700 kiosk.

00:58 So now that's plugging back to the equation to determine the critical pressure.

01:03 So the critical pressure is equal to 0 .52 .83 multiplied by 700kcal, which is equal to 370kcal.

01:27 So now that's determining the critical temperature.

01:30 So when the initial temperature is given as 400 kelvin.

01:33 So now that's plugging back to the equation to determine the critical temperature, which is equal to 0 .8.

01:40 3, 3, 3, multiplied by 400 kelvin.

01:44 And this will give us the critical temperature is about 333 kelvin.

01:52 So now we can determine the critical velocity, which is eventually equal to mark number times square root krt asterisk.

02:01 So when the mark number is given as 1, the specific ratio for the nitrogen gas is 1 .4...

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