The 40 -mm-diameter pipe has a friction factor of f=0.015 ....

The 40 -mm-diameter pipe has a friction factor of $f=0.015 .$ A nozale on the large tank $A$ delivers nitrogen isentropically to the pipe at section 1 with a velocity of $200 \mathrm{~m} / \mathrm{s}$ and temperature of $460 \mathrm{~K}$. Determine the velocity and temperature of the nitrogen at $L=2 \mathrm{~m}$ if $L^{\prime}=3 \mathrm{~m}$.

Key Concepts

Isentropic Process

An isentropic process is characterized by constant entropy, generally implying that the process is both adiabatic and reversible. In fluid dynamics, it provides simplified relationships linking pressure, temperature, and density, making it easier to predict changes in flow properties when no energy is lost to irreversibilities.

Fanno Flow

Fanno flow describes the behavior of a compressible fluid flowing in a constant-area duct under the influence of friction. The friction affects the distribution of velocity, pressure, and temperature along the duct, even though the process remains adiabatic. This concept is crucial for analyzing real-life duct flows where frictional effects cannot be ignored.

Darcy-Weisbach Friction Factor

The Darcy-Weisbach friction factor is a dimensionless parameter used to quantify the frictional losses in pipe or duct flows. It relates the pressure drop due to friction to the flow velocity, pipe length, and diameter. Understanding this factor is essential in predicting how friction influences the energy of the fluid as it moves through a conduit.

Compressible Flow

Compressible flow deals with fluid motion where variations in density are significant, such as in high-speed gas flows. In these situations, changes in pressure and temperature can substantially affect the flow characteristics, requiring specialized analyses that account for the compressibility effects in the energy and momentum balances.

Transcript

00:01 Okay, in this problem, we have to consider a 40 millimeter diameter pipe with a friction force of 0 .015.

00:10 The nozzle at a, here's one and here's two.

00:14 We verify that that's 1 and 2.

00:18 Yes, and this is 40 millimeters.

00:28 It delivers nitrogen isoanthropically to the pipe at section 1 with a velocity 200 meters per second and a temperature of 460 kelvin.

00:38 Or has to find the velocity and temperature of nitrogen at l equals 2 meters.

00:49 And this should be at l equals 2 meters.

00:55 Okay, i think we can begin.

01:00 Let's get our initial mach number, and we're going to first solve for speed of sound as follows.

01:09 1 .4, and then since we've got nitrogen, this will be 296 .8j over kilogram kelvin times, believe it was 460 kelvin, and this will equal 437 .20 meters per second, and m1 will equal a, excuse me, v1 over a1.

01:39 This will be 200 meters per second divided by 437 .20 meters per second, which will equal 0 .4575.

01:54 And we're going to find our final flow print.

01:57 So, let's see, friction force times l max over d for mach 1 equals 4 .0 .45.

02:33 This will equal 1 minus m squared over this times m squared.

02:40 Let's take the natural log of.

03:11 So let's go ahead and do...

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