Compressible Flow Assignment 5 — Flow with Friction  
Instructions
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
05.09.14
Question #1
Air flows steadily in an adiabatic duct of constant area that is 0.3 m in diameter and 3.5 m long. The Mach number at the entrance to the duct is 2.0 and the pressure 101.33 kPa. If the back pressure is 100 kPa and the friction factor is $0.005$, what is the exit Mach number and the exit pressure?
Question #2
Consider an essentially frictionless converging-diverging nozzle connected to a pipe in which friction effects take place:
pipe.png  ./download/file.php?id=1685&sid=f264c7dc69a086188b64ddfc348aef40  ./download/file.php?id=1685&t=1&sid=f264c7dc69a086188b64ddfc348aef40
Then, what is the Mach number distribution along the pipe if the nozzle exit Mach number is $M_{\rm e}=1.65$ and the discharge tank pressure is 220 $\rm kN/m^2$?
Question #3
Consider the following problem:
question3.png  ./download/file.php?id=1686&sid=f264c7dc69a086188b64ddfc348aef40  ./download/file.php?id=1686&t=1&sid=f264c7dc69a086188b64ddfc348aef40
(a)  Estimate the maximum flow rate of air through the passage shown assuming that the friction coefficient of the duct is 0.005.
(b)  For what range of back pressures will this maximum flow rate be achieved?
Question #4
A converging-diverging axisymmetric (circular cross-section) nozzle has an exit-to-throat area ratio of about 1.44. The nozzle exit diameter is 10 cm and is connected to a length of pipe $L=3.0$ m of the same diameter (i.e. $D=10$ cm) whose friction factor is known to be about $f\approx 0.005$. The pipe discharges into a dump tank whose pressure is 340 kPa. Calculate and sketch out the Mach number and pressure distribution along the nozzle and pipe, if the nozzle supply conditions are 600 kPa and 293 K. Calculate the mass flux as well.
Question #5
A stream of air flows in an insulated pipe of constant area having a cross-sectional area of 0.10 $\rm m^2$. At a section 1 of the pipe the pressure is 0.65 bar, the temperature $5^\circ$C, and the rate of mass flow per unit area is 1.2 $\rm kg/s\,m^2$. The pressure in the space to which the tube exhausts is so low that a “choking” condition prevails, i.e., the alterations in the pressure of the exhaust region have no effect on the flow through the pipe.
(a)  Calculate the Mach number at section 1.
(b)  Calculate the Mach number, temperature, and pressure at the exit of the pipe.
(c)  Calculate the total force in the axial direction which must be exerted to hold stationary the section of the duct between section 1 and the exit.
Question #6
Consider the following problem:
question6.png  ./download/file.php?id=1687&sid=f264c7dc69a086188b64ddfc348aef40  ./download/file.php?id=1687&t=1&sid=f264c7dc69a086188b64ddfc348aef40
(a)  Estimate the maximum flow rate of air (kg/sec) that can flow through the passage shown, assuming that the friction coefficient of the duct is 0.005.
(b)  For what range of back pressures will this maximum flow rate be achieved?
Question #7
An isentropic nozzle having an area ratio of 2 discharges air into an insulated pipe of length $L$ and diameter $D$. The nozzle is supplied at $650$ $\rm kN/m^2$ and $20^\circ$C and the duct discharges into a space where the pressure is 275 $\rm kN/m^2$. Calculate $4fL/D$ of the pipe and the mass flow per unit area in the pipe $\rm (kg/s\, m^2)$ for the cases where a normal shock stands:
(a)  in the nozzle throat,
(b)  in the nozzle exit plane,
(c)  in the duct exit plane.
Question #8
It is proposed to measure the chemical composition of the gases approaching the nozzle of a turbojet engine by withdrawing a sample of gas through a capillary tube 1.25 cm long and 0.25 mm in internal diameter. The capillary will be attached to a much larger tube “A”, as shown in the following sketch, and the gas will be pumped to the chemical apparatus by a vacuum pump.
question10.png  ./download/file.php?id=1688&sid=f264c7dc69a086188b64ddfc348aef40  ./download/file.php?id=1688&t=1&sid=f264c7dc69a086188b64ddfc348aef40
One of the practical questions is whether or not the time required to draw a sample of reasonable size is excessive, and it is therefore desired to estimate the maximum rate at which gas can be drawn through the capillary. The stream in which the probe is placed is at 200 $\rm kN/m^2$ and $650^\circ$C and flows at a speed of 300 m/s. For purposes of estimating, it may be assumed that the gas has properties $\gamma=1.4$ and $R=287$ J/kgK. The viscosity of the gas may be taken as $4 \times 10^{-5}$ kg/ms. Since the flow in the capillary is likely to be laminar, the friction coefficient is given by the Poiseuille relationship $f=16/{\rm Re}_D$.
(a)  Estimate the maximum rate at which gas may be sampled in kg/s.
(b)  What is the pressure at the inlet of the capillary for the condition of maximum flow?
Answers
1.  1.32, 190.45 kPa.
2.  252.2 kPa.
3.  2.55 kg/s, 0 — 313 kPa.
4.  7.74 kg/s.
5.  0.0044, 261 Pa, 6438 N.
6.  3.25 kg/s, 0 — 3.87 bar.
7.  754 kg/m$^2$s, 4.95, 761 kg/m$^2$s, 0.43, 0.153.
8.  $8.86 \times 10^{-6}$ kg/s, 214440 Pa.
11.10.17
Due on Thursday November 16 at 16:30. Do Questions #4, #5, #6, #8 only.
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