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Unified Engineering Spring 2004 Fluids Problems F11–F
F11+12. Air is drawn at high speed out of a large reservoir through a duct of constant area A, which contains a radiator delivering a known Q˙^ to the flow (in Watts). The heating and friction of the duct walls are negligible.
. Q
ho � o po^ ho^^1 �^1 p^^1 V^1 ho^^2 �^2 p^^2 V^2
p
p
h
The known flow quantities are: o =^ ho 1 reservoir^ total^ enthalpy o reservoir^ total^ density o reservoir^ total^ pressure 2 outlet^ pressure^ (drives^ the^ flow)
p
V
V
The remaining six unknown quantities inside the duct are: ho 2 outlet total enthalpy 1 inlet^ velocity 2 outlet^ velocity 1 inlet^ density 2 outlet^ density 1 inlet^ pressure
A total of six equations are needed to solve for the six unknowns. One of these equations is the isentropic relation between the reservoir and station 1,
ho 1 − 2 V 2 ��/(�−1) p (^1) =^11 po ho
and two additional ones are the state equations at stations 1 and 2.
p^ �^ −^1 1 =^ � 1 ho 1 −^2 V 12
p^ �^ −^1 2 =^ � 2 ho 2 −^2 V 22
Write down the remaining three equations by constructing a suitable control volume and applying the integral mass, momentum, energy equations. (Do not try to solve the six equations — it gets very messy!)
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F13. You stick your hand out the window of a car traveling at 75 mph, palm into the wind. The ambient air conditions are � = 1. 2 kg/m^3 , p = 105 Pa, T = 300K�^. For air, p =^1004 J/kg^ K�^.^ Determine^ the^ conditions^ on^ the^ stagnation^ point^ on^ your^ palm.
F14. An aircraft is flying at speed V�, in an atmosphere with p�, and �.
a) What is the flight Mach number M�? Give in terms of the quantities above.
b) Determine the stagnation pressure po at the nose of the aicraft in two ways: i) The exact full compressible equation. ii) The incompressible Bernoulli equation, pretending � = � is constant.
Plot po/p� versus M for the two equations. Also plot the “Bernoulli error”
(po/p�)exact − (po/p�)Bernoulli
versus M. What would you judge to be a reasonable upper Mach limit on the validity of the Bernoulli equation?
Unified Engineering Spring 2004
Problem M A solid steel shaft of length 3 m is required to transmit a torque of 200 kNm. The maximum allowable shear stress that the steel can support, ty , is 200 MPa (shear yield stress).
(a) If the shaft has a solid circular cross section, determine the minimum diameter of the shaft to transmit the torque without yielding.
(b) If the shaft is instead hollow with a ratio of external to internal diameter of 5/4, what is now the required diameter of the shaft, what is the weight saving over case (a)?.
(c) What is the ratio of (i) the angle of twist between the ends of the shafts and (ii) the torsional stiffness in the two cases?.
(d) What is the maximum extensional stress acting in the shaft in (b), in what direction(s) does it act?.
Problem M The diagram below shows a rigid bar, length L, pinned at one end, which is initially held vertical. The bar is stabilized by the pair of linear springs, stiffness k, orientated at the angle q to the vertical. Find the critical value of the load P which will cause collapse of this assembly. Hint: Consider the equilibrium of the bar at a small displacement from the equilibrium position shown.
P
q q k (^) k
L
Note. Although the design objective is to minimize the mass of the structure, the credit for the
question will be based on demonstrating a logical approach to selecting a material and a truss
configuration, and then obtaining an estimate for the mass of the truss. Do not spend more than
an hour on this question, and do not analyze multiple truss configurations. A useful exercise is
to estimate what you think the mass of the truss will be before you do any analysis - developing
an intuition for the correct size for structures is a useful skill to cultivate.
