merged_document problems 1-7

Department of Mechanical EngineeringME333 Introduction to Fluid Mechanics

Homework #1, assigned 4/05/13, due 04/12/13

Problem 1

A gas may be considered rarefied (it cannot be studied using the continuum hypothesis)if it contains less than 1012 molecules per cubic millimeter. Remembering that Avagadrosnumber tells you that a gas contains 6.0231023 molecules per mole, how low would thepressure need to be such that air can be considered rarefied at a temperature of 18 C?

Problem 2

Measurements of temperature and pressure in the Martian atmosphere show that T = 50 Cand P = 900 Pa. Calculate:

The density under those conditions assuming the value of the gas constant, Rg, onMars can be approximated with that of CO2.

The the density of the air on Earth under the same pressure and temperature. The density of the CO2 under terrestrial conditions, T = 18 C andP = 101.6 kPa.

Problem 3

A rigid tank contains helium gas at 600 kPa absolute pressure and 20 C. What is the changein pressure if the temperature is increased to 40 C?

Problem 4

A 25 mm diameter shaft is pulled through a cylindrical bearing as shown in the figure below.A lubricant with a kinematic viscosity of 8104 m2/s and density of 910 kg/m3 fills the0.3 mm gap between the shaft and the bearing. Determine the force, P, required to pull theshaft at a constant velocity of 3 m/s. Assume that the velocity inside the gap varies linearlybetween the shaft and the stationary bearing casing.

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Department of Mechanical Engineering

ME333 Fluid Mechanics Homework #1. Assigned 4/1/11, due 4/8/11

Problem 1

A gas may be considered rarefied, that is, it can not be studied with theories based on the con-tinuum assumption, if it contains less than 1012 molecules per cubic millimeter. Rememberingthat Avogadros number tells you that a gas contains 6.023 1023 molecules per mole, how lowwould pressure need to be so that air can be considered rarefied at a temperature of 18C?

Problem 2

Measurements of temperature and pressure in Mars atmosphere show values of T = 50Cand P = 900 Pa.

Calculate:

(a) The density under those conditions assuming the value of the gas constant, Rg, inMars atmosphere can be approximated by the value of CO2.

(b) The value of the density on Earth under the same conditions, and under more typicalconditions on Earths surface T = 18C and P = 101.6 kPa

Problem 3

A rigid tank contains air at 600 kPa absolute pressure and 18C. What is the change inpressure if the temperature is increased to 40C?

Problem 4

A 25 mm diameter shaft is pulled through a cylindrical bearing as shown in figure . A lubricantwith kinematic viscosity equal to 8 104 m2/s and specific gravity 0.91fills the 0.3 mm gapbetween the shaft and the bearing. Determine the force, P, required to pull the shaft ata velocity of 3 m/s. Assume that the velocity inside the gap varies linearly between themoving shaft and the stationary bearing casing, and that the shear stress is proportional to thelubricants dynamic viscosity and the slope of the velocity profile: = dV

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ME333 Introduction to Fluid Mechanics


Problem 1

The basic elements of a hydraulic press are shown in the figure. The plunger has an areaof 0.0012 m2, and a force F1 can be applied to the plunger through a lever mechanism thatmultiples the force by a factor of 4. If the large piston has an area of 0.2 m2, what load, F2,can be raised by a force of 1000 N applied to the lever?




Problem 1

A 0.3 m diameter pipe is connected to another pipe, the second one witha 0.02 m diameter, and both of them are rigidly held in place. Bothpipes are horizontal with pistons at each end. If the space between thepistons is filled with water, what force will have to be applied to thelarger piston to balance a force of 80 N applied to the smaller piston.

Problem 2

The basic elements of a hydraulic press are shown in the figure. Theplunger has an area of 0.0012 m2, and a force F1 can be applied to theplunger through a lever mechanism that multiples the force by a factorof 4. If the large piston has an area of 0.2 m2, what load, F2, can beraised by a force of 1000 N applied to the lever?

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Problem 2

A cylindrical tank with hemispherical ends contains a volatile liquid and its vapor (see figureat the top of the next page). The liquid density is 800 kg/m3, and its vapor density isnegligible. The pressure in the vapor is 120 kPa and the atmospheric pressure is 101 kPa.Determine:

the gage pressure reading on the pressure gage the height h of the mercury manometer

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Problem 3

A 1 m wide, 3 m long rectangular gate weighs 5.0 kN. It is held in place by a horizontalflexible cable, as shown in the figure. Water exerts pressure on the gate, which is hinged atpoint A. Determine the tension in the cable and the reaction force at the hinge.

Problem 4

A homogeneous 1 m wide, 3 m long rectangular gate weighs 500 kg. It isheld in place by a horizontal flexible cable, as shown in the figure. Waterexerts pressure on the gate, which is hinged at point A. Determine thetension in the cable.

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Problem 4

The Ballard Locks, or Hiram M. Chittenden Locks, raises or lowers boats between betweenLake Union and Puget Sound. A top view of the locks is diagramed in the figure below. Thewidth across the locks is 24 m, and the angle between the gate and the lock wall, , is 15.The water depth on the Lake Union side is 16.7 m and the water depth on the Puget Soundside is 8.8 m. Calculate:

Force exerted by the water on a single gate Contact force between the two gates Reaction force at hinges

Problem 5

A rectangular gate, 8 meters in height and 3 meters in width (into the paper) is located andthe end of a rectangular passage that is connected to a large open tank filled with water.The gate is hinged at its bottom and held close by a horizontal force FH applied at the centerof the gate, as shown in the figure. The maximum value for FH is 3500 kN .

Determine the maximum value of the depth, h, above the center of the gate, beforethe gate starts to open

Would the value be the same if the gate was hinged at the top? Show the calculationsand explain physically what the result means.

Problem 6

The U tube in the figure is partially filled with water and rotates around its axis of symmetry,a-a. Determine the angular velocity that will cause the water to start vaporizing at thebottom of the tube, point A.

Problem 6

A rectangular gate, 8 meters in height and 3 meters in width (into thepaper) is located and the end of a rectangular passage that is connectedto a large open tank filled with water. The gate is hinged at its bottomand held close by a horizontal force FH applied at the center of thegate, as shown in the figure. The maximum value for FH is 3500 kN .

Determine the maximum value of the depth, h, above the centerof the gate, before the gate starts to open

Would the value be the same if the gate was hinged at the top?Show the calculations and explain physically what the result means.

Problem 7The U tube in the figure is partially filled with water and rotates aroundits axis of symmetry, a-a. Determine the angular velocity that will causethe water to start vaporizing at the bottom of the tube, point A.

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Problem 7

A barge has a trapezoidal shape and is 22 m long into the paper. If the total weight of thebarge and cargo is 300 tons, what is the draft, H, of the barge in seawater.




Problem 1

Air at atmospheric conditions is drawn into a compressor at a steady rate of 15.0 ft3/s.The compression ratio is Pout/Pin = 10, and the evolution of the gas can be assumed to beisentropic P/ = constant, where is the ratio of specific heats for the gas and is equalto 1.4. Note that the isentropic assumption gives the best possible theoretical performanceof the compressor. If the design criteria is that the velocity at the outlet does not exceed70 ft/s, what is the minimum diameter for the round pipe at the outlet?



Problem 1

Air at atmospheric conditions is drawn into a compressor at a steady rate of 0.5 m3/s. Thecompression ratio is Pout/Pin = 10, and the evolution of the gas can be assumed, as a first orderapproximation, to be isentropic P/ = constant, where is the ratio of specific heats for thegas and is equal to 1.4). If the design criteria is that the velocity at the outlet does not exceed30 m/s, what is the minimum diameter for the round pipe at the outlet?

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Problem 2

Calculate the force necessary to hold the plug in place at the exit where there is a flow rate of0.1 m3/s of oil coming out of 0.5 m diameter pipe. The pressure difference between the insideof the pipe and the outside is 5 MPa.

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Problem 2

When a 2-D liquid jet hits an inclined flat jet, it splits into two streams with equal speedbut uneven thickness. Assuming the shear stresses on the plate is negligible, calculate theresulting thicknesses, h2 and h3, as a function of the plate angle, . Also calculate the forceon the plate necessary to keep it in place.

Problem 3

When a liquid jet hits an inclined flat jet, it splits into two streams with equal speed butuneven thickness. Assuming the shear stresses on the plate is negligible, calculate the resultingthicknesses, h2 and h3, as a function of the plate angle, . Comment on the limiting cases = 0o and = 90o. Calculate the force on the plate necessary to keep it in place.

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Problem 4

To evaluate the terminal velocity of a water droplet in air )the velocity at which it falls undergravity in steady state), we measure the air velocity behind a droplet as it falls in a windtunnel. This velocity if well approximated by the expression: V [1 (r/H)2]2, where r is thedistance from the droplets center of mass trajectory (axis of symmetry) and H and V areknown values.Calculate the terminal velocity of the droplet and the drag that the air exerts on it.Note: Its helpful to study the problem relative to a reference frame fixed to the droplet. In thisreference frame, the problem, and the appropriate control volume, can be sketched as follows:

Problem 3

A 10 mm diameter jet of water is deflected by a homogeneous rectangular block (15 mm x200 mm x 100 mm) that weighs 6 N. Determine the minimum volume flow rate needed totip the block. Neglect any splashback and neglect shear forces on the surface of the block.

Problem 2

When a liquid jet hits an inclined flat jet, it splits into two streamswith equal speed but uneven thickness. Assuming the shear stresses onthe plate is negligible, calculate the resulting thicknesses, h2 and h3, asa function of the plate angle, . Comment on the limiting cases = 0

and = 90. Calculate the force on the plate necessary to keep it inplace.

Problem 3

When a liquid jet hits an inclined flat jet, it splits into two streams with equal speed butuneven thickness. Assuming the shear stresses on the plate is negligible, calculate the resultingthicknesses, h2 and h3, as a function of the plate angle, . Comment on the limiting cases = 0o and = 90o. Calculate the force on the plate necessary to keep it in place.

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Problem 4

To evaluate the terminal velocity of a water droplet in air )the velocity at which it falls undergravity in steady state), we measure the air velocity behind a droplet as it falls in a windtunnel. This velocity if well approximated by the expression: V [1 (r/H)2]2, where r is thedistance from the droplets center of mass trajectory (axis of symmetry) and H and V areknown values.Calculate the terminal velocity of the droplet and the drag that the air exerts on it.Note: Its helpful to study the problem relative to a reference frame fixed to the droplet. In thisreference frame, the problem, and the appropriate control volume, can be sketched as follows:

Problem 3

A 10 mm diameter jet of water is deflected by a homogeneous rectan-gular block (15 mm x 200 mm x 100 mm) that weighs 6 N. Determinethe minimum volume flow rate needed to tip the block.

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Problem 4

Water flows through a 90 pipe bend. The pipe cross section is constant 0.10 ft2. The flowrate through the pipe is a constant 280 gpm (US gallons per minute). The pressure at thetop and left side are 24 psi and 26 psi (absolute), respectively. The total volume inside the

bend is 2 ft3. Calculate the net force, ~R, required to hold the pipe bend in place.

Problem 5

You are conducting a wind tunnel test to measure the aerodynamic drag (air resistance)over a cylinder. Apply the momentum principle for a control volume to relate the dragforce per unit width, w, in terms of the upstream and downstream velocities and pressures.The upstream velocity is a 15.6 m/s. The downstream horizontal velocity measurements areshown in the table below. Upstream pressure is 133.5 Pa (gage) and downstream pressure is0.0 Pa (gage). The diameter of the cylinder is 100 mm. Total wind tunnel height is 240 mm.Neglect any forces on the wind tunnel walls. The horizontal velocity profile is symmetricabout the y axis.

Table 1: Wind Tunnel Velocity Profile

Vertical Position (mm) Horizontal Velocity (m/s)

0 5.010 5.820 7.030 9.240 12.150 15.260 18.070 20.280 21.490 22.4100 23.0110 23.6120 23.9




Problem 1

A rocket is held fixed on a test stand by a horizontal force, F . The rocket is powered byliquid oxygen reacting with a liquid fuel. The absolute pressure of the exhaust gas, pe isnot necessarily atmospheric. The exhaust gas can be treated as air and an ideal gas. Alsoassume that the rocket is to be launched for peaceful purposes.

Assuming that the pressure, area, velocity, and density of the exit, pe, Ae, Ve, e, andthe fuel and oxygen flow rates are known, show that the horizontal force is equal toFx = eAeV

2e + Ae(pe patm)

Given that mF = 1.45 kg/s, mO = 7.25 kg/s, pe = 90 kPa, Ae = .015 m2, Te = 860 Kcalculate the speed of the exhaust gas, Ve, and the force, F .

Calculate the rate of energy added by the incoming fuel and oxidizer assuming anadiabatic system. Hint: calculate the rate of enery flux from the exiting exhaust gases.

Problem 2

Consider flow through a pipe of radius R. The axial, or z, velocity does not vary along thelength of the pipe. The axial pipe velocity does vary across the pipe radius according thethe relation:

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First, calculate the volumetric flow rate of the pipe through section 1 in terms of the radius,R, and max velocity. Second, use the integral form of conservation of momentum to relatethe change in pressure, P1 P2, to the in terms of the shear stress on the pipe wall. Is thechange in pressure, P1 P2, greater than, equal to, or less than zero? Third, write the wallshear stress in terms of the flow rate and and therefore relate the flow rate to the change inpressure. Recall that the axial component of the shear stress is z = vr .

Problem 3

Air is blown over a computer processing chip for thermal management. The velocity, tem-perature, and width of the incoming air stream is known; V1 = 10 m/s, T1 = 15

C, H1 =10 cm. The temperature and velocity of the exiting air stream are also known; V2 = 10 m/s,T2 = 75

C. Do not assume that the air is incompressible. The system is at a steady stateand can be treated as 2D. Calculate:

The width of the air flow leaving the control volume, H2. The total rate of heat transfer from the processor to the air, Q.

Problem 4

Water is supplied at 150 ft3/s and 60 psi (absolute) to a hydraulic turbine through a 3 ftdiameter pipe. The turbine discharge pipe has a 4 ft diameter, and the water pressure atthe discharge is 10 psi (absolute). If the turbine develops 2500 hp, determine the turbinesefficiency. Hint: calculate the maximum possible work that could be extracted if the flowwas isentropic and adiabatic.

Problem 5

Air at 20C and atmospheric pressure enters a compressor with an inlet diameter of 12 cmat 75 m/s and leaves at an absolute pressure and temperature of 200 kPa and 345 K,respectively, into a pipe of diameter of 8 cm. Cooling water around the compressor removes18 kW of heat. Determine the power required by the compressor. Also determine theefficiency of the compressor.




Problem 1

A velocity field for a particular flow under consideration is found to be given by:

~v = Ay +Ax

where A is a positive constant. Find: What are the dimensions of A? Calculate the individual components of the acceleration field. Find the equation for the streamline passing through a fixed point (x0, y0). Assume that, in SI units, A=2.0 and make a contour plot of the streamlines on the

range -2.0< x




Problem 1

Consider a thin layer of water flowing down an inclined plane. Assume that the flow issteady, incompressible, fully developed, and 2D. Calculate:

The velocity distribution within the water in terms of the knowns of the problem A simple sketch of the velocity distribution The volume flow rate of liquid down the slope The shear stress from the water onto the plane The maximum speed of the water given that the angle is 30 and the height is 1 mm

Problem 2

Consider two fluids of known density and viscosity in a channel which is subjected to aconstant pressure gradient in the x direction. Calculate the velocity profile in both liquids.Assume steady, incompressible, fully developed and 2D flow.

Problem 3

A steady flow velocity field is given by ~v = 2xy + (x2 y2). Let the density be constantand neglect gravity. If the flow is inviscid, find the pressure, p(x, y), if the pressure at x=0y=0 is po.

Problem 4

A siphon draws water from a larger container. Assuming inviscid flow, derive an expressionfor the flow rate through the tube as a function of h. What is the maximum possible valueof h that can occur before the fluid begins to cavitate? In the case where the water is at20C, H = 3 ft, D = 1 in, what is h?




Problem 1

Flow leaves a water reservoir from a pipe that is at a depth h1. The diameter of the pipechanges from D1 to D2 = 0.75D1 and then to D3 = 2D1. The velocity at the end of thepipe, V3, is known. Assuming the Bernouilli equation is applicable, calculate the pressuresand velocities at points 1, 2, and 3. Take h1=10 m, h2=2 m, h3=13 m, D1=10 cm, V3=3 m/s.



Problem 1

Flow comes out of a water reservoir from a pipe that is at a depth h1 below the free surface. Thediameter of the pipe changes from D1 to D2 = 0.75D1, and then to D3 = 2D2. This happenswhile the pipes change height to h2 and h3. The velocity at the end of the pipe, V3 = 3 m/s.

Assuming that Bernouilli is applicable in the pipes, calculate what are the pressures andvelocities at 1, 2 and 3.

Take h1 = 10 m, h2 = 2 m, h3 = 13 m, and D1 = 0.1 m.

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Problem 2

A closed cylindrical tank filled with water is emptied by a vertical pipe with a known length,htube, and diameter, dtube. Since the tank is closed, the pressure inside is equal to the vaporpressure of water at 20C. Assuming the draining is slow enough such that the Bernoulliequation is applicable, derive an expression in symbolic form for the rate-of-change of theheight of water inside the reservoir, hres, in terms of the other knowns in the problem.Integrate your expression to obtain hres as a function of time, given that hres at time t=0 isknown, say h0. If Dres=10 m, h0=10 m, dtube=0.1 m, htube=8 m, Hres=10 m, how long doesit take for the tank to drain?

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Problem 3

Wind tunnel tests are to be conducted to to evaluate the design of a new wind turbine. Thepower output of the turbine, P , should depend on the local wind speed, U , air density andviscosity, and , turbine blade length, `, and blade rotation rate , i.e.,

P = f (U, , , `,)

How many dimensionless parameters do you expect for this problem? What are the dimen-sionless parameters for this problem?

The actual prototype will have a blade length of 50 m rotating at a rate of 0.1 rev/soperating in air at standard conditions with a speed of 10 m/s. A scaled model beingprepared for wind tunnel tests has a blade length of 0.5 m. Assuming the air in the windtunnel is also at standard conditions and the wind tunnel air speed is 2 m/s, what should bethe rotation rate of the model turbine in order to obtain dynamic similarity? If the powermeasured by the model is 4.0 W, what would be the expected power output for the actualprototype?

Problem 4

A centrifugal pump increases the pressure, i.e. P = Poutlet Pinlet, of an incompressiblefluid with a flowrate Q, viscosity , and density . The impeller blades of the pump havesome diameter D with a rotation rate and require some input work W . The functionaldependence of the flow rate is given as

Q = f (P, W ,, D, , ).

Use dimensional analysis to evaluate the dependency of Q in terms of non-dimensionalgroups.

Problem 3

Use dimensional analysis to estimate what does the drag on a submarine depend on.!

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Problem 4

A centrifugal pump uses electricity to move a set of blades that push a liquid flow rate Q fromthe inlet at a pressure P1 to the outlet at a pressure P2. The rotor (set of moving blades) movesat an angular speed . Use dimensional analysis to evaluate the dependency of Q in terms ofnon dimensional groups.

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merged_document problems 1-7

Documents

gas constant

measurements of temperature

diameter shaft

shaft ata velocity

helium gas

kpa absolute pressure

c andp

themoving shaft