PETE 314 Midterm 2

What is frictional heating?

conversion of energy or external work into internal energy

ℱ=0

frictionless flow

ℱ>0

all real flows

ℱ<0

the assumed direction of the flow is incorrect

static head=

pressure head + gravity head

dynamic head=

static head + velocity head

p/ρg

pressure head

z

gravity head

V²/2g

velocity head

dw_n.f./gdm

pump or turbine head

ℱ/g

friction head loss

List the important types of fluid-flow measuring devices based on the frictionless form of B.E.

pitot tube, pitot-static tube, venturi meter, orifice meter, rotameters

B.E. is the energy-balance equation for steady flow of constant-density fluids

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For constant-density fluids the term (∆𝑢 − 𝑑𝑄 𝑑𝑚) in B.E. represents the friction per unit mass, 𝔉.

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although no fluid has exactly constant density, B.E. can be applied with negligible error to almost all steady flows of liquids and to steady flow of gases at low velocities, because in those flows the effect of changes in density is negligible

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A large number of fluid-measuring devices are based on the frictionless form of B.E. Friction is present in these devices. The commonly used equations for these devices retain the frictionless B.E. form and add empirical correction factors to deal with the effects of friction.

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B.E. can predict negative absolute pressures for some impossible flows. For gas flows, this prediction normally means that the velocities are too high for B.E. to apply. For liquids, it normally means that the fluid will boil, leading to a two-phase flow and a much lower velocity than predicted.

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Although B.E. is a steady-flow equation, it can be used for unsteady flows if the time rate of change of velocity at every point in the system is small compared with the accelerating forces (e.g., the acceleration of gravity). It is not useful for problems involving sudden starting and stopping of flows; those are best solved with the momentum balance.

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Normally we ignore the differences in velocity perpendicular to the flow in applying B.E. to the flow in pipes and channels. This causes negligible errors, except in shallow, gravity-driven flows, like the flows over weirs.

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The classic pressure-drop experiment to determine ___________ is performed on an apparatus like that shown in the figure above. In this experiment, we set the ________________ of the fluid with the flow-regulating valve. We measure the _______________ with the tank or bucket on the scale and a _________. At steady state, we read ________ gauges 𝑃1 and 𝑃2 and record their ________. Usually, we are interested in ____________, so we divide the pressure drop by ________ (the length of the test section) and plot ______________against volumetric flow rate Q.

F, volumetric flow rate, volumetric flow rate, stopwatch, pressure, pressure drop per unit length, change in x, (P1-P2)/dx

Regardless of what ___________ liquid is flowing or what kind of pipe we use, the result is always of the form shown in the figure, and for all gases ___________ the result is the same as that shown.

newtonian, at low velocities

At low flow rates the pressure drop per unit length is proportional to the volumetric flow rate to the ___________ power

1.0

At high flow rates the pressure drop per unit length is proportional to the volumetric flow rate raised to a power that varies from ___________ (for very smooth pipe) to ___________ (for very rough pipes).

1.8, 2.0

At intermediate flow rates there is a region where the experimental results are ___________ easily reproduced. The two curves for the other two regions are shown dotted, extrapolated into this region. The flow can ___________ back and forth between these two curves and take up values between them.

not, oscillate

ℛ = 𝐷𝑉𝜌 /μ

D=

pipe diameter

ℛ = 𝐷𝑉𝜌 /μ

V=

average fluid velocity in the pipe

ℛ = 𝐷𝑉𝜌 /μ

𝜌 =

fluid density

ℛ = 𝐷𝑉𝜌 /μ

μ =

fluid viscosity

for laminar flow, R is

less than 2000

Transition region

2000<R<4000

For turbulent flow, R is

greater than 4000

The ______ the change of diameter, the ______ the pressure losses. The reasons for the losses are as follows.

In a sudden ______, the fluid is slowed down from relatively high velocity and high kinetic energy in the small pipe to relatively low velocity and low kinetic energy in the large pipe.

In a sudden ______, the flow does not come into the pipe entirely in the axial direction. Rather, it comes from all directions.

larger, greater, expansion, contraction

The steady flow of fluids in constant-cross-section conduits can be of two radically different kinds: laminar, in which all the motion is all in the flow direction, and turbulent, in which there is a chaotic crosswise motion perpendicular to the net flow direction. The same is true for unconfined flows like the oceans or the atmosphere.

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In laminar flow the pressure drop per unit length is proportional to the first power of the volumetric flow rate. The entire flow behavior can be calculated simply. The calculation requires the observational fact that fluid clings to solid surfaces, i.e., the velocity at the surface is zero.

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In turbulent flow the pressure drop per unit length is proportional to the flow rate to the 1.8 to 2.0 power. The behavior cannot be calculated without experimental data.

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All experimental data on the turbulent flow of Newtonian fluids in circular pipes can be represented on the friction factor plot.

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The friction factor plot can be replaced by two fairly simple equations. The first, for laminar flow, is simply a rearrangement of Poisueille’s equation and is restricted to laminar flow in a circular tube, for which it is rigorous. The second is simply a satisfactory fit of the experimental data. With these two equations we can completely replace the friction factor plot. But the plot has considerable intuitive content and is still useful for hand calculations.

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All data on turbulent flow through valves and fittings can be correlated by assuming that each kind of fitting contributes as much friction as a certain number of pipe diameters of straight pipe. That number is about the same for one kind of fitting, independent of pipe size, fluid properties, etc. An alternative approach assigns a resistance coefficient, K, to each kind of fitting.

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Laminar flow in a few kinds of noncircular conduits can be analyzed by the same technique used for circular pipes.

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Turbulent flow friction losses in many kinds of noncircular conduits can be estimated by substituting 4 times the hydraulic radius for the diameter in the Reynolds number, 𝜀 𝐷 the friction factor plot, and B.E.

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The economic size for a pipe is the size with the lowest sum of annual charges for the purchased cost of the pipe and pump and the annual power cost of running the pump or compressor needed to overcome friction. For turbulent flow this results in an economic velocity which is practically independent of everything but fluid density; it is about 6 ft /s for most low viscosity liquids and about 40 ft / s for air and other gases under normal conditions.

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The forces of fluids flowing over bodies are ordinarily correlated by the drag equation, in which the drag coefficient plays the same role as does the friction factor in pipe flow.

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