But, First a Word About Units

[Pages:12]Review for Final Exam

Review for Final Exam

Larry Caretto Mechanical Engineering 390

Fluid Mechanics

May 6, 2008

May 6, 2008

Outline

? Properties, statics and manometers ? Bernoulli and continuity equation ? Momentum and energy equations ? Dimensional analysis and similitude ? Pipe flow ? External flow ? Choice of open-channel flow or

compressible flow

2

The Final

? Tuesday, May 13, 5:30 to 7:30 pm ? Open textbook only

? No class notes, homework solutions, classmate conversations, text messaging, etc.

? Will be problems similar to the midterm and quizzes

? Think of it as the midterm times 120/75 or four back-to-back quizzes

3

But, First a Word About Units

Quantity

SI units EE units BG units

Density Pressure & shear stress Velocity Viscosity

Specific weight = g

kg/m3 kPa = kN/m2

m/s

lbm/ft3

slug/ft3

1 psi = 1 lbf/in2 =

144 psf = 144 lbf/ft2

ft/s

N?s/m2 = lbf?s/ft2 =

lbf?s/ft2 =

kg/m?s 32.2 lbm/ft?s slug/ft?s

N/m3

lbf/ft3

Tabulated values at standard gravity

4

Density and Related Summary

? Density: = mass per unit volume with units of kg/m3 or slugs/ft3

? Specific weight: = g with units of N/m3 or lbf/ft3 (varies with local g)

? Specific gravity: SG = /ref = /ref

? Liquid ref: water at 4oC with = 1000 kg/m3 and = 9806.65 N/m3 or water at 60oF with = 1.94 slugs/ft3 and = 62.4 lbf/ft3

? Gas ref: air at 15oC (59oF) with = 1.23 kg/m3 = 0.00238 slugs/ft3 and = 62.4 lbf/ft3

5

ME 390 ? Fluid Mechanics

Pressure Solid Melting line

Boiling line

Transitions Between Phases

Liquid

? For phase transitions pressure and temperature are related

? Vapor pressure is

Gas

the pressure at

which liquid-vapor

Sublimation curve transition occurs

Temperature 6

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Review for Final Exam

May 6, 2008

Ideal Gases

? From chemistry: PV = nRT (V is volume)

? n = m / M is the number of moles

? for mass in kg n is in kilogram moles (kmol); for mass in lbm, n is in pound moles (lbmol)

? R = 8.31447 kJ/kmol?K = 10.7316 psia?ft3 / lbmol?R is universal gas constant

? R = R/M is engineering gas constant that is different for each gas

? Real gases like ideal gases at low pressures

? P = nRT / V = (m/M)RT / V = (m/V)(R/M)T

P = RT

7

Viscosity

Newtonian and non-Newtonian Variation of shear stress with rate of shearing strain for several types of fluids, including common nonNewtonian fluids.

= u y

Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald

Young, and Theodore Okiishi,Copyright ? 2005 by John Wiley &

8

Sons, Inc. All rights reserved.

Surface Tension Rise

? Vertical force balance: R2h = 2Rcos

? Surface tension depends on fluid and

temperature, wetting angle, , depends on fluid

and surface

h = 2 cos R

Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald

9

Young, and Theodore Okiishi,Copyright ? 2005 by John Wiley &

Sons, Inc. All rights reserved.

Pressure Relations

? Pressure is a scalar ? The force exerted by a pressure is the

same in all directions ? Want to see how pressure changes in a

static (nonmoving fluid) ? Look at balance of pressure force and

fluid weight over a differential volume element, xyz

10

Incompressible Fluid

p2 + z2 = p1 + z1

? Which pressure is higher?

z2

p2

p1 = p2 + (z2 + z1)

h

p1 = p2 + h > p2

z1

p1

? Pressure

increases with

depth

11

ME 390 ? Fluid Mechanics

Gage and Absolute Pressure

Figure 2.7, Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi Copyright ?

12

2005 by John Wiley & Sons, Inc. All rights reserved.

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Review for Final Exam

May 6, 2008

Barometric Pressure

? Mercury barometer used to measure atmospheric pressure

? Top is evacuated and fills with mercury vapor

? Patm = h + pvapor ? pvapor = 0.000023 psia =

0.1586 Pa at 68oF (20oC)

? h = 760 mm = 29.92 in for standard atmosphere

Figure 2.8, Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi Copyright ?

13

2005 by John Wiley & Sons, Inc. All rights reserved.

Solving Manometer Problems

? Basic equation: pressures at two depths open

in same fluid: p2 = p3 + (z3 ? z2) = p3 + h

? "Open" means p = patm

3

? patm = 101.325 kPa = 14.696 psia

? For gage pressure, patm = 0

? Same pressures at same level on two sides of a manometer with same fluid

? p1 = p2

14

Solving Manometer Problems II

? Write equations for (1) pressures at two open depths in same fluid and (2) equal pressures at same level (with same fluid) 3 at all branches in manometer.

? Eliminate intermediate pressures from equations to get desired P

? Watch units for length, psi or

psf, N or kN

? For gases z 0

15

Figure 2.17, Fundamentals of Fluid

Mechanics, 5/E by Bruce Munson, Donald

Young, and Theodore Okiishi Copyright ?

2005 by John Wiley & Sons, Inc. All rights reserved.

Slanted Surface

FR = pdA

A

= hdA

A

= y sin dA

A

16

Resultant Force

FR = sin ydA = yc Asin

? Center of pressureA, not yc, is location of resultant force (in diagram yc = ystart + a/2)

yCP

=

Ix Ayc

=

I xc Ayc

+ yc

y

Ix = Ixc + AyC2 (in

diagram Ixc = ba3/12 and A = ab

ystart

x

Figure 2.18(a), Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, 17 Donald Young, and Theodore Okiishi Copyright ? 2005 by John Wiley & Sons,

Inc. All rights reserved.

Buoyancy

? Buoyant force, FB, due to difference in pressure between top and bottom

? FB = fluidVb ? Passes through the

centroid of the submerged body

Figure 2.24(b,c), Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, 18 Donald Young, and Theodore Okiishi Copyright ? 2005 by John Wiley & Sons,

Inc. All rights reserved.

ME 390 ? Fluid Mechanics

3

Review for Final Exam

May 6, 2008

Streamlines

? A line everywhere tangent to a the velocity vector is a streamline (s, n) = distance (along, normal to) streamline

Normal direction, n, faces center

Radius of curvature

Figure 3.1, Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald 19 Young, and Theodore Okiishi Copyright ? 2005 by John Wiley & Sons, Inc. All

rights reserved.

Bernoulli Equation

? Limited to steady, inviscid, streamline flow

? Restriction to steady flow comes from assumption that velocity changes in space, but not with time (V/t = 0)

? Have to know -p equation to integrate

? Simplest relation is constant density

( ) For constant

density

g(z2-z1 )+

p2- p1

+

V22

- V12 2

=0

Bernoulli Equation

20

Bernoulli Forms and Dimensions

? Energy ? dimensions of L2/T2

( ) g(z2-z1)+

p2- p1

+

V22

- V12 2

=0

? Head ? dimensions of L

( ) z2-z1 +

p2- p1

+

V22 -V12 2g

=0

( ) ?

Pressure ? dimensions of

(z

2

-

z1

)

+

p

2

-

p1

+

V22 - 2

V12

=0

F/L2 = M/LT2

21

Forces Normal to Streamline

? Similar force balance involving pressure

and weight as forces

? Result is

dz dn

+

p n

+

V 2

=0

is radius of

curvature

? For negligible density p = - V 2 n

? For swirling flows, pressure decreases towards the center

22

Dynamic and Total Pressure

? Bernoulli equation analysis of stagnation point flow shows stagnation pressure, p2 = p1 + V12/2

? Call V2/2 the dynamic pressure ? Call p + V2/2 the stagnation pressure ? Call pressure, p, static pressure to

distinguish this from other pressures ? Total pressure is p + z + V2/2

23

Static and Stagnation Pressure

? Parallel streamlines

? h is static pressure at (1)

? H is stagnation pressure at (2)

Figure 3.4, Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi

Copyright ? 2005 by John Wiley & Sons, Inc.

All rights reserved.

24

ME 390 ? Fluid Mechanics

4

Review for Final Exam

May 6, 2008

Incompressible?

? Accurate results for gas flows can be found from incompressible flow equations provided that the Mach number is less than 0.3

? Ma = 0.3 gives relative error of 0.13% for computation of stagnation pressure

? Results less accurate for Ma > 0.3 ? Catastrophic errors for Ma > 1

? 1145% relative error for Ma = 4 ? Shock wave for Ma > 1 changes equation

25

Continuity Equation

dm

=

dt control

m& in -

m& out =

in AinVin -

out AoutVout

volume

? For steady flow dm/dt = 0

Figure 4.13 Fundamentals of

Fluid Mechanics, 5/E by Bruce

Munson, Donald Young, and

Theodore 2005 by

JOokhinishW2i Ci6leoyp&yriSgohnt s?,

Inc. All rights reserved.

Cavitation

? Change from small to large velocity (V1 4100 to 10,000 is turbulent

? For flat plate flow transition is at about Re = 500,000

39

Laminar vs. Turbulent Flows

? Laminar flows have smooth layers of fluid

? Turbulent flows have fluctuations

Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 38 Theodore Okiishi. Copyright ? 2005 by John Wiley & Sons, Inc. All rights reserved.

Energy Equation Head Loss

? Energy equation between inlet (1) and

outlet (2)

z2

+

p2

+ V22 2g

=

z1 +

p1

+ V12 2g

+ hs

- hL

? Previous applications allowed us to compute the head loss from other data in this equation

? Call this the measured head loss

? We can compute it, but we have no way of knowing its cause

40

Head Loss

? Computed head loss in pipe flows from equation like the following

hL

=

f

l+ D

K

L

V 2

2

g

? Call this the design head loss

? Allows us to determine the major head loss from empirical relations among Re, f, and /D, and minor head loss from emprical KL

? Multiple straight pipes have multiple f values

41

Energy/Head Loss

? Combine energy equation with f and KL

z2

+

p2

+ V22 2g

=

z1 +

p1

+ V12 2g

+ hs

- hL

z2 +

p2

+ V22 2g

= z1 +

p1 + V12 2g

+

hs

-

f

l+ D

K

L

V2 2g

? Top equation: compute hL if you know all other terms in the equation

? Bottom equation: find flow properties accounting for calculated head loss

42

ME 390 ? Fluid Mechanics

7

Review for Final Exam

Energy/Head Loss II

z2 +

p2

+ V22 2g

= z1 +

p1 + V12 2g

+

hs

-

f

l+ D

K

L

V2 2g

? Simplest case: z1 ? z2 =V1 ? V2 = hs = 0

p1 -

p2

=

f

l+ D

K

L

V 2

2

g

=

f

l+ D

K

L

V 2

2

? Often have V2 = V1 so head loss is

p1 + z1 - ( p2

+

z2

)

=

f

l D

+

K

L

V 2

2

43

May 6, 2008

Moody Diagram

Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi. Copyright ? 2005 by John Wiley & Sons, Inc. All rights reserved.

44

Moody Equations

?

Colebrook equation 1

(turbulent)

f

=

-2.0

log10

D 3.7

+

2.51 Re f

? Haaland equation (turbulent)

1 f

-1.8

log10

6.9 Re

+

D 3.7

1.11

? Laminar

f

=

1

p l V 2

=

128lQ

D 4 1 l V 2

=

256 D3

V

4

V 2

D2

=

64 VD

=

64 Re

2D

2D

45

Pipe Flow Calculations

? Not for partially filled pipes ? Find minor loss coefficients from various

tables and charts ? Parabolic laminar velocity profile

? Adjustment coefficients for momentum and kinetic energy flows terms

? Flat turbulent velocity profile

? No adjustment terms required

? Non-circular pipes use Dh = 4A/P

46

More Pipe Flow Calculations

? Wall shear stress related to pressure

drop: p = 2w/R = 4w/D (for horizontal pipe)

?

Entry lengths for

? laminar le = 0.06 D

developing

Re turbulent

flloew=s4.4

D

Re1

6

? e 10D for turbulent flows

? hL varies as D-4 for laminar flows and D-5

for fully turbulent flows with fixed Q

47

ME 390 ? Fluid Mechanics

Head Loss Problems

? Find the pressure drop given fluid data, pipe dimensions, , and flow (volume flow, mass flow, or velocity)

? Get A = D2/4 ? Get V = Q/A or V = m& /A if not given V ? Find and for fluid at given T and P ? Compute Re = VD/ and /D ? Find f from diagram or equation

? Laminar f = 64/Re; Colebrook for turbulent

? Compute hL = f (/D) V2/2g

48

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