Instructions:

UNIVERSITY OF TORONTO

FACULTY OF APPLIED SCIENCE AND ENGINEERING

FINAL EXAMINATION: December 14, 2017 DURATION: 2 and½ hrs (9:30 pm - 12:00 pm), SF3202

CME 270 - FLUID MECHANICS

Calculator Type: 2 (all non-programmable electronic calculators)

Exam Type: A (closed book)

Examiner: Ron Hofmann

1. Your answers must go in the answer booklets provided. 2. If you don't understand a question, write down any assumptions that you make. The

professor/invigilator will not answer any questions during the exam to ensure fairness among all students.

3. This is a closed book exam. A formula sheet, aid tables and figures, and a Moody Diagram are provided at the back of the question booklet.

4. When appropriate, draw a sketch of the system in your answer booklet that is labeled in such a way as to clearly explain your answer to avoid any ambiguity that might result in loss of marks.

5. Rulers are an allowable aid. 6. Use the Moody diagram (provided) to determine when flows are laminar or turbulent. (i.e. Re

greater than about 4000 means turbulent, and less than about 2000 is laminar)

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Question 1: (4 marks)

For each of the following statements, indicate whether they are true or false (remember to put the answers in the answer booklet):

(a) Streamlines, pathlines, and streaklines are identical under steady flow. (b) At every point in the flow field, a streamline is perpendkular to the velocity vector. (c) Skywriters use small airplanes equipped with smoke machines to create patterns and messages

visible from the ground. If we think of the airplane as a single particle, the pattern in the sky

would represent a pathline.

(d) The smoke trace seen around an active chimney on a windy day is an example of a streakline.

Question 2: (5 marks)

A horizontal water jet (p = 1000 kg/m3) with a velocity of 10 m/s impinges normally on a vertical flat

plate and splashes off the sides in the vertical plane. The plate 's moving toward the oncoming water jet with a velocity of 5 m/s. If a force of lON is required to maintai, the plate stationary (i.e. before the piate starts to move), how much force is required to move the plate toward the water jet at 5 m/s? Assume that the jet is uniform, and that the cart experiences negligible friction.

- 1--------, ''

10 mis -. ,,-2

1 ·#ikl~t =J=-~========= " '' Water jet ;j

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.)

Question 3: (9 marks)

In the pipe system shown below, water in a large aquarium is circulated steadily through a valve and a

filter using a pump (marked "P"), in a clockwise direction on the diagram. The aquarium has a constant

water depth of 10 m above the pipe entrance while water is circulating. The pump, located 10 m below

the valve, supplies a constant head of SO m. All pipes have a roughness, E, of 0.01 mm and are 10 cm in

diameter. The total length of all piping is 100 m. Several individual pipe lengths and dimensions are

given in the figure below, however these are included in the 100 m of total piping. Water enters the pipe

at a valve, with a loss coefficient of 1.5. There is one elbow joint, which has a loss coefficient of 0.5. The

filter has a loss coefficient of 16. The water hasp= 1000 kg/m3, µ = 1.0 x 10-3 Ns/m2•

(a) What is the flow rate through the pipes (m3/s)?

(b) Draw the EGL/HGL from the Aquarium (A) to just after the filter, in the direction of flow. Label

the drawing with values for the total and piezometric heads just before and after the valve, the

pump, and the filter. Set the elevation datum (i=O) at the pump elevation.

Aquarium

E 0 T""

{ ~~ '---~ ~---~~ ~ T""

~~0 /p·----1 ~-~/l~-1 - )

' ..____ _______ s_m __ 1_o_m ____ _

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Question 4: (6 marks)

A jet of fluid having a specific gravity of 0.79 strikes a vertical plate as shown in the figure below. A force

of 400 N is required to hold the plate stationary. Assuming that there are no losses in the nozzle, and

that the density of water is 998 kg/m3, determine:

a) The mass flow rate of the fluid through the nozzle.

b) The absolute pressure at point (1), P1.

Question 5: (2 marks)

Pl

l :

!! ' ..

F=400 N ..,_

A siphon is shown in the figure below. Siphons are convenient devices used to remove fluid from a

container entirely by gravity. Assume that the cross-sectional area of the tube is constant, and that the

tank is large enough so that the fluid velocity anywhere a little. bit away from the tube's inlet is

essentially zero. Is the velocity at the outlet of the tube, V4, a function of the density of the fluid? Justify

your answer with mathematical proof.

Z1= 50 cm !

Zi= 30 ~; I

Zo= 0 en_, ___________ ~

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Question 6: (4 marks)

A prototype (model) train is doing a wind tunnel test. The model train has a frontal cross-sectional area

of 0.4 m2 and a width of 0.5 m. The characteristic area that controls drag force is the frontal area. The

Reynolds number is controlled by the width (i.e. Re = pvD where D = 0.5 m). Test data is shown in the µ

table below (pa;,= 1.22 kg/m3, µa;,= 2.2x10·5 Ns/m2 ):

Air Speed (m/s) 6 8.5 11.5 15 19

Drag Force (N) 6.15 9.52 15.48 27.4 44.0

a) Plot a simple graph of drag coefficient (Co) on the Y axis, versus Reynolds number (Re) on the X axis. Make it a linear-linear graph for simplicity (not log-linear).

b) Estimate the drag force for the prototype train at a speed of 80 km/hr.

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Question 7: (S marks)

Fluid A and a slurry mixture are separated by a hinged gate, which is designed to open and flush away the

slurry once it lowers to a certain depth. A = 3.2 m, B = 0.5 m, W = 1.0 m, Ywate,= 9810 N/m3, SGA = 1.00,

SGs1urrv = 1.50. At what depth (H) of slurry will the gate open?

Helpful tip: Since you will be taking moments about the hinge, and both force and distance will be a function of H, you may end up with a polynomial equation that needs to be solved for H. It is acceptable to make a little table of "guess values" for H at increments of, say, 0.25 m, and then interpolate to find the solution to within an acceptable margin of error.

. :c· .. . . . .

Question 8: (3 marks)

Hinge

Fluid A w

Gate: Side View

A pneumatic piston with mass, mp, can slide up and down in a 2m diameter cylinder that contains a gas.

There is a manometer pipe that extends from the cylinder as shown, and contains water (y = 9810

N/m3), shown by the hatched markings, and it is open to the atmosphere at the end. The water level on

the right-hand side of the "U shape" of the pipe is higher than the level on the left-hand side, extending

a distance L, as shown. The angle ofthe pipe is 9=30°.

Find the mass of the piston, mp, Assume that the specific weight of the gas is negligible.

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z 0 r+ (I)

"C ii (I) -, 0 C

ao ~ ::::, (I) Vl Vl

-h OJ n r+ 0 ,-.

,M

iii n OJ

ro 0. ~

:?-" ~ (I) -, ,ro OJ ::::, 0.

~ ;;;· r+ ~ (I)

0 OJ -, n -<

I

:;E (I)

Vl 0-OJ n ~ -+, -, ;::;· r+ ;5 ::::, n 0 (I)

=+: ;::; · ro· ::::,

-....J ~ 0 -h

I-' N

<.....

c .I!! -~ ...... ...... Q) 0 u Q) u C

2 -~ (/) Q)

a::

0.100

0.090

0.080

0.070

0.060

0.050

0.040

0.030

0.025

0.020

0.015

0.010

0.009

0.008

\ , '\ ,

" I I ' \ \I \ \ ~ 1 ' , 1 1 1 \ -, ' Ful ly rough turbulent flow

, I\.

.... I'--- ·, , , 1 , ..... \ .....

' \. -

~ ....

~I'-- ' ..__ >- -..... .... ' .....

..... ~ .... ...... i-... .. _ -~

.. ........ - ~ .... -- ~ '· ..... 'I'-,~ :-,., ~

..... ~t::::: ~ ::::-- - '~ - 6) ~ ... ~t---.:::r--- - I' , .....

~ ~~~ - ~ ~ - , .. .... /= Re , ~~r-,..... ~,.. - •

~;~ '" - \ ~~ -- \ ~ ...... __ I',

- . ... "' ~ --- . - ....... .......... ' - ~ ', - ~ ~::::: ~~::::---- ... -

'" ---~~ .... -- ' , ........... ~ ...... . ---. .... ...

~~ r,,. ~-~ .... r----. ,,

.:::I."" ......... -.! ' ..... .1'~~r--.- , ....

Smooth pipes "" ~~

- ~F==:::::l:'.---, I I I I I I I I I I I I I I I I I I I I I I I I I "'1-J."" ~ti- I

103 2 4 6 8 104 2 4 6 8 105 2 4 6 8 106 2 4 6 8 107 2

Re= VD V

0.05

0.04

0.03

0.02

0.015

0.01 0.008 -.:.e" IQ

0.006

0.004

0.002

0.001 0.0008 0.0006 0.0004

0.0002

0.0001

0.00005 .....

'i... \ \ 0.0000 1

4 6 8108

(/) (/) Q) C .c 00 :J

e Q)

.2: ..... ro Q)

a::

~

'

A-10...,_ ____________________________ ~A~PP=E .... N .... DC'\ .... •

TabkA.5 APPROXJMAtE PHYSICAL PROPERTIF.S OF WATER• AT ATMOSPHERIC PRESSURE

Temperature Density SpedOc Weight Dynamic: Viscosity Kinematic Viscosity Vapor Pressure

ks/mi N/m3 . N ••11»1 '· oa1/s ,: N/m1 abs

O"C 1000 9810 l.79x Hr3 ·l.79 x JO-' 611

5'C 1000 9810 I .St x Jcr,l I.SJ x 10'"' 872 l.31 ~ 10-3 1.JJ X 10~

' ,- . ·- -- --lO"C 1000 9810 1,230 isic; ' 99~ 9800 .l.14 >< i<rl 1.I4x 101 1,700 ,.

l.OOx 10-3 ., ·1.00 )( to-6 20"C 998 9790 2,340

2S°C m 9781 8.91 X Jo-4 8.94>< ur7 3,170

30"C 996 9771 7.97 X JO◄ 8.00 x 10-7 4,250

35-C 994 97S1 7.20x Jo-' 7.24x 10-7 S,630 . ' "

6.58 ~ 10-1 40"C 992 9732 6.53 X 10 4 7,380

so-c 988 9693 S.47 x 10~ S.S3 >< 10-7 12,300 60°C 983 9643 4.66x ta-4 4.74 X }0"7 20,000

'

4.04 x. io-' 4.l3x 10-7 70'C 978 9594 31,200

80"C 972 9535 3.54 X 10-4 3.64 X 10"7 47,400 . •

'9467: 3.ts x 10 ... .3.26>< 10-1 90"C: 96$ 70,100

lOO"C 958 9398 2.82 x llr°' 2.94 X t0•7 101,300 llllp/ft3. lbflft' . lbf-tlft1 fr/1 psia

40"F 1.94 62.43 3.23 X }0-S l.66x ur5 0.122

507 1,94 62.40 :Z.73~ Hi; J.41 x 10-s 0.178 ' , . .:, ~~-"

:Z:36 ~1~· 60°F 1.94 62.37 1.22 X 10' S 0.2S6 2.0Sx 10-S 1.06 x I 0-5

.. ,

0.363 700f J.94 62j0

80"F 1.93 62.22 l.80x 10-s 0.930 x 10-5 0.506 100°F 1.93 62.00 l.42x·10-s 0.739x urs 0.949 '.

·1:17 x 1Cf5 . 0.609 X 10-5 .

120"F 1.92 61.72 1.69

~~!1Wf~-~· 1.91 . 61.38 0.98t"St l~ . 0.5(4:)( 104 2.89 . -~ .. r· ' .. 0.838 X tiS t60"F 1.90 61.00 0.442 X 10-S 4.74

1.88 60.58 0.726x10-5 0.385 x 10-' 7.'1 200"F l.87 60.12 0.637 )( 10-5 Oj41 x 10-s 11.53

lfi1~•J'tJ . .:i,S~.~*.sl'-t!!~-........ ~.;I,_1, 1.86 · $9.83 · o.s93 x io"' 'o.lJ9x 10-s 14.70

• Notes:{l)Bulkmodul\lsE.ofwa1crisapproximatcly2.2GPa{J.2 X lo'psi);(2}wala-ilirsurfacetensionisapproxlmately7.3 x ur1N/m(5 x lo-'lbf/1> from IIJC to 50"C. SOIJIUl!: Reprinted with permiaion from R. E. BolJ and G L. Tuve, Handbook of Tabla for App/kd Enginffr/JfR Sc/mt,:~. CRC Press, Inc., <.:lewlaL 1973. a,pyrig!d O 1973 by The Chemical Rubber Co., CRC PRss, Inc.

8 of 12

=-=AP=P:....,E=N....:.:D=I=X,..__ ______ _ ________ _____ ____ _ __________ A-7

cP

J s KJnematic: R, kgK utherland's

Densl!J VI c:orltv Gas Consta.nt C Con tant kg/ m m2/ s · J/ kgK ( Btu ) k = ..!. K

Gas (slugs/W) (Wi s) (ft-lbf/ slug-0 R) lbm-0 R c. {°R)

1004 ' ' J:40 111 (0.240) (1 9)

Carbon dioxide 1.30 222 (400)

... Helium

Hydr g 0 .0851 J.01 X lo-' 4127 14,223 1.41 96.7 (0.00017) (1.()() X 10-3) (24,677) ( .40) (174)

/,';~ ~ •fL _,'1"""'...,,._ ,i I ' ; .

2208 .

M~e(nat'U11il ~) · 0.618 1.59 X 10"1 . 198 . ~ (0.00t3) (l.72 X 10-') (0.S28 ( 6)

Nitrogen 1.18 1.40 107 (192)

!ilOURi : V. L. treeter (ed.), Handbook of Fluid Dynamics, McGraw-Hill Book Company, New York, 1961; also R. E. Bolz and G L. Tuvc, Handbook of1ables Ju, Applied Engin ring Science, CRC Press, Joe. evcland. 1973; and Handbo<Jk of Chemistry and I'hysic.3, hemicaJ Rubber Company, 195 l .

N .--t '+-0 en

=--=AP~P:...;:END==--=IX=--=---______________________ _ __________ _ A -9

Liquid and Temperature

~ -akobol<'lCJ) /' ~(68°F) 1-M<. '

Carbon tetrachloride()> 20°C (68°P)

}

Kerosene< ll(ll

20°c (68°F) ~ . ~ -fj~(~~~ Sea water 10°c

at 3.3% salinity

~~1)00°F)

SAE I OW-3()C'•l

Density

kg / ml

(slugs / fl")

799 (US)

1,590 (3 .09)

814 (1 .58)

1,026 (1.99)

870 ( f.69)

880 ( 1.71)

, 38() . ( t.71)

"'Liquid-air surface teruiion values.

Spedftc Grulty

0.79

1.59

0.81

1.03

0.88

Spttlflc Weight

N/ nr

()bf/rt') (

7,8ffl (50.0) '

lS,600 (99.5)

10,070

8,630 (55.1)

Dyumic KlnematJc ViJJcoslty Viscosity

N • s / m2 m1/ s

(lbf-s / ft1) (Wis)

l:2 X tu-3 l.5x to-' (2.5 x u,5) ( 1.6 X !(,5)

9.6 X lo-' 6.0 X 10-7

(2.0 X lo-~) (6.S x 10-6) -· t.41

(2.95 X l !r)

l.9 X 10-(4.0 X 10-S) .

s IS X 10-3 ' •

(3.1 )( to·'r

,: , .. ,, 1.12 X 10-3

(1.22 )( Hr2)

2.37 X 10-6

(2.55 x I o-5)

1.2 X 10-7

~ · ( 1.3 X 10-,

1.4 X )~

(1.5 X 10 ... 5) ,,

7.6 X 10-5

(8.2 X Jo-') ·"'""''"""..,_,,_, ·-'""''!"r~,.. ,,_,,,,,,,,,.

Surface Tension

Ni m*

()bf/ft)

2.9 X 10-2

(2.0 X J(,3)

tOIJ!lc~: (1) V. L. Stn:eler, Handbool of Fluid DynamiC$, McGraw-Hill, Ne York. 1961 : (2) V. L. Sttccter, FTu;d Mecl,anics, 4th ed., McGrnw-Hill, New York, 1966; (3) A. A. Newman, Glycerol, CRC Press Cleveland. 1968: (4) R. E. Bolz and O L. Tuve. Handbook of Tables for Applied Engineering Sdencas, CRC Press, Cleveland, 1973.

N .-i .... 0

0 .-i

,I

EQUATION PAGES -b+✓b 2 -4ac

Quadratic equation: given ax2 + bx+ c = 0, X = ------2a

p

F P=­

A

PV = mRT, 273K = 0°C

PV = nRT

Gas constants:

Runiversal = 8.314 J/{mol·K)

Rair = 287 J/(kg-K)

Rhelium = 2,077 J/{kg·K)

- + Z = constant for static fluid y

Volume of cylinder = nr2 x height

Area of circle = rrr2

Volume of sphere= {4/3) rrr3

P1 a1 vf P2 a2vf - + Z1 +-- = - + Z2 +--+ h1 + ht - hp y 2g y 2g

a = 1 for turbulent, 2 for laminar flow

Archimedes' principle: F8 = yV

m = pAv = £ pv • dA

d f- f- - -L F acting on CY = - V ref pdV + vref pv cs . d A dt CV cs

L F aclingonjlow = I vref,011/ pvcs,011/A- L vref,in pvcs,in A

F = mv = (pv 2 A under some conditions)

P1A 1 V1 = P 2 A2V2

p =yZ

dv r=µ­

dy

F = f PdA

F = PA

- I ydA y =-­

A

- + fxx Yep= Y yA

- - lxy Xcp - X + Ay

_ bh3

I xx = 12 for a rectangle of base b, height h 4

l xx = ~fora circle of radius r 4

0.25 f = 2 for turbulent

[ ( E 5.74)]

log10 3.7D + Reo.9

64 f = - for laminar

Re

R VD pVD f . 1 . d e = - = - or c1rcu ar pipes, an Re > V µ

2000 is turbulent in round pipes

L L v 2

h1 = f--+ . D2g

pipes components

p P Qh d

output = y an 11=

Pinput

v2 K-

2g

11 of 12

(4r 4r) Centroid of a quarter circle:. (x, y) = - , - where the origin is at the center of the circle.

3rr 3rr

3 Centroid of half a sphere is - R away from the centre (see below).

. 8

/----, '

/ H-R \

Centroid of a right triangle is 1/3 distance along either edge away from the 90 degree angle.

Drag force: F0 = ~ C0 pv 2 A where A=cross-sectional area for calculating Fo for a sphere 2

24 ( S o 687) 0.42 Drag coefficient of a sphere= C0 = - 1 + 0.1 Re · + 4 1 Re 1+4.25X10 Re- ·

,:,)

100

80 ~o

40

2U

11)

8 t,

4

2

1.0 0.8 ll.6

ll.4

- \ I I ... \ I I

\ Disk ,-.

Vo it~ ... ' _,, "r,(jJ? ')' I ::...::..:.----,

~ C .1:!.-I\ \ Vo ~)~ e " Re _ .,, e:, ?('

\ I -'- <> 6 G

\ I \ Ct1p

- ' \ ,\ l'o 1 j --" c~.,c;~i~

\ \ , I --

~ v---,.. r,...,_ \ I :- Sphere _:_---::

"" =-~10Eli .- ,,.._ )

0.2 ...

0.1 0,08 (1.06

"\. I

' .. ). -..../ ~ '\

- body ,1 , -0.04

$orcumlincd (!!. s) ~-'- /

~ '1~ . -,-

~- I. =---.i o.oa

O.ol w-a 10~1

I 10

I t

"rJI Heynold, number. Ke• --;;-

I I 11 I > I I 111

107

12 of 12