120
en: Incompressible flow in a circu nnel and Re = 1800, where D = 10 d: (a) Re = f (Q, D,) (b) Re = f(dm/dt, D,) (c) Re for same Q and D = 6 mm

Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D, ) (c) Re for same Q and

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Page 1: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm.

Find: (a) Re = f (Q, D,) (b) Re = f(dm/dt, D,) (c) Re for same Q and D = 6 mm

Page 2: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Incompressible flow in a circular channel. Re = 1800, where D = 10mm

Find: (a) Re = f (Q, D, ); (b) Re = f(dm/dt, D, );

Equations: Re = DUavg/ = DUavg/ Q = AUavg dm/dt = AUavg A =

D2/4

(a) Re = DUavg/ = DQ/(A) = 4DQ/(D2) = 4Q/(D)

(b)Re = DUavg/ = (dm/dt)D/(A) = (dm/dt)D4/(D2) = 4(dm/dt)/(D)

Page 3: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Incompressible flow in a circular channel. Re = 1800, where D = 10mm

Find: (c) Re for same flow rate and D = 6 mm

(a) Re = DUavg/ = DQ/(A) = 4DQ/(D2) = 4Q/(D)

Q = Re (D)/4

(c) Q1 = Q2

Re1D1/4 = Re2D2/4 Re2 = Re1(D1/D2)

Re2 = 1800 (10 mm / 6 mm) = 3000

Page 4: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 5: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

For most engineering pipe flow systems turbulence occurs around Re = 2300. On a log-log plot of volume flow rate, Q, versus tube diameter, plot lines that cor-respond to Re = 2300 for standard air and water at 15o.

Q = ReD/4 Q = 2300 D/4

Air: = / = 1.46 x 10-5 m2/s at 15oC Table A-10Water: = / = 1.14 x 10-6 m2/s at 15oC Table A-8

Page 6: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Re = 2300

Re = Q/(A) = 4DQ/(D2) = 4Q/(D)Q = ReD/4

Why gethigher Q’s

for same Reand D in airthan water?

air ~ 13 water

Page 7: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 8: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

What is the direction and magnitude (lbf/ft2) of shear stress on pipe wall ???

Page 9: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

(P2-P1 = -P)

Page 10: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Pipe wall exerts a negative shear on the fluid. Consequently the fluid exerts a positive shear

on the wall.

Page 11: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 12: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Given: Fully developed flow between two parallel plates, separated by h. Flow is from left to right.

y = h/2

y = 0

y = h/2

Plot: xy(y)

Net Pressure Force

u(y) = [h2/(2)][dp/dx][(y/h)2 – ¼]

Page 13: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

u(y) = [a2/(2)][dp/dx][(y/a)2 – ¼] Eq. 8.7

(y=0 at centerline & a = h)u(y) = [h2/(2)][dp/dx][(y/h)2 – ¼]

xy = du/dy

xy = [h2/(2)][dp/dx][2y/h2] = [dp/dx][y]

xy = [dp/dx][y] (for y=0 at centerline)

Net Pressure Force

xy = [dp/dx]{y–[a/2]}(for y=0 at bottom plate)

0

h/2

-h/2

Page 14: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

xy(y) = [dp/dx][y] dp/dx < 0

So xy is < 0 for y > 0

And xy is > 0 for y < 0

|Maximum xy| = y(+h/2) and y(-h/2)

u

Shear stress forces

xy

y??? Sign of shearstress and directionof shear stressforces “seem”contradictory???

Page 15: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

xy

xx

xzx

y

z

Page 16: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

sign convention for stress (pg 26): A stress component is positive when the direction of the stress component and the normal to the plane at which it acts are both positive or both negative.

Stresses shown in figure are all positive

Page 17: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

+

+

Shear force

Shear forceShear sign convention

xy

yPlot: xy(y)

Page 18: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

1507 by Leonardo in connection with a hydraulic project in Milan

Page 19: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

D = 6mmL = 25 mmP = 1.5MPa (gage)M = ?, SAE 30 oil at 20o

Q = f(p,a), f =?Velocity of M = 1 mm/mina = ?

Q

Page 20: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

D = 6mmL = 25 mmP = 1.5MPa (gage)M = ?SAE 30 oil at 20o

Velocity of M = 1 mm/mina = ?

M = ?

Fully DevelopedLaminar Flow

Between InfiniteParallel Plates

Page 21: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Q= f(p,a) ?Q pQ a3

l

Q l

Fully Developed Laminar Flow Between Infinite Parallel Plates

l = 2R = D

Page 22: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

D = 6mmL = 25 mmP = 1.5MPa (gage)M = ?SAE 30 oil at 20o

Velocity of M = 1 mm/mina = ?

If V = 1 mm/min, what is a?

v

Q = a3pl/(12L) Q = VA =UavgDa

Page 23: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 24: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

1500

1500

Page 25: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

1500

1500

Re = Va/

Page 26: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 27: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

What is velocity profile, u(y),for a plate moving verticallyat Uo through a liquid bath?

Uo

x

y

Shear Forces on Fluid Element

Shear stress

B.C.zx

y

Page 28: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

What is velocity profile, u(y),for a plate moving verticallyat Uo through a liquid bath?

dy

Uo

x

y

Shear Forces on Fluid Element

-(zx+[dzx/dy][dy/2]dxdz+(zx-[dzx/dy][dy/2]dxdzgdxdydz = 0

-dzx/dy = - g

zx = gy + c1

ASSUME FULLY DEVELOPED:

mg

Page 29: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

What is velocity profile, u(y),for a plate moving verticallyat Uo through a liquid bath?

dy

Uo

x

y

Shear Forces on Fluid Element

zx = gy + c1

zx = du/dyu(y) = gy2/[2] + c1y/ + c2

u(y=0) = U0

So c2 = Uo

ASSUME FULLY DEVELOPED:

mg

Page 30: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

What is velocity profile, u(y),for a plate moving verticallyat Uo through a liquid bath?

dy

Uo

x

y

Shear Forces on Fluid Element

zx = gy + c1

zx = du/dyu(y) = gy2/[2] + c1y/ + Uo

du/dy (y=h) = 0du/dy = zx / = 0 at y=h0 = gh/ + c1/c1 = -gh

ASSUME FULLY DEVELOPED:

mg

Page 31: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

What is velocity profile, u(y),for a plate moving verticallyat Uo through a liquid bath?

dy

Uo

x

y

Shear Forces on Fluid Element

zx = gy + c1

zx = du/dyu(y) = gy2/[2] + c1y/ + c2

ASSUME FULLY DEVELOPED:

mg

c2 = Uo

c1 = -gh

u(y) = gy2/[2] + -ghy/ + Uo

u(y) = g/{y2/2 –hy} + Uo

Page 32: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

velocity profile for water film on vertically moving plate

-2000

-1500

-1000

-500

0

500

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

distance from plate to water surface

velo

city

At y = 0velocity at edge of film 0

but du/dy = 0

Page 33: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 34: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

What is the maximum diameter of a vertical pipe

so that water running down it remains laminar?

D

mgg(D2/4)(dx)

dx

rz

ReD = VD/ = uavgD/

D = ? = 2300()/V

Page 35: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

What is the maximum diameter of a vertical pipe so that water running down it remains laminar?D

mgg(D2/4)(dx)

dx

rz

Page 36: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

What is the maximum diameter of a vertical pipe so that water running down it remains laminar? Assume: Fully Developed

D

g(D2/4)(dx)+ rz2rdx=0

dx

Page 37: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

What is the maximum diameter of a vertical pipe so that water running down it remains laminar? Assume: Fully Developed

D

g(D2/4)(dx)+ rz2rdx=0

dx

For fully developed, laminar, horizontal,pressure driven, Newtonian pipe flow:u(r) = -{(dp/dx) /4}{R2 – r2}

Page 38: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

What is the maximum diameter of a vertical pipe so that water running down it remains laminar? Assume: Fully Developed

D

g(D2/4)(dx)+ rz2rdx=0

dx

Page 39: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

What is the maximum diameter of a vertical pipe so that water running down it remains laminar? Assume: Fully Developed

D

g(D2/4)(dx)+ rz2rdx=0

dx Re = uacgD/ = VD/D = Re /uavg

Re = 2300= 1.0 x 10-6 m2/s at 20oC

D = 1.96 mm for water

Page 40: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 41: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 42: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

uavg/umax = (y/R)1/n

Not accurate aty=R and y near 0

Page 43: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 44: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 45: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 46: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

u / Umax = (y/R)1/n

Page 47: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

from Hinze –Turbulence, McGraw Hill, 1975

n = 1.85 log10ReUmax –1.96

Page 48: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Turbulent Flow

0.720.740.760.780.80.820.840.860.88

1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07

Re=UmaxD/nu n uavg/umax4000 4.703811 0.74544120000 5.996905 0.79111950000 6.733095 0.810498100000 7.29 0.82293110000 7.366576 0.824514200000 7.846905 0.833835500000 8.583095 0.8463461000000 9.14 0.8546291100000 9.216576 0.8556982000000 9.696905 0.8620653200000 10.07453 0.866689

n = 1.85 log10(ReUmax) –1.96

Uavg/Umax = 2n2/((n+1)(2n+1))

ReUmax

Uavg/Umax

For F.D. laminar flow uavg = ½ umax

Page 49: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Turbulent Flow

0.720.740.760.780.80.820.840.860.88

1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07

Uavg/Umax

ReUmax

Page 50: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Uavg/Umax = (1 – r/R)1/n = (y/R)1/2; n = f(Re)

Power Law Velocity Profiles for Fully-Developed Flow in a Smooth Pipeu/U = Uavg/Umax

close to wall

Page 51: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Eq.1: u/u* = yu*/; u*=(wall/)1/2;

note: wall = du/dy ~ (u/y) = u*2 u/u* = yu*/

Eq.5: u/u* = 2.5ln(yu*/) + 5.5 or u/u* = 5.75l0g(yu*/) + 5.5

Page 52: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 53: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Consider fully developed laminar pipe flow.

Evaluate the kinetic energy coefficient, .A(u2/2)udA = A (u2/2)VdA

= (dm/dt) V2/2u = u(r) = V = V(r); uavg = V

= A u3dA / ((dm/dt) V2)

Question: What is for inviscid flow?

Page 54: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Eq. 8.14

Eq. 8.13e

Page 55: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

3

Page 56: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 57: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 58: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Fully developed turbulent pipe flow.

Evaluate the kinetic energy coefficient, .A(u2/2)udA = A (u2/2)VdA

= (dm/dt) V2/2u = u(r) = V = V(r); uavg = V

= A u3dA / ((dm/dt) V2)

Question: What is for inviscid flow?

Page 59: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 60: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 61: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

= f(n)

Page 62: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 63: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07

Re = UmaxD/

Velocity profile getting flatterIf no viscosity completely flat

and = 1

Page 64: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 65: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

u(y)/Umax

y = (R-r)

Page 66: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.00E+00 1.00E+06 2.00E+06 3.00E+06 4.00E+06 5.00E+06 6.00E+06

Re = UmaxD/

Uavg

/Umax

Page 67: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 68: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Given: Smooth pipe, fully developed turbulent flow, Avg velocity = 1.5 m/s, diameter = 50mm, Re = 75,000,p1=590 kPa (gage), z1 = 0, p2 = atmosphere, z2 = 25m

Find: Head loss between 1 and 2

25 m(1) (2)

Dimensions of L2/t2 (energy per unit mass)

Page 69: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Given: Average velocity = 1.5 m/s, Diameter = 50mm, Re = 75,000, p1=590 kPa (gage), z1 = 0, p2 = atmosphere, z2 = 25m

25 m(1) (2)

Page 70: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 71: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

[2-3]

[3-4]

Page 72: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Assumptions: Incompressible 2uavg2

2 = 3uavg32

z2 = z3

energy/mass

energy/weight

PUMP HEAD [ 2 – 3 ]

Page 73: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Assumptions: Incompressible 3uavg3

2 = 4uavg42

p4 = atm; z3 = 0

PIPE LOSSES [ 3 – 4 ]

(p3/ + 3Vavg2/2 + gz3) - (p4/ + 4Vavg

2/2 + gz4) = hLT hLT = hl + hlm = (fL/D + K)Vavg

2/2

(p3/ + 3Vavg2 + gz3) - (p4/ + 4Vavg

2 + gz4) = hLT

hLT = p3/ –gz4 = 50 (lbf/in2)(ft3/1.94 slug)(144 in2/ft) – 32.2(ft/sec2)90(ft)(lbf-sec2/slug-ft)

hLT = 813 lbf-ft/slug

0 0

or H = hLT/g = 25.2 ft

Page 74: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

From a History of Aerodynamics by John Anderson

Page 75: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

REMEMBER ~

Page 76: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Fully Developed turbulent Flow4000 < Re < 105; f = 0.316/Re1/4

Page 77: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Boundary Layer Theory – Schlicting, 1979

This = our fDarcy

= 64/ReD

= 0.3164/ReD1/4

1/1/2 = 2.0 log(ReD 1/1/2) – 0.8

Page 78: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Leonardoda Vinci

1452-1519

Page 79: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 80: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

= 16/Re

Re = UavgR /

Be careful that you know if using Darcy or Fanning friction factor and if Re is bases on D or R

Page 81: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

From a History of Aerodynamics by John Anderson

Page 82: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

D1 =50mmD2 = 25mmp1-p2=3.4kPa

Q = ?

(p1/ + Vavg12/2 + gz1) = (p2/ + Vavg2

2/2 + gz2) + hLT

hLT = hl + KVavg22/2

Vavg1 = Vavg2(A2/A1) = Vavg2AR

p1/ + Vavg22AR2 /2 = p2/ + Vavg2

2/2 + KVavg22/2

0

Page 83: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

D1 =50mmD2 = 25mmp1-p2=3.4kPa Q = ?

AR = ¼

K = 0.4

Page 84: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

p1/ + Vavg22AR2 /2 = p2/ + Vavg2

2/2 + KVavg22/2

(p1 – p2) / = (Vavg22 /2)(1 – 0.0625 + 0.4)

D1 =50mm; D2 = 25mm; p1-p2=3.4kPa; = 999 kg/m3

Q = ?

Page 85: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 86: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 87: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

K = 0.78; Table 8.23ft

Page 88: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 89: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

z1 = 3 ft

Page 90: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Could be greatly improved by rounding entrance and

applying a diffuser.

(about 30% increase in Q)

Page 91: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 92: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

= ?

Page 93: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 94: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 95: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Eq. 8.43

Page 96: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

N/R1 = 0.45/(.15/2) = 6

Cp 0.62

AR 2.7

Pressure drop fixed, want to max Cp to get max V2

Page 97: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 98: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

The end

Page 99: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 100: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 101: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 102: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

Given: Laminar, fully developed flow between parallel plates

= 0.5 N-sec/m2; dp/dx = -1200 N/m3

Distance between the plates, h = 3mm

Find: (a) the shear stress, yx, on the upper plate(b) Volume flow rate, Q, per unit width, l.

(a) yx = du/dy

(b) Q = u(y)dA = u(y)ldy

y = 0 at centerline

Page 103: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

(a)

Shear stress on plate = 1.8 N/m2?

Page 104: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

(a)

Page 105: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

(b)

Page 106: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 107: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 108: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 109: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 110: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 111: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

8.63OLD Consider fully developed laminar flow of water between infinite plates. The maximum flow speed, plate spacing, and width are 6 m/s, 0.2 mm, and 30mm respectively. Evaluate the kinetic energy coefficient, .

= 999 kg/m3

= 1 x 10-6 m2/s

Page 112: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

8.63

Page 113: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

8.63

Page 114: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

8.63

Page 115: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 116: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and
Page 117: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

8.63OLD Consider fully developed laminar flow betweeninfinite plates.

Evaluate the kinetic energy coefficient, .

Page 118: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

8.63

1/2

1/2

Page 119: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and

8.63

Page 120: Given: Incompressible flow in a circular channel and Re = 1800, where D = 10 mm. Find: (a) Re = f (Q, D, ) (b) Re = f(dm/dt, D,  ) (c) Re for same Q and