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Department of Metallurgical Engineering and Materials
Science
IIT BombayFormulas/Charts for Transport Phenomena Course
Newton’s law of Viscosity
τ yx = −µdvx
dy (1)
Mechanical Energy Balance
1
2
v22β 2− 1
2
v21β 1
+ g(z2 − z1) +
2
1
1
ρ
dP + Ŵ +
Ê f = 0 (2)
ˆE f = 2
L
Dv
2
f F (3)
Fanning friction factor chart for pipe flow [1]
Figure 1: Fanning friction factor chart for pipe flow [1]
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Expressions: Fanning friction factor for pipe flow
Laminar Flow (approximately upto Re 2000)
f F = 16
ReD(4)
Turbulent Flow
Colebrook correlation
1√ f F
= −4log
1
3.7
D +
1.255
ReD√ f F
Turbulent flow, for ReD > 4000
(5)
Churcill Correlation
1√ f F
= −4log
0.27
D +
7
ReD
0.9 Turbulent flow, for ReD
> 4000(6)
Blasius equation for hydraulically smooth pipes
f F = 0.0791
Re0.25DTurbulent flow, Smooth Pipes, for ReD
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Equivalent Pipe Length to Diameter Ratio for some Common Pipe
Fittings
(Turbulent Flow) [2]Pipe Fitting LeD
Globe Valve, wide open ∼ 300Angle Valve, wide open
∼170Gate Valve, wide open ∼ 7
3/4 open ∼ 401/2 open ∼ 2001/4 open ∼
900
90o Elbow, standard 30
long radius 20
45o Elbow, standard 15
Tee, used as elbow, entering the stem 90
Tee, used as elbow, entering one of two side arms 60
Tee, straight through 20180o close return bend 75
Ordinary entrance (Pipe flush with the wall of the vessel)
16
Borda entrance (Pipe protruding into vessel) 30
Rounded entrance, union, coupling Negligible
Sudden enlargement from d to D
Turbulent flow in d 14f F,ind
1− d2
D2
2Sudden contraction from D to d
Turbulent flow in d 110f F,ind
1.25− d2
D2
Flow Through Packed Bed
∆P
L =
150µ(1− ε)2D2 pε
3 v0 +
1.75ρ(1− ε)ε3D p
v02
D p = φ ×DsphDsph = Diameter of
sphere having the same volume as that of the particle
φ = Surface area of the sphere having equivalent
volume of the particle
Surface area of the particle
Fluidized Bed
∆PA = AL(1− εmf )(ρs − ρ)g
= AL
150.0µ(1− εmf )2
D2 pε3
mf
vmf + 1.75ρ(1− εmf )
ε3mf D pvmf
2
At minimum fluidization, voidage in the bed is given by
1
φε3mf ≈ 14 or 1− εmf
φ2ε3mf ≈ 11
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Dsphvmf ρµ
=
33.72 + 0.0408D3
sphρ(ρs − ρ)gµ2
0.5 − 33.7Voidage in the bed beyond minimum fluidization is
given by
v0 = εm
where the exponent, m is obtained from a plot
of ReDp vs. m and is given in
Figure 3
[3].
Figure 2: Exponent m in correlation for bed voidage
in particulate fluidized bed [3]
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Flow Pass a particle
beginequation* FDrag = A 12ρv
2
∞f
Figure 3: Fanning friction factor for flow pass sphere
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References
[1] R. Perry and D. Green, Chemical Engineers
Handbook . McGraw-Hill, New York,USA, 8th ed., 2008.
[2] O. Levenspiel, Engineering Flow and Heat Exchange.
Plenum Press, New York,
USA, 1984.
[3] M. Leva, Fluidization. McGraw-Hill, New York, USA,
1959.
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