CALCULTION 0F'FRICTION FACTORS
by
Donald C. Spencer
Submitted in Partial Fulfillment of the Requirements for
the Bachelor of Science Degree in General Engineering.
f rom the
MIassachusetts Institute of Technology
1936
( Signature
Acceptance:
Instructor in Charge of Thesis
I -1
,IC
I y 9, 1956
- 'rIi,
av.1A
L,
4V,
)
Riverbank Court Hotel
IT~ass.
Cambridge,
'..Iy 19, 1936
Professor
George ,. Swett
of the Faculty
Secretary
Institute of Techlolosr
;Massachusetts
Cambridgee,
Tass.
Dear Sir:
I respectfully submit this thesis entitled
"Calculation of Friction Factors" for consideration
as
rartial fulfillment
of the requirements
for the
'N
of Science
in -eneral Engineering.
t,
deg-ree of Bachelor
v
I
I hope that it satisfies the requirements in this
KI,
V.
regard.
Yours respectfully,
-
-
v-
A
(
Donald C. Spencer.
Acknowledgement is gratefully made
to Dr. H. Peters for his supervision of this
thesis and for his many suggestions, and to
Dr. R. H. Cameron for suggesting a method of
evaluating the definite integral occurring in
the solution of the problem of the elliptical
pipe.
TABLE OF CONTENTS
PACE
I
Introduction ..............................
II
Smooth Circular Pipes .....................
II
Rough Circular Pipes .......................
9
IV
Parallel Plates............................
11
V
Elliptical Pipes of Small Eccentricity.. ...
15
VI
Conclusions ..............................
22
1
TA3ULATION OF S
0OLS
c = temporal mean velocity at any point in a circular pipe.
c = mean velocity over the pipe section.
~1 = thickness
of
the laminar
).
(e =
e = effectivity
sublayer.
f = friction factor for pines (f =
D
2)
h = distance between the parallel plates.
K = constant of the turbulent exchange (K = 0.40 approx.).
absolute roughness height.
k
(>= a (h) 1
plates
= friction factor for parallel
1
mixing length.
= dynamic coefficient of viscosity.
= kinematic coefficient of viscosity.
distance to any point from the center of the pipe.
r=
R = pipe radius.
= density.
Rey = Reynolds Number.
shearingstress.
T=
o
= shearing stress at the wall.
u = temporal mean velocity in direction of flow.
u' = component
in the direction
of flow of the difference
between u and the velocity at any instant of time.
v' = component
normal to the flowr of the difference
between u and the velocity at any instant of time.
Yo
=
distance from the wall of a rough pipe to points at
which the mean velocity is zero.
Note - Figures in parentheses designate references
listed in the appendix.
The purpose of this investigation is to illustrate
a met-loc for the calculation of friction
.e sipler
methods.
which is thouCrihtto
nd more idely applicable than the existing
Von Karman's similarity
flow pattern
concept of the turbulent
(1) has produced many satisfactory
However, the logarithmic velocity distribution
results.
developed
from this concept is not valid, for example, in the center
of a pipe line or at the outer part of the boundary layer of
a flat
plate.
Here the correlation
velocity components is not perfect.
between the u and v'
This equation would
require that 1 become equal to zero since
T =1l2 (u)2
and the slope does not vanish for y finite.
The mixing
length is zero, of course, only at the wall.
It has been pointed out by H. Peters and C. G.
Rossby (2) tat
the solution of many problems involving
boundary layer theory might be simplified by the assumption
of a maximum value for the mixing length.
Nikuradse's
measurements in pipe lines indicate a maximum value for 1
of 0.07 of the diameter, a value varying sliglhtlywith the
Reynold's
Number.
Tollmien obtained the same constant in
the mixing zone of a to
b te
dimensional jet with 1 = 0068b,
vridth of the mixing zone.
The value
for the circularjet with d the diameter.
the result 1 = 0.065,
as 0.073
Rossby obtained
for the atmosphere, where H is the
-2tthickness of the layer of variable stress.
This constant
therefore appears to have considerable significance.
A few simple problems have been solved by makning
use of this constant in an attempt to establish the validity of this assumption.
Near the wall the stress was
assumed constant with linear increase of the mixing length
with normal distance from the wall.
Further out, the
mixing length was given the constant value 1 = cd, where c =
0.07 and d is some length dimension
of the flow; the stress
was assumed to diminish rapidly in this region.
At the
boundary of separation between these two layers continuity
in velocity, stress, and mixing length was satisfied.
velocity gradient is zero at the center line.
;··
:I
-rq
:
'*'
`r·
·fi
J
-ii
.7
j
a
The
SmoothCircular Pipes.-
In the pure lninar
distribution
layer next to the pine wall, the velocity
subcan
be obtained from the equation of Navier-Stokes in cylindrical coordinates:
1 ~Cz
2cz
~1 ) Cz
r
-7
)'c
12
'%,~2
r" t'P
Dc
d
d
dZ2d
The flow is assumed to be radially symmetrical, hence,
I
z-
=
0
r
+
+=
Al so,
II
r
Equation of Continuity
= O and c r =0O
Since c
C
_
dZ
0
=
Therefore, for steady flow,
(c dt
((+
rr +
Hence,
2
L A dr-
dc
rdrzd'
1
or,
dr) r
z
r
d2 c z
dr2
dr
L
dC
+ drZ
-a-P Z cZ)
C
=
z
=
-4-
do rdr
dz AK
dc
- dr
Since
-t dc
rrdr
dz ,k
=
d(r
Hence, integrating from 0 to R,
o
dz 2
Also, the velocity is given by
cdz
dp
Since
F
4
the fluid
+Kllnr +
A(
clings
wall must be zero.
velocity
to the pipe wall, the velocity
Also, from symmetry considerations, the
is a maximum
at the center of the pipe.
Therefore,
putting
c = O when r = R
dc
dc = 0
hen r = 0
the velocity equation becomes
c c. -
1
R2
(P-
r2
LettingR - y = r
I
=_d (y2_ Ry)
4 dz
Neglecting the squared term and substituting for
terms ofT
I
I
Ii
Ia
I
O'
at the
in
-5-
(I)
C
=
r"11.5
The value y -=
.5/ has been fix ed by
!
(-)
ex-erlment by Nikuradse as a reasonable limit for the region
influenced by lamxinar friction.
There
is theoretical
that
derivation,
a simple
ustification, as is shown by
the stress varies as given by the
equation
See, Append'x
r
Takringthis variation into account and assurminglinear increase of the mixing length with normal distance from the
pipe .all, the velocity equation in the reg;ion (1_yzO.35
R)
tales the form
Cmos'
:
C
-
n
where C is a constant.
However, the complication
of this equation as compared writh
that of the ,r-ell-knowmsimpler forr derived on tle assumption
of constant stress did not seem to justify its use in this
case.
Therefore, assuming aain
1 -
that
ky
but thcat
T
=To
See
Appeydcx
the velocity equation becomes
To-erc
(
since
dc
2
2
Integrating this equation bet-een the limits
1 and
Y,
II
C-C!n
In
SV
Hence
jTo
(ZI)
- 5.4 + 5.75 logo
c
Let 1 be assuriled constant over th.e center section
of the pipeand equal to 0.14
,
nd let the stress
be !iiven
by
T =t°
See AppendyX
.35Ro
Then, sinceT =
dc
4,a
1
12()
1
o.1R/2 006R
014R
.
Inte-ratin0 betwreen
the limits 0.,5 R and y,
0.35R- 3.00 - 5.s (--)
Fror
, the velocity ecuation is
(II), at y = 0.35 R,
-7R0
- 2.8+ 5.75 logl0
vc~~~~~~C1
iY$
Hence,
5.8 + 5.75
(III) c
- 5.88(RR )
gl10R
C20,
I
.1
The mean velocity
I
of the cross
section.
e-
l
2
-
iven by the integration
is
R
t rdr
u
o
T1/(T
0
2 2~(11;~2$1 (1,5~$2
zz.
9
-9.0
I'
I
I
of
ancn (III) over the respective portions
equations (I), (II),
c~
R
y
0.35R
R
(zz5)'
11.5
+ 0. 14 R~
AI2
+ 51 (11.5) 'i
o
)
(-()
(T.
r,
-1.21 R10o 10
P
+ 0.85
I
i
-.
r
R
+
VI
R + 1.21
"
5.1
-
R + 2.88 R21og10g
2
R
4
log10 0
i
-
R
J
R
)
1
1
1
+ 11.5R _ .O
11.5/'+ s.33R'
61
(-)
lo
To
Elt
R
*
V)
-8-
SuICtt 'i
c by
Oefor.
r;e.n.s o' tbe formula
aion
t
AF~
~A-
give
I1
- 0.30 +
Af
5810
_
150
f~ Rey
+
lor0,f
The euQatton of von Koi'rman-Ni
-0.
I,
8otz
IogofF Rey
Rey
AS
isee
A pehdLX
-9-
Rou--;h Circular
Pipes
- In this case there is no
pure laminar sublayer.Asstumin, as for smooth pines,
- to , 1 - Vy or
dc
1
dy
Ky
c = K -nn
=
c = 0
5.75
hen y = yo,
log10 Y
f
u
0.35R,
+ c
Since
C
y
Y
which is customarily written as,
=5.5 75log10
(I)
in order
ta
c = 0 iwhen y = 0.
As for smooth pipes (y ~ 0.35 R, 1 = 0.14R)
C-C0.35
3.00
5.88 (ML)3/2
-?"
R
From (I),
c0.35R
C ?
3 5
-
5.75 o10
g 0.35R
Yo
substituting,
__-_
(IT)
c
r
0.37
+ 5.75
Og10 Y
5.ss
5.8
R-[z 3/ 2
that
-0The mean velocibty is obtained
by the integration
of
equations (I) and (II) over the cross section.
1.26-
Lo
-
3 56
y-
5.75 loO
2
o1o
10 e
vlhee k = eyo
The best value of the effectivity for sanded
from measured values of the friction
as determined
surfaces,
factor substituted
in the above equation, is 30. HIence, the
equation is
=
'Sog10
+ 1.74
For sanded surfaces
See Appendix
Infinite Parallel Sooth Plates.-
The velocity
equations are the se as those for the circular pipe, nd ar
obtained in the same mannrerby making similar assumptions
concerning the variation of the mixing lenth
and the stress.
In the pure laminar sublayer next to the plates, the velocity distribution can be obtained from the equation of NavierStokes in rectangular coordinates.
a t+
f
V +r,
+
+2u
UKxu
+
2u
Also,
+
+
= 0
Equation of Continuity
Let u = f(y), where y is measured normal to a plane midway
between the two plates.
Let
Let x be in the direction of flow.
v = O
w=O
- = 0 and the continuity is satisfied.
Then
Hence,
d~u
PIday 2
dp
dx
But
d-
Hence
du -
= O, since u = f(y)
- a constant.
1
2
1
D_
__ r-R
++ ely +
d-T)r
U=
ez
siderations,
the velocity is a mimum midwaybetween the
two plates.
O
u
Putting
Y =+
rhen
-du
-d 0
h
h
when y= 0
the velocity equation becomes
u - d
y
(h)
h
Therefore,
au _ d_ 1y
du
|y =
A dy
d= d y
Assumingconstant stress near the wall,T
!
,
=o
fl ?)~r~
_
Hence,
u-
V
,
y + C
Putting u = 0 when y = +
u =T
.
u =
Since
,
(y - h) for y positive
o (y
Co
is
TO
negative y,
h
+
) for y neative.
negative for positive y, and positive for
e
h
(i_
2 - Y)
1~~
2
4
Co
(I)
TO
---A1
hff31
(% + y)
0
Theoretically, the stress varies as
iven by
owev
ing,
er,
ass
as or circula
ppes, tat
assuming, as for circular pipes, that
-owever,
= co
1
1a
= K (h - y) for y positive
1 = K (
+ y) for y negative
In the ran~e (
(2
y
O)
let t
As sumin
y
"
+ Y
a1 -
2 y,
y
h + y.
1 = 0.07 h for the range
0.175h),
and let the stress be assumed to
y---
=o 0.65h
Then,
A/\
0.175h
and in the range
vary as
iIII) u _
0.175h
log O
5.4
+5.75
(II)(-0.175hA
2h) let t
y
2-
h - 16.65 (i)
4.00 + 5.75 logo0
9
-14The above develoment
the velocity
is a function
Therefore,
ci-rcular pipe.
is almost unnecessary since
of only one variable
it is obvious
as for the
the velocity
equations must be the same with y replacing r and h re-
placing R.
The mean velocity is given by
52
u
1
dy~
h
u.
dy
AN
= - 0.74 + 5.74
Let
-
= TC
2
IIu
Then,
1
J
(h)
2
74.6
log
1 0
ru
-1
S -u
-~X~Re
Ajf
= - 1.13
+ 4 log1 0 Rey'
-
74._6
-
Li
-
5--
Fllliptical Pipes of Sall
Eccentricity.-
The
equation of the ellipse in polar coordinates with the pole
at the center is
r
2
b
1 - e
wrhere
e
cos2-G
is the eccentricity.
b
t
For small values of the eccentricity, the velocity
can be defined in terms of distance out from the wall along
the radius vector.
This is a good approximation for small
eccentricity only.
With this approximation the velocity
eauations will be of the same form as for the circular pipe,
and continuity in velocity, stress, and mixing length
ill
be satisfied at all points of the elliptical section.
The mean velocity is obtained by integration over
the ellipse.
Hence
r~r r
c
c
-V
TabJo
uy
O
C)
rdrd -
uy~-
-16-
The integration with respect to r does not differ from that
over the circular pipe section. Therefore,
c
1
-Trab ,{r
f2
__
r
74
+ 168-
8 r
r
.88 2 log
.82V
2(168)
fr
+-wab
2.88
(
iTab
r d+
+-
+
37.4a
92
()
( C)
ab
4-
o
l
r2lnrd:'-
-ab
2
r2d
ITab
loaLb
r
3
d0
°10
I
where
de
rd = 4bb
2
= 4b F(e,
2
)
1 - e sin4and
,0
-
Z.rrbL
= b2
=r251--
Also,
- e
2
co s
2
l-
40-
e2
4r-
Y
r 2 n rr drl
b
!
cosn
l-e
4-
n
n
b!
l-e"cos -de
Or
-b
2 -n
b
0
te:'
b
,
02
2
io
0
l-e cos -
-17-
roL-2
!
JA
j-e cooOJ 2Tn(-~cs-_de=4i
This integral
I
2 '-)
(1-ecsin
-o sin
-d
L-e
sin2
can be expressed in terms of
p
(z) by
integra1 for the Leogendre
transforming Laplace's first
function Pn (z).
(z + ~ z°-
Pn(Z)
Differentiating
j (z +\ Tz2
_ cos+)dG--
integrand is continuous in both 4- and n.
- = 24'
b
TP n2
(z)
+
cos
PC+
z
cos(z2,0)nln(z +
(z Z+ ~D
v_
Substituting
cos )n in(z+
n~~
resultin,
since tile
Then letting
d-
vrith respect to n,
- Pn(Z)
1-1
n
1 cos)
sin2 -
cos 24= 1-
Pn(z) =
in (1
I
-r
2-1)
(Z +
ubK bV,)UtV'
I
14z
2
O A2,
(1-
1
_
z
+sn)z2
-I
-,' n
n
sinJ+Z)d
+ z27
err
+
'Tr
(z
+
Z
)
(,
1--0 -
z
F
br
in
(z + z2-)da4-
in)n
sin
x
Let
C-i-
21,-
+
-_
e
2C
~~z
-- -)
Thlen,
z
1
2-e-
2__e
-
+
1
-Tee--r=e
Therefore,
1 2-e
d
dn n 2)
2
2
=
(1-e PIsin2
-2
ni~-
1sin2,0-dg-
a)
(l-e 2 sin2e_)n d --
J~~A
Tfl-e 2
or
rT2
'I
10
2
(1-e sin--
n
n
n
(-
n
1-
2- ~c
si -)d
-
J
/1
-..
2
+ n
Putting n = - 1
d
dn
1 2-e
n
2
1-e 2
0
v/L
'/0
rn(l-esin2
- WF-e2
J(1-esin)
sni
2
le
d ! 1 2-e2
dn
2 .
-lr/- 2 1 -i- e
+ n i 1-e
l-e2 sin20
n=-l
-19-
r00
(n+) (n+2) ..
pn(z)
function,
of pn(Z) as a hypereometric
In iUurphyts expression
-
(rn+)
r=0
(-)
(-L-n)
..
(l-n).
-
1
1
n; 1;
1
1
1
2 z)
-
2(1-4) 2,
when \ -z| I
let r = (2
0 -e & --
z)
2
Then
( n+(n+2) --- (n+r).(-) (1-n)---(r-l-n) w
Pn (z)= r=O
r=O
(r) 2
--- (n+r)).(-n)(-n) -- (r-l-n)
(-n+2)
L[nPn(Z) =
ince
ters
te
since the terms
b
n
12
z)'
z
(r:)2
= F(n+1,
I
1
-
e factor(
containing
the factor
r=O 2---(r-
p-1
--------
r=O
r=O
= -
1
r
(1) (2)
r.r
r
V!
(n+!) vani sh.
--- (r) wr
r
n (1-w) = -
2 C
(1 + z)
n
I -e2
2
1\-
Hence,
ln(l-e sin2&.
(l-e2sin
6
..
O)
F1
d. &-=
2
-e
T
1 -e '-
2-e 23
ln 12 -t1O" AF-L
-e
-20-
Therefore, the expresoion
I
or the mean velocity
is
336
c
'
-
ab
g-b
9
- /02
T1o
+ 2b
)
'pj-
T
C
F(e,
7ra-c
2.
1
) n
(1-e IY
s'
( 1+
2)e
+
y ,>-
-4b
5.75 b
lotF:S0
V
Letting
lhen
I
I
1
I
- 0.80
+
b
,aI
e
a+ loe
I
3810
f (Rey)
blog-o
+2
+ai
e
2
=
'jiey
~- Rey
{(12e2)1n
where Rey
Since
b )2
a
150
b
(1 + 2
2
)-
u
a2 - b
2
a
I
1
0. 0 +
3810
b
f(Rey)a
(over)
150
(b)
TfRey a2
)F(e
)
e
-c1
-
+ 2 log10 P-Rey
I L'
+ .3(
Lo 1 0 2
· )) ~-oml
I
I
c
2 b
+ a j }a - 3".1
2. 3 loS°10 ba
-22-
For
satisfactory
these cases the above method
results.
The velocity
seems to give
gra.dient is zero at the
cent-erline, as it obviously should be from symmetry considerations.
In particular, the equation defining the
friction factor for the circular pipe obtained by this
method agrees very closely
ith the experimental data and
with the equation of von Karmtnm-Nikuradse. Except for small
values of the Reynolds Nunber where the flow is laminar, the
difference between the two equations is negligible (see
;rcaphin Appendix).
The friction factor for the elliptical pipe is
smaller than that for the circular pipe at the same Reynolds
NTumber as is seen from a comparison
This is to be expected
of the two equations.
since the minor aLxis 2b of the ellipse
was used in defining the Reynolds NiuTber. The equation for
the elliptical pipe becomes the equation for the circular
pipe, as
t obviously must, as the eccentricity approaches
I
T
zero since F(e,
) approaches
.
It may be concluded from the
ood agreement of the
equation for the circular pipe with experimental data and
with the equation of von Krm6n-Nikuradse
that Peters and
Rossby's assumption of a maximum value for the mixing length
is justified.
Further, it constitutes a definite imDrove-
lment in the method
of calculating
friction
factors.
The
velocity gradient is now zero and.the mixing length finite
"ii
at the center line, thus bringing
tie equations into better
- 2S-
aereeraent ith the pisicai
facts.
Tieconsideration oz the
pralle1 flat lates an6.of tile elliptical pi-e indictes
th..at te ethod. nimihtbe etended rather enerally to probleim-s in -which thle velocity
maaybe defined
;fhich the stress is unidirectional.
as above and in
The eistence
of
secondairy flows, as, for example, in the corners of the
rectangular
duct, would require a more general method of
solution, but ,vould in no ay affect the validity
oriinal assumption.
This method miLht be extended further
plate.
of the
to the flat
But here, although much isknovrn concerning the
frictional drag, little is known concerning the velocity
distribution.
It is therefore improbable that much add.i-
tional information concerning, the validity of this assumiption
would be obtained
rom a consideration of this problem.
JAPPENDIX
L
0.40 II-- --- --- -
I---
:I . i -
.
:
.
of
-TVriri~nttm
.. .
I
. .
.
&
..
.
. ...
:... ...
;....
. ,.......
.... ..-
-- ----
.
. .
-
-I
I
.
-.....
.
.
-_
------- i
I
i
.
-
-
-
i
!j
'-:..-
Ij4hcnni i---- nnth
'''II
-*·llrLPUW,sr-lrUICUill·II+L
C-·-
l
-·----
r
S.L1~t±Ln
~
:'
--
II.
____
And
/ I
,,
/
.
...
?..
.
_
..
034.-- -:
':
'
!,~,,
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References.
(1) Tl%.von Krrnman, Journ. Aero. Sciences, Vol.l,o.L
(I3)H. Peters arndC.G. Rossby, Journ. Aero. Sciences,
Vol. 2, No. 2
(3) J. Nikuradse, Forschungsheft 356, 361
(4)
Wien-Harms, Ilancbuchder Experimental Physik,
Band IV,
Tell 4
(5) Prandtl-Tietjens, Fund. Hydro- and Aeromechanics