ORE
3b
4./ulletin No. 41
July 1968
.
r:f' I
LLL, 1
p.
19o8
I
(I
LL E C
Engineering Experiment Station
Oregon State University
Corvallis, Oregon
STTEL8RA
0:4
b fEGON
Engineering Calculations of
Momentum, Heat and
Mass Transfer
Through Laminar Boundaries
E. ELLY
and
G. A. MYERS
THE Oregon State Engineering Experiment Station was established
by act of the Board of Regents of Oregon State University on
May 4, 1927. It is the purpose of the Station to serve the state in
a manner broadly outlined by the following policy:
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(2) To serve the industries, utilities, professional engineers, public departments, and engineering teachers by making investigations
of interest to them.
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investigations, and research as will be of greatest benefit to the people of Oregon, and particularly to the State's industries, utilities, and
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(1) BULLETINS covering original investigations.
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Single copies of publications are sent free on request to residents of Oregon, to libraries, and to other experiment stations exchanging publications. As long as available, additional copies, or
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price of this publication is Si .50.
For copies of publications or for other information address
OREGON STATE UNIVERSITY ENGINEERING EXPERIMENT STATION,
CORVALLIS, OREGON
100 Batcheller Hall
ENGINEERING CALCULATIONS OF
MOMENTUM, HEAT AND MASS TRANSFER THROUGH LAMINAR BOUNDARY LAYERS
by
Eugene Elzy, Assistant Professor
Department of Chemical Engineering
and
G. A. Myers, Research Assistant
Department of Chemical Engineering
BULLETIN NO. 41
JULY 1968
Engineerinq Experiment Station
Oregon State University
Corvallis, Oregon
ACKNOWLEDGMENT
The authors wish to acknowledge the cooperation and financial
support given this project by the Oregon State University
Computer Center and by the Department of Chemical Engineering.
Gary Myers was supported by a NASA traineeship during the
course of this work.
1
TABLE OF CONTENTS
Pag e
ACKNOWLEDGMENT
i
INTRODUCTION
1
FORMULATION OF PROBLEM
1
DEVELOPMENT OF SOLUTION
7
GRAPHICAL PRESENTATION OF SIMILAR SOLUTIONS
15
APPLICATION OF FIRST-ORDER APPROXIMATION
16
MOMENTUM TRANSFER RESULTS OF THE FIRST-ORDER
APPROXIMATION AND COMPARISONS WITH OTHER WORK
28
HEAT AND MASS TRANSFER RESULTS OF THE FIRST-ORDER
APPROXIMATION AND COMPARISONS WITH OTHER WORK
41
EXAMPLES
52
SUMMARY
71
REFERENCES
72
NOMENCLATURE
75
APPENDIX - A
APPE)IX - B
SIMILAR SOLUTIONS
FUNCTION
1.
INTRODUCTION
The calculation of heat, mass and momentum transfer is of major
importance in many problems of engineering interest. A large
number of methods for this calculation have been proposed for
the steady two-dimensional or axisymmetrical laminar boundary
layer with arbitrary external pressure distribution and constant
physical properties. The formally exact power series expansion
methods [1,2,3,4,5] and the finite difference technique [6]
produce reasonably accurate solutions, but require considerable
computation.
The approximate methods of Spalding [7,8],
Pohihausen [9] , Merk [10,11] and Eckert [10) all require
This reduction in labor
significantly less numerical labor.
is unfortunately accompanied by a great loss of accuracy in
many flows, such as decelerating flow.
A new method has recently been proposed by Sisson L12] which
appears to be a reasonable compromise between ease of solution
and accuracy of results. The present report presents the
application of this method to a wide range of problems. The
figures necessary for this general type of calculation are also
included.
An extensive graphical presentation of the similar solutions
to the boundary layer equations is to be found in Appendix A.
A wide range of mass transfer rate K, pressure parameter
and dimensionless property ratio A has been covered.
2.
2.1
FORMULATION OF THE PROBLEM
The laminar boundary layer equations
The boundary layer equations may be derived from the
general equations of change express ing the conservation
For two-dimensional
of momentum, mass and energy [13]
or axisymmetrical laminar flow of a pure or binary
fluid, the steady constant-property boundary layer
equations are given by:
Equation of continuity of mass
By
u
+
= 0
Bx
By
for two-dimensional flow
1
(1)
+
for axisymmetrical flow
= 0
(2)
Equation of motion
ldp
B2u
dx
Equation of energy
U+V
-
2T
(4)
cx
Equation of continuity of a species in a binary mixture
BXA
u+v------
D
2XA
AB ay2
(5)
subject to the boundary conditions,
at y = 0:
U = 0
(6)
V
(7)
Vw(X)
T=Tw
()
XA
(9)
XA
and as y
u = U(x)
(10)
T=T
(11)
XA
Viscous dissipation, radiation and chemical reaction
within the fluid, and the Soret and Dufour effects are
The coordinate system used is shown in
neglected.
Fig. 1.
(12)
FLOW
r
TWO-DIMENSIONAL WEDGE
/
FLOW
ARBITRARY TWO-DIMENSIONAL BODY
FLOW
ARBITRARY AXISYMMETRICAL BODY
FIGURE 1.
COORDINATE SYSTEMS
3
2.2
Transformation of variables
A more convenient representation of the boundary layer
equations can be obtained by employing Meksyn's [11]
transformation of variables with Mangler's [14]
transformation included to reduce the equations for
axisymmetrical flow to the same form as for twodimensional flow.
A stream function, IL, is defined
such that:
L
u=-r
y
v = -
(13)
L
iJ
r
x
(14)
This definition of p immediately satisfies the equation
of continuity (1) and (2). Here, L is an arbitrary
reference length on the body in the flow system.
The
factor r/L implies Mangler's transformation from
axisymmetrical coordinates.
For two-dimensional flow:
r/L = 1
(15)
The x and y coordinates are transformed by:
=rx U(x)
(16)
L2 L
Uo:
0
Re
1
2
U(x) r
LL
(17)
where U(x) is the velocity of the fluid at the edge of
the boundary layer, tJco is a reference velocity and
UL
Re=
(18)
The stream function is now written:
1
i(x,y) = UL2 () f(,)
4
(19)
Substitutions of the definitions (16)
(17) and (19)
into equation (3) yields a third order, non-linear partial
differential equation
,
f
+ f
f +
f f
(1
f
= 2
)
(f
f )
(20)
where the prime indicates differentiation with respect to
The parameter 3 is defined by:
n.
dU(x)
2
(21)
d
U(x)
The velocity components in terms of the transformation
variables are:
u(x,y) = U(x)f(,)
v(x,y) =
r
L
(22)
U(x)
(2Re)½
(f + 2
+
(
1)f')
(23)
The boundary conditions on equation (20) may now be
written:
at r
= 0,
f =0
(24)
vL (2Re)½
f + 2
and as
r
U(x)
(25)
Ti
f
-*1
(26)
Equations (4) and (5) will be transformed by introducing
the following dimensionless profile functions:
T(x,y)
T -Tw
T
w
(27)
XAW
xA(x,y)
=
(28)
-x Aw
x
and defining:
A
A
(29)
T
c
(30)
AB
DAB
The equations for thermal energy and mass diffusion are then
transformed to an identical form:
II
+ AfTT
= 2A (f
I
fl
(31)
II
With the assumption that T, T, XAW and xA
are
independent of x, the boundary conditions on equation (31)
become:
at r = 0,
as
II = 0
(32)
II
(33)
fl
1
These non-linear partial differential equations (20) and
(31), then, with their boundary conditions (24), (25), (26)
and (32), (33) , respectively, describe momentum, heat and
mass transfer within a steady two-dimensional or axisyrnmetrical constant-property laminar boundary layer. The problem
in any practical application is that of finding an adequate
solution to these equations for a particular external
velocity distribution.
DEVELOPMENT OF THE SOLUTION
3.
3.1
A series solution of the momentum equation [12)
The calculation of the momentum transfer in laminar boundary
layer flows involves the substitution of two series into
the momentum equation (20).
The following series are defined:
an(n(0)
=
(34)
ni
and
f(n,)
a()f(,0)
f0(n,0) +
(35)
For convenience, a1 is defined as
d130
a1 = 2
(36)
At the present time, the an for n > 2 are not of interest.
However, they will have similar arbitrary definitions.
Substitution of series (34) and (35) into equation (20)
produces the ordinary different ial equations:
,
t
I
f
0
+f
0+(1-ff)=0
00
I
I
I
I
f
(37)
0
and
III
f
1
II
I
II
I
+ff
-2(3 +l)ff +3f
01
0
01
0
I
I
=f0---.
-
(1
0
1
çB0
f
I
f f
0_U
)
The function i may now be defined by setting the
boundary conditions on equation (38) such that
7
(38)
at
n = 0,
=
0
(39)
=
0
(40)
f'= 0
(41)
f
(42)
f
and as
0
Similarly, if the an are chosen so that the ri appear as
free constants in the equations defining the n' it will be
possible to define the 13n by setting the boundary conditions
for the n equations such that, for n > 1,
at
= 0,
and as
fl
=
0
(43)
=
0
(44)
f1= 0
(45)
0
(46)
--
.
Thus it may be seen that the f for n
1 make a zero
contribution to the boundary conditions (24), (25) and (26)
on the moimntum equation (20) .
They also make no contribution to f'' at
= 0, so momentum transfer calculations may
be made using only f' at n =
0.
The boundary conditions on equation (37) may now be
written:
at
= 0,
f
0
fl
= -K
r
=
and as
(2Re)½
=
U(x)
0
(47)
(48)
--
fo
1
(49)
It may also be shown that
f'(0,0) = f(0,)
3.2
(50)
Extension to heat and mass transfer
Equation (31) describing heat and mass transfer may now
be considered.
The following series is defineth
all(,0)
(51)
nl
Substituting equations (35)
(36) and (51) into
it is found that
equation (31)
,
,
+ Af0T[0 = 0
lb
(52)
and
+ Af I
H
0 1
1
2Af
= A(f
II
0
1
-) - 3Allf1
II
0
13O
(53)
0
Boundary conditions on these and the higher order
equations are
at
= 0,
and for n >
110 = 0
(54)
= 0
(55's
1
(56)
0
(57)
1,
11
n
and as
11
and for n >
1,
I-f
n
is defined implicitly by equation
The function
and, with knowledge of the an, the 13n and the function
it would be possible to evaluate the function
(34) ,
Equation (35) could then be used, with knowledge of the
to calculate the exact values of f(,)
3.3
Various simplifications
Similar solution
When the external velocity distribution satisfies the
relationship [71:
dU(x)
dx
= C1U n(x)
(58)
the substitution of the transformed variables into
equation (3) yields equation (37) and its corresponding
boundary conditions. Upon transformation both the energy
equation (4) and the continuity of species eauation (5)
reduce to equation (52) with its transformed boundary
conditions.
The solutions to equations (37) and (52) , with K and o
constant, are known as "similar solutions" because the
velocity profiles at all points within the boundary layer
Tabulations
differ only by a constant scale factor [15] .
of the numerical solutions are available for large ranges
of the parameters 13, K and A [16,17,18,19]. Appendix A
contains a graphical presentation of these solutions.
This is discussed more fully in part 4 of this paper.
Flow over a wedge [20] , where
(i
U(x) =
2m
m + 1
(60)
m-1
and
v (x)
w
C2 x
(61)
Using
comprises an important class of similar solutions.
it
may
be
shown
that
the
potential flow theory [21] ,
corresuonds to a wedge with an included angle
parameter
of (fl) , as illustrated in Fig. 1.
10
Merk similar method
The zero-order approximation is the truncation of
series (34) after the first term, so that
= i()
(62)
This method was proposed by Merk [11]. The Merk similar
method unfortunately gives poor results for decelerating
flow.
First order approximation
A first-order approximation, which will also be called
the present method, is made by truncating series (34)
after the second term to obtain
d130
+ 2
a:---
(63)
Rearrangement of equation (63) produces an ordinary
differential equation
d
(64)
2i
with a boundary condition
at
= 0
=
(65)
= 0 the right side of equation (64) is indeterminBy invoking L'Hospital's rule the boundary condition
can be found in the form of the derivative with respect
to
Consequently,
ate.
.
at
= 0
d0
1
1 + 213i
d
w
can be determined by equation (21) and since
and K, equation (64) may be solved
l is a function of
to find an approximation of
Since f3(F)
11
The extension to the calculation of heat and mass
transfer rates is simplified by the introduction of
another approximation. The series (51) defining H is
truncated after the first term to obtain:
H(ri,,K,A)
110(n,1301K,A)
(67)
Thus II' may be evaluated from the similar solutions at
the proper value of 3ci. just as f'' has been.
Since
will in most cases be evaluated using the
approximate equation (64)
there will be two approximations
used to evaluate II'.
However, if values of ll for n > 1
were available, II' could be evaluated more accurately by
At the
including additional terms in the series (51)
present time no values of H11 for n > 1 are available.
It
is anticipated that the calculation of heat and mass
transfer coefficients using the approximate equations (64)
and (67) will be of sufficient accuracy for most practical
applications.
,
.
3.4
The
l
(130,K) function
and K and is determined
function depends only on
by solution of equations (37) and (38) with boundary
conditions (39) to (42) and (47) to (49). These equations
have been solved by Sisson l2.J and the l function
A graphical and
tabulated for a wide range of
and K.
function is provided in
tabular presentation of the
Appendix B.
The
3.5
fB-
Definitions
Transfer coefficients
The local friction coefficient, c, defined by
.
11
w
\) (_)
yy=o
cf
½u,
r'c7
ow be written in terms of the transformed variables
c
U2(x)
= 2
L U2(2Re)½
Co
12
(O,,K)
(69)
The local Nusselt number, NU, defined by
)y=0L
.
Nu
=
T
for
(T
-T)
w
=
k
heat transfer
(70)
and
A
(_)y0L
kL
Nu
=
AB
(x A
CDAB
for mass transfer
XA
w
(71)
may be written as
r
Nu
U(x)
(0,,K,A)
II
j:j
(72)
u(c)2
Re
These expressions (69) and (72) , then, relate the
dimensionless boundary layer functions f(,,K) and
1T(,,K,A) to the dimensionless functions cf and Nu,
which are commonly used in momentum, heat and mass transfer
calculations.
Flux ratios
Other frequently used quantities are the dimensionless
flux ratios, R,
NA MA + NB 4B
R
(NA MA + NB ?IB) rj
w
w
V
w
-pU
c
2
i
+ NB C0
NA C
w -B
w A
(NA
w
+ NB C )
p
w -B
w -A
h'
p=
NA
+ NB
w
(73)
(T
T)
4
w
q
XA
xA
w=
w
AB
N
x
(75)
A
-x
+ NB
NA
w
13
where
3u
T
=
w
(76)
Byy=0
-k
(77)
y y=O
When all physical properties are constant and equal
for each species, the flux ratios may be written as:
pv (x)
w
K
(78)
pUcof
c/2
f''(0,,K)
A
PCpVw(X)
KAT
(79)
RT
h
ll'(O,,K,AT)
KA
cv (x)
w
AB
(80)
RAB
k
ll'(O,,K,AAB)
The last two equations can be summarized in the sinqie
expression:
I
II
(81)
(O,,K,A)
Rate factors
The rate factors, 4, are the values of the corresponding R
with the transfer coefficients evaluated for the case of
no mass transfer, K = 0.
pv(x)
=
K
=
f
PtJcoCf/2
v (x)
pw
T
KA
h
T
(83)
II' (O,,O,AT)
cv(x)
AB
(82)
(0,,0)
KAAB
(84)
kx
ll'(0,,0,AAB)
14
U
The general expression is
K A
(85)
H' (O,,O,A)
Correction factor
The correction factor, 0, is the ratio of the transfer
coefficient for mass transfe r to the transfer coefficient
for low or no mass transfer:
S
cf
h'
cf' 0T
0V
k
h
AB
x
k
and by inserting the definition of the transfer
coefficients,
H (O,,K,A)
(86)
o
H' (0,,O,A)
4.
R
GRAPHICAL PRESENTATION OF SIMILAR SOLUTIONS
To facilitate the calculation of heat, mass and momentum transfer
through laminar boundary layers, the similar solutions have been
presented in various graphical forms in Appendix A. Most of the
The solutions,
results were obtained by Elzy and Sisson [16.1
near or at the separation of the boundary layer, were taken from
the papers of Stewart and Prober [173 and Evans [18]. Some
The
extrapolation was done near the regions of separated flow.
extrapolated regions are indicated by dashed lines.
.
The following figures are available in Appendix A:
Figure
Axes
Variable Parameter(s)
A-i
f'
A- 2
(O,130,K) vs K
ll'(O,30,K,.7) vs K
-1<130<5
A-3 thru A-il
A/3H' (0,0,K,A) vs K
A- 12
A- 13
eV
A-l4
A-l9
vs 1 + RV
vs
-1<00<5
-1<00<5 and .l<A<20
-. 1<00<5
-. 05<13<5
0.<13o<5 and
.l<A<
thru A-18
thru A-23
A-24 thru A-32
K vs 1 + R
0.<i3<5 and .l<A<co
and .l<A<20
A-33 thru A-41
ll'A/3
-.15<130<5 and .1<A<20
o
o
vs 1 + R
vs 1 + R
15
APPLICATION OF THE FIRST--ORDER APPROXIMATION
5.
5.1
Taylor series solution
Evaluation of the local transfer coefficients requires a
local knowledge of
and K.
The solution of the firstorder differential equation (64) provides the function j.
Since
is given in tabular or graphical form, numerical
solution of equation (64) is necessary. However, it is
possible to expand
= 0.
in a Taylor series about
(87)
=
n= 0
The bn are found by repeated use of L'Hospital's rule to
be:
b0 =
(88)
= 0
b
1
b
2
1
=(
1
=(
b3 =
(89)
+ 213i d
(1
1
1 + 4
d2IB
2b2 _i
___) )
d
l
6b1b2
(
b1
1 + 6i
4b2
B
dK
1
BK
(90)
BK
d
d3
1
1
2b1
1
2 d
dh
b1 (()
d
2
B2
2
0
d2K
(91)
BK2
d
BK
etc.
I,
(0,,K) and II
The functions f
expanded in a Taylor series about
(0,,K)
= 0.
(92)
n=0
16
(0,,K,A) can also be
00
ll'(O,,K,A)
(93)
n= 0
where
,
I
= f
I
Bf
c1 = (b1
2
2
I
I
dK Bf
(OlK))
B
d
BK
(0,,K)
+
22''
f
B
d
d
=
II
Bf
+ 2b2
(b
-S-
I
(O,,K)
B2 f
1
C
(94)
(O,,K) J
(95)
(0,,K)
B0
(0,,K)
2
dK
Bf
(0K))
BK2
d2
BK
(0,,K,A)
(96j
(97)
0
Bll(O,,K,A)
d1
2
2
dK Bfl'(0,
B0
l
l
dK
B211
+ 2b2
Bil (O,,K,A)
B0
B02
B211
()
BK
d
(0,,K,A)
PKA))
d2K Bil' (O,K?A))
(0,,K,A)
d
BK2
(99)
BK
The coefficients rapidly increase in difficulty of
computation, which, coupled with the slow convergence of
the series, limits the range of an exact solution of
The first few
= 0.
equation (64) to the region near
terms are valuable in certain approximations as will be
Table 1 contains some of the
shown in Section 4.4.
partial derivatives necessary for the evaluation of the
above coefficients. The partials are tabulated for several
The evaluation of
K and A.
of the most commonly used
the partials was carried ou numerically with a differentiated Stirling's interpolation formula.
,
17
Table 1.
Derivatives for Taylor series coefficients
*
B2
1
-
0
-
.5
.5
.5
1.0
1.0
1.0
1.0
-.272
-.0515
-.0636
-.0671
-.0214
-.0257
-.0287
-.0285
-1.
- .5
.0
.5
Bil
K
(.01)
B0
0
.O859
.5
1.
0
.00237
.00125
*
**
II
(.01)
B02
0
-.0395
-.00386
-.00128
is
f'' is f''(0,0,K)
I
I
II
(.7)
is
II
1.00
1.30
.652
.705
.725
.482
.514
.537
.541
2
B
(0,0,K,.7)
BTI
2
(.05)
B0
.0339
.00981
.00530
f
Bil
(.7)
II
B
B2
.429
1.34
.0850
.125
.148
.0224
.0304
.0376
.0392
-.l49
.5
.0
.5
.0
.5
0
B
B2
K
0
2
,
Bf
1
2
B
B2
-1.29
-3.05
- .389
- .507
- .580
- .160
- .197
- .228
- .239
(.05)
B02
-.150
-.0154
-.00524
BTI
(0.1)
B0
.0580
.0172
.00945
-.332
-.858
-.0509
-.0948
-.152
-.0134
-.0221
-.0341
-.0455
.102
.209
.0468
.0674
.0871
.0220
.0300
.0389
.0455
2
'
II
(.7)
B
II
(.1)
B02
-.252
-.0264
-.00912
Bil
(10.)
B0
.754
.258
.158
B
2
II
(10.)
B02
-3.04
- .332
- .122
5.2
Numerical solution
Discussion of differential equation (64)
The most accurate values of the local coefficients are
obtained by solving equation (64) numerically to obtain
the function I3ü(). Since the right side of equation (64)
= 0 difficulties are sometimes
is indeterminate at
For the case of ideal
incurred in numerical solution.
potential flow around a cylinder with no mass transfer,
not even such simtde techniques as Euler or Runga-Kutta,
give a stable solution. However, the external velocity
It is
profile U(X)/Uc,, = 1 - x/L offers no problem.
possible to avoid the instability by using the Taylor
for a short distance and then
series expansion of
continue on to separation with numerical integration.
Sisson 112] circumvented the instability by calculating
at station n+l from
h(3
n+½
O
(100)
+
n+l
o
fl
h
+ 2n+½1 n+i
1
where
n+1
n
+ h
(101)
+
(102)
and
n+½
The accuracy of equation (100) has been compared with
Runga-Kutta and Euler's integration methods for the case
The results
x/L.
of no mass transfer and U(x)/U = 1
The
close
agreement
of
the methods
are given in Table 2.
indicates that equation (100) provides sufficient accuracy
for most engineering applications.
Table 2.
U(x)/U
Comparison of Sisson's integration
formula (100) with Runga-Kutta and
Euler's methods
= 1
x/L, K = 0 and A
= .002
13
Sisson's formula (100)
Runga-Kutta
Euler
.02
-.03247
-.03247
-.03246
.04
-.06623
-.06623
-.06621
.06
-.10109
-.10109
-.10105
.08
-.13663
-.13663
-.13661
.1
-.17197
-.17197
-.17200
.11
-.18890
-.18889
-.18900
.116
-.19828
-.19827
-.19850
20
Procedure for solving differential equation (64)
An outline of the integration procedure is given below
For the first
for a general single integration step.
will be determined by the boundary
and
step,
condition (65)
i3
Given:
values of
and
Determine:
values of
n+l and
Step 1
Choose a step length h and calculate
using equation (101)
n+l
Step 2
Use equation (21) to calculate
n+½' which is 3 evaluated
at
+
Step 3
at
For the first trial, evaluate
n+½
n+½
n+l
and Kfl+½.
n
evaluated
using
For successive trials, evaluate
from
+
o
n+½
o
)
0n+½
(103)
Step 4
using equation (100)
Calculate
n+1
Steo 5
is within
If the chanqe in the calculated values of
n+1
last
the desired error limit, accept the
Otherwise, repeat Steps 3 and 4.
value of
.
n+l
The procedure may now he repeated for the next integration
step by starting at Step 1.
Procedure for calculation of local coefficients
The present method may be applied to the practical
calculation of momentum, heat and mass transfer coefficients
using the stepwise procedure outlined below.
Given:
U(x), Vw(X) and the fluid properties
Determine:
the local transfer coefficients
Step 1
Determine
(x) using equation (16) and U(x).
a convenient method of determining x()
b.
analytically
as a successive approximation
c.
olotxvs
a.
Then find
Step 2
Determine 1?() using equation (21) and U(x).
Step 3
Determine K() using equation (47) , v(x) and the fluid
properties.
Step 4
With 1() , K() solve equation (64) using integration
formula (100) to obtain
Step 5
Evaluate the dimensionless oradients f0(O,,K) and
and K()
ll(O,10,K,A) at the calculated values of
using Fig. A-1 through Fig. A-ll or tables of similar
solutions [16,17,18,191.
Step 6
Calculate the transfer coefficients c
equations (69) and (72)
5.3
and Nu
using
Average coefficients for heat and mass transfer
The average heat or mass transfer coefficient is given by
the integral expression:
22
( NudA
(104)
fdA
Expressing the differential area in terms of x and then
substitution of equation (72) for Nu yields:
,_(
)
U(x) ll(0,,K,A)
(r)
d()
L
L
=
(105)
x
r2
L d
I
J (X)
L1
or introducing the transformed variable
2
(0,,K,A)
fl
:
d
J1
(106)
(.)
I
d()
L
1
Since the step-by-step procedure of Section 5 .2 leads to
a tabular function for II' (0,,K,A) equation (106) must be
evaluated numerically.
Replacing II' (0,,K,A) in ecuation (106) with its Tayloi'
series, equation (93) , and then integrating in the
numerator gives:
23
(d
x
2
r
X)
L
L
(l/2
l/2)
02
1
L1
l(3/2
3/2
d2
5/2
+ 52
5/2
+ .)
(107)
Truncation of this series after one term produces good
This approximation is compared with numerical
results.
integration of the local coefficients in the examples in
The two methods usually differ
Sections 8.2, 8.3 and 8.4.
It is also shown how the
from three to ten per cent.
addition of two more terms reduces the difference to
around one per cent.
A modification of the Taylor series approach to average
calculations is the truncation of equation (93) after the
third term and the addition of an empirically fitted term.
ll'(0,,K,A) = d0 + d1
+
d22
+
d55
(108)
The coefficient d5 and exponent s may be found by forcing
equation (108) to meet two specifications:
1)
Equation (108) must qive the correct II' (0,,K,A)
for
2)
sep
When inserted into equation (106) , equation (108)
= 0
for the range
must yield the correct Nu/V
to
sep
Values of the exponent s for various cases are listed in
It is suggested that s be taken as equal to ten
Table 3.
if no better information is available. Then, knowing
d5 can be comnuted from:
II'
(II
d0
dlsep
d2ep)/p
(l09
Combininq equations (106) and (100) and integrating gives:
24
(d0(2
2
=
r'L
2
I
J()
L1
d1
1/2
l
3/2
+
3/2
+
rd()
L
L
5/2
+ 52
d
5/2
l
21/2
2l/2))
(110)
+
Equation (110) has been used in the examples in
In all three cases the error
Sections 8.2, 8.3 and 8.4.
was reduced to less than one per cent. However, when
= 0, the addition of the
dil' (0,,K,A) is positive at
empirical term does not greatly improve the
d
It can even give slightly worse results as is
results.
shown in the example of Section 8.4.
5.4
Calculations involving R,
and 0
In many engineerinq problems R(x) is known instead of
The following step-by-step procedure may he used.
v(x) .
P(x) and the fluid properties
Given:
U(x) ,
Determine:
the local transfer coefficients
Step 1
(x) using equation (16) and U(x).
Determine
a convenient method of determining x()
b.
analytically
as a successive approximation
c.
aplotofxvs
a.
Then find
Step 2
Determine 0(g) using equation (21) and U(x)
Step 3
Determine R() from R(x) and the relationship between
x and
found in Step 1.
25
Table 3.
Exponent s for various cases
r(x)/L
1
1
x/L
x/L
l-x/L
1
x/L
l-x/L
2 sin x/R
2 sin x/R
2 sin x/R
2 sin x/P.
2 sin x/P.
2 sin x/R
2 sin x/R
1.5 sin x/R
1.5 sin x/R
1.5 sin x/R
1.5 sin x/R
1.5 sin x/R
K
A
1
1
1
1
1
0
.01
.05
1
1
1
1
1
1
1
0
0
0
0
sin
sin
sin
sin
sin
0
0
0
0
0
.5
-.5
x/
x/R
x/R
x/P
x/P
0
0
0
.5
-.5
26
.1
.7
10.
.01
.05
.1
.7
10.
.7
.7
.01
.7
10.
.7
.7
s
10.1
8.7
8.3
7.2
7.8
11.5
10.7
9.6
8.8
8.9
8.4
9.2
27.4
11.1
10.8
8.8
13.3
Step 4
Choose a step lenqth h and calculate n+l and
equations (101) and (102) respectively.
n+½ from
Step 5
n+½' which is
Use equation (21) to calculate
at
1
evaluated
Step 6
Estimate Kn+½ by extrapolating from the values of K
at the previous stations. Newton's formula for 3-point
extrapolation would be:
K+½
3K.½
= 3Kn
(111)
+ Kn_l
In order to begin at station n=0 use:
K½ = K0 +
h dK
2
(112)
dth=0
Step 7
For the first trial, evaluate
For successive trials, evaluate
calculated from equation (103)
at
fl2
On and Kfl½.
by usinq IBn
fl+-
Step 8
Calculate
On+l
from equation (100)
Step 9
is within
n+l
the desired error limit, accept the last value of tBo
n+l
If the chanqe in the calculated values of 13o
Otherwise, repeat Steps 7 and 8.
Step 10
Locate K+i in Fiq. A-24 through A-32 using R(1) and
27
Step 11
Interpolate K+ from:
K-K +--K -K1
3
3
n+1
(113)
n
If this Kfl+½ agrees with the Kn+½ used in Step 7 within
the prescribed error limits, accept the last
calculated in Step 8. Otherwise, use the Kfl+½
calculated from equation (113) and repeat Steps 7-li
until agreement for Kfl+½ is attained.
Step 12
Evaluate the dimensionless gradients f01(0,o,K) and
and
IT' (0,301K,A) at the calculated values of
Kn+1 using rig. A-i through A-li or tables
of similar solutions [16,17,18,19]
Step 13
S
Calculate the local transfer coefficients cf and Nu
using equations (69) and (72)
S
Repeat Steps 4-13 until separation occurs or the region
of interest has been covered.
6.
6.1
MOMENTUM TRANSFER RESULTS OF THE FIRST-ORDER
APPROXIMATION AND COMPARISONS WITH OTHER WORK
The circular cylinder with U(x)/U. = 2 sin x and K =
0
Fig. 2 shows the results of momentum transfer calculations
made using the first order approximation with the external
velocity distribution of U(x)/U = 2 sin x, which is
predicted by ootential flow theory for a circular cylinder
in crossflow. Also shown are the results of calculations
made by the approximate methods of 1erk [lii, Eckert [10],
Spalding [711 and Pohlhausen [15], by the series methods of
Blasius [15] and Gortler [3] and by the continuation
method of Gortler and Witting [22] . The separation points
predicted by these different calculations are presented in
Table 4.
The present method shows good agreement with the Blasius
series and the calculations of Gortler and Wittinq. The
1.2
POHLH4USEN METHOD
I.0
MERK SIMILAR METHOD [I
0.8
\
2
0.6
1*
SPALDING METHOD [vI0.4
GORTLER SERIES
\
ECKERT SIMILAR METHOD
\\\
ALONE [5]
\\ \
BLASIUS SERIES [151
GORTLER AND WITT
PRESENT METHOD
0.2
0
0
0.2
0.4
0.6
0.8
X
FIGURE2
1.0
1.4
1.6
1.8
2.0
RADIANS
COMPARISON OF SEVERAL METHODS FOR MOMENTUM TPANSFER WITH U(x)/U
= 2 sin x.
Table 4.
Separation points with U(x)/U
= 2 sin x
and K = 0
Separation Point
Method
Eckert [10]
1.626
Merk
1.661
[ii]
Spalding (7]
1.768
Blasius (15]
1.898
Gortler and Wittina [22]
1.90
Present method
1.902
Pohihausen [15]
1.912
Gortler series, 6 terms 13]
2.052
30
Merk and Eckert similar methods are reasonably close out
to x = 1.4, where they fall off rapidly.
Spalding's
method is in reasonable agreement out to x = 1.6, where
it also falls rapidly.
Pohihausen's method is low in
the region near the stagnation point and somewhat high
in the region near separation, although it predicts
separation closely. With six terms, Gortler's series
does not converge adequately for x > 1.6. Thus, for
this case, the first-order approximation appears to qive
substantially better results for momentum transfer
calculations than do other approximate methods.
6.2
The velocity distribution U(x)/U
= 1 - x
The results of momentum transfer calculations by the
present method for the external velocity distribution
= 1
x are shown in Fig. 3. Also shown are the
formally exact calculations of Clutter and Smith [61 and
Gortler and Witting [22]
Calculations made using the
Merk similar method are also shown, for they represent
the zero-order approximation and thus are of special
interest.
Table 5 compares the predicted separation
point with that predicted by other investigators. See
the example in Section 8.2 for sample calculations.
.
Calculations made by the present method and by Gortler
and Witting [24] for U(x)/U = 1
xr, with n = 2, 3, 4,
and 5, are shown in rig. 4. Table 6 gives the separation
points predicted by several methods.
Except for Spaldinq's
method, the Merk similar method and the present method,
all are considered formally exact.
For n = 1, 2, and 3, there is reasonably good agreement
between the present method and the formally exact methods,
although it appears that the present method is slightly
high in the region just before separation. For n = 4 and
5, the present method is siqnificantly hiqher than the
calculations of Gortler and Witting. However, the latter
calculations are less reliable for these values of n
because only two terms could he used in (ortler's series
although Witting's numerical continuation was started
relatively early [23]
6.3
The velocity distribution U(x)/U
=
(1 +
y-enturn transfer calculations for the external velocity
distribution U(x)/U = (1 + x)" have been carried out
using the present method. Fig. 5 shows the results for
31
Separation points with U(x)/U
Table 5.
x
Separation Point
Investigator
0.0867
Merk [ii]
Spalding
= 1
0.1073
[71]
Leiqh [23]
0.1198
Clutter and Smith [6]
0.1200
Present method
0.1241
Gortler and Witting [241
0.125
32
Table 6.
Separation points with U(x)/U
= 1
x
Exponent
n
Method
2
Tani [25]
Present method
Gortler and Witting
[24]
0.271
0.288
0.290
Present method
Gortler and Witting
[241
0.408
0.409
3
4
5
Table 7.
Separation Point
Tani [25]
Gortler and Witting
Present method
[24]
0.462
0.485
0.495
Gortler and Witting [243
Present method
0.552
0.560
Separation points with U(x)/U
= (1 + x)
Exponent
n
Method
1
Merk [11]
Clutter and Smith [61
Present method
Gortler and Witting [22]
0.1045
0.1450
0.1534
0.161
1.5
Present method
Gortler and Witting [22]
0.0984
0.101
2
Present method
Gortler and Witting [22]
0.0725
0.075
2.5
Present method
Gortler and Witting [221
0.0573
0.058
3
Present method
Gortler and Witting [22]
0.0474
0.048
3.5
Present method
Gortler and Witting [22]
0.0404
0.041
4
Present method
Gortler and Witting [22]
0.0353
0.036
Separation Point
33
0.6
0.5
K=O
0.4
t)
g
0.3
PRESENT METHO
0.2
MERK SIMILAR METHOD [ii]
cxi
CLUTTER ANDSMITH [6]
GORTLER AND...
t21
0
WITTI11G
0
0.02
x
FIGURE 3.
0.10
0.08
DIMENSIONLESS DOWNSTREAM DISTANCE
Q06
0.04
COMPARISON OF SEVERAL METHODS FOR MOMENTUM TRANSFER WITH U(x)/U
0.12
= 1 - x.
I
I
I
1
0.5
0.4
fl3
fl2
flz5
flz4
0.3
U.'
cc
I.-
0.2 - G:GORTLER AND WITTINGE2s]
\\.- p
P; PRESENT METHOD
G
ni
\\p
I
0
0.1
0.2
X
FIGURE 4.
0.3
0.4
.- p
G
G
I
0.5
DIMENSIONLESS DOWNSTREAM DISTANCE
COMPARISON OF THE PRESENT METHOD WITH THE CALCULATIONS
OF GORTLER AND WITTING WITH U(x)/U= 1 - x".
0.6
n = 1.
Also shown are the calculations made by Clutter
and Smith [161, by Gortler and Witting [22] and by the
Merk similar method [111. Predicted separation points
for n = 1, 1.5, 2, 2.5, 3, 3.5 and 4 are given in
Table 7.
For n = 1, the first-order approximation compares quite
well with the calculations of Clutter and Smith, and
Gortler aid Witting. In fact, the results at every point
lie on or between these formally exact calculations. The
relatively sharp curvature in the calculations of Clutter
0.13, however, indicates that they may
and Smith at x
have experienced difficulties with their numerical methods
and that their results past this point may not be reliable.
For the higher values of n, the present method agrees
quite well wjth the calculations of Gortler and Witting
in prediction of the separation points.
6.4
Axisymmetrical flow on a sphere with U(x)/U
= 1.5 sin x
An example of axisymmetrical flow, the sphere with
= 1.5 sin x, has also been treated using the present
The results are shown in Fig. 6. Also shown are
method.
the calculations by Clutter and Smith [6] and by the Merk
similar method [11]. Table 8 gives the separation points
predicted by four different methods.
In the region just before separation, 1.6 < x < 1.9, the
present method gives results significantly higher than the
formally exact calculations of Clutter and Smith. However,
the Blasius-tvpe series employed by Schlichting predicts
separation after the first-order approximation does. Since
the Blasius series gives good results on the cylinder, the
calculations of Clutter and Smith may not be reliable in
It may be noted that in this mostly accelerated
this region.
and, for 0 < x < 1.2, reasonably similar flow the Merk
similar method gives reasonably good results.
6.5
A cylinder with injection at the surface
The present method was used to calculate momentum transfer
on a cylinder with several different rates of injection at
These results are comDared in Fig. 7 with
the surface.
calculations made using the series method of Sparrow,
Torrance and Hung [4]. The velocity distribution used was
a seventh-deqree polynomial correlation of Elzy's [10]
0.36276(x/R)
data and is given by U(x)/U = l.79(x/R)
0.010l53(x/)7.
+ 0.02323(x/R)5
36
0.6
0.5
0.4
0.3
9
.4-
0.2
GÔRTLER SERIES
ALONE 13]
MERK SIMILAR METHOD [I ii
CLUTTER AND
0.I
SMITH
GbRTLER AND -
[6)
< WITTING [ul
PRESENT METHO
0
0
0.2
0.4
0.8
0.6
X
FIGURE 5.
1.0
12
1.4
16
DIMENSIONLESS DOWNSTREAM DISTANCE
COMPARISON OF SEVERAL METHODS FOR
MOMENTUM TRANSFER. WITH U(x)/U
= (1 + x)1.
r
Table 8.
Separation points on a sphere
Separation Point
1Iethod
Merk [ii]
1.693
Clutter and Smith E.6
1.846
Present method
1.894
Blasius-type series [151
1.913
Table 9.
Separation points on a cylinder
with injection at the surface
Present
Method
Sparrow, Torrance
and Hung [4]
0.137
1.482
1.479
0.697
1.434
1.428
1.771
1.350
1.349
1.0
0.8
w
0
-
PRESENT METHOD
0.6
9-
CLUTTER AND SMITH
0.4
[s1
MERK SIMILAR METHOD [iii
0.2
0
0
0.2
0.4
0.6
0.8
1.0
12
14
1.6
X RADIANS
FIGURE 6.
COMPARISON OF SEVERAL METHODS FOR MOMENTUM TRANSFER
WITH AXISYMMETRIC FLOW AND U(x)/U, = 1.5 sin x.
1.8
2.0
P: PRESENT METHOD
3.0
fwd=0.137
S SPARROW, TORRANCE
AND HUNG SERIES 41
P
2.5
0.697
S
2.0
S
1wd= 1.771
1.5
C.)
1.0
P
S
0.5
01'
0
I
0.2
0.4
I
0.8
0.6
X
I
I
I
1.0
1.2
1.4
1.6
RADIANS
COMPARISON OF THE PRESENT METHOD WITH SPARROW, TORRANCE AND HUNG
FIGURE 7.
SERIES FOR MOMENTUM TRANSFER ON A CIRCULAR CYLINDER WITH INJECTION
AT SURFACE FOR U(x)/U = 1.79x - .36276x3 + .02323x5 - .010153x7.
The parameter
is a measure of the normal velocity
at the surface and is defined by:
WU
f=
pv
w w
pU
(114)
In this case
is positive, and consequently the momentum
transfer coefficients are reduced.
The separation points predicted by these methods at various
rates of injection are given in Table 9. The results
obtained with the present method compare quite well with
the formally exact calculations made using the series of
Sparrow, Torrance and Hunq.
7.
7.1
HEAT AND MASS TRANSFER RESULTS OF THE FIRST--ORDER
APPROXIMATION AND COMPARISONS WITH OTHER WORK
Introduction
Few accurate analytical solutions are available for heat
and mass transfer in non-similar constant-property
incompressible laminar boundary layer flow, so the
calculations of heat and mass transfer by the present
method are also compared to experimental data.
7.2
Analytical solutions for the circular cylinder
8 shows the comparison between the first-order
approximation and Newman's Blasius-series type calculations
[26] which are formally exact. The external velocity
profile was U(x)/U = 2 sin x, K = 0 and A = .7. It is
en that both methods agree very well. It is instructive
to also consider the Schmidt-Wenner [27] velocity profile,
U(x)/U = l.8l6(x/R)
0.4094(x/R)3
0.005247(x/R)5, so
that the effect of the form of U(x) can be demonstrated.
Fig. 9 shows the results.
Blasius series methods require
coefficients in the polynomial expansion for U(x) so that
a fifth order polynomial does not allow Newman's [26]
method to include higher order terms. l\s a result Newman's
results are not accurate in the range 80-90 degrees, while
the present method performs quite well.
Fig.
The external velocity orofile U(x)/U = 2 sin x was used
for the calculation of the heat and mass transfer
coefficients for the followinq cases:
41
NEWMAN'S METHOD
xi
[.26)
PRESENT METH
NJ
z
A =0.7
KzO
0.2
20
40
60
80
tOO
120
e DEGREES
FIGURE 8.
COMPARISON OF THE PRESENT METHOD WITH NEWMAN'S SERIES FOR
HEAT AND MASS TRANSFER ON A CIRCULAR CYLINDER WITH U(x)/U
= 2 sin x.
FE.]
Ii
NEWMAN'S BLASIUS
()
,,'-SERIES
Z04
A
0.2
M
PRESENT METHOD
0.7
K=O
20
40
60
80
100
0 DEGREES
COMPARISON OF THE PRESENT METHOD WITH NEWMAN'S SERiiS
FIGURE 9.
FOR HEAT AND MASS TRANSFER ON A CIRCULAR CYLINDER WITH
U(x)/U = 1.816x - .4094x3 - .005247x5.
120
= .01
K
0,
A
4)
K=-.5,A=.7
K=0, A=.7
K=.5, A=.7
5)
K
0,
A = 10
1)
2)
3)
The results are given in Fig. 10.
7.3
Experimental heat and mass transfer data for a circular
cylinder
Heat and mass transfer data for a circular cylinder in
crossflow taken from five investigations [27, 28, 29, 30,
311 is compared in Fig. 11 with calculations made using
The velocity distribution of
the present method.
Schmidt and Wenner [271, correlated by a fifth degree
0.4094(x/R)3
= 1.8155(x/R)
polynomial,
The
- 0.005247(x/R)5, was used in the calculations.
parameter Nu/A°4Re°5 is recommended by 50gm and
Subramanian [29] for the correlation of heat and mass
transfer data taken at different values of A.
U(X)/Ux,
The present method shows reasonably good agreement with
the data up to the point of separation, which was calculated
to be 81.0° and which appears to vary in the data from
about 77° to 90°.
The mass transfer data of Schnautz [28], taken on a
Shown for comparison are
cylinder, is shown in Fig. 12.
calculations made using the iMerk [ii] and the Eckert [101
0.7.
similar methods, and the present method, all with A
Here the Chilton-Coihurn analogy [131 has been used to
convert Schnautz's data to A = 0.7. The velocity distribution of Schmidt and Wenner as given above was used in the
calculations. An additional calculation was made with the
present method using a fifth degree polynomial velocity
0.2935(x/P)3
distribution, U(x)/U = 1.737(x/R)
0.0593(x/R)5, measured by Elzy [101 in the same wind
tunnel used by Schnautz.
Both the Merk and the Eckert similar methods agree well wifh
the data until shortly before they predict separation, at
x = 1.231 and 1.253 respectively. However, separation does
not occur until about x = 1.45. The present method agrees
well with the data. The velocity distribution of Elzy
1.355, than
predicts separation slightly earlier, at x
does the velocity distribution of Schmidt and Wenner,
which indicated separation at x = 1.416. Since the velocity
distributions were actually quite similar, it appears that
good correlation of velocity data may be quite important.
44
12
0
U)
d,.
L1
1.0
.
4D0
0U
0.8
0.6
122,000
O 32,800
U)
0.4
c
® 39,800
o 39,000
e 110,000
9 2,900-97,000
SOGIN AND SUBRAMANIAN [2J
WINDING AND CHENEY
SCIMIDT AND WENNER [2i1
ZAPP
ZAPP
.
[31)
[31]
9
SCHNAUTZ [28] -AVERAGE OF 7 RUNS
0
z
0.2
10
20
30
40
50
60
70
e DEGREES
COMPARISON OF THE PRESENT METHOD FOR
FIGURE 11.
=
1.816x - .4094x3 - .005247x5 WITH
U(x)/U
HEAT AND MASS TRANSFER DATA.
80
90
iw-.
'I
I:
fl 6
I.
DATA OF SCHNAUTZ 1281
TURBULENT INTENSITY< 1%
MERK SIMILAR METHOD
[iiJ-
o 2,900
2,600
28,570
-I
4
z
ECKERT SIMILAR METHOD
37,100
Q 44,050
97,000
_-'
PRESENT METHOD VELOCITY
DISTRIBUTION
FROM:
k>\
[io]-'
ELZY [IO1
a
/
SCHMIDT AND WENNER [21J-"
0.
)c
p
INDICATED SEPARATION POINT
0.2
0.4
0.6
0.8
1.0
.2
X RADIANS
FIGURE 12.
COMPARISON OF PRESENT METHOD, AND MERK AND ECKERT
SIMILAR METHODS WITH SCHNAUTZ'S MASS TRANSFER DATA.
1.4
7.4
Experimental transoiration coolinc data on a circular
cylinder
The series method of Sparrow, Torrance and Hung 4] and
the present method are compared in Fjq. 13 with the heat
transfer data of Elzy lOj taken on a cylinder with
transpiration cooline. The injection parameter, 1wd'
Two velocity distributions
defined by equation (114).
were used. The velocity profile U(x)/U = l.665(x/R)
+ 0.0393(x/E.)
0.1977(x/R) was used in the region near
the stagnation point.
The seventh decTree nolvno:'ial
correlation of Elzv's L10 data,- discussed earlier, was
used over the rest of the cylinder. See the example in
Section 8.3 for samnle calculations of a case with a
different velocity orofile and suction.
7.5
results for deceleratinc flow
The external velocity profile U(x)/U
used in the followinci cases:
1)
2)
3)
K = 0,
K = 0,
K = 0,
= 1
x has been
A = .01
A = .7
A = 10
The heat and mass transfer results are plotted in Pie. 14.
See the example in Section 8.2 for sample calculations
of case 2.
7.6
Results
or a sphere
The external velocity profile U(x)/U
used in the following cases:
1)
2)
3)
4)
5)
K=0,
K
K
K
K
= 1.5 sin x has been
A=.0l
= -.5, A = .7
A = .7
= 0,
A = .7
= .5,
A = 10
= 0,
The het and mass transfer reults are plotted in Fig. 15.
See the example in Section 8.4 for sample calculations
of case 4.
I
I
I
I
I
I
S: SPARROW, TORRANCE AND HUNG
SERIES
L41
0
d°697
0.6
'a,
z,°
0.4
f
1.771
S
0.2
10
20
30
40
50
60
e DEGREES
COMPARISON OF THE PRESENT METHOD AND SPARROW, TORRANCE
AND HUNG SERIES WITH ELZY'S TRANSPIRATION COOLING DATA
FOR INJECTION AT THE SURFACE OFA CIRCULAR CYLINDER.
FIGURE 13.
70
EXAMPLES
8.
8.1
Similar solution of flat plate with injection
Ekample 1.
Air is flowing over a flat plate through which the
injection rate is proportional to (Y½. Determine the
L
local Nusselt number and local
friction factor as a function of the blowing parameter,
(2Re)½,
for A = .7.
c,-1
As rn = 0,
1.
= 1, and U(x)/U
= 0,
For a flat
ecuation (61Y requires that v(x) = C2x½ for a similar
Since it is given that the injection
solution to exist.
rate is proportional to ()½ or expressing this in terns
L
of v(x),
v(x)/U
= C(),
the problem has a similar solution.
needed are:
The rther expressions
x
x
r
d()
d()
=i:
v(x)
K
U..
J2
L (2Pe)
v(x)
V2Re
r U(x)/U
(U(x) )2 ft
c
L
U
(0,,K)
(2Re)
or
c/
v' f(O,,K)
52
2
Re
(0,,K)
x
r x
Nu=Nu
=-X
L
L L
½
U(x)(Re
Rex ½
,
H (0,,K,A) =
(0,,K,A)
(-i----)
or
(0,,K,A)
II
It
I
For the desired K look up f
(0,c,K) and TI (0,,K,A)
in Fig. A-i and Fig. A-2 respectively.
The local Nusselt
number and local friction factor can now be calculated.
The results are provided in Table 10.
8.2
Example 2.
Decelerating flow
A system with decelerating flow has an external velocity
function, U(x)/U = 1
x/L.
There is no mass transferred
through the solid surface an A = .7. Calculate the local
friction factor, local Nusselt number, separation point
and averaqe Nusselt number for the reg ion from
= 0 to
separation.
Solution
The functions
and B are derived for this two-dimensional
case, i.e., r/L = 1.
x
x
U(x)
r
d() =
(1
)
1
d()
X
J0
L
= 1
- 2
and
2
U(x)/U
dU(x)/U
d()
d()
-
d
2
(-1)
i -
53
1
2
=
x
2
1
Table 10.
K = j/2Re
Results of calculations for Example 1
f'(O,,K)
c/
ll'(0,,K,,.7)
Nu//
0
.4696
.6641
.4139
.2927
.1
.3986
.5637
.3618
.2558
.2
.3305
.4674
.3108
.2198
.3
.2658
.3759
.2610
.1846
.4
.2049
.2898
.2126
.1503
.5
.1485
.2100
.1656
.1171
.6
.09747
.1378
.1201
.08492
.8
.01757
.02485
.03399
.02403
54
Since there is no mass transfer, K = 0. Table ii shows
the results of Steps 4, 5 and 6 of Section 5.2 using a
step increment of A = .002.
The solution may be approximated with the Taylor series
(92) and (93).
The following derivations and limits are
needed:
=131
0
2
=
d2i3
= -
=0
so that
2)2
(1 -
=0
8
so that
(1 - 2)3
= -2
d213
d2 =0
= -8
Tables B-i and 1 and Fig. A-i and A-2 supply the
following:
= .129105
13
1
= -.272
(0,,K)
= .4696
f'' (O,,K)
= 1.30
(0,,K)
= -3.05
3132
II' (0,,K,A)
311' (0,,K,A)
3211'
(0,,K,A)
2
J=
= .4139
= .209
= -.858
55
Table 11.
x
Results of calculations for Example 2
f0
0.0
(cfv')1
ll(0,,K,A) (Nu//)1 (cfV)2 (Nu//)2
0.4139
0.4696
0.0
0.0
0.01
0.01005
-0.01606
0.4483
6.213
0.4104
2.873
6.21
2.87
0.02
0.02020
-0.03247
0.4257
4.086
0.4066
1.992
4.09
1.99
0.03
0.03046
-0.04920
0.4015
3.082
0.4024
1.593
3.09
1.59
0.04
0.04083
-0.06623
0.3755
2.443
0.3978
1.349
2.45
1.35
0.05
0.05131
-0.08354
0.3476
1.979
0.3925
1.178
2.00
1.18
0.06
0.06192
-0.1011
0.3173
1.612
0.3866
1.047
1.64
1.05
0.07
0.07264
-0.1188
0.2840
1.306
0.3797
.9410
1.36
.951
0.08
0.08348
-0.1366
0.2472
1.038
0.3716
.8514
1.11
.866
0.09
0.09446
-0.1544
0.2053
0.7937
0.3617
.7720
.991
.795
0.10
0.1056
-0.1720
0.1555
0.5563
0.3487
.6973
.723
.733
0.11
Q.1168
-0.1889
0.0914
0.3039
0.3297
0.6209
.561
.678
0.116
0.1236
-0.1983
0.0109
0.0348
0.3000
0.5459
.471
.648
0.1164
0.1241
-0.19883
0.0002
0.0005
0.2957
0.5368
.466
.646
1
2
Results of numerical solution.
Results of truncation of Taylor series after third term.
Calculation of the proper coefficients, equations (94-99),
gives as approximate results:
f' (0,,K) = .4696
2.07
-
6.l02
II' (0,,K,A) = .4139
.332
-
l.442
The Nusselt number and friction factor predicted by these
The expansion for
expansions are presented in Table 11.
II' (0,,K,A) is also compared with the numerical integration
results in Fig. 16.
It is now possible to calculate values of iSii /v'
from
= 0 to the separation point calculated earlier
Numerical integration of equation (105)
using H' (O,,K,A) calculated by numerical means gives
= 1.53. The percent error listed after the
Nu //
following approximate averaqe Nusselt numbers uses this
as the true value.
Nu //
('sep = .1164) .
d %/2
0
i //
sep
% error = 5.2
= 1.61
()
Rsep
d1
V'(do/sep +
i:1 /V
d
3/2
sep
=
sep
+
= 1.54
x
sep
% error = .7
Using sep = .1164 in equation (109) yields,
Equation (110) cTives:
= -1.32 x 108.
ü /i
8.3
Example 3.
= 1.53
Flow around a cylinder with suction
A cylinder of radius R rests in a stream of air which is
moving perpendicular to the axis of the cylinder. Assume
that the external velocity profile is that of Elzy [10],
.0593(). The velocity
U(x)/U = l.737() - .2935()
R
R
R
of the air being pulled through the surface of the cylinder
is given by v(x)
=-.9319.
Calculate the local friction factor, local Nusselt number,
separation point and average Nusselt number for the region
from x = 0 to separation. Assume that A = .7.
57
0.45
0.40
I!
Ui
0.35
0.30
0
- NUMERICAL
- --TAYLOR SERIES - THREE TERMS
--TAYLOR SERIES WITH EMPIRICAL
TERM
0.06
0.07
0.01
FIGURE 16.
0.02
0.03
0.04
0.05
0.08
Q09
0.10
0.11
0.12
DIMENSIONLESS HEAT AND MASS TRANSFER PROFILE FOR EXAMPLE 2 WITH U(x)/U,, = 1 - x.
c!,-1 .,1-.
= 1 and the functions
The flow is two-dimensional so
and K become:
x
x
U(x) r
x
x
x
u3()
1u1()
=
x
u5()
] d()
Jo
Jo
= .8685()
.O73375()
.0098833()
dU(x)/Uco
2
x2
.O73375()
2(.8685()
.0098833()6)
-
=
(l.737() - .2935()
.O593()5)2
(1.737
R
-
R805(X)2
v (x)
K =
(2Re)2
w
U
.2965())
r
(.8685()2
= -.9319
.O73375()
-
.0098833(2)6)½
R
R
1.737(e)
x3
.2935(i)
R
Q593()5
The values needed to start the solution are
= 1
Klo = -.5
Proceedino with Steps 4, 5 and 6 of Section 5.2 with
= .05 orovides results which are qiven in Table 12.
Substitution of the aooropriate results into ecuation (105)
= 1.25 for
and then inteqratinq numerically gives Nu//i
the range from stagnation to separation.
Table 12.
x
0.0
0.0
esults of calculations for Example 3
K
f0'
c/
ll!(0,,K,A)
(Nu//)1 (Nu//)2
-.5000
1.0000
1.542
0.0
.7410
1.381
1.38
.1
.3410
-.5077
.9707
1.532
3.264
.7441
1.366
1.37
.2
.4849
-.5163
.9361
1.519
4.427
.7476
1.349
1.35
.3
.5972
-.5260
.8950
1.504
5.171
.7514
1.331
1.33
.4
.6938
-.5370
.8462
1.485
5.658
.7557
1.311
1.31
.5
.7808
-.5494
.7884
1.462
5.946
.7603
1.290
1.28
.6
.8614
-.5637
.7194
1.432
6.064
.7656
1.266
1.26
.7
.9375
-.5803
.6370
1.396
6.023
.7714
1.239
1.23
.8
1.011
-.5997
.5379
1.349
5.827
.7780
1.209
1.19
.9
1.082
-.6227
.4184
1.288
5.473
.7855
1.175
1.15
1.0
1.152
-.6508
.2739
1.208
4.955
.7939
1.137
1.11
1.1
1.222
-.6856
.0998
1.100
4.262
.8034
1.092
1.06
1.2
1.292
-.7305
-.1072
.9479
3.380
.8139
1.038
1.00
1.3
1.365
-.7910
-.3447
.7370
2.333
.8272
.9745
.928
1.4
1.442
-.8788
-.5951
.2401
.6389
.8097
.8587
.840
1.422
1.460
-.9044
-.6516
.0008
.0020
.7875
.8114
.817
1
2
Results of numerical solution.
Results of truncation of Taylor series after third term.
In order to calculate
from the Taylor
series the following limits are needed:
d
3u3
=
=O
= -. 2918
u1
=
d2I=O
21
4-(40u1u5
39u3
V
u3
w
= -.5478
20 2lu
dI=0
3
16u1u5
(2u1)/2
= .0421643
-
= -.0257
= .740987
II' (0,,K,A)
all' (0,,K,A)
.0300
l=0
I3o
B211' (0,,K,A)
=
= .07296
(2u1) 5/2
d2K
311t
)
3 u1
= -6
3K
2
-.0221
.00448
=0
(0,,K,A)
= -.524
3K
3211' (0,,K,A) I
-.119
3K
61
v
( /) = -.04517
The II' (0,,K,A) expansion for three terms becomes
This expansion
II' (0,,K,A) = .74099 + .0302 + .003842.
is compared graphically in Fig. 17 with that calculated
can now be
The following values of Nu/i/
numerically.
The percent error listed after te results
calculated.
uses the numerically integrated value of Nu//ij calculated
previously as the true value.
2do/sep
-
% error = 3.
1.21
x
sep
3/2
2(do/sep +
Nu//i
sep)
=
5/2
d2
+
sep
= 1.24
x
()
sep
% error = .8
Now that the separation point is known equation (109)
yields, d5 = -.000123. Equation (110) gives
= 1.24
8.4
Example 4.
Ideal flow around a sphere with injection
A sphere of radius P. is falling at a constant velocity
Potential theory gives U(x)/U = 1.5 sin()
through air.
for ideal flow around a sphere. Assume that K remains
Calculate the local Nusselt number,
constant at .5.
friction factor, separation point and average Nusselt
0 to separation. Assume A is
number for the region x
constant at .7.
c,-1
-.4--
,-.',-'
or axisymmetrical flow around a sphere:
r
The functions
sin
and 13 become:
X
x
U(x) r
1.5
d() =J
=
x
.5[2
sin3()d()
cos() (sin2() + 2)1
62
0.84
INHNuIIHIllHfflOHImHI1
HNI
0.82
Ii!ilhIO1Ni1Ii
-
0.80
II
1=
_IiNuuuuiuiiJoi
0.78
()
11111
0.76
0.74
1iIgIIiIII1uiIp.ui
1IU!iUII1
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
DIMENSIONLESS HEAT AND MASS TRANSFER PROFILE FOR EXAMPLE 3 WITH SUCTION
FIGURE 17.
= 1.737x - .2935x3 - 0593x5.
AT THE SURFACE OF A CIRCULAR CYLINDER AND
.
U(X)/Uco
cos() (sin2() + 2)1 cos()
R
[2
dU(x)/Uc,,
2
U(x)/U
X
d
1.5 sin4 ()
The necessary starting limits are
= .5
K(0 = .5
Carryina out Steps 4, 5 and 6 of Section 5.2 with
= .01 gives the results which are tabulated in
Calculation of the average Nusselt number
Table 13.
= .456
gives Nu//P
The following values are needed in order to compute
the Taylor series coefficients:
urn
d1
0
lirn
0 d2--p
dE2
= .0637851
= -.0671
II
= .262224
(0,,K,A)
all' (0,,K,A)
0871
L=0
2' (0,,K,A)
II
B0
2
-.152
I
shows that a representative
vs
A plot of
A
qood representation of
= T.
attained at
can be acquired by letting
64
is
Table 13.
f0
x
0.0
0.0
Results of calculations for Example 4
cv'
TT(0,,K,A)
(Nu//)1
(Nu/v1j)
.5000
.6594
0.0
.2622
.6423
.642
.1
.7535
.4477
.6206
2.828
.2574
.5717
.576
.2
.9168
.4172
.5972
3.004
.2544
.5375
.541
.3
1.034
.3877
.5739
2.993
.2513
.5085
511
.4
1.131
.3574
.5492
2.895
.2480
.4815
.484
.5
1.216
.3254
.5224
2.741
.2441
.4554
.457
.6
1.294
.2910
.4923
2.545
.2397
.4294
.432
.7
1.367
.2535
.4582
2.314
.2343
.4028
.407
.8
1.436
.2124
.4186
2.050
.2278
.3751
.382
.9
1.504
.1670
.3717
1.752
.2193
.3452
358
1.0
1.571
.1167
.3153
1.419
.2082
.3123
.333
1.1
1.638
.06138
.2444
1.042
.1921
.2734
.309
1.2
1.705
.00065
.1498
.5987
.1660
.2232
.284
1.290
1.768
-.05017
.0003
.0010
.0948
.1204
.262
1
2
Results of numerical solution.
Results of truncation of Taylor series after third term.
2
= e0 + e1
where
+
e22
e0 = .5
e1 =
d3
=
= -.3672
e2 = -.06803
aoproxiration
The value of e2 was selected so that the
In
order
to
0.1.
at
qives the true value of
calculate d1 and d2 let the 3 derivatives needed be
given by
1
= e1 = -.3672
d2i3
2e
----s-
2
= -.13606
,K,A) expansion becomes:
The f[' (0,
.0284
II' (0,,K,A) = .262224
-
.0ll82
This is comoared with the numerically computed II' (0,,K,A)
in Fia. 18.
Application of equation (107) to the above expansion
with both one and three terms gives the following
approximations for Nu//. The percentaqe error
listed after the results uses the previously found
average for the true value.
Nu//j
2dOV'sep
% error = 9.2
.498
=
1
cos
sep
+
sep
5/2
d
3/2
d1
1/2
2(d0
+
seo
=
1
cos ()5p
% error = 2.4
'ig. A-2 provides the value of H'(0,,K,A) = .09459 at
separation when K = .5. Eauation (109) then cTives
d5 = -.008702. Now equation (110) gives:
.457
s
:s
8.5
Example 5.
Injection from a cylinder with constant RAB
A gaseous species B is moving perpendicular to the axis of
a cylinder. Assume the external velocity profile is the
same as in Example 3. The conditions are such that the mole
fraction of species A remainS .544 over the entire surface
of the cylinder.
The Schmidt number is equal to .7.
The
solubility of B in A is neqliqible. Calculate the local
friction factor, local Nusselt number, separation point
and average Nusselt number for the region from x = 0 to
separation.
Solution
The functions
and
are the same as in Example 3.
Assuriinq ideal gas behavior, equilibrium at the surface
of the cylinder and complete insolubility of species B
in species A, we have:
XAw =
x
A
NB
544
=0
= 0
Substitution of these values into eauation (75) gives
XA
AB
.544 -
N
NA
1
= 1.19
.544
X
+ NB
It can be shown that:
= 1.0
This value of
found from Fjq
and the value of RAB allow K0 to be
A-27.
KI0 = .5
In order to start the computation,
Since R
is constant:
.8
dK
is needed.
RAB dli'
dK
d
A
and also
.JL
-
d
Jl'
dK
K
d
Eliminatingdli'
- gives
d30A
dK
d
=0
d
all')
K
RAB
1
d13
=0
ll/(AAB
1 + 23l
o
RAB
all)
K
=0
The followina auantities required to evaluate
dK'
are:
= .0419968
= .0455
=o
=0
K
d'
-t-
= -.354
=
2Ql8
Therefore,
d,
= -.0130
Carryinq out Steps 4-13 of Section 5.3 gives results which
are tabulated in Table 14. The average Nusselt number is
= .475.
found by numerical inteqration to be
Table 14.
Results of calculations for Example 5
x
0.0
0.0
cfv"
f0
1.000
ll(0,,K,A)
Nu/V
.9692
0.0
.2933
.5467
0.1
.3410
.9707
.9540
2.033
.2925
.5369
0.2
.4849
.9361
.9357
2.726
.2915
.5261
0.3
.5972
.8950
.9136
3.141
.2902
.5141
0.4
.6938
.8464
.8869
3.379
.2886
.5009
0.5
.7808
.7888
.8545
3.477
.2866
.4862
0.6
.8614
.7205
.8147
3.449
.2841
.4697
0.7
.9375
.6391
.7653
3.303
.2808
.4509
0.8
1.011
.5420
.7031
3.038
.2763
.4294
0.9
1.082
.4265
.6233
2.648
.2700
.4041
1.0
1.152
.2902
.5183
2.126
.2606
.3732
1.1
1.222
.1332
.3751
1.453
.2451
.3331
1.2
1.292
-.0355
.1552
.5535
.2055
.2622
1.243
1.324
-.1032
.0000
.0000
.1703
.2103
SUMMARY
9.
1.
The method of calculation proposed by Sisson [121 works
well because the Bi function is defined such that it
forces f' = 0 at r = 0. Thus the second derivative of
the stream function, which is given by the expansion
f'' (0,)
= f0(O,B0) + 2
f
(OB) +
can be approximated very well by similar solutions at
the value o' i.e., f'(0,Bü).
The
function has been
shown to be a function of the parameters B and K; if
equation (35) is substituted into equation (25)
the
exact form for equation (39) is given by
,
1
f
(0)
or f (0) = i
= -0
=0
0 n=0
is also a function of the parameter (K/aB0) J=O
Since in most practical problems K is nearly
constant or (K/Bi30) n=0 is not large, the boundary
condition was replaced by f1(0) = 0 as in equation (39)
so that B1 will deend only on B and K. This procedure
produces little error in the B1 runction as shown by
calculations for high rates of injection at constant
velocity from a cylinder where (K/BB)max
Ficr.
0.5. (See
7.)
2.
The most accurate results are obtained by solving
Sisson's formula (100)
equation (64) by numerical means.
is recommended as it is both accurate and provides a
stable solution near
= 0.
3.
When the separation point is unknown, the Taylor series
expansions (92) and (93) give accurate solutions near
Ecuation (107) gives very good approximations of
= 0.
Nu//. The results for truncation after one term are
in error by no more than ten per cent.
4.
If the separation point is known, equation (110) provides
an approximation of Nu//
which is in error by no more
than one er cent.
71
REFERENCES
1.
H. Blasius, Grenzschichten in Flussigkeiten mit kleiner
Reibung, Z. Math. Phys. 56, 1-37 (1908)
(Translated in
N.A.C.A. TM 1256, 1950.)
.
2.
N. Frosslinq, Evaporation, heat transfer, and velocity
distribution in two-dimensional and rotationally symmetrical
laminar boundary layer flow, N.A.C.A. TM 1432 (1958)
3.
H. Gortler, A new series for the calculation of steady
laminar boundary layer flows, J. Math. Mech. 6, 1-66 (1957).
4.
E. M. Soarrow, K. E. Torrance and L. Y. Hung, Flow and heat
transfer on bodies in crossflow with surface mass transfer,
Paper tresented at the Third mt. Heat Transfer Conf.,
A.S.M.E., Chicago, August 7-12 (1966).
5.
E. Wraqe, Ubertragung der Gortlerschen Reihe auf die
Berechnung von Temperaturrenzschichten. Teil I, DVL
Bericht 81 (1958).
6.
D. W. Clutter and A. M. 0. Smith, Solution of the incomnressible boundary layer equation, Doucrias Aircraft Co. Aircraft
Division Encrnq. Paner no. 1525 (1963)
7.
D. B. Spalding, Mass transfer through laminar boundary
layers
1.
The velocity boundary layer, mt. J. Heat Mass
Transfer 2, 15-32 (1961)
8.
D. B. Spalding and S. W. Chi, Mass transfer through laminar
boundary layers
4.
Class I methods for predicting mass
transfer rates, Tnt. J. Heat Mass Transfer 6, 363-385 (1963)
9.
K. Pohlhausen, Zur naheruncrsweisen Integration der
Differentialqleichung der laminarer Reibunqsschicht, Z.
Angew. Math. Mech. 1, 252-268 (1921)
10.
E. Elzy and C. E. Wicks, Transoirational heat transfer from
a cylinder in crossflow including the effects of turbulent
intensity.
To be published in CEP Symposium Series, Heat
Transfer
Seattle (1968)
11.
H. 3. Merk, Rapid calculations for boundary layer transfer
using wedge solutions and asymptotic expansions, 3. Fluid
Mech. 5, 460-480 (1959)
72
12.
R. M. Sisson, A new method for the calculation of momentum,
heat and mass transfer in laminar boundary layer flow,
M.S. thesis, Oregon State University, Corvallis, Oregon
(1967)
13.
R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport
Phenomena, Chapters 18, 19, 21, Wiley, New York (1960)
14.
w. Mangler, Zusammenhang zwischen ebenen und
rotationssymmetrischen Grenzschichten in kompressiblen
Flussiqkeiten, Z. Anciew. Math. Mech. 28, 97-103 (1948).
15.
H. Schlichting, Boundary Layer Theory, 4th ed., McGrawHill, New York (1960)
16.
E. Elzy and R. M. Sisson, Tables of similar solutions to
the equations of momentum, heat and mass transfer in
laminar boundary layer flow, Engnq. Exp. Station Bulletin
no. 40, Oregon State University, Corvallis, Oregon,
February (1967)
17.
W. E. Stewart and R. Prober, Heat transfer and diffusion
in wedge flows with rapid mass transfer, mt. J. Heat
Mass Transfer 5, 1149-1163 (1962)
18.
H. L. Evans, Mass transfer through laminar boundary
layers
7.
Further similar solutions to the b-equation
for the case B = 0, mt. J. Heat Mass Transfer 5, 35-57
(1962).
19.
D. B. Spalding and H. L. Evans, Mass transfer through
laminar boundary layers - 3.
Similar solutions to the
b-equation, mt. J. Heat Mass Transfer 2, 314-341 (1961)
20.
W. H. Kays, Convective Heat and Mass Transfer, McGrawHill, New York (1966)
21.
J. G. Knudsen and D. L. Katz, ''1uid Dynamics and Heat
Transfer, Chapter 3, McGraw-Hill, New York (1958)
22.
H. Gortler and H. Wittinc, Einiqe laminare
Grenzschichtstromunqen, berechnet mittels elner neuen
Reihenmethods, Z. Anoew. Math. Phys. 9, 293-306 (1958).
23.
D. C. Leigh, The laminar boundary layer equations. A
method of solution by means of an automatic computer,
Proc. Cambridge Phil. Soc. 51, 320-332 (1955)
24.
H. Gortler and H. Witting, Zu den Tanischen Grenzschichten,
Osterreichisches Ingenieur-Archiv 11, 111-122 (1957)
73.
25.
I. Tani, On the solution of the laminar boundary layer
equations, J. Phys. Soc. Japan 4, 149-154 (1949).
26.
J. Newman, Blasius series for heat and mass transfer,
mt. J. Heat Mass Transfer 9, 705-709 (1966)
27.
E. Schmidt and K. Wenner, Warmeabgabe uber den Umfang
Forschung auf
eines angeblasenen geheizten Zylinders.
dem Gebiete des Ingenieurwesens 12, 65-73 (1941).
(Translated N.A.C.A. TM 1050, 1943)
28.
J. A. Schnautz, Effect of turbulence intensity on mass
transfer from plates, cylinders, and spheres in air
streams, Ph.D. thesis, Oregon State University, Corvallis,
Oregon (1958)
29.
H. H. Sogin and V. S. Subramanian, Local mass transfer
from circular cylinders in crossflow, J. Heat Transfer,
C83, 483-493 (1961)
30.
C. C. Windinq and A. J. Cheney, Jr., Mass and heat
transfer in tube banks, md. Engng. Chem. 40, 1087-1093
(1948)
31.
G. M. Zanp, The effect of turbulence on local heat
transfer coefficients around a cylinder normal to an
air stream, M.S. thesis, Oregon State University, Corvallis,
Oreqon (1950)
74
NOMENCLATURE
Dimensions are given in terms of mass (M), length (L)
and temperature (T).
,
time (t)
area of the surface, L2;
a
coefficients in the series (34)
dimensionless;
bn
coefficients in the series (87), dimensionless;
C1
constant in equation (58);
C ,
constant in equation (61);
cn
coefficients in the series (92)
C
heat capacity at constant pressure for species i,
p
,
,
(35) and (51)
dimensionless;
L2t2T;
C
total molar concentration, moles L3;
c
local friction coefficient defined by equation (68)
or (69), dimensionless;
d
n
coefficients in the series (93)
,
dimensionless;
D
binary diffusivity, L2t1;
f
stream function, dimensionless, see equation (20);
similar stream function, dimensionless, see
equation (35);
f
n
auxiliary stream function of order n, used in
equation (35)
dimensionless;
,
heat transfer coefficient, (ML2t2)L2tT', see
h
equation (70);
k
thermal conductivity of the fluid, (ML2t2)L-t-T-;
K
dimensionless mass transfer rate defined by
equation (47);
x
mass transfer coefficient, moles L2t(mole fraction),
see equation (71);
75
L
arbitrary reference length, L;
m
molecular weight of species i, M(moles)-;
m
exponent in equation (59) , dimensionless;
molar flux of species i with respect to stationary
coordinates, moles L2t-;
NU
local Nusselt number defined in equation (72),
dimensionless;
energy flux at wall, Mt3, see equation (77);
r(x)
radius of axisymmetrical body, L, see Fig. 1;
R
flux ratio, dimensionless, see equation (76);
Re
Reynolds number, UL/v, dimensionless;
s
exponent in eauation (108) , dimensionless;
t
time, t;
T
temperature, T;
u
longitudinal velocity, Lt;
coefficients in external velocity distribution,
see Section 8.3;
U1
constant in equation (59);
U(x)
mainstream velocity, Lt;
reference velocity, Ltl;
v
transverse velocity, Lt1;
x
longitudinal coordinate, L;
mole fraction of species A, dimensionless;
y
transverse coordinate, L;
thermal diffusivity, L2tl;
flow parameter defined by equation (21)
di:'ensionless;
76
0
flow parameter defined by equation (60) or (64),
dimensionless;
parameters in the f
see equation (34);
equations, dimensionless,
n
similar coordinate defined by equation (17),
dimensionless;
U
correction factor, dimensionless, see equation (81);
diffusivity ratio defined in equations (29) and
(30) , dimensionless;
v
11
ll
kinematic viscosity, L2t-;
profile function defined by equations (27) and (28)
dimensionless;
similar profile function defined by equation (36),
dimensionless;
auxiliary profile function of order n, used in
equation (51), dimensionless;
coordinate defined by equation (16), dimensionless;
p
mass density, ML3;
Tw
shear stress at solid surface, Mt2L1, see
equation (76);
rate factor, dimensionless, see equation (80);
Subscripts
d
the diameter is used as the reference length L;
sep
evaluated at the separation point;
V
quantity relative to momentum transfer;
w
evaluated at wall or surface conditions;
evaluated at
=
A
evaluated for species A;
T or A.B
quantity relative to energy or mass transfer,
respectively;
77
Supers cripts
denotes a quantity evaluated at the prevailing
mass transfer conditions;
Over lines
-
denotes an average quantity;
1.
per unit mass;
per mole.
APPENDIX A
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FIGURE A-32
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APPENDIX B
121
TABLE B-la.
-1.
THE FUNCTION
-0.5
(30,K)
-0.2
-0.15
-0.1
-0.05
K
-5.
-4.
-3.
-2.
N)
-1.6
-1.2
-1 0
-o 8
-0.6
-0.5
-0.4
-0.3
-0.2
-0 1
0.0
0.1
0.2
0,3
0,5
0.6
0.8
0. 2 33256-1
0.203698-1
0.357854-1
0. 294184-1
0.6 1995 9-1
0.456076-1
0.787118-1
0.102821
0.141754
0.173171
0 226494
0.138539
0. 237153
0.189121-i
0.265207-1
0.391344-1
0.61 5372-1
0.753552-1
0.93 7507-1
0.105363
0.119229
0.136262
0s 146486
0.158386
0.172783
0.191479
0.219983
0.186881-1
0.260887-1
0. 382170-1
0.593264-1
0 720556-1
0.886192-1
0.988205-1
0.110700
0.124758
0.132856
0 141896
0.152176
0.164225
0.179138
0.199812
0.240398
0 184689-1
0.256696-1
0.373384-1
0 572567-1
0.690122-i
0.839889-1
0.930090-i
0 103286
0.115076
0.12 16'l
Os 128769
0.136534
0.145105
0.154755
0 166005
0.180071
0.200886
0.182546-1
0.252628-1
0.364963-1
0.553157-1
0 .6619Th-i
0.797909-1
0.878128-i
0.967779-1
0.106792
0.112231
0 117980
0.124061
0. 130 504
0 137349
0 144667
0. 152 584
0.16137 7
0.171768
0 186396
0.252467
.0
'.4
2.
3.
VALUES IN TABLE MUST BE MULTIPLIED BY THE POWER OF TEN INDICATED AFTER THE NUMbER
TABLE B-lb.
K
-4.
-3.
-2.
-1 .6
NJ
-1 2
-1.0
-0.8
-0.6
-0.5
-0.4
-0 3
-0.2
-o 1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.8
1.0
1.4
2.
3.
THE FUNCTION
l (0,K)
0.0
0.05
0.180448-1
0.248679-1
0.356885-1
0.534921-1
0 635873-1
0.759685-i
0.831394-1
0.910157-1
0.996132-1
0.178396-1
0i 176388-1
0, 244844-1
0.241117-1
0.341683-1
0 501590-1
0. 104177
0. 108910
0.3
0.2
0.1
0.4
13o
0,113796
0.118814
0.123933
0. 129105
0.134260
0 139297
0.144058
0 148299
0.151607
0.153217
0 140773
0.349131-1
0.517762-1
0.611610-1
0.724744-1
0.789144-i
0.858777-1
0.172495-1
0.233973-1
0i 327638-1
0.47 1899-1
0 168762-1
0.227215-1
0.314627-1
0.445305-1
0. 7 50770-1
0.548155-1
0.635955-1
0.683725-1
0.812674-1
0
0. 9332 46-1
0 .87 7673-1
0 70 7641-1
0.972016-1
0.101155
0.105156
0.109165
0.113126
0.116964
0.120574
0.123811
0.126469
0.128255
0.128747
0.127329
0.115014
0.910931-1
0.944365-1
0.977626-1
0.783841-1
0.808911-1
0.833511-1
0.657293-1
0.b79827-1
0.900592-1
0.918956-1
0.934157-1
0.945292-1
0.951302-1
0.950981-1
0 942 996-1
8 36340-1
0 83 3402-1
0.810704-1
0.795784-1
0 745180-1
0.666022-1
0.448593-1
0
Os 589005-1
0.692691-1
0. 101025
0.104166
0.107107
0.109746
0. 111952
0.113552
0.114326
0.113991
0 112200
0.102653
0. 3 36 160-1
7 33363-1
0.925961-1
0.859796-1
0.747354-1
0.438260-1
0. 130303-1
5 12269-1
0 587 341-1
0
0.627146-1
0.667606-1
0 727028-1
0 745669-1
0 763260-1
0 7 79446-1
0 7938 13-1
0 80588 3-1
0.165179-1
0.220814-i
0.302546-1
0.421366-i
0.480525-i
0 .4526 1-i
0 .78803-1
0 .61223-1
0.644535-1
0.659832-1
0 .b74279-i
0.687622-1
0.699576-1
0 .7098 lb-i
0.815113-1
0.717990-1
o 723700-1
0 82 0894-1
0 .72652t-i
0.822559-1
0.726030-1
0 721766-1
0.713314-1
0.700292-1
0.659507-1
0.8 19402-1
0 17412 1-1
0 367 946-2
0 59894 8-1
0 .43432-1
0 19996 b1
0.477801-2
VALUES IN TABLE MUST BE MULTIPLIED BY THE POWER OF TEN INDICATED AFTER THE NUMBER
TABLE B-ic.
THE FUNCTION
0.5
0,6
0.161737-i
0.214743-1
0.291301-1
0.399717-1
0.452266-1
0.508515-1
0.537055-1
0.565009-1
0.591437-i
0.603704-1
0.615101-1
0.625420-1
0.634430-1
0.641881-1
0.647501-1
0.651001-1
0.652083-1
0.650443-1
0.645788-1
0.637851-1
0.626404-i
0.592433-1
0.543824-1
0.413202-1
0.213745-1
0.575461-2
0.158427-1
0.208978-1
0.280812-1
0.380055-1
0.426967-1
0.476173-1
0.500667-1
0.524277-1
0.546161-1
0.556133-1
0.565254-1
0.573356-1
0.580253-1
0.585746-1
0.589628-1
0.591682-1
0.591688-1
0.589432-1
0.584712-1
0.577351-1
0.567205-1
0.538255-1
0.497976-1
0.390919-1
0.219971-1
0,659711-2
(,K)
0.7
0.8
0.9
1.0
0.152180-1
0.198282-1
0.261826-1
0.345727-1
0.383615-1
0.421946-1
0.440387-1
0.457665-1
0.473121-1
0.479924-1
0.485968-1
0.491135-1
0.495304-1
0.498346-1
0.500134-1
0.500537-1
0.499433-1
0.496703-1
0.492243-1
0.485966-1
0.477810-1
0.455766-1
0.426308-1
0.349474-1
0.220264-1
0.789331-2
0.149229-1
0.193313-1
0.253212-1
0.330669-1
0.364922-1
0.399019-1
0.415173-1
0.430114-1
0.443263-1
0.448957-1
0.453943-1
0.458122-1
0.461394-1
0.463655-1
0.464804-1
0.464737-1
0.463358-1
0.460578-1
0.456319-1
0.450518-1
0.443133-1
0.423559-1
0.397796-1
0.331132-1
0.217242-1
0.836983-2
0.146385-1
0.188574-1
0.245116-1
0.316801-1
0.347860-1
0.378349-1
0.392578-1
1<
U,
-5.
-4.
-3.
-2.
-1.6
-1.2
-1.0
-0.8
-0.6
-0,5
-0.4
-0.3
-0.2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.8
1.0
1.4
2.
3.
0.155244-1
0.203498-1
0.271008-1
0.362130-1
0.404200-1
0.447510-1
0.468690-1
0.488809-1
0.507119-1
0.515316-1
0.522704-1
0.529142-1
0.534482-1
0.538564-1
0.541227-1
0.542304-1
0.541632-1
0.539053-1
0.534426-1
0.527627-1
0.518563-1
0.493468-1
0.459319-1
0.369455-1
0.221545-1
0.730684-2
0.4055801
0.416844-1
0.421643-1
0.425784-1
0.429185-1
0.431761-1
0.433429-1
0.434103-1
0.433704-1
0.432155-1
0.429386-1
0.425339-1
0.419968-1
0.413242-1
0.395707-1
0.372922-1
0.314384-1
0.213180-1
0.875079-2
VALUES IN TABLE MUST BE MULTIPLIED BY THE POWER OF TEN INDICATED AFTER THE NUMbER
TABLE B-id.
K
-5.
-4.
-3.
-2.
-1.6
-1.2
-1.0
-0.8
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.?
0.3
fl.s
13o
THE FUNCTION
(,K)
1.2
1.4
1.6
1.8
2.0
5.0
0. 14099 5-1
0.135970-1
0.171634-1
0.131276-1
0.164206-1
0.205262-1
0.252358-1
0.270820-1
0.287603-1
0 294887-1
0.126882-1
0.157367-1
0.194591-1
0.236127-1
0.122761-1
0.820571-2
0.934832-2
0 104769-1
0.179726-1
0.230307-1
0.292128-1
0.317966-1
0.342601-1
0. 3 5380 3-1
0 363810-1
0.372221-1
0.3 75692-1
0.378596-1
0.380876-1
0.382471-1
0.383323-i
0.383375-1
0.382574-1
0 380870-1
0.378221-1
0 374590-1
0.369952-1
0 164292-1
0.34991 2-1
0.331612-1
'.4
0. 285167-1
2.
0.203547-i
0 928026-2
0. 2 17 102-1
0.270861-1
0.292591-1
0 312799-1
0.321768-1
0.329612-i
0 336016-1
0.338573-1
0.340643-1
0.342182-1
0.343148-1
0.343497-1
0.343191-1
0 342193-i
0.340470-1
0.337992-i
0.334739-i
0.330695-1
0 32 5852-1
0.313788-]
0.298681-1
0.260727-1
0.193335-1
0.957621-2
0. 301130-1
0.25 1953-1
0.266042-1
0.272030-1
0.277063-1
0 306082-1
0.307992-1
0
0 30948 1-1
0. 310516-1
0. 28345 5-1
0.311064-1
0.311096-1
0 284403-1
0 284231-1
0.283598-1
0 282486-1
0.280878-1
0. 3 10582-1
0 3 09497-1
0 307817-1
0 305523-1
0 302601-1
0.2 99041-1
0 294840-1
0.284537-1
0.271802-1
0 240076-1
0.183363-1
0 970946-2
2 80940-1
0.282381-1
0. 284137-1
0. 27876 1-1
0.276125-1
0.272965-1
0 269280-1
0.260353-1
0 249436-1
0.222439-1
0 173956-1
0.973059-2
0. 151051-1
0 184928-1
0.221782-1
0.235458-1
0.247398-1
0. 252 372-1
0.2 56473-1
0.259539-1
0.260631-1
0.261403-1
0.261834-1
0.261906-1
0. 11426 1-1
0. 11730 s-i
0.118989-1
0 119597-1
0.119934-1
0 119983-1
o 11989 3-1
0 119726-1
0 119480-1
0 11915 2-1
0.26 1601-1
0. 11874 2-1
0.260903-1
0.259798-1
0.258274-1
0.256322-1
0.253934-1
0. 118249-1
0.247842-1
0.240014-1
0 117672-1
0.117010-1
0.116265-1
0.115435-1
0 114521-1
0.113524-1
0 1 i124-1
0. 2 305 28-1
0. 10872 7-1
0.207217-1
0 102732-1
0.919167-2
0 712688-2
0.25 1107-1
0 16522 3-1
0.967508-2
VALUES IN TABLE MUST BE MULTIPLIED BY THE POWER OF TEN INICAT
AFTEk Ti-ft. NUMDER
OREGON STATE UNIVERSITY
ENGINEERING EXPERIMENT STATION
CORVALLIS, OREGON
LIST OF PUBLICATIONS
Bulletins-
No.
1.
I'relitriiirary Report err tIre ('oritrol of Stream l'i,llution ur ( )regon, by C. V. I .angtun
No.
2.
A Sanitary Survey of
No.
3.
No.
4.
No.
5.
and u.S. Rogers.
I '428. 15g.
lie \'ilIanrette
1430.
Valley, by 11. 5. Rogers, C. A. .\Iiiekrioire,
40.
anti C. 1). Adams.
TIre Properties of Cenisn -Sawdust Mortars, I ']ajn and with Various Ailini stOres,
by S. H. Graf and R. Ii. Johnson. 1930. 40f.
Interpretation of Exhaust Ga.. A rialysea, by S. II. Graf, ( . \V. I deeyoii, on! \V. II.
Paul. 1934. 25.
a! Reference to I 'rol it inis
Boi ler-\Vater 'Iroubi es anril Treat rneiits with Specr
Vestern Oregon, liv B. F. Sttnnrriers. 1(135. Noni-av anlalile.
0
A Sanitary Survey of tIle \Villaruetle River from Seitwood Bridge to the (olunilea,
by G. \V. Gleeson. 1936. 25ç'.
Gli-cson arid
No. 7. Industrial and l)otriestic \Vastes of the \Vitlariictte Valley, by G. \V.
F. Merryfield. 183). 50g.
Statistical Study of the l'rg,lictivc
No. 8. An Investigation of Sonic Oregon Sanils with aIi.
Graf. 1)37. 50.
Values of 'ft liv C. E. Tlionias an,! S.
1939 Progress Report ott tile lost Farm,
No. 9. Preservative Treatments of Fence Posts.
by T. J. Starker. 1938 25. Yearly progress retiorts, 9-A, 9-It, 5-C, 9.1),
').E, 'IF, '4-4.. tan'.
Aircraft, lie
No. 10. Precipitation-Static Baum interference I'hieneriienr,i Originating on
}. C. Starr. p43'). 75g.
No. 11. Electric Fence Controllers with Stuecial Referetice to Equitiment Developed for
No.
6.
No. 12.
Measuring Their Characteristics, by F..\. Everest. 1939. 404'.
Matlrenratics of Alignment Chart Construction \\ithout the Use of Determinants,
by J. R. Grittithn. 1940. 254'.
No. 13. Oil-Tar Creosote for \Vood Preservation, by Glenn Voorhies. 1940. 254'.
No. 14. Otitirn uini l'o,s-er arid Fconrotniy Air- Fuel Bali ii.. fur 1.0! iretSeil I 'ctrolc-unn loses, by
\V. H. Paul and M. I'ojiovich. 1941. 254'.
No. 15. Rating and (are of I)oiiiestie Satc,hitst Btnrners, by F. C. \Villey. t'14t. .254'.
No. 16. The Intrhirovennent of Reversible Dry Kiln Paris, liv A. I). }ingliea. 1441. 2 S.
No. 17. An 1 nverilory of Sasvoti II \Vaste in Oregon, by Glenn Vuuorli irs. 1442. 25g.
No. 18. The Use of the Fourier Series in the Solution of Beam l'roblcms, by B. 1'. Rufiner.
1944.
No. 19.
No. 20.
No. 21.
No. 22.
No. 23.
50g.
1945 Progress Report ott Pollution of Oregon Streams, by Frc,l lIIerrylielil soil
\\'. G. \Vilmot. 1945. 40g.
The Fishes 0f the \\illamrtette River Systeor in Relation to Pollution, by B. E.
l)ii,ick Sri,! Fcc,! Merrvtiel,t. 1945. -40c.
TIre U-ic of the Fourier Series in tire Solution of Beam-Column l'roblems, by
B. F. Ruffner. 1945. 254'.
hn,Iurtrial and Crty \Vastes, by Fred Merryficlil, \\'. B. Bollen, and F. C. Kacltelhioffer. 1947. 40f.
1 err-Year Mortar St rengtli lest s of Sonic 4 regon Sartils, by C. F. 'l'lioiiia.. sri,!
S. 11. Graf. 1948. 25g.
No. 21.
Space Iteattitg by Electric Railiantt Panels Sri,! by Reverse-Cycle, by Louis SIegel.
No. 25.
The Banki Water 'l'urliirre, by C..\.
1148.
SOc.
40 ç'.
iIocbrn,,r,-
,riul
1-re,l i\lerrs lie),!.
if Various tapers, \Voo,ls, and Fabric...
liv S.
I-el, 1849.
II. Graf.
No. 26.
Igniution lerirtleratures
Mar- 194'). 60g.
No. 27.
Cylinrlcr Ileaul 'l'elrltwratures ins Four Airplanes witit Contirtental A-65 Fb,ginrcs, lip
No. 28.
ltielesl,-,c Properties of 1totinias l-,r at 1tib t-rciiier,cies. lv I. 4. \Vittkupf any]
M. I). Macilori,tl,l. litiv 1849. 4Oc.
I)iele, Inc t'rotiertiv.. of h',,ui,Iero, l'ic it Itiab !-r,'titerrci,, bs I. I. ((itikojuf ii,,]
No. 29.
No. 30.
No. 31.
S. II. I.ouvv.
hirtv
1') I').
\I. 1). Macdor,al, I - Set 'tcrinlicr 1549. 4(1g.
l-:x;srolei! Shale .\ynregate ii, Stnttctinrah (orrerete,
(raf.
Aug 1951. 604'.
Improvements in the Field Distillatiorr
Aug 1552. ('Of.
127
liv I). I). Ritcltie ;,n,l S. II.
of Peppermint Oil, by A. P. Hughes.
No. 34.
A Gage for the Measurement of Transient Hydraulic l'resiures, by N. F. Rice.
Oct 1952. 40&
as
The Effect of Fuel Sulfur and Jacket Tenrlle rature on I ston Iti rig I\Vear
'ctersr,n.
I )ete united by ii adjoact cc Tracer, by M. I opovrch and If.
July 1953. 4O.
Pozzolanic Properties of Several Oregon lumicites, by C. 0. Heath, Jr. and N. R.
35.
Model Studies of Inlet Designs for Pipe Culverts on Steep Grades, by Malcolm
No. 36.
A Study of Ductile iron and Its Response to \Velding, by IV. R. Rice and 0. G.
No. 32.
No. 33.
'i
.
No.
No. 37.
No. 38.
Brandenburg.
50.
1953.
H. Karr and Leslie A. Clayton. June 1554.
40f.
I'aasclic. Mar 1955. (Se.
Evaluation of Typical Oregon Base-Course Materials by Triaxial Testing, by
M. A. Ring, Jr. July 1956..(0.
Bacterial Fermentation of Spent Sulfite l.iquor for tue Production of l'rotein
En. 39.
Conceu teat e Air oral Feed Suppli-men by 1 lerman R. Arnberg. Oct. 1956. 50 f.
\Vood Waste Disposal ann 1 tili cation, by R. W. ltouhel, M. Nrnrtlicraft, A. Van
Vliet, M. i'opovich ..-ug. 1958. $1 WI
N,. -0.
Tables
No. 41.
Er,giirecririg ('alculatinirs of Xlorirnrrturri, I In-at arid Mass Trairsier 'llir,,ugh l.arn,rrar
Ilniurrilary I_ayers, nv F. Flcy-.nr,i (. .1. Myers. July 19(8. $1.50.
1,
.\lllnin-ntnrrr, linat and Mass l'rarrsferof Sinular- Slnlutioris Ii, tile I<qualiinris
in la,r,irrar Itniurinlary l.ayer Flu,,, by F. Flzv arid if. Xi. Srssori. ic!,. 1967.
Ii
CircularsNo.
I.
No.
2.
A Discussion of tIne l'rnrperties atr,l Economics of Irteis Use,1 in Oregon, I,y C. E.
'l'hornas anil G. D Keerirts. 1929. 25.
.\djirsriirerrt of Autorriotive Carburetors for Econoir,y, by S. If. Graf annl (. II'.
N0.
3.
Elements of Refrigeration for Snrall Conrniercial I'lants, h IV. 11. Martin.
No.
4.
Some Eirgiireerirrg Aspects of Locker arid Home Cold-Storage Plants,
No.
5.
No.
N0.
6.
7.
No.
S.
(;le,on.
1930.
Noise avai lalile.
Noire available.
Martin. 193S. 25ç'.
Re fri ge rintioii Aptil real ions to Certain Oregon I n,lustri es, by
2Sf.
\\'.
nv
IV, H.
H. Mart in.
t '(40.
The Use of a Technical Library, by XV. E. Jorgenson. 1942. 25.
Saving Fuel in Oregon Honres, by E. C. \\'illey. 1942. 25ç'.
Technical Ap1iroacli to tire Utilizatiorr of \Varnirise Motor Ftiels, by II'.
1944.
1935.
11.
l'aul.
2Sf.
No. 9.
No. 10.
No. Ii.
Electric and Other Types of House Heating Systenrs, by 1.ouis SIegel. 1946. 2H.
Economics of Personal Airplane Operation, by IV. 3. Skinner. 1947. 25.
Digest of Oreginrr l.aiiri Surveying Laws, by C. A. Mockorore, M. P. Coopey,
No. 12.
No. 13.
K.
No. 14.
No. 15.
Ii. B Irving, and NA. Bucklrorn. 191S. 25.
'I'he .-\lnnrtinirrrr I iinluvt rv of the Nortirwest, by J. Grarrville Jensen. 1950. 25c.
Fuel Oil 1fecuirrrriernts of Oregon and Sorrtlierra Washington, by Chester
Sterrett. 1950. 25
Market for Glass Coritairrers in Oregon and Soutirern \Vashington, by Chester K.
Sterrett. 1951. 25g.
Proceenliirgs of the 1951 Oregorr State Conference on Roads and Streets. April
1951.
flOg.
No. 16.
No. 1 7.
No. 18.
No. 19.
\Vater \Vorks Operators' Manual, try Warren C. \Vesigarrh. Mar 1953. 75g.
1 'mcccl hugs of tue 11153 Xi, ri hive r Con fe rerrce on lfoz,,l Bui liii rig. July 1983. 60 f.
No. 23.
Researclr aird Testing irr the Sclio,,l of Eirginieeririg, by M. Popoviclr. May 1957.
24.
N0. 25.
No. 26.
Proceenliiigs of the 1957 Xortlirrest Conference nut Roanl Buildiirg, July 1957. $1.00.
i'roceedirrgs of tire 1959 Nortlrsvest Conference orr Road Building, Aug 1959. $1.00.
Ifvsni,ircli Act ivitin-s in lie School of Engineering, by 1. C. Kniuulsrin Nov 1960.
Proc ceIlings of the I 55 N,,rilnnvest Conference ,lir Roail Building. Jtrrre 1955. 60f.
Review for Eingineerirrg Registration, I. Fundanrentals Section, by Leslie A. Clayton. Dec 1955. 60.
No. 20. Digest of Orcgorr 1.1,1,11 Surveying l.aws, by Kennetlr J. O'Connell. June 1956. 75g.
Review
for Engineering Registration, 2. Civil Engineering, by Leslie A. Clayton
No. 2t.
arid Marvin A. If.nig. _Iirly 1956. $1.25.
Revie,r for Eiigirreerirrg Registration, 3. 7,1ei,airicai Engineerirug, by Clrarles 0.
N0. 22.
Heath, Jr. Feb 1957.
No.
$1.25.
,
I
No. 27.
i'noceeilinigs of tIle 1960 N,,rtliwcst Higlirray Eirginieeririg C,,nfercrice, Die 1960.
No. 28.
l'roceeilinigs of tIle 1962 North,, cot Roinnis arid Street Conference, Mar 1962. $1.0)).
$
1.1)0.
128
No. 29.
No. 30.
No. 31.
N
.12.
No. 33.
No .14.
No. .15.
No. .11,.
NIl. .17.
I'rocee dingo of tile I';lentii l'iicific Nortllsvest lioltistrial \Vastc Conference- 1963,
Sept 1963. $10).
I 'roceedl ii gs of the 1964 N orliiwest Road'. and Streets Conference, J nec 1 064. $1.00.
1964.
Activities iii tile Scilloli of Eiigiiieeritig, Ily I. (. I<iiu,lseii, Seth.
Research
Tile toe ot a lccliiiical I.ihtrary, Ily If. K. \ValIlroii, Oct. 1064. 35ç'.
I
Irllceel hug'. 0f tIle lOpi Surveying tool 2iIaljIing Confereuice, Itils 10)
ole 9ls. 1sf.
If. %V.
\'tonl Resi1Iute iulcolertltull,u iii L_1lee IllirlIers,
kilillislil,
Research ACtivIties in the Sellout of Engiuleeriilg, t064-6(, Iw I. I
IllIItIIl_I,
lI)
N uveuuier 191,1,. 7Sf.
liuduistry, ily \V. F. Engesw'r. ,\llg. 1067. (''lI.
lillIustriat F;ilgilleeritlg
lilt- 1(66 NllrtIllcest Ifllalhs 51111 Street (IliltereilCe, Nov. I '(67. $1.25.
I'ullceel lfllgs
II
Ill
Reprints-
No.
No.
1.
I,Ire I ,nc I luslulatllr Ie.stlllC
Inst ruuuleil t5, 1 y I'. ( ). McM iliac. ]feIlri
Assoc. 1027. 2l).
MetlIolls of
No.
tell from I 'roe N \V E ICC
I .1
I
\VaIer ('Ileuilistry, by It. K.
Some Anoullalies of Siliceous Matter in Boiler
lIly.
(il 1)_is,
('I,lllilllsilllil.
SIllIllIlers. kept 1111111
I 1. Mc 11 ian.
3. Asphalt Fmiiulsiou Ti-cat neil 1 1 'resents Ralilo Interference, by I.
None
avtlihaille.
ReprInted from Electrical \Vest. Jan 19.15.
kelirillted
1,Ic7ifii;Iml.
4. Some (Ilaracleristics (If A-C Coillluctllr Corlimla, by F. 0.
from I'. cci rIcat EngIneering. Mar 1033. None available.
IlarlIcIl.
SIc Mihlaut an,I 11.
5. A Rallio lilterfereilce 3leasIlring Iulstrluulleiht, hr F. 1)
Rellrinicli frllulu Electrical Eiigiiceriiig. Aug. 1935. tOf.
2.
I 111111
No.
tfesllits (If 'I' sts n iti1 1)lfferelit
'owcr
11111
111,1
II
1
I
NIl.
No.
6.
Water-Ga-- ItetlehiolI .\i'iarcmutiy ('omltrol
1(111
\\. II. l'atil, Reprimltel
111.1111
1
elI
No.
7.
C
.51.
'I,
No. II).
Nile
I 'Ill,.
>teauli ( ,euler,Il ion
.11111
NIl.
\elll
ii,hlilIO
IV
Irllmlu
Iliiri1iuig
..\,r
I '1)1,
l,Illl,
II
II),.
If.
SuIuIlIIlers,
11.
No.
1.1.
No
I I.
Repriiitell from
llerltiuug
K, If. EllIreligI'. IfelIrhiltell
It. Ii. I';Ilil
'IS-Co ElectrIc Emigimie Jilliicatlli-, i,
front ( )rcgnn State Techmuical Recorli. Nov t')35. lOg.
1.1111
tclutllerature, iIy \\'. II. Xt;Irtlmu and F. C. \\'illey. Ifellrinteul
I-lnuuillllt)
front Power 'lam Euigineering, Feli 10.17. NIlile available.
heat Tramisft-r Efllciel,cv of Raii',u- 'nit'., IV \\'. .1. \\'ai-,II. Ifejlriulleli frolli
Electrical EuIgIIeeriIIg Aug 1937. None avaitaille.
I'llc
.11111
11111
1.
F. \\'aIernlan. Rellruited from Concrete. Nov
111,17 .N OilIS a caiblilic.
12.
by
Natio,ual I'etroleuuui Nests.
IcO 1,11,115.
No. it. Design of Concrcte Mixtures, ily
".11.
.
Engue Exhaust Gas ('onullositlon,
\\'aler-\\'Ise Refrlgeralloil, ily IV. II. \ltlrtiil
Power. Italy 1939.
11111
10.
It. E. Summers. Itellrumltell from
I 'Illarity Limits of tIle Sphere Gall, ly I". (I. SIcMmllan. Reprinted front Al RE
10 u
-r ran sad ions, hal 50. Mar 19,11),
Heat Transfer, by It'. i Short. ltellriltei front Electrical
Influence of ('tell-lb
.
lIlt
Engumicering. NIIv 1039. I0.
If. F. Sumtiuuicrs. Iti-lhrilute-Il ft
of Metals,
No. I. ('iIrrosiIIiI Sill1 Self- I'rlltedtiouio,l Oct
19.15. III e.
industrial I 'ower. Sri It
ily
11. I". lfltffnl'r. tfIllriltell fillill
II,. Monoc Illilie I"tiseitigc Circular fin6 .-\il;IiV'.ls,
NI,.
Journal of tile Aeronautical ScIence-.. Liii 103'). 100,
No. I 7. 'I'lie l'ltotoclaatic SletlIlIll as 501 Aill iii Stre-s Analysis 11111 Sirnctnrai I)esigii, IV
Ruliin-r. Rc1,rluiie,l (cIlia .',er,I ItIg 1551. .'lllr 1')3'). l))i.
"nd hIlls l,f I IiIi'(;rllll-llI v-._ SI-s-,lil,i. CIII Ill I)llIugb,I'. 'ii', by I_s-c Gablie. Itellrintell
NI,. 16.
froni Tin' IiiIiII'rnlaul. I hue 111,10, 1 I),-.
I". It.
Nil. t'I. Stlllcllilllllelrle (;Iiclilsltiuils (If Exbl;III'.t (eN, 1,1 1;. II. (;iee'.11i, 1,111 100.
\\'ooIltielIl, J m'. Repri tIed frooi National I'l'trl,]euuin News. Nov 10.10.
-h.
Fyeru'sI
I-'
_\iulllll
fit-cs,
Lv
(1111
putt
161' AlllIlIcauiolI of t'ce,lllaek III \\'i,lc- Ilsn,i
II, 2)1.
If till luisiutuie of Ifadio Engineers,
Still 11. IC ioiln-tlhil. RelIrliutell from 19-Il,III
1,111
II
Ii.
I'.
'I-I)).
No. 21.
I Ill,
of
SIres-es l)uie to Sr-eIIi,,ltIr\ Ill'illllile. l,v It. I-. IflIhlIwi-. Ifellciutl-Il fill0, Fl-lIeAla,'
I"trt Nl,rlll,,c'.t l'lll,t,Il-lastRItI' (',,ill'crcl Ill. tlIill'r"ll% II) \V;I'.hiuigill,u.
1040,
lOg.
It'sfl Ileal Lw,s Ilack (If tttIIliall, rs, he F. 1. \\,hlt-y. IfeiII'lntell fi',lili IIciIthilg 11111
Venlilating. Nov 19411. IOç'.
i's
No. 23. Stress ('Ilnci-ntrsltlon i-actor,, in Main ),leiiti,er, l)ne I,, \\'eillell Siiffeulers,
\V. If. C berry, Rel,rinted from tile \\'el,ling J annual, If eseareli Slu)ll,lrnueni,
ICc 1041. 1515.
'i.
No. 24. Ifl(i,,00tal.I'IIi;Ir-I';Ittei,] hiaeei' for I )ii'I-cli,,il;Ii It 51511 le,Ist ,Aillellulas. 115' F.Ifs,!,,,
I',vercst 5111,1 II, S. l'riiellctt. tfellriltc,I fr,,iii i'is,c If The Institute (If
Eiugiueers. Slay 1042. 1 0l.
76(1. .73,311,, lIre SIelll,,,is of MinI' S,IIIIIhiiug, ill Ii. I'. .SIeII'it'. Ifeilci lit-li frlhiul hlle (OiIiihiI's
No. 22.
(If Siguuua Gattuna Epsilon. .1511 1042.
129
tOe.
No. 26.
llro;ide;ust ,\iitetit,as
Ifellit iiii.Iii)is. 0
I
Ni' avai lalile.
Sctt 1144
No. 27.
Ari'av,, C';,lciilatioii of Ra,I,atio,i Patterns; Impedance
\\'tl,ao, ritelit-it. I)ciiriiilcd (rim, ('iu,iiii,iiiicatiohis. Aug
1),,)!
Heat l,osses Th'-ouli Wi-tied \\'alIs, be F. ('. \Villey. Reprinted from ASIIVE
I0.
of,1 Au' Conilitionitug. June 1946.
Section of Heiiliite, Ii
1) .Me 7tlillao. Reprinted front Electrical Engineer-
j oiiriial
No. 2$. Electric lower in China, by I'.
No. 20.
inn. au l')47. lUc.
Tbc 'I'ratu',icuut Euuergv hletlo,il of Calculating Stal,ilitv, by I'.
No, .14.
l)iffusion Coefficients of Urgiit uc I .i, iuiils
C'.
Magnusson.
Reiwiiteil froiii A I FE lrauiaetii,ns, Vol 1,6. 047. 106.
No. 30, Observations Ott ,\rc 1 tiseliarges it low Ire' si,rc, by 7,1. T. Kofoid. Reprinted
10,.
fr,uuui li,itrital of Atiplicil l'li .ics . A10 ii 1049.
No. 31. lone- Rauir' l'lauuuiing for l'owi r Sttl,tlls , liv I. 0. McMillan. Reprinted from
Flectricil Eneiiuee rifle, l)ce 1045. lUç.
No. .13. heat Transfer Coefficients in Beds if Moving Solids, by 0. Levctus1iiel and
S. \Valton. Ret ,rinteil fruit, l'roe of the H eat 'Fran sfer and FIn)! ?iI celia,, ics
Institute. 1940. 10$.
.13.
('atalvtie lIe!,, ilrogcnation of Etliane liy Selective ()xiilation, by J. I'. Mc('ulloueli
.1. S. \Valtuo. Reprinted front lii,lustrial mi! Eutgitsecriutg Cltentistrv. July
1949.
Xi,. 35.
10g.
'1' raflsac lions, I') 9)).
No. 36.
No. 37.
in
Solution from Sue face 'I ensiun
Measure,uciits, liv It. I.. ())s,,uu aol J . S. \\'alton. Reprinteil fri,m Industrial
Engineerin p Clicutnstrv. Mar 1051. br.
'l'ruu'ic,its in Coupled l,ulitc'tanee-Caj:icitance Circuits Analyzeil in Tcriiis of a
Rolling-Bail .-hivalopuc. lv I', ('. 7ilapoit'son. Retirioteil from Vol (9, $51EE
11)6.
Geometric Mean l)istaitce of Anglc.Shatie,l Conductors, by I'. C. Magnussoii. Re'
iriiiteil from Viii 7)).. \I ER 'l'uoosieto,os. 1951. tl).
Energy Cltoose It \V,selv 'I'iolav for Safety Tontorrosv, by G. \V. Glceson. Re'
tin ut,'d from 'S lIVE ,l ouiinal Section iii I eating, l'uio,,g, awl Air Condition.
ing. Aug 105!. lii.
An Analysis of Co,i,htctor Vibratii,n Field I )ata, by R. F. Steidel, Jr and M. B.
Elton. Al FE conference taller tirescotel at l'acific General Meetutup, t',irtlanil.
Oregon Aug 23, 11)51. tO
N. .1'). 'Hi- lluniiliri'vs ('o,u'tal'I-( snipe, _uu,ui Enuie. liv V,. II, 'aol ui,! I. It, lIon,tdu'eys. ltclirinteil fi'i,ou 5.\E t)uartcnlv 'l'co,sactwuu. .\tlril 1992. lUg.
P
No. 4)) (is.fioliil Film Foe (lie cots hf I [eat Transfer io 1"ltmu,l,zeiI ('oil Ileuls, be
\Valton, R. 1.,. Olson, iii,! Octave Levecspiel. Retirinteul from industrial an,!
No. 38.
J
No. 41,
No. 42.
.
Emtgmnceriug ('liemnistry. June 1092. lOg.
Rr'st,,mr:,nt \'euutil,tion, liv \\'. ii. Martin. Reloinuel (rub The Sa,uitaria,m, Vol
14, No. 6, Mav-Jumue 1952. 106.
the lucille Northwest, liv l osu-phu Scluulcuo. Reprinted from
Electrochemistry
Jomirnal of In' Eiectrochetuieal Society lone 109.1. 2)10.
it,
Model Stn,bc,- of Tatiered In lets for Box ( 'olvcrts, liv hoe H. Shot-maker until
I es! e A. ('layton. Iteprmnlcil frottu Re'ea reh Re1 or t 15-B, Iii gli lvi)' Research
Board, \Vuisliu,ton, I). C. 1953. 206.
No. 44. Bed-Wall Heat Transfer in Fluidized Systems, by 4). I .cvei,sbocl and .1. 5. \\'attomu.
Ye,. 43.
IH'priuits from,, Heat 'I ram,sfcr-lteseareh Studies. 1994,
No. -iS.
No. 46.
FAll'!.
No. 47.
1)16,
Shunt ('atiar itors in L.arge Transn,ission Networks, liv E. C. Starr at,,! F, j
10g.
I tuirrumigtiiu. RcI,ri,iteil frotti l'ii,ver Apparau,m mi1 5, stems, 1)ee 1953,
The Design and Effectiveness of an Underwater Diffusion 1,ioe for bIte Disposal
if P cot u,liuliits t.iiuor, by It. F. A,nher an,! li. t Strang. lieturiuited fronu
.
July
I
us-),
[06.
Compare- Your lcuiiuds with this Setrvey, Iw .\rtliur I.. Roberts and l.yle F.
\Veatlierluce. Rciui'ioteul frooi \Vestern lnilu'trv. Icc 195.1. 10$.
Sonme Stream PolI,,tiom, I'rol,lem, and Abatement Measures l'u,dertalen in the I,'
cube Northwest, by ii. 11. Aml,crg. Retiri,ute,l from 'I'AI'I'I. Feb. 1955. 10g.
No. 49. Fabrication of a Zircon inn).! .i me,! Reae tio,t Vessel, by 0. Cli Paasclme anl A.
No. 48.
No. 50.
No. 51.
No. 52.
No. 53.
Killiu. Repr,tmueul fr,,u,i '['he \Velding Journal. Feb t'991. 20.
Heat Tramtsfer Betsvccuu It,tmniscuhle 1,i,jvtids, by S. S. Grover anil J. G. Emiodsen.
Reiirumueul (nun CIuenu,eal Engineering, No. 17. Vol 51. 106.
How Oil Viscosity Affects I'istsn Ring Wear, by H. Potioviclu atmil 1,. F. Job,,sott. Reprittmeil from Atttomotive Industries. January 1956. 10.
Intermittent Discharge of Stient Smilfite I.iituor, by Hernuan R. Amberg and Robert
Elder. April 1956. 10g.
Hv,lraulics of Box Culverto with Fush-T.a,lder Bafiles. by Roy H. Shoemaker, Jr.
I'roeeeclings of the Flieluway Researclt Board, Vol 35. 1956.
Riut,rututcd from
No. 54.
A Nutmuerical Solutioit 10 J)iittensiotual Analy'is. liv 0. Levenspiel. N. J, \Veinsteitt,
J. C. F. Li. Retirititeil frotu Imtdtmstrial attd Eutgineering Clmeniistry, Vol 48.
Fe!, 1956. 25c.
No. 55.
Tlicrnmal Conuluctivity of Liquid-Lii1uid Etutlsions, by F. H. Wang attul
Jatumes C.
Ktui! sen. Re printeil from I tudustri a I atol Engineermn p Cltentislry, Nov 1059. 106.
13C
No. 5. l,i,crtl Shell-Side Ilerit 'I'raitster (.oel)icients in the Vicinity rrt Segmental Raffles in
tubular Heat Enelratigers, by M. S. Guriisltanluariah anti I. G. Knudsen. lIetrinteil from Iletit Tr;uitr-fer-Chieago i'sue if AIChE Svntposiuin Series. 1050.
10 f.
No. 57.
I lard-Water Scaling of Finned Tubes at Moderate Temperatures, by II. K.
C. Kitudse it. Reprinterl from I leat Tranr,fer-Clticago issne of
itic(luer rind
A1ChE Svinposiuitt Series. 1930. lOg.
heat 'l'ransfer front Isotliernial Flat 'bites: his Exteio.ion of l'olilliausr-ns Suit,tion to loss aitI lR0h I'ranrlti Nttnilrer l"ltn,ls, by F. I). Fisher am1 .1. 1'.
J
y0.
s,
.
Knudsen. Reprinted horn (lienircal Engineering Progress Ssniposutin Ser,e.s
No. 2'). 1959.
lOf.
interference Attcitriatirrn on Energize! Ilipli-\oltage 'l'ransniission lutes:
hir-ri'rmreirrr-nt sort .\rjrIiertion, is I.. N. Slrritr . II. II. a-brig, urd It. S. (Sn,.
No. 5').
Itrtilii,
No.
tteprinteil from l'usver ;'ttrirtmiiii_ms S stettis. I lee I 95'). 1 lIf.
El 17)' Single tutu '[tent liutirlls- (onrluctnrs- -Influence of (ottrinetor l)ianteter unl
I
(it).
No. 6 I
-
Sir;umirl I )irnteier on Rrdiu I nflnencr- \'oliage and (nrona Initiation 7)'oltags-,
liv I.. N. Stumie. Itepriuterl front l'osver Atrparrtos aol Svr'tents. l)ec 1959. tOe.
local Shell-Side 1 feat Transfer Coefficients mr Pressure Prop in a Tubular 1 lest
C. Knudsen. It ci rioter I
Excltatuger with 0 ri flee I iaffle-,, by K. S. I .ee a nil
front .\ 1(1, K Journal, Dec 19(10. 109.
Itates of I teat l'ransfer front Short Sections of art Isotirernuat Pipe, by A. A.
laruuqrn and 1. C. Knudsemu. Reprinted front 1 leat Tramusfs-r-Bnlfato issue of
.
No.
No.
2.
1.1.
AICItE Svtuuirosiuunu Series. Mar 11)61. 1Dm'.
Drag- (',reltigie,tts it huts Iteyttolils Nttntbers lot Flute last Jitinterserl Studies, he
.5. 7)1. _l,rtres amtrl - C. Ktturisett. Reirritterl trortt .5 I('l,E _l'rurtt,l, hiam- 1)61. tIle.
i't-rst,if,rte ( )xirliz;rhle (arlorru hurl Ill) I) is :r Olmursture ri ( )rgatie l'r,llntir,tu itt Water,
miii I. K. (arthr I. 'ii. Crititirtur. I'. Ole rrvfit-lrl, I-. I. ltnrgess, I,. l'ttrlei-sr,ut,
'rite
f t Stir i't,rtlue I itrlits(ri;ml 'thistu.' Ct,itf, Olty
steP. Itelrrinte,l t erritu
J
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1,4.
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-
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trait lrriirmi \\ater l'r,lluuirrit (mmii-
liv
itirtriiu-. Iteirrotterl
tirrt Is.rlerrmi,rrit, _-\ug 10)11. tIlt.
E)Iect if Neuritis-c ('rrerrtttt I rr,t, Irrrtttatirrmt nI I'rr.itis-e Crania, Ire I. -h. ltearstit.
Itt'iii unit-ri fir iii l'rrss er .\trirarrtiis & Svsteirrs, I Icc 11)1,1
ES1AC .\lgelrrare (rrtrtirrtter, re I. (, I,rrrrmtev tori itI. I ,.7)lrrrgmtr.
16,Itusirlir
Itetri tiller1 irurtri I It 1': I'rait-.rtctirriis ru l',lectrrrmtic ( rrtntrotet-s, Sept I 9ij 100,
Ilrerr,rrl SIre-ut--, rut iii Itle.rlizerl \Virrs Structure, try 7sirrk l,eviitsrrit. Retrr,mtterl
rrturrt ri .\ernrspace Sciences. 101,!. tilt-.
C. I_ninnies-. Reirri,tted frmnirr
lire 1.55 I _\I. I'rniettiirtl-I'Iaiie Air,tlrrg ('rrnttpttter. ire
'i
.
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'sri. Sri.
l)eer 'l'rickttttg tillers,
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(nrittrrnl, hits
101,1.
lilt'
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I-. Ihrrritlrrtt's,lr. Re1rritrnn.'rl 1r,rirr Is-sIr Its- (ri. I .51,1-5, Itrtlrt'rt .0, Irtit Samittury
S. Dept if Fir_-altit. Knits-turn, anti
I-diet- (ctrir'r. I'ttirllt- Iterritir Sr'usirr \\eiirrrt', I '11,1 tilt-.
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of II, ri 'I r.rit'fs'c intl1 l'rr-sstirr- l.rr'e rir tile Vicimiti
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iii
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'the (atta,lirr,r
mrrrvuctir,it hen lrrtit-fer Frn,rtr ir'atrss ens,- Frirmtenl'I,tits-s, by it. 13. I'm rut,l I. C.
Kititrlseit. Itr-irriirtr fiorit 1 lreio,errl Eo-.rirns-ertttg I'rrrgress, ui 191,.). lOg.
run I'. It.
(rriss-fmrrrl. Its-lrrititerl fruit I-des'um-i,-al I-irgrr leeritig Nrrv 1963. 106.
C. Smlagtry
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1
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cml Orreiet , I )es I 'ilu.i. till.
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131
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F.
'tie us,
THE ENGINEERING EXPERIMENT STATION
ADMINISTRATIVE OFFICERS
CHARLES R. HOLLOWAY, Jr., President, Oregon State Board of
Higher Education.
ROY E. LIEUALLEN, Chancellor, Oregon State System of Higher Education.
JAMES H. JENSEN, President, Oregon State University.
G. W. GLEESON, Dean and Director.
J. G. KNUDSEN, Assistant Dean in charge of Engineering Experiment
Station.
J. K. MUNFORD, Director of Publications.
STATION STAFF
GORDON WILLIAM BEECROFI, cE., Highway Engineering.
J. RICHARD BELL, Ph.D., Soil Mechanics and Foundations.
RICHARD WILLIAM BOUBEL, Ph.D., Air Pollution.
FREDRICK JOSEPH BURGESS, M.S., Water Resources Engineering.
HANS JUERGEN DAHLKE, Ph.D., Stress Analysis.
EDWARD ARCHIE DALY, M.S., Nuclear Engineering.
WILLIAM FREDERICK ENGESSER, M.S., Industrial Engineering.
GRANT STEPHEN FEIKERT, M.S., E.E., Radio Engineering.
CHARLES OSWALD HEATH, M.S., Engineering Materials.
ARTHUR DOUGLAS HUGHES, M.S., Heat, Power, and Air Conditioning.
JOHN GRANVILLE JENSEN, Ph.D., Industrial Resources.
JAMES CHESTER LOONEY, M.S., E.E., Solid State Electronics.
PHILIP COOPER MAGNUSSON, Ph.D., Electrical Engineering Analysis.
THOMAS JOHN McCLELLAN, M.Engr., Structural Engineering.
ROBERT EUGENE MEREDITH, Ph.D., Electrochemical Engineering.
ROBERT RAY MICHAEL, MS., Electrical Materials and Instrumentation.
JOHN GLENN MINGLE, M.S., Automotive Engineering.
ROGER DEAN OLLEMAN, Ph.D., Metallurgical Engineering.
OLAF GUSTAV PAASCHE, M.S., Metallurgical Engineering.
DONALD CHARLES PHILLIPS, Ph.D., Sanitary Engineering.
JAMES LEAR RIGGS, Ph.D., Industrial Engineering.
JOHN CLAYTON RINGLE, Ph.D., Nuclear Engineering.
JEFFERSON BELTON RODGERS, A.E., Agricultural Engineering.
JOHN LOUIS SAUGEN, Ph.D., Control Systems.
ROBERT JAMES SCHULTZ, M.S., Surveying and Photogrammetry.
MILTON CONWELL SHEELY, B.S., Manufacturing Processes.
LOUIS SLEGEL, Ph.D., Mechanical Engineering.
CHARLES EDWARD SMITH, Ph.D., Applied Mechanics.
LOUIS NELSON STONE, B.S., High Voltage and Computers.
JESSE SEBURN WALTON, B.S., Chemical and Metallurgical Engineering.
LEONARD JOSEPH WEBER, M.S., Communications Engineering.
JAMES RICHARD WELTY, Ph.D., Transport Phenomena.
CHARLES EDWARD WICKS, Ph.D., Chemical Engineering.
Oregon State University
CORVALLIS
RESIDENT INSTRUCTION
Undergraduate Liberal Arts and Sciences
School of Humanities and Social Sciences (B.A., B.S. degrees)
School of Science (B.A., B.S. degrees)
Undergraduate Professional Schools
School of Agriculture (B.S., B.Agr. degrees)
School of Business and Technology (B.A., B.S. degrees)
School of Education (B.A., B.S. degrees)
School of Engineering (BA., B.S. degrees)
School of Forestry (B.S., B.F. degrees)
School of Home Economics (B.A., B.S. degrees)
School of Pharmacy (B.A., B.S. degrees)
Graduate School Fields of Study
Agriculture (M.S., M.Agr., Ph.D. degrees)
Biological and Physical Sciences (MA., M.S., Ph.D. degrees)
Business and Technology (M.B.A., M.S. degrees)
Education (M.A., M.S., Ed.M., Ed.D., Ph.D. degrees)
Engineering (M.A., M.S., M.Bioeng., M.Eng., M.Mat.Sc., A.E.,
Ch.E., C.E., E.E., I.E., M.E., Met.E., Min.E., Ph.D. degrees)
Forestry (M.S., M.F., Ph.D. degrees)
Home Economics (M.A., M.S., M.H.Ec., Ph.D. degrees)
Pharmacy (M.A., M.S., M.Pharm., Ph.D. degrees)
Summer Term (four, eight, and eleven week sessions)
Short Courses, Institutes, Workshops
RESEARCH AND EXPERIMENTATION
(Date indicates year established)
General Research (1932)
Agricultural Experiment Station (1888)
Branch stations at Astoria, union, Kiamath Falls, Ontario, Hood River, Aurora,
Pendleton, Moro, Medford, Burns, Hermiston, and Redmond.
Computer Center (1965)
Engineering Experiment Station (1927)
Forest Research Laboratory (1941)
Genetics Institute (1963)
Marine Science Center at Newport (1965)
Nuclear Science and Engineering Institute (1966)
Nutrition Research Institute (1964)
Radiation Center (1964)
Science Research Institute (1952)
Transportation Research Institute (1960)
Water Resources Research Institute (1960)
EXTENSION
Federal Cooperative Extension (Agriculture and Home Economics)
Division of Continuing Education, State System of Higher Education