Engineering Calculations of Momentum, Heat and

3b4./ulletin No. 41. July 1968

(I I

LL E C 0:4

bfEGON

Engineering Experiment Station

Oregon State University

Corvallis, Oregon

ORE STTEL8RA

r:f' I p.LLL, 1 19o8

Engineering Calculations of

Momentum, Heat and

Mass Transfer

Through Laminar Boundaries

E. ELLY

and

G. A. MYERS

THE Oregon State Engineering Experiment Station was establishedby 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 ina manner broadly outlined by the following policy:

(1)To stimulate and elevate engineering education by develop-ing the research spirit in faculty and students.

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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 ProfessorDepartment of Chemical Engineering

and

G. A. Myers, Research AssistantDepartment of Chemical Engineering

BULLETIN NO. 41

JULY 1968

Engineerinq Experiment StationOregon State University

Corvallis, Oregon

ACKNOWLEDGMENT

The authors wish to acknowledge the cooperation and financialsupport given this project by the Oregon State UniversityComputer Center and by the Department of Chemical Engineering.Gary Myers was supported by a NASA traineeship during thecourse 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-ORDERAPPROXIMATION AND COMPARISONS WITH OTHER WORK 28

HEAT AND MASS TRANSFER RESULTS OF THE FIRST-ORDERAPPROXIMATION AND COMPARISONS WITH OTHER WORK 41

EXAMPLES 52

SUMMARY 71

REFERENCES 72

NOMENCLATURE 75

APPENDIX - A SIMILAR SOLUTIONS

APPE)IX - B FUNCTION

1. INTRODUCTION

The calculation of heat, mass and momentum transfer is of majorimportance in many problems of engineering interest. A largenumber of methods for this calculation have been proposed forthe steady two-dimensional or axisymmetrical laminar boundarylayer with arbitrary external pressure distribution and constantphysical properties. The formally exact power series expansionmethods [1,2,3,4,5] and the finite difference technique [6]

produce reasonably accurate solutions, but require considerablecomputation. The approximate methods of Spalding [7,8],Pohihausen [9] , Merk [10,11] and Eckert [10) all requiresignificantly less numerical labor. This reduction in laboris unfortunately accompanied by a great loss of accuracy inmany flows, such as decelerating flow.

A new method has recently been proposed by Sisson L12] whichappears to be a reasonable compromise between ease of solutionand accuracy of results. The present report presents theapplication of this method to a wide range of problems. Thefigures necessary for this general type of calculation are alsoincluded.

An extensive graphical presentation of the similar solutionsto the boundary layer equations is to be found in Appendix A.A wide range of mass transfer rate K, pressure parameterand dimensionless property ratio A has been covered.

2. FORMULATION OF THE PROBLEM

2.1 The laminar boundary layer equations

The boundary layer equations may begeneral equations of change expressof momentum, mass and energy [13]

or axisymmetrical laminar flow of afluid, the steady constant-propertyequations are given by:

Equation of continuity of mass

derived from theing the conservationFor two-dimensionalpure or binaryboundary layer

u By+ = 0 for two-dimensional flow (1)

Bx By

1

+ = 0 for axisymmetrical flow (2)

Equation of motion

ldp B2u

dx

Equation of energy

2TU+V cx- (4)

Equation of continuity of a species in a binary mixture

BXA 2XAu+v------ D

AB ay2(5)

subject to the boundary conditions,

at y = 0:

and as y

U = 0 (6)

V Vw(X) (7)

T=T ()w

XA XA (9)

u = U(x) (10)

T=T (11)

XA(12)

Viscous dissipation, radiation and chemical reactionwithin the fluid, and the Soret and Dufour effects are

neglected. The coordinate system used is shown in

Fig. 1.

FLOW

/

FLOW

FLOW

TWO-DIMENSIONAL WEDGE

ARBITRARY TWO-DIMENSIONAL BODY

ARBITRARY AXISYMMETRICAL BODY

r

FIGURE 1. COORDINATE SYSTEMS

3

2.2 Transformation of variables

A more convenient representation of the boundary layerequations can be obtained by employing Meksyn's [11]

transformation of variables with Mangler's [14]

transformation included to reduce the equations foraxisymmetrical flow to the same form as for two-dimensional flow. A stream function, IL, is definedsuch that:

Lu=-- (13)r y

L iJ

v = - (14)r x

This definition of p immediately satisfies the equationof continuity (1) and (2). Here, L is an arbitraryreference length on the body in the flow system. Thefactor r/L implies Mangler's transformation fromaxisymmetrical coordinates. For two-dimensional flow:

r/L = 1 (15)

The x and y coordinates are transformed by:

=rx U(x)(16)

L2 L0

Uo:

1

Re2

U(x) r(17)LL

where U(x) is the velocity of the fluid at the edge ofthe boundary layer, tJco is a reference velocity and

ULRe= (18)

The stream function is now written:

1

2i(x,y) = UL () f(,) (19)

4

Substitutions of the definitions (16) , (17) and (19)into equation (3) yields a third order, non-linear partialdifferential equation

ff + f f + (1 f f ) = 2 (f f ) (20)

where the prime indicates differentiation with respect ton. The parameter 3 is defined by:

2 dU(x)

U(x) d(21)

The velocity components in terms of the transformationvariables are:

u(x,y) = U(x)f(,) (22)

r U(x)v(x,y) = (f + 2 + ( 1)f') (23)

L (2Re)½

The boundary conditions on equation (20) may now bewritten:

at r = 0,

and as Ti

f =0 (24)

vL (2Re)½f + 2 (25)

r U(x)

f -*1 (26)

Equations (4) and (5) will be transformed by introducingthe following dimensionless profile functions:

T(x,y) Tw

T -Tw(27)

and defining:

=xA(x,y) XAW

x -xAw

AT c

AAB DAB

(28)

(29)

(30)

The equations for thermal energy and mass diffusion are thentransformed to an identical form:

I fl

II + AfTT = 2A (f II (31)

With the assumption that T, T, XAW and xA areindependent of x, the boundary conditions on equation (31)become:

at r = 0,

as fl

II = 0 (32)

II 1 (33)

These non-linear partial differential equations (20) and(31), then, with their boundary conditions (24), (25), (26)

and (32), (33) , respectively, describe momentum, heat andmass transfer within a steady two-dimensional or axisyrnmetri-cal constant-property laminar boundary layer. The problemin any practical application is that of finding an adequatesolution to these equations for a particular externalvelocity distribution.

3. DEVELOPMENT OF THE SOLUTION

3.1 A series solution of the momentum equation [12)

The calculation of the momentum transfer in laminar boundarylayer flows involves the substitution of two series intothe momentum equation (20).

The following series are defined:

=

nian(n(0) (34)

and

f(n,) f0(n,0) + a()f(,0) (35)

For convenience, a1 is defined as

d130

a1 = 2 (36)

At the present time, the an forHowever, they will have similarSubstitution of series (34) andproduces the ordinary different

, t I I I

and

n > 2 are not of interest.arbitrary definitions.(35) into equation (20)

ial equations:

I I

f +f f0

0+(1-ff)=0 (37)0 00

III II I I II

f +ff -2(3 +l)ff +3f f1 01 0 01 0

I II

- (1 f f ) (38)1 0_U

=f0---.0 çB0

The functioni may now be defined by setting the

boundary conditions on equation (38) such that

7

at n = 0,

and as

= 0 (39)

f = 0 (40)

f'= 0 (41)

f 0 (42)

Similarly, if the an are chosen so that theri

appear asfree constants in the equations defining the n' it will bepossible to define the 13n by setting the boundary conditionsfor 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 zerocontribution to the boundary conditions (24), (25) and (26)on the moimntum equation (20) . They also make no contribu-tion to f'' at = 0, so momentum transfer calculations maybe made using only f' at n = 0.

The boundary conditions on equation (37) may now bewritten:

at = 0,

f =(2Re)½

= -K (47)0 r U(x)

= 0 (48)

and as fl--

fo 1 (49)

It may also be shown that

f'(0,0) = f(0,) (50)

3.2 Extension to heat and mass transfer

Equation (31) describing heat and mass transfer may nowbe considered. The following series is defineth

nlall(,0) (51)

Substituting equations (35) , (36) and (51) intoequation (31) , it is found that

lb + Af0T[0 = 0 (52)

and

H + Af I 2Af II = A(f II -) - 3Allf1 (53)1 0 1 0 1 0

13O0

Boundary conditions on these and the higher orderequations are

at = 0,

and for n > 1,

and as

and for n > 1,

110 = 0 (54)

11 = 0 (55'sn

111 (56)

I-f 0 (57)n

The function is defined implicitly by equation(34) , and, with knowledge of the an, the 13n and the function

it would be possible to evaluate the function

Equation (35) could then be used, with knowledge of theto calculate the exact values of f(,)

3.3 Various simplifications

Similar solution

When the external velocity distribution satisfies therelationship [71:

dU(x) ndx

= C1U (x) (58)

the substitution of the transformed variables intoequation (3) yields equation (37) and its correspondingboundary conditions. Upon transformation both the energyequation (4) and the continuity of species eauation (5)reduce to equation (52) with its transformed boundaryconditions.

The solutions to equations (37) and (52) , with K and oconstant, are known as "similar solutions" because thevelocity profiles at all points within the boundary layerdiffer only by a constant scale factor [15] . Tabulationsof the numerical solutions are available for large rangesof the parameters 13, K and A [16,17,18,19]. Appendix Acontains a graphical presentation of these solutions.This is discussed more fully in part 4 of this paper.

Flow over a wedge [20] , where

U(x) = (i

2mm + 1

(60)

and m-1v (x) C2 x (61)w

comprises an important class of similar solutions. Usingpotential flow theory [21] , it may be shown that theparameter corresuonds to a wedge with an included angleof (fl) , as illustrated in Fig. 1.

10

Merk similar method

The zero-order approximation is the truncation ofseries (34) after the first term, so that

= i() (62)

This method was proposed by Merk [11]. The Merk similarmethod unfortunately gives poor results for deceleratingflow.

First order approximation

A first-order approximation, which will also be calledthe present method, is made by truncating series (34)after the second term to obtain

d130

+ 2a:---

(63)

Rearrangement of equation (63) produces an ordinarydifferential equation

- (64)d 2i

with a boundary condition

at = 0

= (65)

= 0 the right side of equation (64) is indetermin-ate. By invoking L'Hospital's rule the boundary conditioncan be found in the form of the derivative with respectto . Consequently,

at = 0d0 1

1 + 213i d wSince f3(F) can be determined by equation (21) and since

l is a function of and K, equation (64) may be solvedto find an approximation of

11

The extension to the calculation of heat and masstransfer rates is simplified by the introduction ofanother approximation. The series (51) defining H istruncated after the first term to obtain:

H(ri,,K,A) 110(n,1301K,A) (67)

Thus II' may be evaluated from the similar solutions atthe proper value of 3ci. just as f'' has been.

Since will in most cases be evaluated using theapproximate equation (64) , there will be two approximationsused to evaluate II'. However, if values of ll for n > 1

were available, II' could be evaluated more accurately byincluding additional terms in the series (51) . At thepresent time no values of H11 for n > 1 are available. Itis anticipated that the calculation of heat and masstransfer coefficients using the approximate equations (64)and (67) will be of sufficient accuracy for most practicalapplications.

3.4 Thel

(130,K) function

The fB- function depends only on and K and is determinedby solution of equations (37) and (38) with boundaryconditions (39) to (42) and (47) to (49). These equationshave been solved by Sisson l2.J and the l functiontabulated for a wide range of and K. A graphical andtabular presentation of the function is provided inAppendix B.

3.5 Definitions

Transfer coefficients

The local friction coefficient, c, defined by

11

\) (_). yy=ocf½u,

wr'c7 ow be written in terms of the transformed variables

c = 2U2(x) (O,,K) (69)

L U2(2Re)½Co

12

and

The local Nusselt number, NU, defined by

.

Nu = =

)y=0Lfor heat transfer (70)

T k (T -T)w

kL A

Nu =

(_)y0LAB CDAB (x

for mass transfer (71)A XAw

may be written as

r U(x)Nu

j:j

u(c)2II (0,,K,A) (72)

Re

These expressions (69) and (72) , then, relate thedimensionless boundary layer functions f(,,K) and1T(,,K,A) to the dimensionless functions cf and Nu,which are commonly used in momentum, heat and mass transfercalculations.

Flux ratios

Other frequently used quantities are the dimensionlessflux ratios, R,

NA MA + NB 4B (NA MA + NB ?IB) rj

Rw w w

(73)V -pU c i2 w

NA C + NB C0 (NA p + NB C ) (T T)w A w -B w -A w -B

4h' qw

NA + NB xA XA(75)ww w=p=

AB Nx A -x

NA + NBw

13

where

3uT = Byy=0 (76)w

-k (77)y y=O

When all physical properties are constant and equalfor each species, the flux ratios may be written as:

pv (x)w K

pU c/2 f''(0,,K)(78)

cof

APCpVw(X) KAT

(79)RTh ll'(O,,K,AT)

cv (x) KAw AB

RABk ll'(O,,K,AAB)

(80)

The last two equations can be summarized in the sinqieexpression:

I

II (O,,K,A)(81)

Rate factors

The rate factors, 4, are the values of the corresponding Rwith the transfer coefficients evaluated for the case ofno mass transfer, K = 0.

pv(x)K

= = (82)PtJcoCf/2 f (0,,0)

v (x) KApw T(83)

T h II' (O,,O,AT)

cv(x) KAAB(84)

AB k ll'(0,,0,AAB)x

14

U

The general expression is

Correction factor

K A

H' (O,,O,A)

The correction factor, 0, iscoefficient for mass transfefor low or no mass transfer:

Scf

h

0V cf' 0T h'

(85)

the ratio of the transferr to the transfer coefficient

kx

AB k

and by inserting the definition of the transfercoefficients,

oH (O,,K,A)

(86)H' (0,,O,A) R

4. GRAPHICAL PRESENTATION OF SIMILAR SOLUTIONS

To facilitate the calculation of heat, mass and momentum transferthrough laminar boundary layers, the similar solutions have beenpresented in various graphical forms in Appendix A. Most of theresults were obtained by Elzy and Sisson [16.1 . The solutions,near or at the separation of the boundary layer, were taken fromthe papers of Stewart and Prober [173 and Evans [18]. Someextrapolation was done near the regions of separated flow. Theextrapolated regions are indicated by dashed lines.

The following figures are available in Appendix A:

Figure Axes

A-iA- 2

A-3 thru A-ilA- 12A- 13A-l4 thru A-18A-l9 thru A-23A-24 thru A-32

A-33 thru A-41

f' (O,130,K) vs Kll'(O,30,K,.7) vs K

A/3H' (0,0,K,A) vs K

eVvs 1 + RV

o vso vs 1 + RK vs 1 + R

ll'A/3 vs 1 + R

15

Variable Parameter(s)

-1<00<5-1<130<5

-1<00<5 and .l<A<20-. 1<00<5-. 05<13<50.<13o<5 and .l<A<0.<i3<5 and .l<A<co

and .l<A<20

-.15<130<5 and .1<A<20

5. APPLICATION OF THE FIRST--ORDER APPROXIMATION

5.1 Taylor series solution

Evaluation of the local transfer coefficients requires alocal knowledge of and K. The solution of the first-order differential equation (64) provides the function j.Since is given in tabular or graphical form, numericalsolution of equation (64) is necessary. However, it ispossible to expand in a Taylor series about = 0.

=n= 0

(87)

The bn are found by repeated use of L'Hospital's rule tobe:

b0 = (88)= 0

1(89)b =(

1 1 + 213i d

1 (1 d2IB 2b2 _i 2b1 ___) ) (90)b =(2 1 + 4 2 d 1 d BK

1 1 d3 ld

b3 = (

1 + 6i6b1b2 b1

B2

0

2 B2d2K

(91)dK 1 dh

4b2 b (()BK 1 d BK2 d BK

etc.

I,The functions f (0,,K) and II (0,,K,A) can also be

expanded in a Taylor series about = 0.

(0,,K)n=0

(92)

16

00

ll'(O,,K,A) (93)

n= 0

where, I

= f (O,,K) J (94)

I I I I

Bf (O,,K) dK Bf (OlK)) (95)c1 = (b1B d BK

(b

B21 f (0,,K) Bf (0,,K)C2 2

+ 2b2 B0

22'' 2

+ -S-B f (0,,K) dK Bf (0K))

(96jd BK2 d2 BK

d = II (0,,K,A) (97)0

Bll(O,,K,A) dK Bfl'(0, PKA)) ()d1l B0 d BK

2 B211 (0,,K,A)+ 2b2

Bil (O,,K,A)

2 l B02 B0

dK B211 (0,,K,A) d2K Bil' (O,K?A)) (99)BK2 d BK

The coefficients rapidly increase in difficulty ofcomputation, which, coupled with the slow convergence ofthe series, limits the range of an exact solution ofequation (64) to the region near = 0. The first fewterms are valuable in certain approximations as will beshown in Section 4.4. Table 1 contains some of thepartial derivatives necessary for the evaluation of theabove coefficients. The partials are tabulated for severalof the most commonly used

,K and A. The evaluation of

the partials was carried ou numerically with a differenti-ated Stirling's interpolation formula.

17

Table 1. Derivatives for Taylor series coefficients

*

1B2

1,

Bf2

B f Bil (.7)2

B II (.7)

K B2 B2 B2

0 - .5 -.l49 .429 1.00 -1.29 .102 -.3320 .0 -.272 1.34 1.30 -3.05 .209 -.858.5 - .5 -.0515 .0850 .652 - .389 .0468 -.0509.5 .0 -.0636 .125 .705 - .507 .0674 -.0948.5 .5 -.0671 .148 .725 - .580 .0871 -.152

1.0 -1. -.0214 .0224 .482 - .160 .0220 -.01341.0 - .5 -.0257 .0304 .514 - .197 .0300 -.02211.0 .0 -.0287 .0376 .537 - .228 .0389 -.03411.0 .5 -.0285 .0392 .541 - .239 .0455 -.0455

Bil (.01)2

B II (.01) BTI (.05)2 '

B II (.05) BTI (0.1)2

B II (.1) Bil (10.)2

B II (10.)

K B0 B02 B0 B02 B0 B02 B0 B02

0 0 .O859 -.0395 .0339 -.150 .0580 -.252 .754 -3.04.5 0 .00237 -.00386 .00981 -.0154 .0172 -.0264 .258 - .332

1. 0 .00125 -.00128 .00530 -.00524 .00945 -.00912 .158 - .122

* is

** f'' is f''(0,0,K)I I

II (.7) is II (0,0,K,.7)

5.2 Numerical solution

Discussion of differential equation (64)

The most accurate values of the local coefficients areobtained by solving equation (64) numerically to obtainthe function I3ü(). Since the right side of equation (64)is indeterminate at = 0 difficulties are sometimesincurred in numerical solution. For the case of idealpotential 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 velocityprofile U(X)/Uc,, = 1 - x/L offers no problem. It ispossible to avoid the instability by using the Taylorseries expansion of for a short distance and thencontinue on to separation with numerical integration.

Sisson 112] circumvented the instability by calculatingat station n+l from

h(3n+½ O

o+ (100)

n+l fl h + 2n+½1 1n+i

where

n+1 n+ h (101)

and

n+½ + (102)

The accuracy of equation (100) has been compared withRunga-Kutta and Euler's integration methods for the caseof no mass transfer and U(x)/U = 1 x/L. The resultsare given in Table 2. The close agreement of the methodsindicates that equation (100) provides sufficient accuracyfor most engineering applications.

Table 2. Comparison of Sisson's integration

formula (100) with Runga-Kutta and

Euler's methods

U(x)/U = 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 belowfor a general single integration step. For the firststep, and i3 will be determined by the boundarycondition (65)

Given: values of and

Determine: values of n+l and

Step 1

Choose a step length h and calculate n+lusing equation (101)

Step 2

Use equation (21) to calculate n+½' which is 3 evaluated

at+

Step 3

For the first trial, evaluate at o and Kfl+½.n+½ n

For successive trials, evaluate using evaluated

from n+½ 0n+½

+ o )(103)

n+½ n+l

Step 4

Calculate using equation (100)n+1

Steo 5

If the chanqe in the calculated values of is within

the desired error limit, accept the last n+1

value of .Otherwise, repeat Steps 3 and 4.

n+l

The procedure may now he repeated for the next integrationstep by starting at Step 1.

Procedure for calculation of local coefficients

The present method may be applied to the practicalcalculation of momentum, heat and mass transfer coefficientsusing 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). Then finda convenient method of determining x()

a. analyticallyb. as a successive approximationc. olotxvs

Step 2

Determine 1?() using equation (21) and U(x).

Step 3

Determine K() using equation (47) , v(x) and the fluidproperties.

Step 4

With 1() , K() solve equation (64) using integrationformula (100) to obtain

Step 5

Evaluate the dimensionless oradients f0(O,,K) andll(O,10,K,A) at the calculated values of and K()using Fig. A-1 through Fig. A-ll or tables of similarsolutions [16,17,18,191.

Step 6

Calculate the transfer coefficients c and Nu usingequations (69) and (72)

5.3 Average coefficients for heat and mass transfer

The average heat or mass transfer coefficient is given bythe integral expression:

22

( NudA(104)

fdA

Expressing the differential area in terms of x and thensubstitution of equation (72) for Nu yields:

,_( )

=

(r) U(x) ll(0,,K,A)d()

L L

x

r2 dI LJ (X)

L1

or introducing the transformed variable :

2 fl (0,,K,A)d

J1

(.)

Id()

L1

(105)

(106)

Since the step-by-step procedure of Section 5 .2 leads toa tabular function for II' (0,,K,A) equation (106) must beevaluated numerically.

Replacing II' (0,,K,A) in ecuation (106) with its Tayloi'series, equation (93) , and then integrating in thenumerator gives:

23

x

2 r X)

L L

L1

(d(l/2 l/2)

02 1

3/2d2 5/2 5/2l(3/2

+ 52 + .) (107)

Truncation of this series after one term produces goodresults. This approximation is compared with numericalintegration of the local coefficients in the examples inSections 8.2, 8.3 and 8.4. The two methods usually differfrom three to ten per cent. It is also shown how theaddition of two more terms reduces the difference toaround one per cent.

A modification of the Taylor series approach to averagecalculations is the truncation of equation (93) after thethird 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 forcingequation (108) to meet two specifications:

1) Equation (108) must qive the correct II' (0,,K,A)

for sep

2) When inserted into equation (106) , equation (108)must yield the correct Nu/V for the range = 0to sep

Values of the exponent s for various cases are listed inTable 3. It is suggested that s be taken as equal to tenif no better information is available. Then, knowingII' d5 can be comnuted from:

(II d0 dlsep d2ep)/p (l09

Combininq equations (106) and (100) and integrating gives:

24

=

r'L 2

Ird()

J() L L

L1

(d0(2 1/2d1

3/2 3/22 l + +

5/2 5/2d

21/2+ 52 l +

2l/2))(110)

Equation (110) has been used in the examples inSections 8.2, 8.3 and 8.4. In all three cases the errorwas reduced to less than one per cent. However, whendil' (0,,K,A) is positive at = 0, the addition of the

d empirical term does not greatly improve theresults. It can even give slightly worse results as isshown in the example of Section 8.4.

5.4 Calculations involving R, and 0

In many engineerinq problems R(x) is known instead ofv(x) . The following step-by-step procedure may he used.

Given: U(x) , P(x) and the fluid properties

Determine: the local transfer coefficients

Step 1

Determine (x) using equation (16) and U(x). Then finda convenient method of determining x()

a. analyticallyb. as a successive approximationc. aplotofxvs

Step 2

Determine 0(g) using equation (21) and U(x)

Step 3

Determine R() from R(x) and the relationship betweenx and found in Step 1.

25

Table 3. Exponent s for various cases

r(x)/L K A s

1 x/L 1 0 .01 10.11 x/L 1 0 .05 8.7l-x/L 1 0 .1 8.31 x/L 1 0 .7 7.2l-x/L 1 0 10. 7.8

2 sin x/R 1 0 .01 11.52 sin x/R 1 0 .05 10.72 sin x/R 1 0 .1 9.62 sin x/P. 1 0 .7 8.8

2 sin x/P. 1 0 10. 8.9

2 sin x/R 1 .5 .7 8.4

2 sin x/R 1 -.5 .7 9.2

1.5 sin x/R sin x/ 0 .01 27.41.5 sin x/R sin x/R 0 .7 11.11.5 sin x/R sin x/R 0 10. 10.81.5 sin x/R sin x/P .5 .7 8.8

1.5 sin x/R sin x/P -.5 .7 13.3

26

Step 4

Choose a step lenqth h and calculate n+l and n+½ fromequations (101) and (102) respectively.

Step 5

Use equation (21) to calculate n+½' which is 1 evaluatedat

Step 6

Estimate Kn+½ by extrapolating from the values of Kat the previous stations. Newton's formula for 3-pointextrapolation would be:

K+½ = 3Kn 3K.½ + Kn_l (111)

In order to begin at station n=0 use:

h dKK½ = K0

+ 2 dth=0Step 7

For the first trial, evaluate at On and Kfl½.

For successive trials, evaluate by usinq IBnfl2 fl+-calculated from equation (103)

Step 8

Calculate On+l from equation (100)

Step 9

(112)

If the chanqe in the calculated values of 13o is withinn+l

the desired error limit, accept the last value of tBon+l

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:

3 3K-K +--K -K1 (113)n+1 n

If this Kfl+½ agrees with the Kn+½ used in Step 7 withinthe prescribed error limits, accept the lastcalculated in Step 8. Otherwise, use the Kfl+½calculated from equation (113) and repeat Steps 7-liuntil agreement for Kfl+½ is attained.

Step 12

Evaluate the dimensionless gradients f01(0,o,K) andIT' (0,301K,A) at the calculated values of andKn+1 using rig. A-i through A-li or tablesof similar solutions [16,17,18,19]

Step 13

S SCalculate the local transfer coefficients cf and Nuusing equations (69) and (72)

Repeat Steps 4-13 until separation occurs or the regionof interest has been covered.

6. MOMENTUM TRANSFER RESULTS OF THE FIRST-ORDERAPPROXIMATION AND COMPARISONS WITH OTHER WORK

6.1 The circular cylinder with U(x)/U. = 2 sin x and K = 0

Fig. 2 shows the results of momentum transfer calculationsmade using the first order approximation with the externalvelocity distribution of U(x)/U = 2 sin x, which ispredicted by ootential flow theory for a circular cylinderin crossflow. Also shown are the results of calculationsmade by the approximate methods of 1erk [lii, Eckert [10],Spalding [711 and Pohlhausen [15], by the series methods ofBlasius [15] and Gortler [3] and by the continuationmethod of Gortler and Witting [22] . The separation pointspredicted by these different calculations are presented inTable 4.

The present method shows good agreement with the Blasiusseries and the calculations of Gortler and Wittinq. The

1.2

I.0

0.8

0.621*

0.4

0.2

00

POHLH4USEN METHOD

MERK SIMILAR METHOD [I

\GORTLER SERIES

\ ALONE [5]ECKERT SIMILAR METHOD

\\\SPALDING METHOD [vI- \\ \BLASIUS SERIES [151

GORTLER AND WITT

PRESENT METHOD

0.2 0.4 0.6 0.8 1.0

X RADIANS

1.4 1.6 1.8 2.0

FIGURE2 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

Method Separation Point

Eckert [10] 1.626

Merk [ii] 1.661

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 outto x = 1.4, where they fall off rapidly. Spalding'smethod is in reasonable agreement out to x = 1.6, whereit also falls rapidly. Pohihausen's method is low inthe region near the stagnation point and somewhat highin the region near separation, although it predictsseparation closely. With six terms, Gortler's seriesdoes not converge adequately for x > 1.6. Thus, forthis case, the first-order approximation appears to qivesubstantially better results for momentum transfercalculations than do other approximate methods.

6.2 The velocity distribution U(x)/U = 1 - x

The results of momentum transfer calculations by thepresent method for the external velocity distribution

= 1 x are shown in Fig. 3. Also shown are theformally exact calculations of Clutter and Smith [61 andGortler and Witting [22] . Calculations made using theMerk similar method are also shown, for they representthe zero-order approximation and thus are of specialinterest. Table 5 compares the predicted separationpoint with that predicted by other investigators. Seethe example in Section 8.2 for sample calculations.

Calculations made by the present method and by Gortlerand Witting [24] for U(x)/U = 1 xr, with n = 2, 3, 4,and 5, are shown in rig. 4. Table 6 gives the separationpoints predicted by several methods. Except for Spaldinq'smethod, the Merk similar method and the present method,all are considered formally exact.

For n = 1, 2, and 3, there is reasonably good agreementbetween the present method and the formally exact methods,although it appears that the present method is slightlyhigh in the region just before separation. For n = 4 and5, the present method is siqnificantly hiqher than thecalculations of Gortler and Witting. However, the lattercalculations are less reliable for these values of nbecause only two terms could he used in (ortler's seriesalthough Witting's numerical continuation was startedrelatively early [23]

6.3 The velocity distribution U(x)/U = (1 +

y-enturn transfer calculations for the external velocitydistribution U(x)/U = (1 + x)" have been carried outusing the present method. Fig. 5 shows the results for

31

Table 5. Separation points with U(x)/U = 1 x

Investigator

Merk [ii]

Spalding [71]

Leiqh [23]

Clutter and Smith [6]

Present method

Gortler and Witting [241

32

Separation Point

0.0867

0.1073

0.1198

0.1200

0.1241

0.125

Table 6. Separation points with U(x)/U = 1 x

Exponentn Method Separation Point

2 Tani [25] 0.271Present method 0.288Gortler and Witting [24] 0.290

3 Present method 0.408Gortler and Witting [241 0.409

4 Tani [25] 0.462Gortler and Witting [24] 0.485Present method 0.495

5 Gortler and Witting [243 0.552Present method 0.560

Table 7. Separation points with U(x)/U = (1 + x)

Exponentn Method Separation Point

1 Merk [11] 0.1045Clutter and Smith [61 0.1450Present method 0.1534Gortler and Witting [22] 0.161

1.5 Present method 0.0984Gortler and Witting [22] 0.101

2 Present method 0.0725Gortler and Witting [22] 0.075

2.5 Present method 0.0573Gortler and Witting [221 0.058


3.5 Present method 0.0404Gortler and Witting [22] 0.041


33

0.6

0.5

0.4

t) 0.3

g0.2

cxi

00

MERK SIMILAR METHOD [ii]

K=O

PRESENT METHO

CLUTTER AND-SMITH [6]

GORTLER AND...

WITTI11Gt21

0.02 0.04 Q06 0.08 0.10 0.12

x DIMENSIONLESS DOWNSTREAM DISTANCE

FIGURE 3. COMPARISON OF SEVERAL METHODS FOR MOMENTUM TRANSFER WITH U(x)/U = 1 - x.

0.5

0.4

0.3cc

U.' -I.-

0.2

ni

1 I I I

fl2 fl3 flz4 flz5

- G:GORTLER AND WITTINGE2s]

P; PRESENT METHOD \\.- p \\p .- p

G G G

I I

0 0.1 0.2 0.3 0.4 0.5 0.6

X DIMENSIONLESS DOWNSTREAM DISTANCE

FIGURE 4. COMPARISON OF THE PRESENT METHOD WITH THE CALCULATIONS

OF GORTLER AND WITTING WITH U(x)/U= 1 - x".

n = 1. Also shown are the calculations made by Clutterand Smith [161, by Gortler and Witting [22] and by theMerk similar method [111. Predicted separation pointsfor n = 1, 1.5, 2, 2.5, 3, 3.5 and 4 are given inTable 7.

For n = 1, the first-order approximation compares quitewell with the calculations of Clutter and Smith, andGortler aid Witting. In fact, the results at every pointlie on or between these formally exact calculations. Therelatively sharp curvature in the calculations of Clutterand Smith at x 0.13, however, indicates that they mayhave experienced difficulties with their numerical methodsand that their results past this point may not be reliable.For the higher values of n, the present method agreesquite well wjth the calculations of Gortler and Wittingin 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

method. The results are shown in Fig. 6. Also shown arethe calculations by Clutter and Smith [6] and by the Merksimilar method [11]. Table 8 gives the separation pointspredicted by four different methods.

In the region just before separation, 1.6 < x < 1.9, thepresent method gives results significantly higher than theformally exact calculations of Clutter and Smith. However,the Blasius-tvpe series employed by Schlichting predictsseparation after the first-order approximation does. Sincethe Blasius series gives good results on the cylinder, thecalculations of Clutter and Smith may not be reliable inthis region. It may be noted that in this mostly acceleratedand, for 0 < x < 1.2, reasonably similar flow the Merksimilar method gives reasonably good results.

6.5 A cylinder with injection at the surface

The present method was used to calculate momentum transferon a cylinder with several different rates of injection atthe surface. These results are comDared in Fig. 7 withcalculations made using the series method of Sparrow,Torrance and Hung [4]. The velocity distribution used wasa seventh-deqree polynomial correlation of Elzy's [10]data and is given by U(x)/U = l.79(x/R) 0.36276(x/R)+ 0.02323(x/R)5 0.010l53(x/)7.

36

0.6

0.5

0.4

0.39.4-

0.2

0.I

0

MERK SIMILAR METHOD [I ii

CLUTTER ANDSMITH [6)

PRESENT METHO

GÔRTLER SERIESALONE 13]

GbRTLER AND -

< WITTING [ul

0 0.2 0.4 0.6 0.8 1.0 12 1.4 16

X DIMENSIONLESS DOWNSTREAM DISTANCE

FIGURE 5. COMPARISON OF SEVERAL METHODS FOR

MOMENTUM TRANSFER. WITH U(x)/U = (1 + x)1.

r

Table 8. Separation points on a sphere

1Iethod

Merk [ii]

Clutter and Smith E.6

Present method

Blasius-type series [151

Separation Point

1.693

1.846

1.894

1.913

Table 9. Separation points on a cylinder

with injection at the surface

Present Sparrow, TorranceMethod and Hung [4]

0.137 1.482 1.479

0.697 1.434 1.428

1.771 1.350 1.349

I

1.0

0.8

0.6w 0

-9-

0.4

0.2

00

PRESENT METHOD

CLUTTER AND SMITH [s1

MERK SIMILAR METHOD [iii

0.2 0.4 0.6 0.8 1.0

X RADIANS

12 14 1.6 1.8 2.0

FIGURE 6. COMPARISON OF SEVERAL METHODS FOR MOMENTUM TRANSFER

WITH AXISYMMETRIC FLOW AND U(x)/U, = 1.5 sin x.

3.0

2.5

2.0

1.5

C.)

1.0

0.5

P: PRESENT METHOD

S SPARROW, TORRANCE

AND HUNG SERIES 41

fwd=0.137

1wd= 1.771

0.697 S

S

S

P

P

01' I I I I I

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

X RADIANS

FIGURE 7. COMPARISON OF THE PRESENT METHOD WITH SPARROW, TORRANCE AND HUNGSERIES FOR MOMENTUM TRANSFER ON A CIRCULAR CYLINDER WITH INJECTIONAT SURFACE FOR U(x)/U = 1.79x - .36276x3 + .02323x5 - .010153x7.

The parameter is a measure of the normal velocityat the surface and is defined by:

pvw wf=WU pU (114)

In this case is positive, and consequently the momentumtransfer coefficients are reduced.

The separation points predicted by these methods at variousrates of injection are given in Table 9. The resultsobtained with the present method compare quite well withthe formally exact calculations made using the series ofSparrow, Torrance and Hunq.

7. HEAT AND MASS TRANSFER RESULTS OF THE FIRST--ORDERAPPROXIMATION AND COMPARISONS WITH OTHER WORK

7.1 Introduction

Few accurate analytical solutions are available for heatand mass transfer in non-similar constant-propertyincompressible laminar boundary layer flow, so thecalculations of heat and mass transfer by the presentmethod are also compared to experimental data.

7.2 Analytical solutions for the circular cylinder

Fig. 8 shows the comparison between the first-orderapproximation and Newman's Blasius-series type calculations[26] which are formally exact. The external velocityprofile was U(x)/U = 2 sin x, K = 0 and A = .7. It isen that both methods agree very well. It is instructiveto 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, sothat the effect of the form of U(x) can be demonstrated.Fig. 9 shows the results. Blasius series methods requirecoefficients in the polynomial expansion for U(x) so thata fifth order polynomial does not allow Newman's [26]method to include higher order terms. l\s a result Newman'sresults are not accurate in the range 80-90 degrees, whilethe present method performs quite well.

The external velocity orofile U(x)/U = 2 sin x was usedfor the calculation of the heat and mass transfercoefficients for the followinq cases:

41

NJ

z

xi

0.2

NEWMAN'S METHOD [.26)

A =0.7KzO

PRESENT METH

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

Z04

0.2

PRESENT METHOD

A 0.7K=O

20 40 60

0 DEGREES

NEWMAN'S BLASIUS

,,'-SERIES M

80 100 120

FIGURE 9. COMPARISON OF THE PRESENT METHOD WITH NEWMAN'S SERiiSFOR HEAT AND MASS TRANSFER ON A CIRCULAR CYLINDER WITH

U(x)/U = 1.816x - .4094x3 - .005247x5.

7.3

1) K 0, A = .012) K=-.5,A=.73) K=0, A=.74) K=.5, A=.75) K 0, A = 10

The results are given in Fig. 10.

Experimental heat and mass transfer data for a circularcylinder

Heat and mass transfer data for a circular cylinder incrossflow taken from five investigations [27, 28, 29, 30,311 is compared in Fig. 11 with calculations made usingthe present method. The velocity distribution ofSchmidt and Wenner [271, correlated by a fifth degreepolynomial, U(X)/Ux, = 1.8155(x/R) 0.4094(x/R)3- 0.005247(x/R)5, was used in the calculations. Theparameter Nu/A°4Re°5 is recommended by 50gm andSubramanian [29] for the correlation of heat and masstransfer data taken at different values of A.

The present method shows reasonably good agreement withthe data up to the point of separation, which was calculatedto be 81.0° and which appears to vary in the data fromabout 77° to 90°.

The mass transfer data of Schnautz [28], taken on acylinder, is shown in Fig. 12. Shown for comparison arecalculations made using the iMerk [ii] and the Eckert [101similar methods, and the present method, all with A 0.7.Here the Chilton-Coihurn analogy [131 has been used toconvert Schnautz's data to A = 0.7. The velocity distribu-tion of Schmidt and Wenner as given above was used in thecalculations. An additional calculation was made with thepresent method using a fifth degree polynomial velocitydistribution, U(x)/U = 1.737(x/R) 0.2935(x/P)30.0593(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 wifhthe data until shortly before they predict separation, atx = 1.231 and 1.253 respectively. However, separation doesnot occur until about x = 1.45. The present method agreeswell with the data. The velocity distribution of Elzypredicts separation slightly earlier, at x 1.355, thandoes the velocity distribution of Schmidt and Wenner,which indicated separation at x = 1.416. Since the velocitydistributions were actually quite similar, it appears thatgood correlation of velocity data may be quite important.

44

12

U)

d,. 1.0

4D0

0U0.8

0.6

U)

c

0.4

z 0.2

0

L1

.

122,000 SOGIN AND SUBRAMANIAN [2JO 32,800 WINDING AND CHENEY® 39,800 SCIMIDT AND WENNER [2i1o 39,000 ZAPP [31)e 110,000 ZAPP [31]

9 2,900-97,000 SCHNAUTZ [28] -AVERAGE OF 7 RUNS

.

90

10 20 30 40 50 60 70

e DEGREES

FIGURE 11. COMPARISON OF THE PRESENT METHOD FORU(x)/U = 1.816x - .4094x3 - .005247x5 WITH

HEAT AND MASS TRANSFER DATA.

80 90

'I

I:

fl

I.-I

z

0.

iw-.

DATA OF SCHNAUTZ 12816

TURBULENT INTENSITY< 1%

o 2,9002,600

4 28,57037,100

Q 44,05097,000

p

MERK SIMILAR METHOD [iiJ-

ECKERT SIMILAR METHOD [io]-' k>\ a

PRESENT METHOD VELOCITY _-'DISTRIBUTION FROM: ELZY [IO1 /

SCHMIDT AND WENNER [21J-"

)c INDICATED SEPARATION POINT

0.2 0.4 0.6 0.8 1.0 .2 1.4

X RADIANS

FIGURE 12. COMPARISON OF PRESENT METHOD, AND MERK AND ECKERT

SIMILAR METHODS WITH SCHNAUTZ'S MASS TRANSFER DATA.

7.4 Experimental transoiration coolinc data on a circularcylinder

The series method of Sparrow, Torrance and Hung 4] andthe present method are compared in Fjq. 13 with the heattransfer data of Elzy lOj taken on a cylinder withtranspiration cooline. The injection parameter, 1wd'defined by equation (114). Two velocity distributionswere used. The velocity profile U(x)/U = l.665(x/R)+ 0.0393(x/E.) 0.1977(x/R) was used in the region nearthe stagnation point. The seventh decTree nolvno:'ialcorrelation of Elzv's L10 data,- discussed earlier, wasused over the rest of the cylinder. See the example inSection 8.3 for samnle calculations of a case with adifferent velocity orofile and suction.

7.5 results for deceleratinc flow

The external velocity profile U(x)/U = 1 x has beenused in the followinci cases:

1) K = 0, A = .012) K = 0, A = .73) K = 0, A = 10

The heat and mass transfer results are plotted in Pie. 14.See the example in Section 8.2 for sample calculationsof case 2.

7.6 Results or a sphere

The external velocity profile U(x)/U = 1.5 sin x has beenused in the following cases:

1) K=0, A=.0l2) K = -.5, A = .7

3) K = 0, A = .7

4) K = .5, A = .7

5) K = 0, A = 10

The het and mass transfer reults are plotted in Fig. 15.See the example in Section 8.4 for sample calculationsof case 4.

0.6

'a,

,° 0.4z

0.2

I II

I I I

S: SPARROW, TORRANCE AND HUNG

SERIES L41

0d°697

f 1.771

S

10 20 30 40 50 60

e DEGREES

FIGURE 13. COMPARISON OF THE PRESENT METHOD AND SPARROW, TORRANCEAND HUNG SERIES WITH ELZY'S TRANSPIRATION COOLING DATAFOR INJECTION AT THE SURFACE OFA CIRCULAR CYLINDER.

70

8. EXAMPLES

8.1 Ekample 1. Similar solution of flat plate with injection

Air is flowing over a flat plate through which theinjection rate is proportional to (Y½. Determine thelocal Nusselt number and local L

friction factor as a function of the blowing parameter,

(2Re)½,

for A = .7.

c,-1

For a flat = 0, = 1, and U(x)/U 1. As rn = 0,ecuation (61Y requires that v(x) = C2x½ for a similarsolution to exist. Since it is given that the injectionrate is proportional to ()½ or expressing this in ternsof v(x), L

v(x)/U = C(),

the problem has a similar solution.needed are:

x x

=i:

rd() d()

J- v(x)v(x)L (2Pe) 2

V2ReKU.. r U(x)/U

The rther expressions

c(U(x) )2 ft (0,,K) 2

(0,,K)L U (2Re) Re

or

c/ v' f(O,,K)

52

x r x U(x)(Re ½ , Rex ½Nu=Nu =--X L L L H (0,,K,A) = (-i----) (0,,K,A)

or

II (0,,K,A)

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 Nusseltnumber 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 anfunction, U(x)/U = 1 x/L. There isthrough the solid surface an A = .7.friction factor, local Nusselt number,and averaqe Nusselt number for the regseparation.

Solution

external velocityno mass transferredCalculate the localseparation pointion from = 0 to

The functions and B are derived for this two-dimensionalcase, i.e., r/L = 1.

x x

U(x) rd() = (1 ) d()

1 X

J0

= 1 - 2L

and

2 dU(x)/U d()2 1 2

U(x)/U- (-1) =

d() di - x 2 1

53

Table 10. Results of calculations for Example 1

K = j/2Re 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 showsthe results of Steps 4, 5 and 6 of Section 5.2 using astep increment of A = .002.

The solution may be approximated with the Taylor series(92) and (93). The following derivations and limits areneeded:

=131 =00 =0

2=

(1 - 2)2 so that = -2

d2i3 8 d213= - so that = -8(1 - 2)3 d2 =0

Tables B-i and 1 and Fig. A-i and A-2 supply thefollowing:

13 = .1291051

= -.272

(0,,K) = .4696

f'' (O,,K)= 1.30

(0,,K)= -3.053132

II' (0,,K,A)J=

= .4139

311' (0,,K,A)= .209

3211' (0,,K,A)= -.858

2

55

x

0.0 0.0

0.01 0.01005

0.02 0.02020

0.03 0.03046

0.04 0.04083

0.05 0.05131

0.06 0.06192

0.07 0.07264

0.08 0.08348

0.09 0.09446

0.10 0.1056

0.11 Q.1168

0.116 0.1236

0.1164 0.1241


f0 (cfv')1 ll(0,,K,A) (Nu//)1 (cfV)2 (Nu//)2

0.0 0.4696 0.4139

-0.01606 0.4483 6.213 0.4104 2.873 6.21 2.87

-0.03247 0.4257 4.086 0.4066 1.992 4.09 1.99

-0.04920 0.4015 3.082 0.4024 1.593 3.09 1.59

-0.06623 0.3755 2.443 0.3978 1.349 2.45 1.35

-0.08354 0.3476 1.979 0.3925 1.178 2.00 1.18

-0.1011 0.3173 1.612 0.3866 1.047 1.64 1.05

-0.1188 0.2840 1.306 0.3797 .9410 1.36 .951

-0.1366 0.2472 1.038 0.3716 .8514 1.11 .866

-0.1544 0.2053 0.7937 0.3617 .7720 .991 .795

-0.1720 0.1555 0.5563 0.3487 .6973 .723 .733

-0.1889 0.0914 0.3039 0.3297 0.6209 .561 .678

-0.1983 0.0109 0.0348 0.3000 0.5459 .471 .648

-0.19883 0.0002 0.0005 0.2957 0.5368 .466 .646

1 Results of numerical solution.2 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 theseexpansions are presented in Table 11. The expansion forII' (0,,K,A) is also compared with the numerical integrationresults in Fig. 16.

It is now possible to calculate values of iSii /v' from= 0 to the separation point calculated earlier

('sep = .1164) . Numerical integration of equation (105)using H' (O,,K,A) calculated by numerical means givesNu // = 1.53. The percent error listed after thefollowing approximate averaqe Nusselt numbers uses thisNu // as the true value.

d %/2

i // 0 sep= 1.61 % error = 5.2

()Rsepd1 3/2 d

V'(do/sep + sep + sepi:1 /V = = 1.54

x

sep

% error = .7

Using sep = .1164 in equation (109) yields,= -1.32 x 108. Equation (110) cTives:

ü /i = 1.53

8.3 Example 3. Flow around a cylinder with suction

A cylinder of radius R rests in a stream of air which ismoving perpendicular to the axis of the cylinder. Assumethat the external velocity profile is that of Elzy [10],U(x)/U = l.737() - .2935() .0593(). The velocity

R R Rof the air being pulled through the surface of the cylinderis given by v(x)

=-.9319.

Calculate the local friction factor, local Nusselt number,separation point and average Nusselt number for the regionfrom x = 0 to separation. Assume that A = .7.

57

0.45

0.40

I!

Ui 0.35

0.30

- NUMERICAL- --TAYLOR SERIES - THREE TERMS--TAYLOR SERIES WITH EMPIRICAL TERM

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Q09 0.10 0.11 0.12

FIGURE 16. DIMENSIONLESS HEAT AND MASS TRANSFER PROFILE FOR EXAMPLE 2 WITH U(x)/U,, = 1 - x.

c!,-1 .,1-.

The flow is two-dimensional so = 1 and the functionsand K become:x x

U(x) r x x x x= 1u1() u3() u5() ] d()

Jo Jo

= .8685() .O73375() .0098833()

2 dU(x)/Uco

x22(.8685() .O73375() - .0098833()6)

= (1.737

R.O593()5)2(l.737() - .2935()

- R805(X)2 .2965())

v (x)K =

(2Re)2w

U r

(.8685()2 .O73375() - .0098833(2)6)½R R R

= -.9319 x31.737(e) .2935(i)

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)and then inteqratinq numerically gives Nu//i = 1.25 forthe range from stagnation to separation.

Table 12. esults of calculations for Example 3

x K f0'c/ ll!(0,,K,A) (Nu//)1 (Nu//)2

0.0 0.0 -.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


In order to calculate from the Taylorseries the following limits are needed:

3u3- = = -. 2918d =O u1

21 2= 4-(40u1u5 39u3 ) = -.5478d2I=O 3 u1

u V3 w

= -65/2

= .07296(2u1)

d2K 20 2lu 16u1u5 v( /) = -.04517

dI=0 3 (2u1)/2

= .0421643

- = -.0257

II' (0,,K,A) = .740987

all' (0,,K,A).0300

I3o l=0

B211' (0,,K,A)-.0221

= .004483K =0

311t (0,,K,A) = -.5243K

3211' (0,,K,A) I -.1193K

61

The II' (0,,K,A) expansion for three terms becomesII' (0,,K,A) = .74099 + .0302 + .003842. This expansionis compared graphically in Fig. 17 with that calculatednumerically. The following values of Nu/i/ can now becalculated. The percent error listed after te resultsuses the numerically integrated value of Nu//ij calculatedpreviously as the true value.

2do/sep- 1.21 % error = 3.

x

sep3/2 d2 5/2

2(do/sep + sep) + sepNu//i = = 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 velocitythrough air. Potential theory gives U(x)/U = 1.5 sin()for ideal flow around a sphere. Assume that K remainsconstant at .5. Calculate the local Nusselt number,friction factor, separation point and average Nusseltnumber for the region x 0 to separation. Assume A isconstant at .7.

c,-1 -.4-- ,-.',-'

or axisymmetrical flow around a sphere:

r xsin

The functions and 13 become:

x X

U(x) rd() =J 1.5 sin3()d()

= .5[2 cos() (sin2() + 2)1

62

0.84

0.82

0.80

1=

0.78

()

0.76

0.74

HNI INHNuIIHIllHfflOHImHI1

Ii!ilhIO1Ni1Ii-

II _IiNuuuuiuiiJoi11111

1iIgIIiIII1uiIp.ui1IU!iUII10 0.2 0.4 0.6 0.8 1.0 1.2 1.4

FIGURE 17. DIMENSIONLESS HEAT AND MASS TRANSFER PROFILE FOR EXAMPLE 3 WITH SUCTIONAT THE SURFACE OF A CIRCULAR CYLINDER AND

U(X)/Uco

= 1.737x - .2935x3 - . 0593x5.

2 dU(x)/Uc,,[2 cos() (sin2() + 2)1 cos()

R

U(x)/U d1.5 sin4

X()

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

Table 13. Calculation of the average Nusselt numbergives Nu//P = .456

The following values are needed in order to computethe Taylor series coefficients:

urnd1

0

lirn d20 --pdE2

= .0637851

= -.0671

II (0,,K,A) = .262224

all' (0,,K,A)0871

L=0

2'II (0,,K,A)

-.1522 I

B0

A plot of vs shows that a representative is

attained at = T. A qood representation ofcan be acquired by letting

64


x f0 cv' TT(0,,K,A) (Nu//)1

0.0 0.0 .5000 .6594 0.0 .2622 .6423

.1 .7535 .4477 .6206 2.828 .2574 .5717

.2 .9168 .4172 .5972 3.004 .2544 .5375

.3 1.034 .3877 .5739 2.993 .2513 .5085

.4 1.131 .3574 .5492 2.895 .2480 .4815

.5 1.216 .3254 .5224 2.741 .2441 .4554

.6 1.294 .2910 .4923 2.545 .2397 .4294

.7 1.367 .2535 .4582 2.314 .2343 .4028

.8 1.436 .2124 .4186 2.050 .2278 .3751

.9 1.504 .1670 .3717 1.752 .2193 .3452

1.0 1.571 .1167 .3153 1.419 .2082 .3123

1.1 1.638 .06138 .2444 1.042 .1921 .2734

1.2 1.705 .00065 .1498 .5987 .1660 .2232

1.290 1.768 -.05017 .0003 .0010 .0948 .1204


(Nu/v1j)2

.642

.576

.541

511

.484

.457

.432

.407

.382

358

.333

.309

.284

.262

= e0 + e1 + e22

where e0 = .5

d3e1 = = = -.3672

e2 = -.06803

The value of e2 was selected so that the 1 aoproxirationqives the true value of at 0.1. In order tocalculate d1 and d2 let the 3 derivatives needed begiven by

= e1 = -.3672

d2i3----s- 2e = -.13606

2

The f[' (0, ,K,A) expansion becomes:

II' (0,,K,A) = .262224 .0284 - .0ll82

This is comoared with the numerically computed II' (0,,K,A)in Fia. 18.

Application of equation (107) to the above expansionwith both one and three terms gives the followingapproximations for Nu//. The percentaqe errorlisted after the results uses the previously foundaverage for the true value.

2dOV'sep= .498 % error = 9.2Nu//j

1 cos

1/2 d1 3/2 d 5/22(d0 sep + sep + seo

=

1 cos ()5p

% error = 2.4

'ig. A-2 provides the value of H'(0,,K,A) = .09459 atseparation when K = .5. Eauation (109) then cTivesd5 = -.008702. Now equation (110) gives:

.457

8.5 Example 5. Injection from a cylinder with constant RAB

A gaseous species B is moving perpendicular to the axis ofa cylinder. Assume the external velocity profile is thesame as in Example 3. The conditions are such that the molefraction of species A remainS .544 over the entire surfaceof the cylinder. The Schmidt number is equal to .7. Thesolubility of B in A is neqliqible. Calculate the localfriction factor, local Nusselt number, separation pointand average Nusselt number for the region from x = 0 toseparation.

Solution

The functions and are the same as in Example 3.Assuriinq ideal gas behavior, equilibrium at the surfaceof the cylinder and complete insolubility of species Bin species A, we have:

XAw = 544

x =0A

NB = 0

Substitution of these values into eauation (75) gives

XA

AB N

XNA + NB

It can be shown that:

.544 -= 1.19

1 .544

= 1.0

This value of and the value of RAB allow K0 to befound from Fjq A-27.

KI0 = .5

dKIn order to start the computation, is needed.Since R is constant:

.8

and also

dK RAB dli'

A d

- .JL Jl' dK

d K d

dli'Eliminating - gives

dKd30A

all')

d =0 d RAB K =0

1 d13 ll/(AAB all)

1 + 23l o RAB K =0

dK'The followina auantities required to evaluate are:

Therefore,

= .0419968

= .0455=o

= -.354K =0

d'-t- = 2Ql8

= -.0130d,

Carryinq out Steps 4-13 of Section 5.3 gives results whichare tabulated in Table 14. The average Nusselt number isfound by numerical inteqration to be = .475.


x f0cfv" ll(0,,K,A) Nu/V

0.0 0.0 1.000 .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

9. SUMMARY

1. The method of calculation proposed by Sisson [121 workswell because the Bi function is defined such that itforces f' = 0 at r = 0. Thus the second derivative ofthe stream function, which is given by the expansion

f'' (0,) = f0(O,B0) + 2 f (OB) +

can be approximated very well by similar solutions atthe value

o' i.e., f'(0,Bü). The function has beenshown to be a function of the parameters B and K; ifequation (35) is substituted into equation (25) , theexact form for equation (39) is given by

1f (0) = -- or f (0) = i

0 =0 0 n=0

is also a function of the parameter (K/aB0) J=OSince in most practical problems K is nearlyconstant or (K/Bi30) n=0 is not large, the boundarycondition was replaced by f1(0) = 0 as in equation (39)so that B1 will deend only on B and K. This procedureproduces little error in the B1 runction as shown bycalculations for high rates of injection at constantvelocity from a cylinder where (K/BB)max 0.5. (SeeFicr. 7.)

2. The most accurate results are obtained by solvingequation (64) by numerical means. Sisson's formula (100)is recommended as it is both accurate and provides astable solution near = 0.

3. When the separation point is unknown, the Taylor seriesexpansions (92) and (93) give accurate solutions near

= 0. Ecuation (107) gives very good approximations ofNu//. The results for truncation after one term arein error by no more than ten per cent.

4. If the separation point is known, equation (110) providesan approximation of Nu// which is in error by no morethan one er cent.

71

REFERENCES

1. H. Blasius, Grenzschichten in Flussigkeiten mit kleinerReibung, Z. Math. Phys. 56, 1-37 (1908)

. (Translated inN.A.C.A. TM 1256, 1950.)

2. N. Frosslinq, Evaporation, heat transfer, and velocitydistribution in two-dimensional and rotationally symmetricallaminar boundary layer flow, N.A.C.A. TM 1432 (1958)

3. H. Gortler, A new series for the calculation of steadylaminar boundary layer flows, J. Math. Mech. 6, 1-66 (1957).

4. E. M. Soarrow, K. E. Torrance and L. Y. Hung, Flow and heattransfer 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 dieBerechnung von Temperaturrenzschichten. Teil I, DVLBericht 81 (1958).

6. D. W. Clutter and A. M. 0. Smith, Solution of the incomnress-ible boundary layer equation, Doucrias Aircraft Co. AircraftDivision Encrnq. Paner no. 1525 (1963)

7. D. B. Spalding, Mass transfer through laminar boundarylayers 1. The velocity boundary layer, mt. J. Heat MassTransfer 2, 15-32 (1961)

8. D. B. Spalding and S. W. Chi, Mass transfer through laminarboundary layers 4. Class I methods for predicting masstransfer rates, Tnt. J. Heat Mass Transfer 6, 363-385 (1963)

9. K. Pohlhausen, Zur naheruncrsweisen Integration derDifferentialqleichung der laminarer Reibunqsschicht, Z.Angew. Math. Mech. 1, 252-268 (1921)

10. E. Elzy and C. E. Wicks, Transoirational heat transfer froma cylinder in crossflow including the effects of turbulentintensity. To be published in CEP Symposium Series, HeatTransfer Seattle (1968)

11. H. 3. Merk, Rapid calculations for boundary layer transferusing wedge solutions and asymptotic expansions, 3. FluidMech. 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, TransportPhenomena, Chapters 18, 19, 21, Wiley, New York (1960)

14. w. Mangler, Zusammenhang zwischen ebenen undrotationssymmetrischen Grenzschichten in kompressiblenFlussiqkeiten, Z. Anciew. Math. Mech. 28, 97-103 (1948).

15. H. Schlichting, Boundary Layer Theory, 4th ed., McGraw-Hill, New York (1960)

16. E. Elzy and R. M. Sisson, Tables of similar solutions tothe equations of momentum, heat and mass transfer inlaminar boundary layer flow, Engnq. Exp. Station Bulletinno. 40, Oregon State University, Corvallis, Oregon,February (1967)

17. W. E. Stewart and R. Prober, Heat transfer and diffusionin wedge flows with rapid mass transfer, mt. J. HeatMass Transfer 5, 1149-1163 (1962)

18. H. L. Evans, Mass transfer through laminar boundarylayers 7. Further similar solutions to the b-equationfor the case B = 0, mt. J. Heat Mass Transfer 5, 35-57(1962).

19. D. B. Spalding and H. L. Evans, Mass transfer throughlaminar boundary layers - 3. Similar solutions to theb-equation, mt. J. Heat Mass Transfer 2, 314-341 (1961)

20. W. H. Kays, Convective Heat and Mass Transfer, McGraw-Hill, New York (1966)

21. J. G. Knudsen and D. L. Katz, ''1uid Dynamics and HeatTransfer, Chapter 3, McGraw-Hill, New York (1958)

22. H. Gortler and H. Wittinc, Einiqe laminareGrenzschichtstromunqen, berechnet mittels elner neuenReihenmethods, Z. Anoew. Math. Phys. 9, 293-306 (1958).

23. D. C. Leigh, The laminar boundary layer equations. Amethod 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 layerequations, 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 Umfangeines angeblasenen geheizten Zylinders. Forschung aufdem Gebiete des Ingenieurwesens 12, 65-73 (1941).(Translated N.A.C.A. TM 1050, 1943)

28. J. A. Schnautz, Effect of turbulence intensity on masstransfer from plates, cylinders, and spheres in airstreams, Ph.D. thesis, Oregon State University, Corvallis,Oregon (1958)

29. H. H. Sogin and V. S. Subramanian, Local mass transferfrom circular cylinders in crossflow, J. Heat Transfer,C83, 483-493 (1961)

30. C. C. Windinq and A. J. Cheney, Jr., Mass and heattransfer in tube banks, md. Engng. Chem. 40, 1087-1093(1948)

31. G. M. Zanp, The effect of turbulence on local heattransfer coefficients around a cylinder normal to anair stream, M.S. thesis, Oregon State University, Corvallis,Oreqon (1950)

74

NOMENCLATURE

Dimensions are given in terms of mass (M), length (L) , time (t)and temperature (T).

a

bn

C1

C ,

cn

Cp

C

c

dn

D

f

fn

h

k

K

x

area of the surface, L2;

coefficients in the series (34) , (35) and (51)dimensionless;

coefficients in the series (87), dimensionless;

constant in equation (58);

constant in equation (61);

coefficients in the series (92) , dimensionless;

heat capacity at constant pressure for species i,L2t2T;

total molar concentration, moles L3;

local friction coefficient defined by equation (68)or (69), dimensionless;

coefficients in the series (93) , dimensionless;

binary diffusivity, L2t1;

stream function, dimensionless, see equation (20);

similar stream function, dimensionless, seeequation (35);

auxiliary stream function of order n, used inequation (35) , dimensionless;

heat transfer coefficient, (ML2t2)L2tT', seeequation (70);

thermal conductivity of the fluid, (ML2t2)L-t-T-;

dimensionless mass transfer rate defined byequation (47);

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 stationarycoordinates, 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

flow parameter defined by equation (60) or (64),0 dimensionless;

parameters in the f equations, dimensionless,see equation (34);

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 kinematic viscosity, L2t-;

11 profile function defined by equations (27) and (28)dimensionless;

ll similar profile function defined by equation (36),dimensionless;

auxiliary profile function of order n, used inequation (51), dimensionless;

coordinate defined by equation (16), dimensionless;

p mass density, ML3;

Tw shear stress at solid surface, Mt2L1, seeequation (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 prevailingmass transfer conditions;

Over lines

- denotes an average quantity;

1. per unit mass;

per mole.

APPENDIX A

79

.

S

S

ho-<

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82

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APPENDIX B

121

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0.00.10.20,3

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TABLE B-la. THE FUNCTION (30,K)

-1.

0. 2 33256-10.357854-10.6 1995 9-10.1385390. 237153

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0.203698-10. 294184-10.456076-10.787118-10.1028210.1417540.1731710 226494

-0.2

0.189121-i0.265207-10.391344-10.61 5372-10.753552-10.93 7507-10.1053630.1192290.1362620s 1464860.1583860.1727830.1914790.219983

-0.15 -0.1

0.186881-10.260887-10. 382170-10.593264-10 720556-10.886192-10.988205-10.1107000.1247580.1328560 1418960.1521760.1642250.1791380.1998120.240398

0 184689-10.256696-10.373384-10 572567-10.690122-i0.839889-10.930090-i0 1032860.1150760.12 16'lOs 1287690.1365340.1451050.1547550 1660050.1800710.200886

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0.182546-10.252628-10.364963-10.553157-10 .6619Th-i0.797909-10.878128-i0.967779-10.1067920.1122310 1179800.1240610. 130 5040 1373490 1446670. 152 584

0.16137 70.1717680 1863960.252467

VALUES IN TABLE MUST BE MULTIPLIED BY THE POWER OF TEN INDICATED AFTER THE NUMbER

NJ

K

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TABLE B-lb. THE FUNCTION l (0,K)

13o0.0

0.180448-10.248679-10.356885-10.534921-10 635873-10.759685-i0.831394-10.910157-10.996132-10. 1041770. 1089100,1137960.1188140.1239330. 1291050.1342600 1392970.1440580 1482990.1516070.1532170 140773

0.05

0.178396-10, 244844-10.349131-10.517762-10.611610-10.724744-10.789144-i0.858777-10. 9332 46-1

0.972016-10.1011550.1051560.1091650.1131260.1169640.1205740.1238110.1264690.1282550.1287470.1273290.1150140 8 36340-1

0.1

0i 176388-10.241117-10.341683-10 501590-1Os 589005-10.692691-10. 7 50770-10.812674-10 .87 7673-1

0.910931-10.944365-10.977626-10. 101025

0.1041660.1071070.1097460. 1119520.1135520.1143260.1139910 1122000.1026530 83 3402-10. 3 36 160-1

0.2

0.172495-10.233973-10i 327638-10.47 1899-10.548155-10.635955-10.683725-10 7 33363-10.783841-10.808911-10.833511-10.657293-10.b79827-10.900592-10.918956-10.934157-10.945292-10.951302-10.950981-10 942 996-10.925961-10.859796-10.747354-10.438260-10. 130303-1

0.3

0 168762-10.227215-10.314627-10.445305-10 5 12269-10 587 341-10.627146-10.667606-10 70 7641-10 727028-10 745669-10 763260-10 7 79446-10 7938 13-10 80588 3-10.815113-10 82 0894-10.822559-10.8 19402-10.810704-10.795784-10 745180-10.666022-10.448593-10 17412 1-10 367 946-2

0.4

0.165179-10.220814-i0.302546-10.421366-i0.480525-i0 .4526 1-i0 .78803-10 .61223-10.644535-10.659832-10 .b74279-i0.687622-10.699576-10 .7098 lb-i0.717990-1o 723700-10 .72652t-i0.726030-10 721766-10.713314-10.700292-10.659507-10 59894 8-10 .43432-10 19996 b10.477801-2

VALUES IN TABLE MUST BE MULTIPLIED BY THE POWER OF TEN INDICATED AFTER THE NUMBER

U,

TABLE B-ic. THE FUNCTION (,K)

0.5 0,6 0.7 0.8 0.9 1.01<

-5. 0.161737-i 0.158427-1 0.155244-1 0.152180-1 0.149229-1 0.146385-1-4. 0.214743-1 0.208978-1 0.203498-1 0.198282-1 0.193313-1 0.188574-1-3. 0.291301-1 0.280812-1 0.271008-1 0.261826-1 0.253212-1 0.245116-1-2. 0.399717-1 0.380055-1 0.362130-1 0.345727-1 0.330669-1 0.316801-1-1.6 0.452266-1 0.426967-1 0.404200-1 0.383615-1 0.364922-1 0.347860-1-1.2 0.508515-1 0.476173-1 0.447510-1 0.421946-1 0.399019-1 0.378349-1-1.0 0.537055-1 0.500667-1 0.468690-1 0.440387-1 0.415173-1 0.392578-1-0.8 0.565009-1 0.524277-1 0.488809-1 0.457665-1 0.430114-1 0.4055801-0.6 0.591437-i 0.546161-1 0.507119-1 0.473121-1 0.443263-1 0.416844-1-0,5 0.603704-1 0.556133-1 0.515316-1 0.479924-1 0.448957-1 0.421643-1-0.4 0.615101-1 0.565254-1 0.522704-1 0.485968-1 0.453943-1 0.425784-1-0.3 0.625420-1 0.573356-1 0.529142-1 0.491135-1 0.458122-1 0.429185-1-0.2 0.634430-1 0.580253-1 0.534482-1 0.495304-1 0.461394-1 0.431761-1

0.641881-1 0.585746-1 0.538564-1 0.498346-1 0.463655-1 0.433429-10.0 0.647501-1 0.589628-1 0.541227-1 0.500134-1 0.464804-1 0.434103-10.1 0.651001-1 0.591682-1 0.542304-1 0.500537-1 0.464737-1 0.433704-10.2 0.652083-1 0.591688-1 0.541632-1 0.499433-1 0.463358-1 0.432155-10.3 0.650443-1 0.589432-1 0.539053-1 0.496703-1 0.460578-1 0.429386-10.4 0.645788-1 0.584712-1 0.534426-1 0.492243-1 0.456319-1 0.425339-10.5 0.637851-1 0.577351-1 0.527627-1 0.485966-1 0.450518-1 0.419968-10.6 0.626404-i 0.567205-1 0.518563-1 0.477810-1 0.443133-1 0.413242-10.8 0.592433-1 0.538255-1 0.493468-1 0.455766-1 0.423559-1 0.395707-11.0 0.543824-1 0.497976-1 0.459319-1 0.426308-1 0.397796-1 0.372922-11.4 0.413202-1 0.390919-1 0.369455-1 0.349474-1 0.331132-1 0.314384-12. 0.213745-1 0.219971-1 0.221545-1 0.220264-1 0.217242-1 0.213180-13. 0.575461-2 0,659711-2 0.730684-2 0.789331-2 0.836983-2 0.875079-2

VALUES IN TABLE MUST BE MULTIPLIED BY THE POWER OF TEN INDICATED AFTER THE NUMbER

K-5.-4.-3.-2.-1.6-1.2-1.0-0.8-0.6-0.5-0.4-0.3-0.2-0.10.00.10.?0.3

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TABLE B-id. THE FUNCTION (,K)

13o1.2

0. 14099 5-10.179726-10.230307-10.292128-10.317966-10.342601-10. 3 5380 3-10 363810-10.372221-10.3 75692-10.378596-10.380876-10.382471-10.383323-i0.383375-10.382574-10 380870-10.378221-10 374590-10.369952-10 164292-10.34991 2-10.331612-10. 285167-10.203547-i0 928026-2

1.4

0.135970-10.171634-10. 2 17 102-1

0.270861-10.292591-10 312799-10.321768-10.329612-i0 336016-10.338573-10.340643-10.342182-10.343148-10.343497-10.343191-10 342193-i0.340470-10.337992-i0.334739-i0.330695-10 32 5852-10.313788-]0.298681-10.260727-10.193335-10.957621-2

1.6

0.131276-10.164206-10.205262-10.252358-10.270820-10.287603-10 294887-10. 301130-10 306082-10.307992-10 30948 1-10. 310516-10.311064-10.311096-10. 3 10582-10 3 09497-10 307817-10 305523-10 302601-10.2 99041-10 294840-10.284537-10.271802-10 240076-10.183363-10 970946-2

1.8

0.126882-10.157367-10.194591-10.236127-10.25 1953-10.266042-10.272030-10.277063-10 2 80940-10.282381-10. 28345 5-10. 284137-10 284403-10 284231-10.283598-10 282486-10.280878-10. 27876 1-10.276125-10.272965-10 269280-10.260353-10 249436-10.222439-10 173956-10.973059-2

2.0

0.122761-10. 151051-10 184928-10.221782-10.235458-10.247398-10. 252 372-10.2 56473-10.259539-10.260631-10.261403-10.261834-10.261906-10.26 1601-10.260903-10.259798-10.258274-10.256322-10.253934-10.25 1107-10.247842-10.240014-10. 2 305 28-1

0.207217-10 16522 3-10.967508-2

5.0

0.820571-20.934832-20 104769-10. 11426 1-10. 11730 s-i0.118989-10 119597-10.119934-10 119983-1o 11989 3-10 119726-10 119480-10 11915 2-10. 11874 2-10. 118249-10 117672-10.117010-10.116265-10.115435-10 114521-10.113524-10 1 i124-10. 10872 7-10 102732-10.919167-20 712688-2

VALUES IN TABLE MUST BE MULTIPLIED BY THE POWER OF TEN INICAT AFTEk Ti-ft. NUMDER

OREGON STATE UNIVERSITYENGINEERING 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 .angtunand u.S. Rogers. I '428. 15g.

No. 2. A Sanitary Survey of lie \'ilIanrette Valley, by 11. 5. Rogers, C. A. .\Iiiekrioire,anti C. 1). Adams. 1430. 40.

No. 3. TIre Properties of Cenisn -Sawdust Mortars, I ']ajn and with Various Ailini stOres,by S. H. Graf and R. Ii. Johnson. 1930. 40f.

No. 4. Interpretation of Exhaust Ga.. A rialysea, by S. II. Graf, ( . \V. I deeyoii, on! \V. II.Paul. 1934. 25.

No. 5. Boi ler-\Vater 'Iroubi es anril Treat rneiits with Specr a! Reference to I 'rol it inis 0

Vestern Oregon, liv B. F. Sttnnrriers. 1(135. Noni-av anlalile.

No. 6. A Sanitary Survey of tIle \Villaruetle River from Seitwood Bridge to the (olunilea,by G. \V. Gleeson. 1936. 25ç'.

No. 7. Industrial and l)otriestic \Vastes of the \Vitlariictte Valley, by G. \V. Gli-cson aridF. Merryfield. 183). 50g.

No. 8. An Investigation of Sonic Oregon Sanils with a Statistical Study of the l'rg,lictivcValues of 'ft liv C. E. Tlionias an,! S. Ii. Graf. 1)37. 50.

No. 9. Preservative Treatments of Fence Posts. 1939 Progress Report ott tile lost Farm,by T. J. Starker. 1938 25. Yearly progress retiorts, 9-A, 9-It, 5-C, 9.1),').E, 'IF, '4-4.. tan'.

No. 10. Precipitation-Static Baum interference I'hieneriienr,i Originating on Aircraft, lie

}. C. Starr. p43'). 75g.

No. 11. Electric Fence Controllers with Stuecial Referetice to Equitiment Developed forMeasuring Their Characteristics, by F..\. Everest. 1939. 404'.

No. 12. 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. 50g.

No. 19. 1945 Progress Report ott Pollution of Oregon Streams, by Frc,l lIIerrylielil soil\\'. G. \Vilmot. 1945. 40g.

No. 20. The Fishes 0f the \\illamrtette River Systeor in Relation to Pollution, by B. E.l)ii,ick Sri,! Fcc,! Merrvtiel,t. 1945. -40c.

No. 21. TIre U-ic of the Fourier Series in tire Solution of Beam-Column l'roblems, byB. F. Ruffner. 1945. 254'.

No. 22. hn,Iurtrial and Crty \Vastes, by Fred Merryficlil, \\'. B. Bollen, and F. C. Kacltel-hioffer. 1947. 40f.

No. 23. 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.

1148. SOc.

No. 25. The Banki Water 'l'urliirre, by C..\. iIocbrn,,r,- ,riul 1-re,l i\lerrs lie),!. I-el, 1849.

40 ç'.

No. 26. Igniution lerirtleratures if Various tapers, \Voo,ls, and Fabric... liv S. II. Graf.Mar- 194'). 60g.

No. 27. Cylinrlcr Ileaul 'l'elrltwratures ins Four Airplanes witit Contirtental A-65 Fb,ginrcs, lipS. II. I.ouvv. hirtv 1') I').

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.

No. 29. I)iele, Inc t'rotiertiv.. of h',,ui,Iero, l'ic it Itiab !-r,'titerrci,, bs I. I. ((itikojuf ii,,]\I. 1). Macdor,al, I - Set 'tcrinlicr 1549. 4(1g.

No. 30. l-:x;srolei! Shale .\ynregate ii, Stnttctinrah (orrerete, liv I). I). Ritcltie ;,n,l S. II.(raf. Aug 1951. 604'.

No. 31. Improvements in the Field Distillatiorr of Peppermint Oil, by A. P. Hughes.Aug 1552. ('Of.

127

No. 32. A Gage for the Measurement of Transient Hydraulic l'resiures, by N. F. Rice.Oct 1952. 40&

No. 33. The Effect of Fuel Sulfur and Jacket Tenrlle rature on I 'i ston Iti rig \Vear asI )ete united by ii adjoact cc Tracer, by M. I opovrch and If. . I 'ctersr,n.July 1953. 4O.

No. 34. Pozzolanic Properties of Several Oregon lumicites, by C. 0. Heath, Jr. and N. R.Brandenburg. 1953. 50.

No. 35. Model Studies of Inlet Designs for Pipe Culverts on Steep Grades, by MalcolmH. Karr and Leslie A. Clayton. June 1554. 40f.

No. 36. A Study of Ductile iron and Its Response to \Velding, by IV. R. Rice and 0. G.I'aasclic. Mar 1955. (Se.

No. 37. Evaluation of Typical Oregon Base-Course Materials by Triaxial Testing, byM. A. Ring, Jr. July 1956..(0.

No. 38. Bacterial Fermentation of Spent Sulfite l.iquor for tue Production of l'roteinConceu teat e Air oral Feed Suppli-men 1, by 1 lerman R. Arnberg. Oct. 1956. 50 f.

En. 39. \Vood Waste Disposal ann 1 tili cation, by R. W. ltouhel, M. Nrnrtlicraft, A. VanVliet, M. i'opovich ..-ug. 1958. $1 WI

N,. -0. Tables of Sinular- Slnlutioris Ii, tile I<qualiinris Ii .\lllnin-ntnrrr, linat and Mass l'rarrsfer-in la,r,irrar Itniurinlary l.ayer Flu,,, by F. Flzv arid if. Xi. Srssori. ic!,. 1967.

No. 41. Er,giirecririg ('alculatinirs of Xlorirnrrturri, I In-at arid Mass Trairsier 'llir,,ugh l.arn,rrarIlniurrilary I_ayers, nv F. Flcy-.nr,i (. .1. Myers. July 19(8. $1.50.

Circulars-

No. I. 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.

No. 2. .\djirsriirerrt of Autorriotive Carburetors for Econoir,y, by S. If. Graf annl (. II'.(;le,on. 1930. Noise avai lalile.

N0. 3. Elements of Refrigeration for Snrall Conrniercial I'lants, h IV. 11. Martin. 1935.Noire available.

No. 4. Some Eirgiireerirrg Aspects of Locker arid Home Cold-Storage Plants, nv IV, H.Martin. 193S. 25ç'.

No. 5. Re fri ge rintioii Aptil real ions to Certain Oregon I n,lustri es, by \\'. H. Mart in. t '(40.2Sf.

No. 6. The Use of a Technical Library, by XV. E. Jorgenson. 1942. 25.N0. 7. Saving Fuel in Oregon Honres, by E. C. \\'illey. 1942. 25ç'.

No. S. Technical Ap1iroacli to tire Utilizatiorr of \Varnirise Motor Ftiels, by II'. 11. l'aul.1944. 2Sf.

No. 9. Electric and Other Types of House Heating Systenrs, by 1.ouis SIegel. 1946. 2H.No. 10. Economics of Personal Airplane Operation, by IV. 3. Skinner. 1947. 25.No. Ii. Digest of Oreginrr l.aiiri Surveying Laws, by C. A. Mockorore, M. P. Coopey,

Ii. B Irving, and NA. Bucklrorn. 191S. 25.No. 12. 'I'he .-\lnnrtinirrrr I iinluvt rv of the Nortirwest, by J. Grarrville Jensen. 1950. 25c.

No. 13. Fuel Oil 1fecuirrrriernts of Oregon and Sorrtlierra Washington, by Chester K.Sterrett. 1950. 25

No. 14. Market for Glass Coritairrers in Oregon and Soutirern \Vashington, by Chester K.Sterrett. 1951. 25g.

No. 15. Proceenliirgs of the 1951 Oregorr State Conference on Roads and Streets. April1951. flOg.

No. 16. \Vater \Vorks Operators' Manual, try Warren C. \Vesigarrh. Mar 1953. 75g.No. 1 7. 1 'mcccl hugs of tue 11153 Xi, ri hive r Con fe rerrce on lfoz,,l Bui liii rig. July 1983. 60 f.No. 18. Proc ceIlings of the I 55 N,,rilnnvest Conference ,lir Roail Building. Jtrrre 1955. 60f.No. 19. Review for Eingineerirrg Registration, I. Fundanrentals Section, by Leslie A. Clay-

ton. Dec 1955. 60.No. 20. Digest of Orcgorr 1.1,1,11 Surveying l.aws, by Kennetlr J. O'Connell. June 1956. 75g.No. 2t. Review for Engineering Registration, 2. Civil Engineering, by Leslie A. Clayton

arid Marvin A. If.nig. _Iirly 1956. $1.25.N0. 22. Revie,r for Eiigirreerirrg Registration, 3. 7,1ei,airicai Engineerirug, by Clrarles 0.

Heath, Jr. Feb 1957. $1.25.No. 23. Researclr aird Testing irr the Sclio,,l of Eirginieeririg, by M. Popoviclr. May 1957.

No. 24. Proceenliiigs of the 1957 Xortlirrest Conference nut Roanl Buildiirg, July 1957. $1.00.N0. 25. i'roceedirrgs of tire 1959 Nortlrsvest Conference orr Road Building, Aug 1959. $1.00.No. 26. Ifvsni,ircli Act ivitin-s in I lie School of Engineering, by 1. C. Kniuulsrin , Nov 1960.

No. 27. i'noceeilinigs of tIle 1960 N,,rtliwcst Higlirray Eirginieeririg C,,nfercrice, Die 1960.$ 1.1)0.

No. 28. l'roceeilinigs of tIle 1962 North,, cot Roinnis arid Street Conference, Mar 1962. $1.0)).

128

No. 29. I'rocee dingo of tile I';lentii l'iicific Nortllsvest lioltistrial \Vastc Conference- 1963,Sept 1963. $10).

No. 30. I 'roceedl ii gs of the 1964 N orliiwest Road'. and Streets Conference, J nec 1 064. $1.00.

No. 31. Research Activities iii tile Scilloli of Eiigiiieeritig, Ily I. (. I<iiu,lseii, Seth. 1964.

N .12. Tile toe ot a lccliiiical I.ihtrary, Ily If. K. \ValIlroii, Oct. 1064. 35ç'.

No. 33. Irllceel hug'. 0f tIle lOpi Surveying tool 2iIaljIing Confereuice, Itils 10) I

No .14. \'tonl Resi1Iute iulcolertltull,u iii L_1lee IllirlIers, lI) If. %V. IllIItIIl_I, ole 9ls. 1sf.

No. .15. Research ACtivIties in the Sellout of Engiuleeriilg, t064-6(, Iw I. I kilillislil,N uveuuier 191,1,. 7Sf.

No. .11,. lillIustriat F;ilgilleeritlg II liuduistry, ily \V. F. Engesw'r. ,\llg. 1067. (''lI.NIl. .17. I'ullceel lfllgs Ill lilt- 1(66 NllrtIllcest Ifllalhs 51111 Street (IliltereilCe, Nov. I '(67. $1.25.

Reprints-

No. 1. MetlIolls of I,Ire I ,nc I luslulatllr Ie.stlllC 111,1 tfesllits (If 'I' sts n iti1 1)lfferelitInst ruuuleil t5, 1 y I'. ( ). McM iliac. ]feIlri II tell from I 'roe N \V E ICC I .1 11111 I 'owcrAssoc. 1027. 2l).

No. 2. Some Anoullalies of Siliceous Matter in Boiler \VaIer ('Ileuilistry, by It. K.

SIllIllIlers. kept 1111111 I 111111 ('I,lllilllsilllil. 1 (il 1)_is, lIly.

No. 3. Asphalt Fmiiulsiou Ti-cat neil 1 1 'resents Ralilo Interference, by I. I 1. Mc 11 ian.ReprInted from Electrical \Vest. Jan 19.15. None avtlihaille.

No. 4. Some (Ilaracleristics (If A-C Coillluctllr Corlimla, by F. 0. 1,Ic7ifii;Iml. kelirilltedfrom I'. cci rIcat EngIneering. Mar 1033. None available.

NIl. 5. A Rallio lilterfereilce 3leasIlring Iulstrluulleiht, hr F. 1) SIc Mihlaut an,I 11. I . IlarlIcIl.Rellrinicli frllulu Electrical Eiigiiceriiig. Aug. 1935. tOf.

No. 6. Water-Ga-- ItetlehiolI .\i'iarcmutiy ('omltrol Engue Exhaust Gas ('onullositlon, by

1 111.1111 1(111 \\. II. l'atil, Reprimltel Irllmlu Natio,ual I'etroleuuui Nests.elI I 'Ill,. Nile IcO 1,11,115.

No. 7. >teauli ( ,euler,Il ion IV Iliiri1iuig l,Illl, II If. 11. SuIuIlIIlers, Repriiitell from llerltiuug

.11111 \elll ii,hlilIO..\,r I '1)1, II),.NIl. C I'llc 'IS-Co ElectrIc Emigimie Jilliicatlli-, i, It. Ii. I';Ilil .11111 K, If. EllIreligI'. IfelIrhiltell

front ( )rcgnn State Techmuical Recorli. Nov t')35. lOg.

.51. 'I, I-lnuuillllt) 11111 1.1111 tclutllerature, iIy \\'. II. Xt;Irtlmu and F. C. \\'illey. Ifellrinteulfront Power 'lam Euigineering, Feli 10.17. NIlile available.

No. II). heat Tramisft-r Efllciel,cv of Raii',u- 'nit'., IV \\'. .1. \\'ai-,II. Ifejlriulleli frolliElectrical EuIgIIeeriIIg Aug 1937. None avaitaille.

No. it. Design of Concrcte Mixtures, ily 1. F. \\'aIernlan. Rellruited from Concrete. Nov111,17 .NOilIS a caiblilic.

".11. 12. \\'aler-\\'Ise Refrlgeralloil, ily IV. II. \ltlrtiil 11111 It. E. Summers. Itellrumltell fromPower. Italy 1939. 10.

No. 1.1. I 'Illarity Limits of tIle Sphere Gall, ly I". (I. SIcMmllan. Reprinted front Al RE

-rran sad ions, hal 50. Mar 19,11), 10 u

No I I. Influence of ('tell-lb lIlt Heat Transfer, by It'. i . Short. ltellriltei front ElectricalEngumicering. NIIv 1039. I0.

No. I. ('iIrrosiIIiI Sill1 Self- I'rlltedtioui of Metals, III If. F. Sumtiuuicrs. Iti-lhrilute-Il ft 1,111

industrial I 'ower. Sri It II o,l Oct 19.15. III e.

NI,. II,. Monoc Illilie I"tiseitigc Circular fin6 .-\il;IiV'.ls, ily 11. I". lfltffnl'r. tfIllriltell fillillJournal 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

Ii. I'. Ruliin-r. Rc1,rluiie,l (cIlia .',er,I ItIg1551. .'lllr 1')3'). l))i.

NI,. 16. "nd hIlls l,f I IiIi'(;rllll-llI v-._ SI-s-,lil,i. CIII Ill I)llIugb,I'. 'ii', by I_s-c Gablie. Itellrintellfroni Tin' IiiIiII'rnlaul. I hue 111,10, 1 I),-.

Nil. t'I. Stlllcllilllllelrle (;Iiclilsltiuils (If Exbl;III'.t (eN, 1,1 1;. II. (;iee'.11i, 1,111 I". It.\\'ooIltielIl, J m'. Repri tIed frooi National I'l'trl,]euuin News. Nov 10.10. 100.

II, 2)1. 161' AlllIlIcauiolI of t'ce,lllaek III \\'i,lc- Ilsn,i (1111 putt _\iulllll fit-cs, Lv I-' -h. Fyeru'sIStill 11. IC ioiln-tlhil. RelIrliutell from 19-Il,- If till luisiutuie of Ifadio Engineers,

'I-I)). I Ill,

No. 21. SIres-es l)uie to Sr-eIIi,,ltIr\ Ill'illllile. l,v It. I-. IflIhlIwi-. Ifellciutl-Il fill0, Fl-lIe ofI"trt Nl,rlll,,c'.t l'lll,t,Il-lastRItI' (',,ill'crcl Ill. tlIill'r"ll% II) \V;I'.hiuigill,u. Ala,'1040, lOg.

No. 22. It'sfl Ileal Lw,s Ilack (If tttIIliall, rs, he F. 1. \\,hlt-y. IfeiII'lntell fi',lili IIciIthilg 11111

Venlilating. Nov 19411. IOç'.

No. 23. Stress ('Ilnci-ntrsltlon i-actor,, in Main ),leiiti,er, l)ne I,, \\'eillell Siiffeulers, i's\V. If. C berry, Rel,rinted from tile \\'el,ling J annual, If eseareli Slu)ll,lrnueni,ICc 1041. 1515.

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. 'i.

I',vercst 5111,1 II, S. l'riiellctt. tfellriltc,I fr,,iii i'is,c If The Institute (If Ifs,!,,,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(If Siguuua Gattuna Epsilon. .1511 1042. tOe.

129

No. 26. llro;ide;ust ,\iitetit,as 1),,)! Ari'av,, C';,lciilatioii of Ra,I,atio,i Patterns; ImpedanceIfellit iiii.Iii)is. 0 \\'tl,ao, I ritelit-it. I)ciiriiilcd (rim, ('iu,iiii,iiiicatiohis. AugSctt 1144 Ni' avai lalile.

No. 27. Heat l,osses Th'-ouli Wi-tied \\'alIs, be F. ('. \Villey. Reprinted from ASIIVEj oiiriial Section of Heiiliite, Ii of,1 Au' Conilitionitug. June 1946. I0.

No. 2$. Electric lower in China, by I'. 1) .Me 7tlillao. Reprinted front Electrical Engineer-inn. au l')47. lUc.

No. 20. Tbc 'I'ratu',icuut Euuergv hletlo,il of Calculating Stal,ilitv, by I'. 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. Reprintedfr,uuui li,itrital of Atiplicil l'li .ics . A10 ii 1049. 10,.

No. 31. lone- Rauir' l'lauuuiing for l'owi r Sttl,tlls , liv I. 0. McMillan. Reprinted fromFlectricil Eneiiuee rifle, l)ce 1045. lUç.

No. .13. heat Transfer Coefficients in Beds if Moving Solids, by 0. Levctus1iiel andS. \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. July1949. 10g.

No, .14. l)iffusion Coefficients of Urgiit uc I .i, iuiils in Solution from Sue face 'I ensiunMeasure,uciits, liv It. I.. ())s,,uu aol J . S. \\'alton. Reprinteil fri,m IndustrialEngineerin p Clicutnstrv. Mar 1051. br.

Xi,. 35. 'l'ruu'ic,its in Coupled l,ulitc'tanee-Caj:icitance Circuits Analyzeil in Tcriiis of aRolling-Bail .-hivalopuc. lv I', ('. 7ilapoit'son. Retirioteil from Vol (9, $51EE'1' raflsac lions, I') 9)). 11)6.

No. 36. 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).

No. 37. 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.

No. 38. 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.

No. 4)) (is.fioliil Film Foe (lie cots hf I [eat Transfer io 1"ltmu,l,zeiI ('oil Ileuls, be J . P\Valton, R. 1.,. Olson, iii,! Octave Levecspiel. Retirinteul from industrial an,!Emtgmnceriug ('liemnistry. June 1092. lOg.

No. 41, Rr'st,,mr:,nt \'euutil,tion, liv \\'. ii. Martin. Reloinuel (rub The Sa,uitaria,m, Vol14, No. 6, Mav-Jumue 1952. 106.

No. 42. Electrochemistry it, the lucille Northwest, liv l osu-phu Scluulcuo. Reprinted fromJomirnal of In' Eiectrochetuieal Society lone 109.1. 2)10.

Ye,. 43. Model Stn,bc,- of Tatiered In lets for Box ( 'olvcrts, liv hoe H. Shot-maker untilI es! e A. ('layton. Iteprmnlcil frottu Re'ea reh Re1 or t 15-B, Iii gli lvi)' ResearchBoard, \Vuisliu,ton, I). C. 1953. 206.

No. 44. Bed-Wall Heat Transfer in Fluidized Systems, by 4). I .cvei,sbocl and .1. 5. \\'attomu.IH'priuits from,, Heat 'I ram,sfcr-lteseareh Studies. 1994, 1)16,

No. -iS. Shunt ('atiar itors in L.arge Transn,ission Networks, liv E. C. Starr at,,! F, j

I tuirrumigtiiu. RcI,ri,iteil frotti l'ii,ver Apparau,m mi1 5, stems, 1)ee 1953, 10g.No. 46. 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 fronuFAll'!. July I us-), [06.

No. 47. 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$.

No. 48. 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.Killiu. Repr,tmueul fr,,u,i '['he \Velding Journal. Feb t'991. 20.

No. 50. 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.

No. 51. How Oil Viscosity Affects I'istsn Ring Wear, by H. Potioviclu atmil 1,. F. Job,,-sott. Reprittmeil from Atttomotive Industries. January 1956. 10.

No. 52. Intermittent Discharge of Stient Smilfite I.iituor, by Hernuan R. Amberg and RobertElder. April 1956. 10g.

No. 53. Hv,lraulics of Box Culverto with Fush-T.a,lder Bafiles. by Roy H. Shoemaker, Jr.Riut,rututcd from I'roeeeclings of the Flieluway Researclt Board, Vol 35. 1956.

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 intubular Heat Enelratigers, by M. S. Guriisltanluariah anti I. G. Knudsen. lIe-trinteil 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.itic(luer rind J . C. Kitudse it. Reprinterl from I leat Tranr,fer-Clticago issne ofA1ChE Svinposiuitt Series. 1930. lOg.

y0. s, 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'.Knudsen. Reprinted horn (lienircal Engineering Progress Ssniposutin Ser,e.sNo. 2'). 1959. lOf.

No. 5'). Itrtilii, interference Attcitriatirrn on Energize! Ilipli-\oltage 'l'ransniission lutes:hir-ri'rmreirrr-nt sort .\rjrIiertion, is I.. N. Slrritr . II. II. I a-brig, urd It. S. (Sn,.tteprinteil from l'usver ;'ttrirtmiiii_ms S stettis. I lee I 95'). 1 lIf.

No. (it). El 17)' Single tutu '[tent liutirlls- (onrluctnrs- -Influence of (ottrinetor l)ianteter unlSir;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.

No. 6 I - local Shell-Side 1 feat Transfer Coefficients mr Pressure Prop in a Tubular 1 lestExcltatuger with 0 ri flee I iaffle-,, by K. S. I .ee a nil . C. Knudsen. It ci rioter Ifront .\ 1(1, K Journal, Dec 19(10. 109.

No. 2. 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 ofAICItE Svtuuirosiuunu Series. Mar 11)61. 1Dm'.

No. 1.1. Drag- (',reltigie,tts it huts Iteyttolils Nttntbers lot Flute last Jitinterserl Studies, he.5. 7)1. _l,rtres amtrl J - C. Ktturisett. Reirritterl trortt .5 I('l,E _l'rurtt,l, hiam- 1)61. tIle.

No. 1,4. i't-rst,if,rte ( )xirliz;rhle (arlorru hurl Ill) I) is :r Olmursture ri ( )rgatie l'r,llntir,tu itt Water,hr I. 'ii. Crititirtur. I'. Ole rrvfit-lrl, I-. I. ltnrgess, I,. l'ttrlei-sr,ut, miii I. K. (art-steP. Itelrrinte,l t erritu 'rite f t Stir i't,rtlue I itrlits(ri;ml 'thistu.' Ct,itf, Olty

'((i - tilt-.No. 15. Evalrirmtirrit ('citeriri uric l)eer 'l'rickttttg tillers, liv F, J. lhtirgess, F. 7)lerryfiehrl,

- K. ('rirusuell, rut1 C. 7)1. 1 itirtriiu-. Iteirrotterl trait lrriirmi \\ater l'r,lluuirrit (mmii-tirrt Is.rlerrmi,rrit, _-\ug 10)11. tIlt.

"m,r. (1 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

's,. aS Itusirlir 'i 16,- ES1AC .\lgelrrare (rrtrtirrtter, re I. (, I,rrrrmtev tori itI. I ,.7)lrrrgmtr.Itetri tiller1 irurtri I It 1': I'rait-.rtctirriis ru l',lectrrrmtic ( rrtntrotet-s, Sept I 9ij 100,

'sr. r,. Ilrerr,rrl SIre-ut--, rut iii Itle.rlizerl \Virrs Structure, try 7sirrk l,eviitsrrit. Retrr,mtterlrrturrt ri .\ernrspace Sciences. 101,!. tilt-.

N,. 3). lire 1.55 I _\I. I'rniettiirtl-I'Iaiie Air,tlrrg ('rrnttpttter. ire . C. I_ninnies-. Reirri,tted frmnirr

.\ittnrtttrntrn (nrittrrnl, hits 101,1. lilt'

'sri. Sri. \'trrrrri Wa-is- ltteriterrrtrrrit, ire hi. I'rrirrrr tell, hI. Nrrrtl,crrrft, it. \\. llrrultet, rittrlI-. Ihrrritlrrtt's,lr. Re1rritrnn.'rl 1r,rirr Is-sIr Its- (ri. I .51,1-5, Itrtlrt'rt .0, Irtit Samittury

I-diet- (ctrir'r. I'ttirllt- Iterritir Sr'usirr - I. S. Dept if Fir_-altit. Knits-turn, anti\\eiirrrt', I '11,1 tilt-.

Nt). SI. lie Itt,1,' rrf (itrrr inn l)istril,ittin,it itt I_'.rtlrrrrlii l'rn,tes-tinnuu. irs It. K . ,\iereilrili. It,r.inluitsi trrriti 'uirnierirri', In, tectiolt, Eel, tilt_S. 111g.

52. l,rrcrrl Itrg of II, ri 'I r.rit'fs'c intl1 l'rr-sstirr- l.rr'e rir tile Vicimiti iii .Autnitlar)r'iftce-., It,- I', S. \\rllir,t,ts r,r,,l I. C. Ki,irrlsett. It,-1,riitterl frnmit 'the (atta,lirr,r

ri) (irs-trite_ri t-dtiriiiet-nrit,r .;\j'r ri trill, tile-.No. 5.). 1 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 1 fiorit 1 lreio,errl Eo-.rirns-ertttg I'rrrgress, ui 191,.). lOg.,Nrr. s-i . Stile icctric Strirree l'rrtnmttirtlu fur hiriclririe (rntttriil. lry It. K. Stliel,rrel run I'. It.

(rriss-fmrrrl. Its-lrrititerl fruit I-des'um-i,-al I-irgrr leeritig Nrrv 1963. 106..'tSt. 55. \trirrltle '10 rIOt l'rti-.r- Ct-trn_-rrriirrtt l'.'tt,g .\t-rrlrnttcite h'rrttt-.i'tors, try \V. C. Smlag-

iiitsrnii. Itept-irrtr-ll (nun IEEE Irrrt sic (tInts, Semi l'i(,.l. tilt.

N r,. sir, Evrilttrrtiitg tire Effect is_i- l.is-'.rr,t,rit us-' rh l)iairttr;igirts ut Elecirni, tie Seirarrttrrr,, Ire

It. F. Stie ret bib stir,! 1. \\. 'l'oirirs. ten it tell front Jirtitmisil of ilte Eiectrmtchettt-cml Orreiet , I )es I 'ilu.i. till.

'sri. id I (sin,--, t' f Xi mrrrsst' .rmir I ('turn citeil lIsP rrr,rtrlrrruis l'irrr lucid rittriirg (rritirn,!lcrl(Srir,Irristiirtt, irs- It. Oh'. I ,ritirel 'itni IS. Itt irius-rtr,it. Itr irri,itrrl fmr,itr ltritrm,stl n,fih, hi, l'rrilnrturmi (r,ittrrrl )ms-'ricn;mttrrir. hut'- t'li,. IlIf.

Nm. 7.3. Sn,itie llnin-i-iimnmt nrf lmnin-rnttI ltr-e,r, irr \\rr,rr i'rrlts_ ir\' C. ('bust-n, lteinrimtls'rl firm,ilrt- Sa-s,rmurl Sn it ir rrir,r unit (ii, Noir-Il, -,srticti,i- 'le'.tittg rn I \\oorl. \\i,lt,ttgisttStttr' I mnrsr,ttn . l'lir.i. ill,-.

>mrr. 7'). l'rr sr-tmcc-Vr,Iirir,r' I lrrmr,,,-t r-rrstiesrnf l'lr,slrc Irma,. in It, \V. lirrolrel ltr-t,rrmutert fimrrrr

.Arrienie,mn Ir,rlrisirtril I-Iyyun-ire ,\s-.rrei;rtrr,mm ir,utrr,ril ,.\i,rt lute t')l,S. 1110.

Xnn. .00. Irlent,hesirtnrmt of i.,rsv-I'trrsv .\rrei,rr triani,,ir Ite,1utmr.-rir, ills (run \',rt,-r )utrrlit, ('rririrmrlIn, (_'rinttltiir_-r 'Iem.liuni,1r_ies. Ire P. I.. Oh ml,,, 'tV OS. 'I'rrrcit,-_ rumnl I'. I. lb rmrr-ss.

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1110-.

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Su-arlers. I5eir. 1')(,(,. i (10-.

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's,. 9.1. .5 IIigIt-Tr-itterrrttrre Sn,lirl Sir,- lln,llers. ire I). I'. I 6c1, urn It. F. 'tie us,Its'irrittts-ti furrrru Elennt-r,s-lt,-tinrcrni [i-c i,i rlul,lc,- Sept.-) let. I 01,7 ill,.

131

THE ENGINEERING EXPERIMENT STATION

ADMINISTRATIVE OFFICERS

CHARLES R. HOLLOWAY, Jr., President, Oregon State Board ofHigher Education.

ROY E. LIEUALLEN, Chancellor, Oregon State System of Higher Edu-cation.

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 UniversityCORVALLIS

RESIDENT INSTRUCTION

Undergraduate Liberal Arts and SciencesSchool of Humanities and Social Sciences (B.A., B.S. degrees)School of Science (B.A., B.S. degrees)

Undergraduate Professional SchoolsSchool 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 StudyAgriculture (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)

EXTENSIONFederal Cooperative Extension (Agriculture and Home Economics)Division of Continuing Education, State System of Higher Education

Engineering Calculations of Momentum, Heat and

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Transcript of Engineering Calculations of Momentum, Heat and