DIMENSIONLESS GROUPS

Analysis of variables in heat, mass, and momentum transfer as well as kinetic rela- tionships yields. DIMENSIONLESS. GROUPS which may be used as prima...
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J O H N P. C A T C H P O L E GEORGE FULFORD

Analysis o f variables in heat, mass, and momentum transfer as well as kinetic relationships yields

DIMENSIONLESS GROUPS which may be used as primary tools to solve design and development problems of dimensionless groups in analyzing chemiThecalvalue engineering problems has long been recognized. Even before Reynolds did his classical work, Helmholtz derived groups in 1873 which are now called the Reynolds, Froude, and Mach numbers in the course of solving a fluid flow problem (57). Later on, Rayleigh (48) and Buckingham (9)established the principles of dimensional analysis that led to the development of many dimensionless groups of practical use. Many standard references explain the technique of dimensional anal)& (29). Weber was the first to name the Reynolds, Froude, and Cauchy numbers (62), and the precedent that he 46

INDUSTRIAL A N D ENGINEERING CHEMISTRY

set for naming groups for pioneering workers was continued,and notably 80, when a capillarity parameter was later named for him (47). Unfortunately, however, a standard naming system was never developed, and the result is that with the number of groups increasing rapidly, several have more than one name, and several names apply to more than one group. (Catfinucdmpagc 59)

Catchpole and George Fulford are Research Engitucrs for the Pkoto Products Department of E. I. du Pont de Nmours 61 Co. The a u t h s acknowledge the he& of G. E. Alves, Senior Consultant to Du Pont's Engiruning Department.

AUTHORS John P.

TABLE 1. Serial No. __

Name

AI

Acceleration number

-KB

A2

Alfven number

NA1

A3

Anonymous group (1)



A4

Anonymous group (2)

K1

A L P H A B E T I C A L L I S T OF N A M E D GROUPS

Symbol

Dejinition

Signaycance Group dependent only on physical properties Ratio of Alfven wave velocity/fluid velocity

A6

Arrhenius group

B1

Bagnold number

Origin of Name

Magneto-fluid dynamics

Transfer processes

temp.+diff. [TI ; An = conc. diff.

[-I Archimedes number

Reference

Transfer processes ’

A5

Field of U s e Accelerated flow

NAT

...

*g

(P

- Po);

PO

=

fluid density; p = particle density (cf. N c a 4

NRe, gravitational force/viscous force

Fluidization, motion of liquids due to density differences

Archimedes of Syracuse (287-212 B.C.)

Svante August Arrhenius (1859-1927) (Swedish)

Ea/@T

Activation energy/ potential energy of fluid

Reaction rates

3capoV2/4d~pe;P V =

Drag force/gravitational force

Saltation studies

Heat transferred by radiation/thermal capacity of fluid

Radiation

Ratio of yield stress/ viscous stress Midplane thermal internal resistance/ surface film resistance Mass transfer rate a t interface/mass transfer rate in interior of solid wall thickness L Inertial force/viscous force

Flow of Bingham plastics Unsteady state heat transfer

gas density; p p = particle density kTAw/Vmc; kp =radia n t heat transfer coefficient; AW = wall area of channel; (cf. N s t ) ruL/ppV ( L = channel width) hLm/k (in French literature, “Biot No.” = NNU)

B2

Bansen number

B3

Bingham number

B4

Biot number (heat transfer)

B5

Biot number (mass transfer)

kcL/Dfnt; L =

B6

Blake number

V P / [ P (~ e)$]

E7

Bodenstein number

V L / D a = Npem; L

thickness of layer, D6nt = diffusivity a t interface

Jean Baptiste Biot (1774-1 862) (French)

Mass transfer between fluid and solid Beds of particles Diffusion in reactors

= reactor length, Da = axial diffu-

sivity (effective)

(L’/?)

B8

Boltzmann number

IThring radiation

B9

Bond number

(P

group

- P’)L2S/U

=

Nwei/NFri if P. p‘ .v p (gas In

Zravitational force/ surface tension force

Atomization, motion of bubbles and drops

liq.) ; p = drop or bubble density; p’ = medium density

E10

Aouguer number

E11

Boussinesq number

E12

Brinkman number

3cDx?/4pDR; CD

Radiant heat transfer to dust-gas streams

wt. dust/unit bed volume ( M / L 8 ) , = mean path for radiation ( L ) . p~ = dust density, R = mean particle radius. Also N B = ~ kL; L = characteristic dimension, k = absorption coefficient of medium v / ( 2 g R H ) ‘ j 2 (cf. N F T Z ) (Inertia force/gravitational force) 112

Wave behavior in open channels

pV2/kAt;

At =

temp. diff.

Heat generation/ heat transferred

Viscous flow

temp. of medium, to = init. temp. of body

Heat for vaporization/heat to bring liquid to boiling point

Heat transfer during evaporation

Capillarity number

Depends only on physical properties

c2

Capillarity-buoyancy number (physical properties group) (film N 0 . p

Depends only on physical properties and g

c3

Capillary number

Viscous force/surface-tension force

Action of surface tension in flowing media Effects of surface tension and acceleration in flowing media (twophase flow) Atomization, twophase flow

B13

__ c1

a

Bulygin number

Joseph Boussinesq (1842-1 929) (French)

Very similar to H u and Kintner’spH factor for drops and bubbles [A.I.Ck.E. J. 1, 42 (19591.

(Continued on next p a g e )

VOL. 5 8

NO. 3

MARCH 1966

47

T A B L E

I.

Seria,

No.

Name

c4

Carnot number

Symbol

Definition

Signifcance

(TZ - T J / ( T J ; TI, TZ= abs. temp. of

Theoretical efficient) of Carnot cvcle operating between T I and T2 Inertia force/compressibility force

two heat sources or

c5

Cauchy number

C6

Cavitation number

c7

Clausius number

[(P - P d / P l / ( V 2 / 2 ) p = local static

pressure (abs.) ; Po = vapor pressure

V3Lp/kAT; A T

Excess of local static head over vaporpressure head/ velocity head

C8

Colburn number Condensation number (1)

Same as Schmidt number ( h / k ) ( p 2 / p Z g )113

Arc, =

(viscous force) Nh'u (gravity force) >

[

1

c10

Condensation number (2)

c1i

Cowling number

C12

Craya-Curtet number

C13

Crispation group

ci4

Crocco number

L3p2gr/krAt; r

mean velocity, v d = dynamic mean velocity pol/u*L; 'v* = undisturbed surface tension; L = layer thickness

+

Cavitation

Condensation

Condensation on vertical \.calls

=

v k / ( V d 2 - vk2/2)liZ; v k = kinematic

V/Vm'max = [l

Compressible flow

1113

latent heat of condensation ( VA/ V )2 s2 (Alfven number)

Magneto-fluid dynamics Radiant heat transfer

Convection currents

Velocity/maximum velocity

Compressible flow

Chemical reaction rate/bulk mass flow rate Chemical reaction rate/molecular diffusion rate Heat liberated/bulk transport of heat

Chemical reaction, momentum, and heat transfer Chemical reaction, momentum, and heat transfer Chemical reaction, momentum, and heat transfer Chemical reaction, momentum, and heat transfer

2]-1/2

- 1)(lV,Ma) V,,, = maximum velocity of gas expanding adiabati(Y

-

tally

-

D1

Damkohler group I

DaI

D2

Damkohler group I1

DaII

D3

Damkohler group I11

DaIII

D4

Damkohler group I V

DaIV

D5 D6

Damkohler group V Darcy number

DaV

D7

Dean number

h'n

D8

Deborah number

D

D9

Delivery number

9

Di0

Deryagin number

De

Di1

Diameter group

6

( R / 4 ) '/2(2H)1/4 d/(V,)lI2, d = im-

Flow machines

peller diam. = [pressure No.] 1 / 4 x [delivery No.]- 112 De/Lm2; D = diffusivity of solute through stationary solution contained in solid; cf. ,VpOrn

Mass transfer

= ( N R ~

see Fanning friction factor (centrifugal ( VLP/d ( L / 2 R) '1'; L = pipe diam.; force/inertial R = radius of force) curvature of bend 97/&; Bo = observaRelaxation time/ tion time observation time V f / A w ; A = imDeller area = ~ d 2 / 4 = [Diameter No.]-3 [Speed No. 1 L(pg/&; L = Film thickness/ film thickness capillary length 4f;

D12

Diffusion group

B

Di3

Drag coefficient

cd =

CD

48

Heat liberated/conductive heat transfer

(P

- P,')LQ/PV'"?.P

density of object; p' = density of medium; cf. f, $, Ne

=

INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY

Gravity force/inertial force

Reference (27)

Heat conduction in forced flows

=

temp. diff.

c9

Field of Use

Flow in curved channels Rheology Flow machines

Coating

Free settling velocities, etc.

Origin

sf Name

Nicolas Leonard Sadi Carnot (17961832) (French) Augustin Louis Cauchy (1789-1 851) (French)

Rudolf Julius Emanuel Clausius (18221888) (German)

CONTINUED Serial No.

Name

D14

Drew number

Symbol

Definition

Significance

ND

mol. wt. of com onents A and B; Mw.= mol. wt. of mixture in vapor and a t wall; YAW = mole fraction of A a t wall; Za = mole fraction of A in diffusing stream V2/CpAT = Eckert No.

Gv,

D15

Dulong number

El

Eckert number

E2

Ekman number

E3

Elasticity number (1)

E4

Elasticity number (2)

E5

Ellis number

E6

Elsasser number

E7

Entry Reynolds Number

E8

Eotvos number

E9

Euler number (1)

E10

Euler number (2)

Ell

Evaporation number

E12

Evaporation number (2)

E13

Evaporation-elasticity number

F1

Fanning friction factor

F2

Fedorov number (1)

Du, ND

v,~/c,AT,

Pierre Louis Dulong (1785-1838) (French)

vm =

velocity of fluid far from body (= 2/recovery factor, 9.v. = Dulong No.) (p/2puL91/2 = (ivRo/NRe)1/2 er#/pL2, L = pipe radius PCP/@ [Gay Lussac No.] X [Hooke No.] + [Dulong No.] wOV/ZTI~~R; PO = zero shear viscosity, r1 - shear stress wienp = p0/2 [M/LI32]< R = tube radius p/wepe NRe/ p g n e t i c Reynolds

Zompressible flow

(Viscous force/ Coriolis force)V2 Elastic force/inertial force Depends on physical properties only

APF/PV~; APF=

pressure drop due to friction d( yd#/dL)/pVa; d = pipe diam., dp/dL = pressure radient E 2 X anning friction factor V2/r [ r = heat of vaporization (L2/@)1 Cp/rp (r as in E l l ) (Gay Lussac No.) > (Ell)/(Dulong NoJ E/rp = Kr/Hooke number

bfagneto-fluid dynamics Viscoelastic flow

Vagn Walfrid Ekman (1874-1954) (Swedish)

Effect of elasticity in flow processes Flow of non-Newtonian liquids

Magneto-fluid dynamics

x VP

x/dh.NRe = y ; x = entrylength ( p - p’)L?q/u Bond No., 4.u.

Origin of Name

Field of Use Boundary layer mass transfer rates; velocity profile distortion; drag coefficients for binary system

Friction head/2 X velocity head

Entry or inlet processes

Roland von Etitvos (1848-1 91 9) (Hungarian)

Fluid friction in conduits

Leonhard Euler (1707-1783) (Swiss)

Fluid friction in conduits

Bi

Evaporation processes

=

dApF/2pvaL, d = dimension of cross section; L = length (cf. resistance coeff., Ne)

Evaporation processes Evaporation processes Shear stress a t wall expressed as number of velocity heads

? h i d friction in conduits

?luidized beds de = equiv. particle diam.; y M = sp. gr. of particles; yg = sp. gr. of gas (cf. VAT)

F3

Fedorov number (2)

F4

Fenske number

F5

Fineness coefficient

L / W D ” ~ ; WD = volume displacement [L8]

Mass transfer analogy of Posnov number Number of stages in separation process

rransport processes

ship modeling

(Continued on next p a g e )

VOL. 5 8

NO. 3

MARCH 1966

49

TABLE I. Serial

No.

E F8

i

Name Fliegner numbers

Definition

Symbol

I

Signijicance

Field of Use

Reference

Origin of Il'arne

Functions of ratio of specific heats and mach number V m ( c T ) ' V A ( p s pvn: = -,/(7 - 1)"2. ?Ma2 X Ma/l (Y - 1 ) / 2 .

+

+

+

Ma2]1'2

= impulse Fliegner number; y = ratio of specific heats, M a = mach number, A = flow area V f / N d , d = impeller diam.

F9

Flow coefficient

F10

Fluidization number

V/Vinit, Vinit

F11

Fourier number (heat transfer) Fourier number (mass transfer) Froude number (1)

te/pcpLm

F12 F13

velocity for initial fluidization

Reech No., Bo& sinesq KO., Vedernikov No.) dgT'=_ ('VFTJ1/2 (cf. Boussmesq No.)

F14

Froude number (2)

v/

F15

Froude numbers (rotating)

DM*/g; D diam.

GI

Galileo number

L 3 g p 2 / p 2 (cf. N A T ,

G2

Gay Lussac number

UBAT

G3

Goucher number

R ( p g / 2 ~ ) " ~R; = wall or wire radius

G4

Graetz (Gratz) number

VmcplkL

G5

Grashof number

G6

Gukhman number

=

of hbt gas stkam,' tm = temp. of moist surface (wet bulb temp.) Given by equation relating volumes of reacting gases and reaction products

Guldberg-U'aage group

H1

Hall coefficient

H2

Hartmann number

H3

Hatta number

H4

Head coefficient

H5

Heat transfer number

H6

Hedstrom number

H7

Helmholtz resonator group

H8

Hersey numbers

F b / p u a (cf.

Hodgson number

wjApF/Pr.pa

Inertial force/gravitational force Velocity of open channel flow/speed of very small gravity wave

impeller

Nusselt thickness group)

G7

Fluid velocitv in fluidized bed/that at start of fluidization

Power required by fans, etc. Fluidization

Unsteady state heat transfer Unsteady state mass transfer Wave and surface behavior Open channel flow; free surfaces Agitation

IVRe X gravity force/viscous force

XGa, =

Gravitational force/ surface tension force112 Thermal capacity fluid/convective heat transfer ~ G = T N R (buoy~ ancy force/viscous force) Thermodynamic criterion of evaporation under isobaric adiabatic conditions

Circulation of viscous liquid, thermal expansion Thermal expansion processes Coating Streamline flow Free convection Convective heat transfer in evaporation

Chemical reaction in blast furnaces

~

H9

50

f c J ( f c = cyclotron frequency, J = av. free path/av.

Magneto-fluid dynamics Magnetically induced stress/ hydrodynamic shear stress (magnetic body force/viscous force) 1 / 2

y/tanhy; y = (rCD)"2/kc, r = reaction rate constant [L3/Me] [a modifled Hatta number has also

truncation

number)

INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY

Jean Baptiste Joseph Fourier (1786-1830) WWrench) illiam Froude (1810-1879) (English)

Magneto-fluid dvnamics

Gas absorption with chemical reaction

Proportional to frequency X residence time Load force/viscous force Time constant of system/period of pulsation

Flow in pumps and fans Heat transfer in stream Flow of Bingham plastics Pulsating combustion Lubrication Pulsating gas flow

Galileo Galilei (15641642) (Italian) JoseDh Louis GavLLssac (1778-' 1850) (French)

CONTINUED Serial No.

Name

HI0

Iomochronous number

H11

3ooke number

51

Jakob modulus

52 53

]-factor (heat transfer)

54

Joule number

K1

KdrmPn number (1)

Significance

Definition V8/L (8 = time for

liquid to move characteristic distance L ) pVZ/E 3 Cauchy No., q.v.

Duration of process/ time for liquid to move through L

Maximum bubble radius/thickness of superheated film

teference

Field of Use 2hoice of time scales

73)

Slasticity of flowing media

25)

Heat, mass and momentum transfer theory

J-factor (mass transfer)

K d r m i n number (2)

K3

Kirpichev number for heat transfer

K4

Kirpichev number for mass transfer

K5

Kirpitcheff number

K6

Knudsen number (1)

K7

Knudsen number ( 2 )

K8

Kossovich number

K9

Kronig number

i o b e r t Hooke (16351703) (English)

3oi1in g

Joule heating energy/magnetic field energy

pd3(-dp/dL)/p2 (d = pipe diam., dp/dL

Magneto-fluid dynamics

.amesPrescott Joule (1818-1 889) (English)

Fluid friction in conduits

rheodor von K d r m i n (1881-1963) (Hungarian)

= pressure gradi-

K2

Origin of Name

e ent) 3 2 ( N ~ djjz V/ VA (see Alfven No.)

q*L/kAt ( C f . NBlh, N N U)

Intensity external heat transfer/internal heat transfer intensity Intensity external mass transfer/internal mass transfer intensity Length of mean free path/characteristic dimension Bulk diffusion/Knud, sen diffusion Heat used for evaporation/heat used in raising tempera. ture of body ( N z e )(electrostatic force/viscous force)

Magneto-fluid dynamics Heat transfer

Mass transfer

Flow around obstacles Low pressure gas flow Gaseous diffusion in packed beds Convective heat transfer during evaporation Convective heat transfer

ES = electric field at surface, N = Avogadro's Number,

-

a = polarization

coefficient, 0 molecular &pole moment, k = Boltzmann's constant, M = molecular weight IEL/pVu'; I = current density [Q/L20], E = voltage [ML/QOz] u' = enthalpy (L2/02] ru/cp(to tro), (to, tro = stream, wall temp.)

Electric arcs in gas streams

Agitation

K10

Kutateladze number (1)

K11

Kutateladze number ( 2 )

L1

Lagrange group (1)

~ / p L 3 N 2 ;L =

L2

Lagrange number ( 2 )

(D

L3

Lagrange number (3)

AP R/pV

L4

Larmor number

L L / L ; (LL = Larmor

L5

Lava1 number

V / P y / ( y +1).RT11'2;

-

characteristic dimension of agitator = N R e . N p

+W / D

Leverett function

Combined molecular and eddy mass transfer rate/molec ular mass transfer rate

radius)

y = ratio of specific

L6

Combined heat and mass transfer in evaporation

Linear velocity/ critical velocity of sound Characteristic dimension of surface curvature/characteristic dimension of pores

Mass transfer in turbulent systems

Joseph Louis Lagrange (1736-1813) (French/ Italian)

Magneto-fluid dynamics Magneto-fluid dynamics Compressible flow Two-phase flow in porous media

(Continued next page) ~~

~~~~

VOL. 5 8

NO. 3 M A R C H 1 9 6 6

51

T A B L E 1. Serial No.

Dejnition

Name

Significance

k/pcpD = a / D 3 NSC(NPIP7 (N.B:: Lewis number is sometimes defined as reciprocal of this quantity) V/Vl; (VI = velocity of light)

Field of Use

Lewis

L8

Lorentz number

L9

Luikov (Lykov) number

kcL/a

LID

Lukomskii number

a/am; urn = potential conductivity of mass transfer [L2/19]

L11

Lundquist number

L12 L13

Lyashchenko number

Fluidization

Lykoudis number

Maqneto-fluid d;namics

M1

XlcAdamq group

hdLpAt/k3pzgr

M2

Mach number

VI?',; ( V S = velocity

kcLpCD/k

ueHepe3'2 L/p'/2

M3

Magnetic force parameter

M4

Magnetic mach number

IM5

Magnetic Oseen number Magnetic pressure number

M6

M7

Magnetic Reynolds number

M8

Maievskii number

M9

Marangoni number

Fluid velocity/velocity of light Mass diffusivityi thermal diffusivity; rate of extension of mass transfer field/rate of extension of heat transfer field

=

pe2He2ueL/PV

uepeLV ( c f . velocity number)

L

Condensation

Magnetic body force/iner tia force ; resistance time of fluid in field/relaxation time of lines of force

Magneto-fluid dynamics

Magnetic force/ inertia force Magnetic pressure/ 2 X dynamic pressure Mass transport diffusivity/magnetic diffusivity

Compressible flow

laver thickness

Magneto-fluid dynamics Magneto-fluid dvnamics Magneto-fluid dynamics Magneto-fluid dynamics Compressible flow

Forced convection Cooling towers, liquid-gas contact

Mi0 M11

Margoulis number Merkel number

M12

Miniovich number

iM13

Mondt number

Convective/conductive heat transfer

Heat transfer

N1

Naze number

N2

Newton inertial force group Newton number

Velocity Alfven wave, velocity of sound Imposed force/ inertial force Resistance force/ inertia force Imposed head/velocity head Heat flow for phase change/superheat (supercooling) of one of the phases Total heat transfer/ conductive heat transfer Intensity of mass flux at interface/specific flux by pure molecular diffusion in layer of thickness, L = ( . 4 ' ~ ~ ) 1 (gravita/3 tional force/viscous force) 1 1 3

Magneto-fluid dynamics Agitation

N3

N4 N5

Number of velocity heads Number for similarity of phys. and chem. changes

N6

Nusselt number

N7

Nusselt number for mass transfer

N&

Nusselt film thickness group

_____~

52

~

Mass of water transferred in cooling per unit humidity differenceimass of dry gas

~~

INDUSTRIAL A N D ENGINEERING CHEMISTRY

LV. H. McAdams (American) Ernst Mach (18381916) (Austrian)

Cellular convection

=

SR/e; R = pore radius

Hendrick Antoon Lorentz (18531928) (Dutch) A. V. Lykov (Russian)

Combined heat and mass transfer

Constant for given surface orientation Linear velocity/sonic velocity

eM a

Au At _ A~ _ ALLz/Pu;

Magneto-fluid dynamics Combined heat and mass transfer

Magneto-fluid dynamics

M H ( RIM/ e ) ( L = thickness of fluid layer)

of sound in fluid) v / l / E X (Eo =bulk modulus of fluid) ( c f . Sarrau number)

Origin of iliame

Combined heat and mass transfer

L7

=

Refe7ence

Drying

Isaac Newton (16421727) (English)

Friction in fluid flobv Friction in conduits Changes of phase

Forced convection Mass transfer

Falling films

Ernst Kraft \Vilhelm Nusselt (German) (1882-1 957)

CONTINUED Serial

No.

Name

SigniFcance

Definition

3mbol

(Fb/pvsVs)(a/R)2(D/b)2;,oad force/viscous

Field of Use

6)

5)

01

kvirk number

02

)hnesorge number

riscous force/ (inertia force X surface tension force) 112

itomization

Pi

'Cclet number (heat)

Porced convection

P2

'tclet number (mass)

P3

'ipeline parameter

VwV0/2 Hs'; (VW =

5ulk heat transfer/ conductive heat transfer 3ulk mass transfer/ diffusive mass transfer vfaximum pressure rise in water hamrner/2X static pressure

P4

'oiseuille number

D2(

=32 for laminar flow in round pipe

Laminar fluid friction

P5

Pomerantsev number

P6

Posnov number

-d,b/dL)/pV ( D = pipe diam., dp/dL = pressure gradient) iL*/k(tm - to) (tm, to = temp. of medium, initial temp. of body) (cf. Damkohler Group IV) B A ~ / ( A ~(cf.~ ) Fen)

P7

Power number

n/LhpN3

P8

Prandtl number

Cpp/k = Da IV/Da I11 X Da V

P9

Prandtl number (mass transfer)

p/pD = N S C ,u (used

Drag on (agitator impeller) or inertial force Momentum diffusivi ty/ thermal diffusivity See Schmidt number

Pi0

Prandtl velocity ratio

V/(TW/P)lj2 ( V =

Pi1

Prandtl dimensionless distance

PI2

Predvoditelev number

Pi3

Pressure number (1)

Pi4

Pressure number (2)

P15

Psychrometric ratio

Ri

Radiation number

E,

R2

Ramzin number

Ra

R3

Ratio of specific heats

Y

R4

Rayleigh number (1)

NRal

R5

Rayleigh number (2)

R2'

Free convection

R6

Rayleigh number (3)

Ras

Combined free and forced convection in vertical tubes

(us = shaft surface velocity; R = shaft radius; D = shaft diam.)

velocity waterhammer wave, V, = initial velocity, Hs' = static head x gl L2/e21

in Russian, German literature)

local fluid velocity) L(prW)1/2/p( L = distance from wall, etc.)

temp.

-

P")

1

density of liquid gas) H/'!P Us2 (Us .= circumferential velocity) E [ diameter No.]h c J z e e d Iio2.]'2 ( p ' , p".=

Mass transfer Water hammer

lean Louis Poiseuille (1799-1869) (French)

Heat transfer with heat sources in medium

Inertial force/wall shear force112

Combined heat and mass transfer Power consumption by agitators, fans, pumps, etc. Forced and free convection

Ludwig Prandtl (1875-1953) (German)

Turbulence studies Turbulence studies

Absolute pressure in system (pressure jump on interface)

Heat transfer by convection/heat transfer by mass transfer

Heat transfer

Aleksandr Savvich Predvoditelev (1893-) (Russian)

Flow machines (turbines, pumps, etc.) Wet and dry bulb thermometry

Radiant transfer

kE/tluT3 = (Nwel)/ (Hooke No.) X (Stefan No.)

cbp

Origin of Name

force

Rate of change of temp. of medium/ rate of change of temp. of body

P/(g.(p'

'eference

.ubrication

Molar mass transfer

-

c,o(Bulygin No.: (Kosovich No

C p / C , (specific heats

Compressible flow

a t constant vresSee Nwe

Breakup of liquid jet

John William Strutt, Lord Rayleigh (1842-1 919)

(English)

(Continued on next page)

VOL. 5 0

NO. 3 M A R C H 1 9 6 6

53

TABLE I. Serial

No. -

Name Recovery factor

R7

Dt$nition

SigniJication

adiabatic wall temp. tm = temp. of moving medium. (cf. Eckert No.) = 1 / ( N ~ dq.v.

R8

Reech number

R9

Resistance coefficient (1)

F R / ' / z pV2L2 (cf. drag

Resistance coefficient (2)

Ap. D H / ' / ~pV2L (Ap

R11

Reynolds number

LVP/P

R12

Re nolds number 8otating) Richardson number

L z N p / p ; L = im-

R10

R13

Romankov number

R15

Rossby number

R16

Roughness factor

SI s2

Sarrau number Schiller number (1)

s3

Schiller number (2)

Actual temp. recovery/theoretical temp. recovery

Inertia force/viscous force

Wave and surface behavior

- ( g / p ) (dp/dL)/

height oT1iquid layer, (dV/'d& velocity gr,adient a t wall] T D /T P R O D V/2 weL sin A ( w e = angular velocity of earth's rotation [l/S]; A = angle between axis of earth's rotation and direction of fluid motion I-] j

Dynamic similarity

Gravity force/inertial force

Stratified flow of multilayer systems

Dr bulb temperaturi &bs.)/product temperature (abs.) Inertia force/ Coriolis force

Drying

mach number, q.u.

Effect of earth's rotation on flow in pipes

V[;

.

1

'

ljS.

V = velocity in fluidized bed; y m , y,%f = specific gravity of medium and material in bed p/pD (6. iVprrn) (= DaII/DaI.D aV)

Semenov number

S6

Senftleben number

s7

Sherwood number

S8

Sommerfeld number (1)

( P N / p b ) (,D/a) ( D = shaft diam., ( c j . ) Ocvirk number)

s9

Sommerfeld number (2)

( F d f i V ' s ) ( ~ / R( V) s~ = veloc. of shaft surface; R = shaft radius). (A'#, =

si0

Spalding function

k J K ; K = reaction rate constant [L/6'] A'Es2 [a 2 / 3 ( p o 2 / k T ) 1 . PI4LMPl Kronig number, q n . k c L / D = Arum (also termed Taylor number)

Kinetic viscosity/ molecular diffusivity

Diffusion in flowing

-(&Iu* =

(T

=o;

- T,) x

(Tw - T w ) , T w = wall temperature, T , = free stream temp., u + = Prandtl velocity ratio IV(Vf) 1 1 2 / ( g " ) 3 / 4 (H' = head of liquid produced by one stage) (cf. speed number) (4 q ) 1 / 2 ( V f ) V N/(2 H)3/4 = (delivery number) 112 X pressure numberj-al4 (cf. specific speed)

INDUSTRIAL A N D ENGINEERING CHEMISTRY

Ernst Heinrich \Whelm Schmidt (1892- ) (German)

Reaction kinetics

+

Convective heat transfer Mass diffusivity/ molecular diffusivitv Viscois force/load force

Mass transfer

Viscous force/load force

Lubrication

Dimensionless temp. gradient at wall

Convection

Lubrication

4/7riV&7j

54

Carl Gustaf Arvid Rossby (1895-1 957) (Swedish/ American)

Compressible flow Flow around obstacles Fluidization

L V ( p 2 / f i F 1~/ 3)

e

Osborne Reynolds (1842-1 912) (English)

Fluid friction

dL

s5

Speed number

Ferdinand Reech, (1805-1880) (Alsatian)

Agitation

peller diam.

Schmidt number

s12

Origin of Name

Fluid friction in conduits

=

s4

Specific speed

1

Flow resistance

pressure drop over length, L ) ( c j . R9)

gp(rAtr p ym rm)

SI1

Reference

coeff., Newton number, Fanning factor)

(dV/dL)wZ[ L =

R14

Field of Use Convective heat transfer in compressible flow

Cp(taw - tm)/V2; taw = attained

Pumps and compressors

Flow machines

Thomas K. Sherwood (American)

C O N T I N U ED

Serial No.

Name

Dejnition

iymbol

Significance

Field of Use

Referena

Stanton number

s14

Stefan number

Heat radiation

s15

Stokes number

Particle dynamics

SI6

Strouhal number

SI 7

Suratman number

Vortex streets; unsteady-state flow Particle dynamics

S18

Surface elasticity number

si9

Surface viscosity number

T1

Taylor number (1)

T2

Taylor number (2)

T3

Thiele modulus

T4

Thoma number

T5

Thomson number

T6

Thring radiation group

Heat transferred/ thermal capacity of fluid

of surfactant in undisturbed state, DS = surface diffusivity, L = film thickness p s / p L ; ps = surface viscosity, [ M / B ] ,L = film thickness

Truncation number

V1

Valensi number

v2

Vedernikov number

v3

Velocity number

Convection

= angular velocity of cylinder; Ra = mean radius of annulus) ( 2 uL2p/p) 2 [ w = rate of spin (i/e) ; &d;reight of fluid 112

112 / k l / 2 t l i 2

Lu(Corio1is force/ viscous force)2

=

(DaII) l / a (Ha - H E - H d / H ( H = total head; Ha = atm. pressure head; Ha = suction head; Hy = vapor pressure head) sV/L; o = characteristic time (cf.

NSI) pCp V/e*q T*(cf.Boltz-

cells

Stability of flow pattern in annulus with rotating cylinder

U C ( R ~ ) ~ W W( w P c ;

Thring-Newby criterion

T8

Net positive suction head/total head

W1

Weber number (1)

W2

Weber number (2)

w3

Weber number (rotating) Weissenberg number

w4

Geoffrey Ingram Taylor (1866- ) (English)

Effect of rotation on free convection Diffusion in porous catalysts Cavitation in pumps

Dietrich Thoma (1881-1 943) (German)

Fluid flow Bulk heat transport/ heat transport by radiation

Radiation Combustion of

nozzle fluid and surrounding fluid [ M / O ] ; R = equivalent nozzle radius; L = furnace half width p + / P (cf. Hersey number)

Shear stress/normal stress

Viscous flow ~~

- ____

Josef Stefan (18351893) (Austrian) George Gabriel Stokes (1 819-1 903) (Irish)

Convection cells

mann number)

T7

Origin of Name

Forced convection

s13

iLZp/p; w = circular oscillation frequency w henp = 0 [l/O] {*[*V/(V, - V ) b*f*(Nm*) (r* = exponent of hydraulic radius in formula [-I; t* = shape factor of channel section; V , = absolute veloc. ity of disturbance wave = M a netic Reynolds num%er, 9.u.

V2pL/u

(NweB)'

~

)scillations of drops and bubbles 2eneralized Froude number

Inertia force/surface tension force

V. V. Vedernikov (Russian)

Instability of openchannel flow

Moritz G. Weber (1871-1951) (German)

Bubble formation, etc.

V(pL/u)'12 =

( N w d' I 2 L3N2p/cr; L = impeller diameter r)aV/wiL; w3 =

Agitation Viscoelastic force/ viscous force

Viscoelastic flow

P(s)ds,w =

so-

G(s)ds, G = re-

laxation modulus of linear viscoelasticity, s = recoverable elastic strain

(Continued on next $age)

VOL. 5 8

NO. 3

MARCH 1966

55

TABLE I. ADDENDA

Thefollowing groups came to the attention of the writers too late to be included in the main tables: G70Up

Symbol

-1

Notes

Referenct

NO ( N = mixer r.p.m., O = mixing time to reach given degree of dispersion)

Prochazka, J., Landau, J., Coll. Czech. Chem. Commun. 26, 2961 (1961) Rothfeld, L. B., A.I.Ch.E. J . 9, 19 (1963)

~

Homochronicity number

HO

Knudsen number for diffusion

N K ~ A = 3eDaa/46K0,& (&A = Knudsen flow permeability constant, ;A = equilibrium mean molecular speed of species A ) , (cf. group K7). P Alternative name for Bingham number

Plasticity number

=

= (Hartmann number)2/Reynolds number

Stewart number

Wilkinson, W. L., “Non-Newtonian Fluids,” Pergamon Press (1960) Tsinober, AB., ApPI. Mech. Rev. 18, 505 (1965) ; Rev. No. 3751.

i

T A B L E I I . TABLES FOR I D E N T I F Y I N G D I M E N S I O N L E S S G R O U P S PHYSICAL PROPERTIES

Electrical and Magnetic Proberties

General Physical Properties Parameter

EX-

Parameter

Symbol

Coefficiertt of bulk expansion

Dimensions

__

Groub

___.__

1/T

MIL3

Densits

Joneni -1 +1 -2

E4, 12, G2 G5, R5, K9, 6

1

EXSymbol

Dimensions

I

Current density Electrical conductivity

ponen __

2:

ne

+1/2

Field strength

He

Magnetic permeability

pe

-1

+1 -2 +1 +2 -1 +1 +S/Z

Voltage

I

E

Group K10 E6 H2

L11,M 3 , 7 $2, L11 K9, M3, 6, S 6 111

+2

+I

M3 K10

-1

P15

-2/a

-112

Thermal Properties

+‘/a +‘/2

Humid heat Latent heats of phase change

+2/a

+I

S

-1

A,

+1 Ratio of specific heats Specific heat

Y

*1/2

c, c

-1

+2

MIL4 LS/e

Density gradient Diffusivity (molecular unless noted otherwise)

+1 -1

R13 -l/a

B5, B7(4), D2, K4(1), K7(8), L2, 7, 10, N7, P2, P9,Sa, 1 3

+1/2

-1/2

Diffusivity (surface)

ML/@

Diffusion tortuosiiy Molecular weight

... ...

Permeability (packed bed) Porosity (voidage)

L2

+1 -1 -1 +1 +‘/z

...

-1 +I

B4 L4 K7

*‘/a

F2

-3

c2

-1/2

...

Specific weight Surface tension

M /82

L9(‘1 S18 K7 D14, K9, S6 D14 L4

-2 -1

-’ / 2 +1 Coeflcient of polential diffusion in mass transfer. Knudsen dz’fusion coeficient. Binary bulk diffusion coeficient. (4) Efecfive di’usiuify (D rn) (molecular eddy transfer).

(I)

(9) ($)

56

+

+

I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY

Surface emissivity Temperature conductivity (thermal diffusivity)

e*

Thermal conductivity

k or A

ff

B2, 13, D3, 15, E l , F11,J2, K8, 11, L7, M10, N5,R3, 513

E, 7,

8

+1

2?3,

-1 -1 +1

T6 L9, P i , 12, R5 C13, L7,lO

-3 -2 -1

Mi R6 B4. 12. C7. 9.

+2/3

-2/3

+I

15

E4, 12, G4, J1, 8, R3, 4, L9, 5, 6 Pi, ,7

514 Rheological and Elastic Behauior Modulus of elasticity Rate of shear

E

-1 +1 +3 +1

C5, E4, H11 C1,E13, R1 A1 T8

TABLE II.

CONTINUED

-

1

EXParameter Viscosity (in all cases kinematic viscosity has been written as

Symbol

Dimensions

P

MIL9

anent

P/P)

Parameter

Group A i , 5 , G1, 5 , K1,9, S17, T2 B6, CIO, D5, 7, E6, 7, H8, L l , 3, 9, 01, P4, 11, R5, 6, 11, s9, 19, T i , V1 F2, K5,N8

Symbol

1

Dimensions

I

I E k t

Group

Areas

1 1

Area Area/unit volume

HZ s2, 3

A

L2

S, A*

1/L

I

+I

1

$1

-1 +1 -1

D9, F6, 7, 8 B2 B6 M11, 12

E2

~ ~ ! 2 ~ 8 3E3, ~13, 5, MI, 0 2 , P8, 9, S4, 8, 15, T8

Volumes

c1

Viscosity (surface)

c2 SI9 B3, H6

M/e

I

Volume

...

I

La

I

A5, H9, M11

TIMES AND FREQUENCIES LENGTHS, AREAS AND VOLUMES Cha racteristic Linear Dimensions Time

(Zn all cases kinematic viscosity has been written as

p/p)

Frequency General characteristic linear dimension

Various

L

-5 -3 -2

+3

P7 F9, L1 D9, 12,E3, F11, 12, H4, 5 , N2, 3, 4, 01, R9, S8, 9, 15 B1, 10, E2, 5 , 10, FI, 13, G4, H10, K6, L4, R6, 10, 15, 16, S6, 19, T5,W4 B11, D7, F14, 02 R4, Ti, WZ B3, 4, 5 , 7, 10, C7, D l , 3, 5 , 11 13,E7 10, F1: 2, 5 , 6 3 , H2, K3, 4, 6, 10, L3, 4, 9, 11, M i , 3, 7, N6, 7, 8, 01, P i , 2, 11, R8, 10, 11, 16, 52, 7, 14, 16, 17, 18, T3, W1 D7, H7, TI B9, D2, 4, E8, H6, K9, 01, P4, 5 , 12, 58, 9, vi A5, C10, G1,5,

+4

T2

-1

-l/a +'/2

+1

+3/2

+2

Dimension of agitator, impeller, etc. Film thickness Furnace half-width Larmor radius Mean free path Particle dimension Pore or nozzle radius Reactor length

Various

L

+5 -5

-3 -2 +1 +3

.+I

de

L L L L L

R

L

+I

L

L

Lf L LL h

-1

+l +1 -2 +1 f 3

I I +I

-1

Thickness of liquid layer

L

L

TEMPERATURES AND CONCENTRATIONS (DRIVING FORCES) Concentrations and Related Quantities Dimensional concentration

M/L3

-1

...

$1 -1 +I

-1/2

Dimensionless concentration-e.g., wt./wt. inert material, etc. Mass capacity Mole fraction Specific mass content, mass/unit mass Surface concentration

L3/M

-1

...

Zt1

...

-1

RZ D14 K4

M/L2

fl

Si8

+l

B13, R2

LP/M

Va or capacity &orow body)

D l , 2, R2 T3 €310 P6 A4, K 8

__

K l , R5

R6 P7 F9 D7,H4 D11,F15 w3 N8 T7 L4 K6 Si5 F2 A5,Gl M12,T7 B7 C3,L9

Temperatures, Temperature Dtyeerences Temperature, temperature difference

T , AT

T

RI*, T6* A4, 6*, B12, 13, c4*, 7, 10, D3* 4* 15, E l , &2,'6*, K3, 8, 9, 11, L9, N5, P5*, 12*, R14*, S6 L5 F6*, 7*, 8* C4, G5, 6, J 1 , 4 , L9, M i , P6, R5, 7, 14* S14*

Rate of temperature change

* Absolute temperature;

PI2

others-temperature differences. (Continued on next page)

VOL. 5 8

NO. 3

MARCH 1966

57

TABLE I I . CONTINUED VELOCITIES, RATES, FLUXES, TRANSFER COEFFICIENTS

1

Parameter

Symbol

Velocities

Parameter

Symbol

Dimensions

Angular velocity (rate of rotation)

N

1/ e

V

LIB

EXonen, __ -3 -2 -1

+l +2 -3 -2

-1

+I

+2

Impeller or agitator circumference Light Sound Waves Velocity gradient Velocity of Alfven waves

+3 -2 -1

US VI vs

Vw d V/dL VA

P7 H4, L i F9, R i 5 E2 S8, 11, 12, T1 F15, T2, W3 H5 C6 11 D13 E9,

Heat transfer coefficient

h

Mass transfer coefficient

kC

IEkzt

N2, 3, 4, R7, 8, 9, 10 A2 B3,C12 14, 3, Fli), H7, 8, K10, L3, Ikf3, P4, S13, 16 A2, B6, 7, 11, c 3 , 12) 14, D5, 7, E5, 7, Fi0, 14, H i 0 , K2, L5, 8, M2, 4, 7, P i , 2, 3, 10 R4, 11, 15, S1, 2, 3, T5, 6, V2, w2, 4 B l , 12, C5, 11, D15, E l , 11, F13, H i l , W1 c7 P14 D9

bi,

-1 -2 -1 +1

P3 R12 A2, K2, M 4 A2 N1 Cli. H8, 01, S9

M/TB3

Grou.4

~

B2, 4, C9, 52, N6, P15, S13

4-1

(H/L$TS) L/O

+4

M 1

4-1

B5, 53, L9, N7, s5, 7

FORCE, HEAD, POWER, PRESSURE

io, h,$6,

L8 M2, N1, S1

+1

us

Group

-1 -1 -1

+2

Velocity of bearing surface

Dimensions

Transfer Coeflcients

-1/2

Fluid velocity

1

Forces Force (resistance)

Force/unit length



H8, 01, S9

Fb

Heads, Power Fluid head

-3/4

+I Head (energy per unit mass of fluid = gH’)

-1

1

si1 H4 P3,T4

Power

vi

Pressures

:

Pressure

Pressure drop

1

AP, dP AP/L,

Pressure gradient

1

I

M/LB M/LW

1’

+I

4-1



1 B13,P13,C6,L6, R2 ’1

111

E9, 10, F1, H9, L3,RlO EiO,Fl,Kl,P4, R10

Flow Rates ( M a s s Fluxes)

Mass flow rate (mass flux) Mass flux density (mass flux/unit area) Mass flux/unit volume (reaction rate) Reaction rate constant

Vm

Volumetric flow rate

Vf

+1 -1

G

+l

U

52, 3, P15, $13 K4, M11

+1

T3 D l , 2, 3, 4

-1

s5

+‘/a

K

B2, M11, T 7 F6, 7, 8, G4, T7

CONSTANTS AND MISCELLANEOUS QUANTITIES Graaity Acceleration Gravity acceleration

A1 Bl,F13,15, M i , 56 -l/a

+‘/a

+1 +I

D9, F9

Other Quantities

Heat Fluxes Heat flux (heat flow/unit time) Heat flux/unit area Heat liberated/ unit mass Rate of heat liberation/unit volume (heat source power)

58

4

IML”B3

H5

4*

M/B3

K3, R6 D3, 4

M/LB3

P5

Q j

Sll B11, F14, P i 3 c9, s3 F2 N8 Ad, B9, C2, 10, D13, E8, G1, 5, H4, R5, 6, 8, 1 3

-”4 -1/2

I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY

Avogadro’s number Boltzmann’s constant Dufour coefficient Energy of activation Gas constant Shape factor Soret coefficient Stefan constant

1 1 A’

1/1m

k

I$

I

1

G* 7

1

L;02

L2/PT

... d/&S3

1 1 +l

-1

~

K 9 S6 K9iS6 A3, 4, F3 AG

-1

A6

-I/.,

Th

1 2:

+ i ” Vi ~

F3, P6 R1,T6

Attempts have been made to list the many dimensionless groups in English (5, s),German (25), and Russian (73) literature, This article lists 210 groups, 79 of which were not included in the most recent compilation (6). The groups are listed alphabetically in Table I, with their symbols, definitions, fields of application, and references. The Russian groups listed should be particularly helpful, as they are seldom defined when they are used in the Russian literature. Dimensionless groups are frequently generated in the analysis of a complex engineering problem. The more common groups thus generated are easily recognized, while the less common ones are not. Unless the less common existing groups are recognized, an already named group could unknowingly be renamed. Table I1 provides a tool that may be used to avoid this occurrence, by listing the groups by the variables of which they consist. These variables-Le., length, density, diffusivity, viscosity, etc.-are further subdivided into their exponents to which they are raised in the groups in question. Thus, Reynolds number is listed under the exponent +1 for the variables, length, fluid velocity, and density, and the exponent - 1 for viscosity. To illustrate the use of the tables in the analysis of a problem, the group (kE/qgT3)might be generated in the solution of a complex heat transfer problem. From Table I1 the groups containing the constituent variables are checked and the groups are listed :

R1

(W) (E+')

F11, L7,

Stefan-Boltzmann coefficient

(7-1)

Surface tension

(u-1)

Temperature

(T-3)

RI, T6 B9, C3, E8, L6, R1, W1 R1, T6

Thermal conductivity

Modulus of elasticity

C1, E13,R1

I t is immediately apparent that the only group common to all the categories listed is the Radiation number, Rl, which is equivalent to the previously unidentified group. The symbol assigned to a dimensionless group is usually the first two letters of its names. Several groups, however, have nonstandard symbols, particularly in the groups which are named after persons. These symbols are listed in the nomenclature. N 0 M ENC LATU RE annulus or clearance width, L area, L a cooling area/unit volume, 1 / L bearing breadth, L groups B6, B11 specific heat, L2/B2T = concentration, M / L a = specific vapor capacity (mass/unit mass/unit pressure change), L02/M = group D13

= = = = = =

= group R9

= group H 4 = mass capacity, La/M = specific heats a t constant pressure and volume,

L2/ev= heat capacity, La/OZT

CQ CS

C

dh

D

= group F9 = group S11 = group C10, dimensional concentration, M/L3 = groups C11, C4 = diameter, L = equivalent diameter (of particles, etc.), L = hydraulic diameter, L = diffusivity (molecular, unless noted otherwise),

LZ/e binary bulk diffusion coefficient, L2/0 Knudsen diffusion coefficient, L a / @ voidage; porosity ( -) surface emissivity ( -) modulus of elasticity, M/L02 activation energy, L2/@ = bulk modulus, M / L P = frequency, l / O , or Group F1 = group F11 = force, M L / P = force per unit length of bearing, M/O2 = group F6 = resistance force in flow, M L / P = acceleration due to gravity, L/02 = mass velocity (mass flux density; mass transfer coefficient), M/eLZ := heat transfer coefficient, M / T 0 3 := convective heat transfer coefficient, M / T O 3 := energy change per unit mass (= g X head), LZ/ee -- fluid head, L = field strength, Q/Le =i homochronicity number (see Addendum) = group F8 = heat liberated per unit volume per unit time, M/LO3 = groups J2,J3 = average free path/average velocity, 0, or group L6 = thermal conductivity, M L / TO3 = mass transfer coefficient, L/B = groups K2, K10, N5 = group A4 = group E4 = group C2 = group P13 groupH5 group E l l = group A1 = group E4 =i group E12 = group E13 =i group R1 group C1 =E group C2 = characteristic dimension (except as noted), L =E distance from midpoint to surface, L =i group T3 = group M10 = group H2 = concentration, wt./wt. ( -) =r specific mass content, mass/mass ( -) = moisture content, wt./wt. bone dry gas (-) =2 rate of rotation, l/O, and groups M3, N4 = groups B7, B8 = group C5 = group C10 = groups D7, D14 == group E l group F4 = groups H1, H9 = group K1 = Knudsen number for diffusion (see Addendum) = group N2 = group P7 = group R7 =: group S8 =ii group S9 =E group N9 = pressure, M / L P = plasticity number (see Addendum) = = = = = =

g

G

-

VOL. 5 8

NO. 3

MARCH 1966

59

b e a r i n g pressure, M / L @ static pressure, MILO2 vapor pressure, M / L V capillary pressure, MIL82 frictional pressure d r o p , MIL82 h e a t flux ( h e a t flow/unit time), MLZ/O3 h e a t flux density ( h e a t f l u x / u n i t a r e a ) , M/83 h e a t l i b e r a t e d / u n i t mass, L2/P l a t e n t h e a t of phase c h a n g e , L2/8* h e a t of vaporization, L2/82 radius, L h y d r a u l i c radius, L g r o u p R5 g r o u p M7 g r o u p V3 gas c o n s t a n t , L2/82T humid heat, L2/Q2T p a r t i c l e a r e a / p a r t i c l e volume, L 2 / L a ,a n d g r o u p

M6 g r o u p SI4 temperature, T absolute t e m p e r a t u r e , T t e m p e r a t u r e difference, T g r o u p P10 reaction rate, M/L3B velocity of surface (solid), LIB fluid velocity, L/B, a n d g r o u p V1 velocity of Alfven magnetic waves, LIB volumetric flow rate, L3/8 velocity of light, LIB mass flow r a t e , MI8 velocity of sound, LIB circumferential velocity, LIB voIume of system, L s gross volume, L3 e n t r y l e n g t h ; distance f r o m e n t r a n c e , L g r o u p P11 group 0 2 t h e r m a l diffusivity ( t e m p e r a t u r e conductivity),

L2/B coefficient of bulk expansion, 1/ T , a n d g r o u p D12 D u f o u r coefficient, T specific gravity ( -) a n d g r o u p R3 r a t e of shear, 1/8 r a t e of c h a n g e of t e m p e r a t u r e of m e d i u m , 7 / e Soret o r t h e r m o g r a d i e n t coefficient, 1/ T , a n d g r o u p D11 difference i n q u a n t i t y height of roughness, L a n d g r o u p A3 e d d y mass diffusivity, L2/0 diffusion tortuosity ( -) radiation coefficient (Stefan-Boltzmann coefficient), M/T4B3 time, 0 relaxation time, 8 m e a n free p a t h , L d y n a m i c viscosity, MILO magnetic permeability, M L / Q 2 rigidity coefficient, M/LB permeability, L2

3.1416 . . . .

power t o a g i t a t o r o r impeller, ML2/B3 densitv. ,, M /. L 3 g r o u p P3 surface tension, M/B2a n d g r o u p S12 g r o u p 66 electrical conductivity, Q2B/L3iM group T 4 group T8 = wall shear stress, M / L @ = yield stress, M/LOZ = g r o u p D9 = groups N8, . P14, . R10 = a n g u l a r velocity (of fluid, unless noted otherwise), 1/0 = mass transfer potential (concn.), MIL3 - ( b a r o v e r ) = m e a n value

- .

Y~

N.B.:

60

(F)

=

(F); (y) (H)=

INDUSTRIAL A N D ENGINEERING CHEMISTRY

REFERENCES (1) Adrianov, V. N., Shorin, S. N., AIAA J . 1, 1729 (1963). (2) Ahlstrom, H. G., J. Fluid Mech. 15, 205 (1963). (3) Becker, H. A., Hottel, H. C., Williams G . C “Ninth Svmposium (International) on Combustion,” p. 7, Academic’Press, gew York, f963. (4) Beer, J. M., Chigier, N. A., Lee, K. B., Zbid., p. 892. (5) Boucher, D. F., Alves, G. E., Chem. Eng. Progr. 55 (91, 55, 1959. (6) Zbzd.,59 ( 8 ) ,75 (1963). (7) British Standard 1991, “Recommendationsfor Letter Symbols, Signs and Abbreviations. Part 2. Chemical Engineering Nuclear Science, and iZpplied Chemistry,” British Standards Institution, Lbndon, 1961. (8) Brown, G. G., et al., “Unit Operations,” Wiley, New York, 1950. (9) Buckingham, E., Phys. Rev. 4, 345 (1914). (10) Berg, J. C., Acrivos, A,, Chem. Ens. Sci. 20, 737 (1965). (11) Chukhanov, Z . F., Intern. J. Heat Mass Transfer 6 , 691 (1963). (12) Dallavalle, J. hf., “Micromeritics,” 2nd ed., Pitman, New York, 1948. (13) El’perin, I. T., Inzh. Fiz. Zh. Akad. Houk Belarus&. SSR 4 ( l ) , 131 (1963). (14) El’perin, I. T., Intern. J . Heot M a s s Transfer 5 , 349 (1962). (15) Engel, F. V. A., Z.V.D.I. 107, 671, 793 (1965). (16) Faller, A. J., J . Fluid iMech. 15, 560 (1963). (17) Fedorov, B. I., Inzh. Fir. Zh. Akad Nauk Belorursk. SSR 7 ( l ) , 21 (1964). (18) Gardner G. O., Kestin, J., Intern. J . Heal 2dosr Transfer 6 289 (1963). ( 1 8 ) Gel‘peri;, I. T., Alnshtein, V. G., Goikhman, I. D., Inzk. biz. Zk. ilkad. .Vauk Belorursk. S S R 7 (7) 15 (1964). (20) Grassmann, P., &hem. Ing.-Tech. 31, 148 (1959). (21) Grassmann, P., Lemaire, L. H., Zbid., 30, 450 (1958). (22) Greene, D. F., Ph.D. Thesis, Columbia Univ., 1961 [Dissertation Abstr. 24, ( E ) , 3248 (1964)l. (23). Gukhman, A. A,, “Introduction to the Theory of Similarity,” Academic Press, hew York, 1965. (24) Gutfinger, C., Tallmadge, J. A,, A.I.Ch.E. J . 10, 774 (1965).

(25) Hahnemann, H. W., “Die Umstellung auf das internationale Einheitensystem

In Mechanik und Warmetechnik,” VDI-Verlag, Dusseldorf, 1959. (26) Holt, M., “Dimensional Analysis” in “Handbook of Fluid Dynamics,” V. L. Streeter, ed., McGraw-Hill, New York, 1961. (27) Hottel, H. C., Sarofim, A . F., Intern. J. Heat Mass Transfer 8 , 1153 (1965). (28) Hottel, H. (2;. Williams, G. C., Jensen, M’.P., Tobey, A. C,, Burrage, P. M. R., p. 923 in Ninth Symposium (International) on Combustion,“ Academic Press, New York, 1963. (29) Huntley, H. E., “Dimensional Analysis,” MacDonald & Co., London, 1952. (30) Johnson, S. P., “Survey of Flow Calculation Methods” p. 98 Preprinted Papers & Program, Aeronautic Br Hydraulic Divisions,’ A.S.M,’E. Summer hleeting, June 19-21, Univ. of Calif. and Stanford Univ., 1934. (31) Kafarov, V. V., Zh. P r M . Khim. 29, 40 (1956). (32) Kay, J,, A I . , “An Introduction to Fluid Mechanics & Heat Transfer,” Cambridge Univ. Press, 1957. (33) Kestin, J., Persen, L. N., Intern. J . Heat M a r s Transfer 5 , 143 (1962). (34) Klinkenberg, A., hlooy, H. H., Chem. En!. Progr. 44, 17 (1948). (35) Koide, K., Kubota, H., Shindo, M., Chem. Ent. (Japan), 28 (8), 657 (1964). (36) Lykov, A. V., hlikhailov Yu. A . “Theory of Energy & Mass Transfer,” Prenrice-Hall, Englewood Clhs, N.J.,’1961. (37) McCabe, W. L., Smith, J. C., “Unit Operations of Chemical Engineering,” McGraw-Hill, New York, 1956. (38) Matsuhisa, S., Bird, R. B., A.I.Ch.E. J . 11, 588 (1965). (39) Mikhailov, Yu. A,, Bornikova, R. M., Znzh. F i z . Zh. Akad. .&‘auk Belorursk. SSR 6 (lo), 45 (1963). (40) Mikhailov, Yu. A,, Romanina, I. V., Ibid., 7 ( l ) , 49 (1964). (41) Miyauchi, T., Nakano, K., Obata, K., Kimura, S., Chem. Eng. (Japan) 26 ( 9 ) , 999 (1962). (42) Mkhitaryan, A. M “Hvdraulics & Fundamentals of Gas Dynamics,” Israel Program for Scientific’?Translations, Jerusalem, 1964. (43) Mordell, D. L., Wu, J. H. T., Con. Aeronaut. Space J. 9 (41, 117 (1963). (44) Motulevich, V. P., EFosipko, V. M., Petrov, Yu. P., in “Physics of Hear Exchan e & Gas Dynamics, A. S. Predvoditelev, ed., Consultants Bureau, New York, 1863. (45) Nagata, S., Chem. Eng. (Japan) 27 (8), 592 (1962). (46) Kield, D. A,, J . Fluid M e c h . 1 9 , 341 (1964). (47) Potter, J. M. F., B.Sc. Thesis, Dept. of Chem. Engrg., Univ. of Birmingham, England, 1959. (48) Rayleigh, Lord, Phil. Mag. 48, 321 (1899). (49) Reiner, M., Phys. Today 17 (l), 62 (1964). (50) Rouse, H., (ed.), “Engineering Hydraulics,” Wiley, New York, 1950. (51) Rouse, H., Ince, S., “History of Hydraulics,” Iowa Institute of Hydraulic Research, Stare LTniversity of Iowa, 1957. (52) Sazhin, B. S., Znzh. F i z . Z h , Akad. Nauk Belorussk. SSR 5 (6), 1 3 (1962). (53) Sazhin, B. S., Miklin, Yu. A,, Ibid., 6 (101, 57 (1963). (54) Schlichting, H., “Boundary Layer Theory,” 4th ed., McGraw-Hill, New York, 1960.

(55) Scriven, L. E., Sternling, C. V., J . Rluid Mech. 19, 321 (1964). (56) Sillem, H., Z.V.D.I. 106, 398 (1964). (57) Szebehelv V. G., p. 771 in “Proc. 2nd U.S. Kat. Congress of Appl. Mech.,” Ann Arbor,‘hich., June 1954; A.S.M.E., N e w York, 1955. (58) Tallmadge, J. A,, Labine, R. A., Wood, B. H., IND. CNO. CHEM.FUSDAMENTALS 4, 400 (1965). (59) Tamarin, 4 . I., Inzh. F i z . Zh. Akad. Nauk Belorussk. SSR 6 (7), 19 (1963). (60) Tartakovskii, D. F., Zbid., 7 ( l ) , 71 (1964). (61) Vedernikov, V. V., Compt. Rend. Acad. Sci. U.R.S.S. 48, 239 (1945); 52, 207 (1946). (62) IVeber, M., Jahrb. Scha/.fbautechn.Ges, 20, 355 (1919). (63) White, J. L.,J. Aflfil. Pol~mer.Ski. 8, 2339 (1964). (64) Yamaguchi, I., Yabuta, S., Nagata, S., Chem. Eng. (Japan) 27 ( E ) , 576 (1963). (65) Yas’ko, 0. I., Znzh.-Fiz. Zh. Akad. Nauk Belorussk. SSR 7 (12), 112 (1964). (66) Zibrodskil, S. S., “Flow & Heat Transfer in Fluidized Beds,” to be published shortly by MIT Press. (67) Zhuravleva. V. P., Inrh.-Fir. Zh. Akad. Xauk Belorussk. SSR 6 ( 9 ) , 73 (1963).

Readers’ Information Service CARBONE and CLARSIL (pg. I) ........................ .Circle No. 503

ADVERTISED PRODUCTS INDEX

Aryl Mercaptans-captive raw materials and unique refining processes enable us to offer them in commercial volume at low cost. Send for more particulars, samany technical assistance ples YOU may require (pg. 2). .Circle No. 47

EQUIPMENT

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I on Exchange Resins-D U 0LITE

ion-exchange and adsorbent res. ins have a world market, a world of applications: conditioning, refining, purifying (pg. OBC) ..Circle No. 504

Pumps-pump several different liquids simultaneously; feed and mix; meter additives (pg. 8) Circle No. 30

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Safety equipment-Send for Haws, “First Aid on Tap” catalog for information on the entire line of eye/face-wash fountains and emergency drench showers (pg. 16). Circle No. 24

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Mixing Propellers-guaranteed absolute accuracy of these propellers makes them a “must”for agitating, mixing or pumping (pg. 8) .Circle No. 508

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lsobutyrate plasticizer-Replace dioctyl phthalate with TEXANOL Isobutvrate. This economical plastidzer permits a reduction of the amount of plastisol charge up to 31% without decreasing wall rigidity or increasing dispersion .Circle No. 505 viscosity (pg. IFC).

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Rubber ingredients-Each of the many products Esso offers the rubber industry has its own special way of being helpful: Raw Materials, Aliphatics, Aromatics, Ketones, Acetates, Alcohols, Resins, Elastomers (pgs. 6-7). Circle No. 506

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Separator-spectacular, new concept in material sizing with the Hi-ProbSizer. The Hi-Prob Sizer may be used for size separations from 2” to 150 mesh; making single or multiple separations; with capacity up to 350 tons per hour .Circle No. 27 (pg. 15).

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Spray Nozzles-Choice of capacities from 0.07 to 1055 GPM at 20 psi. Choice of spray angles from 50” to 130” depending upon ca.Circle No. 4 pacity (pg. 8).

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I I I INSTRUMENTS Gages-Send for Catalog DH-57 on diaphragm gages for your fluid processing jobs (pg. 4) Circle No. 501

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pH instrumentation-Beckman bulletin of 60 pages to acquaint you with every facet of pH-pH meters, blood pH systems, accessories, electrodes, supplies, titrators-and the in-depth service which accompanies every Beckman product (pg. IBC) .Circle No. 502

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CHEMICALS AND MATERIALS Active Carbon-bleach, refine, retrieve and purify with ACTI-

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MARCH 1966

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