DIMENSIONLESS GROUPS - Industrial & Engineering Chemistry

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A new listing that supplements and extends the authors' compilation in B E C , March 7966

DIMENSIONLESS GROUPS GEORGE D. FULFORD

nother 75 more dimensionless groups have come to light in the two years since our previous list of 210 groups (5) was completed. T o keep the earlier list as useful as possible, Table I lists corrections to Table I, reference 5, and Table I1 (alphabetical list of groups) lists the new groups in the same format as before, numbered to permit interpolation into Table I, reference 5. Table I11 of this compilation (tables for identifying dimensionless groups) incorporates Table I1 of reference 5 and, therefore, provides an index to all 285 groups in both parts. The nomenclature throughout, as well as the method of using Table 111, remains unchanged. With only a few exceptions, the groups contained in this listing are fairly recent. Many of them reflect the current burgeoning research interest in transient or cyclic effects and the effects of turbulence on transfer processes-e.g., groups D8a, FlZa, FlZc, L7a, P8a, P12a, P15a, S ~ CS4d, , T7c, and W4a. Similarly, the more specialized groups FlZd, ROa, Rob, ROC,Rod, and WOa are concerned with cyclic effects in packed or

JOHN P. CATCHPOLE

A

AUTHORS George D. Fulford

is Assistant Professor of Chemical Engineering at the University of Waterloo, Ontario. John P. Catchpole is at present on the staf of the Admiralty Materials Laboratory, Holton Heath, Dorset, England. The authors wish to thank the readers of their original listing of dimensionless groufs [IND.ENG.CHEM.58 (3) 46 (7966)lforpointing out additional groups and for locating errors.

TABLE 1.

CORRECTIONS TO TABLE I, REFERENCE 6 ~~

Serial No.

Error Occurs in

Corrected Form

E7 P2

Definition

F6-8

Definition

F9 Fll Fi2 Fl5 H4 Ha H9 K1 K5

Definition Definition Definition Name Definition Name Definition

Origin

Should refer to

E8

y

K7

Definition Definition Definition

K9

Definition

Kli

. Ma

Vf/Nd' ke/pCpLm2 (cf. D12)

Froude No. (rotating) gH'/N2d2 Hersey No. wfAPF//fi?a 5

Z(NllsP

*

v7

q

m

cDAB/DKAr

L2

Reference Definition Definition

L5

Definition

v/

L7

si2

Name Origin Definition Definition Definition Definition

Lewis No. Should refer to group N2 r/CpAt ll/LspN' 0 = (T Tm)/(T, - T m ) (4 a ) 1 I s ( V f ) ' I z N / ( 2 H ) a / 4

si8

Definition

T1 TS page 41

Definition Footnote

L1

Nl N5

P7 s10

Origin

(77) U/&aN*

(D f ~ D ) / D

(A

R T)'"

-

(1886) = (DaIV)1'2

P factor

REPRINTS WILL SHORTLY BE AVAILABLE A T 10 CENTS PER COPY TO COVER POSTAGE AND HANDLING

V O L 6 0 NO. 3

MARCH 1968

71

TABLE I I .

ALPHABETICAL

I-

Serial NU.

A0

Name Absorption No.

DeJinifion

Symbol

Significance Dimensionless mass transfer coefficient

Ab

Field u j Use

Referencc

Gas absorption in wetted wall column

(40)

Propeller studies

(27)

Compressible flow

(27)

koL Ala

Advance ratio

Alb

Aeroelasticity parameter

J ,

..

I

I

Anonymous group 3

c

D x / V / L f (symbols as in Absorption No.).

A4b

Anonymous group 4

l/a

rwR/V,#; R = cylinder radius; V , = velocity outside bound- Frictional force/viscous ary layer force (dimensionless skin friction)

Bla

Bairstow No.

...

B2a

Batchelor No.

.. .

Buoyancy parameter

,

V/Vsw VSw = velocity of sound a t wall (cf. Mach No.)

Darcy No. (2)

Daz

D8a

Generalized Deborah

Nz

Magnetofluid dynamics

(27) (27)

Free convection

Inertia force/permeation force

Flow in porous media

(70)

Generalization of group D8

Rheology

(2)

iRiBni;/c, 8 = thermodiffusion constant = (DT/D)/nlonzo [ - 1 ; DT = thermal diflusioncoefficient[LZ/el; nio’, nlo’ = n:/n’, nn’/n’; n’ t o t a i ~ o ofmolecules . = n; n,: n; = No. of molecules of components 1, 2, in binary mix-

Heat of isothermal mass transfer/enthalpy of unit mass of mixture

Thermodiffusion

(22)

V / V i (Vi = speed of light) (cf. Lorentz No.)

Fluid velocity/velocity of light

.

V L / D ’ . D‘ = permeability coefficient of porous medium [LVBI

,/mw e,, ,

-

Is = invariant of rate of strain tensor (sec.-z); Iw DUZ

...

Buoyancy force/inertia force

variant of vorticity tensor (sec. - 2 ) ;

Dufour KO.

...

VLa,/V12€e = electrical permittivity [ Q W / L a M ] .

D6a

Gas absorption in wetted (40) wall column Laminar boundary layer (72, 28, 35) flow

Previouslv used for Mach ko., now largely obsolete

ee

B13a

D14a

e

,

KO.

en

-

I

-

Special form of Strouhal No. Inertia force/compressibility force ( A b ) Q / ( N s h ) z Dimensionless diflusivity

A4a

(I/@)

II

= liquid side mass transfer coefficient; x = length of wetted surface; L f = film thickness; V i = volume flow rate per wetted perimeter [L2/S] V/ND V = forward speed; D = propeller diameter = Cauchy No,, q.v.

= innatural time (sec.)

+ A;

ture; also Dill = (DT/D)p/pc,Tnzo

..,

Ela

EinsteinNo.

E4a

Electric field parameter

RE

E/VpSHe

E4b

Electrical characteristic No.

El

Nu

p(dx/dT)LZ A T E:/p2 E1 = electrical field strength [ M L / Q 0 2 ]; x = dielectric susceptibility [ Q 2 0 2 / M L ~ l V L / D * ; D* =, 1/z,(D+ D y ) [L2/0], D +, D - = diffusion coefficients of ions (cf. group P2) reV/Q’Lb’ ae = electrical permittivity [ Q W / L a M ] ’ Q‘ = s ace charge density [Q/La]; b‘ = carrier mobilitir [ Q e / M f Alternate name for group E4c, q.v.

E4c

Electrical Kusselt No.

E4d

Electrical Reynolds No. ( 1)

E4e

Electrical Reynolds No. (2 1 Modified Euler No.

ElOa

.. . .. . Eu’

.

.

+

Convection current/ diffusion current

HL‘ = head of liquid on tray [ L ] ; P L ,

Pa = densities of liquid, vapor; VG = vapor velocity based on free area,

(27) (26)

(6)

Electrochemistry

(70)

Electrical eflects in flow

(27) (70)

Friction head/velocity

H ~ ’ p ~ d V ~ 2 P ~

Magnetofluid dynamics Magnetohydrodynamics Electrical eflects on transfer processes

Flow of vapor across mass transfer trays

(73)

l / N p n X density ratio

Rise of bubbles

( 73)

Dimensionless heat effect of reaction

Unsteady state flow (77) problems Heat transfer in reacting ( 2 4 ) systems

head

I

E13a

Expansion No.

Ex

( $ ) ( P y ? )

d = bubble diam., V = bubble veloc., PL, P G of liquid, gas F12a

Fourier No. (flow)

F12b

Frank-Kamenetskii No.

Fof

densities

Ye/pL2 Q”

8

-

Ea L2k,exp(-EEo/aT)

k CRT:

I

.

F12c

Frequency parameter

F12d

Frequency No. (2)

Nf

Fl2e

Frassling No. (heat transfer)

Fsn

Fllf

Frossling No. (mass transfer)

FSm

72

, ,

Q“ = heat liberated per unit mass of material reacting/unit volume [ l / L 0 2 ] ;k = thermal conductivity of reacting mixture [ M L I T P ] ; ku = preexponential constant in Arrhenius equation [ M / B ] w’L/V = 271. X Strouhal No.; Z a / w ‘ = period of motion [e] w,L/V; L = packin element diameter [ L ] ; V = interstitial fluid velocity [ L / O f ; uv = radial frequency (radians/sec.) [ i / e ] (cf. groups HIO, Si6, T5) (NN,

(NSh

- 2)/(NRE1~2Nprl 1’)

-

~)/(NRB~~~N~$/’)

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

Special form of Strouhal Unsteady state flow, etc. No. (cf. also T5) Special form of group Flow in packed or fluidized beds F12c Special dimensionless heat transfer coefficient Special dimensionless mass transfer coefficient

(27) (23, 30)

Heat transfer to spheres in turbulent streams

(Sa)

Mass transfer to spheres in turbulent streams

(sa)

~~

L I S T O F NEW GROUPS

Serial No.

Name

Defnition

Symbol

Gba

Diffusional Grashof No.

GrAa

H8a

Hess No.

Ga

Significance

Lb = boundary layer momentum thickness, [ L ] ; Ro = radius of curvature [ L ] LapggpiAnA/pz nA mass fraction of species A, [ - I ; fiL = coefficient of density change with n i , [- ]

-

H9a Hila

-

H08

I1

Ilyushin No.

I

K7a

Knydsen No. for diffusion

N X , , ~3 ~SDAB/4 ~ K , Aira

K7b

Kondrat’ev No.

Kn

( V d p / p ) . ? r 0 / 3 Vap = 4 r ~ / 3 Vap slip resistance [ M / L 6 2 ]

Laplace No.

La4

L5a

Lebedev No.

Lea

L5b L7a

Leroux No. Turbulent Lewis No.

LCT

L7b L7c

Lewis-Semenov No. Lock No.

M3a

Magnetic Grashof No.

M4a

Magnetic No.

M5a

Magnetic Prandtl No.

P3a p7a

Plasticity No. Modified Power No.

., .

...

...

... R,

.

, , P N i

p~

-

Characterizes development of interfacial area per unit area of tray

. N R ~ T; D = max. dynamic

-

Koa = Knudsen flow permeability constant; ZA equilibrium mean molecular speed of species A qNBih; N ~ i n= heat transfer Blot No.; = temp. field nonuniformity parameter = ($8 ta)/(f to); ta = temp. of surrounding medium; t s = body surface temp.; Z. = body mean temp.

-

L3a

En-8

[a Fil

NB ( N mixer r.p.m., B = mixing time) A ~ ~ / P = L ~We4 L X Laplace No. = NE"^ X Nprl Ap, = pressure drop across li uid on tray [ M / L P ] ; liquid density; L = depth ofqiquid on tray

Homochronicity No. Hydraulic resistance group

rc

1

Reference

Buoyantforces X inertia Interphase transfer by forces/(viscous forces)z free convection (density changes caused by concentration differences)

(KLZ/um) (C0)n-I n = order ofreaction [ - I ; Co = initial concn. [ M / L . P ] ; am = mass transfer conductivity of reaction roducts [LZ/B]; K = reaction rate constant,

Field of Use

-

Differs from K7 by numerical constant

Molar expansion flux/ molar vapor transfer flux

(2.2)

Mixing, agitation Pressure drop in distillation columns

(37)

(34)

Heat transfer between fluid and body

(27)

Interfacial behavior on distillation trays

(32)

Drying of porous materials

(22)

(27) . , (27)

Magnetofluid dynamics

(7)

do.

cf. Magnetic Grashof No., group M3a Bingham No. D’ (AW)-I/~ LOpNa (“,)a.al

(6)

-

(26) (27)

(47)

6

2

(27) (II)

Rotor blade dynamics

See group M3 Magnetohydrodynamics Magnetic R e nolds No./ Magnetofluid dynamics Reynolds (properties of fluid)

uepe If_

(75)

Gaseous diffusion in packed beds

Combined turbulent heat and mass transfer

.

4 7 r c e p e b ~ ) Nar Nur group G5 p a e (ue L / p V ) l / Z = (magnetic force parameter)l/Z

Heat and mass transfer with chemical and phase changes

Flow of viscoplastic fluids (ZOa)

-

= rc/WeI ”&,*pressure drop across liquid on tray [ M / L @ ] ; L = depth of liquid on tray ebT(ta td/cbpps bT = intensity of vapor expansion in capillaries of body on heating [ M / L a T ] ; la = temp. of surroundin medium; to = initial temp.; pa = density of solid [ M / % ] 6 cavitation No. C6 C , P ~ D / ~=T I D / ~ T = eD/eT k~ = eddy thermal conductivity [ L M / T B a ] ; l ~ l~, = mixing len ths for mass, heat transfer [ L ] ; ST = eddy thermal digusivity [La/S] = ~ / N L * NL@ ; group L7 pR4ia’/I p = fluid density. a’ = rotor lift curve slope [ L a / M ] ; i = blade chord [ L l ; ’ R = rotor radius [ L ] ; I = moment of inertia of blade about hinge [L4]

(37)

-

-

(3‘4

Agitation

0: effective agitator diameter [ L l ; L : effective a itator height; Aw = wall proximity factor; Nb No. of blades on agitator; N e effectiveNo. of blade edges

P8a

Total Prandtl No.

Pr

€M+ ST

b/d -!-

d~

-

P

P

eT

eddy transfer coefficient for momentum, heat

Total momentum diffusivity/total thermal diffusivity

Heat transfer in combined turbulent and laminar flows

(25)

[ L ~ I

P8b

Turbulent Prandtl No.

P ~ T

. M / ~ T = I/IT S M , ST = eddy viscosity, eddy thermal diffusivity [ L a / @ ] ;

I , IT

P12a

Predvoditelev No. (mass transfer)

Pdm

P15a

Pulsation No.

NPU

-

mixing lengths for momentum, heat transfer (FmL2/am) ( N F ~ r m = rate of change of mass transfer potential of medium, ( m a d u n i t mass)/time [1/81; am = mass conductivity of material [ L z / B ] ; N F ~ ,= Eroup - - F12

?$e

equiv. diam. of channel

Eddy momentum Heat transfer in turbudiffusivity/eddy lent flow thermal diffusivity Rate of chan e of concn. Mass transfer of mediumTrate of change of concn. of bodv Transfer to pulsed fluid

(9,25) (22)

(20)

(Continued on next page;

VOL 60

NO. 3

MARCH 1968

73

TABLE II. ALPHABETICAL LIST OF NEW GROUPS (CONTINUED)

Serial h'o.

I

Name

Symbol

. .,

Definition

ROa

Radial frequency parameter ( 1 )

Rob

Ria

Radial frequency parameter (2) Radialfrequency parameter (3) Radial frequency parameter (4) Radiation parameter

R6a

Reaction enthalpy No.

R15a

Rossby No.

S4a S4b

Schmidt No. (2) Schmidt No. (3)

(Sc)a

P X / P 4 2

S4c

Total Schmidt No.

sc

6M

ROC

Rod

.. , ... , . ,

NH

... ...

-

-

CD

I II

+ Gulp) +D

-

eddy viscosity [LZ/O] cM/fD=- l/lo r~ eddy viscosity, [LZ/S], I , I D momentum, mass transfer [ L ] = ~ / N , L(see & group L7) L/X = ~ / N K , NA-" group K6 87120' (definitions as group D14a)

-

S4d

Turbulent Schmidt No.

S5a S7a

Semenov No. (2) Smoluchowski No.

S9a

Soret No.

So

SlOa

Spalding No.

B'

e, A T / ( r , q , / V , ) : qv = radiant heat flux [MLIIBa]; V , rate of mass transfer [ M / 6 ]

S13a

Stark No.

Sk

?TaL/k L = thickness of layer [ L ] ;

S14a

Stewart No.

S15a

Stokes No. ( 2 )

.. .

...

.. . SIz

SlBa

Surface tension No.

T5a

Thompson No.

T7a

Thrust coefficient

Te

T7b

Torque coefficient

QE

WOa

Wave No.

Ts

k

Weber No. (3)

Wca

-

W4a

Generalized Weissenberg No.

NI

74

Dimensionless thermodiffusion coefficient Ratio (sensible heat/ latent heat) for evanorated material

-

(Hartmann No.)%/ Reynolds No.

radius [ - I pz/h*up: h* = surface area of packing element/perimeter [ L ]

FTjpVzd1: FT = thrust force [ M L / P ] ; V = forward speed [ L / 6 ] ; d = tip diameter [ L ] F:/pVzda F E propeller torque [ M L Z / @ ]

L(wr/20r)lf* w, radial frequency (radians/sec.) [1/8]

'Fdkitions

density [ M / L a ] : L = depth of liquid on tray

Heat transfer analog of group Rod Surface tension force/ gravity force

as group DBa)

Generalization of group w4

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

Packed and fluidized beds

(20, 39)

Packed and fluidized beds Packed and fluidized beds Eflect of radiation on convective mass transfer Interphase transfer with chemical reaction

(78,33, 39) (39)

(74) (37)

(27)

(27) Electrochemistry

(70)

Mass transfer in combined laminar and turbulent flows

(25)

Mass transfer in turbulent flow

(25)

(27) Coupled heat and mass transfer Droplet evaporation

(22) (33n)

Radiant heat transfer

(7, 30)

Stefan No.)

1.042 mfgp(1 - p/pf)R**//rZ; p , g = density, viscosity of fluid: mj, pf = mass, density of float: R* tube radius/float

g/yfliquid [Ll

Reference (78, 33, 39)

(27)

E

WSa

Field of Use Packed and fluidized beds

See group K6

-

P : ~ $ e ~ / ~ p

Change in reaction ener y/change in sensi%leenergy More general form of group R15

Eddy momentum diflusivity/eddy mass diffusivity

mixing lengths for

-

(3

(Group ROa)Z (Group P2) Analog of Wave No.

No longer used Diflusivity of vorticity/ mass diffusivity of ions Total momentum diflusivity/total mass diflusivity

Semenov No. (2)

x as in group E4b

CM SCT

Significance

wrD/V2 wru/VZ D = 'diffusivity or dispersion coefficient of packed bed [ L Q / 6 ]W, I = radial frequency (radians/sec.) [1/6] wrL2/u wr as in group ROa; L tube radius [ L ] w:DL/V~ (Quantities as in groups ROa and Rob) L(07/2D)1'2 = [!/z(group ROb)]l'z (quantities as in groups ROa and Rob) e + q T L dh/k e + = function of mean surface emissivity of walls, [-I; Tw = wall temp. (abs.) [ T I (Au)A(AnA)/c,(AT) ( A u ) A = enthalpy of reaction per unit mass of A produced [ L 2 / 6 z ] ,nA mass fraction of species A [- ] V/oL

Magnetofluid dynamics

(38)

Calibration of rotameters

(loa)

Mass transfer in packed columns

(50)

Cellular convection

(3)

Propeller studies

(27)

Propeller studies

(27)

Transient flow behavior

(77)

Cyclic heat transfer

(4)

Interfacial area determination in distillation equipment

(32)

Rheology

(2)

TABLE 1 1 1 . TABLES FOR I D E N T I F Y I N G D I M E N S ION LESS G R O U P S These tables incorporate those from t h e earlier publication ( 5 ) Basic dimensions are taken to be: Length [ L Mass [MI Electrical char e [ Q ] Temperature ['%I Time [SI

PHYSICAL PROPERTIES General Physical Propcrties Parametrr

Symbol

Dimensions

Electrical and Maenetic Properties

Groups

Exponent

~

Coefficientof thermal bulk expansion Cdefficient of density change with concn. n.4 Density

B @A'

P

1/T

... M/La

-1 +1

E4 E12 GZ Gi, K9,'M3a, R5, R6

+l

G5a

-2 -1

M1

-a/s -&/a

+'/a

+l

+Z

Density gradient dp/dL Diffusivity (molec- D , am ular unless noted otherwise)

M/La

4-1

A1 B l B10 C l C2 C6, b3 b l 3 b l d a 6 3 E 9,' E l 0,' E loa, 'E 1i, E l l a F1 F l l Fl2a H5 kll; 11'Jl Kh, K i b L5a'Ldb i 7 M3 ' M 3 2 M5.' Mk NZ'N3 k 4 P i Pa;, P9'ROa' R9' R l b R l b S4'S4d S4c1S13 914; S i 5 Si5a h a : T7a 'T7b: T7c,'W3a CS,~S' E2 11 M4a, 0 2 , P i 3 3 ' K 5 'S3 h 0 d 3 P10, P11, R4, &a k 2 FZ, N E : s2 Alb, B1, B6, B9, C5 C7, D5, D7, D13, 'Ea, E4b E6 E7 E8 ElO;, Ei3a,'GZi, H6, H11, 11, 1 J3 4, K1 L7a b b d c , L9'M3; P1 'P8a P l k a Rdb $5 R i i , R13 'S4a 85a h 5 a S l 7 ' T l +5a, i 6 , T j c , Vl,'Wi, w3 A5, ClO, G1, 6 5 , Gba, K9, R5, R6, S15a, T 2 R13 ~... B5 B74 D2 D14a E4c k4e kea: K41'K72: LZ ' ~ 7L i o "7 PZ, ~ 9 P: l i a l , si, S9a

ME,

Jdo Rod A4h, FlZ', K 7 5 K7a E D , e=

-1 +I

LZ,'L~~

Diffusivity (surface) Diffusion tortuosity Dispersion (permeability) coefficient Molecular weight

Ds

-1

518

D'

-1

K7, K7a

-1

D6a ROa, ROC

+1

-1 +1

M

Permeability 6 (packed bed) Porosity (voidage) e

+I/%

-1 -l/a

fl

Specific gravity Surface tension

*'/a

y

-3

u

-2

-

-1 -1/g

Carrier suspectibilitv Current density Dielectric susceptibility Electrical conductivity

Symbol

Dimenrions

D14, K9,S6 D14 L6 B6, M12 L6 K7, K7a, L5a F2

-1

E4d

I

Q/.W

x

QW/LaM

+1 4-1

KlO E4b, S4b

us

QV/L'M

-1

E6, S4b H2 M4a B2; L l l M3 M3a, G 5 a , k 7 , d14s B2a E4d

+'/a

+l Electrical permittivity Field strength

rhermal diffusion coefficient

DT

La/e

1 Coejkient ofpotential diffusion in mass transfer. 2 Knudsen do j'in +@cient. 8 Binary bul dzffusion coejicienl. 4 Effectiue diffwivity (molecular and eddy)

=D

+1

Groups

Q@/M

QW/LaB

ee

Ha

-1 +1

Q/Le

1 : +1 +Z -1 +1

%a H2 L l l M4a K9: M3,'M6, S6, S14a E4a E6 J4 H2 'M3& M4a, M5a, Md. M7

M a netic permeafility

pa

Space charge density Voltage

Q

Q/La

-1

M3,S14a E4d

E

ML/@Q

$;

K10

ML/Qa

+SIz L i i +2

Thermal Properties Eddy thermal conductivity Eddy thermal diffusivity Humid heat Latent heats of phase change

kT

-1

L7a

fT

-1

L7a, Pea, P8b

s h, r

-1

Ratio of specific heats Specific heat

y

P15 A3 E l l El2, E13, Jl t l SiOa B13, 610, K8, K l l , N5 L5

-1 +1

C,

*I/%

-1

cp

BZ B13 D3 D14a 615 k l 411 K8, i2ROa, K11: R9, L7: R6a, Mlb,S13 d5,

Ji

-Ya +'/a

+z/s +I

Surface emissivitv Temperature conductivity (thermal diffusivity ) Thermal conductivity

e*

a

k PCP

-

T6 Rla L7b L9 P1 P8a P12, Rbb, k5, 85a, +5a WOa C13, L7, L10, ROa

-3 -2 -1

n2 -_

Cl B9 C3, C13, E8, L3a k6 R1 S18a W l W i D10,'G3,' 0 2 , 519,' R4,

4l;h:

A3, E4, E12, G4, 1 4 L7a L7b L9 P8a: Rob: R3: R5, R6, R7, Sda, Stoa, T5a

-1

-l/a

k, h

6, F7, Fa, WOa JS

+I -1 +l

M1 R6 B4, B12, C7, C9, C10, D4 F l 2 b G4 K 3 K7k L 7 d L d N6' P i fi5 P i Rbb Ala, Sia, d a , S14, T5a

~5:

-a/a

w9

+1

Exponent

6'

L24 L7b L91 ROa), RO;, ~ 4 a :s5a' S4c S4d

Diffusivity (eddy)

r

Parameter

-1/a

M9; g17, S18, T5a, W3a D14a,S9a

+1

&, WOa F l l , L7, ROa, R l

+ CD.

(Continued on next paze)

VOL 6 0

NO. 3

MARCH 1968

75

TABLE Ill. TABLES FOR IDENTIFYING Rheolo+al Parameter

Symbol

and Elastic Behavior Dimensions

Characteristic Lengths

Exponent

Groups PEa, P8b, Slc, S4d DSa, W4a

Eddy viscosity Invariants of rate-of-strain and vorticity tensors Maximum dynamic slip resistance Modulus of elasticity

I1

A l b C5 E 4 , H l l C1, k13,'Rl Ai T8 H6 B3, PSa E5 P10 Pll A4 b

Rate of shear Rigidity coefficient Shear stress

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

Paramefer

Symbol

Dimensions

Exponent

ID

Mixing lengthmass Mixing lengthheat Mixing lengthmomentum Particle dimension

IT

S4d L7a L7a, P8b

I

4-1

P8b, S4d

de

-2

S15

+1

F2,FlZd A5.Gl B7

C3

Pore or nozzle radius Thickness of liquid layer

Groups

-1 $1 -1

R

+1

L

+1 -1

H2,'Lll C13 M9, T5a

+2 4-3

R5

+4

TZ

M9, T5a

Areas

-1

Area

A

L2

Area/unit volume

S, A*

L -1

+1

-1

P/P)

D9, F6, F7, F8 B2 B6

+1

M11,M12

+1

A5, H9, M11

Volumes

w,W*

Volume

Time

CZ

Frequency f', w' Radial fre uencyl wI (radians/ time) (= 2 Tf9) 1 Also see angular velocity.

si9 B3, H6, P3a

Viscosity (surface) Yield stress

~

Ei eG J2 5 3 B l i C's C13 E3 E5

FiSa,'Mi,k3;, h h a , 02, pa, m a , ~ 9 54, , S4b, S ~ CS8, , S15, T7c, m Cl;Si8a

LENGTHS, AREAS, AND VOLUMES

e

e

Various

L

-1 +1

1/ e 1/8

+1 +'/a

+l

4-2

D8 T7c D8' D8a D1Z E3 F11, $12, F i l a , k9a, HiO, P12a, 515, T5, W4a ~.. FlZc, H1, P15a, S16, Vi Rod, WOa FlZd, ROa, Rob ROC

kl,

TEblPERATURES .ASD CONCESTI