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
