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-
VOL 58
NO. 3
MARCH 1966
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