02 BOPs / Woods D.R 2008 rules-of-thumb-in-Engineering-practice (epdf.tips)

Appendix B:

Dimensionless Groups

Sometimes the dimensionless group is describing heat transfer and sometimes mass transfer. For example, the Biot number and the Peclet number have forms for both mass and heat transfer.

The symbol [=] means “has dimensions of”.

Some terms, such as reaction rate and gaseous mass transfer coefficient, can be expressed in different units. For a dimensionless number to be dimensionless, the units used for the terms must ensure that that happens.

Dimensionless		Word definition	Equation	Range	So what?
number
Archimedes no.,	Ar	(inertial-gravity/	(Re2/Fr) q dimen-	1 to 107	Fluidization: Ar: 1 to
= Ga (Dr/r)		viscous2) q density	sionless density		106. For CFB, 1–100;
		ratio	ratio =		for transported bed,
		= Re (gravitational/	(rG2 Dp3 g)/m2) q		0.5–120; for fixed bed,
		viscous)	((rs–rG)/rG)		106–107;
					Particle settling/flui-
					dization
Arrhenius no.	Arr	relative activation	E/R T	gas solid	Reactions: Use: to tell
		energy	E = activation	reactions:	which is the control-
			energy, kJ/mol	5 to 40.	ling mechanism in a
		(molar activation		liquid	reaction: external
		energy)/		solid	mass transfer (Arr
		(potential energy of		reactions:	very small), transition,
		fluid)		5 to 40	pore diffusion, transi-
					tion, kinetic regime
					(Arr very large).
					Determines rates of
					chemical reactions

Rules of Thumb in Engineering Practice. Donald R. Woods

Copyright c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 978-3-527-31220-7

362 Appendix B: Dimensionless Groups

Dimensionless		Word definition	Equation	Range	So what?
number
Biot no. (for heat	BiH	external heat trans-	hD/k	0.001–10	Unsteady state heat
transfer)		fer to particle or			transfer. high Bi =
		solid/internal con-		10–1000	internal conduction
		duction;			controls
		internal thermal
		resistance/surface
		resistance
Biot no. (for mass	BiD	convective trans-	kG D/D	100–
transfer)		port/molecular	where kG [=] L/T	100 000
		transport
			kG D/D
		Mass transfer at in-	D = diam.
		terface/mass trans-	D = diffusivity
		fer in solid

Biot mass/	BiD/
Biot heat	BiH

Bird no.	Bir	mass of solids
		within a e 0.1 den-
		sity variation from
		the critical or cut
		density used to se-
		parate solids from
		solids

Bi ratio = Bi mass/

Bi heat = [Dc/DT] internal/[Dc/DT]

external

ranges from 10–500 for gas–solid reactions meaning that heat transfer is the controlling resistance externally and mass transfer resistance controlled internally

Reactions: range 10–1000 for gas-solid reactions

Bi ratio = 10 to 104 for gas solid

= 10–4 to 10–1 for liquid solid. To identify the controlling mechanism in a gas– solid reaction: external mass transfer, transition, pore diffusion, transition, kinetic regime

Need density distri-	I 15 then	Solid-solid
bution data for the	jigs,	classification:
feed	tables,	0–7 easy separation
	sluices	7–10 moderate
	OK. i 15	10–15 difficult
	use DMS	15–20 very difficult
		20–25 exceedingly
		difficult
		i25 usually not
		possible

			Appendix B:	Dimensionless Groups		363
Dimensionless		Word definition	Equation	Range	So what?
number
Bodenstein no.	Bd	Flow velocity along	Ivi D/D or height		Diffusion in a reactor.
= Pe for mass		a length/axial mix-	of catalyst bed		RTD. If approaches T
transfer		ing along the	Ivi/Daxial		there is negligible
		length;			backmixing and plug
= Re Scaxial		Special case of Pe			flow occurs; Bd = 0;
		no. for mass trans-			then complete mix-
		fer describing diffu-			ing. Bd/2 = no. of
		sion in a packed bed			ideally mixed cells in
					a catalyst bed of
					height H
Boltzmann no = Thring no.
Bond no.	Bo	gravitational forces/	rgDp2/g	0.05–1	Coating, surface
(Eotvos no.)		surface tension	where D = diameter		shape of curved fluid
		forces	of the tube or dis-		surface:
			tance between		Bo I 0.5 for slot
		Often called the	plates or diameter		coating
		Eotvos no., Eo,	of particle.		Bo i 0.5 for dip
		when applied to			coating
		particles or drops	(rL–rG) gDp2/g		For coating, usually
			where Dp = drop		Bo I 1.
			diameter		Radii of curvature of
					meniscus, the shape
					is not affected by sur-
					face tension if Bo
					large; shape is not af-
					fected by hydrostatic
					pressure if Bo is
					small.
					Bo large when surface
					forces low.
					Surface effects impor-
					tant in particulate
					systems: Bo I 1.
					Atomization, solvent
					extraction, trickle bed
					reactors
Boussinesq no.	Bq	inertial forces/grav-	Ivi/(2 g rH)0.5		Waves in an open
see Froude no.		itational forces	where rH = hydrau-		channel
			lic radius

364 Appendix B: Dimensionless Groups

Dimensionless		Word definition	Equation	Range	So what?
number
Bulk/film volume	d+	relative volume of		1–10 000	Reactions: gas–liquid;
ratio		bulk liquid to the			characteristics of gas–
		mass transfer film			liquid contactors.
		at the surface			Thin film = 1; bubble
					column 4000 to
					10 000
Capillary no.	Ca	viscous forces/sur-	mv/g viscosity,	0.01–10	Two phase flow: in
		face tension forces	velocity, surface		packed beds.
			tension		Free surface flows: as
					in coating.
					Coating: For forward
					roll. coating, predicts
					stagnation line down-

stream of gap;

For forward roll coating: predicts onset of ribbing at higher values (as a function of gap/diameter)

For premetering: predicts dynamic contact angle u proportional to Ca0.22 but independent at Ca I 5 q 10–3. For slot coating: Minimum wet thickness as a function of Ca: for values of Ca of 0.1 to 0.8. There is a critical Ca above which the minimum wet thickness is constant; the critical Ca is linear function of gap; at 1000 mm gap, critical Ca = 0.7

			Appendix B:	Dimensionless Groups		365
Dimensionless		Word definition	Equation	Range	So what?
number
DamkohlerI no.	DaI	(chemical reaction	equation depends	0.1–100	Reactions: If Da I
		rate [=] 1/T)/bulk	on the kinetics:		critical value, a flame
		mass flow rate	r* D/Ivi		is extinguished. For
		or	where r* = reaction		first order reactions
		(time for fluid to	rate [=] 1/T		and CSTRs in series,
		flow a distance)/	D = length		most reaction occurs
		(time to complete	r* = kA Cn-1		in first reactors if Da
		the reaction)	for zero order		i 1; for second order
			= kA D/(Ivi [A])		Da i 20
			where [A] = bulk
			concentration of
			reactant
			for first order :
			= kA D/Ivi
			for second order:
			= [A] kA D/Ivi
DamkohlerII no.	DaII	(chemical reaction	(eq. depends on		Reactions: large
		rate in flowing gas-	reaction kinetics)		values for high tem-
		eous system,	r* D2/D		perature reactors;
		[=] 1/T)/(molecular			approaches 0 for low
		diffusion rate)	for first order:		temp. bioreactors. For
					CSTR Da = 1 conver-
			kA D2/D		sion I 10 %
					= 10 conversion i
					90 %
					Da small = reaction
					controlled by mass
					transfer
					For reactive distilla-
					tion, Da 10 to 20
DamkohlerIII no.	DaIII	(heat liberated via	DH r* D/(cp Ivi T)		Reactions and heat
		chemical reaction)/			transfer
		(bulk transport of	where DH = heat
		heat)	liberated/unit mass
			T = temperature
			above the datum
			r* = reaction rate [=]
			1/T
DamkohlerIV no.	DaIV	(heat liberated via	r DH r* D2/(kT)		Reactions and heat
		chemical reaction)/			transfer
		(conductive or
		molecular heat
		transfer)

366 Appendix B: Dimensionless Groups

Dimensionless	Word definition	Equation	Range	So what?
number
Damkohler heat	Daheat heat liberated/inter-	–DHreact R/a h Tinlet		Reactors
	phase heat transfer	a = area/vol. of
	coefficient	catalyst
		h = interphase heat
		transfer coefficient
		R = ideal gas
		constant

Darcy coefficient:	4f	(head loss/velocity	2 g hf D/(v2 L)
= 4 Fanning		head) q (diameter/	where hf = heat loss
friction factor		length)	due to friction
Deborah number	De	Relaxation time of	De increases with
		system/flow charac-	increase in speed of
		teristic time	coater
		characteristic re-
		laxation time of the
		coating liquid or
		dispersion/charac-
		teristic residence
		time in the process
		flow or time of ob-
		servation
Deborah number		rate of drying/rate
for drying		of stress relaxation

Fluid flow in pipes/ fittings/conduits

Coating: High De makes processing in high speed blade coater problematic De II 1 equilibrium thermodynamics; any work done isothermally is immediately dispersed as heat.

De ii 1 work done is stored in material Viscoelastic behavior

Drying of coating: when De is high, coating cracks upon drying crack formation in coating

Elasticity, surface	ES	viscous/change in	RT GT/m Ivi	0–110	Coating
		capillary force be-
		cause of surface	(1/m Ivi)
		tension lowering	(dg/d ln G)
		because of surfac-
		tant
Elasticity, Gibbs	E	surface tension var-	(dg/d ln G)
surface		iation/surface con-
		centration
Eotvos no. related to Bond no.
fanning friction		wall shear stress/	2 twall/(r v2)	0.005 in	Fluid flow in pipes/
factor = Darcy/4		velocity head		turbulent	conduits; relatively
				flow	constant for
					Re i 2000.
					16/Re in laminar flow

			Appendix B:	Dimensionless Groups		367
Dimensionless		Word definition	Equation	Range	So what?
number
Fourier no.	Fo	time (thermal con-	kt/r cp D2	0.1–20	Heat transfer: unstea-
		duction/inertial		Usually	dy. Extent to which
		heat)		0.1–1.5	heating or cooling pe-
					netrates into a solid
Froude no.	Fr	inertial forces/	v2/gD		Coating; surface con-
see also Bous-		gravitational forces	where D = length		figuration in swirling
sinesq no., Bq					flows.
			N2D/g		Mixing: vortex forma-
			where N = rpm		tion for a free surface
			D = impeller dia-		in a mixing tank
			meter
Froude no.	Fr*	inertial forces/	(v2/gD) (r/Dr)	10–7 to	Gas holdup in bubble
density weighted =		gravitational forces		10–4	column
Fr (r/Dr)
Galileo no.	Ga	intertial-gravity/	(Re2/Fr) =	see Ar	Fluidization, circu-
= Re x(gravita-		viscous2	(rG2 Dp3 g)/m2)		lation of fluids
tional/viscous)
Graetz =	Gz	thermal capacity of	F cp/kD	10–104	Heat transfer; coating
Re Pr (D/D)		the fluid/conduc-	where F = mass		manifold
		tion heat transfer	flow
			D = length of heat
			transfer path
Grashof no.H	GrH	free convection	D3 g r2 b DT/m2		Heat transfer under
		buoyancy force/	where		gravity flow; natural
Ga (b DT)		viscous force	b DT = density dif-		convection
similar to Ar		= Re q (buoyancy	ference caused by
		forces/viscous	thermal difference.
		force)	b =thermal expan-
		= inertia-buoyance/	sion
		viscous force2
Grashof no.D	GrD		D3 g r Dr/m2		Mass transfer under
			where		gravity flow
			Dr =density differ-
			ence caused by con-
			centration differ-
			ence
Gr Pr					= 1000 heat transfer
					by conduction
					= 104 to 106 = heat
					transfer by natural
					convection

368 Appendix B: Dimensionless Groups

Dimensionless	Word definition	Equation	Range	So what?
number

Hatta no.	Ha	reaction in the film/	equation depends
		reaction in the bulk	on the kinetics
		for gas–liquid reac-	(–n1 k1 D1L )0.5/k1L
		tions
			or { (DA kmn (2/(m +
			1)[A]* (—1) [B]bn)0.5 }/
			kL
			or l/tanh l
			where
			l = D(kn [B]n–1/DA)1/2
			m order of the reac-
			tion of reactant A
			n order of the reac-
			tion of reactant B
			kmn = volumetric
			rate constant for re-
			action of orders m
			and n between re-
			agents A and B.
			D = diffusion path
			length = (m2/r2 g)1/3
			for packed column

For gas–liquid reactions: Ha I 1: regime 1: reaction occurs exclusively in bulk and is controlled by either chem. reaction, k (regime 1) overall temp effect positive or (regime 2) diffusion across the liquid film controls, kL slow reaction; overall temp effect negative regime Ha II 1 or kinetic regime; reaction is controlled by film diffusion. kinetics control

regime 3 fast reaction: is so fast that occurs wherever A is reaction entirely in film: diffusion and reaction in film with negligible reaction in the bulk Ha ii 1

and Ha II {[B]bulk/eB

[A]bulk } (DB/DA) 0.5;

mass transfer through liquid film (increasing the temp. decreases the overall rate unless increasing the temp. increases the solubility)

regime 4: instantaneous reaction in both film and bulk Ha ii

{[B]bulk/eB [A]bulk } (DB/DA) 0.5 effect of

temperature increase is negligible

			Appendix B:	Dimensionless Groups		369
Dimensionless		Word definition	Equation	Range	So what?
number
j-factor for heat	jH	St Pr2/3 = Nu/	(h/r cp Ivi)	0.001–	Heat transfer: for tur-
transfer		(Re Pr0.3)	[cp m/k]2/3	0.02	bulent 0.003 decreas-
					ing to 0.001
j-factor for mass	jD	StD Sc2/3	(kG/Ivi)[m/rD]2/3	0.002–	Mass transfer: for gas
transfer				0.007	or liquid 15 I Re I
			where kG [=] L/T		1500
					jH = 1.2 jD
Lewis no.	Le	Sc/Pr: mass trans-	k/cp r D	0.0001–	Mass and Heat trans-
		fer/heat transfer		0.11	fer: Material proper-
		properties			ties ratio of mass
		= thermal diffusiv-			transfer relative to
		ity/molecular diffu-			heat transfer
		sivity

Luikov no.	Lu	mass diffusivity/	kG Dr cp/k
		thermal diffusivity	where D = length
			kG [=] L/T
Mach no.	M	velocity/sonic	v/vsonic
		velocity	for ideal gas
			v/[(cp/cv) (RT/M)]0.5
Marangoni no.	Ma	surface tension gra-	thermal [dg/dT]
		dient forces/viscous	[dT/dy] d2/[m r cv/k]
		stablizing forces	d = film thickness; k
			= thermal conduc-
			tivity
			[dg/dT] DT d/
			[m r cv/k]

Mass and Heat transfer

Compressible flow, gas flow through a nozzle, aircraft speed

Coating surface defectsBenard cells; cells can occur for Ma i 80; roll cells for d I 1 mm, usually always occur because of Ma effects and not Rayleigh (density driven effects). For layers d I 2 mm, convection cells appear at Rayleigh and Marangoni values I the critical values listed above.

Mass transfer: indicative of when mass transfer might be increased by rolls cells

Margoulis no. see Stanton no.

Nusselt no.	Nu	total heat transfer/	h D/k	10–1000	Heat transfer
= St Re Pr		conductive or mo-
		lecular heat transfer

370 Appendix B: Dimensionless Groups

Dimensionless	Word definition	Equation	Range	So what?
number

Nusselt no.AB, NuAB, see Sherwood No.

Ohnesager no.	Z	viscous/(inertial q	m/(r D g)0.5	for G–L:	Spray, drop breakup
= We/Re		surface tension)0.5		10–3 to 10	water = 0.001 to 0.01;
					castor oil = 5
Peclet no. = Re Pr	PeH	bulk heat transport	D/F cp/kA;		Heat transfer: forced
for heat transfer in		by convection/	Ivi D r cp/k		convection
pipes		transport by con-
		duction – diffusion
Peclet no.	PeD	total momentum	Ivi D/Dax		Mass transfer, mixing.
= Re Sc		transfer/(molecular	where D = diam. of		For turbulent, homo-
for mass transfer		mass transfer axial	tube		geneous fluid
in pipe flow		direction)			Re i 104 Peaxial = 2;
					Peradial = 600; Pe = 0
					means complete mix-
					ing; Pe = T means
					plug flow
Peclet no.	PeD	(total momentum	Ivi Dp/Daxial	0.01–105	Mixing. For packed
= Rep Sc for mass		transport)/(molecu-			beds Re i 10; Pe axial =
transfer in packed		lar diffusion in axial			2–3; Pe radial = 10.
beds		direction)	Ivi Dp/Dradial		Flow through a cata-
					lyst bed of particles of
(Bodenstein no.,			or for packed beds		diameter, Dp, and
for axial)		(total momentum	D = Dp diam. of		depth H. Recom-
		transport)/(molecu-	particle		mended range of H/
		lar diffusion in			Dp is 5 to 50. If 50
		radial direction)			then plug flow.

For packed columns PeL

= 0.4–2; PeG = 4–20. If a column H I 0.2– 0.3 m, then backmix causes problems