2.5 PARTICULATE FOULING OF HEAT EXCHANGERS
2.5.1 MECHANISMS OF TRANSPORT TO THE SURFACE
Of the three steps mentioned above, theproblemof particle transport to surfaces is the most well understood, with both experimentaland theoretical work available in the literature. In a general situation, the mechanisms ofparticletransportmay include inertial impaction, convective diffusion, sedimentation,thermophoresis, diffusiophoresis, electrophoresis,and vapordiffusion/condensation. Although any or allof these processes may be present in an industrialheat exchanger, this work is focused on inertial impaction.
Asa particlein amoving fluid approaches an obstacle in the flow, its behavior will be influenced both by the nature of the flow field and by the inertial propertiesof the
particle. A useful parameter to characterize thisparticle-obstacleinteractionisthe Stokes number, which can be thought of asa ratio of the inertial forces on the particle tothe viscous forces itexperiences. Thus, for large values of the Stokesnumber,inertial forces willpredominate and impaction will occur, while forsmall values viscous forces will enable the particle tofollowthe fluid streamlines and movepast the obstruction.
Consideraparticle suspended in an air stream. As theairstream is diverted around an obstruction, thefluidstreamlines bend around the object. But iftheparticle has sufficient inertia, itwill be unable to followthe fluid streamlines exactly and insteadwillmove relativetothe fluid. The motion of the particle canbe described by aforce balance (38):
du (2.13)
where m is themassofthe particle, ⅛ is thevelocityofthefluid, u is thevelocityof the particle, t is time, and f is the drag on theparticle. Forthe simplest case, where the motionofthe particle and the fluid do not differ greatly, thedragmay be expressed by the Stokeslawdrag:
(2.14)
Hereμ is the dynamic viscosity ofthefluid and dpis thediameterofthe particle. If one substitutes theStokesformof thedragin Equation 2.13, assumes a sphericalparticle, and non-dimensionalizes with respecttoUoo,thefreestream fluid velocity,and L, a
characteristiclengthof the obstruction encountered bytheparticle (e.g.,the radius ofthe heatexchanger tube), theresulting expression is
(2.15)
where St is the Stokesnumber, θ=Q0 t/L is the non-dimensional time, and u’ and uf, are the respective velocities non-dimensionalized with Uθ0. The Stokes number is definedas
(2.16)
The density oftheparticle ispp, and thedensityof the fluidis p.
In most real cases, however, especially if the Stokes number is large,the relative speeds of the particle and fluiddifferenoughforthe Stokes law drag tobeaninadequate
description ofthe situation. In this situation the most common solution is to parameterize thedragontheparticle,f, in terms of a drag coefficient, CD:
^gC0(Re,,-faJ.Re,-^t3π^ (2,17)
It should be noticed that hereCdand f are functions ofthe local particle Reynolds number:
Re„ -local Jpl U~Uf∖
(2.18)
where v is thekinematic viscosity of the fluid. Since theparticle Reynolds number, Rep, is givenby
⅛U- v
it can be seen by combining (2.18) and (2.19) that
¼-tocβz = ¼∣m'-m∕∣∙
(2.19)
(2.20)
A typical correlationforCd, applicablefor Re<1000 is
Cd (Re) =~(1 + 0.158 Re273). (2.21)
If one substitutes Equation 2.17 into Equation 2.13 and non-dimensionalizes again, the resulting expression is
du' _ Cp (RCp -ιocaj) ∙ -local J_
dQ~~ 24 St (2.22)
This reduces correctly toEquation 2.15for Re^locjl<<l. However, forcases where the relative motion between the particle and the fluid is large and the Stokes law drag does notapply,
du'
~dQ
Cβ (Rep∣ m,-m∕ I)∙ Rep
24 u'-uf'∖(u'-uf'), (2.23)
and it can beseen thatparticleimpaction is determinedby the Stokes number, theparticle Reynolds number, and the nature of the flow field, uf', above.
Brun et al. (39)have calculated theoretical capture efficiencies for impaction of particles in an inviscid flow fieldarounda cylinder. Captureefficiency,t∣r, is the fraction ofthe particlesintheupstream cross-sectional area of theobstacle, which, in fact, impact on the collector. Thesecalculations result in afamilyof curves, Figure2.5, which present
βcτIon
e ff ic ie n c y ,
Figure 2.5. Collection efficiencyfor cylinders inan inviscid flow with point particles. The parameter, P, isdefined as Re 2∕St. The Stokes numberis based oncylinder radius as the characteristic dimension ofthe collec
tor. Adapted fromBrun et al. (39).
collection efficiencyas a function of the Stokes number andthe parameter P=Rep2∕St. In these calculations it has been assumedthatall particles that come in contact with the cylinderare captured.
Itwould be convenientto be ableto reduce thedependenceof ηκfromtwodimensionless parameters,Rep and St, to a single dimensionlessgroup. By acknowledging thatthe conventional definition of the Stokes number underestimatesparticledrag athigh Rep, Rosneret al. (40) have developed a generalized or effective Stokes number, Stcfp expressed by
sV∕=^4 f⅛)
3 kJN
rκe^ dRe'
'0 Cz>(Re')∙Re' (2.24)
If one designates the function Ψ(Rep) as:
Ψ(Rep) = 24 Re,
Γκe^ t∕Re,
Jo Cp(Re')∙Re', (2.25)
then
S⅛ = Ψ(Re,)St, (2.26)
and one can seethatΨ(Rep) represents themodification ofthe Stokes number to allow for non-Stokesian drag. Figure2.6 showsΨ(Rep) as a function of Rep. It can be seen that for Rep<<l, Ψ(Rep) approaches unity as required.
Rosneret al. haveusedthe effective Stokesnumbertoreplot a modified collection efficiency graph fromBran’s numerical simulations. It can be seen,Figure 2.7, thatthe familyofcurvesinFigure 2.5 has been collapsedonto a singlecollection efficiency curve. Rosner et al. suggest anempirical fit tothecurveinFigure 2.7 ofthe form:
η (St A = [1 + 1.25 st4 + (5.08 × 10^^5)
(1.4 × 10-2)∣ Stejj∙-
-3
Γ1.
ιγ2 8J
(2.27)
fs⅛-p
V 8 J
As before, allparticlesthat were predicted to impacton the cylinder were assumed to stick.