208
Ind. Eng. Chem. Fundam. 1985, 24, 208-214
from lignin with supercritical water. Acknowledgment This work was supported in part by the U.S.Department of Energy under Grant No. DEFG 2282PC50799 and in part by the Ngtional Science Foundation under Grant No. CPE 8204440. The valuable assistance of C. B. Baker and R. J. Phillips in the analysis of the reaction products is gratefully acknowledged. Registry No. Guaiacol, 90-05-1. Literature Cited Allan, G. G.; Matllla, T. I n “Lignins: Occurrence, Formation, Structure and Reactions", Sarkanen, K. V.; Ludwig, C. H.. Ed.; Wlley-Intersclence: New York. 1971. Ameatlca, L. A.; Wolf, E. E. Fuel 1984, 63, 227. Barton, P. Ind. Eng. Chem. RocessDes. Dev. 1983, 22. 589. Bbsslng, J. E.; Ross, D.S. Am. Chem. SOC. Symp. Ser. 1978, 77, 171. Bredenberg. J. Bson; Ceylan, R. Fuel 1983, 62, 342. Cavitt, S. B.; Sarraflzadeh, H. R.; Gardner, P. D. J. Org. Chem. 1962, 2 7 .
1211. Ceylan, R.; Bredenberg, J. Bson Fuel 1982, 61, 377. Connors. W. J.; Johanson, L. N.; Sarkanen. K. V.; Wlnslow, P. Holzforchung 1980, 34, 29. Eckert, C. Ann. Rev. Fhys. Chem. 1972, 23, 239. Fong, W. S.; Chan, P. C. F.; Plchaichanarong. P.; Cwcoran, W. H.; Lawson, D. D. "Experimental Obsecvatlons on a Systematic Approach to Supercritical Extraction of Coal”; I n “Chernlcal Englneerlng at Supercritical Fluid Conditbns”, Paulaltis, M. E.: Penninger, J. M. L.; (;ray. R. D., Jr.; Davldson, P.; Ed.; Ann Arbor Science: Ann Arbor, MI, 1983. Freudenberg, K.; Nelsh, A. C. ”Constitution and Biosynthesis of Llgnin”; Springer-Verlag: New York, 1968. Gardner, P. D.;Sarrafizadeh, H. R.; Brandon, R. L. J. Am. Chem. SOC. 1959. 87, 5515. G l a w , W. G.; Glasser, H. R. Macromolecules 1974, 7 , 17. Harkln, J. M. I n ”Oxidative Coupllng of Phenols”, Taylor, W. L.; Battersby, A. I.; Ed.; Marcel-Dekker: New York, 1967.
Iatridis, B.; Gavalas, G. R . Ind. Eng. Chem. Prod. Res. Dev. 1979, 18,
127.
Jegers, H. E. MChE Thesis, University of Delaware, 1982. Jezko, J.; Gray, D.; Kershaw, J. R. Fuel Process. Technol. 1882, 5 , 229. Klein, M. T.; Vlrk, P. S. MIT Energy Lab Report MIT-EL81-005 (1981). Klein, M. T.; Virk, P. S. Ind. fng. Chem. Fundam. 1983, 22, 35. Koll, P.; Metzger, J. Angew. Chem. Int. Ed. Engl. 1978, 77, No. 10. Kravchenko, M. I.; Kiprlanov, A. I.; Korotov, S. Ya. Nauch. Tr. Leningrad Lesoteckh. Akad. 1970, 735(2), 60. Maddocks, R. R.; Gibson, J.; Williams, D. F. Chem. Eng. Prog. June 1979,
49. Modell, M. “Reforming of Glucose and Wood at the Critical Conditions of Water”, presented at ASME Intersociety Conference on Environmental Systems, San Franclsco, CA, July 11-14, 1977. Olcay, A.; Tugrul, T.; CaHmli, A. “The Supercritical Gas Extraction of Llgnites and Wood”, I n "Chemical Engineering at Supercritical Fluid Conditions“, Paulaltls, M. E.; Pennlnger, J. M. L.; Gray, R. D., Jr.; Davidson, P.; Ed.; Ann Arbor Science: Ann Arbor, MI, 1983. Paulaitis, M. E.; McHugh, M. A.; Chal, C. P. “Solid Solubllities in Supercritical Fluids at Elevated Pressures”, I n Paulaitls, M. E.; Penninger, J. M. L.; Gray, R. D., Jr.; Davidson, P., Ed.; “Chemical Engineering at Supercritlcal Fluid Condltions”: Ann Arbor Science: Ann Arbor, MI, 1983; p 139. Ross, D. S.; Nguyen, Q. FluMPhase Equlllb. 1983, lO(2-3), 319. Scarrah, W. P. ”Liquefactlon of Lignite Using Low Cost Supercritical Solvents”, I n “Chemical Engineering at Supercrltical Fluid Conditions”; Paulattis, M. E.; Penninger, J. M. L.; Gray, R. D., Jr.; Davidson, P.; Ed.; Ann Arbor Science: Ann Arbor, MI, 1983. Shaposhinikov, Yu. K.; Kosyukova, L. V. Khim. Pereabotka Drev., Ref. Inform. 1965, No. 3, 6. Smith, R. D.; Wright, B. W.; Udseth, H. R . ACS Div. Fuel Chem., Prepr. 1983, 235. Squlres, T. G.; Aida, T.; Chen, Y.; Smith, B. F. ACS Div. Fuel Chem.. Prepr. 1983, 228. Vasihkos. N. P.; Dobbs, J. M.; Parlsi, A. S. ACS Div. Fuel Chem., P r e p . 1983, 212. Vuori, A.; Bredenberg, J. &son. Holzforschung 1984, No. 3 , 133. Whitehead, J. C.; Willlams, D. F. J. Inst. Fuel 1975, 48, 182.
Received for review May 23, 1984 Accepted November 5 , 1984
Thermophoretically Enhanced Mass Transport Rates to Solid and Transptration-Cooled Walls across Turbulent (Law-of- t he-Wall) Boundary Layers Siileyman A. Goko$ut and Danlel E. Rosner’ Hlgh Temperature Chemical Reactlon Engineerlng Laboratory, Yale UnlVersi@, Department of Chemical Engineering, New Haven. Connecticut 06520
Convectivedlffusion mass transfer rate predictions are made for both solid wall and transpiration-cooled “lawof-the-wall’’ nonkthermal turbulent boundary layers (TBLs), including the mechanism of thermophoresis, i.e., small particle mass transport “down a temperature gradient”. Our present calculations are confined to low mass-loading situations but span the entire particle size range from vapor molecules to particles near the onset of inertial (“eddy”) impaction. I t is shown that, when Sc >> 1, thermophoresis greatly increases particle deposkion rates to internally cooled solid wails, but only partially offsets the appreciable reduction in deposition rates associated with dust-free gas-transpiration-cooled surfaces. Thus, efficient particle sampling from hot dusty gases can be carried out using transpiration “shielded” probe surfaces.
1. Introduction Accurate engineering predictions of mass transport rates in turbulent, nonisothermal forced convection systems are necessary in many technologies (including the production *Professor and Chairman, Department of Chemical Engineering; Director of HTCRE Laboratory. To whom inquiries concerning this paper should be sent. NASA Lewis Research Center, M.S. 106-1, Cleveland, OH 44135.
of fumed chemicals from combustion or arc-jet sources, gaseous fuels from coal, high temperature gas cleaning) and in research (e.g.,the design of gas sampling systems). Until relatively recently many such calculations have been made assuming that Fick or Brownian diffusion (i.e. concentration diffusion) is the dominant nonconvective contributor to the instantaneous mass flux toward (or away from) the surface. However, a significant augmentation in small particle diffusional transport rates due to thermophoresis (drift of particles down a temperature gradient) in laminar boundary layers (LBLs) has been predicted, for both solid 0 1985 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
and transpiration-cooled surfaces in Goren (1977),Gokoglu and Rosner (1983) and Gokoglu and Rosner (1984a) and experimentally confirmed in Rosner and Kim (1985) for low mass loading, “dusty” combustion products. An Eulerian-continuum treatment of small particle thermophoresis in turbulent boundary layer flows, as well as a critical discussion of the current literature on the subject, is given in Rosner and Fernandez de la Mora (1982) for cooled nonporous walls, along with results confined to the important Sc >> 1 asymptote. Here we investigate the appreciable effects of thermophoresis on mass transport rates across TBLs for both solid and transpiration-cooled surfaces covering the suspended particle size range from vapor molecules (Sc = 00))up to Sc >> 1 particles, large enough to be near the onset of inertial impaction. Illustrative results are presented for the expected dependence of deposition rate on particle size (via Sc), mainstream and surface temperatures, Reynolds number (or Cf coefficient), and dimensionlesswall blowing (transpiration) rate, . u , / ( u , ~ C ~ / ~ ) ’ As / ~ . will be seen (sections 3 and 4), our interest in the effects of transpiration is motivated by its effectiveness in reducing fine particle deposition on surfaces necessarily cooled to temperatures lower than that of the dust-laden mainstream-a factor potentially important in the long-term operation of engineering equipment in such environments (e.g., utility turbines) or in aerosol sampling for downstream analysis in the field, or laboratory. 2. Mathematical Formulation 2.1 Boundary Layer Equations. For constant property turbulent boundary layers (TBLs) with a fully developed Couette-flow type approximation, and a mixing length law modified by blowing or suction, a generalized “law-of-the-wall”representation for the TBL structure can be obtained as follows (Black and Sarnecki, 1965; Stevenson, 1963; Simpson, 1967). Using nondimensional variables (with T, the time-averaged wall shear stress prevailing at the corresponding blowing velocity u,) defined as
Uf
= U/(T,/P)’/2
(2)
u+
= u/(rw/p)’/2
(3)
the time-averaged streamwise momentum balance, integrated once, gives the ordinary differential equation (Gokoglu, 1982)
e~p(-y+/A’)]~)’~~)/2(1 + uWfu’) (4)
12 is Prandtl’s mixing-length constant (usually taken to be about 0.40 by fitting available TBL velocity profile data) and A+ (the constant of the widely used Van Driest damping function) is given (Kays and Moffat, 1975) by A+ = 24.0/[auW++ 11
(5)
a = 7.1 if u,+ > 0; otherwise a = 9.0. The time-averaged streamwise velocity profile within a TBL, obtained by integrating eq 4, is shown in Figure 1 as a function of the blowing or suction parameter ~w/[ue(Cf/2)01= uw+/[~e+(Cf/2)01 (6) Figure 1 may therefore be regarded as a generalization of the familiar “univeral velocity profile” for fully developed E
Bmom
,
-3.3 0. 0
0.3 1. 0 1.5
40
. 15
loo 0
20
10
Ut 5
30
“llr,lpll’2
Figure 1. Universal time-averaged streamwise velocity profile in a turbulent boundary layer in the vicinity of a smooth wall with blowing or suction.
turbulent flows near smooth walls. Consistent with eq 4, we neglect the streamwise variation of the dependent variables in the energy and particle “phase” mass conservation equations, as in the early examples of Prandtl(l928) and Deissler (1955). However, the mass conservation equation now must include the thermophoretic flux term, and, upon time-averaging, the simultaneous fluctuation of the thermophoretic speed and particle concentration fields produces a new “correlation” term. But this latter term, which may be called “eddy thermophoresis”, can be neglected based on arguments given in Rosner and Fernandez de la Mora (1982) and Fernandez de la Mora (1980). Moreover, since the suspended particles are assumed to be present with a negligible mass fraction (“loading”),we neglect the reciprocal diffusion thermo-effect (Dufour) term in the energy equation. Therefore, using the previously defined nondimensional variables, the time-averaged energy and particle mass conservation equations can be written as the second-order ordinary differential equations
for energy and P
-dY+ - - (1 + {l+ 4(1 + ~,+u+)k2y+2[1 du+
209
for suspended particle mass. Here, aT is the thermophoretic diffusion factor (dimensionless), defined such that ( a ~ D ) p ~-(u)[-(grad l ?“)/!TI is the local therpophoretic particle mass flux relative to the mass-averaged velocity of the (pseudo-) binary mixture (see, e.g., Rosner (1980) and Talbot (1981)). (In the notation of Bird et al. (1960), aT is kT/[x(l- x)], where kT is the thermal diffusion ratio and x is the mole fraction of the transported species.) If, for simplicity, we assume that the turbulent Prandtl and Schmidt numbers are both unity (i.e., all turbulent momentum, energy, and mass diffusivities are locally equal) then, in eq 7 and 8 we can write
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Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
StqO = (ue+@e,o+)-l At this point, for convenience, we adopt u+ (rather than y+) as our new independent variable (see eq 4,9, and 10). After integrating eq 7 and 8 once, and introducing the following nondimensional dependent variables
(20)
In the case of transpiration cooling (u,+ > 0) we assume that clean (“dust-free”)gas (with Pr = 0.7, in this study) from a “reservoir” at temperature Tois blown through the (porous) wall at a (“superficial”) velocity u,. As discussed in Gokoglu and Rosner (1984a), the surface temperature T , is then coupled to u, and is obtained from an energy balance struck for a control volume of unit projected area spanning the wall and reservoir, i.e. ~ u w ( h w- h ~=) P U e S t h ( h e - hw)
Assuming constant fluid thermophysical properties, eq 21, in nondimensional variables, reduces to
.,+[I);()$(
we obtain the “first integrals” d U+ -r‘
-dT+ du+
-
-(1 du+
(21)
+ u,+T+)
1-
=
;
ue+sth( - 1)
(22)
Therefore, for any reservoir-to-mainstream temperature ratio (TOIT,)and Reynolds number, T, as a function of u,+ is determined via eq 22. For determining the upper limit of integration, ue+,in the case of nonzero transpiration, we use a more general criterion than in the solid wall case. This criterion, applicable to either transpiration or solid walls (Stevenson, 1968) is similar to a suggestion by Rubesin (1954) and is based on the observation that
for energy and -dw+ -du+
u,+ye+ = u y ef v
for suspended particle mass. Note that, in the absence of thermophoresis, eq 14 reduces to ri W +
where wo+(u+, ...I denotes the correspondingvalue of w+ (eq 12) for the same fluid dynamic and heat transfer conditions, but with thermophoresis “turned off”. 2.2 Boundary Conditions. These TBL equations (4, 13-15) are solved subject to the following boundary conditions: at the wall: u+ = 0, u+ = u,+ (= 0 for “solid” (nonporous)wall) and Te = 0, w+ = 0. The latter condition is equivalent to the presumption that the particle sticking probability is large enough for the steady-state concentration w, to satisfy the inequality w, > 1, and the “suction” effect of thermophoresis therefore increases linearly with Te/Tw.However, the departure from linearity of the thermophoretic deposition rate enhancement is explained by the thermophoretic “pseudo-sink” effect, which also increases as T e / T , departs from unity (as discussed more fully in Goren (19771, Gokoglu and Rosner (1980), and Rosner (1980), and examined experimentally for laminar flow in Rosner and Kim (1985). For a typical particle/gas combination (say, particles in air with a density level low enough such that q L e 0.4) at Re = 5.8 X lo5, the augmentation in deposition rate due to thermophoresis is shown in Figure 3 as a function of temperature ratio T,/T,, covering all particle sizes from vapor molecules (Sc = O(1)) up to the possible onset of inertial effects (Sc = O(105);see, e.g., Rosner and Fernandez de la Mora (1982). [For aerosol particles of diameter small compared to the prevailing host gas meanfree-path, 1, the magnitude of the particle Schmidt number, Sc v / D p , is insensitive to temperature and increases roughly as d,2. Thus, the (20 “C, 1 atm air, 1 g/cm3 particle) values tabulated in Friedlander (1977, Table 2.1) are indicative: Sc(d, = pm) N 2.9, Sc(d, = pm)
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985 211 200 lo3
120
101
0
r
c
E’ E
c
Sc
5
WID P
Figure 4. Enhancement of deposition rate due to thermophoresis as a function of Schmidt number (particle size) in a turbulent boundary layer for different ratios of particle thermophoretic diffusivity to host gas heat diffusivity. ’03
1
1.0
I
I
I
1.5
2.0
2.5
r
I 3.0
Figure 2. TJT, dependence of thermophoretic mass transfer augmentation factor for various particle sizes (via Sc); TBL flow past an internally cooled solid wall; constant thermophysical properties.
lo3
E
/
aTLe = 0.40
R e = 5 . 8 ~ld
0.40 Sc
1
ulDp
Figure 5. Thermophoretic enhancement of deposition rate as a
ld
function of Schmidt number (particle size) and Reynolds number for a law-of-the-wallturbulent boundary layer. 0
+E‘
eE z
IO1
100 100
101
ld
ld
sc Figure 3. Schmidt number dependence of thermophoretic mass transfer augmentation factor for various cold, solid wall TBL conditions. N 2.9 X lo2,Sc(d, = lo-’ pm) N 2.2 X lo4 (NB. l(20 “C, 1atm) N 0.65 X lo-’ pm). For values pertaining to higher temperatures and pressures, and other materials, see, e.g., Rosner and Fernandez de la Mora (1982), and the references cited therein. Typical computed “eigenvalues” are given in Table I. It is remarkable that even without active cooling, if T , J T ,
N 0.98 (say, due to modest heat losses by radiation, or conduction along the wall) the local deposition rate of particles with Sc N lo6 is still increased by more than an order of magnitude! In actual “dusty gas” situations the value of the ratio, cqLe, of the particle thermophoretic diffusivity to the heat diffusivity of the carrier gas is of order unity, and usually near 0.4 for particles small compared to the prevailing gas mean-free-path. Figure 4 shows the effect of this diffusivity ratio parameter on the deposition rate augmentation for different particle Schmidt numbers (sizes), for a typical case (T,/T, = 0.8 and Re = 5.8 X lo5). aTLe values of the order of lo-’ (or less) are of interest in situations for which the particle diameter exceeds the prevailing gas meanfree-path; see, e.g., Rosner and Fernandez de la Mora (1982). As the Schmidt number (particle size) increases, the mass transfer (Brownian diffusion) BL becomes thinner, and, ultimately becomes completely embedded in the viscous sublayer (Sc 5 lo2;see, e.g., Rosner and Fernandez de la Mora, 1982) of the TBL. The effect of changing the Reynolds number on the temperature and particle concentration profiles is, therefore, expected to be different. Since a change in the temperature profile directly affects the thermophoretic mass flux, we demonstrate in Figure 5 how increases in Re result in reductions in the effect of thermophoresis on particle wall arrival rates for a typical
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Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
Table I. Eigenvalues for Solid Walls sc 100 10' 102 103 104 106
Stm 1.600 X 5.649 X 1.315 X 2.523 X 4.583 X lo4 8.211 X
St,SCZI3 1.600 X lom3 2.622 X 2.834 X 2.523 X low3 2.127 X 1.769 X low3
St, 1.801 X
CfP 1.600 X
25
100 10' 102 103 104 105
1.614 X 6.132 X 2.191 X 1.611 X 1.608 X 1.608 X
1.614 X 2.846 X 4.719 X 1.611 X loe2 7.463 X 3.464 X lo-'
1.801 X lo3
1.600 x 10-3
5.8 X lo6
25
100 10' 102 103 104 105
1.633 X 6.873 X 3.790 X 3.646 X 3.643 X 3.643 X
1.633 X 3.190 X 8.166 X 3.646 X 1.691 X lo-' 7.848 X lo-'
1.801 X
1.600 X low3
0.1
5.8 X lo6
25
100 10' 102 103 104 105
1.603 X 5.769 X 1.514 X 5.219 X 4.313 X 4.312 X
1.603 X 2.678 X 3.262 X 5.219 X 2.002 x 10-2 9.289 X
1.801 X
1.600 X
0.8
1.0
5.8
lo6
25
100 10' 102 103 104 106
1.634 X 6.866 X lo4 3.730 X 3.580 X 3.579 x 10-4 3.578 X
1.634 X 3.187 X 8.037 X 3.580 X 1.661 X lo-' 7.709 X lo-'
1.801 X 10"
1.600 X
Loa
0.4
9.6 x 103
16
100 10' 102 103 104 105
3.906 X 1.012 x 10-3 2.118 X 3.965 X 7.169 X lo4 1.283 X lo4
3.906 X 4.699 X 4.564 X 10" 3.965 X loT3 3.327 X 2.765 X 10"
4.722
X
3.906 X
0.8
0.4
9.6 x 103
16
100 10' 102 103 104 105
3.987 X 1.177 X 4.732 X 4.155 X lo-' 4.151 X 4.150 X
3.987 X 5.464 X 1.020 x 4.155 X 1.927 X 8.941 X
4.722
X
3.906 X
100 10' 102 103 104 105
8.651 X 3.685 X 9.394 x 1.845 X 3.367 X 6.036 X
9.425 X lo4
lo4 lo-'
8.651 X lo4 1.711 X 2.024 X 1.845 X 1.563 X 1.301 X
100 10'
8.690 X 3.885 X 1.356 X 8.528 x 10-5 8.465 X 8.464 X
8.690 X 1.803 X 2.921 X 8.528 X 3.929 X 1.824 X lo-'
9.425 X
Tw/TO
Re
ffTLe
u0+
1.00
0.4
5.8
lo6
25
0.8
0.4
5.8 x 105
0.6
0.4
0.8
1.0"
0.8
0.4
0.4
X
X
3.1 x 107
3.1 x 107
34
34
lo2 103 104 105
lo4
10-5
10" 10-2
lo-' lo-'
lo4
8.651
X
8.651 X
Pertains to conditions without thermophoresis (e.g., Stm,J
case (TWITe= 0.8 and aTLe = 0.4). As can be seen from Figures 3,4, and 5, the thermophoretic enhancement for very small particles and vapor molecules (Sc < 10) is not as great as for larger particles (with Sc >> 1). Figure 6 depicts the effect of transpiration cooling and thermophoresis on the concentration profile for two different particle sizes (corresponding to Sc = 25 and 1000). As expected, the BL thickness for the larger particle is thinner than that of the smaller particle. Therefore, the same amount of wall blowing affects the larger particle concentration profile to a greater extent (note the shift between the solid and the dotted curve). However, the surface is cooler due to blowing (TWITe= 0.96) and the
effect of thermophoresis on the concentration profile of the larger particles is also greater (cf. the shift between the dashed and the dotted curve). Note that the corresponding temperature profile is practically unaffected by the same blowing rate, and that the heat transfer BL is much thicker than either of the mass transfer BLs shown. In Figure 7 we demonstrate the reduction in the deposition rate due to transpiration cooling ("blowing") for different particle Schmidt numbers (sizes). As discussed above, the larger particles are more easily "blown away". However, as in the corresponding laminar boundary layer case (Gokoglu and Rosner, 1984a), thermophoresis noticeably offsets the advantageous reduction in deposition
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
---
;v o(T,ITe * 11 v i =0.003WITH THERMOPHORESIS UwKe* 0.%I V$ =0.003WITHOLilTHERMOPHORESlS
}
.7
.6 +al
E3
.5
.1
3 4 5 Y+ Figure 6. Effect of transpiration on profiles of particle concentration and temperature within the viscous sublayer; Re = 5.8 X lo5,
0
1
2
aTLe = 0.4,To/Te= 0.4.
v;.v,.(
“2.
c,12)-1’2
Figure 7. Deposition rate reduction due to transpiration cooling for a turbulent boundary layer. Effect of thermophoresis in offsetting the fouling rate advantages of transpiration as a function of blowing rate and Schmidt number (particle size).
rate that accompanies transpiration, especially for larger particles and for higher blowing rates (cooler surfaces). The transpiration cooling calculations shown in Figures 6 and 7 were done for aTLe = 0.4 and Re = 5.8 X lo5,with the ratio, To/T,, of the coolant gas reservoir temperature to the mainstream temperature taken to be 0.4. 4. Concluding Remarks Numerical calculations performed for low particle mass-loaded “law-of-the-wall”turbulent boundary layers show that thermophoresis dramatically increases the deposition rate of small particles to cooled, smooth walls. (Very small particles are not capable of inertially impacting and therefore, treatable like “heavy molecules”). The
213
deposition rate enhancement depends on such dimensionless parameters as the relevant Schmidt number, (which, in turn, depends on particle size), the particle thermophoretic diffusivity-to-carrier gas heat diffusivity ratio (aTLe),the surface-bmainstream temperature ratio (TWITe), and the Crequivalent Reynolds number (Re). It is shown that transpiration cooling especially reduces the deposition rate of larger particles; however, due to the thermophoresis the reduction is also offset more for larger particles and higher blowing rates (cooler surfaces). Rational yet simple engineering correlations have recently been developed (Gijkoglu and Rosner, 1984b Rosner et al., 1984) to make engineering predictions of the enhancement of deposition rates due to thermophoresis in the presence of transpiration cooling. Their accuracy has been checked against the numerical “self similar” variable-property laminar boundary layer calculations of Goren (1977) and Gokoglu and Rosner (1983) and the constant (thermophysical) property “law-of-the-wall” TBL calculations presented here and in Rosner and Fernandez de la Mora (1982). In all TBL cases the average absolute error of the correlation was only 2% over the parameter range: lo4 IRe I3 X lo’, 0 IaTLe I 1.5, 0.5 ITWIT,I 1, B I5. The success of these correlations, even including developing BL’s (Gokoglu and Rosner, 19844, and allowing the local transition to turbulence, suggests that in future engineering design/optimization calculations the need for expensive and time-consumingcomputer TBL “structure” simulations can be sharply reduced, without incurring an appreciable loss in the accuracy of predicting simultaneous momentum, energy and mass transfer rates across the gas/“solid” interface. It is interesting to note that, in view of our explicit restriction to the low particle mass loading asymptotic limit (here, in Gokoglu and Rosner, 1983, and in Rosner, 1980) the recent criticism of Shaeiwitz (1984) regarding the necessity to consider cross-coupling and associated temperature field modifications is fundamentally incorrect. Moreover, in turbulent boundary layer situations of engineering importance, such as that quantitatively treated here and in Rosner and Fernandez de la Mora (1982), Shaeiwitz’s “constant diffusivity” matrix formulation of thermophoretically augmented convective mass transfer is practically useless. Shaeiwitz’s formulation need not be discarded totally, however, since it may, in some cases, prove qualitatively useful. Unfortunately, this does not even include high mass loading laminar boundary layer situations, since Shaeiwitz’s neglect of momentum coupling (via deposition-induced wall ”suction”) is then unacceptable (Rosner et al., 1985. In closing, we note that El-Shobokshy (1981) has recently proposed and experimentally demonstrated the reduction in deposition rate for dusty gases in turbulent flow through ducts whose impermeable walls are heated above the bulk fluid temperature. Unfortunately, this technique is impractical for sampling the high temperature systems of greatest interest in our laboratory. Therefore, we have emphasized here “cold-wall” cases for which only transpiration (blowing of dust-free gas through the wall) is able to reduce the deposition rate to acceptable levels, even in the presence of “thermophoretic suction” (Gokoglu and Rosner, 1984a; Rosner, 1980). This transpiration technique has, in fact, been used successfully in recent experimental studies of inorganic particle formation during pulverized coal combustion (Quann et al., 1982). Acknowledgment In developing the mathematical models and correlations described here we have benefitted from many helpful
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Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
discussions with Drs. J. Fernandez de la Mora and R. Israel, as well as the numerical calculations of A. Zydney and S. Vaziri (Yale Engineering Summer Research Program). Nomenclature a = constant in eq 5 A+ = effective laminar sublayer thickness; eq 5 B = blowing (suction) parameter; eq 6 c p = specific heat at constant pressure (per unit mass of
mixture) Cf = skin friction coefficient, rW/((1/2)pu,2) D = Brownian (Fick) diffusion coefficient ("concentration" diffusion) j " = diffusion mass flux k = Prandtl mixing length constant Le = Lewis number (ratio of particle Brownian diffusivity to carrier gas thermal diffusivity) 1 = carrier gas mean-free-path Pr = Prandtl number (ratio of carrier gas momentum diffusivity to heat diffusivity) 4" = heat flux Re = Crequivalent smooth wall pipe flow Reynolds number Sc = Schmidt number (ratio of gas momentum diffusivity to particle Brownian diffusivity) S t , = Stanton number for heat transfer S t , = Stanton number for mass transfer T = time-averaged absolute temperature u = time-averaged fluid velocity in x direction (parallel to wall) u = time-averaged fluid velocity in y direction (normal to wall) u, = time-averaged wall blowing velocity y = distance normal to surface Greek Letters cyT
= thermal diffusion factor of particle
momentum diffusivity (kinematic viscosity) of gas density of mixture T = time-averaged shear stress w = time-averaged mass fraction of particles in gas mixture u =
p =
Subscripts
e = outer edge of the boundary layer ("local mainstream") eff = effective h = pertaining to heat transfer m = pertaining to mass transfer mom = pertaining to momentum transfer p = pertaining to particle w = at surface, pertaining to "wall" condition 0 = pertaining to reservoir conditions for (transpiration) coolant gas, or to conditions without thermophoresis, or without blowing (suction) Superscripts
+ = pertaining to "law-of-the-wall" quantities
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Received for review August 1, 1983 Revised manuscript received May 10, 1984 Accepted September 17, 1984
This research has been supported, in part, by the U.S. Air Force Office of Scientific Research (Contract F49620-82K-0020)and NASA-Lewis Research Center (Grant NAG3-201).
