Ind. Eng. Chem. Rmess Des. Dev. lQ83, 22, 203-208
203
Analysis of Thermal Regeneration of Fibrous Diesel-Particulate Filters Farhsng Shadman' Depertment of Chemical Engineering, Unlvmlty of Arizona, Tucson, Arizona 85721
Edward J. Blsbett Mathematics Depertment, Ganeral Motors Research Laboretorbs, Warren, Michmn 48090
Fibrous filters are currently being considered for the removal of exhaust particulates from the exhaust of diesel-powered vehicles. One approach under development is to regenerate these filters on-board by IntermHtently elevating the exhaust temperature and ignlting the particulate deposits. An analytical model which simulates this regeneration process has been developed. The model predicts that the process consists of three stages. During the first stage, regeneration is slow while a temperature peak develops inside the filter. The second stage starts with the onset of a raw combustion around the temperatwe peak. Dwhg tMs stage,two reaction fronts are formed inside the filter and move toward the opposite ends of the fitter. The downstream peak grows with time and is responsible for the maximum solid temperature (hot spot) in the filter. The final stage is a slow combustion and starts after the downstream section is regenerated. A nonuniform distributlon of fiber size along the filter is suggested as an effective technique for lowering the hot spot temperature.
Introduction Filtration of exhaust gases is currently being considered as a technique for reducing the particulate emission from diesel-powered vehicles. Low-density fibrous filters consisting of metallic or ceramic fibers are effective for this purpose. A possible technique for disposing of the particulates collected in the filter is to periodically alter the exhaust properties (for example, temperature, flow rate, and oxygen concentration) into the filter in such a way as to induce incineration of the collected particulates. The increase in feed temperature can be achieved by various techniques and is a possible way of incinerating the particulate deposit on the fibers. The present study will focus on the characteristics of such a regeneration process. Experiments have shown that the burn-off of the particulate deposit is nonuniform and often produces hot spots in the filter which melt or damage the fibers. In practice, a controlled and predictable combustion is required, considering that these filters should last through a large n u m b r of regeneration cycles. An analytical model would be helpful in understanding the details of this complex process and in improving the filters and the regeneration process. The incineration of particulate deposit in fibrous filters is similar to the regeneration of coked fixed beds in many ways. There is a wealth of information available on the latter subject. However, the existing models for coked fured beds cannot be applied to the regeneration of fibrous filters. This is because, in addition to the differences in the nature of the beds, the existing models are based on one or more of the following assumptions which do not hold for the regeneration of fibrous fiikrs. (1) The reaction rate is constant (Van Deemter, 1953,1954) or the rate is fully mass-transfer controlled (Olson et al., 1968; Johnson et al., 1962). In the case considered here, the significant variations in temperature and oxygen concentration suggest that a more general rate expression should be used. (2) The effect of depletion of carbon on the reaction rate is negligible or the carbon depletion is very slow compared to other temporal changes (Johnson et al., 1962). This 0196-4305/83/ 1122-0203$01.50/0
simplification is not valid in the present case because the carbon depletion rate can be fast. (3) The reactor is adiabatic (Van Deemter, 1953,1954, Gonzalez and Spencer, 1963; and Olson et al., 1968). (4) The variations in the gas density and velocity are negligible (Gonzalez and Spencer, 1963; Van Deemter 1953,1954; Johnson et al., 1962; Hano et al., 1976). This assumption is not valid in the present case because of large and rapid variations in temperature. (5) Solid and gas temperatures are equal (Van Deemter, 1953,1954; Sampath and Hughes, 1973; Hano et al., 1976; Johnson et al., 1962; Schulman, 1963; Ozawa, 1969; Zhorov et al., 1967). Our preliminary study shows that the difference between these two temperatures can be significant in the system considered here. (6) The initial physical properties of the bed are constant along its length. This assumption is not valid for the system studied here. Considering the above factors, a model will be presented here which takes into account the unique features of fibrous filters and their regeneration. Analytical Model A schematic diagram of the system is shown in Figure 1. The casing is assumed to be cylindrical or, in general, equivalent to a cylinder with diameter D. Due to the large convective flow rate along the filter, the axial dispersion in the gas phase is neglected. Moreover, heat conduction in the solid phase is neglected because of the very small solid packing density (normally less than 10%)and poor contact between the fibers. Radial changes in temperature and concentration are neglected and average values over a cross section are used to model axial variations. The mechanism of reactions involved in the burning of diesel particulates is complex and not well understood. In this study, the overall reaction is assumed to be represented by the heterogeneous noncatalytic oxidation of carbon to C02. The reaction at any point consits of two steps in series: (1) transport of oxygen from the bulk gas to the solid phase, and (2) reaction on the surface. The expression for the overall rate per unit area of fiber is R = kC (1) 0 1983 American Chemical Society
204
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983 Filter Wall
where (k)-l
+ (k,y)-l
= I-),@
(2)
k, is the mass transfer coefficient and k, is the intrinsic rate coefficientfor carbon oxidation per unit surface area. The parameter y represents the fraction of the fiber surface covered by particulates. The method of evaluation of these parameters will be given later. The overall mass conservation equation is
a
-
Fibers Coated with Paniculate Deposit
/
Exhaust
Gas
Figure 1. Schematic diagram of a fibrous particulate filter.
-(tpu)
ax
\
=0
(3)
t
where v is the interstitial gas velocity. The conservation equation for oxygen, the key gaseous species, is
a
-(&)
ax
= -sR
(4)
The energy balance for the gas phase is
tpuc
aT, ax
4
- sMT, - Tg)+ 5hW(Tw- T g )
where the last two terms account for the heat transferred from gas to solid and from gas to the walls, respectively. In the above equations, the time derivativesrepresenting accumulation of heat and mass in the gas phase and the second spatial derivatives representing diffusion have been neglected. These terms are small compared to convective terms. The energy balance for the solid phase is
sh(Tg- T,) + sR(-AW
4 + -a(TW4 - T:) D
-
(5)
(6)
+ x 0
1
Figure 2. Schematic diagram of a typical solution mesh.
pendence on p in favor of the gas temperature, T The derivative in eq 4 is expanded, and appropriate sutstitution of eq 5 is made into eq 4. This results in the following set of differential equations.
aT, G(t)C - = s h(TJ (T, -T,) p g ax gax
4 + Dh,(T,
- TJ
(12)
cp, s h(TJ ( T ,- T g )+
where the last term accounh for the radiative heat loss to the walls. In calculating the radiative heat exchange, it is assumed that all surfaces are black bodies. Assuming that the filter wall is thin and axial conduction of heat is negligible, the energy conservation for the wall is given by
aTW
w ~ w c p w a= t
hw(Tg - Tw) + a(T,4- Tw4)+ h,(Ta - Tw) (7)
The mass conservation equation for the solid deposit is aq
- = -M$
at The relationship between the density and the temperature of the gas is given by the ideal gas law PTg= POT0
(9)
The initial conditions at t = 0 are
T,
TO;
= qo; T , = To
(10)
The boundary conditions at x = 0 and t > 0 are
T g = T,(t); CT, = Y(t);
t p = ~
G(t)
(11)
Method of Solution The mathematical solution of the above model equations is briefly described in this section. Equation 3 is integrated, subject to its boundary condition from eq 11, to give the velocity, u, as a function of the density, p. Then the equation of state, eq 9, is used to eliminate all de-
Note that the system is hyperbolic, with characteristics x = constant and t = constant. The initial and boundary
conditions are straightforward. Equations 14-16 are structurally unchanged from their previous form. The precise form of the functions h(TJ and R(T,,C,q,T,) will be clear after discussion of heat and mass transfer coefficients in the model parameters section. This reduced system is nondimensionalized and then solved numerically. During the computation, a succession of discrete points in time, fj, j = 0, 1, 2, ... is determined at which an approximate sol_Uicn is calcula-ted. If y is any of the dependent variables, T,, C, T,, Q, or T,, then at each such point of time an approximate solution is calculated at each spatial mesh point, Z i ( f j ) , i = 0 , 1, 2, .... Note that the spatial mesh may change with time (Figure 2).
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983 205
To allow for the possibility of discontinuous boundary or initial conditions or spatially discontinuous physical properties of the filter, a low-order, compact, finite difference approximation was chosen for the derivatives in eq 12-16. The equations were integrated as ordinary differential equations along the characteristics between mesh points. A forward Euler step was made in each characteristic direction to obtain a predictor at each mesh point. The trapezoidal rule was then applied as a corrector. The differences between the predictors and correctors, e, and e,, for the spatial and temporal integrations, respectively, are saved for all mesh points at a given time level to help adjust the mesh. The norm of the values of e, for all meah points at each time level was used to predict the largest value (adjusted down somewhat for safety) for the next time step to maintain a prespecified error bound. A prespecified error bound is also supplied for the spatial integrations. If e, is too far below this bound, then a point in the mesh is discarded. If e, exceeds this bound, a mesh point is inserted between the two currently being considered, and solution values at this new mesh position at the previous time level are obtained by interpolation so that integration at the current time level can proceed with the new mesh point. Effedive use of this approach to dynamic mesh refining and coarsening requires careful choices of tolerances, safety factors, tests for exceptional conditions, and so forth. However, the method in turn provides a very efficient way to maintain numerical accuracy when the solution possesses regions of rapid change that move with time through the spatial mesh. For a typical calculation, the time steps vary between an imposed maximum of 2 and a minimum of approximately 0.007. Also, the mesh spacing in Z varied from an imposed maximum value of l/m to a minimum of 1/610,but no more than 40 additional spatial mesh points were ever needed beyond the original 80. Model Parameters The following typical values were used for the model parameters. (a) Filter Characteristics. L = 17.8 cm; D = 13.4 cm; w = 0.5 cm; C,, = 0.11 cal g-l K-l; C,, = 0.11 cal g-’ K-l; t = 0.93; p, = 7.1 g cmm3; pw = 7.1 g ~ m - s~=; 4(1 - t)/df, assuming cylindrical fibers; df = 0.0188 cm, uniform case; df = 0.012 + 0.031 ( x / L )cm, nonuniform case. (b) Deposit Characteristics. M,= 12 g atom-l; y = q / q * (q < q*); y = 1 (q > q*), where q* is a monolayer coverage of surface packed with particles and is estimated to be 4 X lo* g cm-2 based on a typical value of 0.1 pm for the diameter of individual particles, and a bulk deposit density of 0.4 g ~ m - ~In. the present study, qo = 3.48 X 10-~ g cmS2. (c) Inlet Gas Properties. Before regeneration, the gas and solid phases are in thermal equilibrium. The regeneration considered here is triggered by a step change in the inlet gas temperature from Toto Tf.After this step change, the boundary _conditionsare held constant: Tf(t) = 900 K, Y(t)= 0.15/R g-mol K ~ m - G ~ (;t ) = 0.143 g cm-2
:
S-1.
(a) Local Transport Coefficients. The correlations on heat and mass transfer coefficients around the fibers given by Satterfield and Cortez (1970) were used to estimate k, and h. h = 1 . 1 4 9 p ~ C , ( d ~ p u / p ) ~cal ~ ~cm-2 l ~ K-’ s-l k, = 1 . 1 4 9 ~ ( d ~ p u / pcm ) ~ s-l .~~~ where, from Bird et al. (1960) p = 1.28 X (Tg)1/2g cm-l
5-l
. .,.-...._....
-1
....................................................................................
- - - - - - - _ _------___ _
-
4
0.0
0.2
0.4
0.6
0.8
1.0
Distance Figure 3. Profiles in a uniform filter a t 37 a.
-
:
........
L-
208
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983
:
0-
- Ts --E
........
.......
-------__ ----__
I
a
01
0.0
0.2
0.4
0.8
1.0
0.s
Distance Figure 5. Profiles in a uniform filter at 38.9 s.
I
I
0.0
02
0.4
0,s
0.8
1.0
Distance Figure 8. Profiles in a nonuniform filter a t 41.7 s. 1
1 - .T;
-1 - ---I--
................................................. --.*.a.
-
-..7
0 ;
I
0.0
0.2
0.4
0.6
0.8
1.0
Distance Figure 6. Profiles in a nonuniform filter a t 10 8.
0.0
02
- - -,L.....- - i '
I
1 .
0.4
0.6
0.s
1.0
Distance Figure 9. Profiles in a nonuniform filter a t 42.4 s.
I
I
-i
..............................................................................
.........
..........
I
--------- -----___--
a ! 00
02
.'..
I
04
06
0.8
-1....................... - - - _-.-............ .-.-,-
1
........-
4
- - - - - - i.
!I
1.0
Distance Figure 7. Profiles in a nonuniform filter a t 40 E.
in a rapid sweeping motion. The predicted maximum temperature after about 39 s goes well above 1500 K which is the maximum operating limit for the fibers. The simulation is terminated when T,exceeds approximately 1500
-F s 0-
-- c
A
K. Although the system has a large number of operating parameters which can be varied within a practical range, lowering of the peak temperature was found to be very difficult. In general, operating conditions which can give lower maximum temperatures require a longer regeneration time and/or less initial particulate coverage (more frequent regeneration); both effects are undesirable. The results of a parametric study on this subject will be published in a separate report. A poasible scheme which allows lower peak temperatures without appreciable penalty in performance involves using a filter in which the fiber surface area decreases toward the filter outlet. The fiber surface area can be decreased by increasing the porosity or increasing the fiber size. As
-I--
...............2".*.U.-.-,.-
I
a / 0.0
-.....- - - - - - - - - ........ ....
i I
I
0.2
0.4
0.6
0.8
1.0
Distance Figure 11. Profiles in a nonuniform filter a t 48 s.
an example, the regeneration results for a filter with variable fiber size are shown in Figures 6 to 11. All other conditions including the initial coverage, qo, are kept the
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983
same as before. The fiber size is assumed to increase linearly through the depth of the filter, with the value of df from the uniform case equal to the harmonic mean of the fiber sizes a t the two ends of the filter in the nonuniform case. Since qo is unchanged, note that the nonuniform case possesses approximately 30% less total mass of carbon initially, and the initial carbon concentration is also nonuniform on a grams per unit volume basis because of the effect of df on the specific area, s. In practice a continuous variation cannot be obtained but can be approximated by stacking layers of different fiber sizes. With the above scheme, the temperatures can be kept within a reasonable limit. The profiles shown in Figures 9 and 10 indicate that shortly after the onset of rapid regeneration, two reaction fronts are formed on two sides of a completely regenerated zone. The downstream front moves rapidly in the direction of flow, and its temperature grows with time. The upstream front moves slowly opposite to the flow direction, and its temperature decays with time. Because of the large change in heat flux across the combustion fronts, the neglect of heat transport by conduction in the model equations requires some discussion. First, heat conduction in the gas phase is estimated a posteriori to be more than four orders of magnitude smaller than typical convective fluxes. The gas temperature profiles, while not shown here, display behavior similar to the solid temperature, although the gas temperature peaks are substantially smoother and shifted slightly downstream. The difference between T,and Tgis maximum around the reaction fronts and reaches as much as 70 K. In principle, heat conduction in the solid phase cannot be neglected, since the profiles generated here without conduction show an infinite gradient of heat flux across the reaction fronts. At all other positions in the filter, conduction can be shown to play an insignificant role, but at the reaction fronts it would certainly make the T, profiles more smooth and thereby moderate the temperature maximums somewhat. However, this effect is so highly localized that it can have little effect on the rest of the solution. Moreover, the results generated without conduction can only overestimate the temperature maximums, so that model parameters deemed acceptable on the basis of model calculations cannot lead to higher temperatures if conduction were included. It should be noted that the movement of reaction fronts observed here is different from the phenomenon of creeping reaction zones in fixed-bed reactors which was first reported by Frank-Kamenetskii (1969) and later analyzed theoretically by several investigators (e.g., Rhee et al., 1973,1974). The creeping reaction zone is caused by an imbalance between the overall heat and mass transport mechanisms. However, the depletion of the solid reactant (residue)is the dominant factor which causes the formation and movement of reaction fronts in fibrous filters. T, is also not shown. It displayed a gradual increase with time to a maximum of approximately 620 K after 2 min. A t all times its spatial gradients were quite small, justifying the neglect of axial conduction in the wall, which could only have made the T, gradients smaller. The overall extent of regeneration can be shown by the fractional regeneration defined as
Time,
207
s
Figure 12. The overall extent of regeneration with time in a nonuniform filter.
initial slow preheating, the onset of rapid regeneration,and finally a period of slow regeneration. In summary, the regeneration of fibrous filters involves a complex temperature excursion. The successful utilization of these filters requires a reliable method of controlling the regeneration process. In particular, two procew variables are critical: the maximum solid temperature and the total regeneration time. The present model shows that highest solid temperature occurs at the filter exit as a result of the growth of the downstream reaction front during the second stage of regeneration (rapid combustion). On the other hand, the total regeneration time is mainly controlled by the motion of the upstream reaction front during the third stage of regeneration. There seems to be a trade-off between lowering the peak temperature and reducing the regeneration time. This study shows that a desirable control on both variables can be obtained by using a nonuniform axial distribution of fiber size in the filter. In general, the optimum design of the filter and the regeneration process requires a systematic parametric study using the model; this study is under way. Acknowledgment The work of the first author was performed while he was a consultant to the Physical Chemistry Department of the General Motors Research Laboratories. The authors express their appreciation to the Physical Chemistry management and technical personnel for their support and assistance in this research. Nomenclature
C = oxygen concentration, g-mol/cm3
6 = dimensionless oxygen concentration, C/Co C, = heat capacity, cal/(g K) D = internal diameter of filter, cm df = fiber diameter, cm E = activation energy, cal/g-mol G ( t ) = function defined in eq 11 h = heat transfer coefficient around the fibers, cal/(cm2K S)
ha = heat transfer coefficient at exterior of the filter wall,
cal/(cm2K s)
h, = heat transfer coefficient at interior of the filter wall,
cal/(cm2K s)
Its variations with time, shown in Figure 12, depict the
(-aH)= heat of reaction, cal/g-mol k , = mass transfer coefficient around the fibers, cm/s k , = intrinsic rate coefficient of carbon oxidation, cm/(K s) k = overall rate coefficient defined in eq 2, cm/s L = filter length, cm
208
Ind. Eng. Chem. Process Des. Dev. 1983, 22, 208-211
M, = atomic weight of carbon, g/atom
coverage of surface by carbon deposit, g/cm2 = monolayer coverage, g/cm2 gas constant, atm cma/$-mol K) R = reaction rate, g-mol/cm 8 ) s = total surface area of fiber in a unit volume of filter, cm-’ t = time, s E = dimensionless time, (G(O)/p&)t T = temperature T = dimensionless temperature, T / To T f ( t )= function defined in eq 11 u = interstitial gas velocity, cm/s w = wall thickness, cm x = distance along the filter, cm f = dimensionless distance, x / L Y ( t ) = function defined in eq 11 Greek Letters y = fraction of the fibers surface covered by the particles e = filter void fraction I.( = viscosity, g/(cm s) p = density, g/cm3 u = Boltzmann constant Subscripts g = gas phase m = mass transfer s = solid phase; solid surface w = filter wall q =
$=
0 = characteristic values at t = 0 a = ambient
Literature Cited Bird, R. B.: Stewart. W. E.: Lbhtfoot. E. N. “TransDort Phenomena”; Wiley: New York, 1960; p 402. FleM, M. A.; 0111, D. W.; Morgan, B. 8.; Hawksley. P. 0. W. “Combustion of Pulverized Coal”; Brltlsh Coal UtlRzatlon Research Association, Leatherhead Surrey, England, 1967; p 344. FrankXamenetskU, D. A. “ D W h and Heet Transfer In Chemlcal Klnetlcs”, 2nd ed.: Plenum Press: New Ywk, 1969 pp 501-503. Gonzalez, L. 0.;Spencer, E. H. Chem. Eng. S d . 1963, 18, 753. Hano, T.; Nakashio, F.; Kusundtl, K. J . Chem. Eng. Jpn. 1976, 9(3), 209. Johnson, B. M.;Froment, (3. F.; Watson, C. C. Chem. Eng. Scl. 1962, 17, 835. Olson. K. E.; Luss, D. Amundson, N. R. Ind. EM. Chem. proceSS Des. Dev. 1968, 7, 96. Ozawa, Y. I d . Eng. Chem. pToc(#KI Des. Dev. 1969. 8. 378. Rhee. H.; Fley, D.; Amundson, N. R. Chem. Eng, Sci. 1973, 28, 607. Rhee, H.; Lewis, R. P.; Amundson, N. R. Ind. Eng. Chem. Fundam. 1974, 13. . . 317. - .. . Robertson, G. F. “A Study of Thermal Energy Conservatbn In Exhaust Pipes”; General Motors Research Publication GMR-2883R: Warren, MI, 1979. Sampath, B. S.; Hughes, R. Recess Techno/. Int. 1973, 18(1/2), 39. SatterReld, C. N.; Cortez, D. H. Ind. Eng. Chem. Fundem. 1970, 9 , 613. Schulman, B. L. Ind. Eng. Chem. 1968, 55(12), 44. VanDeemter, J. J. I d . €ng. Chem. 1953, 45(6), 1227. VanDeemter, J. J. Ind. Eng. Chem. 1954, 46(11), 2300. Zhorov, Y. M.; Panchenkov, G. M.; Lazyan, Y. I. Rws. J . phvs. Chem. 1967, 4 1 (7), 842.
.
Received for review February 12, 1982 Accepted August 25, 1982
Extractive Purification of Activated Carbon. 1. Effect of Acid Normality, Volume, and Repeated Contact David 0. Coomy’ and Anthony L. Hlnes Chemical Englneering Depamnent, UniVerSity of Wyoming, Laramie, Wyomlng 82071
A representative activated carbon was treated with hydrochloric acid solutions of various volumes and normaHtles at room temperature in order to extract inorganic matter. It was found that, for a given volume of acid, the amount of material extracted was not a strong function of acid normality. For example, whereas 0.10 N acid extracted about 0.46 wt % of the solid material, 1.0 N acld extracted about 0.51 wt % and 10.0 N acid removed only 0.53 wt % . When different volumes of 2 N acid were used, the total amount of inorganic matter extracted was found to be nearty independent of acid vdume. Repeated washings of an actlvated carbon sample with 2 N acid indicated that many cycles would be requked In order to reduce the extractables content of the carbon to a truly low level. These results have Important implications for skuations where activated carbons are used to treat acidic streams (e.g., refinery wastewaters).
Activated carbons have long been used as adsorbents for a wide variety of industrial and medical applications. Examples include the treatment of wastewaters (from chemical plants, textile mills, and refineries), the production of clean municipal (potable) waters, the pretreatment of industrial process waters, the adsorption of flotation agents in mining operations, the recovery of valuable materials from liquid streams, the decolorization of various liquids, the purification of pharmaceuticals (e.g., vitamins, antibiotics), the purification of many industrial products (fish and vegetable oils, nylon monomers, beverages, glycerin, etc.), and the use of activated carbon as an oral antidote for poisonings. It is well-known that activated carbons generally contain substantial amounts of acid-extractable inorganic matter, sometimes as much as 8% by weight. This inorganic matter consists of a wide variety of components, but the 0198-430518311122-0208$01.50/0
major ones are usually iron, calcium, sodium, copper, sulfates, chlorides, phosphates, lead, zinc, and arsenic. In many liquid-phase applications where activated carbons are used, the liquids are acidic (e.g., wastewater streams from refineries and chemical plants often contain various organic acid compounds and are therefore acidic), and release of inorganic matter into the liquid can easily occur. Even when the liquids being treated are not acidic, some extraction of inorganics still can occur, and in certain applications where contamination must be avoided (e.g., in food and pharmaceutical applications) such inorganic release is intolerable. The conventional means for minimizing the throw-off of inorganics by activated carbons is to pretreat them by acid washing, usually with hydrochloric acid, to extract much of the inorganic matter. The carbons are then normally washed with plain water to remove most of the 0 1983 Amerlcan Chemical Society
