Combustion and Emission Phenomena in Incinerators

Burning Rates and Operational limits In a Solid-Fuel Bed

M. KUWATA, 1. J. KUO, and R. H. ESSENHIGH The Pennsylvania State University

University Park, Pennsylvania

INTRODUCTION

The objectives of this paper have been outlined in the introductory paper [1] of a set of papers to which this paper belongs. They are: (1) to develop a theoret­ical relationship predicting the variation of the rela­tive carbon saturation (RCS) factor with bed depth and (2) to establish guidelines for the operational limits (maximal and minimal air rates) during steady operation.

DIVISION I: RELATIVE CARBON SATURATION (RCS) FACTOR

Abstract

Analysis of reaction of air in a carbon char bed with combustion to form CO, gas-phase reaction of CO to CO2, and reduction of CO2 to CO leads to equations for the variation of all three gases with distance through the bed. Applying these equations to the definition of the relative carbon saturation (RCS) factor (f res) generates the relation: f res = 1 - K exp (-aLI d) for the variation of Ires with dis­tance L through the bed, where d is the particle diameter and a is a constant . This equation had previously been obtained empirically. Comparison of predicted and experimental values of a are con­sistent with physically realistic values of bed dimensions and similar parameters. Derivation of the RCS factor enables calculation of the area firing rate from first principles.

272

Nomenc lature

a average radius of refuse particles

A ash mass fraction of refuse

A s average pore surface area per ft3 of bed

d average particle diameter

D diffusion coefficient for active gas

Ires relative carbon saturation factor

F A area rate of char gasification ,

G A air supply rate per unit grate area

k, k' reaction rate constants

K,K' constants in I rc s factor

L

m

M

n

Pm

v

bed depth

index of CO - O2 reaction

mass of i-th species

moisture mass fraction of refuse (or total num­ber of moles of gas species)

molar concentration of i-th species

oxygen mass fraction of feed air

time

average air velocity through bed

velocity in the pores of bed (v p = vi f)

V volatile mass fraction of the dry-inert-free (DIF) refuse

oV s volume element in solid bed

X molar concentration in pores

Y j mole fraction of i-th species

a a constant in Ires factor

o diffusion layer thickness

f porosity

Statement of Problem

The relative carbon saturation (RCS or Ire s) fac­tor was originally defined by Thring [2] to represent the degree of potential saturation of oxygen by car­bon, and it appears in the following expression developed [3] for the area rate of char gasification (F A) in a solid-fuel or refuse bed:

FA = (3/4)lre s (PmG A)/(l - V)(l - A - M) (1) ,

where Pm is the oxygen mass fraction in air, G A is the air-supply rate per unit area of grate surface, V is the volatile fraction of the Dry-Inert-Free (OIF) refuse, and A and M are the ash and moisture frac­tions of the raw refuse. The fraction (3/4) is the molecular weights ratio (2M elM O2) where the factor 2 appears in the numerator since the expression is based on conversion to carbon monoxide (at Ire s= 1). The interest in knowing how Ire s varies with depth arises from the appearance of this factor in Eq. (1) since bed depth can therefore affect FA.

The RCS factor is defined as derived l ater in this paper by the following expression;

Inasmuch as the RCS factor increases with depth, due to increased gasification (increased CO), the variation of bed depth can chan�e the allowable bed loading FA for a given air rate GA' The possible range can be a factor of 2 or more. If the bed is so thin that only combustion to CO2 occurs, with no gasification, then Ires = 0.5, as may be deduced from Eq. (2). If the bed is thinner than that limit, the air is in excess, and Ires lies below 0.5. As the bed depth increases, the degree of gasification in­creases, and CO rises with falling CO2 , The limit is achieved when the gas composition is all CO, and from Eq. (2) Ires is unity.

This variation of the RCS factor with bed depth could, therefore, create substantial operational problems. Suppose, for example, a unit is operating satisfactorily, but, during an emergency, it is used as a storage hopper and/or is overfed. The increased bed dept,h would then allow a higher gasification rate unless G A is cut back, and the increased gasification rate would then overload the overbed combustion zone. Therefore, it is important to know how sensi­tive the RCS factor is to bed depth and on what para­meters it depends. An empirical expression exists of the form [2, 4, 5]

(3)

where L is the bed depth, d is an average particle diameter, and K and a are arbitrary constants whose values are only known for a few special situations [2, 4, 5]. Inasmuch as it is always undesirable to have to rely on empiricism, particularly where the systems examined empirically differ considerably

Ires = [(C02%) + (CO%)]

1 - [(02%)/(02% initial)] -([(C02%)/(C02% initial)] [1 - (02% initial)/100]) 1 + [(02 %) + (C02 %)J/100

(2)

where O2%, CO%, and CO2% are the gas percentages in a sample taken at any level L in a fuel bed, re­ferred to a common basis (wet or dry analysis), and (02% initial) is the initial oxygen percentage in the input gas, which equals 21 percent for air. This formulation differs in presentation although not in substance from that given by Thring [2] and is given in this form be show the explicit influence of (02 % initial). If hot combustion products are used to assist drying, (02% initial) may be lower than 21 percent. The factor represents the ratio of the mass of carbon being carried in a reactive gas stream to the maximum it could carry if all the available oxygen were com­pletely saturated with carbon (as CO).

273

from the systems the expressions are to be applied to (as in this case), it is an obvious course of action to attempt a theoretical derivation of the empirical expression. This is the objective of Division I of this paper.

Ana lysis of Solid-Bed Gasification

Analysis of the behavior of reacting gases (02, CO2) in a solid-fuel bed, generally considered as carbon on char, has been attempted many times [2, 6-8], and numerical solutions do now exist. The problem is so complex that, if the problem is stated rigorously, analytical solutions are impossible and

numerical solutions are essential. Numerical solu­tions, however, suffer from the usual drawback that the influence of many relevant parameters is then obscure or impossible to determine. Reasonably based physical approximations that lead to analytical solu­tions are, therefore, of major value in providing indications as to the most important physical factors likely to dominate a system and their relationships. This is the justification for the following analysis.

Physical Model and Assumptions

The solid-fuel bed being analyzed is presumed to consist of solid particles that are mainly carbon and inerts, packed randomly on top of a grate, with air supplied from below and fresh fuel from above. Such a bed normally packs down as the burning material moves down since the particles diminish in size and generally become more friable. For this analysis, however, a uniform porosity ( through the bed is assumed. Requirements for uniformity to particle size, shape, and packing are not otherwise explicitly required. The other significant property assumption is that the temperature through the bed is uniform. Reactions in the spaces between the particles are then, according to the "Three Zone Theory" [9]: (1) oxygen heterogeneously with carbon to produce some CO2 but mainly CO; (2) CO2 heterogeneously with carbon, being reduced to CO; and (3) CO and oxygen homogeneously in the gas phase, to oxidize back to CO2,

To proceed with the analysis, we require sufficient information to be able to write down kinetic equations for these three reactions. For a full, detailed analy­sis, this would require information on the diffusion, absorption, and desorption processes involved, and, although much of this material is now available [10-12], its use would introduce far too much complica­tion to allow analytical solutions of the equations developed. Fortunately, it would seem to be a valid approximation to assume that the velocity of flow through the pores of the bed is sufficiently slow for the heterogeneous reaction to be dominated by diffusion. In practice, there is likely to be some faster mixing than only by molecular diffusion be­cause of the changes in direction of flow through the tortuous pores in the bed. In general regard, it is pertinent to note that, although any bed is an agglom­eration of "particles" (including newspapers, books, or tin cans), when agglomerated the problem changes, as Frank-Kamenetskii has pointed out [13], from an "external" to an "internal" one, so that

274

critical flow velocity through the pores could be important.

To substantiate the velocity assumption consider 1 ft2

of a burning bed at 10000C and fired at a rate 50 lb/h fe. If the heat of combustion of the waste is 10,000 Btu/lb, the total air rate per hour is about 5000 ft3 (cold) for stoichiometric supply or 1.4 fe / sec at a bed porosity of 0.5 the average pore velocity is about 3 ft/sec. At 10000C the kinematic viscosity of the gases is about 0.002 ft/sec units. If the pore diameter is about that of the mean particle size, say 0.1 ft, the Reynolds number in the pore is (3 x 0.2/ 0.001) = 600. This should be an upper limit since stoichiometric combustion of dry refuse has been assumed, and, if the bed is to gasify, the air rate must be less than this. A fairly generous pore size has also been assumed. It would seem that the flow will not generally be turbulent.

Kinetic Equations for Solid-Bed Reactions

Consider the elemental volume of the bed re­garded simply as a porous solid of porosity (. The kinetic equations may now be developed by the application of phenomenological reaction kinetics. In this application, we assign a velocity constant k to each of the three reactions listed in the previous section.

For the two heterogeneous reactions (02 and CO2 with solid carbon), the velocity constant k' deter­mines the rate of reaction per uni t of the pore sur­face. Incorporating the described mechanistic con­clusion that reaction is diffusion dominated, the reactant-gas concentration (02 or CO2) at the solid surface can be assumed to be nearly zero; therefore, the specific rate of reaction will be proportional to the mainstream or average concentration across a pore. If this concentration, written as moles per unit volume, is X, then, assuming As is the average pore surface area ft3 of bed (and will be a function of porosity and particle size, etc.), the molar rates of carbon removal (dm/ dt) by the two reactions in a volume element oV s are given by:

(1) For C reacting with O2 to give CO (mainly) by the reaction

2C + O2 = 2CO (4)

(dm,ldt) = 2 (k�As)X ,.oV s = 2k;X ,oV s (5)

(2) For C reacting with CO2 to give CO by the reaction

CO2 + C = 2CO (6)

where, by this formulation, the subscripts 1 and 2 refer, respectively, to the C/02 reaction and to the C/C02 reaction and k', denoting reaction with respect to unit surface area, becomes k by multiplica­tion with A s, denoting reaction with respect to unit volume of the bed.

For the gas phase reaction given, overall, by

CO + (112)02 = CO2 (8)

the details of the kinetic scheme are involved [14], requiring reaction with OH as a key intermediate step, but fortunately there is evidence summarized by Johnson [15] that the rate of CO2 formation can be represented in phenomenological kinetics by the operational equation

(9)

where Y; indicates the mole fraction of CO2 formed by this reaction alone and m is an index apparently lying between 0 and 1.0 [15]. The choice of index is discussed in greater detail later in the discussion.

Molar Balance in Gas Phase

As the result of reaction, as a volume element of gas moves up through the bed, the gas concentrations change, and the gas volume increases (at constant temperature and pressure) due to CO formation. Sup­pose at some point in the bed we have a volume element of bed V s containing a volume V of gas. This gas volume is assumed to remain in this posi­tion for a small period of time ot, during which reac­tion takes place and at the end of which the gas volume moves on, into the next adjacent volume ele­ment of bed. Suppose the total number of moles in the gas volume oV II is Mo at the start of reaction then

(10)

where N is the number of inert moles (nitrogen, etc.) and the subscripts 1, 2, and 3 refer as before to O2, CO2, and CO. If om, moles of carbon form CO by reaction with oxygen and om2 by reaction with CO2,

275

the increase in CO from these two reactions is (om, + 20m2), and the increases in N, and N 2 are (--Om,/2) and (--Om2), respectively. There is also an increase in CO2, to the extent (oN;) moles, due to the gas-phase reaction, and corresponding increases to O2 and CO to the extent, (--ON;/2) and (--ON;), respectively. The net molar increases of the three gases are therefore

oN, = (--Om, /2) - (oN;/2) (l1a)

(l1b)

(Uc)

The total molar increase oM is the sum of the oN�, i.e.,

(12)

or

(13)

if N 3 = 0 when the air enters the bed (at time t = 0) and M = Mo. If Y is the mole fraction of inerts in the gas volume in the bed, then, since N remains un­changed, N = YoM 0 = YM; therefore,

(14)

Similarly, defining Y 3 = N 31M, then substituting for N 3 by Eq. (13) and for M by Eq. (14), we obtain

Y = Yo (1 - Y 3/2) (15)

Since the total of all mole fractions must be unity, we have for the general condition and for the input condition, respecti vel y:

(16a)

Y� = (1 - Yo) (16b)

Eliminating Y and Yo between these last three equa­tions yields

which is the general molar balance required. It ena­bles calculation of any one of the three gases if

either of the other two are measured. It would, of course, give the dry analysis since it is assumed in the derivation that the number of moles of inerts in the volume \dentified remains constant. For material that is dried and pyro1yzed at the top of the bed this should be true.

Derivation of Rate Equations

Let us now consider a horizontal thin element of bed of unit horizontal area and thickness I1x. Its volume (oV s ) is therefore I1x, and the void volume (oV g) containing the gas is (l1x. If reaction proceeds for a time ot, the mass of carbon removed in that time by the two heterogeneous reactions Om is given by multiplying the rate equations (5, 7) by ot. There­fore, multiplying Eq. (l1a) by two and adding to Eq. (l1b), we have

20N1 + oN2 =-Om1 - om2

Dividing by oV s = oV gl ( converts oN into a change in number of moles per unit volume oX. Dividing again on both sides by the number of moles per unit volume (which is a constant at constant temperature and pressure) yields Y, the mole fraction, in place of X. Dividing by ot and taking limits, gives, after rearranging,

[2( dY /dt) + 2k1Y 1 ] + [(dY 21dt) + k2Y 2] = 0 (19)

Dividing through by f and writing (kif) = n, the equation becomes

Each of the two expressions in the brackets are, individually, simple standard forms, and each is equal to some function I(t), which is determined by the gas-phase reaction. Since conversion of N to Y by the divisions described above converts N; to Y; in Eq. (l1a), then combining Eq. (l1a) with (9) yields

(21)

Simultaneous solution of this with Eq. (20) will gener­ate the equations required.

Approximate Solutions Generating ReS Factor

Derivation of ReS Factor

276

Although the RCS factor is quoted in the Introduc­tion, it is not in the form originally obtained by Thring [2]. It is, however, very easily derived. If a gas stream contains mole fractions Y 11 Y 2' and Y 3 of °2, CO2, and CO, respectively, then the moles of carbon already picked up by the gas equal (Y 2 + Y 3)' The maximum pickup is therefore what is already in the gas stream (Y 2 + Y 3)' plus the additional possible pickup, which is 2Y 1 for the oxygen and an additional mole of carbon for each mole of CO2; therefore, the additional total pickup is (2Y 1 + Y 2)' Adding this to the total already carried, the maximal pickup is (2Y 1 + 2Y 2 + Y 3)' From Eq. (17), this can be written alter­natively as [y� (2 - Y 3)]' or as [1 + Y 1 + Y 2]' The RCS factor is therefore defined as the [(actual C pickup)/(total possible C pickup)]; therefore,

[Y2 + Y3] 1 - (y /Y�) - (Yj2Y�)(1 - yn Ires = (2 - Y 3)Y� = 1 + Y 1 + Y 2

(22) which is identical with Eq. (2) with mole tractions in place of percentages. The RCS factor is therefore obtainable as a function of time or distance through the bed if Y 1 is Y 2' and/or Y 3 can be obtained separately as functions of time or distance by simul­taneous solution of Eqs. (20) and (21).

Initial Solution for (2Y 1 + Y J

It is evident from inspection of Eqs. (20) and (21) that simultaneous solution must lead to very complex functions and, in fact, analytical solution does not seem to be possible. There is, however, a potential simplification possible that reduces the complexity substantially. This follows from the mechanism indicated in the discussion of analysis of solid-bed gasification, i.e., that k 1 and k2 depend primarily on diffusion. If that statement is true, they should differ only by the difference between the diffusion coeffi­cients for oxygen diffusing through nitrogen and for CO2 diffusing through nitrogen (and neglecting any Stefan flow effects, which is valid at the high dilu­tion levels encountered in air). To a first approxima­tion or better, these diffusion coefficients are the same (0.18 cm2 1 sec for oxygen, compared with 0.14 for CO2), and the temperature coefficients are also about the same. Therefore, we may write

(23)

Equation (20) now integrates directly, giving

Y 1 + (y 2 /2) = y�. exp (-nt) (24 a)

and

Y 2 = 0 at t = O. (24b)

This is a very substantial simplification. With this alone, it is possible to develop an initial approxima­tion for the RCS factor as a function of time or dis­tance through the bed.

First Approximation Solutions for Y 1 and Y 2

The intial approximation for Eres with time that is possible using Eq. (24a) may be improved if Y 1 and Y 2 can be obtained as separate functions of time. In principle, this is possible by additional solution of Eq. (21), but in practice this is, self-evidently, a formidable problem. Once again, however, a substan­tial simplification is possible if we consider at least the solution for the limiting condition in the gasifica­tion zone where Y 3 is substantial and further changes become decreasingly significant. Let us suppose, therefore, that a range of conditions exists such that Y 3 is substantially constant with a mean value Y�. We can therefore write

dY /dt = -n[Y 1 + (Y�n3/2n)Y�] (25)

which is, of course, a standard form for a linear equa­tion of the first order. If, therefore, we know the value of the index m is Eq. (25) we can obtain a solution for Y l' As remarked earlier, the possible values found in the literature for reaction at or near atmospheric pressure range from 0 to 1 [15]. Phenomenologically, we should expect 0.5, which is close to the value given by Hottel and Williams [16]. Mechanistically, a collision-determined reaction involving an activated intermediate could double the phenomenological value [17], giving unity as found by Fenimore and Jones [18]. There is, in addition, some evidence [15] for a small or zero value, but this clearly cannot be true when the oxygen itself is near zero, since it leads to the absurdity of CO2 being generated without any available oxygen (although this does not necessarily eliminate the possibility of CO vanishing at very low oxygen). Solutions have, therefore, been developed only for m = 0.5 and 1.

and

(1) m = 1: The solutions have the form

[- -n (1 + Y � n 3 I 2 n) I] Y 2 = 2Y� e-nl - e (27)

277

showing that Y 1 decays exponentially at an acceler­ated rate over the value for (2Y 1 + Y 2)' The equation for Y 2 is typical for the rise and fall found with competing reactions. The rate of rise at small t is dominated by the second term, and the rate of fall at high t is dominated by the first term.

(2) m = 0.5: The solution is, naturally, more complex.

Y ,!Y� = (1 + K)2e-nl _ 2K(1 + K)e-nIl2+ K2 (28a)

= [(1 + K)e-n 112 - KF (28b)

where

(28c)

and

Y 2 = 2Y� [K(l + K)e-n II 2 _ K(l + K/2)e-n t - K2/2]

(29) which has substantially the same form but with some significant numerical differences compared with Eq. (27).

Variation of Y 3

If the expressions for Y 1 and Y 2 are obtained, then Y 3 is obtained by substitution for Y 1 and Y 2 in Eq. (17). The expressions obtained, however, are involved and add nothing to clarification of any solution since, if the agreement with prediction for Y 1 and Y 2 is adequate, then adequate agreement with prediction for Y 3 is a truism.

Comparative Analysi s and Experimental Testing

Comparison of the solutions obtained for (Y 1 + Y 2/2) and for Y 1 [(24a, 26, 28)] indicates a potential method for determining the index m by experimental comparison of slopes. According to Eq. (24a), a log­linear plot of (Y 1 + Y 2/2) against time should be a straight line with a slope n. A similar plot for Y 1

alone should yield either: (1) a straight line of slope greater than n, if Eq. (26) is obeyed, or (2) a curve with the extremes of the slopes lying between n and n12, if Eq. (28) is obeyed. Finally, a similar plot of Y 2 should generate a curve that should approximate to a straight line at high values of t, with a slope approximating to n for m = 1 and (nI2) for m = 0.5. We should, of course, remember that the solutions should really hold only at high values of Y 3 when this can be approximated as a constant, but for

want of anything better analytically, it is worth considering extension in the cases where Y 3 is not a constant since inspection indicates that for some­what involved reasons the solutions still seem to be reasonably valid for Y 1 and Y 2'

Experimentally, the equations have been tested using some widely reproduced curves obtained by Mayers [6] . The results were quite clear cut. The appropriate plots (not reproduced) using distance in place of time gave a valueof 5 ft- 1 for n, with values of 16 ft- 1 for the Y 1 plot and again about 5 ft - 1 for the Y 2 plot at high values of distance. These results are consistent with a choice of unity for m as the best approximation. The probable kinetic significance of this, involving a collision activated intermediate, has already been mentioned.

Further theoretical development of the analysis is possible. If a porous bed is considered equivalent to a set of no parallel pores per unit area, of radius a,

then the pore volume in unit volume of bed is no (l7a2) = f. The internal surface of these pores is then no(217a) = As, and, hence, (As/£) = (2/a). If the aver­age particle diameter in the bed is d and if this approximates to the pore diameter, we have (A s/ f)

= (4/d). We can therefore write, for the index n,

n = k/f= k'(As/f) = k'(4/d) (.30)

where k', as has already been defined, is the velo­city constant for a diffusion-dominated reaction with respect to unit surface area of the macropore walls in the bed. If the diffusion coefficient for the active gas is D (assumed to be the same for both 02 and CO2) then k' can be written as (D/a) where a is some effective diffusion-path length or diffusion-layer thickness. The value of a may be estimated for check as a realistic quantity in the following manner. Since n is 5 ft - 1 in length units, this is equal to

JO�-------------------------------------------------------------------__ ---=��

20 �

� " 0 ..... u U '" ... .... ... '" .-< 0 :.::

10

o 1 2

/ /

./ ./

/

;' ",­

",-",-

..-..­..-

---

/ _____ Mayers' experimental data

/ /

________ Present theory

h

� '" "

.......

" .............. ----------------------------------.......... -"""-- Y2

---- ---

J 4 5 6 Bed Depth, L (in.)

Fig. 1 Comparison of Mayers' Gas Analysis Data in a Solid Bed with Solutions of Present Theory Given by Eqs. (26) and (27), with Y l' Y 2' Y 3 as Molar Fractions of 02' CO2, and CO. Respecti vel y

278

8

---

9

11 sec- 1 at a cold velocity of 0.55 ft/sec or a hot velocity (at 900°C) of 2.2 ft/sec. Since D for oxygen diffusing through nitrogen is 2 x 10- 4 fe/sec at OOC and about 3 x 10- 3 fe/sec at 900°C, then (d8) = (4Dln) � 10- 3 ft'. If the pore diameter is about 0.5 in. or about 0.05 ft, then the effective diffusion layer thickness is about 0.02 ft or about 0.25 in. (i.e., a little less than the pore radius). Since the fuel particle size used was 1 in. or more, these values are realistic, and they usefully substantiate the model and general argument developed.

Using the values for the constants obtained from the plot, Fig. 1 shows a back plot to show the rea­sonable agreement obtained between the original curves (full lines) and the calculated curves (dotted lines). Considering the approximations involved, the fit is adequate.

Prediction of ReS Factor

We have now devised the necessary time-depend­ent equations for Y 1 and Y, for substitution in the RCS factor Eq. (22). The result is obviously involved, but we can develop reasonable approximations for practical use by appropriate neglect of small terms. In the first instance, Eq. (22) may be rearranged by adding (Y 1 + Y 2 - Y 1 - Y ,) to the top line on the right-hand side, which, using Eq. (24a), enables us to write without approximation

{res = 1- [(1 + Y� ) / (l + Y 1 + Y,)] e-nt (31)

Now consider behavior of this equation at small t. In the limit, of course, at t = 0, Y, ,= 0 and Y 1 = Y� (giving {res = 0), but, over the combustion zone where Y 1 is falling and Y, is rising, (Y 1 + Y 2 ) is slowly falling from an initial value of 0.21 to about 0.15 or 0.10. The total group of terms [(l + Y� ) / (1 + Y 1 +

Y,)]' therefore, rises from a value of unity to a value of 1.10. If we take a mean value through this range of 1.05 ± 0.05, we would be within 5 percent of the real value, assuming Eq. (30) to be absolutely accurate, and such variation would probably be obscured by ex­perimental scatter.

Similarly, at higher values of t, Y 1 is near zero, and Y, is declining from about 0.1 to zero. The group of terms before the exponential, therefore, rises from about 1.1 to 1.2, so, in this range, adopting a value of 1.15, or 10 percent higher than at low t, should be adequate. The theoretical expression, therefore, may be written

{res = 1 - K' exp [-(n/v)L] (32)

279

where v is the average velocity through the bed con­verting time to distance and K' is 1.05 for small L and 1.15 for large L. This compares very well with the empirical expression obtained by Thring [2] from analysis of Kreisingers curves [19]

{res = Ko [1 - exp (-aL/d)] (33)

where Ko is 0.7 to 0.9 increasing to 0.97 with air preheated to 300°C; it is also high if wall losses are low (see Division II of this paper) and with higher reactivity fuels. This suggests that the principal rea­son for Ko being less than unity is due to such cool­ing by the endothermic reaction at the higher levels of the bed that the CO,/C gasification reaction is so reduced in rate that it reduces the maximum CO value for a given bed depth. This conclusion is supported by the effect of air preheat, which will raise all temperature levels and maintain the gasification reaction in the upper section of the bed. Thring's equation [2], of course, has the advantage that {re S is zero at L = 0, whereas Eq. (32) does not give quite this result.

Comparison of the exponential indices, using also Eq. (30), yields

aid = nlv = 4k'/vd (34a)

or

a = 4k'dvp (34b)

where v p is the velocity in the pores of the bed, given by v p = vi (. In general, k' is also an increasing func­tion with increasing velocity; and, since Thring [2] states that a rises with air rate, we conclude that (k' /v p) also rises with velocity (i.e., k' rises faster than linearly with velocity). Since a also increases with air preheat [2], we have here a combined tem­perature and velocity effect. If we suppose that the effect of air preheat to 300°C will be to raise all temperatures through the bed by this amount, then the diffusion coefficients, which determine the tem­perature coefficient of k', will rise by a factor roughly of 1.6 for a bed initially at 600°C and by a factor 1.4 for a bed initially at 1200oC. The change in diffusion coefficient alone could, therefore, cause an increase in a from 0.6 (a value given by Thring [2]) to 0.9, as found experimentally [2] with preheat.

The value of a may be calculated from Eq. (34) either using the experimental value of n 1 or accept­ing the values of D, 8, etc. calculated in the previous section. Since a = (n/ v)d and (nl v) is 5 ft - 1 , then

for 0.5- to I-in. diameter pores, a will vary from about 0.25 to 0.5. This is in remarkable agreement with Thring's [2] value ranges of 0.3 to 0.6. This also serves to substantiate the model and analysis developed.

There is now one final point of significance developed by this analysis. Thring [2] states that a

increases with fuel reactivity. Taken at its face value, this is one point on which conclusions from the analysis are apparently at variance with experi­ment, since k' in Eq. (34b) is a diffusional velocity constant that is quite unaffected by the intrinsic reactivity of the material. This is a point emphasized by the experimental results of Kuwata et al. [20], who found effective breakdown of diffusional-control round spheres of about I-in. diameter did not occur until the flow velocity was about 6 ftl sec. It is, therefore, difficult to see how intrinsic fuel re­activity can affect a. It should be remembered, how­ever, that "reactivity" measured for industrial cokes generally uses some type of combustion cell or combustion pot where the conditions are very sim­ilar to those being analyzed here. What is therefore being measured in "reactivity" tests could very well be an alternati ve to a itself, and what this analysis indicates is that, if this is so, industrial reactivity may be primarily a function of the coke density and shape and the way that it packs so that, most probably, "reactivity" is primarily a function of the bed porosity f.

Conc lusion

It follows from this analysis that the empirical formulation for the variation of the RCS factor with bed depth, in the modified form, i.e.,

[res = 1 - K' exp (-aLI d)

is a good approximation to the theoretically based equation. K' is close to unity, and a will lie be­tween 0.1 and 1.0, depending primarily on tempera­ture and bed porosity. This is, therefore, a valid operational equation with a theoretical basis.

Substantiation of the analysis by test against existing experimental data also indicates that the assumption of diffusional dominance of the reactions in bed is very probably correct. Under these condi­tions, as discussed by Kuwata et al. [20], internal reaction inside the particles themselves should not take place.

The data are also consistent with the rate of CO oxidation to CO2 being first order with respect both to CO and to O2, in agreement with the results of Fenimore and Jones [18].

280

This section of this paper, therefore, completes the initial analysis necessary for calculation of the rate of refuse gasification from fundamentals. References

[1] R. H. Essenhigh and T. J. Kuo, "Combustion and"

Emission Phenomena in Incinerators: Development of

Physical and Mathematical Models. Part I: Statement of

the Problem," pp. 265-275, this volume.

[2] M. W. Thring. "The Degree of Interaction between

Air and Solid Fuel: The Effect of Fuel Size," Coal Research, p. 70, September 1944.

[3] R. H. Essenhigh and T. J. Kuo, "Development of

Fundamental Basis for Incinerator Design Equations and

Specifications," paper to be presented at Third Mid-Atlantic

Industrial Waste Conference. University of Maryland,

College Park, Md., Nov. 1969. l4J M. W. Thring, The Science of Flames and Fur­

n�ces, John Wiley & Sons, Inc., New York, N.Y., 2nd ed.,

1962, chapt. 3, 4. [5] M. W. Thring, and R. H. Essenhigh, Supplement

to Chemistry of Coal Utilization" H. H. Lowry (ed.), John

Wiley & Sons, Inc., New York. N.Y .• 1963. [6] M. A. Mayers, "Temperatures and Combustion

Rates in Fuel Beds," Trans. ASME, vol. 59, no. 279, 1937. [7] R. S. Silver, "Combustion in Fuel Beds," Fuel ..

vol. 32. no. 2, 1953, p. 138. [8] D. B. Spalding, " The Calculation of Mass Transfer

Rate in Absorption, Vaporization, Condensation and Combus­

tion Process," Proc. Inst. Mech. En&., vol. 168, p. 545, 1954. [9] M. W. Thring, Fuel, vol. 31, no. 355, 1952. [10] P. L: Walker, F. Rusinko, and L. G. Austin,

"Gas Reactions of Carbon," Advances in Catalysis, vol.

11, no. 3, 1959. [11] R. W. Froberg, "The Carbon-Oxygen Reaction:

An Experimental Study of the Oxidation of Suspended

Carbon Spheres," Ph.D. TheSis, Dept. of Fuel Science,

The Pennsylvania State University, University Park, Pa.,

June 1967. [12] L. Kurylko, "The Unsteady and Steady Combus­

tion of Carbon," Ph.D. TheSiS, Fuel Science Section, The

Pennsylvania State UniVersity, University Park, Pa.,

September 1969. [13] D. A. Frank·Kamenetskii, Diffusion and Heat

Exchan&e in Chemical Kinetics, Princeton University

Press, 1955, (Russian 1st ed., 1947). [14] R. M. Fristrom andA. A. Westenberg, Flame

Structure, McGraw-Hill Book Co., Inc., New York, N.Y.,

1965, pp. 328-330, 344-350 [15] M. L. Johnson, "Combustion in the Afterburning

Zone of Propane Flames: A Study of the Air Pollution

Potential of CO from Hydrocarbon Fuels," Ph.D. TheSiS, The Pennsylvania State University, University Park, Pa.,

September 1969. l16j H. C. Hottel, G. C. Williams, N. M. Nerheium,

and G. R. Schneider, "Kinetic Studies in S tirred Reactors:

Combustion of Carbon Monoxide and Propane," Te'nth Sym­posium (International) on Combustion, Combustion Institute,

Pittsburgh, Pa., 1965, p. 111.

[17] R. A. Strehlow, Fundamentals of Combustion, International Textbook Co., Scranton, Pa., 1968, pp. 101, 102.

[18] C. P. Fenimore and G. W. Jones, "The Water

Catalyzed Oxidation of Carbon Monoxide by Oxygen at

High Temperatures," J. Phys. Chem., vol. 61, no. 651, 1957.

[1 9] H. Kreisinger, F. K. Ovitz, and C. E. Augustine,

"Sampling and Analyzing Fuel Gases," U. S. Bureau of Mines Paper 137, 1916.

L20J M. Kuwata, J. P. Stumbar, and R. H. Essenhigh,

"Combustion Behavior of Suspended Paper Spheres,"

Twelfth Symposium (International) on Combustion (Poiters

France, July 1968), The Combustion Institute, P i�tsburgh, •

Pa., 1969 p. 663.

DIVISION II: O PE RATIONAL LIMITS FO R

MAINTENANCE O F IGNITION

Abstract

A heat balance on a char bed leads to an equation predicting the equilibrium bed temperature. The same equation predicts minimum airflow rates required for maintenance of reaction if the system temperature at extinction is known. This can be predicted by appli­cation of the thermal-analysis theory. Insertion of appropriate values for the coefficients and para­meters in the equation predicts air rates at minimal flow that lie in the range found experimentally. The minimal flow rate is quite strongly influenced by wall losses. If these are negligible, maintenance of reaction in a bed of Type 2 refuse with 50 percent moisture should be possible at a feed rate of 2 lb/h ft2 and an airflow rate of 0.16 lb/h ft2. The moisture margin for maintenance of reaction would seem to be about 70 to 75 percent.

Nomenc lature

A grate area

B calorific value of fuel

cp s pecific heat

Do diffusion coefficient of feed air

f res relative carbon saturation (RCS) factor

F A area firing rate ,

G total air rate ,

G A rate of feed air per uni t grate area

Ii film coefficient between bed and surroundings

L bed depth

Pm mass fraction of oxygen in feed air

QJ heat production in bed

Qn heat loss from bed

281

r gas analysis ratio

s grate perimeter

T temperature

To input temperature

o diffusion-layer thickness

( bed porosity

e effective burning fraction of solid bed

Po feed-air density

Statement of Problem

The operational limits for maintenance of ignition in a fuel bed are the minimal and maximal air rates temperatures, bed depths, and other factors, outsid; which ignition fails. The thermal basis for predicting these limits has been outlined in a paper [21] of a set to which this paper belongs. The objective of Division II of this paper is to consider application of the principles outlined to the solid bed.

Theory

Basic Heat-Gain and Heat-Loss Equations

As explained elsewhere [21], the theory of ignition and extinction is based on developing, separately, the equations for heat generation by reaction (I) and heat loss (II) and by equating these, or determining the tangency of the slopes, for predicting the steady states and critical conditions, respectively. The heat generation rate QJ for the whole of the solid bed is given by the product of the firing rate, the calorific value of the fuel, and the factor allowing for only partial combustion of CO to CO2, If the grate area is A, then we have from Ref. [21]

(35)

For the heat loss from the bed Qn, there are two sources: (1) the convective loss from the top of the bed due to flow-through of the gases and (2) wall losses. For the con,vective loss, if the input mass flow rate of air is G A lb/h ft2, the carbon gasifica­tion rate is [21] (3/4)frcs (PmGA), and the total mass flow rate is the sum of the two. If the specific heats of air, char, and combustion products are all about the same, at cp Btu/lb OF and if the temperature le,aving the top of the bed is T, the convective loss is (GAA) [l + (3/4)Pmfres]cp (T - To), where To is the

input temperature. If the grate perimeter is s ft and the bed depth L ft, the contact area between bed and wall is sL. If the heat transfer coefficient between bed and surroundings is h, and the rate of wall loss is (sL'/i (T - To). The total heat loss from the bed is, therefore,

Qu = [(Acp) [1 + (3/4)Pmfrcs]GA + (shL)]. (T - To) (36)

In this formulation, the somewhat sweeping assumption has been made that the bed temperature is reasonably uniform both vertically and horizontally. In fact, gradients must exist, and the analysis of the system with thermal gradients and internal heat flow is possible in principle but cumbersome and relative­ly uninformative. If T is regarded as an average, and appropriately chosen, the gradients may be neglected, and Eq. (36) is reasonably applicable to a first approximation.

Initial Evaluation

To compare QI and QIl' these must be formulated in terms of temperature. QII' given by Eq. (36), already has this formulation. To do similarly for Q we must substitute for e and FA' The fraction e I' has been given elsewhere [21] in terms of Y 2 and Y 3' If gas analysis shows little CO, e is about unity. As the CO rises and the CO2 falls, e falls to

about 0.28. If e were constant at unity, Qr would be given approximately by

Qr = (3/4)(PmB)(AGA) � 1 - exp

[-(4PoD oAI (AG A (0) (LI d) (TIT 0)0.75]f (37)

where the exponent nt has been written out in terms developed previously in this paper, with distance L , in place of time, and mass flow rate G A combined with density in place of velocity. In terms of tem­perature, with other parameters constant, this equa­tion has the typical form of a (1 -,exp), with a high limit for QI given by (3/4) (PmB)G. As the limit is approached, however, the parameter e starts to de­cline from unity down to the lowest value given [21] by e = (1 - O. 71�), when r is unity; therefore, e = 0.281. The resultant curve for QI is illustrated graphically in Fig. 2.

The heat-loss curve, on the other hand, is a straight line according to Eq. (36). This is also plotted in Fig. 2. The intersection of the two curves is then the stability point or bed temperature. This is, of course, very much a first-approximation approach since the heat-sink effect of the gasifica­tion process has been ignored. To take this into account, a modification of the reactor sequence used by Vulis [22] and elaborated elsewhere [23] can be used, but this refinement is being left for the future

0' e = 1 (Combustion only)

'" '" o ,..J ... o " o ..... u C1l ... OJ C OJ t.:>

Heat generation rate --

" ---------

I-Heat loss rate QIl «:) - Stable operating point or bed

TO System Temperature - T

(Gasification included)

Fig. 2 Qualitative Illustration of Heat-Generation and Loss Curve (01 and OIl) for a Solid Bed (Semenov Plot) (Intersection of two curves gives effective bed temperature ond heat-generation rate under steady-state conditions. This pair of lines will be valid for any given air-feed rate.)

282

when there is greater certainty about the coefficient values involved and the additional work is justified.

Effect of Air Rate

An alternative method of plotting to obtain data or stability points is to plot Q against the total air rate G. This is illustrated in Fig. 3. This should not be misunderstood; there is a nominal but somewhat misleading similarity to the Semenov (Q - T) plots. In Fig. 2, the abscissa-axis parameter T is a depen,­dent parameter; in Fig. 3, the same axis parameter G is independent, with the dependent parameter T deter­mining the precise location of the lines.

This plot illustrates two significant points. The first is that there are two intersection points, and the one at lower G exists because of the QIl inter­cept on the ordinate axis at G = 0, which is due to wall losses. Without these wall losses, the heat-loss line would go through the origin, and only the upper intersection would exist. This is, evidently, of major significance in determining the lowest air rates re­quired for maintenance of combustion, as will be evident later in discussing data by Moles and Thring [24].

The second point partly illustrated by the Fig. 3

plot is the effect of changing the air rate. The net effect will be a change in temperature, so that new intersections are formed on new Qr and QIl lines

OJ '-' '" e>: Ul Ul o H .... o c: o OM '-' '" .... OJ c: OJ '-'

Effect of T increasing

Curves for heat

(shown dotted). An increase in air rate, however, will not necessarily lead to an increased tempera­ture. From Eq. (36), an increased air rate will always increase Qu' The reciprocal of air rate in the ex­ponent of Eq. (37), however, will ultimately lead to the opposite, so that Qr will start to fall above a certain air rate at any given temperature. This is illustrated explicitly in Fig. 3. If, therefore, Qu in­creases from a stability point but QI decreases, the temperature is bound to fall. The decreased heat­generation rate at the higher velocities, incidentally, is explained physically by saying that faster through­put means a shorter contact time (or less time to react). Ultimately, an infinite quantity of air moving at infinite velocity through the bed must mean there is no time at all to react, and Qr goes to zero.

Conversely, at the lower intersection point, an increase in air rate should always increase the burn­ing rate. The heat loss will increase similarly, but the net effect will be an increase in temperature of the system.

Both these aspects have been illustrated in Fig. 3. What is not shown in Fig. 3 is the effect of reach­ing an extinction temperature. The principle of the extinction calculation has been given previously and will be amplified later in this discussion. At this point, however, we may emphasize that, given the existence of an extinction temperature, the equations developed predict that this can be achieved

Air Rate - G

Fig. 3 Qualitative Illustration of Alternative Method of Determining Stability Values of G and Q at Given Temperatures

-----Tl ------ T2

283

either (1) at a very high air rate, designated by Vulis [22] as "adiabatic extinction, " where the con­vective losses are dominant, or (2) at a very low air rate, designated by Vulis [22] fiS "thermal extinction, " where the wall losses are dominant. In our system of a solid-fuel bed, our prime concern is with thermal extinction, at very low air rates.

Critical Operation and E xtinction

Reaction at Lo w Temperatures

Up to this point in the paper, the reaction condi­tions required for extinction have been neglected. They must now be introduced. To do this, we must examine the special conditions of reaction behavior at low temperatures, where "low" in this context is the range 400 to 800oe.

In the equations thuf:! far developed, only the diffusional mode of reaction has been considered, for the reasons given in Division I of this paper. We do know, however, that, when the temperature drops below some determinable value, the whole reaction mode changes from diffusional to kinetic and the reaction rate then drops off very sharply with temperature. Examining Fig. 2 and comparing this Fig. 1 of Ref. [21] , we see that such a change in reaction rate overriding the diffusional kinetics

OJ w co <>: Ul Ul o ...., .... o

, Effect of increasing G near thermal

conditions

is necessary if we are to have a region where critical conditions (extinction and ignition) can be defined. This is illustrated schematically in Fig. 4, with a particular shape chosen for the kinetic curve based on past work [25-30].

The original experimental basis for the picture of a rapidly rising rate curve, with a high tempera­ture coefficient for the reaction, through a region of chemical-kinetic dominance, passing fairly abruptly into a high temperature region of diffusional­kinetic dominance, was developed in the classic paper by Tu, Davis, and Hottel [25]. Since then, the experimental behavior has been substantiated and elaborated in many experiments adequately reviewed elsewhere (e.g., Refs [26] and [27]). Theoretically, there have been some changes, with Hottel 's original developments now shown to be somewhat oversimplified. In one development [28], it was shown that a high velocity at high temperatures could shift the reaction dominance from diffusional to chemical absorption. In other theoretical develop­ments [26], the elaboration of the Zone concept showed the importance of internal reaction inside the particles themselves by micropore diffusion, so the available surface area for reaction can be substan­tially greater than the superficial area. The most recent experimental results [29, 30] have determined the reaction order in this low-temperature region

at extinction

Heat gene rati on c urve

c o ..... w co .... OJ C OJ <!> w co OJ :r:

Section of Qr curve with chemical kineti c behav10r overriding diffusional kinetic behavior

T o

Fig. 4

Extinction temperature

System Temperature - T

Qualitative Illustration of Heat-Generation and Loss Curves (Or and OIl) for a Solid Bed (Semenav Plot) with Change in Reaction Mode Included and Heat- Loss Curve at Extinction

284

(Zone I) to be zero (although CO burnup can change the bed temperature) in contrast to Hottel 's first­order assumption.

These data account for the primary shape of the kinetic-dominant region of the curve in Fig. 4, which should also be velocity independent; however, its location can be expected to change with variation in real intrinsic reactivity, which is a function jointly of decomposition-activation energy, frequency factor, and active internal surface. As the temperature rises, the Zone-I mode of reaction, with total oxygen pene­tration into the solid, changes to Zone II, with only partial penetration. The reaction order is then 0.5, and the activation energy is halved. The Zone-II region, however, is often quite narrow particularly at low velocity (although Froberg [29] found that this was the region where extinction would occur) so to a first approximation we may regard the reaction changing discontinuously from Zone I to diffusion� dominated Zone III. Under these conditions, the change has been shown theoretically to be discon­tinuous [28], and this accounts for the discontinuity of slope in the Fig. 4 illustration.

Extinction

Figure 4 provides the anticipated qualitative data for prediction of extinction. We have only to note that the peculiar shape of the heat-generation curve, with its discontinuity of slope at the intersection of Zone I (or Zone II) with Zone III reaction, requires a slight modification to the second of the critical points criteria. We cannot obtain a tangency of two curves. In place of this, we require only that the heat-loss curve pass through the Zones intersection point, with the whole of the Qr reaction curve lying to the right of the Qu loss curve.

Now consider a bed just maintained on or barely abov� the extinction point. If the air rate is decreased, the QIl line will rotate clockwise, allowing, initially, an intersection on the Qr line if this does not move at a higher temperature. At the same time, however, the Qr line will move down, so that the intersection will vanish and extinction will result. If the air rate is in­creased instead, the QIl line will rotate counterclock­wise. The Qr line will initially move up, maintaining an intersection between the two at an initially in­creasing temperature. Ultimately, however, the air-rate term in the exponent of Eq. (36) will become over­riding. The heat-production rate will drop (as shown by Fig. 3), and again extinction will ultimately result. The temperature will be somewhat higher than at the previous point (low-velocity or thermal extinction)

285

and will occur on the extension of the kinetic-domin­ance curve. At the higher velocities, however, the Zone-II reaction mode may be significant, but extinc­tion is still likely to occur when the Qu line passes through the Zone II and Zone III intersection on the Qr line.

Experimental Behavior

Experimental test of the equations developed is still somewhat embryonic, but useful operational con­clusions may be drawn from experimental data devel­oped by Moles and Thring [24] using coke beds. Test against refuseochar beds has yet to be carried out since this was self-evidently redundant until exist­ing data had been analyzed.

Existing data are nonetheless sparse. Review of previous relevant work by Moles and Thring [24] sug­gested that this was inadequate for determining mini­mum burning rates, and their own work was designed to provide these. Their unit was a cylindrical com­bustion pot of 0.5 fe grate area with water-cooled walls. A 6-in. deep bed was used with particles that ranged mostly from 0.5 to 1 in. in size. The bed was lit by gas underneath and raised to about the same temperature in each case. The gas was then turned off, and the air rate, adjusted to a fixed level for a given test. At high air rates, the bed temperature would then rise rapidly. At l ower rates, it would rise gradually. At lower rates still, it would fall initially and then very gradually start to rise again. At rates very near the minimum, the temperature could be main­tained more or less constant for as long as 5 h or more. Below the minimum, the temperature would stay constant for an appreciable period and then quite suddenly drop. The wall loss, measured by the water­temperature rise, was also indicative of the critical air rate. Just above the critical rate, the wall loss would initially fall, then recover after 1 or 2 h, and then rise again. Below it, the wall loss fell continuously.

In this way, minimal air rates were measured for four different cokes: a highly reactive semi-coke and three oven cokes, one of which was "unreactive." The critical air rates for the four ranged from less than 1 to 6 lb/h. For the three oven cokes, the wall loss at the critical air rate was about the same at 1000 CHU/h or 800 CHU/ft2 h of cooling surface. In agreement with that result, the bed temperatures were all about the same. Temperatures on the bed axis were measured with thermocouples and were about 725, 650 and 500°c at 1.5, 3, and 4.5 in. above the

grate. These values are consistent with mainly com­bustion and some initial gasification, and this was supported by the gas analysis of the top gas with mostly CO2 , at 14 to 18 percent, some 02

' 2 to 6

percent, and less CO, 0.5 to 2 percent. A particularly interesting qualitative result was

the observation, from preliminary experiments, that a combustion pot with refractory walls provided too little thermal load for extinction. In Fig. 3, this corresponds to an intersection very near the origin, and, apparently, there was always sufficient air inleakage to maintain the bed temperature with the bed in a smoldering condition at the very least. An equivalent situation was also observed in larger (2 or 3 fe ) combustion pots with water-cooled walls where reaction of the coke adjacent to the walls was extinguished, but this provided a thermal insulation for the center of the bed, which stayed lit. In passing, we may note that burning refuse heaps from coal mines are parallel cases where adequate combustion for high temperatures is maintained by fairly slow diffusion of air into the heap.

Comparison of Theory and Experiment

The data obtained by Moles and Thring are suffi­cient to provide an approximate check between calcu­lation and experiment. For this purpose, the equations developed may be substantially simplified. The gas analyses indicate that e is about unity and {res is about 0.5. Taking B as 7000 CHU/lb and Pm as 0.23, we may write QI = 605 G. Taking cp as 0.25 CHU/lb °C the temperature rise as 700°C, and the wall loss as

'1000 CHU/lb, we find that QII = 188.5 6 + 1000.

Equating and solving for 6 yields 6 = 2.6 lb/h, which lies right in the range obtained experimentally, of 1 to 6 lb/h. This broadly substantiates the general approach. A more precise check is possible in princi­ple but requires additional measurements that are unavailable at present.

An interesting consequence of the calculation is the order-of-magnitude of the values obtained for the total heat generation and loss. The total generation rate is roughly 1500 CHU/h, of which two-thirds is lost through the walls and only one-third is lost by convection. If the wall loss were minor, the air rate might drop to about 0.8 lb/h. Since l Ib of coke requires about 10 lb of air, this would allow a minimal burning rate of coke of about 0.08 lb/h or 0.16 lb/h fe . Translating this into refuse com­bustion, assuming a Type 2 waste of 50 percent moisture and 10 percent ash, with 80 percent volatile natter on the f'IF material, it should l:-e possible to

286

maintain burning is the bed at a feed rate in the region of 2 lb/h fe , which is an order-of-magnitude lower than upper limits generally quoted (31].

Cone Ius ions

In conclusion, it does seem that the theory devel­oped here is realistic and capable of predicting order-of-magnitude values of minimum burning rates. More accurate prediction will require better under­standing of the real effect of intrinsic reactivity of the material. There is reason to believe, however, that most carbons will show a change in reaction mode from kinetic to diffusional in the region of 600 to 800°C, the values found by Tu, Davis, and Hottel [25] and by Froberg [29] in both cases lie in this range. This wi11 largely determine the extinction tem­peratures, variations between carbons, as determined by a minimum air rate, will probably depend primarily on the packing of the particles (i. e., bed poros ity) and the particle size. This is more evident from Eq. (37). If the porosity and/or the particle size were to be reduced, comparing one bed with another, then the exponent determining the value of {res at a given temperature, air rate, and bed depth would increase, thus increasing {res' At the same time, however, e would drop and QI would be too small. To correct for this, an increase in G will decrease the exponent, reduce {res to the optimal heat-generating value of 0.5, and increase e to its maximum of unity. In agree­ment, therefore, with the conclusion reached in Division I of this paper, the chief variables responsi­ble for variation in "reactivity" would seem to be porosity and particle size.

In applying these results to refuse beds, it would seem that the minimum air rates are about a factor of Hi lower than would be required to meet normal oper­ating standards. This indicates that, in reasonably dry material at least, there should not normally be a problem in maintaining ignition in the bed. If the waste is wet, this can be a different story, although a waste containing 50 percent moisture and fired at 2 lb/h fe would require roughly 950 CHU to evapor­ate the water. Since the surplus heat, from Moles and Thring's experiments [24] , amounts to about 2000 CHU/h fe , this is more than sufficient for drying and burning, although there could be a problem of overbed burn up because of dilution. The margin other­wise would seem to be about 70 to 75 percent mois­ture, which is the level at which the IIA standards require auxilIary fuel.

In general, however, it is evident from this analy­sis that the primary factors determining the capacity

of an incinerator, if no regard is given to completion of overbed combustion, are bed depth and air rate; and, in general, even with quite wet material, there should be little difficulty in maintaining ignition in the bed.

References

[2 1 ] Op. cit., Ref. [ 1 ]. [22] L. A. Vulis, Thermal Regimes of Combustion,

McGraw-Hill Book Co. , New York, N.Y., 1 96 1 . [23] R . H . Essenhigh, " A New Application o f Perfect­

ly Stirred Reactor (P.S.R . ) Theory to Design of Combus­tion Chambers, " paper presented at Seminar on Combustion Engineering, The Pennsylvania State University, Univer­sity Park, Pa.

[24] F. D. Moles and M. W. Thring, "Minimum Burning Rates in Coke Fue l Beds, " Science in the Use of Coal, The Institute of Fuel, L ondon, 1 958, p. E26.

[25] C. M. Tu, H. Davis , and H. C. Hottel, "Combus­tion Rate of Carbon," Ind. Eng. Chem., vol. 26, 1 934, p. 749.

[26] Op. cit . , Ref. [ 1 0]. [2 7] R. H. Es senhigh, 11 th Symposium (International)

on Combustion, The Combustion Institute, Pittsburgh, Pa . , 1967.

E28] R . H. Essenhigh, R. Froberg, and J. B. Howard, "Combustion Behavior of Sma ll Particles , " Ind. Eng. Chem., vol. 57, no. 9, p. 33, 1 96 5 .

[29] O p . cit., Re f. [ 1 1 ]. [30] Op. cit., Ref. [ 1 2 ]. [3 1] I.I.A. Incinerator Standards, Incinerator Institute

of America, New York, N. Y. , May 1966.

ACKNOWLEDGMENTS

We have pleasure in acknowledging the prime financial support of this research from the Depart­ment of Health, Education, and Welfare (Public Health Service), under Contract No. 5R01 AP00397; in addition, we also acknowledge additional support for one of the investigators (M.K.) from the National Science Foundation, under Grant No. GK 858.

A PPENDIX

Improved Approximation for Gas Ana lyses in Sol id Bed

In the integration of Eqs. (20) and (21) to obtain Y " Y 21 and Y 3 as functions of time or distance through the bed, the approximation was introduced, generating Eq. (25), that Y 3 was constant or nearly so over a substantial range of t or L. This assump­tion enabled the first integrations to be performed. The solutions obtained may be improved by using the results obtained for Y , and Y 2 to form Y 3 (by Eq. (17)) and reintegrating using this result for Y 3 in place of the original assumption that it is constant. By repeating the procedure progressively improved approximations may be obtained and taken to any desired degree of accuracy.

287

The procedure may, however, be shortened by assuming arbitrarily a functional variation of Y 3 with distance that more closely approximates the real behavior than does, assuming that l' 3 equals a constant. Inspection indicated that a good first approximation is

(A I)

where Y� is the limiting value at L = 00 and to is some appropriately selected decay constant. Sub­stituting Eq. (AI) in Eq. (21) assuming m = 1, and using Eq. (22), we obtain a modified form of Eq. (25)

dY , /dt = - n[l + (Y�n 3/2n) (l - e-t/ t o )]y , (A2)

Integrating with limits as before yields

(A3)

This solution will be recognized' as being the same as Eq. (25) except for the addition of the second RHS term. If t is small, this second term can be neglected, and the original solution is a good approxi­mation. If t is large, the term Y � n 3 to/2, which is the residue from the second RHS term, is small compared with the first RHS term, and again the original solu­tion is a good approximation. If t is not small but (tlto ) is small, the exponential may be expanded. This adds a term (Y �n 3 t/ 2) to the original solution; but, if (Y �n3/2n) is reasonably small compared with unity, then the additional quantity is again relative­ly unimportant. This would seem to be the reason for the unexpectedly good agreement between theory and experiment for Y , illustrated in Fig. 1, in spite of the sweeping assumption regarding the constancy of Y3•

A deviation still does exist, however, and this appears more explicitly in the comparison of curves for CO2, The effect of the additional term will be to decrease the ratio of rise of CO2 with time (or dis­tance) so the correction is the right direction. This probably accounts for part of the discrepancy; the rest will be largely due tQ temperature.

Finally, if Eq. (A3) is combined with Eq. (24a) to obtain Y 2 and the two are combined in Eq. (17) to obtain Y 3' the result can be compared with the assumed Eq. (AI) to obtain a value of to in reason­ably fundamental terms.