cenkefa-7

Axial transport and residence time of MSW in rotary kilns:

Part II. Theoretical and optimal analyses

S.-Q. Li a,b,*, Y. Chi a, R.-D. Li a, J.-H. Yan a, K.-F. Cen a

aDepartment of Energy Engineering, Zhejiang University, Hangzhou 310027, PR ChinabDepartment of Thermal Engineering, Tsinghua University, Beijing 100084, PR China

Received 17 October 2000; received in revised form 28 December 2001; accepted 31 December 2001

Abstract

A novel particulate trajectory model (PTM) is developed to predict axial transport and dispersion of municipal solid wastes (MSW), based

on the vector analysis on particle’s gravity-induced axial displacement in a single excursion. Three parts of work are extended with respect to

this PTM. First, the simplified formulas about mean residence time (MRT) and material volumetric flow (MVF) are derived by incorporating

statistic-averaged analysis on all repeated excursions of solids within kiln into PTM. The correctional factors—et for MRT and ef for MVF—

are introduced to improve the model’s validity under such practical cases, i.e. irregular MSW existence or internal-structure presence.

Reasonable agreement is obtained between the empirical formulas and experiments with correlation factor in excess of 90% for all runs.

Second, a stochastic PTM is extended to predict the residence time distribution (RTD) curves of segregated MSW by considering the

probability of the rolling distance of individual particle. As for MSW, the main cause of axial dispersion is the segregation of rolling distance

of solids, due to variation of MSW components, shapes and sizes. Finally, the optimization model for geometry design of a laboratory-scale

rotary kiln pyrolyser of MSW is presented and the corresponding optimum solutions are provided.

D 2002 Published by Elsevier Science B.V.

Keywords: Rotary kiln; MSW; Axial transport; Particulate trajectory model; Optimization

1. Introduction

The experimental studies of mean residence time (MRT)

and material volumetric flow (MVF) of municipal solid

wastes (MSW) in a technical-scale rotary kiln simulator

have been conducted in Part I. Modeling on the transport

behavior of MSW in rotary kiln provides not only the basis

to simulate heat transfer as well as apparent kinetic of a

continuous reaction, but also the fundamentals to achieve

optimal geometry design of processing.

Henein et al. [1] first classified various forms of solid

motion through a rotary kiln as slipping, slumping, rolling,

cascading, cataracting and centrifuging beds. Generally, the

slipping, slumping and rolling beds often occur in practical

kiln used for MSW incineration or pyrolysis, which are

operated at relatively low rotational speed. In a slipping bed,

the solid particles slide against the kiln wall and the bed acts

as a rigid body with poor axial and lateral mixing. In a

slumping bed, the solids are carried upwards along the

inside wall and reach the upper limit of the angle of repose,

then slump downward by periodic ‘avalanche’ as a segment

of solids detaches itself from the topper half of the bed to the

lower half. In a rolling bed, the solids flow as a continuous

layer, and the inclination angle of the solids is just the

dynamic angle of repose. Of all three beds aforementioned,

the rolling one is the desired case for most kiln operation

due to well mixing of materials. So far, nearly all models on

the solid transport as well as the heat transfer in rotary kilns

are developed in assumption of rolling bed.

Existing models developed to describe the flow and

mixing behaviors of particles in rotary kiln can be generally

classified into two types. One is so-called axial distribution

model (ADM) assuming the solids as ideal fluid. The other

is the single particle trajectory model (PTM) considering

solids as granular substances. Danckwerts [2] pioneered an

axial dispersion model with an appropriate Pe number

determined by experiments. Fan and Ahn [3] first applied

the similar ADM to predict the residence time distribution

(RTD) and dispersion coefficient of solid materials in a

continuous rotary cylinder. Rutgers [4] verified the validity

0032-5910/02/$ - see front matter D 2002 Published by Elsevier Science B.V.

PII: S0032 -5910 (02 )00015 -3

* Corresponding author. Tel.: +86-10-62782108.

E-mail address: [email protected] (S.-Q. Li).

www.elsevier.com/locate/powtec

Powder Technology 126 (2002) 228–240

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of ADM within a case of uniform radial mixing, and

determined the axial dispersion coefficients under varied

conditions of loading, rotational speed, inclination, bed

regimes, shapes of the kiln entrance and exit end faces,

and bulk characteristics of solids. Abouzeid and Fuerstenau

[5] adopted the same ADM to demonstrate its applicability

but noted three explicit premises: the constant axial velocity,

the radial mixing large enough to smooth the concentration

at any cross-section, and the constant dispersion coefficient

for fixed operating conditions. Wes et al. [6] further adjusted

the ADM in a rotary drum system with the remarkable large

Pe number. Mu and Perlmutter [7] developed a novel ADM,

which took into account some of key flow features of the

solids. It simulated the flow by N rolling steps, each

consisting of a flow through a tube followed by a well

mixed vessel with by-pass and recycle. The model contains

three parameters, one of which has to be obtained from RTD

experiments. In addition, Sai et al. [8] and Groen et al. [9]

attempted to use the N tanks-in-series model to fit exper-

imental RTD curves.

As for the PTM, Saeman [10] originally proposed a

model from the geometric analysis of a single particle

trajectory to predict MRT in a shallow rolling kiln. Vahl

and Kigma [11] and Kramers and Croockewit [12] further

confirmed the model validity in a horizontal kiln or

inclined kilns, respectively. The calculation of hold-up as

well as MRT by the modified PTM was considered in their

works. Roger and Gardner [13] extended Saeman’s PTM

and adopted a Monte Carlo simulation to successfully

predict the experimental RTD and axial dispersion for

granular solids in a horizontal rotary drum with variable

bed depth. Hehl et al. [14] initially described the material

behaviors in the kiln applying ADM and PTM simulta-

neously. The ADM was used to evaluate the dispersion

coefficients in the experiments, while MRT and hold-up

were calculated by means of an extended model of Vahl et

al. One disadvantage of Heh’s model is that ADM and

PTM are independent. The work of Gupta and Khakhar

[15] can be regarded as a breakthrough of PTM since

Saeman’s. Starting from kinematics of solids, they deduced

the stochastic evaluation equation for the probability of

rolling distance of a particle within active layer, which is

further incorporated into Saeman’s model to predict RTD

curves. The model predictions were compared with the

experimental data from Hehl et al. and the reasonable

agreement is reached. Similarly, Kohav et al. [16] devel-

oped several stochastic algorithms evaluating the axial or

lateral position of a particle in each rolling step, and

employed them into PTM to predict RTD or axial dis-

persion. An important advantage of Kohav’s model is that

the radial segregation of solids is accounted for. However,

the validity of model is limited to the uniform bed depth

or filling level of solids. Generally, PTM has received

more attention within the last decade by many scholars,

e.g., Gupta and Khakhar [15], Kohav et al. [16], Langrish

[17], Lebas et al. [18], Spuring et al. [19], Afacan and

Masliyah [20]. More recently, Wightman et al. [21]

emphasized the discrete dynamics and colliding interaction

of particles and attempted to use the discrete element

method (DEM) to simulate the flow and mixing of solids

in rotating and rocking cylinder.

The deficiencies of previous models are: (1) The physical

basis of the trajectory of individual particle in a single

rolling step is all from Saeman’s geometrical analysis but

not from the kinematics of the granular flow itself. (2) The

extrapolation of a model developed from homogeneous

particles to highly irregular MSW rarely happens. (3) The

model modification under some special conditions such as

internal-structure presence is seldom considered. (4) The

effect of the solid’s segregation, especially for MSW in an

inclined kiln, is scarcely considered in PTM. (5) An attempt

to extend empirical formulas for MRT and MVF to the

optimization of the geometry design of the system is absent

so far.

In present study, at first, the vector analysis is originally

adopted to account for the gravity-induced axial displace-

ment in a single rolling step, which forms the physical basis

of PTM. Furthermore, three parts of work are extended

elaborately with respect to PTM. First, the empirical for-

mulas on both MRT and MVF are deduced by employing

the statistical-averaged analyses on all rolling steps of

particles into PTM. Two correctional coefficients, et for

MRT and ef for MVF, are proposed to improve model

applicability under practical condition such as a case of

internal-structure presence. Secondly, a stochastic PTM

incorporating the segregation characteristics of MSW is

developed originally to predict detailed RTD curves in an

inclined, bed-depth-varied, continuous rotary kiln. The

simulated RTD of MSW will be compared to the exper-

imental one. Finally, the optimizing model for the geometry

design of a laboratory-scale rotary kiln pyrolyser is pro-

posed and the optimal solutions are discussed.

2. Description of particle trajectory model

2.1. Dynamics of the rolling bed

Before developing the model it is worthwhile to exam-

ine the dynamics of solid materials in a rolling bed. As for

the rolling regime, the bed material is characterized into

two distinct regions shown in Fig. 1(a,c), i.e. the thinner

active layer and the thicker stagnant region under the

former. Particles roll down continually and form a constant

layer on the bed surface, and then enter the stagnant region

where they are carried upward by the rotating wall until

they re-enter the active layer and roll again. The dynamics

of the rolling bed can be vividly depicted as that the

energy imparted by the kiln’s rotation is continuously fed

to solids in the stagnant region as potential energy which is

subsequently dissipated in terms of rolling within the

active layer.

S.-Q. Li et al. / Powder Technology 126 (2002) 228–240 229

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2.2. A gravity-induced particle trajectory model

With regard to the phenomena above, a particle trajectory

model (PTM) is proposed. The solid motion in rotary kiln

consists of many repeated excursions. Every excursion is

composed of two processes, i.e., the particle rolling in the

active layer, and the particle rotation around kiln axis in

stagnant region that does not cause axial displacement.

Therefore, the axial motion of solids in rotary kiln is solely

determined by the axial particle rolling in active layer.

Meanwhile, the randomness of rolling motion will inevi-

tably result in both radial and axial mixing. In this study, it is

assumed that the rolling of particles in the active layer is

induced only by the gravity. Thus, the trajectory of individ-

ual particle during each rolling step follows the direction

determined by the gravitational component vector parallel to

the solid bed surface, as shown in Fig. 1(b). Then the axial

displacement in a single excursion can be calculated accord-

ing to the vector analysis on its trajectory. As the time of a

single excursion is considered, it is valid that the solids’

rolling time is much shorter than their rising time in the

stagnant region provided the active layer is thin, which have

been admitted by Saeman [10] and Gupta and Khakhar [15].

So the rising time of solids in the rotation is nearly equal to

the time of single excursion, which implies the rolling time

shall be ignored.

If the aforementioned PTM is to be further formulated

empirically or simulated numerically, several premises are

required, that is:

1. Steady-state solids flow conditions prevail.

2. The kiln is operated in the rolling mode.

3. The upper surface of solid bed is a plane and the exit end

effect is negligible.

4. Compared to the residence time in stagnant region, the

rolling time on bed surface is negligible (it is precise

enough under low rotating speed).

2.3. Axial displacement in single step by vector analysis

In a single rolling step, the rolling displacement of solids

in the active layer has the axial component and the lateral

one. The relationship between these two components plays

an important role in the development of PTM, which attracts

intensive research efforts [10–12,15,16].

In the Cartesian coordinate system shown in Fig.

1(a,b,c), we denote a, b and d by the kiln slope, the bed

slope and the dynamical angle of repose of solids, respec-

tively. The gravity vector of individual particle can be

expressed as (cosa sin, cosa cos, sina), the normal vector

of bed surface as (0, cosb, �sinb), and the vector of the

symmetric line of bed surface as (0, sinb, cosb). Thus, v, theangle between the gravity and X coordinate, can be

expressed as

cosv ¼ cosasinhd ð1Þ

And g, the angle between the gravity and the symmetric line

of bed surface is expressed as

cosg ¼ cosacoshd sinb þ sinacosb ð2Þ

We define Ds as the rolling distance of a particle in each

step and its components in direction of X coordinate

(namely lateral component) and that in direction of the

symmetric surface line are Dx and Dm, respectively. Defin-

Fig. 1. Schematic view of a single particle trajectory according to PTM starting from gravity-induced rolling. (a) Overall view, (b) simplified gravity analysis,

(c) side view at cross-section, (d) planform of bed surface.

S.-Q. Li et al. / Powder Technology 126 (2002) 228–240230


ing u as the angle between Ds and Dx, its tangent function is

obtained by dividing Eq. (2) by Eq. (1),

tanu ¼ cosgcosv

¼ cosbtanasinhd

þ cothdtanb

� �ð3Þ

The particle’s axial displacement is given as Dz=Dmcosb.Since tan u=Dm/Dx, the axial displacement in one excursion

can be expressed as

Dz ¼ Dxcos2btanasinhd

þ cothdtanb

� �ð4Þ

Similarly, Saeman [10] and his successors [11–16] have

obtained this relationship all from the simplified geometric

consideration as

Dz ¼ Dxa

sinhdþ bcothd

� �or

Dz ¼ Dxtanasinhd

� cothddh

dz

� �ð5Þ

In Eq. (4), cosb approaches to 1 in the practice, and tan bis just the alternative term of �dh/dx. It indicates that our

relationship from the vector analysis on the gravity-induced

rolling is not only consistent with that of Saeman’s, but also

more reasonable from a point of view of the kinematics of

granular solids.

3. Development of simplified formulas and its validation

We first focus on the development of the simplified

formulas of both MRT and MVF on the basis of PTM.

3.1. Simplified formula of MRT

When lightly loaded kiln (viz. the fill level of solids is

low) is considered, the statistic-averaged method can be

introduced here to derive the formulation of MRT. If the

number of the excursion of solids through whole kiln is

signified as k, the kiln length is equal to

L ¼Xki¼1

Dzi ¼Xki¼1

cos2btanasinhd

þ cothdtanb

� �Dxi ð6Þ

In any specific ist step, the lateral component of rolling

distance, Dxi, is composed of the upper half, xiV, and the

lower half, xiW, of the bed surface (Fig. 1d). Then,

L ¼ cos2btanasinhd

þ cothdtanb

� �Xki¼1

ðxiVþ xiWÞ ð7Þ

One remarkable feature of the rolling mode is that the

position of particle entering into the stagnant region from

the active layer is stochastic because of the rolling step’s

randomicity, meanwhile the position of particle re-entering

the active layer is certain due to the rotation step’s

symmetry. Thus, it is concluded that the lower part of

Dxi in the i step is just the upper part of Dxi+1 in the i+1

step. That is,

xViþ1 ¼ xWi for i ¼ 1; 2: : :k ð8Þ

As was stated earlier, the residence time of one excursion

is mainly determined by the rotation time of solids in the

stagnant region. For the ist excursion, the residence time is

expressed as

ti ¼/r

2pn¼ sin�1ðxiW=rÞ

pnð9Þ

where n is the rotational speed and r the radius of particle

rotation. With respect to low filling ratio of solid bed, Eq.

(9) simplifies to

ti ¼sin�1ðxWi=rÞ

pnc

xWiprn

ð10Þ

So the overall mean residence time through the whole kiln

can be presented as

MRT ¼Xki¼1

ticXki¼1

xWiprn

ð11Þ

Taking Eqs. (7–8) into Eq. (11), we obtain

MRT ¼ Lsinhd2prnðtana þ coshdtanbÞcos2b

þ 1

2prnðxV1 � xWkÞ ð12Þ

where xV1 and xWk are both the random number at age

½0;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðR� hÞ2

q� . And the second term in the right of Eq.

(12) is far smaller than the first one and can be ignored if k is

big enough ( > 1000). Then

MRT ¼ Lsinhd2prnðtana þ coshdtanbÞcos2b

ð13Þ

The rotating radius of particle in every excursion is a

random variable related to the rolling distance, Dxi, and the

bed depth, h. Since r is nearly equal to kiln radius R for

lightly filled kiln, R is chosen instead of r as a parameter in

Eq. (13) to predict MRT in practice.

3.2. Simplified formula of MVF

The averaged axial velocity of solids in a single excur-

sion can be expressed by the ratio of L to MRT, then we

get

u ¼ 2prncos2btanasinhd

þ cothdtanb

� �ð14Þ



The increment of area in a cross-section in the ist

excursion can be given as

dA ¼ 2rsin�1 xWir

� �dr ð15Þ

Then multiplying Eq. (14) by Eq. (15) and taking account of

Eq. (10), we get

udA ¼ 2prnð2xWiÞcos2btanasinhd

þ cothdtanb

� �dr ð16Þ

where xWi is given in terms of

xWi ¼ ðr2 � ðR� hÞ2Þ1=2 ð17Þ

The materials flow rate at any axial position can be

expressed as

MVF ¼Z R

R�h

udA

¼ 4

3npcos2b

tanasinhd

þ cothdtanb

� �ð2Rh� h2Þ3=2

ð18ÞFor any rotary kiln operated under steady-state condition,

MVF is uniform at any axial position. However, we find that

the last term in Eq. (18), i.e. (2Rh�h2)3/2, decreases non-

linearly with coordinate Z. Thus, it is concluded that the bed

slope angle b is not a constant along the kiln axis, and the

closer the kiln outlet end is, the larger the slope b is.

Consequently, the approximation of b as mono-value is

very helpful to the application of the empirical formulations,

i.e., Eqs. (13) and (18). Generally, b can be substituted by

the averaged bed slope determined by the difference

between the inlet and outlet depth, that is,

b� ¼ ðh0 � hexÞ=L ð19Þ

In addition, the bed depth (h) in Eq. (18) can be replaced by

the inlet bed depth (h0) in practical cases, since MVF in any

axial point is uniform.

3.3. General formulas with corrected factor

The practical operation of rotary kiln may differ from the

assumptions or ideal cases of PTM, of which the most

common case is the application of internal structures. The

studies of Afacan and Masliyah [20] showed that the

predicting results by PTM have a great discrepancy with

the experimental ones for a case with axial ribs and end

constriction. It is because the mechanism of PTM only

considers the particle–particle interaction in terms of

dynamic angle of repose, but ignores the substantial inter-

action between the particles in the rolling bed and the axial

ribs in the bottom of the drum (Langrish [17]). Thus, the

modification on Eqs. (13) and (18) is inevitable in order to

broaden the validity of our model in such a special case.

Otherwise, the desired rolling mode may be taken place by

the slumping modes for a case with a relatively low rotation.

The loading of system may not always meet the requirement

of the lightly filling of solids. These cases can also result in

the prediction of deviations of PTM, although they are less

appreciable than those arising from internal structures.

Therefore, two correctional coefficients et and ef are intro-

duced in our paper, which can be evaluated by the linear

least-squares fit of experimental and theoretical data. Then

the modified formulas are given as,

MRT ¼ etLsinhd

2pRnðtana þ coshdtanb�Þcos2b�

ð20Þ

MVF ¼ 4ef3

npcos2b� tana

sinhdþ cothdtanb

��

� ð2Rh0 � h20Þ3=2 ð21Þ

It seems that et should be the reciprocal of ef from the

derivation of the model. However, it is not the case in fact.

The explanation lies in that the et and ef have other differentcorrecting content besides the joint modification (namely

modification to internal structures, bed modes and bed

slope), e.g., the correction to the substitution of r by R is

considered in et, and that to the replacement of h by h0considered in ef.

3.4. Comparison with experimental results

3.4.1. Cases for the smooth wall without end-constriction

In previous PTM studies, many researchers have adopted

small-size regular materials to verify the validity of the

model [10–13,15]. Here, the importance is attached to the

large-size heterogeneous MSW. Fig. 2 presents the compar-

ison between the experimental and the predicted MRT of

MSW. Reasonable agreement between them is obtained.

The correctional coefficient, et, is about 1.04 and the

correlation factor, R2, is more than 99%. As shown in Fig.

3, similar conclusion on MVF can be drawn. The corre-

Fig. 2. Comparison of MRT by Eq. (20) and experimental one (MSW,

smooth wall).



sponding ef and its R2 are about 1.03% and 93.9%, respec-

tively. By the way, Yang’s [22] investigation also showed

that the equation derived from the single particle trajectory

model of Saeman was applicable to predicting MVF of

irregular MSW with very small modification. We tried to

use our Eqs. (20) and (21) to compare with Yang’s exper-

imental data. Similarly the correctional coefficients are

closed to 1, either. Accordingly, it is concluded that the

empirical formulas, Eqs. (20) and (21), can predict both

MRT and MVF quite well for the case of irregular MSW

only with minimum modification, in spite of the different

kiln geometry size.

3.4.2. Cases for internal-structures existence

Figs. 4 and 5 successively show the comparisons

between experimental and predicted results of MRT and

MVF when an internal-structure group labeled 12n–4n is

employed. Although MSWand sand have distinct difference

in the physical or rheological characteristics, their modifi-

cation et as well as ef are quite proximate, which implies that

the influence of internal structure on solid transport is

insensitive to the variation of the feeding materials.

The correctional coefficients of MRT and MVF and their

correlation factors under the various operational conditions

are presented in detail in Table 1. It is noted that the values

of correlation factors more than 90% in all runs reflect the

good linearity between the predicting and measured data,

which further implies the broad applicability of our model to

any special conditions such as internal-structure employ-

ment. When it comes to the industrial-scale rotary kilns, the

current formulas Eqs. (20) and (21) can be precisely applied

as the fundamental to fulfil the optimization of the system,

provided et and ef are regressed by the linear least-squares fit

of a spot of experiments (particularly, et and ef approach one

for an end-opened system).

Moreover, as presented in Table 1, the influences of

internal structures on both MRT and MVF can also be

theoretically explained, which have already been discussed

in part I only from an experimental point of view. For

example, as the height of axial ribs in a certain internal-

structure group increases from 10 to 20 mm (12n–4n!12b–4n), the correctional coefficient for MRT, et, in-

creases only by 3.8–5.0%. As the number of circular ribs

rises from 4 to 7 (12b–4n!12b–7n), the increment of etis 2.4–4.1% for both MSW and sand. However, as height

Fig. 4. Comparison of MRT by Eq. (20) and experimental one (12n–4n

employed).

Fig. 5. Comparison of MVF by Eq. (21) and experimental one (12n–4n

employed).

Table 1

Correctional coefficients et, ef and corresponding correlation factors in

all runs

Case Material et ef

et Correlation

[%]

ef Correlation

[%]

Smooth wall MSW 1.04 99.6 1.03 93.9

12b–4n MSW 1.64 99.0 0.846 98.0

sand 1.67 96.4 0.865 90.1

12b–7n MSW 1.68 93.2 0.771 97.8

sand 1.74 99.6 0.804 98.1

12n–4n MSW 1.58 96.7 0.930 98.4

sand 1.59 97.9 0.905 93.0

12b–4b MSW 1.81 99.2 0.690 97.7

sand 1.83 98.0 0.752 96.3

Fig. 3. Comparison of MVF by Eq. (21) and experimental one (MSW,

smooth wall).



of circular ribs increases from 30 to 50 mm (12b–4n!12b–4b), et increases remarkably by 9.6–10.4%. Thus, it

can be concluded that the effect of the height of circular

ribs is most significant in an internal-structure group. As

far as ef for MVF is concerned, similar conclusion can be

drawn.

4. Development of stochastic PTM and its simulation

So far, we have presented the simplified formulas of

MRT and MVF by using the statistical-averaged analysis

on PTM. But the detailed residence time distribution is

not yet predicted. As mentioned earlier, the most remark-

able feature of PTM is that the position of particle

entering stagnant region from active layer is stochastic

due to the rolling step’s randomicity, while the position of

particle re-entering active layer is certain due to the

rotation step’s symmetry. As for a ternary-mixture system

here, this randomicity should include not only the random

collision of particles (so-called axial mixing), but also the

selectivity of the trajectory of the different particle com-

ponents of mixture (so-called radial and axial segrega-

tion). From studies of Gupta et al. [23], Boateng and Barr

[24] and Bridewater [25], it is doubtless that the trajectory

segregation of particles will play much more important

role in determining the distribution of residence time than

the randomness of particle collision in such a case. So the

randomness of the collision is ignored here in order not to

introduce the excrescent empirical parameter, which had

been considered by Gupta et al. [15] for uniform par-

ticles. The trajectory segregation of different particles

really means the selectivity of the rolling length of

individual particle in the active layer. Thus, if the prob-

ability distribution function of the end point of rolling

distance (e.g., xiW, see Fig. 1) is given, a random PTM

can be extended to predict RTD curves.

4.1. Calculation of single particle’s residence time

In this paper, the stochastic PTM was developed as a

computer program, see Fig. 6 for the Flow-sheet. The

main idea of random PTM is to introduce the certain

number of tracers into a steady-state rotary kiln, calculate

the residence time of individual tracer one by one and

conduct the statistic analysis on the distribution of all

tracers. The calculation of the residence time of individual

particle plays an important role, which consists of five

main procedures defined as initialization of the particle

position at kiln inlet, judgement of the upper half loca-

tion, selection of probability distribution of rolling dis-

tance, rolling step calculation and rotation step calculation,

respectively.

In our program (Fig. 6), as a tracer enters a steady-

state kiln, its initial axial position (z) as well as radial

position ( y) are a certain value, and they are equal to zero

and R�h0, respectively. However, the initial lateral loca-

tion is a random value, which is given as,

xV¼ x0 unifrndð�1; 1Þ ð22Þ

where x0 is the half length of active layer surface, and

unifrnd (�1, 1) is a uniform random number in the range

(�1, 1). Subsequently, we shall judge whether the initial

location of tracer is on the upper half of bed surface.

Once the tracer locates at the lower half surface, a

rotation step will conduct and the residence time of the

tracer begins to be accumulated according to Eq. (9), and

then particle will arrives to its symmetrical position at the

upper half surface.

Then the tracer will roll on the active layer. The

determination of the end-point of rolling step (i.e., the

position that tracer enters stagnant region from active layer,

xW) is a key element for developing the random PTM.

According to previous studies, xW shall be a random number

subjected to a specified distribution. In order to find the

valid probability distribution for xW, the effect of segregationof materials used must be considered, which is discussed it

in detail as follows.

4.1.1. Probability distribution for xW with ideal materials

If materials used are ideal homogeneous particles of the

same type, size and rheological property, there is no

segregation in kiln. The probability distribution for xW is

easily obtained by considering the volumetric fraction of

particles that enter the stagnant region from the active layer

in the range (xW, xW+dxW) at a particular axial position (see

Fig. 7). As the kiln rotates an angle interval, y/r, we have

pðxW; xWþ dxWÞ ¼ x0dxWdzy/r

ðx20=2Þdzy/r

¼ 2xW

x0

� �d

xW

x0

� �ð23Þ

where the half length of active layer x0 changes with respect

to coordinate z. It is apparent that the probability for xW/x0 isa specified term of Beta(a,b) distribution with factors a=1

and b=2 (Gupta and Khakhar [15]). Thus, xW can be

obtained by,

xW ¼ x0Betarnd ð2; 1Þ ð24Þ

where Betarnd (2,1) is a random number subjected to Beta

(2,1) distribution.

4.1.2. Probability distribution for xW with segregated mate-

rials

For the most general cases, materials used have different

sizes or types, especially for the case of MSW pyrolyser/

incinerator. Therefore, the effect of the segregation on

probability distribution for xW should be considered. See

Fig. 8, Boateng and Barr [24] proposed that the radial

segregation of materials tends to concentrate finer particles

within the core and hence cause the trajectory segregation of

different sized particle. That is, the smaller particles will



Fig. 6. Flow-sheet of numerical simulation of RTD by stochastic PTM.



enter stagnant region with the smaller xW (namely the end of

rolling distance), while larger ones will enter with the large

xW. Due to segregation, xW is dispersed around a point x�Wwith a variance r0 that decreases with an increasing segre-

gation level of MSW. Because of the lack of a predictive

model about relations between the segregation with the

variation of particle properties, we assume that x�W is

symmetric to the particle start point of rolling, xV. So in

our model different xV correspond to particles with different

flow properties. Thus, It can be inferred that the term of

(xW�xV)/r0 can be described as a random number subjected

to a truncated standard Gaussian distribution. The Gaussian

distribution is truncated in order to limit xW in the range

(0,x0). Normalizing variance r0 as rs with respect to x0, then

xW can be obtained as,

xW ¼ xVþ rsx0 Normrndð0; 1Þ að0; x0Þ ð25Þ

where Normrnd(0,1) is a random number subjected to

standard Gaussian distribution.

As long as the start and end of rolling distance, xV and

xW, is formulated, the value of rolling distance can be

determined by Dx=xV+xW. Then the calculation of rolling

step is conducted, in which the axial displacement z of a

particle is obtained by Eq. (4). One advantage of the

model presented here over the previous ones (e.g., Kohav

et al. and Gupta et al.) is that the varied bed depth profile

(i.e., the bed slope) is incorporated into random PTM.

That is, the fill level of solids along kiln axis varies, which

leads to the variation of the trajectory of particles at

different cross-section. According to previous studies on

bed depth profile (Kramers and Croockewit [12] and

Lebas et al. [18]), the bed slope is given in terms of a

differential equation,

tanb ¼ dh

dx¼ tana

coshd� 3MVFtanhd

4pnR3

2h

R� h

R

� �2" #�1:5

ð26Þ

As the fill level of solids is low (e.g., h0/R<0.5), the

formula can be altered by,

tanb ¼ dh

dx¼ h0 � hex

Lð27Þ

As bed slope is solved by above formulas, the particle’s

coordinate y and the half length of active layer x0 at

different axial position z can be accounted for, as given in

Flow-sheet by Fig. 6.

Finally, the calculation of rotation step is conducted, in

which the residence time of single tracer is acquired by

accumulation of Eq. (9). It is noted that only the time during

rotation step is accumulated while the time during rolling

step is ignored in our work, the feasibility of which has

already been proved by other researchers [10,15,16]. After

rotation step, the tracer will reach the symmetric point of xW,namely xV=xW. So far, the tracer completes an intact excur-

sion. Again a new rolling step occurs, so does a new rotation

step. The procedure is repeated until the tracer exits the kiln

(z>L). In common case, the residence time of a tracer

through whole kiln can be gained after experiencing about

more than 1000 excursions.

4.2. Statistic analyses on RTDs

The residence time of each tracer is calculated by

aforementioned method and registered as ti. Denoting the

number of tracers introduced by N, the mean residence time

and relative variance is defined as,

MRT ¼ 1

N

XNi¼1

ti ð28Þ

rr2 ¼ 1

N � 1

XNi¼1

ðti �MRTÞ2=MRT2 ð29Þ

Thus, the detailed RTD curve can be fitted by,

f ðtÞ ¼ 1ffiffiffiffiffiffi2p

prrMRT

exp � ðt=MRT� 1Þ2

2rr2

!ð30Þ

As soon as RTD of solids is determined by numerical

simulation, the dispersion coefficient in terms of Peclet

Fig. 7. Schematic of volumetric fraction of particles entering stagnant

region at xW.

Fig. 8. Schematic of radial segregation of particles within a cross-section

of kiln.



number, namely the ratio of convection to diffusion times,

can be evaluated by the relative variance rr,

rr2 ¼ 2

Pe� 2

Pe2½1� expð�PeÞ� ð31Þ

And for large Pe (>100), the approximation

Pe ¼ 2

rr2

ð32Þ

is applicable. The equation above indicates that rr can

take place of Pe to evaluate the axial dispersion of solids

in kiln.

4.3. Simulation results

Numerical simulation was carried out to determine

detailed RTDs as well as s and rr of RTDs. In most practical

rotary-kiln reactors such as waste incinerator/pyrolyser, the

kiln is usually operated with a slight inclination from the

horizontal and its outlet end is open without dam. In such

case, both the kiln slope and bed slope should be taken into

accounted in stochastic PTM. However, so far the research on

PTM considering both these two aspects simultaneously is

absent. In our simulations, the kiln outlet is open for all runs

(hex/R=0), which implies the bed slope can not be ignored. In

addition, unless otherwise stated, we set L/D=6, a=2.40, n=4rpm and d=48.5j for all runs, among which the last one is in

accordance with the dynamic angle of repose of MSW.

The simulated RTD densities of ideal particles and

segregated particles, as well as the normal distribution fit

curves, are illustrated in Fig. 9. It can be seen that the RTD

of ideal particles is well fitted by the normal distribution

density function with a low relative variance (rr=0.0075).As it comes to the segregated particles, the trajectory

segregation in terms of the variance of rolling distance

(rs) due to variation in particle flow properties will deter-

mine the residence time distribution. The reduction of rs

(namely the increment of radial segregation of solids)

causes not only the great increment of rr of RTD, but also

increasing discrepancy of the solid’s RTD with the normal

distribution fit curve. That is, as rs varies from 0.5 to 0.1,

rr increases by more than 200% while MRT increases by

less than 1%. It is inferred that the solids’ segregation in

kiln has effect on rr but not on MRT. Moreover, the

simulated RTD for the case of rs=0.1 has a relative great

discrepancy with the normal distribution fit curve, as shown

in Fig. 9.

In order to verify the validity of our stochastic PTM for

segregated MSW, the comparison between the numerical

simulation and the experiment is performed. In this simu-

lation, the segregation factor, rs, is regressed as 0.1 by the

linear least-squares fit between experimental rr and simu-

lated rr of RTD. In addition, the results of MRT calculated

by simplified Eq. (20) are also attached. As shown in Fig.

10, the simulated MRT from stochastic PTM agrees quite

well with the experimental one, while the simplified Eq.

(20) predicts a slightly shorter MRT than experiments. It is

because that the simplified Eq. (20) is developed under the

premise of the lightly loaded kiln. Fig. 11 shows the

comparison between experimental and simulated rr or r.The reasonable agreement is reached, especially taking

account of the high measuring error of rr or r due to the

high MSW segregation.

Fig. 9. Residence time distribution obtained from stochastic PTM (a=2.40j;n=4 rpm; h0/R=0.5; hex/R=0; hd=48.5).

Fig. 10. MRT from both PTM and experiments with various rotation

speed.

Fig. 11. Variance from both stochastic PTM and experiments with various

rotation speed.



To further discuss the predicted difference of MRT

between stochastic PTM and simplified Eq. (20), the sim-

ulations under various inlet bed depths are conducted. Fig.

12 shows the comparison between MRT obtained from

various segregated stochastic PTM and that obtained from

simplified Eq. (20). For all models, MRT decreases with the

increment of the inlet bed depth (h0/R). It can be explained

that the increasing h0/R causes the increment of bed slope

(here, kiln outlet keeps open, hex/R=0), and thus the incre-

ment of axial displacement in a single step (see Eq. (4)),

which finally results in the reduction of residence time.

Moreover, the simplified Eq. (20), developed for lightly

loaded kilns, predicts shorter MRT than all stochastic PTM.

The higher the value of h0/R is, the greater the difference

between them is. The high h0/R implies the high fill level of

solids in kiln. Therefore, the simplified Eq. (20), assuming

the kiln to be lightly loaded, will differ greatly from the

stochastic PTM under the case of high h0/R. The conclusion

is drawn again that the solid’s segregation does not influ-

ence MRT apparently, though it exerts a great influence on

the dispersion of RTD (shown in Fig. 13). As rs varies from

0.7 to 0.1, rr increases from 40% to 280%. Furthermore, the

higher the value of h0/R is, the higher increasing rate of rr is

and the more violent dispersion of RTD prevails.

5. Appliance of PTM in optimization

5.1. Optimizing model development

According to simplified Eqs. (20) and (21), the optimiz-

ing geometry design model of rotary kiln pyrolyser can be

developed. The object function of the optimization is the

minimal space occupied or the minimal consumption of

steel. The constraints are imposed by the practical operating

variables (rotational speed and kiln slope), the permits of

desired MVF, the permit of the complete pyrolysis time, the

design of the kiln (exit dam and internal structures), the

limits of heat transfer, and the experiential limitation of L/R

of the kiln. Concretely, the fill ratio of solids in kiln is

generally less than 20% in order to keep the high efficient

heat transfer or energy utilization [26]. The complete

pyrolysis time is about 45–60 min by experiences, which

shall be less than MRT [27]. The ratio of the outlet to inlet

depth, hex/h0, replaces the average bed slope, b�, in the

optimization model. Then the optimizing geometry design

model of rotary kiln is described as,

min p R2 L

s:t: MVFðR; L; a; n; h0; hex=h0Þ ¼ fre

MRTðR; L; a; n; h0; h0=hexÞ ztre

alVaVau

nlVnVnu

Cðh0;RÞV0:2

hex=h0V1

8VL=RV12

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

ð33Þ

where C represents the fill ratio at inlet end and is expressed

in terms of h0 and R,

C ¼ 1

pcos�1 R� h0

R

� �� ðR� h0Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2Rh0 � h20Þ

ppR2

ð34Þ

In Eqs. (33) and (34), the dimensions of MVF, MRT and n

are l/min, min and rpm, respectively. And the international

Fig. 12. MRT obtained from stochastic PTM and empirical formula with

various inlet bed depths.

Fig. 13. rr obtained from random PTM with various inlet bed depths.

Table 2

Optimal solution to the scale design of rotary kiln to dispose solid waste

with various tre (MVF=0.5 l/min)

MRT

(tre)

[min]

n [rpm] a [rad] h0 [m] hex/h0 C L [m] D [m] H

20 1.000 0.0300 0.0741 0.5000 0.2 1.8006 0.2917 0.14

25 1.0000 0.0285 0.0719 0.4221 0.2 1.4206 0.2828 0.14

30 1.0002 0.0300 0.0728 0.4435 0.2 1.6626 0.2864 0.14

35 1.0000 0.0277 0.0753 0.5000 0.2 1.8091 0.2965 0.14

40 1.0039 0.0109 0.0853 0.4961 0.2 1.6132 0.3357 0.14

45 1.0000 0.0106 0.0869 0.5000 0.2 1.7494 0.3419 0.14

50 1.0000 0.0100 0.0885 0.5000 0.2 1.8738 0.3483 0.14

55 1.0000 0.0100 0.0895 0.5000 0.2 2.0125 0.3524 0.14

60 1.0000 0.0100 0.0905 0.5000 0.2 2.1488 0.3563 0.14



standards are adopted for the rest. The upper and lower

limits of a are 0.01 and 0.03, respectively. The rotating

speed is limited from 0.5 to 8 rpm.

5.2. Discussions on the solution of optimization

Eq. (33) is a single-objective, constrained and nonlinear

optimization problem. Subsequent quadratic program (SQP)

in Matlab Toolbox can be employed to solve it [28]. The

optimum geometry design of rotary kiln pyrolyser can be

considered starting from two aspects. On one hand, MSW is

generally composed of various components such as woods,

plastics and papers, each of which has a different tre. When

MVF is given as 0.5 l/min, the optimum solution to the

geometry design of rotary kiln with various tre is presented

in Table 2. On the other hand, rotary kiln is sometimes used

to pyrolyze the unitary wastes such as scrap tyre, which has

the common tre. So the optimum solution to the design of

rotary kiln under varied MVF is presented in Table 3.

From above Tables, it can be seen that the fractional

hold-up (i.e. overall fill ratio) in terms of H=MRT�MVF/VR

keeps as a constant about 0.14 during the optimization (this

value is also calculated as 0.143 by Hehl et al.’s empirical

formula [14]). Thus it is concluded: (1) the reactor volume is

only dependent to the products of MRT and MVF for a

given inlet G(h0,R); (2) a, n and L/D are just the flexible

adjustable parameters to meet the required MRT or MVF.

The conclusion is absolutely much meaningful to the scale-

up of rotary kiln. By the way, we have already designed and

built a technical-scale rotary kiln pyrolyser of waste tyre on

the basis of above results, which is given in detail in

literature [29].

6. Conclusion

The vector analysis is originally adopted to account for

the gravity-induced axial displacement in a single rolling

step, which promotes the physical basis of PTM pioneered

by Saeman. Then, three parts of work based on the PTM are

extended intensively, as follows.

(1) The statistical-averaged analysis on PTM is intro-

duced for lightly filled kiln. The simplified formulas on both

MRT and MVF are deduced, which are presented as Eqs.

(20) and (21), respectively. Two correctional, factors et andef, are proposed to improve model validity under practical

conditions such as internal-structure presence. Good agree-

ment is obtained between the empirical formulas and experi-

ments with correlation factor in excess of 90% in all runs.

(2) A stochastic PTM incorporating the trajectory segre-

gation of MSW (rs) is developed originally to predict

detailed RTD curves in an inclined rotary kiln, in which

bed depth varies. The main cause of the dispersion of RTD

is the segregation of particle rolling distance in a single

excursion, due to variation of MSW flow properties. As rs

changes from 0.5 to 0.1, rr increases by more than 200%

while MRT varies by less than 1%. The simulated RTD

from stochastic PTM fits quite well with the experimental

one, while the simplified Eq. (20) predicts a slightly shorter

MRT than experiments. The predicted difference between

stochastic PTM and simplified equation is related to the fill

level of solids, namely the bed inlet depth h0/R for an end-

opened system.

(3) The optimizing models for the geometry design of a

laboratory-scale rotary kiln pyrolyser are proposed and the

optimal solutions are discussed in this work. These results

have given author many useful references in the design and

development of a technical-scale rotary kiln pyrolyser of

waste tyre.

Acknowledgements

This research was supported mainly by Nation Natural

Science Funds of China (No. 50076037) and partially by

Zhejiang Provincial National Science Funds of China (No.

RC99041). We are grateful to Dr. A.-M. Li for helpful

discussions about rotary kiln transport processes. The

contributions of Dr. J.T. Huang and Z.X. Zhang to this

work is gratefully acknowledged.

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