Powder Technology 230 (2012) 29–35

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Powder Technology

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Exact analytical solution for the bed depth profile of solids flowing in a rotary kiln

Mohammad Danish, Shashi Kumar, Surendra Kumar ⁎Department of Chemical Engineering, Indian Institute of Technology Roorkee, Roorkee‐247667, Uttarakhand, India

⁎ Corresponding author.E-mail address: skumar@iitr.ernet.in (S. Kumar).

0032-5910/$ – see front matter © 2012 Elsevier B.V. Alldoi:10.1016/j.powtec.2012.06.042

a b s t r a c t

a r t i c l e i n f o

Article history:Received 7 March 2012Received in revised form 22 June 2012Accepted 23 June 2012Available online 29 June 2012

Keywords:Granular materialSolid bed depthExact analytical solution

In this communication exact analytical solution of the well known Saeman's model equation, which is used tofind the bed depth profile of the granular material flowing in a rotary kiln, has been obtained. Mathematicallythis model equation is characterized by a first order nonlinear ordinary differential equation and the exactanalytical solution has been obtained in terms of an implicit form of elementary transcendental functionby following the methods of separation of variables and partial fraction decomposition. To the best of theauthors' knowledge, this is perhaps the first attempt towards obtaining the exact analytical solution for theabove model equation. The results obtained by using the exact analytical solution have been successfully ver-ified against those obtained by using the numerical method and are found to be superior to those obtained byusing the existing approximate analytical solutions. Besides, the exact analytical solution has also been fruit-fully employed to simulate various experimental results available in literature. Furthermore, the effects ofvarious parameters on the bed depth profile have also been briefly analyzed.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Rotary kilns are widely used in the chemical and allied industriesfor performing various operations on the granular solids e.g. mixing,heating and gas–solid reactions. Some of the processes in whichthese operations occur are calcination and sintering. Rotary kilns arebasically a type of reactor in which solids travel down the cylinderwith the hot gas flowing co-currently or counter-currently. The heattransfer takes place from the hot gas to the solid particles and theabove different operations take place simultaneously. The heat trans-ferred to the solids in the kiln strongly depends on the residence timeand hold-up in the kiln, and these in turn are dependent on the beddepth profile. Therefore, to design a new system or to judge the per-formance of an existing one it becomes essential to accurately predictthe solid bed depth profile. The solid bed depth profile is influencedby several factors: the dimensions of the kiln (L, R and h0), the oper-ating conditions ( _m, n and β) and the properties of the solids (ρ and θ).Once the bed depth profile is known the other secondary quantitiese.g. fractional hold-up can easily be found.

This paper mainly focuses on obtaining the exact analytical solu-tion of the popular Saeman's model equation [1], which is widelyused for predicting the bed depth profile of solids flowing in a rotarykiln. Saeman's model equation is governed by a nonlinear first orderODE (ordinary differential equation) constituting an IVP (initial valueproblem). Unfortunately, the presence of nonlinearity in the Saeman'smodel equation renders the search towards numerical or approximateanalytical approaches and the exact analytical solution has not been

rights reserved.

found till date. However, in the literature several studies are availablein which different approximate analytical solutions for finding thesolid bed depth profile have been reported. For example, Kramers andCroockewit [2] derived the approximate analytical solution for findingthe solid bed depth profile and the fractional hold-up. However, dueto the high prediction error in certain cases, the approximate analyticalsolution of Kramers and Croockewit [2] is generally not preferred [3].Lebas et al. [4] performed experimental and simulation studies for find-ing the bed depth profile and residence time, however, here too, theexact analytical solution was not given for the solid bed depth profile.Recently, Liu et al. [3] derived a more accurate approximate analyticalsolution for finding the solid bed depth profile by substituting an ap-proximation for the nonlinear terms in Saeman's model equation andsuccessfully simulated the experimental data taken from two differentsources [5,6]. In doing so, these researchers have shown that their ap-proximate analytical solution is superior to the one derived by Kramersand Croockewit [2]. It should, however, be noted that in some cases, theapproximate analytical solution of Liu et al. [3] is also susceptible toerrors and develops deviations in its predictions.

In this work, the exact analytical solution of the Saeman's modelequation has been found in the implicit form of the transcendentalfunctions by following the simple mathematical steps of separation ofvariables and partial fraction decomposition. The results obtained byusing the exact analytical solution have been successfully cross-checkedagainst those obtained by using the numerical method and are found tobe equally good or better than those obtained by using the approximateanalytical solution of Liu et al. [3]; moreover, a sound match is observedbetween the predicted and the available experimental results. Further-more, the effects of various parameters on the bed depth profile havealso been studied and discussed in some details.

30 M. Danish et al. / Powder Technology 230 (2012) 29–35

2. Mathematical model

In this section, Saeman's model equation has been presented.However, the details of the derivation of this model equation havebeen omitted and can be found elsewhere [1]. Consider a rotary kilnwith radius R and length L, inclined with horizontal at an angle βand spinning at an angular speed n, as shown in Fig. 1. The granularmaterial (bulk density ρ, particle diameter d and dynamic angle ofrepose θ) is being fed to the upper end of the kiln with a mass flowrate _m. Due to the inclination and rotation, the solids move down-ward spiraling. Based on the geometry and the mechanism involvedin the movement of the solids, the solid bed depth along the kilnlength can be represented by the following nonlinear first orderODE constituting an IVP [1–3].

dhdx

¼ 3 tan θð Þ4πn

_mρ

R2− h−Rð Þ2h i−3=2

− tan βð Þcos θð Þ ð1aÞ

where h is the solid bed depth at the axial distance x from the dis-charge end (x=0). The associated IC (initial condition) is given bythe depth of the solid bed at the exit (x=0) and is equal to the heightof the dam if present i.e.

IC : h 0ð Þ ¼ h0 height of the damð Þ: ð1bÞ

In case of the absence of dam, the solid bed depth at x=0 is takenequal to the particle diameter, i.e. h(0)=d. It is well known that thesolid bed depth can also be expressed in terms of the local fill angle(ε) and the radius of the kiln (R) as shown below [1–3]:

h ¼ R 1− cos εð Þ½ �: ð2Þ

Substitution of this relation in Eq. (1a) yields:

dεdx

¼ 3 tan θð Þ4πnR4

_mρ

1sin4 εð Þ−

tan βð ÞR cos θð Þ

1sin εð Þ ð3aÞ

solids

m

x

βh

Θ

Rε

h

Cross section

Fig. 1. Schematic diagram

and the allied IC becomes:

IC : ε ¼ ε0 ¼ cos−1 1−h 0ð ÞR

� �¼ cos−1 1−h0

R

� �; if dam is present:ð3bÞ

In case of the absence of dam, the following relevant change ismade in the IC, i.e.

ε ¼ ε0 ¼ cos−1 1−h 0ð ÞR

� �¼ cos−1 1− d

R

� �:

3. Exact analytical solution

Before finding the analytical solution of Eq. (3a), it is convenient toexpress this equation into the following dimensionless form:

dεdX

¼ Hsin4 εð Þ−

1sin εð Þ ð4Þ

where H ¼ ab; a and b are dimensionless constants, and are given by:

a ¼ 3 tan θð ÞL4πnR4

_mρ and b ¼ L tan βð Þ

R cos θð Þ, respectively. X ¼ b xL is the dimensionless

axial distance from the discharge end.On separating the variables in Eq. (4), one finds:

sin4 εð ÞH− sin3 εð Þ

!dε ¼ dX: ð5Þ

After performing the polynomial division of the bracketed term,one obtains:

− sin εð Þdε−Hsin εð Þ

sin3 εð Þ−H

� �dε ¼ dX: ð6Þ

By decomposing the bracketed term, Eq. (6) can further be repre-sented as follows:

− sin εð Þdε−Hsin εð Þ

sin εð Þ−r1ð Þ sin εð Þ−r2ð Þ sin εð Þ−r3ð Þ� �

dε ¼ dX ð7Þ

0

n

D

L

gas

of the rotary kiln.

31M. Danish et al. / Powder Technology 230 (2012) 29–35

where r1, r2 and r3 are the roots of the cubic equation: sin3(ε)−H=0 orsin3 εð Þ− a

b ¼ 0 or r3− ab ¼ 0, and are given by the following expressions:

r1 ¼ ab

� �1=3; r2 ¼ − 1

2þ

ffiffiffi3

p

2i

!ab

� �1=3; r3 ¼ −1

2þ

ffiffiffi3

p

2i

!ab

� �1=3:

ð8Þ

Since a>0 and b>0, it is clear that one of the roots will be real andpositive (say r1), whereas, the remaining two roots will be imaginary(say r2 and r3). This means that all the three roots are distinct, andthereby render the process of partial fraction decomposition verysimple. Now, the partial fraction decomposition of the bracketedterm in Eq. (7) is carried out and the following equation is obtained:

− sin εð Þdε−Hr1

r1−r2ð Þ r1−r3ð Þ sin εð Þ−r1ð Þ� �

dε−Hr2

r2−r1ð Þ r2−r3ð Þ sin εð Þ−r2ð Þ� �

dε

−Hr3

r3−r1ð Þ r3−r2ð Þ sin εð Þ−r3ð Þ� �

dε ¼ dX:

ð9Þ

Eq. (9) is integrated with respect to X between 0 and X, i.e.

∫ε

ε0

− sin εð Þdε−Hr1

r1−r2ð Þ r1−r3ð Þ∫ε

ε0

dεsin εð Þ−r1ð Þ−H

r2r2−r1ð Þ r2−r3ð Þ∫

ε

ε0

dεsin εð Þ−r2ð Þ

−Hr3

r3−r1ð Þ r3−r2ð Þ ∫ε

ε0

dεsin εð Þ−r3ð Þ ¼ ∫

X

0

dX:

ð10Þ

Integration of the above expanded Eq. (10) is quite simple and thefollowing result is obtained:

cos εð Þjεε0−2Hr1

r1−r2ð Þ r1−r3ð Þ1ffiffiffiffiffiffiffiffiffiffiffiffir21−1

q tan−11−r1 tan

ε2

� �ffiffiffiffiffiffiffiffiffiffiffiffir21−1

q264

375jε

ε0

− 2Hr2r2−r1ð Þ r2−r3ð Þ

1ffiffiffiffiffiffiffiffiffiffiffiffir22−1

q tan−11−r2 tan

ε2

� �ffiffiffiffiffiffiffiffiffiffiffiffir22−1

q264

375jε

ε0

− 2Hr3r3−r1ð Þ r3−r2ð Þ

1ffiffiffiffiffiffiffiffiffiffiffiffir23−1

q tan−11−r3 tan

ε2

� �ffiffiffiffiffiffiffiffiffiffiffiffir23−1

q264

375jε

ε0

¼ X:

After substituting the limits, one gets the following exact analyticalsolution for ε.

cos εð Þ− cos ε0ð Þð Þ− 2Hr1r1−r2ð Þ r1−r3ð Þ

1ffiffiffiffiffiffiffiffiffiffiffiffir21−1

q

× tan−1 1−r1 tan ε=2ð Þffiffiffiffiffiffiffiffiffiffiffiffir21−1

q264

375− tan−1 1−r1 tan ε0=2ð Þffiffiffiffiffiffiffiffiffiffiffiffi

r21−1q

264

375

0B@

1CA

− 2Hr2r2−r1ð Þ r2−r3ð Þ

1ffiffiffiffiffiffiffiffiffiffiffiffir22−1

q

× tan−1 1−r2 tan ε=2ð Þffiffiffiffiffiffiffiffiffiffiffiffir22−1

q264

375− tan−1 1−r2 tan ε0=2ð Þffiffiffiffiffiffiffiffiffiffiffiffi

r22−1q

264

375

0B@

1CA

− 2Hr3r3−r1ð Þ r3−r2ð Þ

1ffiffiffiffiffiffiffiffiffiffiffiffir23−1

q

× tan−1 1−r3 tan ε=2ð Þffiffiffiffiffiffiffiffiffiffiffiffir23−1

q264

375− tan−1 1−r3 tan ε0=2ð Þffiffiffiffiffiffiffiffiffiffiffiffi

r23−1q

264

375

0B@

1CA ¼ X

ð11Þ

For the evaluation of ε at each X, the Eq. (11) can be solved numer-ically by using any of the available computational tools e.g. MATLAB,

MATHEMATICA and MAPLE. Once ε is known, the bed depth profilecan easily be found by using Eq. (2).

4. Results and discussions

4.1. Validation of the exact analytical solution

For the validation of the exact analytical solution, the resultsobtained by using it have been compared with those obtained by usingthe numerical method as well as with the existing experimental results.For this purpose several experimental conditions have been selectedfrom various sources [3,4,7,8]. These experimental conditions havebeen presented in Table 1 and the corresponding experimental results(solid bed depth profiles) have been shown (by filled symbols) inFigs. 2–8. These experimental values of the solid bed depth profileshave been read from various figures plotted in the respective referenceswith the help of a user-friendly software “Plot Digitizer” available onlinefor free. For the same experimental conditions, the solid bed depth pro-files have also been calculated by using the exact analytical solution[Eq. (11)] and the numerical method, and the obtained analytical andnumerical results are also plotted in Figs. 2–8. Numerical results havebeen obtainedwith the help of a soft computing tool, i.e.MATHEMATICA.For this a computer code has beenwritten in the programming languageof MATHEMATICA and the use of one of its inbuilt command “NDSolve”has been made. These figures clearly show that the results obtained byusing the exact analytical solution match well with the numericallyobtained results as well as with the experimental results, and thus val-idate the presently obtained exact analytical solution.

4.2. Comparison of the exact analytical solution with the existingapproximate analytical solution

Eq. (4) along with the Eq. (3b) has recently been solved by Liu etal. [3] in an approximate manner to yield the approximate analyticalsolution for the solid bed depth profile. In their work, Liu et al. [3]have approximated ε [in Eq. (4)] by "0.39 sin 2(ε)+0.86 sin(ε)" andobtained the following approximated form of Eq. (4).

0:78 sin εð Þ þ 0:86ð Þ d sin εð ÞdX

¼ Hsin4 εð Þ−

1sin εð Þ ð12Þ

It can be observed that by using the above approximation for ε,these researchers were successful in retaining the dependent variablein the form of “sin(ε)” only, instead of both ε and sin(ε). Due to thismodification, the original Eq. (4) got transformed into the above easilysolvable form. For convenience, we reproduce below the approximateanalytical solution of Eq. (12) as obtained by Liu et al. [3].

0:14Hμln

Y2 þ μY þ μ2� �

μ−Y0ð Þ2

Y02 þ μY0 þ μ2

μ−Yð Þ2

24

35þ 0:26 Y0

3−Y3� �

þ 0:43 Y02−Y2

� �

þ0:26H lnbY0

3−a� �bY3−a

������������

0@

1A

þ0:5Hμ

tan−1 1:15Y0

μþ 0:58

� �− tan−1 1:15

Yμþ 0:58

� �� �¼ X

ð13Þ

where Y=sin(ε), Y0=sin(ε0) and μ ¼ ab

1=3 ¼ Hð Þ1=3 .For the purpose of comparison, the solid bed depth profiles have

also been found by using the above approximate analytical solution forthe same experimental conditions as listed in Table 1. The obtained ap-proximate results have also been shown in Figs. 2–8. It is clearly visiblefrom these figures that in most of the cases, the results obtained byusing the approximate solution of Liu et al. [3], i.e. Eq. (13), successfullymatch with those of the exact analytical solution, the numerical resultsas well as with the experimental results. These cases are present in all

Table 1Experimental data used for simulation.

Data no. Source Mass feed rate Bulk densityof solids

Dynamic angle ofrepose of solids

Radiusof kiln

Length ofrotary kiln

Inclinationof kiln

Damheight

Rotationspeed

μ ¼ r1 ¼ ab

1=3 Presentfig. no.

Reference Fig. no. _m (kg/s) ρ (kg/m3) θ (deg) R (m) L (m) β (deg) h0 (m) n (rev/s)

1 Spurling et al. [7] 3 2.45×10−3 1600 32 0.0515 1 1 0 0.0855 0.9831 22 Spurling et al. [7] 4 0.65×10−3 1600 32 0.0515 1 1 0 0.0855 0.6317 23 Spurling et al. [7] 5 3.1×10−3 1600 32 0.0515 1 1 0 0.119 0.7559 24 Spurling et al. [7] 6 3.15×10−3 1600 32 0.0515 1 2 0 0.0595 0.9574 35 Spurling et al. [7] 7 3×10−3 1600 32 0.0515 1 4 0 0.0862 0.6605 36 Spurling et al. [7] 8 3.05×10−3 1600 32 0.0515 1 2 0 0.0842 0.8436 37 Scott et al. [8] 3 1.5×10−3 1600 32 0.0515 1 1 0.0115 0.097 0.8004 48 Scott et al. [8] 3 1×10−3 1600 32 0.0515 1 1 0.0115 0.097 0.6992 49 Scott et al. [8] 3 0.56×10−3 1600 32 0.0515 1 1 0.0115 0.097 0.5764 410 Scott et al. [8] 3 0.1×10−3 1600 32 0.0515 1 1 0.0115 0.097 0.3246 411 Lebas et al. [4] 11 80.556×10−3 750 37 0.3 6 1 0 0.0333 0.9941 512 Lebas et al. [4] 11 80.556×10−3 750 37 0.3 6 1 0 0.05 0.8684 513 Lebas et al. [4] 11 80.556×10−3 750 37 0.3 6 1 0 0.0667 0.7890 514 Lebas et al. [4] 12 68.333×10−3 750 37 0.3 6 1 0 0.05 0.8220 615 Lebas et al. [4] 12 97.222×10−3 750 37 0.3 6 1 0 0.05 0.9246 616 Lebas et al. [4] 12 133.333×10−3 750 37 0.3 6 1 0 0.05 1.0272 617 Liu et al. [3] 3 1.5×10−3 1600 32 0.0525 1 1 0.0125 0.097 0.7852 718 Liu et al. [3] 3 1.0×10−3 1600 32 0.0525 1 1 0.0125 0.097 0.6859 719 Liu et al. [3] 3 0.56×10−3 1600 32 0.0525 1 1 0.0125 0.097 0.5654 720 Liu et al. [3] 3 0.1×10−3 1600 32 0.0525 1 1 0.0125 0.097 0.3184 821 Liu et al. [3] 4 3.69×10−3 2500 27.4 0.0525 5.9 1.37 0.015 0.0167 1.0081 822 Liu et al. [3] 4 3.69×10−3 2500 27.4 0.0735 5.9 1.37 0.015 0.05 0.6990 8

32 M. Danish et al. / Powder Technology 230 (2012) 29–35

Figs. 2–8, i.e. bed depth profiles for data numbers 1, 2 and 3 in Fig. 2; fordata numbers 5 and 6 in Fig. 3; for data numbers 7–9 in Fig. 4; for datanumbers 12 and 13 in Fig. 5; for data numbers 14 and 15 in Fig. 6; fordata numbers 17–19 in Fig. 7 and for data number 22 in Fig. 8. However,in some cases the solid bed depth profiles obtained by using the approx-imate analytical solution of Liu et al. [3] show some deviation from thetrue values, e.g. the solid bed depth profiles corresponding to the datanumber 1 in Fig. 2, data number 4 in Fig. 3, data number 10 in Fig. 4,data number 11 in Fig. 5, data number 16 in Fig. 6, data number 20 inFig. 7 and data number 21 in Fig. 8. Such discrepancies in the predic-tions of approximate solution appear in case of (i) higher filling ratio(25% corresponding to ε≥65.89° or μ≥0.91) or (ii) lower filling ratio(0.9% corresponding to ε≤20.05° or μ≤0.4). This fact can also be ob-served from the corresponding values of μ shown in Table 1. The formersituation emergeswhen the solid bed depth profile is increasing, whilstthe latter one occurs when the solid bed depth profile is decreasing.It is noteworthy that Liu et al. [3] have also pointed out the first typeof the discrepancy, i.e. the one arising due to the higher filling ratio,and reported that their approximate solution deviates from the numer-ical solution in the case of higher filling ratio (25% corresponding toε≥65.89° or μ≥0.91). Basically, these deviations in the prediction ofthe approximate analytical solution are due to the restricted validity

0

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1

Dep

th o

f so

lid b

ed, h

(mm

)

Axial distance from the discharge end of rotary kiln, x(m)

Data No. 1Data No. 2Data No. 3

Exp. data (Spurling et al. [7])

----- APS (Liu et al. [3])NSAS

Fig. 2. Solid bed depth profiles along the length of the rotary kiln.

of the approximated form of the nonlinearity as assumed by Liu etal. [3], i.e. ε≈0.39 sin 2(ε)+0.86 sin(ε). In fact the percentage rela-tive error in this approximation decreases monotonically from 14%at ε=0°to −2.5% at ε=42.5° and then increases continuously to17% at ε=85.94°. These errors in the approximation of ε are quitesignificant from the view point of accuracy, as the subsequent oper-ation, e.g. integration, may further enhance the errors in some cases.Nevertheless, the optimum range, for this approximation to be valid, isfound to be lying between ε=20.05° (μ≈0.4) to ε=65.89° (μ≈0.91),i.e. the % relative error lies within 3.5%. The absolute % relative error[=(ynumerical−yapproximate)/ynumerical×100%] in the prediction of theapproximate analytical solution of Liu et al. [3] for the data numbers 1,4, 10, 11, 16, 20 and 21 has been shown in Figs. 9–11. From these figuresit can be noted that the error can be as large as 20%. Moreover, for someof the cases the approximate solution of Liu et al. [3] yields only imagi-nary values (see profile for data no. 21 in Fig. 11).

In contrast, the results obtained by using the exact analytical solu-tion are found to be in good harmony with the numerical results aswell as with the experimental results for all the cases consideredherein. Therefore, the exact analytical solution, being valid for theentire range, has wider applicability as compared to the approximateanalytical solution of Liu et al. [3], especially in those cases where ε

0

5

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1

Dep

th o

f so

lid b

ed, h

(mm

)

Axial distance from the discharge end of rotary kiln, x(m)

Data No. 4Data No. 5Data No.

Exp. data (Spurling et al. [7])

----- APS (Liu et al. [3])NSAS

Fig. 3. Solid bed depth profiles along the length of the rotary kiln.

0

5

10

15

20

0 0.2 0.4 0.6 0.8 1

Data No. 7Data No. 8Data No. 9Data No. 10

Exp. data (Scott et al. [8])

----- APS (Liu et al. [3])NSAS

Dep

th o

f so

lid b

ed, h

(mm

)

Axial distance from the discharge end of rotary kiln, x(m)

Fig. 4. Solid bed depth profiles along the length of the rotary kiln.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 1 2 3 4 5 6

Data No. 14Data No. 15Data No. 16

Exp. data (Lebas et al. [4])

----- APS (Liu et al. [3])NSAS

Dep

th o

f so

lid b

ed, h

(mm

)

Axial distance from the discharge end of rotary kiln, x(m)

Fig. 6. Solid bed depth profiles along the length of the rotary kiln.

33M. Danish et al. / Powder Technology 230 (2012) 29–35

(or μ) may lie outside the optimum range set for the approximatesolution (some of the cases in which rotary kiln may operate outsidethis optimum range are also given in Table 1).

Here it is important to mention that we have not considered theapproximate analytical solution of Kramers and Croockewit [2], sinceit is reported in the literature that under certain circumstances it canexhibit up to 50% error in its prediction [3].

4.3. Analysis of the effect of various parameters

Several independent numerical experiments have also been car-ried out to study the effect of various parameters on the bed depthprofile, and a careful analysis of the obtained results is summarizedbelow:

(i) The dimensionless bed depth h�R

�is found to be dependent on

two dimensionless variables, namely H and ε0.(ii) For a fixed value of ε0 [or h(0)], the dimensionless bed depth

h�R

�increases with the increase in H and vice-versa.

(iii) Whether the bed depth first increases or decreases, it asymp-totically reaches a constant value named as asymptotic beddepth. The asymptotic depth reached by the bed is easilyobtained by forcing dh

dx ¼ 0 in Eq. (1a) and simplifying the resul-tant equation. The asymptotic bed depth is given by:

hAsymptotic ¼ R 1−ffiffiffiffiffiffiffiffiffiffiffiffi1−r21

q� �¼ R 1−

ffiffiffiffiffiffiffiffiffiffiffiffiffi1−μ2

q� �: ð14Þ

4.4. Solution of a case study

In this subsection, we illustrate the use of presently derivedexact analytical solution to find the bed depth profile for one of the

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 1 2 3 4 5 6

Data No. 11Data No. 12Data No. 13

Exp. data (Lebas et al. [4])

----- APS (Liu et al. [3])NSAS

Dep

th o

f so

lid b

ed, h

(m)

Axial distance from the discharge end of rotary kiln, x(m)

Fig. 5. Solid bed depth profiles along the length of the rotary kiln.

experimental runs (for data no. 7 in Table 1). Corresponding to thisexperimental condition, the experimentally obtained bed depth pro-file has been shown in Fig. 4. The procedure starts with the evaluationof ε0, which is obtained by using Eq. (3b), and for the selected exper-imental condition (data number 7 in Table 1), it is found that: ε0=39.04°. Thereafter, the dimensionless constants a (=0.204961) andb (=0.399663) are evaluated. Once the values of a and b are available,the roots of the cubic equation r3−H=0 or r3− a

b ¼ 0 are found withthe help of Eq. (8), and are given below.

r1 ¼ 0:800435r2 ¼ −0:400217−0:693197ir3 ¼ −0:400217þ 0:693197i

Substituting these values in Eq. (11), one obtains the followingnonlinear algebraic equation:

− 0:669902−0:590535ið Þ tan−1½ 0:165119þ 0:819124ið Þþ 0:633898þ 0:213367ið Þ tan ε=2ð Þ�

− 0:590535−0:669902ið Þ tanh−1½ 0:819124þ 0:165119ið Þþ tan ε=2ð Þ� þ 2:50211 cos 3½ �

þ1:78293 tanh−1 1:668828 1−0:800435 tan ε=2ð Þð Þ½ �þ −2:16534þ 2:80063ið Þ ¼ X:

ð15Þ

In Eq. (15), the value of ε for a given X can be found numerically.For example, for X=0.3 the value of ε is found to be 45.59°, whichcorresponds to the bed depth h=0.0154601 m. It can be verified

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4Dim

ensi

onle

ss d

epth

of

solid

bed

, h/R

Dimensionless axial distance from the discharge end of rotary kiln, xb/L

Data No. 17Data No. 18Data No. 19Data No. 20

Exp. data (Liu et al. [3])

----- APS (Liu et al. [3])NSAS

Fig. 7. Dimensionless solid bed depth profiles along the dimensionless length of therotary kiln.

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5Dim

ensi

onle

ss d

epth

of

solid

bed

, h/R

Dimensionless axial distance from the discharge end of rotary kiln, xb/L

Data No. 21Data No. 22

Exp. data (Liu et al. [3])

----- APS (Liu et al. [3])NSAS

Fig. 8. Dimensionless solid bed depth profiles along the dimensionless length of therotary kiln.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

0 1 2 3 4 5 6

Abs

olut

e %

rel

ativ

e er

ror

Length of the kiln, m

Data No. 11 (Lebas et al. [4])Data No. 16 (Lebas et al. [4])

Fig. 10. Absolute percentage relative error in the solid bed depth profile as obtained bythe approximate analytical solution [Eq. (13)].

34 M. Danish et al. / Powder Technology 230 (2012) 29–35

that exactly the same value of h is found by using the numerical solu-tion for the same value of X. This is also evident from Fig. 4 for datanumber 7. In a similar way the whole profile for the bed depth canbe obtained.

5. Conclusions

Exact analytical solution of the Saeman's model equation widelyemployed for estimating the solid bed depth profile has beenobtained in terms of an implicit function. The exact analytical solutionhas been validated by showing that the results obtained by using itmatch well with the numerical results. Usefulness of the exact analyt-ical solution has been proven by showing that it successfully simu-lates various existing experimental results. It is revealed that thepresently derived exact analytical solution covers the entire range ofε unlike the approximate solution of Liu et al. [3], which is accurateonly in its optimum range (20.05° ≤ε≤ 65.89° or 0.4≤μ≤0.91) andin some cases may also exhibit a relative percentage error of 20% ormore. Besides, effects of various parameters, e.g. increase or decreasein the bed depth profile, have also been studied and discussed.

Abbreviations

AS exact analytical solutionNS numerical solutionAPS approximate analytical solutionExp. experimental

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

0 0.2 0.4 0.6 0.8 1

Abs

olut

e %

rel

ativ

e er

ror

Length of the kiln, m

Data No. 1 (Spurling et al. [7])Data No. 4 (Spurling et al. [7])Data No. 10 (Scott et al. [8])

Fig. 9. Absolute percentage relative error in the solid bed depth profile as obtained bythe approximate analytical solution [Eq. (13)].

Nomenclaturea dimensionless parameter (=3 tan θð ÞL _m=4πnR4ρ)b dimensionless parameter (=L tan(β)/R cos(θ))d diameter of solid particle, mh0 height of the dam at the discharge end, mh depth of the solid bed, mhAsympt asymptotic depth finally attained by the solid bed, mH dimensionless constant (=a/b)L length of the rotary kiln, m_m mass flow rate of the solids, kg/sn rotational speed of the kiln, revolution/sri ith root given by the Eq. (8)R radius of the rotary kiln, mx axial distance from the discharge end along the kiln length,

mX dimensionless axial distance from the discharge end along

the kiln length ¼ b xL

Y sin(ε) in Eq. (13)Y0 sin(ε0) in Eq. (13)

Greek lettersβ inclination of the kiln to the horizontal, degreeε0 the local fill angle of the solids at the exit of the kiln, degreeε the local fill angle of the solids, degreeμ dimensionless parameter (=μ ¼ a

b

1=3 ¼ Hð Þ1=3 )θ dynamic angle of repose of the solids, degreeρ bulk density of the solids, kg/m3

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1

Abs

olut

e %

rel

ativ

e er

ror

Length of the kiln, m

Data No. 20 (Liu et al. [3])

Data No. 21 (Liu et al. [3])Imaginaryvalues of solid beddepth (and thus in absolute %error) yielded by theapproximate analytical solutionof Liu et al. [3] for length greaterthan 0.73m

Fig. 11. Absolute percentage relative error in the solid bed depth profile as obtained bythe approximate analytical solution [Eq. (13)].

35M. Danish et al. / Powder Technology 230 (2012) 29–35

References

[1] W.C. Saeman, Passage of solids through rotary kilns: factors affecting time of pas-sage, Chemical Engineering Progress 47 (1951) 508–514.

[2] H. Kramers, P. Croockewit, The passage of granular solids through inclined rotarykilns, Chemical Engineering Science 1 (1952) 259–265.

[3] X.Y. Liu, J. Zhang, E. Specht, Y.C. Shi, F. Herz, Analytical solution for the axial solidtransport in rotary kilns, Chemical Engineering Science 64 (2009) 428–431.

[4] E. Lebas, F. Hanrot, D. Ablitzer, J.L. Houzelot, Experimental study of residence time,particle movement and bed depth profile in rotary kiln, Canadian Journal of Chem-ical Engineering 73 (1995) 173–180.

[5] P.S.T. Sai, G.D. Surender, A.D. Damodaran, V. Suresh, Z.G. Philip, L. Sankaran, Resi-dence time distribution and material flow studies in a rotary kiln, MetallurgicalTransactions B 21B (1990) 1005–1011.

[6] R.J. Spurling, Granular flow in an inclined rotating cylinder: steady state and tran-sients, Ph.D. Thesis, Department of Chemical Engineering, University of Cambridge(2000).

[7] R.J. Spurling, J.F. Davidson, D.M. Scott, The transient response of granular flows inan inclined rotating cylinder, Transactions of IChemE 79 (2001) 51–61.

[8] D.M. Scott, J.F. Davidson, S.-Y. Lim, R.J. Spurling, Flow of granular material throughan inclined, rotating cylinder fitted with dam, Powder Technology 182 (2008)466–473.