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Available online at www.sciencedirect.com

Chemical Engineering and Processing 47 (2008) 2061–2065

Markov chain model of residence time distributionin a new type entrained-flow gasifier

Qinghua Guo, Qinfeng Liang, Jianjun Ni, Shouze Xu,Guangsuo Yu ∗, Zunhong Yu

Institute of Clean Coal Technology, East China University of Science & Technology,Shanghai 200237, People’s Republic of China

Received 8 November 2006; received in revised form 7 September 2007; accepted 21 October 2007Available online 20 February 2008

bstract

A continuous-time Markov chain has been used to establish the residence time distribution (RTD) model in a new type entrained-flow gasifier,hich is called entrained-flow gasifier with opposed multi-burner. According to the measurement results of the flow fields in the gasifier, the state

ransfer diagram of Markov chain formed in the case of the flow fields are simplified. The results show that this method is feasible in modeling the

ow system which consists of ideal mixing cells and plug flow regions. The flow pattern of the gasifier is closed to continuous stirred tank reactorCSTR). The simulation results are in good agreement with the experimental data. The established model has been applied to forecast the RTD inhe industrial gasifier.

2007 Elsevier B.V. All rights reserved.

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eywords: RTD; Markov chain; Entrained-flow gasifier

. Introduction

Stochastic approaches have been widely used in analysingnd modeling chemical processes. The significance of thetochastic nature of the residence time in a continuous flow sys-em is first pointed out by Danckwerts in 1953 [1]. Gottschalk etl. revisited Danckwerts’ law in 2006 [2]. The use of stochasticathematics to describe flow systems and, in particular, their

esidence time distributions is well developed. There are manytochastic models employed to determine the RTD of the flu-ds or particulates [3–13]. Based on the stochastic nature of

aterials’ RTD in continuous flow system, a suitable mathe-atical method to handle such a process could be Markov chainodel.Gasification technology offers the cleanest way of obtain-

ng energy from coal. Entrained-flow gasifiers have become the

referred gasifier for hard coals and have been selected for theajority of commercial-sized integrated gasification combined

ycle (IGCC) applications. The principal advantages of using

∗ Corresponding author. Tel.: +86 21 64252974; fax: +86 21 64251312.E-mail address: gsyu@ecust.edu.cn (G. Yu).

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255-2701/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.cep.2007.10.017

ntrained-flow gasification technology are the ability to handleractically any coal as feedstock and to produce a clean, tar-freeas. The two best-known types of entrained-flow gasifiers arehe top-fired coal-water-slurry (CWS) feed gasifier, as used inhe Texaco process and the dry coal feed side-fired gasifier aseveloped by Shell and Krupp-Koppers (Prenflo) [14]. Besides,new type of gasifier, the CWS entrained-flow gasifier with

pposed multi-burner, which was first developed by ECUSTnd Yankuang Group, is also a side-fired gasifier.

In operation and design of entrained-flow gasifiers, it ismportant to know the RTD of syngas in the gasifier. Yu et al. [3]resented a stochastic model describing the RTD of downflowtreams in an entrained-flow gasifier. They simulated the processs a Markov chain, discretized in time and state by dividing theasifier into discrete cells. This method can be applied to Texacoasifier, the feedstock of which is fed with one burner fixed athe top. Since Elperin first suggested the technique of impingingtreams, it gained great development because of its potential formproving the mass and heat transfer [15]. As mentioned earlier,

he techniques of Shell, Prenflo and opposed multi-burner arell based on impinging streams theory. As a result, a stochas-ic model is established to simulate the RTD in the impingingrocess. In this study, a continuous Markov chain approach is

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062 Q. Guo et al. / Chemical Engineerin

mployed to generate the RTD curves for the opposed multi-urner gasifier.

. Model description

Stochastic approaches have been widely employed to simu-ate flow in the reactors because of their probabilistic nature. Inhe approach, fluid flow is considered as a Markov process withontinuous time parameter and countable states. For this pur-ose, we visualize the continuous flow gasifier as an aggregatef states of ideal mixing, each of which exhibits the residenceime behavior of a CSTR and cells of ideal displacement, eachf which has the residence time behavior of a plug flow reac-or (PFR). We assume that the flow rate and flow pattern of theystem were not changed with time.

When the stochastic variable X(t) denotes the position (thetate of region) of one flow element, the {X(t)} can be denoted ascontinue-time Markov chain based on the Markov theory. pij(t)

epresents the standard transition probability of flow elementrom state i to state j, and pii(t) is the probability to remain inhe state i during the transition process. If pij(t) is continuous athe point t = 0, then, for any i, j, k ∈ E:

lim→0+pij(t) =

{1 i = j

0 i /= j(1)

t is supposed that the state of gas flow element {X(t), t ≥ 0} athe time t is i. We select a small time interval �t, the probabilityf flow element remain in state i at the time t + �t is

(X(t + �t) = i|X(t) = i) = 1 − qi�t + o(�t) (2)

rom Eq. (2), the probability of leaving state i at time t + �t isi�t + o(�t). If the pij represents the condition probability ofow element from state i to j (j /= i), pii is the probability ofemaining in state i for a duration of �t, for any i and j /= i, therobability from time t to t + �t is written as

(X(t + �t) = j|X(t) = i) = [qi�t + o(�t)]pij

= qipij�t + o(�t) (3)

ij = qipij (4)

(X(t + �t) = j|X(t) = i) = qij�t + o(�t) (5)

here qij represents the transition probability intensity.The qij is correlated to the volumetric flow rate from state i

o state j and the volume of the region i. For the region of CSTR

ij = Fij

vi

(6)

here Fij is the volumetric flow rate from state i to state j, andi is the volume of region i.

The transition intensity matrix Q is [16]

=

⎧⎪⎨⎪⎩

qij if i /= j

−∑i /= j

qij if i = j (7)

F

Fi

Processing 47 (2008) 2061–2065

he off-diagonal elements qij with i /= j are nonnegative, whilehe diagonal entry is a negative number chosen to make the rowum equal to 0.

For any i, j, k ∈ E, according to the Chapman–Kolmogorovquation

ij(t + h) =∑k ∈ E

pik(t)pkj(h) (8)

ence

pij(t + h) − pij(t)

h=

∑k /= jpik(t)pkj(h) − [1 − pjj(h)]pij(t)

h(9

hen h → 0, then

∂pij(t)

∂t=

∑k /= j

pik(t)qkj + qjjpij(t) =∑

k

pik(t)qkj (10)

t can be written as

′(t) = P(t)Q (11)

The formal solution of Eq. (11) can be expressed as

(t) = P(0)eQt = eQt (12)

We define the matrix

Qt =∞∑

n=0

(Qt)n

n!=

∞∑n=0

Qn tn

n!(13)

The matrix eQt can be calculated by the method of Jor-an transformation. Suppose that the matrix Q has the form= UJU−1, where J is the Jordan standard form matrix, then

he solution of the Eq. (11) is given by

(t) = UeJtU−1 (14)

The theory of residence time distribution is a cornerstonef chemical engineering science and practice, in general, andhat of chemical reactor analysis and design, in particular. Here,he mathematical approach of RTD calculation is presented. Weuppose the initial state is 1, n is the exit state, and T representshe residence time of flow element in the gasifier. The probabilityf T > t can be written as

r(T > t) =n−1∑j=1

p1j(t) (15)

Hence, the probability of the flow element residence time inystem is

n−1∑

(t) = Pr(T ≤ t) = 1 −

j=1

p1j(t), t ≥ 0 (16)

(t) is the probability that the flow element, which is originallyn compartment X1, will leave from the flow system by age t.

g and Processing 47 (2008) 2061–2065 2063

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3.2. Model testing

We take some of the regions combined to model the gasifiereasily. The combined regions are as follows: (1) each burner’s jet

Table 1

Q. Guo et al. / Chemical Engineerin

The probability density function of residence time is

(t) = dF (t)

dt= −

n−1∑j=1

p′1j(t) (17)

Consequently, the mean residence time tm can be deducedrom the above formula as

m =∫ ∞

0tf (t) dt (18)

Variance σ2t also can be deduced

2t =

∫ ∞

0t2f (t) dt −

[∫ ∞

0tf (t) dt

]2

=∫ ∞

0t2f (t) dt − t2

m

(19)

Furthermore, we can define a dimensionless variable thateasures time in units of the mean residence time of the whole

ystem, i.e.

= t

tm(20)

Then from Eq. (20), we have the dimensionless RTD densitys

(θ) = tmf (t) (21)

And the dimensionless variance, σ2θ , may be expressed as

2θ = σ2

t

t2m

(22)

. Model for the gasifier

.1. Flow fields in the gasifier

ECUST and Yankuang Group have developed the new coalasification technology during China’s 10th 5-year plan. Theemonstration plant built in Shandong province of China haseen operating successfully now. This new gasifier is fed withour opposed burners, and the flow fields in the gasifier havelready been tested [17], as shown in Fig. 1. According to theold model test, the flow fields can be divided into the followingegions along the flow direction.

1) Jet stream region. Each burner spouts with high-speed

streams, so four jet stream regions are generated.

2) Impinging region. Four opposed jet streams impinged oneach other tempestuously at the center of the gasifier.

3) Impinging stream region. After the four streams impinged,the streams flow along the axis direction of the gasifier,which have the similar flow characters and opposite direc-tions, then the impinging streams are formed above andbeneath the impinge flow region, respectively.

T

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ig. 1. Diagram of flow regions in cold model gasifier: (1) jet stream region; (2)mpinging region; (3) impinging stream region; (4) jet reflux region; (5) reentrytream region; (6) impinging stream reflux region; (7) tube stream region.

4) Jet reflux region. This region exists at the boundary of jetstream region and the impinging stream region. There arealso four jet reflux regions.

5) Reentry stream region. One of the impinging streams goesup to the top of the gasifier, and then reentries downwardalong the gasifier. This region has been divided into twoparts.

6) Impinging stream reflux region. Another impinging streamgoes down to the gasifier wall and generates this region.

7) Tube stream region. After the fluid passes the above-mentioned processes, it will flow down to the outlet and itsvelocity is almost identical in magnitude. So the flow canbe regarded as plug flow. In the present work, this regionhas been divided into several small regions, each of whichexhibits the residence time behavior of a CSTR and also hasthe same volume.

The volume of each region in the laboratory-scale gasifieras been estimated by the geometry method, which is shown inable 1.

he estimated volume of each region for cold model gasifier

otal volume (m3) 3.1et stream region (m3) 0.02473 × 4et reflux region (m3) 0.07418 × 4mpinging region (m3) 0.07536mpinging stream region (m3) 0.3456 × 2mpinging stream reflux region (m3) 0.5184 × 2eentry stream region (m3) 0.5103ube stream region (m3) 0.3927

2064 Q. Guo et al. / Chemical Engineering and Processing 47 (2008) 2061–2065

Fig. 2. The state transfer diagram of the Markov chain: (1–4) jet stream andreflux region; (5) impinging region; (6 and 7) impinging stream and reflux region;(8) reentry stream region; (9) tube stream region; (10) outlet flow.

Table 2Comparison of simulation results and experimental data

Number of CSTR 1 2 3 4 Exp. data

tσ

sasMso

t

Table 3Simulation results with different volumes

Volume 1.4 times 1.2 times 0.8 times 0.6 times

tσ

a

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m (s) 7.543 7.476 7.389 7.066 7.0452θ

0.370 0.387 0.416 0.544 0.491

tream region and jet reflux region; (2) impinging stream regionnd impinging stream reflux region. All of these regions are con-idered as CSTR and their transient states are time homogeneousarkov chain. The outlet flow may be regarded as an absorbing

tate. According to these assumptions, the state transfer diagramf the Markov chain is shown in Fig. 2.

The parameters used for choosing the appropriate model arehe mean residence time t and the dimensionless variance σ2

θ ,

Fig. 3. RTD density curves.

ttgt

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m (s) 11.809 10.021 6.33 4.2732θ

0.413 0.439 0.6 0.967

nd the criterion which is applied are

model = texp.; σ2θ model = σ2

θ exp.

Here, the reflux ratio in jet reflux region and impinging streameflux region are supposed to be identified. In the experimen-al condition, each burner’s gas flow rate is 370 m3/h; the totalow rate is 1480 m3/h. The changes of the mean residence timend the dimensionless variance with different number of themall CSTR that divided in the tube stream region are shown inable 2. When the tube stream region is considered to be oneSTR, the simulation results deviate from the experimental data.ith the increase of the number of small CSTR, the simulation

esults are close to the experimental data gradually. Then theube stream region can be regarded as a PFR. When the num-er of small CSTR is four, the simulation mean residence timend experimental value are 7.066 s and 7.045 s, respectively. Theimulation dimensionless variance is 0.544 and the experimen-al value is 0.491. The model result of dimensionless residenceime density curve is close to the experimental results, as shownn Fig. 3.

.3. Model for the industrial gasifier

The industrial gasifier with the volume of 41.51 m3 has sim-lar regions compared to the laboratory-scale gasifier. The ratiof each region’s volume is equal to that of the laboratory-scalene. In the industrial process, the flow rate of gas is 17383 m3/h.ccording to the above Markov chain model, the simulation

esults are as follows: the average residence time is 8.232 s andhe dimensionless variance is 0.468. It can be concluded thathere is enough time for the secondary reaction, especially forasification reaction, and the flow type in the gasifier tends tohe CSTR.

Although the demonstration gasifier has been built in Chinaow, it is also important to optimize the gasifier further. Here,he volume of the gasifier has been changed, and the simula-ion results are shown in Table 3. The RTD density curves withifferent volumes are shown in Fig. 3. It can be concluded thathe mean residence times increase and the dimensionless vari-nces decrease as the volumes increased under the conditionf the same flow rate. Both the mean residence times and theimensionless variances change less and less when the volumesncreased at the same ratio.

. Conclusions

Based on the flow fields in the entrained-flow gasifier withpposed multi-burner, the Markov chain model is establishedith the continuous time and discrete states. The results show

hat when the tube region divided into four small CSTR regions,

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he model is more reasonable and the simulation results are accu-ate. It is shown that the present approach is convenient andffective to model the RTD of the impinging stream reactors.

With the help of the established model, the industrial gasi-er’s RTD curve has been simulated. The mean residence timehanges with the gasifier volume correspondingly. The wholeeld tends to the CSTR. The gasifier’s volume is reasonable,nd it is beneficial to the gasificaion process.

cknowledgements

Funding for this work is partly supported by Shanghaiorning Twilight (06SG34) and National Key State Basicesearch Development Program of China (973 Program,004CB217703).

ppendix A. Nomenclature

(t) residence time distribution densityij flow rate of jet from state i to state j (Nm3/h)

Jordan standard form matrixij transition probability from state i to state j(t) matrix of transition probabilityij transition probability intensity from state i to state j

time (s)m mean residence time (s)

t time interval (s)i the region volume of state i

reek lettersdimensionless time

2 variance2θ dimensionless variance

[

[

Processing 47 (2008) 2061–2065 2065

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