Powder Technology 280 (2015) 103–112

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Powder Technology

journal homepage: www.elsevier.com/locate/powtec

Development of a model for the prediction of hydrodynamics of a

liquid–solid circulating fluidized beds: A full factorial design approach
Ritesh Ramesh Palkar, Vidyasagar Shilapuram ⁎
Department of Chemical Engineering, National Institute of Technology, Warangal, Telangana 506 004, India

a r t i c l e i n f o a b s t r a c t

Article history: Hydrodynamics of a liquid–solid circulating fluidized bed (LSCFB) system has a significant impact on the reactor
Received 8 December 2014 design. In the present study, statistical design approach was adopted to model the hydrodynamic behavior of an
Received in revised form 17 March 2015 LSCFB in terms of average solids holdup and solids circulation rate. Primary liquid velocity, auxiliary liquid
Accepted 20 April 2015
velocity, solids inventory and liquid viscosity are the input variables, called factors, which affects the system.
Available online 28 April 2015
Average solids holdup and solids circulation rate are called responses of the system. A full factorial design ap-
Keywords:
proach with four factors and three levels of the factors were considered. Various models such as linear, two factor
Circulating fluidized bed interaction, quadratic, and cubic models were tested for the adequacy. Within the range of experiments conduct-
Statistical modeling ed, for both responses, the quadratic regression model is suggested. The model shows that some of the interaction
Design of experiments effects between the factors are dominant. The developed model was verified by using various statistical tests.
Hydrodynamic behavior Also, the model was validated using various experimental data sets chosen at different conditions. Results
Response based on ‘R’ value, and deviations in parity plot were falling within agreement level. This suggests that the
proposed model can be adopted for various processing applications in the LSCFB unit.
© 2015 Elsevier B.V. All rights reserved.

1. Introduction municipal wastewater by means of simultaneous removal of carbon,

nitrogen and phosphorous [5]. LSCFBs are also used for many green
Fluidized beds are the processing equipment in which particles are technological aspects such as extractive fermentation of lactic acid
fluidized against gravity by the fluid. In general fluid may be liquid or and removal of cesium from radioactive liquid waste [6,7]. Various
gas. Variation in the forces acting on the particles leads in the variation experimental studies by different research groups for different reactor
of fluid velocity which results in dynamic behavior. Circulating fluidized geometries, mode of operation and operating conditions are available
beds (CFBs) are one such fluidized bed reactors, which essentially for the hydrodynamic studies in the liquid–solids and gas–liquid–solid
consists of two columns in the same unit, namely the riser and the circulating fluidized bed reactor systems [8–20]. Average solids holdup
downcomer with solids circulating in between these two columns. and solids circulation rate are the most important hydrodynamic prop-
These reactors have a wide application range wherever the process erties which decide the reactor performance and separation efficiency
requires temperature uniformity, plug flow conditions, avoidance of as a contactor in the LSCFB. In literature, various empirical approaches,
mass transfer limitations, favoring of fast reactions, good heat transfer modeling, simulation and optimisation strategies were adopted by
between the processing materials, and continuous regeneration of cata- different researchers to predict the hydrodynamics [21–24]. Core-
lyst [1]. Gas–solid circulating fluidized bed (GSCFB) is used for product annulus flow structure model used for GSCFBs were extended to
purification and energy optimisation studies [2]. In this, continuous LSCFB [21]. Computational fluid dynamic models were tried to predict
capture of CO2 was studied, where this equipment was used with the flow of solid and liquid phases [22]. Mathematical model based on
sorbent adsorption and regeneration phenomenon [3]. GSCFB also has homogeneous fluidization was developed and extended to predict the
potential all-round applications in many processing industries for fluid performance of an LSCFB for ion-exchange system for protein recovery
catalytic cracking, Fischer–Tropsch synthesis, combustion, environmen- [23]. Same group carried out multi-objective optimisation using the ge-
tal remediation, heterogeneous catalytic applications, solids processing netic algorithm approach and verified with experimentally obtained
i.e. calcinations, reduction of iron ore and gold roasting. This is due to data for continuous protein recovery [24]. A correlation for liquid
higher efficiency, operational flexibility and overall profitability as com- phase axial dispersion coefficient has been reported for liquid–solid
pared to other reactors [4]. Recently, LSCFBs are gaining importance in circulating multistage fluidized bed in which liquid phase residence
various applications. To name a few, LSCFBs are used to purify the time distribution were studied along with CFD validation [25]. Recently
an empirical correlation was developed from the exhaustive experi-
⁎ Corresponding author. Tel.: +91 83329 69463. mental data by considering various factors [19]. In addition, various
E-mail address: vidyasagars@nitw.ac.in (V. Shilapuram). empirical correlations are available for the average solids holdup and

http://dx.doi.org/10.1016/j.powtec.2015.04.045
0032-5910/© 2015 Elsevier B.V. All rights reserved.
104 R.R. Palkar, V. Shilapuram / Powder Technology 280 (2015) 103–112

solids circulation rate with their respective applicability range [26,27]. The factorial design approach was used to determine the interaction
Most of the approaches reported, as discussed above either show signif- effects of all independent variables affecting the response of the LSCFB.
icant deviation or applicable within the specific experimental range Four different independent variables (termed as factors) that are
studied. responsible for the hydrodynamic behavior/variation of the LSCFB are
Design of experiments (DOE) is one of the powerful data mining primary liquid velocity (UL1) in m/s, auxiliary liquid velocity (UL2) in
tools. It is considered as a black box of simulation with potential m/s, solids inventory (L0) in m and liquid viscosity (μL) in Pa∙s. Typically
applications in various research areas. It can be implemented to get there are two ways of presenting the solids inventory. One is in terms of
transparency in the developed model behavior with less time and the height and the other in terms of kgs of solids in the downcomer be-
lower cost for processing [28]. Conventionally, experiments give the fore start-up of operation. Both of these are inter-convertible. In the
output or response value for a given fixed set of input factors that are present method, similar to the reported literature solids inventory is
influencing the system. Upon comparison of responses by varying one presented in terms of height of the packed bed height in the downcomer
particular input keeping the rest of the input factors constant gives an in the cylindrical portion [11,26]. The average solids holdup (εS)
idea about the effect of that particular input factor on the response. and solids circulation rate (GS) are the responses that depict the hydro-
However, in reality, combination of simultaneous variation in input dynamic behavior in the LSCFB. Therefore, in this study, effects of all
factors may result in different responses because of interactions four operating variables (factors) on the two dependent variables
among the input factors. From the observed results, it is difficult to (responses) were studied simultaneously. As the number of factors
notice the effect of interactions among the input factors that are was less (i.e. four), a full factorial design was adopted with four factors
reflected in the observed experimental response. This cannot be visual- and three levels of these factors. Three levels of factors were chosen,
ized in conventional experiments. Therefore, one of the significant ad- as the experimental results showed that the observed responses were
vantages of DOE is to come up with a model which gives the effect of neither monotonically increasing nor decreasing nor constant for the
each individual factor as alone as well as a combination with the other systematic increase in all the input factors.
factor on the response. This model development also gives an idea for The list of independent variables with their actual and coded values
more accurate and realistic model. used in the factorial design is shown in Table 1. The coded values are in-
So far the DOE technique has not been used very extensively in the dicated by −1, 0 and +1. ‘−1’ indicates the lowest level, ‘0’ indicates
field of process engineering. Abbas and Baker studied the influence of center point level and ‘+ 1’ indicates the highest level of variables
operating parameters, i.e., catalyst weight, decomposition temperature used for the modeling. This statistical modeling study consisted of the
and methane partial pressure on rate of methane decomposition [29]. 34 number of experiments, where base value (3) represents the number
Jena et al. provided a statistical analysis for a gas–liquid–solid system of levels and exponent value (4) indicates the number of factors used in
for estimation of gas holdup and liquid holdup based on the factorial the modeling study. Only part of design matrix along with the experi-
design [30]. Hydrodynamic characteristics of three-phase fluidization mental observations and simulated results are presented in Table 2.
of a homogeneous ternary mixture of particles using factorial design The percentage deviation of the simulated observations with respect
has been carried out to study the effect of various operating parameters to experimental is also reported to verify the developed model
like superficial liquid velocity, liquid velocity, initial static bed height, mathematically.
average particle size and column diameter [31]. A detailed study based Design Expert® software, version 9.0.2, was used to determine the
on the full factorial design has been carried out by Al-Hassani et al. in suitable regression from the design matrix created for chosen set of
which various parameters such as temperature of the reaction, reaction experimental data. In the subsequent step model is verified by using
relative time and types of the catalyst used are taken as the operating analysis of variance (ANOVA) which helps to understand effect of
variables which are used to build a statistical model for methane factors and its interactions. The criteria used for choosing the best suit-
decomposition rate [32]. able regression model is based on the higher order polynomial where
In the previous work, experimental results showed that the ob- maximum interacting terms are significant and model is not aliased.
served responses (average solids holdup and solids circulation rate) In this modeling work, several models like linear, two-factor interaction,
were neither monotonically increasing nor decreasing nor constant for quadratic and cubic polynomial were analyzed in order to fit the exper-
the systematic increase in all the input factors that are affecting the imental data and simultaneously to test the best possible outcome in
system [16–19]. This reflects that there is a relation between the input terms of the predicted response from the derived model. The summary
variables of the response. Further details related to the experimental of various models used for predicting average solids holdup and solids
setup, experimental procedure and physical properties and range of liq- circulation rate are reported in Table 3. This table suggests that the
uid velocities are described elsewhere [16–19]. In the present study, a quadratic model can be used for developing a regression model to
full factorial approach is helpful to determine these effects efficiently predict the both average solids holdup and solids circulation rate as
and develop a model which can be extended for a particular application the F-test N p-test and p b 0.05.
to develop an efficient process [33]. Therefore, an attempt has been
made to adopt the DOE approach for modeling the chosen LSCFB
system. As per the best of the authors' knowledge this type of study 3. Development of regression models
has not been reported in the literature for modeling the LSCFBs.
The regression model is used when more than two variables are
2. Experimental design — statistical method significant or responsible for change in the response of the system.

In order to evaluate the hydrodynamics of an LSCFB, factorial design

approach was used. Factorial design method is one of the more efficient Table 1
statistical techniques in regard of time, resources and sample size. In List of independent variables used in the factorial design.
addition to this, where the interaction plays a dominant role; this Variables Actual value (coded value)
methodology is a key factor to get insight over a range of experimental
Primary liquid velocity 0.149 (−1) 0.163 (0) 0.189 (+1)
data. Factorial design method felicitates the relationship between (UL1), m/s
various operating parameters, where influence of the independent Auxiliary liquid velocity 0.03 (−1) 0.052 (0) 0.08 (+1)
variables is collectively expressed in terms of various mathematical (UL2), m/s
and statistical techniques, which ultimately leads to the analysis of the Solids inventory (L0), m 0.15 (−1) 0.25 (0) 0.35 (+1)
Liquid viscosity (μL), Pa∙s 9.03 × 10−4 (−1) 1.55 × 10−3 (0) 3.22 × 10−3 (+1)
degree of influence on the dependent variables [34].
 R.R. Palkar, V. Shilapuram / Powder Technology 280 (2015) 103–112 105

Table 2
Design matrix along with experimental and predicted responses.

Factors Response — 1 Response — 2

Variables Average solids holdup (εS) Solids circulation rate (GS), kg/m2 s

UL1, m/s UL2, m/s L0, m μL (×10−3) Pa∙s Experimental values Predicted % deviation Experimental values Predicted % deviation

0.149 0.03 0.35 3.22 0.069 0.07 −1.45 14.214 14.27 −0.39
0.149 0.052 0.35 0.0903 0.102 0.1 1.96 8.783 9.99 −13.74
0.149 0.052 0.35 3.22 0.084 0.089 −5.95 23.326 22.38 4.06
0.149 0.08 0.35 1.55 0.126 0.13 −3.17 50.753 41.62 17.99
0.149 0.08 0.15 3.22 0.067 0.072 −7.46 19.26 19.36 −0.52
0.149 0.052 0.35 1.55 0.122 0.12 1.64 26.252 24.05 8.39
0.149 0.08 0.15 1.55 0.078 0.092 −17.95 22.601 25.73 −13.84
0.149 0.08 0.35 0.0903 0.105 0.11 −4.76 22.69 27.62 −21.73
0.149 0.08 0.25 3.22 0.127 0.095 25.20 41.093 31.29 23.86
0.149 0.03 0.25 1.55 0.089 0.09 −1.12 20.062 15.48 22.84
0.163 0.03 0.15 3.22 0.043 0.031 27.91 7.964 6.66 16.37
0.163 0.03 0.35 3.22 0.065 0.064 1.54 15.991 14.96 6.45
0.163 0.08 0.15 0.0903 0.077 0.065 15.58 16.717 15.28 8.60
0.163 0.052 0.35 0.0903 0.089 0.094 −5.62 10.731 11.46 −6.79
0.163 0.08 0.15 3.22 0.061 0.066 −8.20 19.229 20.34 −5.78
0.163 0.052 0.25 3.22 0.08 0.077 3.75 13.921 18.03 −29.52
0.163 0.052 0.35 3.22 0.077 0.083 −7.79 26.887 23.06 14.23
0.163 0.052 0.15 3.22 0.058 0.052 10.34 14.323 9.56 33.25
0.163 0.03 0.35 1.55 0.094 0.093 1.06 12.814 17.07 −33.21
0.163 0.08 0.35 1.55 0.122 0.12 1.64 48.215 42.84 11.15
0.189 0.03 0.35 3.22 0.054 0.061 −12.96 17.141 13.3 22.41
0.189 0.08 0.35 1.55 0.114 0.11 3.51 51.475 42.17 18.08
0.189 0.052 0.35 0.0903 0.082 0.084 −2.44 13.652 11.24 17.67
0.189 0.03 0.25 1.55 0.089 0.075 15.73 20.062 16.62 17.16
0.189 0.08 0.25 1.55 0.118 0.1 15.25 50.932 36.42 28.49
0.189 0.08 0.35 3.22 0.093 0.09 3.23 38.338 38.71 −0.97
0.189 0.08 0.35 0.0903 0.088 0.092 −4.55 26.062 28.81 −10.54
0.189 0.08 0.15 0.0903 0.068 0.056 17.65 17.664 15.63 11.51
0.189 0.08 0.25 3.22 0.088 0.087 1.14 25.803 30.68 −18.90
0.189 0.052 0.35 3.22 0.068 0.079 −16.18 25.058 21.36 14.76

ANOVA analysis is performed for the verification of the suggested re- xj are the factors. For the present study x1, x2, x3 and x4 are respectively
gression model to establish the correlation between the experimental UL1, UL2, L0, and μL as presented in Table 1.
and predicted results. These ANOVA results are presented in terms of The minimum total liquid velocity in the riser at which there will be
various statistical parameters. If the statistical parameters evaluated a circulation of solids from the riser to the downcomer and simulta-
are within satisfactory limits, model developed shows the good neously recirculation of solids from downcomer to riser to establish
relationship between these factors for the response. continuous circulation between riser and downcomer is called critical
In general, the regression model is classified as first-order model, transition to circulating fluidized bed regime (Ucr). The total liquid
second-order model, etc. A general form of the regression model is as velocity in the riser is the sum of the primary and auxiliary liquid veloc-
shown in Eq. (1). ities. Therefore, for all the liquid velocities above Ucr, there always exists
a liquid solid circulating fluidized bed. However, the value of Ucr (i.e. the
combination of primary and auxiliary liquid velocity) depends on the
X
n X
n −1 X
n X
n
y ¼ β0 þ βi xi þ βi j xi x j þ βii x2i ð1Þ fluid and particle properties chosen as well as structure of solids feeding
i¼1 i¼1 j¼iþ1 i¼1 system. The critical transitional liquid velocities for circulating fluidized
bed regime are different for different viscosities of liquids [19]. Similarly,
the final total liquid velocity (primary as well as auxiliary liquid veloci-
where, y is the value of the response predicted by the model; β0, βi, βij, ty) to complete circulating fluidized bed regime or to reach unstable
and βii are the coefficients of the model estimated by the ANOVA; xi and operating condition depends on the fluid and particle properties chosen
as well as on the solids feeding system. Hence, operating ranges of
primary and auxiliary liquid velocities for the circulating fluidization
Table 3 regime were different for different viscosities of liquids. Usually with
Model summary used for the fitting average solids holdup and solids circulation rate. low viscous liquids, high window of circulating fluidized bed (CFB)
regime is expected in terms of primary and auxiliary liquid velocity.
Model F-test p-Value R2 Comment
With increasing viscosity operating window under the circulating fluid-
1. Average solids holdup
ized bed regime becomes narrower [19]. Therefore, common ranges of
Linear 34.15 b0.0001 0.6425
2-factor interaction 0.41 0.8725 0.6545 primary and auxiliary velocities under circulating fluidized bed regime
Quadratic 14.57 b0.0001 0.8165 Suggested are identified for various viscosities of fluidizing media as shown in
Cubic 2.16 0.0200 0.8914 Aliased Table 1. Lowest velocities of primary and auxiliary liquid velocities are
2. Solids circulation rate
assigned ‘−1’ as coded value whereas highest values of primary as
Linear 34.78 b0.0001 0.6467 well as auxiliary velocities are assigned ‘+1’ as coded value and center
2-factor interaction 0.65 0.6885 0.6654 values were assigned ‘0’ as coded value. Experimental data considered
Quadratic 26.01 b0.0001 0.8701 Suggested for this study is from Vidyasagar et al. [16,18] and Shilapuram et al.
Cubic 2.50 0.0069 0.9279 Aliased
[17,19].
106 R.R. Palkar, V. Shilapuram / Powder Technology 280 (2015) 103–112

The multiple linear regression models for developing the statistical 3.2. Model term verification by ANOVA
relationship to predict hydrodynamic behavior of LSCFB in terms of
coded factors are represented in Eqs. (2) and (3). Model verification was carried out by using ANOVA test. This is a
method based on the Fisher test which considers the null hypothesis
  of no treatment effects. This is an important statistical tool which
εS ¼ 0:10 − 6:652  10−3  x1 þ ð0:015  x2 Þ þ ð0:017  x3 Þ subdivides the total variation of a set of data into component parts
   
associated with specific sources of variation. This test mainly used to
− 2:543  10−3  x4 − 1:823  10−4  x1  x2
  determine the significance of model terms, effect of individual factors
− 4:369  10−4  x1  x3 and its interaction effects, ultimately leads to the precision of the
 
þ 2:414  10−3  x1  x4 model developed. Fisher's test is used to determine the regression,
  standard error and significance of each term used in the regression
− 1:387  10−3  x2  x3 ð2Þ
  analysis. The value of ‘F’ is calculated by dividing the mean square by
þ 1:681  10−3  x2  x4 the residual mean square. Along with F-test, ‘p-value’ also has signifi-
    cance in ANOVA test. This usually relates to the risk of falsely rejecting
þ 2:193  10−3  x3  x4 þ 3  10−3  x21
    a given hypothesis. Generally p-value less than 0.05 is considered as
− 5:571  10−3  x22 − 9:648  10−3  x23 significant, whereas values more than 0.1 have less significance in the
  model prediction. However, in some of the cases depending on the
− 0:023  x24
knowledge of the process the terms having a p-value from 0.05 to 0.1
were also considered as significant [34].
GS ¼ 30:20 þ ð0:78  x1 Þ þ ð10:30  x2 Þ þ ð6:45  x3 Þ The significance of each term and various interaction terms are
þ ð4:5  x4 Þ−ð0:24  x1  x2 Þ þ ð0:50  x1  x3 Þ evaluated within 95% confidence interval. For the design matrix
−ð0:93  x1  x4 Þ þ ð2:13  x2  x3 Þ   ð3Þ shown in Table 2, ANOVA results obtained are summarized for the
−ð0:69  x2  x4 Þ þ ð0:72  x3  x4 Þ þ 0:45  x21 average solids holdup in Table 5. Constant F value for the average solids
     
þ 1:44  x22 − 1:69  x23 − 15:48  x24 holdup is 20.98. In addition, the p-value of the constant is having value
less than 0.0001. The table also shows that model operating parameters
such as primary liquid velocity, auxiliary liquid velocity, solids inventory
and square effects of solids inventory and liquid viscosity are the most
3.1. Verification of developed model significant terms.
The same procedure as discussed above was followed for the regres-
Various statistical parameters like correlation coefficient (R2), sion model for solids circulation rate. These results are presented in
adjusted R2, predicted R2, adequate precision etc. are employed to get Table 6. From the table it may be observed that constant F-value for
the degree of model fitness. These performance measures adopted in developed model is 31.58 and p-value reported is less than 0.0001
this modeling work for both hydrodynamic properties are presented showing significance of the developed model. In addition, auxiliary liq-
in Table 4. The correlation coefficient is used to determine quantitative uid velocity, solids inventory, liquid viscosity; interaction of auxiliary
relationship between the interacting variables to observe the resem- liquid velocity and solids inventory, and square effect terms of solids
blance of an observed and predicted response. Correlation coefficient inventory and liquid viscosity have a significant effect which is clearly
was calculated from the ratio of the regression sum of squares to the observed from this ANOVA test.
total sum of squares. For developed model of average solids holdup Percentage contributions by the terms involved in the model
(Eq. (2)) and solids circulation rate (Eq. (3)) estimated R2 values are developed for the both average solids holdup and solids circulation rate
0.8165 and 0.8701 respectively. These values are adequate for selection are shown in Table 7. Upon comparison of Tables 5 and 6 with Table 7
of the model. In addition, adjusted R2 and predicted R2 are also used to it may be observed that terms significant in Tables 5 and 6 are showing
determine the suitability of the developed model. Adjusted R2 is the its significant percentage contribution for the predicted response.
correction of the R2 based on the number of terms and the sample
size. Adjusted R2 values show good agreement for statistical model 4. Interaction effect of factors
development if the difference between adjusted R2 and predicted R2 is
less than 0.2 [34]. As observed in Table 4 the differences between adjust- Effects of various factors with their interaction in the hydrodynamic
ed and predicted are below 0.2. Hence, the quadratic model is variation are discussed in the following sections separately. Though an
suggested. Standard deviation and mean noticed in the case of the empirical correlation was developed recently which could help to
average solids holdup were very low. In the case of solids circulation determine the hydrodynamic properties but it was unable to convey
rate the standard deviation and mean noticed were 5.11 and 17.85. interaction effects of the hydrodynamic variables [18]. To overcome
Adequate precision is another statistical term used to compare the this deficiency in the present modeling work a new approach along
range of predicted values. This term relates the signal to noise ratio of with a statistical design has been adopted.
the terms used. Values of this adequate precision are found to be
19.409 and 23.408 for the average solids holdup and solids circulation 4.1. Average solids holdup
rate respectively. Generally, values above four indicate adequate
model discrimination. Since the values observed in Table 4 are well The regression model developed for the current study in terms of ac-
above four, the model further confirms the adequacy and this is strongly tual factors is as follows:
supports the use of quadratic model.

εS ¼ 0:155 −ð3:007  UL1 Þ þ ð1:647  UL2 Þ þ ð0:762  L0 Þ þ ð52:116  μ L Þ

Table 4
Statistical parameters obtained by ANOVA study for hydrodynamic properties. −ð0:36459  UL1  UL2 Þ
Hydrodynamic SD Mean CV R2
R2adj R2pred Adequate −ð0:218  UL1  L0 Þ þ ð104:207  UL1  μ L Þ−ð0:554  UL2  L0 ð4Þ
property (%) precision  
þ ð58:043  UL2  μ L Þ−ð18:931  L0  μ L Þ þ 7:498  U2L1
εS 0.012 0.078 15.04 0.8165 0.776 0.7281 19.409      
GS 5.11 17.85 28.63 0.8701 0.8426 0.8080 23.349 − 8:912  U2L2 − 0:964  L20 − 17070:548  μ 2L :
 R.R. Palkar, V. Shilapuram / Powder Technology 280 (2015) 103–112 107

Table 5
Statistical ANOVA data for average solids holdup model.

Term SS df MS F-value p-Value Comment

Constant 0.041 14 2.915E−003 20.98 b0.0001 Significant

UL1 2.315E−003 1 2.315E−003 16.66 0.0001 Significant
UL2 0.011 1 0.011 79.88 b0.0001 Significant
L0 0.015 1 0.015 111.04 b0.0001 Significant
μL 3.438E−004 1 3.438E−003 2.47 0.1205
UL1 ∗ UL2 1.238E−006 1 1.238E−006 8.91E−003 0.9251
UL1 ∗ L0 7.078E−006 1 7.078E−006 0.051 0.8221
UL1 ∗ μL 2.302E−004 1 2.302E−004 1.66 0.2025
UL2 ∗ L0 6.956E−005 1 6.956E−005 0.50 0.4818
UL2 ∗ μL 1.089E−004 1 1.089E−004 0.78 0.3793
L0 ∗ μL 1.844E−004 1 1.844E−004 1.33 0.2535
(UL1)2 1.302E−004 1 1.302E−004 0.94 0.3366
(UL2)2 5.400E−004 1 5.400E−004 3.89 0.0529 Significant
(L0)2 1.676E−003 1 1.676E−003 12.06 0.0009 Significant
(μL)2 5.750E−003 1 5.750E−003 41.38 b0.0001 Significant
Residual 9.171E−003 66 1.390E−004
Total 0.050 80

In this modeling work, it can be clearly seen that, individual terms of the primary liquid velocity in LSCFB at constant solids inventory and
like auxiliary liquid velocity, solids inventory and liquid's viscosity; liquid viscosity. Thus, within the circulation fluidized bed regime at
interaction terms like primary liquid velocity — liquid viscosity, auxilia- lower primary liquid velocity and higher auxiliary liquid velocity this
ry liquid velocity — liquid viscosity; and square effect term of primary hydrodynamic property shows a peak point.
liquid velocity have a synergistic effect which positively helps to
increase the average solids holdup. Furthermore, the remaining terms 4.1.2. Interaction effect of primary liquid velocity and solids inventory on
like primary liquid velocity, liquid viscosity; interaction effect of average solids holdup
primary liquid velocity–auxiliary liquid velocity, primary liquid veloci- In this study, solids inventory of 0.15 m, 0.25 m, and 0.35 m was
ty–solids inventory, auxiliary liquid velocity–solids inventory, solids considered. A combined simultaneous effect of primary liquid velocity
inventory–liquid viscosity; and square effects of auxiliary liquid velocity and solids inventory at constant auxiliary liquid velocity and liquid
have very less antagonistic effect. viscosity is shown in Fig. 2. The observations in this case are similar to
the combined effect of primary and auxiliary liquid velocity on average
4.1.1. Interaction effect of primary and auxiliary liquid velocity on average solids holdup. However, quantitatively it may be different. The predict-
solids holdup ed average solids holdup region as shown by the color code on the right
The combined effect of primary liquid velocity and auxiliary liquid side of the figure suggests that at lower primary liquid velocity and a
velocity is studied in the developed model. 3-D surface plot for variation higher range of solids quantity in the downcomer contributes to an
of velocities is shown in Fig. 1. This effect is observed at constant solids increased average solids holdup. A confirmation of the positive effect
inventory in the downcomer and liquid viscosity. From this figure it can laden by the solids inventory on the prediction of the average solids
be observed that simultaneous increase in the primary and auxiliary holdup can be observed from this figure.
liquid velocities tend to reduction in the average solids holdup. This is
due to the increase in liquid velocity in the reactor exerts more drag 4.1.3. Interaction effect of primary liquid velocity and liquid viscosity on
on solid material in the riser section, therefore solids tends to go out average solids holdup
from the top of the riser. If the individual effect on the constant auxiliary Fig. 3 shows the effect of primary liquid velocity and liquid viscosity
liquid velocity is considered, the average solids holdup decreases with on the average solids holdup. This study was carried out at constant
increase in primary liquid velocity whereas at constant primary velocity auxiliary liquid velocity and solids inventory in the downcomer. This
increase in auxiliary velocity increases the average solids holdup. This combined effect shows that, at a constant auxiliary liquid velocity and
firmly indicates the synergistic effect induced by the introduction of solids inventory, average solids holdup increases at lower value of
the auxiliary liquid velocity and antagonistic effect due to the presence primary liquid velocity (i.e. 0.15 m/s) and liquid viscosity (i.e. up to a
viscosity range of approximately 0.002 Pa∙s to 0.00225 Pa∙s); afterward
Table 6
Statistical ANOVA data for solids circulation rate.
Table 7
Term SS df MS F-value p-Value Remark Percentage contribution of terms used for statistical modeling.

Constant 11,542.62 14 824.47 31.58 b0.0001 Significant Term Average solids holdup Solids circulation rate, kg/m2 s
UL1 31.86 1 31.86 1.22 0.2733
% contribution
UL2 5483.77 1 5483.77 210.05 b0.0001 Significant
L0 2142.25 1 2142.25 82.06 b0.0001 Significant UL1 6.20 0.27
μL 1078.58 1 1078.58 41.31 b0.0001 Significant UL2 29.45 46.87
UL1 ∗ UL2 2.23 1 2.23 0.085 0.7712 L0 40.15 18.31
UL1 ∗ L0 9.23 1 9.23 0.35 0.5541 μL 0.92 9.22
UL1 ∗ μL 34.52 1 34.52 1.32 0.2544 UL1 ∗ UL2 0.00 0.02
UL2 ∗ L0 164.20 1 164.20 6.29 0.0146 Significant UL1 ∗ L0 0.02 0.08
UL2 ∗ μL 18.16 1 18.16 0.70 0.4073 UL1 ∗ μL 0.62 0.30
L0 ∗ μL 19.66 1 19.66 0.75 0.3887 UL2 ∗ L0 0.19 1.40
(UL1)2 2.90 1 2.90 0.11 0.7400 UL2 ∗ μL 0.29 0.16
(UL2)2 35.95 1 35.95 1.38 0.2448 L0 ∗ μ L 0.49 0.17
(L0)2 51.36 1 51.36 1.97 0.1654 Significant (UL1)2 0.35 0.02
(μL)2 2625.74 1 2625.74 100.58 b0.0001 Significant (UL2)2 1.45 0.31
Residual 1723.06 66 26.11 (L0)2 4.49 0.44
Total 13,265.68 80 (μL)2 15.39 22.44
108 R.R. Palkar, V. Shilapuram / Powder Technology 280 (2015) 103–112

Fig. 1. Effect of primary liquid velocity and auxiliary liquid velocity on average solids Fig. 3. Effect of primary liquid velocity and liquid viscosity on average solids holdup.
holdup.
auxiliary liquid velocity. From this study, it is observed that variation
it decreases significantly. In the previous study, it was noticed that the of the liquid viscosity plays an important role.
average solids holdup was increased up to a certain liquid viscosity
followed by a decrease in average solids holdup with further increase 4.1.6. Interaction effect of solids inventory and liquid viscosity on average
in viscosity. The value of viscosity for maximum solids holdup was solids holdup
dependent also on the value of the solids inventory in the downcomer The interaction effect of solids inventory and liquid viscosity studied
[18]. The same phenomenon is also reflected in terms of interaction for the system is illustrated in Fig. 6. A dome shaped orientation of the
effects as shown in Fig. 3. surface plot signifies the same effect as discussed above.

4.2. Solids circulation rate

4.1.4. Interaction effect of auxiliary liquid velocity and solids inventory on
average solids holdup
Solids circulation rate is an important hydrodynamic variable which
Fig. 4 shows the interaction effect of auxiliary liquid velocity and
decides the intimate contact between the solids and fluid for an adsorp-
solids inventory at constant primary liquid velocity and liquid viscosity.
tion and desorption as well as reaction and regeneration. Also gives an
From this figure, it is evident that the average solids holdup increases
idea about the residence time of solids in the riser. Statistical model
with an increase in the interaction of auxiliary liquid velocity and solids
developed for the prediction of the solids circulation rate in terms of ac-
inventory in the downcomer of the LSCFB. It is also observed that, aver-
tual factors is as follows:
age solids holdup increases individually with an increase in auxiliary
liquid velocity or solids inventory when the rest of the factors are held
GS ¼ −39:695−ð291:575  UL1 Þ þ ð77:489  UL2 Þ þ ð47:146  L0 Þ
constant.
þ ð58027:211  μ L Þ−ð488:211  UL1  UL2 Þ
þ ð249:472  UL1  L0 Þ−ð40350:737  UL1  μ L Þ
4.1.5. Interaction effect of auxiliary liquid velocity and liquid viscosity on þ ð852:220  UL2  L0 Þ−ð23703:612  UL2 μ L Þ ð5Þ
average solids holdup þ ð6181:080  L0  μ L Þ þ 1119:210  U2L1
     
Fig. 5 shows a surface plot for variation of average solids holdup with þ 2299:472  U2L2 − 168:918  L20 − 1:153  107  μ 2L :
auxiliary liquid velocity and liquid viscosity. This result shows that there
is a significant interaction between UL2 and μL which showed a saddle
point. At a certain value of liquid viscosity (which is somewhere In this developed statistical model, for the prediction of the solids
between the range of viscosities chosen) maximum value of the average circulation rate, individual effects like auxiliary liquid velocity, solids
solids holdup is noticed. This maximum value increases at increasing inventory and liquid viscosity as well as in interaction effects like

Fig. 2. Effect of primary liquid velocity and solids inventory on average solids holdup. Fig. 4. Effect of auxiliary liquid velocity and solids inventory on average solids holdup.
 R.R. Palkar, V. Shilapuram / Powder Technology 280 (2015) 103–112 109

Fig. 5. Effect of auxiliary liquid velocity and liquid viscosity on average solids holdup. Fig. 7. Effect of primary liquid velocity and auxiliary liquid velocity on solids circulation.

primary liquid velocity–solids inventory, auxiliary liquid velocity–solids solids circulation rate is seen up to 0.25 m of solids inventory after
inventory, solids inventory–liquid viscosity; and square effect terms of which a sluggish nature of the solids circulation rate is clearly visualized.
liquid velocities are shows positive effect.
4.2.3. Interaction effect of primary liquid velocity and liquid viscosity on
4.2.1. Effect of interaction of primary and auxiliary liquid velocity on solids solids circulation rate
circulation rate An interaction effect of primary liquid velocity and liquid viscosity is
Surface plot for the combined effect of primary and auxiliary liquid shown in Fig. 9. In this study this effect has been observed at constant
velocity on solids circulation rate is shown in Fig. 7. For an individual auxiliary liquid velocity and solids inventory. In this study, it is observed
effect in the same study, it is observed that the solids circulation rate that as the primary liquid velocity and liquid viscosity are increased
increases with an increase in the primary liquid velocity and auxiliary simultaneously solids circulation rate shows a nonlinear behavior in
liquid velocity when the rest of the factors are held constant. As this solids circulation rate. With increasing combined effect, the solids
liquid velocity increases it will try to push the solids into the reactor circulation rate increases and reaches maximum at a certain point and
and form a closed loop. As a result of this interaction effect also showing after that point it tends to decrease as the viscosity effect is dominant.
positive influence results in increase in the solids circulation rate. This shows that the dominance of first factor is higher compared to
the second when the second factor is at a lower value and vice versa.
4.2.2. Interaction effect of primary liquid velocity and solids inventory on
solids circulation rate 4.2.4. Interaction effect of auxiliary liquid velocity and solids inventory on
The interaction effect of primary liquid velocity and solids inventory solids circulation rate
is shown in Fig. 8. Effect of interaction effect is monotonous. For an The interaction effect of auxiliary liquid velocity and solids inventory
individual effect, the solids circulation rate shows nonlinear increasing in the downcomer is shown in Fig. 10. This interaction effect has been
behavior with respect to the primary liquid velocity at a constant carried out at constant auxiliary liquid velocity and liquid viscosity. It
auxiliary liquid velocity and liquid viscosity. However, range of primary is observed that, increasing auxiliary liquid velocity and solids inventory
liquid velocity was chosen from 0.15 m/s to 0.19 m/s. This shows a results in a monotonic increase in solids circulation rate. This trend is
lower deviation in the solids circulation rate because the primary and the same as that in the case of averages solids holdup (Fig. 4). The
auxiliary liquid velocities were chosen such that this range would be main reason for this effect is not only individual but also interaction
common for all the range of viscosities studies. A linear change in the effects are having positive significance on the solids circulation rate.

Fig. 6. Effect of solids inventory and liquid viscosity on average solids holdup. Fig. 8. Effect of primary liquid velocity and solids inventory on solids circulation rate.
110 R.R. Palkar, V. Shilapuram / Powder Technology 280 (2015) 103–112

Fig. 9. Effect of primary liquid velocity and liquid viscosity on solids circulation rate. Fig. 11. Effect of auxiliary liquid velocity and liquid viscosity on solids circulation rate.

4.2.5. Interaction effect of auxiliary liquid velocity and liquid viscosity on 5. Statistical model validation
solids circulation rate
An interaction effect between auxiliary liquid velocity and liquid The performance of the developed statistical model is tested by
viscosity studied at constant primary liquid velocity and solids comparing the predicted response with the experimental response
inventory is shown in Fig. 11. The combined effect of these properties noticed. The experimental data points chosen were other than consid-
was studied at constant primary liquid velocity and solids inventory. ered in the design matrix for model development. Eqs. (4) and (5) are
From this study, it is evident that the solids circulation rate demon- used for the prediction of statistical response. Numbers of datasets
strates simultaneous increasing and decreasing trends. This shows used for the model validation chosen were 447 and 333 for εS and GS
a nonlinear behavior of the system with respect to operating respectively. Figs. 13 and 14 are the parity plots obtained for the valida-
parameters studied. This result is similar to that noticed in case of tion of developed statistical model in terms of average solids holdup and
average solids holdup (Fig. 5). However, the response of solids solids circulation rate respectively. The correlation coefficient (R) for
circulation rate was faster compared to solids holdup. This shows this comparison is found to be 0.89 and 0.92 for average solids holdup
that the significant interaction is between these variables as well as and solids circulation rate respectively. From this statistics it very clear
viscosity effect's dominance on the prediction of the solids circula- that predicted responses are very close to the respected observed exper-
tion rate. imental as the R values closely approaches to 1. Another method
employed to find out the accuracy of the developed model is the %
deviation of the model with the experimental observations. The
4.2.6. Interaction effect of solids inventory and liquid viscosity on solids percentage deviation is defined as the error between experimental
circulation rate and predicted response over experimental response multiplied by
The combined effect of solids inventory in the downcomer and 100. % deviation is found to be ±25% and ±35% for εS and GS respective-
liquid viscosity is shown in Fig. 12. This interaction effect is observed ly. However, larger deviations were noticed in the case of solids circula-
at constant liquid velocities. From this it is observed that, simultaneous tion rate because of the nature of solids circulation rate change noticed
increase in the solids inventory and liquid viscosity increases solids with the liquid velocity. That is at lower velocities changes in solids
circulation rate to a maximum value followed by a decrease. These circulation were rapid and after over particular liquid velocities changes
results are similar to Fig. 6 and same reason holds good in this case as in solids circulation rates were insignificant. Since the developed
well. model has to capture both the variations hence resulted in more
deviation. Therefore, these observations are in good agreement with
the developed statistical model and strongly evident that the developed

Fig. 10. Effect of auxiliary liquid velocity and solids inventory on solids circulation rate. Fig. 12. Effect of solids inventory and liquid viscosity on solids circulation rate.
 R.R. Palkar, V. Shilapuram / Powder Technology 280 (2015) 103–112 111

0.14 Nomenclature
L0 (m), µ L × 10-4 ( Pa.s)
0.15, 0.90 ANOVA analysis of variance
0.12 0.25, 0.90
0.35, 0.90 CV coefficient of variation
0.15, 1.17 + 25% df degrees of freedom
0.1 0.25, 1.17 F-test fishers test value
0.35, 1.17
0.15, 1.55 GS solids circulation rate (kg/m2 s)
S

0.08 0.25, 1.55 LSCFB liquid solid circulating fluidized bed

Predicted

0.35, 1.55
0.15, 3.22 L0 solids inventory in the downcomer (m)
0.06 0.25, 3.22 p-Value probability value
- 25%
0.35, 3.22 R correlation coefficient
0.04 SD standard deviation
R = 0.89 SS sum of squares
0.02 MS mean squares
No. Of Data Points = 447
UL1 primary liquid velocity (m/s)
0 UL2 auxiliary liquid velocity (m/s)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
xi, xj UL1, UL2, L0, μL corresponding factors in the general model
Experimental S
equation
Fig. 13. Model validation parity diagram for average solids holdup.
Greek letters
βij coefficients of the polynomial equation
statistical model gives the best predictions for the hydrodynamics of the εS average solids holdup
LSCFB. μL viscosity of the liquid (Pa∙s)

6. Conclusions
Acknowledgments
The statistical regression model has been developed and subse-
quently model suitability is confirmed by the ANOVA for the hydrody- Help rendered by Mr. Bishwadeep Bagchi, Department of Chemical
namic prediction of the LSCFB. Investigations of the present study are Engineering, National Institute of Technology, Warangal, in refining
summarized as follows: the language of the manuscript in the final stage is thankfully
acknowledged.
1. This study supports the use of the statistical design approach for the
model development in terms of interacting effects contributing to
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