Correlation for Calculating Heat Transfer Coefficient in Conical

Subscriber access provided by Northern Illinois University

Article

CORRELATION FOR CALCULATING HEAT TRANSFER COEFFICIENT IN CONICAL SPOUTED BEDS Juan F Saldarriaga, Roberto Aguado, Aitor Atxutegi, John R. Grace, Javier Bilbao, and Martin Olazar Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b02234 • Publication Date (Web): 17 Aug 2016 Downloaded from http://pubs.acs.org on August 24, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Industrial & Engineering Chemistry Research is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Industrial & Engineering Chemistry Research

CORRELATION FOR CALCULATING HEAT TRANSFER COEFFICIENT IN CONICAL SPOUTED BEDS Juan F. Saldarriagaa, Roberto Aguadob,*, Aitor Atxutegib, John Gracec, Javier Bilbaob, Martin Olazarb

a

Program of Environmental Engineering, Faculty of Engineering, University of Medellin, Carrera 87 Nº 30 -65, Medellin, Colombia.

b

Department of Chemical Engineering, University of the Basque Country, PO Box 644 – E48080 Bilbao, Spain.

c

Deparment of Chemical and Biological Engineering, University of British Columbia, 2360 East Mall, Vancouver, BC V6T 1Z3, Canada

*

corresponding author. Tel.: +34 946 015 394, Fax: +34 946 013 500; e-mail: [email protected]

ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract A correlation has been proposed for predicting local bed-to-surface heat transfer coefficients in conical spouted beds based on an experimental study conducted using beds made up of sand, sawdust and their mixtures under various operating conditions, namely, different static bed heights and gas velocities over minimum spouting. A comparison of the results with those obtained using the correlations in the literature proved that they provide very poor predictions and, furthermore, they are not able to predict local coefficients in the bed. Based on a statistical analysis, the significant moduli have been identified in the relevant literature correlations and two new moduli related to the radial and longitudinal positions have been contemplated. The analysis identified two groups of heat transfer coefficients, those within the bed and those on its surface. The correlation proposed is specifically suitable for ascertaining the best location of heat transfer devices within the annulus of the spouted beds.

Keywords: spouted bed, conical spouted bed, heat transfer coefficient, heat transfer correlation


Page 2 of 35

Page 3 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


NOMENCLATURE Ar:

Archimedes number, gdp3ρ(ρs-ρ)/µ2, dimensionless

Cs:

mass ratio of mass to solution, kg/kg

Do:

inlet diameter, m

DC:

column diameter, m

Db:

diameter at the bed surface, m

Di:

contactor base diameter, m

dp:

mean diameter, mm

γ:

cone angle, °

Gu:

Gukhman number, (Td – Tw)/Td, dimensionless

Ho:

static bed height, m

hi:

instantaneous heat transfer coefficient, W/m2K

ms:

mass flow rate of the solution, kg/s

mg:

mass flow rate of the gas, kg/s

mw:

mass flow rate of the liquid, kg/s

MSR: mean square residual, dimensionless Nu:

Nusselt number, hD/k, dimensionless

q:

instantaneous power dissipated by heat transfer probe, W

ρb:

particle density, kg/m3

ρs:

bulk density, kg/m3

Pr:

Prandtl number, Cpµ/k, dimensionless

Re:

Reynolds number of minimum spouting, ρumsdp/µ, dimensionless

φ:

shape factor, dimensionless

R:

radius of the contactor

Rf :

reference electrical resistance, Ω

Rpb:

electrical resistance of heat transfer probe, Ω

r/R:

radial dimensionless position



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

s:

slot width, m

SSR:

residual sum of squares, dimensionless

t:

time, s

T:

temperature, K

Td :

temperature of the gas, K

Tsusp:

suspension temperature, K

Tpb:

temperature of the probe, K

Tw:

wet-bulb temperature of the gas, K

u:

air velocity referred to the inlet diameter, m/s

ums:

minimum spouting velocity referred to the inlet diameter, m/s

V1, V2: voltages before and after heat transfer probe, respectively, V w:

bed width, m

z:

probe center height, m

z/Ho:

longitudinal dimensionless position


Page 4 of 35

Page 5 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1. Introduction Although spouted bed technology has already been implemented in many applications, being the subject of numerous studies involving hydrodynamics and gas and solid flow patterns, heat transfer is a point that has been scarcely approached in the literature. Thus, certain authors have studied the influence of temperature on hydrodynamics1–5, but heat transfer itself has hardly been addressed, with drying being the main subject approached by most papers related to heat transfer6–11. In fact, drying has so far been the main and more popular application of spouted beds. Nevertheless, conical spouted beds have also performed well in the pyrolysis of biomass, plastics and tyres12. Several authors have developed mechanistic models based on heat and mass balances in order to describe heat transfer in spouted beds, but they require sophisticated calculations for solving the complex sets of differential equations13–18. Other studies approach heat transfer in these beds by monitoring the changes in temperature the gas undergoes when passing through the bed, i.e., by monitoring the temperature at the inlet and at several radial and longitudinal locations in the bed7,11,19,20, but this methodology is not suitable for determining local bed-to-surface heat transfer rate. Another approach used for determining heat transfer in spouted beds involves relating the heat transfer coefficient to the total surface area of the particles in the bed19. Several correlations based on these relationships have been reported as a function of dimensionless moduli. The most relevant ones are set out in Table 1. These equations have been proposed for the drying of particles in spouted beds, and therefore they quantify gas-to-particle heat transfer rate. Eqs. (1), (4) and (6)-(8) include water flow throughout particle drying, either in the Gukhman modulus, eqs. (1) and (6), or in the flow rate modulus, eqs. (4), (7) and (8). Eq. (7), has been proposed for slot rectangular spouted beds, and one of its variables is therefore the width of the slot face.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Page 6 of 35

Table 1. Correlations for heat transfer calculation in spouted beds. Author Reger et al.

Correlation

21

Kmiec22

Kmiec23

Kmiec and Kucharski20

Kmiec and Jabarin24

Englart et al.19

Kudra et al.25

Rocha et al.26

Nu= 0.0597 Re

Ar

− 0.438

Gu

Nu= 0.045 Re Nu= 9.4723 Re

Pr

0.6128

0.333

Pr

Ar

0.333

0.226

Ar

γ tan 2

0.2302

Ho dp

Nu= 2.673 Re 0.516 Pr 0.333 Ar 0.033 tan

γ 2

Nu= 0.0030 Re 0.836 Pr 0.333 Ar0.236 Gu− 2.527

Nu= 1.975 Re

0.64

0.61

γ 2

Nu= 0.897 Re0.464 Pr 0.333 Ar 0.116 tan 0.664

− 1.0

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) () ( ) ( )

2.0

− 0.852

− 1.031

− 2.331

Di dp

− 1.20

Ho dp

− 0.813

Ho dp

− 1.47

Nu= 0.9892 Re1.6421 Pr0.333

Ho φd p

Ho w

− 1.3363

− 1.334

Ho dp

Db dp

0.45

m˙ s m˙ g

φ2.261

Di dp

(2)

0.947

2.304

φ

1− C s Di dp

− 4.121

( ) ( )() ( ) ( )( Ho dp

− 1.19

(3)

0.602

m˙ s m˙g

3.356

Ho dp

(1)

s dp

0.795

φ2.261

(4)

0.602

φ2.102

m˙w m˙ g

(5)

0.600

φ− 0.918

(6)

0.26

0.71

tan

(7)

γ 2

0.1806

)


(8)

Page 7 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The mathematical models for predicting heat and mass transfer require making assumptions involving the structure of the equipment used. Thus, the model developed by Kmiec23,27 for particle circulation and heat transfer between particles and the spouting gas is based on the definition of two zones, namely, a lean phase spout zone at the core of the bed and an annular zone surrounding the spout, while ignoring the fountain. These models usually predict heat and mass flows, as well as temperature distribution in the bed by using correlations based on dimensionless moduli for predicting heat transfer coefficients28. Knowledge of heat transfer rates is mandatory in the drying processes because they are controlled by the gas-to-particle heat transfer. Furthermore, knowledge of heat transfer rates is also required when using spouted beds at industrial scale for the thermal upgrading of polymeric wastes (biomass, plastics, tyres), given that devices should be inserted in these reactors for providing heat to the process (pyrolysis or gasification) or heat should be removed from the reaction environment (combustion), with these devices usually being placed in the annulus of the contactor. In fact, this is the bed zone in which heat recovery devices may be placed without disturbing bed hydrodynamics. To our knowledge there are no correlations in the literature for calculating local bed-to-surface heat transfer rates. Therefore, the paper's main aim is the proposal and validation of an empirical correlation for estimating the heat transfer coefficient at any location in the annular zone of the bed, i.e., from the spout-annulus interface to the wall of the contactor. 2. Experimental 2.1. Heat transfer measurement instrumentation The device for measuring heat transfer coefficients has been detailed in a previous paper5 and consists of a probe with a control circuit designed to keep it at a constant temperature. Accordingly, heat transfer was measured based on the power required to keep the probe's temperature constant.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The probe consists of a thin palladium film and its design allows determining both the heat flux from the probe and its temperature with relative ease and high accuracy5. It is mounted at one end of a guard heater (Figure 1) consisting of a cartridge heater inserted into the core of the rod from its other end. The temperature of this guard heater is measured by a thermistor and kept slightly lower than that of the probe. This minimizes and stabilizes heat loss from the back of the probe. It also stabilizes and limits the temperature variations of the glass support. Throughout the runs, the average temperatures of the probe and guard heater were kept at 83 °C and 80 °C, respectively. This small difference was needed to maintain the stability of the controller. Allowance was made for this temperature difference in determining the heat transfer coefficient. Figure 1 The temperature is controlled by varying the voltage drop across the probe circuit through the programmable power supply. The computer continually monitors the voltage drop across the probe and computes the probe temperature. If the probe temperature is lower than the set point, the probe resistance will also be lower. The computer then increases the programming voltage to the power supply by means of a simple feedback control action, which in turn raises the voltage across the probe circuit. This increases the power to heat the Pd film, restoring its temperature to the set point. 2.2. Experimental unit The experimental unit used is shown in Figure 2. The conical spouted bed contactor was constructed of polymethyl methacrylate with an included cone angle of γ = 28°, an inlet diameter of Do = 0.04 m, a column diameter of Dc = 0.304 m and a bottom diameter of Di = 0.06 m. The static bed heights used are: Ho = 0.10, 0.20 and 0.30 m. The probe was placed at several bed levels, z = 0.10, 0.20 and 0.35 m, and heat transfer was measured at different radial positions. Measurements were made at three gas superficial velocities: minimum spouting


Page 8 of 35

Page 9 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


velocity and 10 and 20% above this value. The air and the bed solids are maintained at ambient temperature Figure 2 Beds made up of sawdust, sand and sawdust/sand mixtures (30/70, 40/60 and 50/50 mass ratios) were used for the heat transfer study. Key properties are provided in Table 2. The Archimedes number was estimated at room temperature. The moisture content was measured following the ISO-589 standard using a halogen moisture analyser (HR83, Mettler Toledo). Particle density was measured by mercury porosimetry29, and the average particle size (mean reciprocal diameter) was calculated from:

d‾p=

1 x

∑ di

pi

(9)

Table 2. Properties of solid particles. Properties Mean diameter, dp, mm Particle density, ρb, (kg/m3)

Sawdust 0.76 496

Sand 0.17 2650

Bulk density, ρs, (kg/m3) Moisture content, (wt%, d.b.) Archimedes number, (Ar) Geldart classification

189 9.0 6.7 · 103 D

1690 9.3 4.0 · 102 A

3. Results and discussion Local heat transfer coefficients were determined by monitoring local heat flow. Knowledge of these coefficients is essential for ascertaining the more suitable locations for the heat removal devices within the bed, especially when scaling up. Thus, as observed in Figures 3 and 4, corresponding to beds made up of sawdust and sand, respectively, the heat transfer coefficient increases as bed level is higher and the positions are closer to the contactor's axis. Therefore,



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

heat transfer coefficients reach peak values in the upper central zone in both cases, with values being as high as 304 W m-2 K-1 for sawdust and as high as 590 W m-2K-1 for sand. It should be noted that although the trends are similar in both cases, heat transfer coefficients are higher and more homogeneous for sand beds. Figure 3 Figure 4 Likewise, Figure 5 shows the experimental results (points) for the heat transfer coefficient registered at the bed level of z = 0.2 m at a radial positions of 3 cm from the wall for the three sand-sawdust mixtures studied. Values are also shown (lines) for the heat transfer coefficient expected based on the linear contribution of the components in the mixture. As observed, the experimental values are lower than those expected according to a linear mass contribution of the components in the mixture, but they are slightly higher than expected for a volume contribution of the components in the mixture. This is probably due to the different hydrodynamic behaviour of the mixture compared to pure sand and pure sawdust. Figure 5 The understanding of these changes in bed hydrodynamics, and therefore the principles and mechanisms involved, would require not only a detailed analysis of these results, but also a specific hydrodynamic study. Nevertheless, from a practical perspective, volume contribution provides a reasonable estimation of the heat transfer coefficient for the mixture. The aim of this study is to delve into the knowledge of heat transfer in conical spouted beds and provide a tool to be used for scaling up these beds. Accordingly, a correlation will be proposed for the prediction of local heat transfer coefficients in these systems. Few studies in the literature deal with the measurement of heat transfer coefficients in beds made up of sand and biomass operating in the spouted bed regime5. Furthemore, all the correlations except for those proposed by Kmiec22,23 and Kmiec and Jabarin24, eqs. (2), (3) and


Page 10 of 35

Page 11 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(5) in Table 1, involve drying and contain moduli related to water or steam flow rate in the process. Therefore, the correlations proposed by Kmiec have been taken as a starting point in this study. Many applications of spouted beds, such as drying, pyrolysis, gasification or catalytic polymerization12,30,31, involve additional heat transfer from the wall to the bed, apart from the usual heat transfer from the spouting gas to the particles. Accordingly, two types of studies have been reported in the literature, namely, those dealing with heat transfer from the wall and those dealing with heat transfer between the gas-particle/ medium and objects submerged in the bed11,22,23. Thus, Mickley and Fairbanks32 stated that the heat transfer mechanism is different depending on the medium and surrounding properties. Thus, whereas wall-to-bed heat transfer occurs simultaneously by conduction and convection, the mechanism governing heat transfer in the bulk of the bed is mainly convection, which is also the case in dilute fluidized beds. This consideration is consistent with the hypothesis that a thermal boundary layer (approximately 1 cm thick) is formed on the contactor wall33,34. Nevertheless, the results of this study do not reflect this clear difference between the zone near the wall and the bulk of the bed. Thus, as observed in Figures 3 and 4, although the lower values of the heat transfer coefficient have been measured at the wall, the differences between these values and the higher ones in the bulk of the bed do not reveal changes in the heat transfer mechanism, as such differences are probably due to a decrease in convective heat transfer coefficients caused by lower turbulence in certain zones in the bed. Therefore, an empirical correlation would be useful for estimating the value of the heat transfer coefficient (h) at any location in the bed, from the spout-annulus interface to the wall, given that this is a zone where devices for heat input of recovery would be placed with very low bed perturbation. In fact, a suitable location for the devices may be the spout-annulus interface, given that they would play a stabilizing role similarly to an open-sided dratf tube. Eqs. (2), (3) and (5) set out in Table 3 have been taken as a starting point for the proposal of a



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 35

correlation for the calculation of heat transfer coefficients in conical spouted beds. Eqs. (2) and (3) provide values close to zero for the Nu number, which means they predict negligible convection compared to conduction under these conditions. The main explanation for the poor predictions of these correlations lies in the experimental conditions used, which are significantly different from those in this study. Thus, Kmiec22 used silica gel and activated coal particles in a 9 cm I.D. column with a 30° and 90° conical base. The features of these materials, as well as well as the geometry of the reactor, are considerably different to those used in this study. His unit is a conventional spouted bed (cylindrical with conical base), whereas our experimental system is a conical spouted bed. Our unit scale is also large than his. All these facts are responsible for the lack of fitting and require retuning the correlations in the literature for spouted beds. The dimensional moduli are valid but the coefficients need to be modified. Eq. (5) provides values of the same order as the experimental ones, Figure 6, but it cannot account for the changes in the heat transfer coefficient within the bed, as evidenced by the data stratification in the parity plot. It should be noted that the properties (heat capacity and conductivity) used for calculating Nusselt and Prandtl numbers are those corresponding to the solid instead of the gas, given that the values predicted by eq. (5) are otherwise much poorer than those shown in Figure 6. This is consistent with the trends observed in this study, i.e., the thermal properties of the solid making up the bed are the conditioning factors for the heat transfer coefficient. Table 3. Correlations chosen as a starting point for the proposal of a new one. Correlation Kmiec22

Eq.

(

Nu= 0.897 Re 0.464 Pr 0.333 Ar 0.116 tan

γ 2

− 0.813

)

− 1.19

( )

φ2.261

− 1.47

0.947

Ho dp

(2)

Kmiec23

Nu= 0.045 Re

0.664

Pr

0.333

Ar

0.226

(

tan

γ 2

− 0.852

)

( ) () Ho dp

Kmiec and Jabarin24


Di dp

2.304

φ

(3)

Page 13 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(

Nu= 2.673 Re 0.516 Pr 0.333 Ar 0.033 tan

γ 2

− 2.331

)

− 1.334

0.602

( ) () Ho dp

Di dp

φ2.102

(5)

Figure 6 The next step was to improve eqs. (2), (3) and (5) by changing the minimum number of significant coefficients in the moduli included in these equations. Accordingly, the significant modulus or moduli should first be identified by regression and discrimination analysis. The fitting of the correlations in Table 3 has been carried out by means of an algorithim written in Scilab. Figure 7 shows the information flow diagram. Figure 7 The fminsearch optimization subroutine (based on the Nelder-Mead algorithm) is used for the fitting of all the equations in Table 3. The procedure is based on a stepwise non-linear regression by minimizing the mean square residual, MSR, defined by the expression: n

MSR=

2 1 [( Nu)exp − ( Nu)cal ] ∑ n i= 1

(10)

where n is the number of experimental data, (Nu)exp is the experimental Nusselt value and (Nu)cal is the value predicted by the corresponding equation. The procedure for ascertaining the coefficients that should be changed has involved fitting only one coefficient (modulus exponent) at a time and choosing the one of higher significance based on a hypothesis testing approach. The Sum of Squares for the Residual with any number of regressors (SSR) is calculated by summing the squared deviations between the experimental Nusselt values, (Nu)exp, and those calculated with the regressors, (Nu)cal. The sum of squares due to the regression of each coefficient (SSRj - SSRi) is calculated, and then compared with the residual sum of squares (SSRi). The F statistics is calculated as follows:



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

SSR j− SSRi i− j F= SSRi n− i

Page 14 of 35

(11)

where j is the number of regressors in the original model and i is the number of regressors in the model whose fit is being checked. In the first step, the original model has no fitting parameters and the coefficients are introduced one by one. Therefore, j=0 and i=1. Accordingly, the number of degrees of freedom for the numerator is one, while for the denominator it is n-i, with n being the total number of experimental data. The value of the F statistics is compared with the critical F value for the corresponding degrees of freedom and 95% confidence interval. A value of the F statistics higher than the critical one means a significant decrease in MSR due to this regressor (coefficient). Once the regressor of highest significance has been determined, the next step involves retaining this regressor and selecting a new one based on the same procedure as that described for the first step. The fitting process continues until no significant regressor is found. This fitting method allows ascertaining the parameters affecting the Nusselt number according to their significance, and the fitting quality of the correlations according to their MSR. Based on this analysis, there is only one coefficient whose change provides a significant improvement in the fit of eqs. (2) and (3). Thus, a change in the coefficient corresponding to the modulus (Ho/dp) is significant in both equations (-0.38 instead of -1.19 in eq. (2) and -0.92 instead of -1.47 in eq. (3)). Concerning eq. (5), there is no change providing better fit. The parity plots corresponding to the improved eqs. (2) and (3) are shown in Figure 8. These improved equations provide a similar fit as eq. (5), with data stratification being especially significant. Figure 8 Data stratification in the parity plots is due to the significant change in the heat transfer coefficient within the bed, and this fact is not explained by any one of the correlations. Therefore, the next step has involved inserting two new moduli within eqs. (2), (3) and (5) and


Page 15 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


proving their significance. These moduli are the radial dimensionless position, r/R, and the longitudinal dimensionless position, z/Ho, with R being the radius of the contactor at the measurement level and Ho the height of the static bed. The radius R at any level is a function of the longitudinal position z as follows:

R= z tan

γ 2

()

(12)

The procedure for ascertaining the coefficients that should be changed to improve the fitting is the one described above. Table 4 shows the correlations of best fit obtained according to this procedure. A comparison of eq. (13) with the original equation (eq. (2)) reveals that three coefficients have been modified in this correlation, which are those corresponding to Pr, tan(γ/2) and (Ho/dp), whose coefficients are 0.13, 1.02 and -0.38 instead of 0.333, -0.813 and -1.19, respectively. Furthermore, the dimensionless longitudinal coordinate should be retained with an exponent of 1.34, but the radial one is not significant. Only two moduli required changing in eq. (14), namely, (Ho/dp) and (Do/dp), whose coefficients are -0.92 and 1.01 instead of the original 1.47 and 0.947, respectively. In this case also the longitudinal coordinate should be retained with an exponent of 1.54. In the case of eq. (15), there is no need to change any coefficient, but both the longitudinal and the radial dimensionless coordinates should be retained, with their coefficients being -0.51 for (r/R) and 4.76 for (z/Ho).

Table 4. Correlations of best fit obtained. Correlation

Nu= 0.897 Re

Nu= 0.045 Re

0.464

0.664

Pr

Pr

0.13

0.333

Ar

Ar

0.116

0.226

γ tan 2

1.02

)

γ tan 2

− 0.852

)

− 0.38

( )

(

(

Eq.

Ho dp

φ2.261

− 0.92

1.01

( ) () Ho dp

1.34

( ) z Ho

Di dp


2.304

φ

(13)

1.54

( ) z Ho

(14)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(

Nu= 2.673 Re0.516 Pr 0.333 Ar 0.033 tan

γ 2

− 2.331

)

− 1.334

0.602

( ) () Ho dp

Di dp

Page 16 of 35

φ2.102

− 0.51

4.76

() ( ) r R

z Ho

(15) Figure 9 compares the experimental values of the Nu number with those calculated using eqs. (13), (14) and (15). Figures 9a and 9b show that the insertion of the longitudinal modulus slightly improves the predictions provided by eqs. (13) and (14), but overall the fit in both cases is poor. Concerning eq. (15), two trends are observed. Thus, a group of data corresponding to the bed surface fit accurately eq. (15), but other data corresponding to locations within the bed follow a different trend. This means that the latter follow the same trend with the moduli as those corresponding to the bed surface, but they need a different multiplying coefficient. Figure 9 Accordingly, these data have been fitted to eq. (15) with the multiplying coefficient being the adjustable parameter. The new value for this parameter is 17.11 instead of 2.67. Figure 10 compares the experimental values of the Nu number with those calculated using eq. (15) with the multiplying coefficient 2.67 (bed surface) and 17.11 (within the bed). Figure 10 As observed, there is an excellent fit between the experimental Nu values and those calculated using two different proportionality constants. The correlation proposed for the heat transfer calculation is therefore as follows:

Nu= k Re

0.516

Pr

0.333

Ar

0.033

γ tan 2

(

− 2.331

)

− 1.334

0.602

( ) () Ho dp

Di dp

2.102

φ

− 0.51

4.76

() ( ) r R

z Ho

(16)

with k = 2.67 on the bed surface and k = 17.11 within the bed. This correlation allows determining the heat transfer coefficient for sand and sawdust beds at any location in the annulus, and therefore may be used in the simulation of conical spouted beds and for optimizing the best location of internal devices for heat recovery in this zone.


Page 17 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Conclusions Local convective heat transfer coefficients have been measured in the annular zone of a conical spouted by means of a specialy designed probe. The coefficients increase slightly as the radial position is closer to the spout-annulus interface, and significantly so as the longitudinal position in the bed is higher. The average values are 203 W m-2 K-1 for sawdust beds and 505 W m-2 K-1 for sand beds. The experimental values of heat transfer coefficient for sawdust/sand mixtures are lower than those expected according to the linear mass contribution of the components in the mixture, but they are slightly higher than expected for a volume contribution of the components in the mixture. This involves a different hydrodynamic behaviour of the mixture compared to pure sand and pure sawdust. The correlations in the literatures are not suitable for predicting local heat transfer coefficients in the bed. Accordingly, and based on a statistical analysis, the significant moduli have been identified in the relevant literature correlations and two new moduli related to the radial and longitudinal positions have been included. The analysis allowed identifying two sets of heat transfer coefficients: those within the bed and those on its surface. The correlation proposed accurately predicts local heat transfer coefficients and is specifically suitable for ascertaining the best location of heat transfer devices within the annulus of the spouted beds. Acknowledgment This work has been carried out with the financial support of the Ministry of Economy and Competitiveness of the Spanish Government (Project CTQ2013-45105-R), the European Regional Development Funds (ERDF), the University of the Basque Country (US14/37) and the University of Medellin (Project 854). Juan F. Saldarriaga is grateful to the UBC Fluidization Research Center for facilitating this work and for a Ph.D. grant from the Administrative



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Department of Science, Technology and Innovation, COLCIENCIAS (Colombia). Aitor Atxutegi is grateful for a Ph.D. grant from the University of the Basque Country (UPV/EHU).


Page 18 of 35

Page 19 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


References (1) (2) (3) (4)

(5)

(6)

(7) (8) (9) (10) (11)

(12)

(13) (14)

(15) (16) (17)

(18)

(19) (20) (21)

Zhao, J.; Lim, C.; Grace, J. Flow Regimes and Combustion Behavior in Coal-Burning Spouted and Spout Fluid Beds. Chem. Eng. Sci. 1987, 42 (12), 2865. Wu, R. Heat Transfer in Circulating Fluidized Beds, University of British Columbia: Vancouver, 1989. Ye, B.; Lim, C.; Grace, J. Hydrodynamics of Spouted and Spout Fluidized-Beds at HighTemperature. Can. J. Chem. Eng. 1992, 70 (5), 840. Olazar, M.; Lopez, G.; Altzibar, H.; Aguado, R.; Bilbao, J. Minimum Spouting Velocity Under Vacuum and High Temperature in Conical Spouted Beds. Can. J. Chem. Eng. 2009, 87 (4), 541. Saldarriaga, J. F.; Grace, J.; Lim, C. J.; Wang, Z.; Xu, N.; Atxutegi, A.; Aguado, R.; Olazar, M. Bed-to-Surface Heat Transfer in Conical Spouted Beds of Biomass-Sand Mixtures. Powder Technol. 2015, 283, 447. Tulasidas, T.; Kudra, T.; Raghavan, G.; Mujumdar, A. Effect of Bed Height on Simultaneous Heat and Mass-Transfer in a 2-Dimensional Spouted Bed Dryer. Int. Commun. Heat Mass Transf. 1993, 20 (1), 79. Freitas, L. a. P.; Freire, J. T. Gas-to-Particle Heat Transfer in the Draft Tube of a Spouted Bed. Dry. Technol. 2001, 19 (6), 1065. Szafran, R. G.; Kmiec, A. CFD Modeling of Heat and Mass Transfer in a Spouted Bed Dryer. Ind. Eng. Chem. Res. 2004, 43 (4), 1113. Prachayawarakorn, S.; Ruengnarong, S.; Soponronnarit, S. Characteristics of Heat Transfer in Two-Dimensional Spouted Bed. J. Food Eng. 2006, 76 (3), 327. Kmiec, A.; Englart, S.; Ludwinska, A. Mass Transfer During Air Humidification in Spouted Beds. Can. J. Chem. Eng. 2009, 87 (2), 163. Makibar, J.; Fernandez-Akarregi, A. R.; Alava, I.; Cueva, F.; Lopez, G.; Olazar, M. Investigations on Heat Transfer and Hydrodynamics under Pyrolysis Conditions of a Pilot-Plant Draft Tube Conical Spouted Bed Reactor. Chem. Eng. Process. 2011, 50 (8), 790. Olazar, M.; Aguado, R.; San, J.; Bilbao, J. Kinetic Study of Fast Pyrolysis of Sawdust in a Conical Spouted Bed Reactor in the Range 400-500 °C. J. Chem. Technol. Biotechnol. 2001, 76 (5), 469. Markowski, A. Drying Characteristics in a Jet-Spouted Bed Dryer. Can. J. Chem. Eng. 1992, 70 (5), 938. Devahastin, S.; Mujumdar, A. S.; Raghavan, G. S. V. Diffusion-Controlled Batch Drying of Particles in a Novel Rotating Jet Annular Spouted Bed. Dry. Technol. 1998, 16 (3–5), 525. Feng, H.; Tang, J.; Cavalieri, R. P.; Plumb, O. A. Heat and Mass Transport in Microwave Drying of Porous Materials in a Spouted Bed. Aiche J. 2001, 47 (7), 1499. Jumah, R. Y.; Raghavan, G. S. V. Analysis of Heat and Mass Transfer during Combined Microwave-Convective Spouted-Bed Drying. Dry. Technol. 2001, 19 (3–4), 485. Heyd, B.; Broyart, B.; Hernandez, J. A.; Valdovinos-Tijerino, B.; Trystram, G. Physical Model of Heat and Mass Transfer in a Spouted Bed Coffee Roaster. Dry. Technol. 2007, 25 (7–8), 1243. Kmiec, A.; Englart, S. Heat and Mass Transfer. In Spouted and spout-fluid beds Fundamentals and applications; Epstein, N., Grace, J. R., Eds.; Cambridge: Cambridge, 2011; pp 161–174. Englart, S.; Kmiec, A.; Ludwinska, A. Heat Transfer in Sprayed Spouted Beds. Can. J. Chem. Eng. 2009, 87 (2), 185. Kmiec, A.; Kucharski, J. Heat and Mass-Transfer During Coating of Tablets in a Spouted Bed. Inzynieria Chem. Proces. 1993, 14 (1), 47. Reger, O.; Romankov, P. G.; Rashkovskaya, N. B. Drying of Paste-like Materials on Inert Bodies in a Spouting Bed. Zhurnal Prikl. Khimii Leningr. 1967, 40, 2276.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(22) (23)

(24)

(25) (26) (27) (28)

(29)

(30)

(31)

(32) (33) (34)

Kmiec, A. Simultaneous Heat and Mass-Transfer in Spouted Beds. Can. J. Chem. Eng. 1975, 53 (1), 18. Kmiec, A. Bed Expansion and Heat and Mass Transfer in Fluidized Beds. In Scientific Papers of Institute of Chemical Engineering and Heating Equipment of Wroclaw Technical University; Monographs No 19; Publishing House of Wroclaw University of Technology: Wroclaw, Poland, 1980. Kmiec, A.; Jabarin, N. A. Hydrodynamics, Heat and Mass Transfer during Coating of Rings in a Spouted Bed. In Proceedings of the 12th International Drying Symposium; Kerkhof, P. J. A. M., Coumans, W. J., Moolweer, G. D., Eds.; Elsevier: Noordwijkerhout, Netherlands, 2000; p 1–13 (Paper No. 19). Kudra, T.; Mujumdar, A.; Raghavan, G. Gas-to-Particle Heat-Transfer in TwoDimensional Spouted Beds. Int. Commun. Heat Mass Transf. 1989, 16 (5), 731. Rocha, S.; Taranto, O.; Ayub, G. Aerodynamics and Heat-Transfer During Coating of Tablets in 2-Dimensional Spouted Bed. Can. J. Chem. Eng. 1995, 73 (3), 308. Kmiec, A. Hydrodynamics of Flows and Heat-Transfer in Spouted Beds. Chem. Eng. J. Biochem. Eng. J. 1980, 19 (3), 189. Hosseini, S. H.; Fattahi, M.; Ahmadi, G. CFD Study of Hydrodynamic and Heat Transfer in a 2D Spouted Bed: Assessment of Radial Distribution Function. J. Taiwan Inst. Chem. Eng. 2016, 58, 107. Saldarriaga, J. F.; Pablos, A.; Aguayo, A. T.; Aguado, R.; Olazar, M. Determinación de la densidad de partícula mediante porosimetría de mercurio para el estudio fluidodinámico de biomasa en lechos móviles. Av. En Cienc. E Ing. 2014, 5, 63. Olazar, M.; Arandes, J. M.; Zabala, G.; Aguayo, A. T.; Bilbao, J. Design and Operation of a Catalytic Polymerization Reactor in a Dilute Spouted Bed Regime. Ind. Eng. Chem. Res. 1997, 36 (5), 1637. Elordi, G.; Olazar, M.; Lopez, G.; Amutio, M.; Artetxe, M.; Aguado, R.; Bilbao, J. Catalytic Pyrolysis of HDPE in Continuous Mode over Zeolite Catalysts in a Conical Spouted Bed Reactor. J. Anal. Appl. Pyrolysis 2009, 85 (1–2), 345. Mickley, H.; Fairbanks, D. Mechanism of Heat Transfer to Fluidized Beds. Aiche J. 1955, 1 (3), 374. Epstein, N.; Mathur, K. Heat and Mass Transfer in Spouted Beds - Review. Can. J. Chem. Eng. 1971, 49 (4), 467. Mathur, K. B.; Epstein, N. Spouted Beds; Academic Press: New York, 1974.


Page 20 of 35

Page 21 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure Captions Figure 1.

Front and section views of the probe assembly consisting of plastic film, palladium-coated glass, and guard heater. All dimensions are in mm.

Figure 2.

Spouted bed reactor.

Figure 3.

Values of the heat transfer coefficient (W/m2K) in sawdust beds. Ho = 0.3 m, u/ums = 1.1.

Figure 4.

Values of the heat transfer coefficient (W/m2K) in sand beds. Ho = 0.3 m, u/ums = 1.2.

Figure 5.

Experimental values of heat transfer coefficient based on mass contribution (grey points), volume contribution (black points) and those expected according to the linear contribution of the components in the mixture (lines). Ho = 0.2 m, z = 0.2 m.

Figure 6.

Comparison of the experimental values of the Nu number with those calculated using eq (5).

Figure 7.

Algorithm developed for the stepwise fitting of literature correlations.

Figure 8.

Comparison of the experimental values of the Nu number with those calculated using improved eqs. (2) and (3).

Figure 9.

Comparison of the experimental values of the Nu number with those calculated using eqs. (13), (14) and (15).

Figure 10.

Comparison of the experimental values of the Nu number with those calculated using the proposed equation, eq. (16).



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 1. Saldarriaga e tal.


Page 22 of 35

Page 23 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 2. Saldarriaga et al.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



Page 24 of 35

Page 25 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60





1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



Page 26 of 35

Page 27 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60





1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



Page 28 of 35

Page 29 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 8a. Saldarriaga et al.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 8b. Saldarriaga et al.


Page 30 of 35

Page 31 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 9a. Saldarriaga et al.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure9b. Saldarriaga et al.


Page 32 of 35

Page 33 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 9c. Saldarriaga et al.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



Page 34 of 35

Page 35 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Graphical abstract


Correlation for Calculating Heat Transfer Coefficient in Conical

Recommend Documents