Simulation of 2-D Temperature Distributions in Fluidized Bed Reactors

782

Ind. Eng. Chem. Res. 1998, 37, 782-792

Simulation of 2-D Temperature Distributions in Fluidized Bed Reactors for Highly Exothermic Reactions Stefan Artlich, Ernst-Ulrich Hartge, and Joachim Werther* Technical University Hamburg-Harburg, D 20171 Hamburg, Germany

Fluidized bed reactors are generally known for their temperature uniformity, but with highly exothermic reactions, significant temperature gradients may occur. The generation of temperature distributions inside the fluidized bed is the result of a competition between two mechanisms: local feeding of reactants creates heat of reaction locally, and this heat must be dispersed inside the bed by solids mixing. The present work investigates these mechanisms for the case of coal combustion in pressurized bubbling fluidized beds. With a simple onedimensional carbon mass balance, the governing parameters for the generation of temperature profiles are deduced. For the quantitative calculation of temperature, carbon and oxygen distributions, a two-dimensional model was formulated. In pressurized fluidized beds, the fuel is often fed as a coal-water slurry, so the drying of the fuel also has been considered in this model. The model has been validated with measurements made in the pressurized fluidized bed combustor (PFBC) pilot-scale test unit of Deutsche Babcock. The results are in good agreement with measurements for full-load as well as part-load conditions. Feeding the coal as a slurry is shown to be a means to flatten the temperature distribution because the slurry agglomerates are dispersed in the fluidized bed while drying. Further simulation results point at and quantify critical parameters that have to be considered in the scaleup of pressurized fluidized bed combustors. 1. Introduction One of the main advantages of fluidized bed reactors for gas/solid reactions or heterogeneous catalytic reactions is their well-known temperature homogeneity (e.g., Kunii and Levenspiel, 1991). The reason for this homogeneous temperature distribution throughout the fluidized bed is the very intense solids mixing and the good heat transfer between gas and solids. However, for highly exothermic reactions, the heat transport by the solids may not always be sufficient to avoid the formation of temperature gradients or even hot spots inside the bed; a reason for this could be the high rate of heat release if the reactant is introduced locally into the fluidized bed. Furthermore, in such reactors, densely packed heat exchanger tube bundles that are needed to withdraw the heat from the reactor may hinder solids mixing and thereby reduce the heat transport across the reactor. In small-scale fluidized beds the absolute temperature difference may even be tolerable under these conditions. However, temperature differences will become significant in fluidized beds with large bed heights and/or diameters and may cause severe operational problems. An example of such a fluidized bed reactor where a highly exothermic reaction is carried out in a large bed that is densely packed with heatexchanger tubes is the pressurized fluidized bed combustor (PFBC). In a PFBC (Figure 1), coal is burned in a bubbling fluidized bed operated at pressures of up to 1.6 MPa. Compared with atmospheric bubbling fluidized bed combustors, the heat release per cross-sectional area is increased roughly proportional to the operating pressure and can thus reach values of up to 17 MWth per m2 bed * Corresponding author. Telephone: +49-40-7718 3039. Fax: +49-40-7718 2678. E-mail: [email protected].

Figure 1. Pressurized fluidized bed combustion test rig of Deutsche Babcock (HGF ) hot gas filter; Dehn et al. (1991)).

area. To make sufficient heat exchanger surface available the PFBC is normally operated with a bed height of ≈4 m. The coal is fed into the lower part of the fluidized bed, which is free from heat exchanger tubes. Two different systems are in use for fuel feeding. Alternatively, either predried coal is fed pneumatically or coal-water paste is pumped into the bed. The latter system is preferred for coals with high calorific values. Due to the high local heat release, the hindered solids mixing and the large bed height temperature differences across the fluidized bed may be expected and have indeed been experienced: inside different test facilities, temperature differences have been measured of up to 170 K between the coal feed point and the top of the

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Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 783

4-m high fluidized bed as well as of 70 K over the width of a bed with a 2 × 2 m cross-sectional area (Smith et al., 1982). This macro-scale nonisothermicity must be distinguished from the micro-scale nonisothermicity that is present in any fluidized bed coal combustor because the temperature of the burning coal particles is typically 50 to 100 K above the temperature of the surrounding ash particles (e.g., Basu, 1977). There are several models of PFBCs available in the literature; most of them, however, are one-dimensional models. In most cases, models of atmospheric fluidized bed combustors have been extended to higher pressure. For example, Miccio (1991) extended the IEA atmospheric fluidized bed model towards higher operating pressures. His model, like most other models for atmospheric fluidized bed combustors, neglects the existence of temperature gradients by simply assuming isothermicity for the whole bed. One-dimensional models that include the competition of heat generation by reaction and heat dispersion by solids mixing have been presented, for example, by Raven and Sparham (1983) and by Werther et al. (1989). Groenewald (1990) extended the latter model to two dimensions, and Itami et al. (1995) presented a generalization to the threedimensional case. However, in the latter two investigations, just the dry feed of coal has been considered. Whereas Itami et al. (1995) present qualitative results only, Groenewald (1990) reports severe numerical difficulties in the solution of his partial differential equations.

dxv of the reactor, MC is the molecular weight of carbon, kC is the reaction rate constant, CO2 is the oxygen concentration, and CC is the carbon concentration. 2.1. A One-Dimensional Model. The generation of temperature distributions inside the fluidized bed is the result of a competition of two mechanisms: local reaction creates heat and this heat must be dispersed inside the bed by solids mixing. If the local heat generation exceeds the amount that can be dispersed by solids mixing, a temperature gradient will build up. A simple one-dimensional model may illustrate the governing parameters. We consider a volume element in the fluidized bed. Inside this element, carbon is consumed by reaction. Across its boundaries, carbon is transported into the x-direction. With the assumptions just described, the carbon balance for this onedimensional model is given by:

0 ) Dx

δ2CC δx2

δ2C′C

Dx

dmC ) -MCkCmCCO2 dt

(1)

mC ) At dxv CC

(2)

(3)

where Dx is the coefficient of dispersion into the xdirection. Introducing a characteristic length L in the direction of carbon spreading (e.g., the spacing between neighboring coal feed points), with x′ ) x/L and a nondimensional carbon concentration C′C ) CC/CC,0 eq 3 can be rewritten as follows:

2. Theory To model the temperature inside the fluidized bed reactor, the heat release by the reaction and the heat transport inside the bed have to be considered. Heat transport may be described by a dispersion process, with the underlying idea being that heat is primarily transported by the motion of solids that can be described by a dispersion model. The fuel transport inside the bed may be treated as a dispersion process, too. In the present fluidized bed model, the gas and the solids phases are treated separately. No distinction was made between bubble and emulsion phases. This approach seems to be justified because in the underlying system of coarse bed particles, the mass transfer between bubbles and suspension phase is very fast. Further basic assumptions are that the gas flows as plug flow, that the temperature of gas is equal to the temperature of the solids at each location, and that fuel is transported by dispersion only. The model is currently restricted to vertical walls of the combustion chamber. For the sake of simplicity the overall carbon oxidation reaction C + O2 f CO2 has been taken into account only; that is, devolatilization processes as well as carbon monoxide formation and oxidation were neglected. A simple first-order rate expression was used for the carbon consumption:

- MCkCCCCO2

L2MCkCCO2 δx′2

- C′C ) 0

(4)

Replacing the effective reaction rate constant kC by the surface reaction rate constant kO (e.g., Avedesian and Davidson, 1973),

kC )

(

6

dp,fuelFfuelwC,fuel

)

dp,fuel 1 + kO(T,Tp) ShDG

-1

(5)

yields

δ2C′C

δx′2 L2MCCO2 Dx

(

dp,fuelFfuelwC,fuel

6

)


C′C ) 0 (6)

The carbon balance eq 6 means that a steep carbon concentration profile in the x-direction will occur if it holds for the following dimensionless quantity:

L2MCCO2 Dx

(

dp,fuelFfuelwC,fuel

6

)


.1

(7)

where

and mC is the carbon mass in the volume element At

Because the carbon concentration profile C′C(x′) is directly related to the temperature profile T(x′), the criterion of eq 7 means that a significant temperature profile may occur if the following conditions are met:

784 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998

as:

δ δ H˙ (x ,x )∆xh H˙ (x ,x )∆xvδxh C,h h v δxv C,v h v δ δ H˙ (x ,x )∆xh H˙ (x ,x )∆xv δxh BM,h h v δxv BM,v h v δ H˙ (x ,x )∆xh + Q˙ R(xh,xv) + Q˙ TB(xh,xv) + δxv RG,v h v Q˙ E(xh,xv) (10)

0)-

Figure 2. Enthalpy balance on a differential volume element of the fluidized bed reactor.

(1) Dx is small (i.e., if solids mixing is, for example, hindered by a dense packing of heat-exchanger tubes). (2) L is large (e.g., if the distance between neighboring coal feed points is too large or if the bed height is large). (3) dp,fuel is small (i.e., fine coals may simply burn too fast before they are dispersed in the bulk of the bed). (4) kO is large (i.e., highly reactive coals that are fed as fine particles may be critical). (5) CO2 is large (i.e., the steepest profiles will occur in the lower part of the combustor); a reduction of the steepness of the temperature profile may in this case be achieved by, for example, flue gas recirculation. Because CO2 is much larger in a PFBC than in atmospheric fluidized bed combustors, the risk of steep temperature profiles is much higher in the former case. 2.2. The Two-Dimensional Model. Whereas the one-dimensional model, eq 3, has been formulated to illustrate the mechanisms involved in the formation of temperature profiles, the following two-dimensional model has been developed for quantitative calculations. The model consists of mass balances for carbon and oxygen, respectively, and an enthalpy balance. The twodimensional mass balance for carbon reads:

0 ) Dh

δ2CC δxh2

+ Dv

δ2CC δxv2

- MCkC(T)CCCO2 + F˘C,F

δCO2 δxv

+ (1 - )kC(T)CCCO2

H˙ C,h ) -∆xvZ(1 - )cjp,CDh

δCC (x ,x )(T(xh,xv) - TB) δxh h v (11)

H˙ C,v ) -∆xhZ(1 - )cjp,CDv

δCC (x ,x )(T(xh,xv) - TB) δxv h v (12)

The constant Z is the depth of the fluidized bed (in the direction perpendicular to the xh, xv plane), T is the local temperature, cjp,C is the specific heat of carbon, and TB is a reference temperature (TB ) 25 °C). The heat transport by dispersion of the bed material can be expressed as:

H˙ BM,h(xh,xv) ) -∆xvZλh

δT (x ,x ) δxh h v

(13)

H˙ BM,v(xh,xv) ) -∆xhZλv

δT (x ,x ) δxv h v

(14)

with the effective thermal conductivity of the bed as:

λh ) (1 - )FBMcjp,BMDh

(15)

λv ) (1 - )FBMcjp,BMDv

(16)

(8) The fluidizing gas transports heat according to:

The first term describes the solids mixing in the horizontal, and the second one describes the solids mixing in the vertical direction. The carbon consumption by reaction is given in the third term, and F˘C,F describes a local carbon feed rate. The oxygen balance is much simpler because of the assumption of plug flow of the gas (i.e., without horizontal or vertical mixing). The balance consists, therefore, of a vertical convection term and a term describing the oxygen consumption by the combustion process only:

0)u

The heat transport by carbon dispersion is given by:

H˙ RG,v(xh,xv) ) u∆xhZFRGcjp,RG(T(xh,xv) - TB)

(17)

The heat generation by reaction is given by:

Q˙ R(xh,xv) ) ∆xh∆xvZ(1 - )MCkC(T)CCCO2(Hu/wC,fuel) (18) The heat extraction by the tube bundle is given by:

ATB Q˙ TB(xh,xv) ) -∆xh∆xvZ ξ(x )k (T - TKM) (19) BHTBZ v TB

(9)

The enthalpy balance is more complex (cf. Figure 2). It takes account of the heat transport by horizontal and vertical dispersion of bed material (index BM) and carbon, the heat transport by the upflowing gas (index RG), the heat generation by the reaction, Q˙ R, the heat extraction by the tube bundle, Q˙ TB, and an additional heat sink, Q˙ E, which will be explained in detail later. From Figure 2, the enthalpy balance can be deduced

where ATB is total surface area of the tube bundle, B is the width of the fluidized bed, HTB is the height of the tube bundle, kTB is the heat transfer coeficient, and TKM is the temperature of the fluid inside the tubes. As is illustrated in Figure 3, the function ξ(xv) describes the presence of the tube bundle; that is, ξ(xv) ) 1 for positions with tubes and ξ(xv) ) 0 for positions without tubes. Finally, the additional heat sink is given by:

Q˙ E(xh,xv) ) -∆xv∆xhZq˘ E(xh,xv)

(20)


At the bed surface it holds that:

λv

|

δT )0 δn Γ3

(26)

because the gas is leaving the bed with a temperature that is equal to the temperature of the bed at Γ3. Because the walls along Γ2, Γ4, and Γ6 are cooled membrane walls, the heat transported horizontally to the wall has to be equal to the heat extracted through the wall:

Figure 3. Sketch of the reactor geometry and definition of the boundaries Γ1-Γ6. The dimensions are those of Deutsche Babcock’s pilot PFBC.

Combining eqs 10 to 20 results in the final form of the enthalpy balance:

(

0 ) (1 - )cjp,C Dh

(

(

)

δ δCC (T - TB) + δxh δxh

))

δ δCC (T - TB) Dv δxv δxv

δ2T δ2T + λh + λv 2 δxh δxv2 δT 1 uFRGcjp,RG + (1 - )MCkC(T)CCCO2 H δxv wC,fuel u ATB ξ(x )k (T - TKM) - q˘ E (21) BHTBZ v TB The boundary conditions that are needed to solve the system of differential eqs 8, 9, and 21 may be explained with the help of Figure 3, where the boundaries are designated by Γ1 to Γ6 around the circumference of the reactor. The boundary conditions for the carbon balance are as follows: At the feed point (Γ5), the dispersive transport of carbon in the horizontal direction has to be equal to the carbon feed flux:

Dh

|

δCC δn

)Γ5

m ˘ fuelwC,fuel (1 - )HFEZ

|

δCC δn

Dv

Γ2,Γ4,Γ6

|

δCC δn

Γ1,Γ3

)0

(23)

CO2,0 )

0.21 F ML L

(24)

The enthalpy flow entering the fluidized bed through the gas distributor (i.e., the enthalpy flow of the incoming air) has to be balanced by the heat transport inside the bed, which gives:

λv

|

δT ) -uFRG(cjp,RGT|Γ1 - cjp,LTL) δn Γ1

|

δT ) δn Γ5 ˘ fuelwfuelcjp,C(T - TB)|Γ5 m ˘ fuelcjp,fuel(Tfuel - TB) - m

λh

HFEZ

(25)

(28)

2.3. The Submodel for Slurry Feeding. In the case of slurry feeding, the coal has to dry prior to its ignition. During this drying process, the wet fuel will already be dispersed in the system. To take this premixing into account, a submodel for the dispersion during coal drying is suggested. This submodel is based on the assumption of a known fixed drying time tdr. It is furthermore assumed that the mass flow of dried and heated carbon that leaves each local element for combustion is proportional to the local carbon concentration. The resulting local mass balance for the drying carbon can be written as:

δ2CC,dr δx2h

+ Dv

δ2CC,dr δx2v

-

1 C (x ,x ) (29) tdr C,dr h v

where CC,dr is the local mass concentration of wet carbon. This submodel is independently solved before solving the combustion model. The models are linked together by the local carbon flow that is leaving the drying model (i.e., the sink term in eq 29) and is entering as a local carbon feed flow:

F˘C,F )

The entry condition for the oxygen is given by:

(27)

The last border for which a boundary condition is needed is the feed point. Here an enthalpy flow is entering the fluidized bed with the coal. This enthalpy flow has to be balanced by the heat transport due to the dispersion of solids and of carbon:

0 ) Dh

)0

|

δT ) -kw(T - TKM)|Γ2,Γ4,Γ6 δn Γ2,Γ4,Γ6

(22)

where n denotes the normal to the boundary that is positive when orientated into the reactor. There is no carbon flow through any of the other boarders:

Dh

λh

1 C (x ,x ) tdr C,dr h v

(30)

into the combustion model. Another link is established by the local heat flux needed for evaporation in the drying model, which is treated as the additional heat sink q˘ E in eq 21 of the combustion model. The heat flow rate q˘ E leaving the combustion model is equal to the evaporation enthalpy flux h˙ H2O of the evaporated water in the drying model:

q˘ E(xh,xv) ) h˙ H2O(xh,xv)

(31)


The local evaporation enthalpy flux h˙ H2O is proportional to the absolute value of the gradient of the dispersive mass fluxes of the drying coal:

h˙ H2O(xh,xv) ∝

x(

) (

δCC,dr (xh,xv) δxh

Dh

2

+ Dv

)

δCC,dr (xh,xv) δxv

2

(32)

The total evaporation enthalpy flux required in the reactor for heating and evaporation of the water is the integral of the local enthalpy fluxes over the reactor volume and is equal to the enthalpy flux needed to heat and evaporate the water m ˘ H2O contained in the coal feed:

∫0H∫0Bh˙ H O(xh,xv) dxh dxv

h˙ H2O,total ) Z

2

Figure 4. Vertical solids mixing in cold model fluidized beds filled with bundles of horizontal tubes (after Groenewald (1990)).

) (cp,H2O(TS - Tfuel) + r + cp,V(T - TS))wH2O,fuelm ˘ fuel (33) Combining eqs 32 and 33 gives the local enthalpy flux: h˙ H2O(xh,xv) )

x( ∫ ∫ x(

Z

B

0

H

0

) ( ) (

δCC,dr (xh,xv) δxh

2

δCC,dr Dh (yh,yv) δxh

2

Dh

+ Dv

) )

δCC,dr (xh,xv) δxv

2

× 2 δCC,dr + Dv (yh,yv) dyh dyv δxv h˙ H2O,total (34)

3. Results and Discussion For the numerical simulation, the finite element method on a triangular mesh has been used. There have been tests with piecewise linear and quadratic basis functions that only differ in the convergence rate. The results that are presented in the following have been computed on a mesh with 8385 unknowns for each of the model components CC and T. The necessary CPU time for the entire model has been ∼13 min on a HP9000/735 workstation with a 40 Mflops PA-RISC processor. In the simulations, the dimensions of the 15 MWth pilot scale PFBC test facility operated by the Deutsche Babcock at their Friedrichsfeld site (cf., Dehn et al., 1991) has been used. The combustor had a bed height of 4.2 m, and a cross-sectional area of 1.45 m2 at the distributor level and of 2.2 m2 at the bed surface. This pilot plant has been chosen because there exist temperature profile measurements that could be used for the validation of the present model. 3.1. Determination of Model Parameters. Model parameters of the described model are the reaction rate constant, kc, and the coefficients Dh and Dv of dispersion in the horizontal and vertical directions, respectively. First of all, the reaction rate constant has been calculated from eq 5 using the kinetics suggested by Field et al. (1967):

(

)

T + Tp -149 470 exp k0 ) 596 2 RTp

(35)

A constant temperature difference (T - Tp) between the burning carbon particle and the bed of 100 K has been assumed.

Figure 5. Temperature profile calculated without consideration of slurry drying.

The horizontal solids dispersion coefficient Dh was calculated from the correlation by Werther et al. (1987), which relates Dh to local bubble sizes in a bed filled with horizontal tubes. The vertical solids dispersion coefficient Dv was taken from a correlation in Groenewald’s (1990) dissertation. Because this reference is not easily accessible, the correlation is given here. For a large three-dimensional bed with a bundle of horizontal tubes it was found that:

Dv ) 0.056(u j - umf)

(36)

j and umf are to be inserted in where Dv is in m2/s, and u m/s. Equation 36 has been derived from measurements with subliming CO2 tracer particles in a cold 1:1 model of Deutsche Babcock’s pilot-scale PFBC. The measured results are presented in Figure 4. Because the PFBC was equipped with inclined walls, an average fluidizing velocity u j was defined for the average height between the distributor level and the bed surface. Also included in Figure 4 are measurements taken in another cold model that was equipped with side-walls of variable inclination. The measurements show that there is no significant influence of the inclination of the walls if the measured dispersion coefficient is related to the average fluidizing velocity u j . The dispersion coefficients mea-


Figure 6. Influence of the drying time (tdr) on the temperature profiles.

Figure 7. Dry carbon feed flux (kg m-3 s-1) and evaporation enthalpy flux distribution (kW m-3) for a drying time (tdr) of 60 s.

sured in the latter cold model are lower than those measured in the PFBC cold model because a thickness of this bed of 0.3 m when compared to the bed height of 4 m results in a significant hindrance of bubble and solids motion in this nearly “two-dimensional” bed. 3.2. Validation of the Model. In a first approach, the model was tested with the operating conditions of the Babcock PFBC pilot plant without considering the actual coal slurry feeding. The resulting two-dimensional temperature distribution is shown in Figure 5. The temperature is seen to exceed 930 °C in the vicinity of the coal feed port. The maximum temperature difference inside the bed is >200 K. Because such temperature differences were never experienced in the PFBC pilot plant and because the steepest temperature gradient is observed near the coal feed point, it must be concluded that the smoothing effect of coal slurry drying has to be considered in the model. The effect of drying the coal slurry is clearly seen in Figure 6 where temperature distributions are plotted for drying times (tdr) of 10, 45, and 100 s. The flattening of the temperature distributions with increasing drying time is obvious. Figures 7 and 8 illustrate the underly-

ing mechanisms for a drying time of 60 s. The local coal feeding source in the case of dry coal feeding is replaced by a field of dry feed sources that are spread all over the area (or volume) of the combustor, as illustrated in the left-hand part of Figure 7 where the dry carbon mass flux distribution is shown which represents the twodimensional distribution of coal feed sources in the combustor model. The corresponding evaporation enthalpy flux distribution that is required in the drying process is shown in the right-hand part of Figure 7. The high heat flux required for drying in the vicinity of the coal feed port helps to keep the temperature down in this area. Figure 8 shows the two-dimensional concentration distributions of carbon and oxygen for the 60-s drying time. The carbon distribution reflects the fact that vertical mixing in the fluidized bed is significantly better than horizontal solids mixing (Kunii and Levenspiel, 1991). Oxygen, on the other hand is evenly distributed at the distributor level but is consumed very fast in the vicinity of the coal feed point. A characteristic oxygen concentration distribution results with a maximum concentration in the lower right-hand corner on the side


Figure 8. Carbon and oxygen distributions for a drying time (tdr) of 60 s.

Figure 9. Influence of the drying time on the calculated maximum in-bed temperature difference.

opposite to the feed point and a minimum level in the upper left-hand corner above the coal feed port. Figure 9 gives a summary of the influence of slurry drying on the temperature distribution by plotting the maximum in-bed temperature difference (Tmax - Tmin) against the drying time, tdr. With increasing drying time, the temperature difference is seen to approach an asymptotical value that under the present operating conditions is ∼25 K. This finite temperature difference results in the model calculation from the fact that the incoming air cools the bed at the distributor level and that heat is extracted by the tube bundle in the upper part of the bed only (cf., Figure 3). For practical PFBC operation, this small temperature difference is insignificant. Simple heat transfer and drying calculations for the particle agglomerates that are released at the coal slurry feed nozzle have shown that drying times of the order of 1 min are not unreasonable. Direct measurements of drying times under PFBC conditions are not available. However, Wirsum (1997) has recently reported results of an experimental investigation where he measured the drying times of sewage sludge agglomerates in an atmospheric fluidized bed combustor. Drying

times between 150 and 700 s were observed for sludge agglomerates with masses between 20 and 400 g. Sewage sludge is generally pumped as a paste into the combustor with piston pumps as they are used for concrete pumping. This procedure explains the presence of large agglomerates in the bed. Coal slurry agglomerates will certainly be significantly smaller. They will probably have masses of the order of several grams depending on the nozzle design of the slurry feeding arrangement. No better information was available, so all further calculations were carried out with a drying time, tdr, of 60 s. The model was validated with measurements of temperature distributions in Deutsche Babcock’s pilot PFBC that were made during a 100% load run. Numerical values of the model parameters used for this calculation are H ) 4.1 m; B ) 1.17 m; Z ) 1.55 m; HTB ) 3.15 m; HFE ) 0.2 m; m ˘ fuel ) 0.57 kg/s; m ˘ air ) 5.69 kg/s; wC,fuel ) 0.65; wH2O,fuel ) 0.14; Hu of the coal ) 38 400 kJ/kgC, Hu of the coal-water paste ) 37 300 kJ/ kgC; dp,fuel ) 0.758 mm; Tfuel ) 305.2 K; Ffuel ) 1,280 kg/m3; FΒΜ ) 2,600 kg/m3, kTB ) 357 W/m2 K; kW ) 30 W/m2 K. The calculation was fitted to the measurements by varying the two dispersion coefficients Dh and Dv. A best fit was obtained for Dh ) 0.011 m2/s and Dv ) 0.059 m2/s. Figure 10 shows the comparison between measurements and calculation. The scatter of the measured temperatures is quite large, but it is understandable. The temperatures had to be measured inside a densely packed tube bundle where the distances between neighboring tubes were

Simulation of 2-D Temperature Distributions in Fluidized Bed Reactors

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