A three-dimensional model for suspension-to-membrane-wall heat transfer in circulating fluidized beds

Chemical Engineering Science 58 (2003) 4247 – 4258

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A three-dimensional model for suspension-to-membrane-wall heat transfer in circulating &uidized beds D. Xie, B. D. Bowen, J. R. Grace∗ , C. J. Lim Department of Chemical and Biological Engineering, University of British Columbia, 2216 Main Mall, Vancouver, BC, Canada V6T 1Z4 Received 4 October 2002; received in revised form 5 May 2003; accepted 15 May 2003

Abstract A three-dimensional model is proposed for both furnace-side and wall-side heat transfer in circulating &uidized beds with membrane walls. Following previous publications (Int. J. Heat Mass Transfer (2003a, b)), a core-annulus &ow structure is employed in the model, with consideration of the membrane wall in&uence on bed hydrodynamics. The model couples radiation, conduction and convection on the furnace side to conduction and convection on the wall side. Radiation in the wall layer is simulated by the moment method. A 9nite-element method is employed to solve the set of non-linear, partial di:erential equations. The solution is demonstrated for a typical example. The model gives predictions of suspension-to-wall heat transfer which show satisfactory agreement with published experimental data. ? 2003 Published by Elsevier Ltd. Keywords: Heat transfer; Modeling; Fluidization; Radiation

1. Introduction Heat extraction in circulating &uidized bed combustors is usually accomplished by using membrane walls composed of parallel tubes connected by 9ns. The correct sizing of these heat transfer surfaces is important to ensure proper operation, load turndown, and optimization of CFB boiler systems. It is therefore essential to understand the mechanisms of heat transfer between the gas–solid suspension and the membrane wall surface, to consider heat conduction in the membrane walls, and to develop an appropriate model to predict the rate of heat transfer. Several models of CFB heat transfer have been proposed and reviewed (e.g., Basu & Nag, 1996; Xie, 2001). In a previous paper (Xie, Bowen, Grace, & Lim, 2003a), a model was proposed which coupled radiation, conduction and convection from the hot core on the furnace side to conduction and convection into the coolant on the wall side. However, that model was two dimensional, treating only risers having smooth walls (including &at surfaces and the wall of a cylindrical riser). None of the previous CFB heat transfer ∗ Corresponding author. Tel.: +1-604-822-3121; fax: +1-604-822-6003. E-mail address: [email protected] (J. R. Grace).

0009-2509/$ - see front matter ? 2003 Published by Elsevier Ltd. doi:10.1016/S0009-2509(03)00310-5

models has considered the three-dimensional geometry of membrane walls. This paper extends the Xie et al. (2003a) model to accommodate the three-dimensional membrane wall geometry. 2. Model development 2.1. CFB hydrodynamics with membrane wall Experiments in CFB combustors (Weimer, Bixler, Pettit, & Wang, 1991) reveal that vertical waterwall surfaces experience very little wear. This suggests either that few particles actually touch the wall or that the particle velocity adjacent to the wall is not very high. Lints and Glicksman (1994) showed that there exists a particle-free gas layer along the wall of thickness 0.3–1.0dp , depending on the overall suspension density. Thus, it is common to assume a stagnant gas gap between the dense layer and the wall. For CFB risers with &at walls, the two-dimensional model of Xie et al. (2003a) assumed a wall layer of particles whose concentration varies with distance from the heat transfer surface, with a thin gas gap separating this layer from the wall. Particles and gas were assumed to descend at di:erent velocities in the wall layer. Particles exchange heat with the gas

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by convection and participate in the radiation heat transfer process from the core to the wall through the wall layer. The gas receives heat from the core by conduction. Heat transferred by the gas through the particle layer is then conducted to the coolant through the stagnant gas gap and the solid wall. Whether or not they have membrane-containing surfaces, CFB risers have a core-annulus hydrodynamic structure, and the heat transfer mechanisms described in the earlier paper (Xie et al., 2003a) still apply. However, the geometry of the membrane has a profound in&uence on the dynamics of gas and particle &ows, especially in the wall layer. Observations and measurements have shown that the time-averaged solids volumetric concentration at the junction between the 9n and the tube is much higher than at the crest or at the center of the 9n (e.g., Lockhart, Zhu, Brereton, Lim, & Grace, 1995; Wu et al., 1991; Zhou, Grace, Brereton, & Lim, 1996). The membrane wall also in&uences the average particle residence length in the 9n and crest regions. Visual observations near the walls of a 12 MWth boiler (Golriz, 1994) show that particles stay in the 9n region longer than near the crest of the tubes. These observations suggest that heat transfer results or models developed for smooth surfaces may fail to adequately represent the behavior in industrial units with membrane walls or rough heat transfer surfaces (Grace, 1990). 2.2. Thermal radiation in wall layer with membrane wall As discussed by Xie et al. (2003a), the particulate suspension in the annular layer near the heat transfer surface constitutes an emitting, absorbing and scattering gray medium. Radiation from the 9n and tube surfaces as well as from the core must be considered together. When all these factors are taken into account, the equation of two-dimensional radiation transfer becomes di
di
cos ’ + cos  sin ’ dx dy = − ai
(x; y) + aib
(x; y) − s i
(x; y) s + 4




4

!i =0

i
(x; y)(!; !i ) d!i ;

(1)

where ’ is the polar angle measured from the positive x-axis and  is the azimuthal angle. By application of the lowest-order moment method, Eq. (1) becomes (for details, see Xie, 2001) @ @x



+

1 1 @G 3 a + (1 + f)s @x @ @y





1 1 @G 3 a + (1 + f)s @y



y

Γ1 1 Bulk

Γ88 Wall Layer Ω1

Ω2

Γ7 7 Γ99

Gas Gap

Ω3

Ri

Ω4

Γ2 2

Γ10 10

Ro

Γ6 6

t

Γ3 3

Membrane Wall

x

Coolant

Γ44 Γ5 5

Fig. 1. Plan view of membrane wall and wall-layer assembly (: domain; : boundary).

The Marshak boundary condition for this method is @G 1 qr = − 3(a + (1 + f)s ) @n =

ew (G − 40 Tw4 ): 2(2 − ew )

(3)

2.3. Governing equations Consider the plan view of the membrane wall and wall layer shown in Fig. 1. Because of symmetry, the solution domain corresponds to a half-tube and a half-9n plus the associated wall layer. Heat balance on gas in wall layer (1): Consider a Cartesian control volume in the wall layer as shown in Fig. 2. The energy balance for the control volume ignoring heat generation (e.g., due to reaction) and conduction in the z-direction can be written as   @Tg @Tg @ g Cpg ug − kg @z @x @x   @Tg @ − − Qpg = 0; kg (4) @y @y where Qpg is the volumetric rate of heat convection from particles to gas given by

= a(G − 40 Tp4 ):

(2)

Qpg = hpg s(Tp − Tg ) = 6 Nupg (1 − ) kg (Tp − Tg )=d2p (5) with the particles assumed to be spheres of uniform size.

D. Xie et al. / Chemical Engineering Science 58 (2003) 4247 – 4258

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Bulk side of wall layer, at z ¿ 0, (x; y) ⊂ 1: Tube

Tg = Tp = Tb = f(z);

Coolant

Fin

Gas Gap z

qr = −

Wall layer

=

y x

dy

dx dz

ug up

Fig. 2. Three-dimensional view of membrane wall and wall layer showing Cartesian control volume.

Heat balance on particles in the wall layer (1): For the same control volume as for the gas, considering the volumetric energy input due to fresh particles having temperature Tb , heat loss due to radiation, and particle-to-gas convection, the energy balance for the particles in the wall layer is p Cpp cup

@Tp + Ex Cpp (Tp − Tb ) + Qpg + ∇ · qr = 0; @z (6)

where the divergence of the radiative &ux is given by
4 ∇ · qr = 4aeb − a i
(!; S) d!

(14)

@G 1 3(a + (1 + f)s ) @n

eb (G − 40 Tb4 ): 2(2 − eb )

(15)

Symmetry boundaries of wall layer, at z ¿ 0, (x; y) ⊂ 2; 8: @Tg @G = = 0: (16) @x @x Membrane wall–wall-layer interface, at z ¿ 0, (x; y) ⊂ 9: @Tg @Tw + qr ; (17) = kg kw @n @n @G 1 qr = − 3(a + (1 + f)s ) @n =

ew (G − 40 Tw4 ): 2(2 − ew )

(18)

Insulated or symmetry boundaries of membrane wall, at z ¿ 0, (x; y) ⊂ 3; 4; 5; 7: @Tw = 0: (19) @n Cooling water–membrane wall interface, at z ¿ 0, (x; y) ⊂ 6: @Tw q = −kw (20) = hc (Tw − Tc ); @n where hc can be calculated from standard correlations.

!=0

= 4aeb − aG = 4a0 Tp4 − aG and Ex (see Xie et al., 2003a) can be obtained from   c(*) dc(*) : Ex = p up + Lar dz Heat conduction through gas gap (2):     @Tg @Tg @ @ kg + kg = 0: @x @x @y @y Heat conduction through membrane wall (3):     @ @ @Tw @Tw kw + kw = 0: @x @x @y @y Heat balance on cooling water (4): dTc Ri c Cpc uc + 2q = 0: dz

(7)

(8)

(9)

(10)

(11)

Boundary and interface conditions: Top of heat transfer zone, at z = 0, (x; y) ⊂ 1: Tg = T p = T b ;

(12)

Tc = Tc; out :

(13)

2.4. Parameter determination For the membrane wall geometry, some parameters need to be adjusted from those used in the two-dimensional model. Except where otherwise indicated, the parameters in Eqs. (6)–(20) are the same as those described by Xie et al. (2003a). 1. Voidage distribution in wall layer: It is commonly agreed (e.g., Wu et al., 1991; Lockhart et al., 1995; Zhou et al., 1996) that the particle concentration in the 9n region is higher than in the crest region. However, no quantitative correlations are available to describe the particle concentration pro9le in the wall layer near a membrane wall. Hence, the correlation of Issangya, Grace, Bai, and Zhu (2001), (−1:5+2:1* (*) = mf + (sec − mf )sec

3:1

+5:0*8:8 )

(21)

for &at walls, is again employed to predict the particle concentration in the wall layer. In Eq. (21), the particle concentration is only a function of the dimensionless distance, * = 1 − (y − t)=R from the center of the riser. Since (y − t) is measured from the 9n surface, the particle concentration predicted for the 9n region is higher than for the crest region, in agreement with experimental observations.

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2. Wall-layer geometry: For membrane wall heat transfer surfaces, the annular layer on the 9n surface will be a little thicker than at the tube crest. For a relatively thin wall layer, i.e., for /=Ro 6 2, we approximate the outer edge of this layer as part of a circle concentric with the tube, with the correlation of Bi, Zhou, Qin, and Grace, (1996),  / = 0:5[1 − 1:34 − 1:30(1 − sec )0:2 + (1 − sec )1:4 ] D (0:0015 ¡ 1 − sec ¡ 0:2)

(22)

selected to predict the mean wall-layer thickness. For a relatively thick wall layer, i.e., /=Ro ¿ 2, the boundary is simply assumed to be &at (i.e., parallel to the x-axis). The sensitivity analysis by Xie (2001) showed that the wall-layer thickness does not in&uence the heat transfer coeOcient, so long as it is signi9cantly larger than the thickness of the thermal boundary layer. 3. Average particle residence length Lar : For &at walls, the average particle residence length was assumed to be 1:5 m (Xie et al., 2003a). The particle residence length in the 9n region is expected to be larger than this value since the membrane tubes prevent particles from exchanging with the core (e.g., Golriz, 1994). On the other hand, the residence length in the crest region is shorter because the tube extends further into the reactor and the particle motion there is more strongly in&uenced by the upward moving particles and gas in the core region. For the present study, we assume  2:0 m in the 9n region; x ¿ Ro ; Lar = (23) 1:0 m in the crest region; x 6 Ro : 4. Sensitivity analysis: The in&uence of di:erent parameters on the suspension-to-wall heat transfer coeOcient was investigated previously with respect to the simple two-dimensional model (Xie, 2001; Xie, Bowen, Grace, & Lim, 2003b). Suspension density and bulk temperature were found to have the most signi9cant in&uence on heat transfer. Particle size, volumetric heat capacity and average residence length also have major e:ects. Gas-gap thickness, particle downward velocity and particle emissivity have moderate e:ects, while gas downward velocity in the wall layer, wall layer thickness, wall-side thermal resistance and water-side heat transfer coeOcient all have insigni9cant in&uence. These trends are also applicable in the present case.

3. Numerical methods Eqs. (6)–(20) couple reactor-side radiation, gas conduction, gas and particle convection, particle-to-gas convection and wall-side conduction. There are fourth-power radiation terms, as well as parameters such as gas conductivity, heat capacity and density which are functions of temperature. Hence, this set of equations is nonlinear and highly coupled, and can only be solved numerically. The Galerkin

9nite-element method was employed to solve the conduction problem in the membrane wall and annular wall layer, as well as the radiation problem in the wall layer. To reduce the number of 9nite elements and increase the calculation accuracy, six-noded triangular elements and quadratic shape functions were employed, i.e., the temperature or irradiance distribution in each element was assumed to be T (x; y) or G(x; y) = C1 x2 + C2 y2 + C3 xy + C4 x + C5 y + C6 :

(24)

The local approximations of the solution over each element were assembled for the entire mesh by adding up the contributions by each element. The gas convection term @Tg =@z in the resulting equation was approximated by a Galerkin 9nite-di:erence method that has an error of O(Pz 2 ) and is unconditionally stable. Since there are no conduction terms in the particle heat balance, equation (6), it was also approximated by a Galerkin 9nite-di:erence method. With these numerical approximations, the governing equations reduce to a set of algebraic equations which can be solved iteratively (see Xie, 2001). Owing to the high thermal conductivity of the membrane wall, the temperature variations there are small, and hence its elements can be relatively coarse. A much 9ner mesh is needed in the layer nearest the wall, where the temperature and irradiance gradients are expected to be large. In the wall layer closer to the bulk, where more uniform temperature and irradiance distributions are expected, relatively coarse elements can once again be employed. A two-level algorithm was used in the vertical direction, with the solution at a particular height z calculated from the temperatures and irradiances at the previous height z − Pz. At the top of the heat transfer surface (i.e., at z = 0), the gas temperature near the wall decreases sharply along the heat transfer surface because of the large temperature di:erence between the gas and the wall. Therefore, Pz needs to be very small near z = 0. With increasing distance below z = 0, the temperature di:erence decreases, so that Pz can be enlarged. To obtain smoothly increasing values of Pz, a Fibonacci series (i.e., Pz(i) = Pz(i − 1) + Pz(i − 2)) was employed until a pre-set maximum step-size Pzmax was reached. 4. Predictions for a pilot-scale CFB unit 4.1. Case description The unit simulated was the pilot-scale CFBC facility previously located in the Pulp and Paper Center at the University of British Columbia. The riser was 7:3 m in height and its cross-section was 0:152 m × 0:152 m. The experimental membrane wall consisted of pipes and connecting 9ns, divided into two adjacent sections. Two stainless steel 347 pipes (21:3 mm OD and 14:1 mm ID) and three 9ns (6:4 mm half-width) of the same material were welded

D. Xie et al. / Chemical Engineering Science 58 (2003) 4247 – 4258

4251

Table 1 Base parameters used in example calculations below Particle diameter Particle heat capacity Particle thermal conductivity Suspension density Gas velocity in wall layer Bulk temperature Conductivity of wall Water outlet temperature Water velocity in tube

286 m 840 J kg−1 K −1 1:9 W m−1 K −1 52:5 kg m−3 0:4 m s−1 1077 K 21 W m−1 K −1 80◦ C 0:5 m s−1

0.85 2610 kg m−3 1:2 m s−1 10:5 mm 77:1 m 0.99 0.90 Fin: 2 m; Crest: 1 m

90

Heat flux normal to surface (kW/m2)

0.02

0.015

0.01

y (m)

Particle emissivity Particle density Particle velocity in wall layer Wall-layer thickness Gas-gap thickness Bulk emissivity Wall-surface emissivity Particle average residence length

0.005

80

Total Radiation

70

Conduction 60

through gas gap

50 40 30

Tube

Fin

20 0

5

10

15

20

25

Distance travelled around surface from crest (mm)

0

Fig. 4. Lateral variation of heat &ux along membrane wall surface at z = 1:5 m for base case conditions listed in Table 1.

-0.005

4.2. Heat :ux distribution -0.01 0

0.005

0.01 x (m)

0.015

Fig. 3. Finite-element mesh and nodes for membrane wall and wall layer.

together to form half of the membrane wall, while the other half was made of stainless steel 316 pipes and 9ns of the same dimensions. The total length of the membrane section was 1:626 m, and the total exposed area of each half of the membrane wall was 0:150 m2 . The membrane wall was cooled by water &owing upward through the tubes. More details are provided by Luan, Bowen, Lim, Brereton, and Grace (2000). To simplify the calculations, the outlet water temperature was 9xed, rather than the inlet temperature. Other key parameters are typical of those encountered in pilot-scale circulating &uidized bed combustors and are listed in Table 1. The 9nite-element mesh generated in the membrane wall and wall layer, and the corresponding nodes are shown in Fig. 3. The grid contained 120 elements and 281 nodes. Pz(1) was 1 × 10−8 m and Pzmax was 0:1 m.

The predicted lateral heat &ux distribution at di:erent parts of the exposed pipe and 9n surfaces at z = 1:5 m is illustrated in Fig. 4. From the crest of the pipe to the junction of the tube and 9n, the conductive heat &ux increases due to the e:ect of particle concentration. The conductive heat &ux is minimal on the 9n side of the junction, and then increases slightly toward the center of the 9n. The radiative heat &ux has its lowest value at the junction since the particles have their lowest temperature there, i.e., the junction is shielded by the cool particles around it. The particles near the crest of the tube are lower in concentration and higher in temperature; hence, the radiative heat &ux is highest there. The predicted conductive, radiative and total heat &uxes, obtained by integrating the local values tangentially over the pipe and 9n surfaces, are plotted as functions of vertical distance in Fig. 5. In general, both the conductive and radiative heat &uxes decrease in descending from z = 0, due to particle cooling along the surface. The conductive heat &ux decreases very rapidly near z = 0 because, when the particles and gas enter the wall layer, they are assumed to have the same temperature as the CFB core, a value substantially higher than the wall temperature. The radiative heat &ux is predicted to increase for a short distance near z = 0

4252

D. Xie et al. / Chemical Engineering Science 58 (2003) 4247 – 4258 2.0

0.0

20 0.5

1.0

through gas gap Radiation Total

0.5

1.5

10

y (mm)

Conduction

1.0

15 z (m)

Height (m)

1.5

5

0 0.0 30

60

90

120

2.0 180

150

Heat flux (kw/m2)

-5

Fig. 5. Vertical variation of integral heat &uxes along the wall for base case conditions given in Table 1.

-10 0

5

10

15

20

25

x (mm)

22

Irradiance (kW/m2)

20

Fig. 7. Gas and wall temperature distributions at z = 1:5 m for base case conditions listed in Table 1.

18

y (mm)

16 14 12 10 8 6 4 2 0

5

10

15

20

25

x (mm)

Fig. 6. Irradiance distribution at z = 1:5 m for base case conditions listed in Table 1.

because the wall temperature initially decreases sharply with distance from the top of the membrane wall.

core, but drops sharply near the membrane wall. The predicted wall temperature is close to that of the water. There has been concern (e.g., Grace, 1990) that radiant exchange between the tube and the 9n might a:ect the overall heat transfer between the suspension and the membrane wall. The predicted wall surface temperature shows that the two portion of the membrane surface have very similar temperatures. Therefore, the radiation exchange between the tube and the 9n is negligible. This surface temperature similarity also suggests that if some parameters, such as an appropriate particle residence time and equivalent wall thickness, can be selected, the corresponding two-dimensional model (Xie et al., 2003a) can give reasonable predictions for many three-dimensional membrane wall cases of practical interest.

4.3. Irradiance distribution

5. Comparison of model prediction with experiments

Fig. 6 shows the irradiance distribution in the wall layer at z = 1:5 m. The irradiance is the sum of incoming radiation &uxes integrated over all solid angles. The irradiance is predicted to be high on the core side of the wall layer where the particle temperatures are high and the radiation from the bulk is strong, and low on the wall side, since particles there have low temperatures and the radiation from the wall is weak. The irradiance is predicted to have its lowest value at the junction of the tube and 9n.

5.1. Comparison with data of Wu, Lim, Chaouki, and Grace (1987), Wu, Grace, Lim, and Brereton (1989)

4.4. Gas and wall temperature distribution Fig. 7 shows the gas and wall temperatures at z = 1:5 m. The gas temperature pro9le is quite uniform close to the

Wu et al. (1987) reported experimental suspension-tomembrane water wall heat transfer coeOcients obtained from the CFBC facility described above. Heat transfer data were obtained from two nearly identical surfaces, each 1:53 m long and 148 mm wide located on one wall of the column and beginning at 1.22 and 4:27 m, respectively, above the distributor plate. One of the central tubes in the upper surface was instrumented with twelve thermocouples, approximately 150 mm apart, to measure the vertical temperature distribution of the cooling water. The cooling water &ow rate was also recorded. This information enabled

D. Xie et al. / Chemical Engineering Science 58 (2003) 4247 – 4258 200

4253

2.0

0.0

1.5

0.5

(A)

50 300

(B)

250

z (m)

100

Height (m)

Heat transfer coefficient (W/m2K)

150

1.0

1.0

(A)

(B)

200

0.5

150

1.5

100

100

50

0 50

100

150

Fig. 8. Comparison of predicted average heat transfer coeOcients (•) with experimental data of Wu et al. (1987) for sand particles (◦). Gas temperature: 150 –400◦ C. A: dp = 356 m; B: dp = 188 m.

the average heat transfer coeOcient between thermocouples 1 and 12 or between any pair of thermocouples to be calculated. The suspension-to-wall heat transfer coeOcient was determined from h=

1 ; 1=U − Ro =(Ri hc ) − Ro ln(Ro =Ri )=kw

(25)

where U=

mc Cpc (Tc; out − Tc; in ) Ao (Tb − Tc )

300

400

200

Suspension density (kg/m3)

(26)

is the suspension-to-water heat transfer coeOcient. The experimental gas temperatures were between 150◦ C and 400◦ C. Fig. 8 compares the model predictions and experimental results for the column average heat transfer coef9cients. For both particle sizes, the model overestimates the heat transfer for low suspension density conditions and underestimates for higher suspension densities. Wu et al. (1989) reported experimental average and local heat transfer coeOcients obtained with the same CFB facility and membrane surface under higher suspension temperature conditions. Fig. 9 compares the model predictions and experimental heat transfer coeOcients averaged over the interval between z = 0 and z for sus = 54 kg m−3 . Since the stainless steel wall emissivity was unknown, two values, 0.5 and 0.9, were tried. The actual value likely lies between these two extremes. The model predicts the experimental average heat transfer coeOcients very well with ew = 0:5. Fig. 10 compares the predicted column-average heat transfer coeOcients with experimental results. A wall emissivity of 0.5 is used in the simulations. The model slightly underpredicts the experimental results.

Fig. 9. Comparison of predicted average heat transfer coeOcients with experimental data (solid circles) of Wu et al. (1989) for sus =54 kg m−3 . Solid line: ew = 0:5; dashed line: ew = 0:9; A: Tb = 860◦ C; B: Tb = 407◦ C.

Heat transfer coefficient (W/m2K)

0

200 300 400 100 200 Heat transfer coefficient (W/m2K)

180 150 120 90 60 30 0 150 120 90 60 30 0 150 120 90 60 30 0

(A)

(B)

(C)

0

10

20

30

40

50

60

70

80

Suspension density (kg/m3)

Fig. 10. Comparison of predicted average heat transfer coeOcients (•) for ew = 0:5 with experimental data of Wu et al. (1989) (◦). A: Tb = 870 ± 14◦ C; B: Tb = 681 ± 18◦ C; C: Tb = 410 ± 15◦ C.

5.2. Comparison with data of Luan (1997) Luan (1997) carried out experiments in the same UBC CFBC facility as Wu et al.(1987, 1989). Like Wu et al. (1989), Luan (1997) reported local heat transfer coef9cients based on the cooling water temperature change measured between adjacent thermocouples. To maintain consistency, experimental average heat transfer coeOcients (i.e., integrated over z) were calculated from Eq. (25) as well. Fig. 11 compares the model-predicted average coeOcients with the experimental results. In these simulations, the wall emissivity was taken as 0.9, a reasonable value for stainless steel 347 oxidized at high temperature. 5.3. Comparison with data of Andersson and Leckner (1992) Andersson and Leckner (1992) reported the results of experiments carried out in a 12 MWth circulating &uidized

2.0

0.0

1.5

0.5

1.0

1.0

(A)

z (m)

D. Xie et al. / Chemical Engineering Science 58 (2003) 4247 – 4258

Height (m)

4254

(B)

0.5

1.5

100

200

300

400

100

200

300

400

2.0

0.0

1.5

0.5

1.0

1.0

(C)

z (m)

Height (m)

Heat transfer coefficient (W/m2K)

(D)

0.5

1.5

100

200

300

400

100

200

300

400

Heat transfer coefficient (W/m2K)

Fig. 11. Comparison of predicted average heat transfer coeOcients (lines) with experimental data (points) of Luan (1997). A: Tb = 804◦ C, sus = 52 kg m−3 ; B: Tb = 706◦ C, sus = 52 kg m−3 ; C: Tb = 804◦ C, sus = 22 kg m−3 ; D: Tb = 706◦ C, sus = 22 kg m−3 .

bed boiler at Chalmers University of Technology in Sweden. The combustion chamber, consisting of membrane tube walls, has a cross-section of 1:7 m by 1:7 m and is 13:5 m tall. The outer tube diameter is 60:3 mm, tube wall

12

thickness 5:6 mm, 9n length 8:8 mm, 9n thickness 6:0 mm and the radius of the 9n-tube junction 4:0 mm. The bed material was silica sand with a density of 2600 kg m−3 , and the fuel was 0 –20 mm bituminous coal. The suspension densities in the case considered here varied from 45 kg m−3 at 2 m to 10 kg m−3 at 11 m above the base. Four methods: (1) heat balance; (2) local heat &ow meters; (3) local water temperature and (4) 9n-tube temperature di:erence—were employed to determine the heat transfer coeOcients (for details, see Andersson & Leckner, 1992). To further analyze the heat transfer in the 9n region, a heat &ow meter was positioned at Z = 3:8 m where the suspension density was about 25 kg m−3 . A 48 mm diameter tubular obstacle was inserted into the combustion chamber 0:1 m from the 9n surface and 0:5 m above the meter. This arrangement increased the heat transfer coeOcients measured by the heat &ow meter by 50% as shown by the solid diamond in panel A of Fig. 12. Fig. 12 compares the predicted heat transfer coeOcients with the experimental results obtained by Andersson and Leckner (1992). Since the wall emissivity was unknown, values of 0.6 and 0.85 were tried. Method 2 actually measured the heat transfer coeOcient at the 9n surface. Hence, only the predicted 9n heat transfer coeOcients are shown in panel A of Fig. 12. When the obstacle was 0:5 m above the heat &ux meter, it disturbed the wall layer near the 9n causing fresh particles to be introduced from the core. Hence, a new heat transfer process starts from the obstacle. Lines 1 and 2 in the 9gure show the predicted heat transfer coeOcients based on the assumption of complete particle renewal at the level of the obstacle. Experimental results from the same boiler

12

(A)

(B)

(total)

10

10

8

8

Height (m)

Height (m)

(fin)

6

6

1 4

4

2 2

2

0

50

100

150

200

250

300

Heat transfer coefficient

0

50

100

150

200

250

300

(W/m2K)

Fig. 12. Comparison of predicted local heat transfer coeOcients with experimental data of Andersson and Leckner (1992). ◦: method 2; +: method 3; ×: method 4; solid diamond: method 2 with obstacle 0:5 m above meter. Dashed lines: prediction for ew = 0:6; solid lines: prediction for ew = 0:85. A: heat transfer coeOcient at 9n; B: total heat transfer coeOcient for 9n and tube. Lines 1 and 2: Predictions assume particle renewal 0:5 m above heat &ux meter.

D. Xie et al. / Chemical Engineering Science 58 (2003) 4247 – 4258

4255

Table 2 Main operating conditions of Andersson (1996) Case s−1 )

Ug (m ut (m s−1 ) Tsb (◦ C) Tst (◦ C) dp (m) Hx (m)

A

B

C

D

E

F

G

H

I

J

1.76 3.47 883 637 440 0.38

1.76 2.38 873 659 335 0.42

1.83 1.25 830 777 220 0.46

3.58 2.26 857 777 323 0.39

4.53 3.33 845 826 425 0.36

2.66 3.43 837 711 435 0.38

2.68 2.39 845 747 335 0.41

2.65 1.32 826 799 227 0.44

3.68 1.90 865 842 288 0.39

6.39 3.45 881 871 438 0.18

12

12

Height (m)

10

(A)

(B)

(C)

(D)

10

(E)

8

8

6

6

4

4

2

2 0

15

30

45

0

15

30

45

0

15

30

45

0

15

30

45

0

15

30

45

60

Suspension density (kg/m3) 12

12

Height (m)

10

(A)

(B)

(C)

(D)

(E)

10

8

8

6

6

4

4 2

2 0

50

100 150 200

0

50

100 150 200

0

50

100 150 200

0

50

100 150 200

0

50

100 150 200 250

Heat transfer coefficient (W/m2K)

Fig. 13. Measured local suspension densities and comparison of predicted (lines) and experimental (points) local heat transfer coeOcients for runs A, B, C, D and E of Andersson (1996). Operating conditions are listed in Table 2.

obtained by Golriz (1996) showed that if the obstacle was 2 m above the heat &ux meter, the in&uence of the obstacle was negligible. Our simulations also predict that, after 2 m, the heat transfer coeOcients in the presence and absence of the obstacle should be very similar. 5.4. Comparison with data of Andersson (1996) Andersson (1996) measured local suspension densities and suspension-to-membrane-wall heat transfer coeOcients in 10 sets of experiments carried out in the same CFB boiler at Chalmers University. Three di:erent narrow-sized fractions of the same silica sand were used as the bed materials. The Sauter mean particle diameter of the active bottom bed material, including sand, fuel and ash, varied between 0.22 and 0:4 mm, and the particle density was 2600 kg m−3 . The bed temperature was measured near the bottom of the bed, just above the primary air distributor, and at the gas exit. For tests in which the top and bottom temperatures di:ered, a linear pro9le was assumed between Z = 2 m and the gas exit, to estimate the core temperature for evaluation of local heat transfer. The key operating conditions for the ten runs

are listed in Table 2. The cross-sectional average suspension densities were evaluated from pressure drop measurements, and are shown in the upper panels of Figs. 13 and 14. The measured local suspension densities with linear interpolation between each pair of adjacent points are used as inputs for the model simulations. In these simulations, the bulk temperatures are again assumed to vary linearly from Z =2 m to the gas exit. A wall emissivity of 0.8 is employed. The lower panels of Figs. 13 and 14 compare the predicted local heat transfer coeOcients with the experimental results. Andersson (1996) also evaluated the variation of heat transfer for three portions of the membrane assembly: the tube crest, tube side and 9n. Average heat &uxes were estimated for each part. In order to compare results from di:erent locations, the local values (qc , qs and qf ) were normalized by the average total heat &ux, i.e., divided by ((qc lc + qs ls + qf lf )=(lc + ls + lf )) where lc , ls and lf are the lateral distances along the crest, side and 9n surfaces, respectively. Fig. 15 compares the predicted lateral variation of relative heat &ux with experimental data at Z = 10:5 and 3:4 m. For most cases, the model gives good predictions in

4256

D. Xie et al. / Chemical Engineering Science 58 (2003) 4247 – 4258 12

12

Height (m)

10

(F)

(G)

(H)

(I)

(J)

10

8

8

6

6

4

4

2

2 0

15

30

45

0

15

30

45

0

15

30

45

0

15

30

45

0

15

30

45

Suspension density (kg/m3) 12

12

Height (m)

10

(F)

(G)

(H)

(I)

(J)

10

8

8

6

6

4

4

2

2 0

50

100 150 200

0

50

100 150 200

0

50

100 150 200

0

50

100 150 200

0

50

100 150 200 250

Heat transfer coefficient (W/m2K)

Fig. 14. Measured local suspension densities and comparison of predicted (lines) and experimental (points) local heat transfer coeOcients for runs F, G, H, I and J of Andersson (1996). Operating conditions are listed in Table 2.

the crest region, but underestimates the relative experimental values in the 9n region.

6. Conclusions The model of Xie et al. (2003a) is extended to circulating &uidized beds with membrane walls, a three-dimensional and more complex geometry. The governing equations are solved by 9nite-element and 9nite-di:erence methods using the moment method for radiation transfer. The solution is 9rst demonstrated for a typical example. The model is then tested against experimental results from the literature. It gives mostly satisfactory predictions of the suspension-to-wall heat transfer. The model predicts that the local suspension-to-wall heat transfer varies both vertically and laterally along the membrane surface. It is a strong function of the suspension density in the adjacent wall layer. Solids exchange between the core and the wall region and the voidage distribution in the wall layer are important in determining suspension-to-wall heat transfer. Further studies are needed to investigate these parameters.

Notation a Ao Cpc

suspension absorption coeOcient for gray medium, m−1 total outside area of membrane surface, m2 cooling water heat capacity, J kg−1 K −1

Cpg Cpp dp D eb ew Ex f G hc hpg H Hx i
ib
kg kw lc lf ls Lar mc n Nupg q qc

gas heat capacity, J kg−1 K −1 particle heat capacity, J kg−1 K −1 particle diameter, m diameter or hydraulic diameter of riser, m bulk emissivity, dimensionless wall emissivity, dimensionless particle exchange rate between core and wall layer, kg m−3 s−1 coeOcient related to phase function, for gray media, large particles and independent scattering, f = 0:4458 (see Xie, 2001), dimensionless irradiance, W m−2 heat transfer coeOcient from tube to coolant, W m−2 K −1 particle-to-gas heat transfer coeOcient, W m−2 K −1 length of heat transfer surface, m height of bottom bed, m radiation intensity, W m−2 sr −1 black body radiation intensity, W m−2 sr −1 thermal conductivity of gas, W m−1 K −1 thermal conductivity of membrane wall, W m−1 K −1 lateral membrane distance along crest, m lateral membrane distance along 9n, m lateral membrane distance along side, m particle average residence length in wall layer, m cooling water &ow rate, kg s−1 normal direction, dimensionless Nusselt number=hpg dp =kg , dimensionless heat &ux through wall, W m−2 local heat &ux at membrane crest, W m−2

D. Xie et al. / Chemical Engineering Science 58 (2003) 4247 – 4258

4257

Relative heat flux (-)

1.5 1.0 0.5

(F)

(F)

0.0 0

10

20

30

40

50

0

10

20

30

40

50

10

20

30

40

50

10

20

30

40

50

10

20

30

40

50

10

20

30

40

50

1.5 1.0 0.5

(G)

(G)

Relative heat flux (-)

0.0 0

10

20

30

40

50

0

1.5 1.0

(H)

0.5

(H)

0.0 0

10

20

30

40

50

0

1.5

Relative heat flux (-)

1.0 0.5

(I)

(I)

0.0 0

10

20

30

40

50

0

1.5 1.0 0.5

(J)

(J)

0.0 0

10

20

30

40

50

0

Distance travelled around surface from crest (mm) Fig. 15. Comparison of predicted lateral variation of relative heat &ux (dashed lines) with experimental data of Andersson (1996) (solid lines). Left panels: Z = 10:5 m; right panels: Z = 3:4 m. Operating conditions are listed in Table 2.

qf qr qs Qpg R Ri Ro s t Tb Tc Tc; in

local heat &ux at membrane 9n, W m−2 radiative heat &ux, W m−2 local heat &ux at membrane side, W m−2 volumetric heat convection rate from particles to gas, W m−3 radius or hydraulic radius of riser, m inner radius of membrane tube, m outer radius of membrane tube, m particle surface area per unit volume, m−1 half thickness of membrane 9n, m bulk temperature, K cross-sectional average coolant temperature, K Tc at inlet of heat exchanger, K

Tc; out Tg Tp Tsb Tst Tw uc ug up ut U

Tc at outlet of heat exchanger, K gas temperature, K particle temperature, K suspension temperature in bottom part of furnace, K suspension temperature in top part of furnace, K wall temperature, K average axial coolant velocity (upwards), m s−1 axial gas velocity (downwards), m s−1 axial particle velocity (downwards), m s−1 particle terminal velocity (downwards), m s−1 overall bed to cooling water heat transfer coeOcient, W m−2 K −1

4258

Ug x y z Z

D. Xie et al. / Chemical Engineering Science 58 (2003) 4247 – 4258

super9cial gas velocity, m s−1 horizontal coordinate, parallel to 9n, m horizontal coordinate, normal to 9n, m vertical coordinate, directed vertically downward, m height above distributor, m

Greek letters / Pz Pzmax  mf sec  :g c g p sus s 0 *  ’ ! !i

wall layer thickness, m grid step length in vertical direction, m maximum grid step length in vertical direction, m suspension voidage, dimensionless loosely packed bed voidage, dimensionless cross-sectional average suspension voidage, dimensionless azimuthal angle, dimensionless gas viscosity, kg m−1 s−1 coolant density, kg m−1 gas density, kg m−3 particle density, kg m−3 suspension density, kg m−3 scattering coeOcient for gray medium, m−1 Stefan–Boltzmann constant, W m−2 K −4 dimensionless lateral distance in the riser (=1 − 2(y − t)=D), dimensionless scattering phase function, dimensionless polar angle, dimensionless solid angle, Sr solid angle for incoming radiation, Sr

Acknowledgements Financial support from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. References Andersson, B. A. (1996). E:ects of bed particles on heat transfer in circulating &uidized bed boilers. Powder Technology, 87, 239–248. Andersson, B. A., & Leckner, B. (1992). Experimental methods of estimating heat transfer in circulating &uidized bed boilers. International Journal of Heat and Mass Transfer, 35, 3353–3362. Basu, P., & Nag, P. K. (1996). Heat transfer to walls of a circulating &uidized bed furnace. Chemical Engineering Science, 51, 1–26. Bi, H. T., Zhou, J., Qin, S.-Z., & Grace, J. R. (1996). Annular wall layer thickness in circulating &uidized bed risers. Canadian Journal of Chemical Engineering, 74, 811–814.

Golriz, M. R. (1994). Temperature distribution at the membrane wall of a 165 MWth CFB boiler. In Proceedings of heat and mass transfer in circulating :uidized beds application to clean combustion. Eurotherm Seminar No. 38, Marseilles, France (pp. 22–24). Golriz, M. R. (1996). Enhancement of heat transfer in circulating &uidized bed combustors by using horizontal 9ns. In Proceedings of 2nd European thermal-science and 14th UIT national heat transfer conference, Rome. Grace, J. R. (1990). Heat transfer in high velocity &uidized beds. In G. Hetsroni (Ed.), Proceedings 9th international heat transfer conference, Vol. 1, Jerusalem (pp. 329 –339). Issangya, A. S., Grace, J. R., Bai, D., & Zhu, J. (2001). Radial voidage variation in CFB risers. Canadian Journal of Chemical Engineering, 79, 279–286. Lints, M. C., & Glicksman, L. R. (1994). Parameters governing particle-to-wall heat transfer in a circulating &uidized bed. In A. A. Avidan (Ed.), Circulating :uidized bed technology, Vol. IV (pp. 297– 304). New York: AIChE. Lockhart, C., Zhu, J., Brereton, C. M. H., Lim, C. J., & Grace, J. R. (1995). Local heat transfer, solids concentration and erosion around membrane tubes in a cold model circulating &uidized bed. International Journal of Heat and Mass Transfer, 38, 2403–2410. Luan, W. (1997). Radiative and total heat transfer in circulating :uidized beds. Ph.D. dissertation, The University of British Columbia, Vancouver, Canada. Luan, W., Bowen, B. D., Lim, C. J., Brereton, C. M. H., & Grace, J. R. (2000). Suspension-to-membrane-wall heat transfer in a circulating &uidized bed combustor. International Journal of Heat and Mass Transfer, 43, 1173–1185. Weimer, A., Bixler, D., Pettit, R. D., & Wang, S. I. (1991). Operation of a 49 MW circulating &uidized bed combustor. In P. Basu, M. Hasatani, & M. Horio (Eds.), Circulating :uidized bed technology, Vol. III (pp. 341–346). Oxford: Pergamon. Wu, R. L., Grace, J. R., Lim, C. J., & Brereton, C. M. H. (1989). Suspension-to-surface heat transfer in a circulating &uidized bed combustor. A.I.Ch.E. Journal, 35, 1685–1691. Wu, R. L., Lim, C. J., Chaouki, J., & Grace, J. R. (1987). Heat transfer from a circulating &uidized bed to membrane waterwall surfaces. A.I.Ch.E. Journal, 33, 1888–1893. Wu, R. L., Lim, C. J., Grace, J. R., & Brereton, C. M. H. (1991). Instantaneous local heat transfer and hydrodynamics in a circulating &uidized bed. International Journal of Heat and Mass Transfer, 34, 2019–2027. Xie, D. (2001). Modeling of heat transfer in circulating :uidized beds. Ph.D. dissertation, University of British Columbia, Vancouver, Canada. Xie, D., Bowen, B. D., Grace, J. R., & Lim, C. J. (2003a). Two-dimensional model of heat transfer in circulating &uidized beds, Part I: Model development and validation. International Journal of Heat and Mass Transfer, 46, 2179–2191. Xie, D., Bowen, B. D., Grace, J. R., & Lim, C. J. (2003b). Two-dimensional model of heat transfer in circulating &uidized beds, Part II: Heat transfer in a high density CFB and sensitivity analysis. International Journal of Heat and Mass Transfer, 46, 2193–2205. Zhou, J., Grace, J. R., Brereton, C. M. H., & Lim, C. J. (1996). In&uence of membrane walls on particle dynamics in a circulating &uidized bed. A.I.Ch.E. Journal, 42, 3550–3553.