Effect of freeboard deflectors on the temperature distribution in packed beds

Applied Thermal Engineering 89 (2015) 134e143

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Research paper

Effect of freeboard deflectors on the temperature distribution in packed beds Babak Rashidian*, Yasir M. Al-Abdeli, Guan H. Yeoh, Ferdinando G. Guzzomi School of Engineering, Edith Cowan University, Joondalup, WA 6027, Australia

h i g h l i g h t s
The influence of freeboard deflectors on the temperature distribution in packed beds is studied.
Methods applied include CFD modelling, validation against experimental data and empirical fits.
The impact of deflectors largely depends on the heating mode (wall versus air stream; 100e400  C).
Stronger effects occur on wall temperature (in the freeboard) compared to packed bed temperature.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 November 2014 Accepted 21 May 2015 Available online 30 May 2015

Freeboard deflectors have been applied in solid fuel combustors but little investigation has been undertaken to understand their impact on packed beds. This paper studies the influence of a deflector above a packed bed by implementation of a three-dimensional Computational Fluid Dynamics (CFD) model of a porous media. Through validating the model against experimental data, the effects of a freeboard deflector on the radial and axial temperature profiles is studied for a temperature range typical for drying and volatile release in biomass combustion (100e400  C). Numerical results indicate that the deflector influences wall temperatures as well as temperatures along the freeboard but this is dependent on the mode of heating and emissivity of the deflector. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Packed beds CFD Freeboard deflectors Modelling Experimental data Validation

1. Introduction High surface-area to volume ratio associated with packed beds exists in a variety of engineering applications [1]. These applications include chemical reactors [2e6], fixed bed combustors [7e10], particle dryers [11e16], air dehumidifiers [17], air conditioning [18], heat storage systems [19e21] and heat exchangers [22,23]. Optimising the heat transfer characteristics of packed beds plays a pivotal role in achieving specific design and performance gains in the abovementioned applications. Packed beds remain the subject of ongoing investigation, where Table 1 presents a summary of several experimental studies to resolve their heat transfer coefficients [24e29]. Nevertheless, a number of unresolved challenges persist. Such challenges include, a better understanding on

* Corresponding author. School of Engineering, Edith Cowan University, 270 Joondalup Drive, WA 6027, Australia. Tel.: þ61 8 63045998. E-mail address: [email protected] (B. Rashidian). http://dx.doi.org/10.1016/j.applthermaleng.2015.05.045 1359-4311/© 2015 Elsevier Ltd. All rights reserved.

the variation of the effective thermal conductivity over a range of Reynolds (Re) numbers and temperatures, the influence of wall effects on the axial temperature distributions (within the bed) and the effects of porosity. The task of resolving these factors in relation to modelling particle drying and fixed-bed combustion is exacerbated by the prevalence of studies with either extremely low Reynolds numbers [30e32]; the use of glass, metallic or other nondrying particles [2]; poorly defined boundary condition data [33,34] and/or that most packed bed reaction models are simulated as porous media [14,35,36]. In combusting packed beds, such as those appearing in high temperature processes involving solid particle combustion, the residence time and radiation effects of the freeboard (space above the packed bed) are important. Biomass combustion has been investigated on laboratory-scale fixed beds to better comprehend the thermal conversion processes [37,38]. In this context, heat transfer rates inside packed beds limit evaporation rates [12] and other sub-processes such as volatile release. In counter-current fixed bed combustion [39], whereby ignition occurs in the top

B. Rashidian et al. / Applied Thermal Engineering 89 (2015) 134e143

135

Table 1 Summary of experimentally derived heat transfer relationships in packed beds. Author (s)

Equation

Calderbank and Pogorski [24] Yagi and Kunii [26] DeWasch and Froment [27] Li and Finlayson [28] Demirel et al. [29] Demirel et al. [29]

Nu Nu Nu Nu Nu Nu

¼ ¼ ¼ ¼ ¼ ¼

Packed bed

4.21Re0.365 15 þ 0.029Re 12.5 þ 0.048Re 0.178Re0.790 0.197Re0.718 0.217Re0.756

Cylindrical packed bed, 8 < dt/dp < 16, aluminum spheres Annular packed bed, 3.9 < dt/dp < 51, glass beads Cylindrical packed bed, dt/dp ¼ 176 Cylindrical packed bed, 3 < dt/dp < 5, celite spheres Rectangular packed bed, 3 < dt/dp < 5, polyvinyl chloride Raschig rings Rectangular packed bed, 4.5 < dt/dp < 7.5, expanded polystyrene spheres,

layer of the packed bed and propagates downwards, the char reaction zone at the surface of the fixed bed is typically 50 mm thick [40]. Therefore, operational or design (geometrical) features that influence the heat transfer in the upper layers of combusting packed beds may also affect the high temperature (post-bed) reactions occurring in the freeboard. In commercial-scale combustors, deflectors placed in the freeboard have been employed to reduce particulate matter and influence gaseous species emissions [17,41]. These devices also enhance performance by reducing flame radiation into the exhaust stack by affecting the heat transfer in the freeboard [17,42]. Fig. 1a shows a freeboard deflector mounted above a packed bed in a laboratory-scale fixed bed combustor. Whilst no systematic studies into the effects of freeboard deflectors has been made, published work [43] indicates that wall temperatures and flow dynamics in the post-bed (freeboard) region affect the migration of dust, fly ash, soot and other Hydrocarbon (HC) formation. In non-combusting packed beds, where drying from heat transfer is the primary objective, the pressure drop can be determined along the axial direction of the packed bed [44,45]. However, other factors which may affect the heat transfer in drying packed beds include the axial and radial temperature distributions. Therefore there is a focus on the requirement to attain suitable effective thermal conductivity [3,25,26,32,46,47] of the heat

dd

t

transfer process; some correlations for the metal-air, glass-air and catalyst-air systems have been proposed. Although numerous numerical studies exist for the heat transfer and flow characterisation of packed beds [2,34,36,48e51], the available literature does not contain analysis into the effect of freeboard deflectors on the axial pressure drop and temperature distribution, particularly in the uppermost bed layers and freeboard regions. Whilst the availability of powerful CFD techniques can be used to effectively predict the performance of thermo-fluid systems, the application of CFD to packed beds remains challenging. The treatment of packed beds in many commercially available CFD codes considers them as porous media, whereby effective thermal conductivity (Ke) is calculated as a weighted average of the solid (Ks) and gaseous phases (Kf) that uses the porosity (void fraction) of these phases [52], respectively. The application of this effective thermal conductivity, albeit for its simplicity, precludes the actual heat transfer processes that may be occurring between contacting solid particles and in the voids of the packed beds where fluid flows around the solid particles. An alternative approach to circumvent the problem associated with the porous-media is to physically track the individual solid particles in packed beds [33,48,53]. The computational demands of such an approach are large and as a result packed beds are mostly treated as relatively shallow layers to limit computational demands. Consequently, flow dynamics and

Un-insulated freeboard 265 mm

Outlet boundary

Deflector

Zone 11

Zone 9

Zone 7 Zone 6

h

Deflector

Wall Freeboard temperature Top layer

H Packed bed

Zone 8 1100 mm

Radiation Flow dynamics cs

dp

Bed temperature

x dt

(a)

Zone 5 Zone 4 Zone 3 Zone 2 Zone 1

100

Un-insulated freeboard

Zone 10

Inlet air

(b)

Inlet air

(c)

Fig. 1. (a) A freeboard deflector mounted above a packed bed; (b) CFD Model-I with 265 mm freeboard used to validate against experimental data [55] and (c) CFD Model-II with 265 mm freeboard and deflector (dd ¼ 36 mm, h ¼ 10 mm).

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B. Rashidian et al. / Applied Thermal Engineering 89 (2015) 134e143 Table 2 Correlations for packed bed effective thermal conductivity. Author

Equation

Experimental conditions

Demirel et al. [29]

ke kf

¼ 2:894 þ 0:068Rep

Rasching rings, 5.6 < dt/dp < 6.6, 0 < Re< 1500, T < 45  C

Demirel et al. [29]

ke kf

¼ 10:433 þ 0:0481Rep

Polystyrene sphheres, 4.5 < dt/dp < 7.5, Re < 1500, T < 45  C

Bunnell et al. [54]

ke kf

¼ 5:0 þ 0:061Rep

Alumina cylinders, dt/dp ¼ 16, Re < 100, 100 < T < 45  C

Votruba et al. [32]

1 Peh

Vortmeyer and Adam [47]

ke kf

¼

Yagi and Kunii [25]

ke kf

¼

¼

k0e =kf Re:Pr

k0e kf k0e kf

þ

14:5 p ð1þðC3 =Re:PrÞÞ

þd

gu 1þhu

þ ðabÞNpeM

Glass, lead, iron and alumina spheres, 3.85 < dt/dp < 11.55, Re < 30, 0 < T<200  C Glass, bronze, brass and steel spheres, 8.6 < dt/dp < 68.33, Re < 12, 0 < T < 50  C Glass spheres, 3.9 < dt/dp < 51, Re < 60, 0 < T < 100  C

heat transfer (at the exit plane and uppermost bed layers) are affected by strong non-homogeneity [4,37], which are expected to have a greater relative influence at higher temperatures (e.g. combusting packed beds), due to stronger radiation effects and higher heat flux. For practical considerations, the porous media approach is adopted in this study to model the effects of freeboard deflectors on packed beds through the utilisation of suitable effective thermal conductivity values based on published empirical models. Table 2 presents a list of equations that have been derived for the effective thermal conductivity based on their respective experimental conditions [25,29,32,47,54]. Numerical assessment is performed to determine which equation would be most appropriate to explore the effect of freeboard deflectors on the temperature of the upper most bed layers and the freeboard, up to 400  C. 2. Methodology 2.1. Model geometry Experimental data of a heated packed bed, detailed in Wen and Ding [55,56], is firstly used to validate the numerical model without a freeboard deflector. This experimental data was acquired in a vertical (stainless steel) cylindrical column (1100 mm high) having an internal diameter dt ¼ 41 mm and operated with external air supplied from beneath the column at Re ¼ 328 and Re ¼ 556. During the experiments, the walls were heated to 100  C, where the walls have a thickness of t ¼ 3.5 mm. The data is obtained with randomly packed glass spheres (dt/dp ¼ 8.2), with temperatures measured at several axial stations (x ¼ 30 mm to x ¼ 1062 mm). For more details, the reader is referred to the original data sets in Ref. [55]. In this study, two CFD models have been developed. Model-I shown in Fig. 1b is used for validation against the experimental data [55]. This model only differs from the original geometry by an extension of the freeboard 265 mm beyond the original length of cylindrical tube (1100 mm). This extension is to accommodate the freeboard deflector. The diameter, height and porosity (ε ¼ 0.416) of the packed bed however, remain identical to those in the experimental data and also features a heated wall (Tw ¼ 100  C). This model is first used to ascertain whether any extension to the freeboard (even without any deflector) imposes changes to the original pressure drop, temperature profiles and heat transfer characteristics within the packed bed [55]. Commensurate with the original experiments [55], the boundary conditions applied to validate Model-I simulations against the original data are: (i) a constant inflow temperature (20  C) at x ¼ 0 and (ii) a constant temperature of 100  C at the exterior walls (x ¼ 0e1100 mm), which are assumed to be adiabatic. Model-II as shown in Fig. 1c, is then employed to study the effects of a freeboard deflector mounted within the 265 mm

non-insulated extension to the cylindrical column. The deflector with a diameter of dd ¼ 36 mm and thickness h ¼ 10 mm, was placed at either H ¼ 10 mm or H ¼ 185 mm above the packed bed. Model-II either features a heated wall or preheated air stream, both at 400  C. The 265 mm extension (height) of the freeboard has been selected to be longer than any potential reattachment length for flow reversals formed behind the deflector. Notably, the deflector essentially constitutes a backward facing step (or bluffbody) for similar geometries and Re. Such reattachment (extension) lengths depend on the Expansion Ratio (ER) and Re [57,58] and their value can be derived based on the Biswas correlation [58]. For a 36 mm diameter deflector at Re ¼ 556, the longest reattachment length in an equivalent backward facing step is then predicted to be 60 mm. Hence, even for a deflector positioned at H ¼ 185 mm, the length of the remaining freeboard downstream of its rear face ensures no flow reversals (at the exit plane) since they can perturb the flow by drawing-in ambient air into the packed column. In both CFD Model-I and Model-II, the packed bed is simulated by eleven porous zones (each 0.1 m high) to accommodate different effective thermal conductivities within the packed bed, which are a function of temperature (downstream distance) [55]. The gaseous phase (air) is assumed to behave as a Newtonian fluid and the two-equation k-ε turbulence model is used with fluid properties that are also considered as temperature dependent. The ideal gas law is used for the temperature dependence of density and Sutherland's law is used to derive the dynamic viscosity of an ideal gas as a function of temperature. Tables 3 and 4 present model parameters used at the inflow boundary (x ¼ 0 mm), within the packed bed (x ¼ 0e1100 mm) and freeboard/deflector (x ¼ 1100e1365 mm), respectively. For the deflector, the characteristics used are typical of those for alumina ceramic due to its thermal resistance. However, in the CFD modelling results to be presented, the effects of varied radiative emissivity have also been studied (Fig. 9). To obtain

Table 3 Packed bed and inflow parameters for Model-I and Model-II (Tw¼100 C). Parameter

Value

Density [kg m3] Heat capacity [J kg-1 K1] Porosity Permeability [m2] Inertial resistance factor [m1] Inlet air velocity (at Re ¼ 328) [ms1] Inlet air velocity (at Re ¼ 556) [ms1] Steel density [kg m3] Steel heat capacity [J kg-1 K1] Steel heat conductivity [Wm1 K1] Air density [kg/m3] Air viscosity [kgm-1 s1] Air heat Capacity [J kg1 K1] Air heat conductivity coefficient [Wm1 K1]

2590 780 0.416 3.181  108 5678.464 1.25 1.8 8030 502.48 16.27 1.225 1.7894  105 1006.43 0.0242

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Table 4 Characteristics of simulated freeboard and deflector. Parameter

Value

Freeboard length [m] Bed height [m] Bed particle diameter [m] Column inner diameter [m] Wall thickness [m] Wall emissivity Wall density [kgm3] Wall heat capacity [J kg1 K1] Wall thermal conductivity [Wm1 K1] Free stream heat transfer coefficient [Wm2 K1] Free stream temperature [ C] Deflector density [kg m3] Deflector heat capacity [J kg1 K1] Deflector thermal conductivity [Wm1 K1]

0.265 1.1 0.005 0.041 0.0035 0.85 8030 502 16.27 10 27 3420 880 15

Fig. 4. Model-I: Comparison between computed pressure drop in the packed bed against experimental data [55] and the Ergun equation (Tw ¼ 100  C).

Fig. 2. Model-I: Validation of computed axial temperature distribution against experimental data [55] at Re ¼ 328 and Re ¼ 556 (Tw ¼ 100  C). Fig. 5. Model-I Effective thermal conductivity derived empirically from the experimental data of Wen and Ding [55] compared to that using the empirical fits from Yagi and Kunii [25] and Vortmeyer and Adam [47] when applied in packed beds up to 400  C.

representative values of the steady-state effective thermal conductivity for each of the eleven zones, preliminary simulations are carried out against the original data [55]. Mesh sensitivity analyses are performed for Model-I and assessed based on the axial temperature distribution. Results from CFD modelling using 68,000 and 110,000 grid nodal points for the models (packed bed with freeboard) are also shown in Fig. 2; it is shown that the mesh of 68,000 is deemed to be sufficient for the current study. 2.2. Numerical approach

Fig. 3. Model-I: Validation of computed radial temperature profiles against experimental data [55] for two axial positions at Re ¼ 328 and Re ¼ 556 (Tw ¼ 100  C).

The steady state heat transfer and fluid flow characteristics inside a packed bed and freeboard are modelled using ANSYS Fluent 14.5. Although the porous media is applied, the effective thermal conductivity in the packed bed is not solely based on the weighted solid phase (ks) and gas phase (kf) thermal conductivities [14,36]. In modelling packed beds, there have been several attempts to derive empirical models for the effective

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B. Rashidian et al. / Applied Thermal Engineering 89 (2015) 134e143

Fig. 6. Comparison between axial temperature distributions for Model-I and Model-II at Tw ¼ 100 and 400  C for Re ¼ 556. Vertical lines denote the axial locations of the freeboard deflector.

Fig. 7. Effects of deflector on the axial temperature distribution with heated inlet air at 400  C. The vertical lines denote the positions of freeboard deflectors.

thermal conductivity, where some are highlighted in Table 2. These models all agree that effective thermal conductivity (keff) can be obtained by combining two terms; thermal conductivity without flow (k0e Þ and a second contribution which depends on

the fluid flow (Reynolds and Prandtl numbers) as given by Equation (1) [37,42,47]. Yagi et al. [25] presented one of the earliest correlations for effective thermal conductivity as depicted by Equation (2), however Vortmeyer et al. [47] later suggested a modified expression to calculate the effective thermal conductivity given by Equation (3), and concluded that the term gRePr is independent of packed bed solid phase thermal conductivity, and presented some modelling parameters for the glass-air system (g ¼ 1060 m1s and h ¼ 96 m1s). With regards to the term ðk0e =kf Þ, this can be calculated according to Equation (4) [59]. In addition, the effective thermal conductivity is more sensitive to variations in flow rate when calculated in the axial direction compared to the radial direction [25]. However, it has also been seen that in addition to particle diameter, the Reynolds number has less influence compared to the “artificial linear velocity” [47]. The empirical model represented by Equation (3) is based on the observation that the Reynolds number does not appear to be an appropriate parameter to use in the rightmost term of Equation (1). Alternatively, u which is an artificial linear velocity has been proposed.

keff k0 ¼ e þ gRePr kf kf

(1)

B. Rashidian et al. / Applied Thermal Engineering 89 (2015) 134e143

139

Fig. 8. Radial temperature profiles for Model-I and Model-II over different axial positions at Tw ¼ 100 and 400  C. The results for Model-I are given to show behaviour predicted without the presence of freeboard deflectors.

ke k0e ¼ þ ðabÞNpeM kf kf

(2)

ke k0e gu ¼ þ kf kf 1 þ hu

(3)

k0e ¼ kf

ks kf

   ! 0:280:757 logðεb Þ0:057 log ks kf

(4)

A significant limitation from all these models is that not all have been derived in relation to packed beds formed from randomly distributed packing of spherical particles. Furthermore, even when

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B. Rashidian et al. / Applied Thermal Engineering 89 (2015) 134e143

Fig. 9. Radial temperature profiles for Model-II over different emissivity (heated air stream of 400  C). The insert shows the effects of deflectors on the flow dynamics in the freeboard at H ¼ 185 mm (a), H ¼ 10 mm (b) in comparison with no deflector (c).

empirical models have been presented for spherical particles [25,32,47], the range of temperatures covered has either been too low or the formulations have been derived at extremely low Reynolds numbers. Both these limitations raise questions about their applicability to higher temperatures, typical of drying, volatile release (300e400  C) or char reaction (800e1000  C) in combusting fixed beds of biomass. The equations governing the transport of mass, momentum and energy for a porous media can be found in Appendix A. In this paper the surface-to-surface (S2S) model has been used to account for radiation exchange in the CFD simulations. Although CFD simulations may not consider the radiation effects between two phases in the porous media, as pointed out by Chen and Churchill [60], the radiant contributions to heat transfer in packed beds at low temperatures (<800  C) is negligible. Furthermore, results derived from the literature indicate that at a temperature of 400  C the contribution of the radiation is less than 10% for 5 mm glass beads in packed beds [60], despite radiation being nominated as one of the heat transfer mechanisms which are independent of fluid flow [46]. These packings are similar to those used in the present study. A similar result has been found by Nasr et al. [61] when using alumina-air packed beds where the contribution of the radiant conductivity decreases with temperature and particles size. However, radiative heat transfer does depend on packed bed materials. For the alumina-air packed bed with 6.64 mm particles, the relative contributions of radiation are about 30% [61]. Due to such generally small contributions of radiation to heat transfer in the packed beds of 5 mm glass spheres at temperature of 400  C [60], in this study the radiative heat transfer between two phases in the porous media is not taken into account in the calculations. However, this maybe an avenue of study in future.

2.3. 2.3 Model validation Model-I, which has no freeboard deflector but captures all other geometrical features, is validated before studying the effects of freeboard deflectors (Model-II). Figs. 2 and 3 show a comparison between the computed axial and radial temperature profiles with measurements at Re ¼ 328 and Re ¼ 556 [55]. Fig. 4 shows a comparison of the CFD computed pressure loss along the packed bed compared to the measured data [55] and the Ergun equation (Equation (A.20)). It can be seen that the results for Model-I agree

reasonably well with the experimental measurements conducted based on wall temperatures at 100  C. In the present paper, analyses are required to be carried out up to 400  C, which covers the temperature range for both particle drying and partially volatile release in fixed bed combustion of biomass [62,63]. Values of effective thermal conductivities in Model-I are varied locally in order to match the original axial and radial temperature distribution data for a wall temperature of 100  C [55]. The derived values are then used to decide on the best empirical formulation to use within Model-II for temperatures up to 400  C. Preliminary analysis showed the optimum values of thermal conductivity such that the temperature profiles in the bed [55] manifest in the CFD models (over the eleven axial zones) are: K1eK3 ¼ 0.4, K4, K5 ¼ 0.6, K6e K8 ¼ 0.7, and K9eK11 ¼ 0.95. Although Model-II explores the effects of freeboard deflectors at higher temperatures (up to a constant wall temperature of 400  C), no experimental data is available to help derive or correlate against the effective thermal conductivity. To arrive at a representative empirical correlation, Fig. 5 shows the calculated effective thermal conductivity for Model-I using three empirical approaches. Considerable discrepancies can be seen among the correlations which may be due to Vortmeyer and Adam's correlation [47] is derived for Re < 12 and 0 < T < 50  C and Yagi and Kunnii's equation [25] was similarly proposed for Re < 60 and 0 < T < 100  C. Notably, these two empirical relationships were originally derived [25,47] with relatively low Reynolds numbers and bed temperatures. It appears the Wen and Ding experimental data reasonably agree with the low temperature (<100  C) behaviourbehaviour predicted by Vortmeter and Adam [47], but leads to an over estimation at high temperatures (300e400  C). Because of this disparity in some of the ensuing CFD results, figures will feature both the empirical fits from Refs. [47,55] to help identify the sensitivity of predicted Model-II behavior in relation to the effects of effective thermal conductivity (keff). Results show that in Model-II, both empirical fits of effective thermal conductivity yield similar qualitative effects (only), within the bed, when studying the effects of deflectors. However, the type of model used does exert a quantitative influence on the results which means temperatures within the bed may vary. This does not affect the freeboard area studied in relation to deflectors.

3. Results and discussion Fig. 6 compares the axial temperature distributions with and without freeboard deflectors over two wall temperatures (Tw ¼ 100 and 400  C) at Re ¼ 556. Results are given for deflectors placed at two downstream locations above the bed (H ¼ 10 and 185 mm), with axial temperatures (T/Tw) being normalized by relevant wall temperature. It can be seen that if heating occurs from the wall, deflectors have no significant effect on the centreline axial temperature profile inside the bed and freeboard, where the provided comparison are based on a similar derivation of effective thermal conductivity. Fig. 7 presents the results for deflectors at the same axial location above the bed (H ¼ 10 and 185 mm) to assess the effects of freeboard deflectors on packed beds when the air stream (not wall) is heated at 400  C, but with other boundary conditions the same as previous results. Two interesting observations are identified. Firstly, the deflector position (H) does not appear to influence the intra-bed temperatures, which is expected. Secondly, for both H ¼ 10 mm or H ¼ 185 mm, the freeboard deflector leads to a ~5% increase in temperature within the freeboard. This temperature change inside the freeboard is likely to be attributed to a convective and radiative effect from the deflector. More

B. Rashidian et al. / Applied Thermal Engineering 89 (2015) 134e143

investigations are warranted to identify the relative significance of both these mechanisms. This indicates the effect of deflectors on the freeboard temperature will be more pronounced but dependent on the mode of heating. This increase in freeboard temperatures is expected to be much more significant if the air stream temperatures within the packed bed are higher. A similar, yet stronger temperature influence is expected from placing a freeboard deflector above combusting fixed beds, where the upper bed layers are not limited to 400  C but can reach 900e1000  C [42]. In this regard, flow dynamics and wall temperatures in the post-bed region, affect fly ash, soot and unburned hydrocarbons [43]. In such instances, the effects of freeboard deflectors on the post-bed combustion of volatiles and species generation are expected to be more significant. More work is warranted to further identify the role of freeboard deflectors at higher (air) temperature or where the heat source occurs at the top of the bed (seen in countercurrent, fixed bed biomass combustors). Fig. 8 presents the deflector effects on the radial temperature distributions for cases where the walls are heated at Tw. Results are given for CFD predictions at axial locations within the combustor, for instance within the packed bed (x ¼ 764 mm), the top layer of the bed (x ¼ 1100 mm) and slightly downstream within the freeboard (x ¼ 1110 mm). The presented model data also shows the effects of two different deflector positions (H ¼ 10 and 185 mm). Results indicate that in stream-heated packed beds, deflectors have minimal effects on the radial temperature distributions inside the bed (x ¼ 764 mm) as presented in the Fig. 8a and d. However, the deflector in the freeboard does increase the temperature close to the wall and changes the radial temperature profiles (Fig. 8bef). At stream temperatures of 100  C, the presence of a deflector leads to an increase of ~5% (Fig. 8c) but this increase to ~7% at 400  C (Fig. 8f). Finally, Fig. 9 shows Model-II but with heated air at 400  C. These results indicate the effects of the deflector appear to be due to both its radiative heat transfer (varied emissivity) as well as the expected longer residence time upstream of the deflector caused by more elongated streak lines, compared to a packed bed with no deflectors.

dp dt Fi G H

141

particle diameter [m] column diameter [m] external body forces component [N] mass velocity of fluid, based on empty column [kgm-2 s] distance between deflector and top of the packed bed [mm] deflector thickness [mm] turbulent kinetic energy [m2 s2] fluid thermal conductivity [Wm-1 K1] effective thermal conductivity [Wm-1 K1] thermal conductivity at zero flow [Wm-1 K1] solid thermal conductivity [Wm-1 K1] Mach number static pressure [Pa] pressure loss [Pa] turbulent Prandtl number Reynolds number Particle Reynolds number [Rep ¼ rudp/m] temperature at any axial or radial location [K] Wall temperature [K] velocity [ms1]

h k kf ke k0e ks Mt p Dp Prt Re Rep T Tw u

Greek letters packed bed permeability [m2] ε dissipation rate of kinetic turbulence energy [m2 s3] εb void fraction [%] l laminar thermal conductivity [Wm1 K1] lt turbulent thermal conductivity [Wm1 K1] m viscosity [Pas] mt turbulent eddy viscosity [Pas] r density [kgm3] sk, sε equivalent parameters for Prandtl number for k and ε

a

Appendix A The steady state equations governing the conservation of mass, momentum and energy for a porous media can be written as:

4. Conclusions

V:ðruuÞ ¼ V:ðtÞ þ rg þ F  Vp

(A.1)

The effect of freeboard deflectors on heat transfer inside nonreacting packed bed columns has been studied. The contribution of the radiative heat transfer inside the packed bed has not been included due to the modest temperatures used, but is worth exploring in future. For the temperature range investigated (100  Ce400  C), deflectors increase wall temperatures but this depends on the heating mode (wall heated versus heated primary air). The impact of deflectors on the temperature profiles inside the packed bed is negligible. By changing boundary conditions and increasing inlet air temperature (instead of constant wall temperature), deflectors more significantly affect both the temperature distributions inside the packed bed and freeboard. Deflector effects are a combination of radiation and changed fluid flow (longer residence time).

V:ðruÞ ¼ 0

(A.2)

Acknowledgements The Edith Cowan University International Postgraduate Research Scholarship (ECU-IPRS) provided is greatly appreciated. Nomenclature cp heat capacity [J kg-1 K1] C2 inertial resistance factor [m1] C1ε, C2ε, Cm model constants

t¼m



  2 Vu þ VuT  VuI 3

(A.3)

    V: rCp uT ¼ V: lT

(A.4)

    V: rCp uT ¼ V: ðl þ lt ÞVT

(A.5)

 m 1 ui þ C2 rVs ui Fi ¼  a 2

(A.6)

a¼

d2p

ε3b

150 ð1  εb Þ2

C2 ¼

3:5 ð1  εb Þ dp ε3b

(A.7)

(A.8)

where p is the static pressure, rg is the gravitational body force, and F is external body forces. The stress tensor (t) is expressed by Equation (A.3), where m is the molecular viscosity, I is the unit tensor, and the second term on the right hand side is the effect of

142

B. Rashidian et al. / Applied Thermal Engineering 89 (2015) 134e143

volume dilation. The quantity, F, in Equation (A.1) represents the external force applied on the fluid and also simulates the flow inside the porous media for a given ith flow direction (Equation (A.6)), where Vs is the apparent velocity in an empty column (unpacked). It has been assumed that the porous medium and the fluid flow are in thermal equilibrium. The permeability, a, and the inertial loss coefficient C2, in each component direction are identified as Equations A.7 and A.8 [52]. In this study, the inertial loss coefficient may be neglected in the laminar flow regime and factor C2 is thus not required. The standard two-equation k-ε turbulence model equations are:

v v ðrkui Þ ¼ vxi vxj

"



mþ

#

mt vk v  ðrkÞ þ Gk þ Gb  rε  YM vt sk vxj

Gk ¼ rui uj

Gb ¼ bgi

v v ðrεui Þ ¼ vxi vxj



mþ

 C2ε r

#

mt vε v ε  ðrεÞ þ C1ε ðGk þ C3ε Gb Þ vt k sε vxj

ε2 þ Sε k (A.10)

    2 V:ðruuÞ ¼ V ðm þ mt Þ Vu þ VuT  VuI þ rg þ F  Vp 3 (A.11) mt ¼ rCm

k2 ε

y



C3ε ¼ tanh

u

(A.13)

YM ¼ 2rεMt2

(A.14)

(A.16)

mt vr rPrt vxi

(A.17)

For the two-equation k-ε models the value of Prt is set to 0.85. The coefficient of thermal expansion is calculated by Equation (A.18):

 1 vr r vT p

(A.18)

The Ergun equation expresses pressure drop due to resistance of the bed. Equation (A.19) shows the relationship between friction factor (in a packed bed) and the Reynolds number [44,65], where fp and Rep are defined by Equations A.20 and A.21. Additionally, Dp is the pressure drop across the bed, L is the bed height, dp is the equivalent spherical diameter of the packing, r is the flowing gas density, m is the dynamic viscosity of the flowing gas, Vs is the superficial velocity (i.e. gas velocity through an empty tube at the same volumetric flow rate) and εb (0.416) is the bed porosity. These values are presented in Table 4.

fp ¼

150 þ 1:75 Rep

fp ¼

Dp Dp L rVs2

(A.12)

where k is the turbulent kinetic energy, ε is dissipation rate, sk is the turbulent Prandtl numbers for k, sε is the turbulent Prandtl numbers for ε, Gb is the generation of turbulence kinetic energy due to buoyancy and YM represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. The source terms Sk as well as Sε are user defined source terms. The turbulent viscosity, mt, is defined by Equation (A.13), whereby Cm ¼ 0.09. Equations A.9 and A.10 contain adjustable constant C1ε, C2ε, sk and sε which have the following default values: C1ε ¼ 1.44, C2ε ¼ 1.92, sk ¼ 1.0 and sε ¼ 1.3. In general, C3ε can be estimated by Equation (A.13) where y is the component of the flow velocity parallel to the gravitational vector and u is the component of the flow velocity perpendicular to the gravitational vector. Since the main flow direction is aligned with the direction of gravity, C3ε for the buoyant shear layers will become unity. For buoyant shear layers that are perpendicular to the gravitational vector, C3ε is zero. Dilatation dissipation, YM, expresses the effect of the compressibility on turbulence which is normally neglected in the simulation of weakly compressible flows [64]. The production of turbulence kinetic energy is defined by Equation (A.15). The generation of turbulent kinetic energy due to buoyancy is specified by Equation (A.16) where gi is the component of the gravitational vector in the ith direction and Prt is the turbulent Prandtl number.

mt vT Prt vxi

Gb ¼ gi

(A.9) "

(A.15)

For ideal gas, the Equation (A.16) reduces to:

b¼

þ Sk

vuj vxi

Rep ¼

(A.19)

ε3b 1  εb

!

Dp Vs r ð1  εb Þm

(A.20)

(A.21)

The average porosity, εb, is calculated by Equation (A.22) [37]:

εb ¼ 0:375 þ 0:34

dp dt

(A.22)

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