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Effect of blockage on drag and heat transfer from a single sphere and an in-line array of three spheres
Powder Technology 168 (2006) 74 – 83 www.elsevier.com/locate/powtec
Effect of blockage on drag and heat transfer from a single sphere and an in-line array of three spheres A. Maheshwari a,⁎, R.P. Chhabra a,⁎, G. Biswas b a b
Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 208016, India Received 20 January 2006; received in revised form 15 April 2006; accepted 12 July 2006 Available online 21 August 2006
Abstract The effect of blockage ratio on the steady flow and heat transfer characteristics of incompressible fluid over a sphere and an in-line array of three spheres placed at the axis of a tube has been investigated numerically. The Navier–Stokes and thermal energy equations have been solved numerically using FLUENT for the following ranges of parameters: for a single sphere, 2 ≤ β ≤ 10; 1 ≤ Re ≤ 100; for the three-sphere system, for two values of sphere-to-sphere distance, namely s = 2 and 4. All computations were carried out for two values of the Prandtl number, i.e., 0.74 and 7, corresponding to the flow of air and water respectively. Extensive results on streamline patterns, wake characteristics (angle of separation and recirculation length), drag coefficient and Nusselt number are presented to elucidate the interplay between the blockage and the Reynolds number and their influence on drag and Nusselt number. © 2006 Elsevier B.V. All rights reserved. Keywords: Nusselt number; Wall effects; Sphere; Three spheres; Prandtl number
1. Introduction The flow of fluids past and heat transfer from spherical particles is encountered in a wide variety of industrial settings including during aseptic processing of food particles in water and polymer solutions in a continuous flow system (e.g., see Refs. [1–5], etc.), processing of suspensions in tubular heat exchangers in mineral, chemical and process engineering applications [6,7], etc. Aside from these wide ranging applications, model studies involving single spheres and arrays of spheres are germane to the development of suitable frameworks for the analysis of multi-particle systems as encountered in real life applications in the processing of liquid–solid suspensions. Notwithstanding the significance of the detailed kinematics of the flow, the temperature and the residence time characteristics, it is readily acknowledged that the hydrodynamic drag force and the convective heat transfer coefficient of a sphere on its own and as a part of an array represent the most important design parameters in such systems [6,8–10]. This study is concerned with the flow over a single sphere and over a linear ⁎ Corresponding authors. Tel.: +91 512 259 7393; fax: +91 512 259 0104. E-mail address: [email protected] (R.P. Chhabra). 0032-5910/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2006.07.007
array of three spheres fixed at the axis of a cylindrical tube for a single sphere and that for the array of three spheres. The effect of the blockage ratio is elucidated on drag and heat transfer by varying the ratio of sphere diameter to that of the tube. However, before presenting the present work in detail, it is useful and instructive to present a concise account of the previous pertinent studies available in this field. 2. Previous work It is customary to report the values of drag force and heat transfer coefficient using dimensionless groups. Thus, for instance, the dimensionless drag coefficient is a function of the blockage ratio and the Reynolds number whereas the Nusselt number shows further dependence on the Prandtl number. Furthermore, additional geometric ratios emerge for the system of multiple spheres. Over the years, voluminous body of information has accrued as far as the flow and heat transfer from a single sphere is concerned, albeit most of it relates to an unconfined sphere [8,11–13]. It is well known that the confining walls give rise to higher drag than that for an unconfined sphere [14], and similarly one would intuitively expect somewhat larger values of the heat transfer coefficient for a
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confined sphere than that for an unconfined sphere. This is simply due to that fact that the steeper temperature gradient exists in the fluid medium due to the presence of the confining walls, and this assertion is supported by the preliminary experimental results of Sastry et al. [15]. On the other hand, very little is known about the flow and heat transfer from a chain of three aligned spheres placed on the axis of a tube. Tal et al. [16] studied the flow and heat transfer from a pair of spheres in the viscous flow regime up to the Reynolds number of 40. Due to the interactions, they found the drag coefficient and Nusselt number of either spheres to be less than that for an isolated sphere otherwise under identical conditions. Subsequently, this study has been extended to the case of three spheres [17] and to a linear array of eight spheres [18]. However, it needs to be emphasized here that all these and other numerical studies have considered the case of unconfined spheres only, a condition which is nearly impossible to realize in practice owing to a range of practical constraints [19]. In an extensive experimental and numerical study, Liang et al. [20] have studied interparticle interactions by way of measuring drag force on four model configurations (including a two-particle system, an inline array of three spheres, a hexagonal and a cubic assembly) made up of equal-size spheres. In particular, the effects of interparticle spacing and the Reynolds number were investigated on the drag behaviour and flow patterns. Their results entail only very minor wall effects. Finally, it is also appropriate to add here that a few studies are also available on the free settling behaviour of chains of spheres falling in Newtonian and non-Newtonian media [21,22]. Both of these studies attempted to link the drag on a chain of spheres to that of the primary sphere. Based on the aforementioned discussion, it is thus fair to summarize the current state of the art as follows: while for a single sphere, limited results are available on the severity of the blockage
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as far as the drag is concerned, and virtually no results are available on the effect of blockage on Nusselt number even for a single sphere. Similarly, as far as known to us, no prior results exist on the extent of blockage effect for a three-sphere system. This study aims to fill this void in the existing literature. In particular, the governing equations are solved numerically in the following ranges of conditions; 1 ≤Re b 100; 2 ≤β ≤ 10 and inter-sphere spacing of 2 dp and 4 dp. All computations have been carried out for two values of the Prandtl number, namely, Pr = 0.74 (air) and Pr = 7 (water). 3. Problem statement and formulation Consider the steady and incompressible flow past a sphere or an array of three spheres as shown schematically in Fig. 1(a) and (b) respectively, together with the details of the domain used for the purpose of computations. Owing to the ϕ-symmetry, the flow is two-dimensional and no flow variable shows dependence on ϕ. The governing equations for the flow and heat transfer are that of continuity, Navier–Stokes and the thermal energy as written below: • Continuity equation jd V ¼ 0
ð1Þ
• Navier–Stokes equation q
DV ¼ −jp þ lj2 V Dt
ð2Þ
• Thermal energy equation qCp
DV ¼ kj2 T Dt
ð3Þ
In writing Eq. (3), the thermo-physical properties have been assumed to be independent of temperature and the viscous
Fig. 1. (a) Schematics of flow over a single sphere. (b) Schematic of flow over a row of three spheres.
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Table 1 Comparison of drag coefficient values for an unconfined sphere
4. Numerical solution methodology
Re
Present value
[28]
[29]
2 5 10 20 50 100 200
14.005 6.565 3.967 2.513 1.516 1.058 0.758
14.00 6.60 4.00 2.50 1.50 1.05 0.75
– – 3.9 2.5 1.50 1.06 0.75
dissipation to be negligible thereby leading to the decoupling of the flow and temperature fields. The physically realistic boundary conditions for this flow are outlined below. • At the surface of the sphere, no slip condition is prescribed, i.e., Vz = 0; Vr = 0 and for heat transfer, the sphere is assumed to be isothermal, i.e., T = Ts. • At the inlet, i.e., uniform axial velocity and temperature of the entering fluid, i.e., T=To, Vz =U, Vr =0 at z= 0 are implemented. • In this work, the walls are assumed to move with the same velocity as the inlet fluid velocity, i.e., at r = dt / 2, Vz = U, Vr = 0 and walls are assumed to be adiabatic, i.e., AT Ar ¼ 0. • At the exit plane, while as such there is no unique prescription for the outflow boundary, but the condition imposed here must not influence the conditions upstream of this plane. The commonly used boundary condition according to Orlanski [23], is equivalent to the so-called zero diffusion flux (for all flow variables) option in FLUENT. • In addition to the aforementioned conditions, owing to the symmetry of the flow, the computations have been carried out only in half of the domain, and thereby using the following symmetry conditions at the axis of the tube:
Thus, Eqs. (1)–(3) together with the above-noted boundary conditions provide the theoretical framework for studying the momentum and heat transfer characteristics over a range of the pertinent dimensionless parameters. These equations have been solved here numerically using FLUENT as described in the next section. Once the fully converged velocity and temperature fields are available, these are further processed to infer values of the integral quantities such as the drag coefficient, recirculation length, separation and Nusselt number as defined here: 2FD qU 2 dAp
hdp Nusseltnumber; Nu ¼ k
5. Results and discussion Dimensional analysis of these flows suggests the drag coefficient to be a function of the Reynolds number, blockage ratio (β) and the dimensionless inter-particle distance whereas the Nusselt number shows additional dependence on the Prandtl number. This study endeavours to establish this functional dependence for a single sphere and for a linear array of three spheres. However, prior to presenting the new results on the effect of the blockage ratio on the flow and heat transfer for the single sphere and for the in-line array of three spheres, it is useful to establish the reliability and accuracy of the results by making a few benchmark comparisons, as described in the next section. 5.1. Validation
AVz AT Vr ¼ 0; ¼0 ¼ 0; Ar Ar
Drag coefficient; CD ¼
The momentum and thermal energy equations have been solved numerically using the segregated solver module of FLUENT (version 6) for a range of conditions. Since detailed descriptions of the solution method and mesh generation, etc. are available elsewhere for the flow of Newtonian fluids past a circular cylinder [24,25] and past a circular disk [26,27], only the salient features are included here. Using GAMBIT, a structured grid was created with triangular cells only. The mesh was continually refined up to the point beyond which any further refinement changed the solution by less than 0.5% at the expense of an enormous increase in CPU time. Similarly, extensive experimentation was carried out to ascertain optimal values of the upstream and downstream distances, Lu and Ld respectively. Therefore, as will be seen below, the results presented herein are believed to be free from domain and grid effects. The accuracy and reliability of the numerical results is ascertained by benchmarking them against the available literature values for a few limiting cases.
ð4Þ ð5Þ
The recirculation length lw is defined as the distance from the rear of the sphere surface up to the point of the closed streamline. This is a measure of the size of the wake region.
Reliable results on drag and heat transfer are now available for an unconfined sphere. In this work, the flow past an unconfined sphere was simulated by using Lu = 150 dp; Ld = 250 dp and β = 400 and the values of drag coefficient and recirculation length were calculated for a range of values of the particle Reynolds number. Tables 1 and 2) show the comparisons between the present results with the literature values; excellent correspondence is seen to exist in both tables without any discernable trends. Similarly, limited results were also obtained for heat transfer from an unconfined sphere for two values of the Prandtl number, namely, Pr = 0.74 and 7.0 corresponding to air and water. The prior numerical and experimental results on Nusselt number for a sphere have been collated by Kaviany [11] who Table 2 Comparison of dimensionless recirculation length for an unconfined sphere Re
Present value
[28]
[29]
50 100 150 200
0.856 1.752 2.423 2.846
0.85 1.750 2.45 2.85
0.86 1.76 2.40 2.84
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Table 3 Comparison of mean Nusselt number for an unconfined sphere for Pr = 7
Table 5 Effect of blockage ratio on lw and θs
Re
Present value
Eq. (6)
Eq. (7)
Re
1 5 10 50 70 100
3.103 4.691 5.859 10.545 12.089 14.155
2.999 4.734 5.938 10.517 11.922 13.939
3.002 4.33 5.36 9.94 11.51 13.53
presented the following equation for estimating the value of the average Nusselt number: Nu−1
1=3 Peþ1 Pe dPr
¼ Re0:41
ð6Þ
A comparison between the present values and the predictions of Eq. (6) is presented in Table 3 for Pr = 7 for a range of values of the Reynolds number. Once again, the two values are seen to be within 1.5% of each other. Included in this table are also the predictions of another experimental correlation according to Whitaker [30]: 1=4 l Nu ¼ 2 þ ð0:4Re1=2 þ 0:06Re2=3 ÞPr0:4 b ð7Þ l0
30 50 70 100
θs
lw β=2
β=6
β = 10
β=2
β=6
β = 10
0.12 0.60 1.04 1.60
0.28 0.90 1.20 1.80
0.31 0.91 1.22 1.81
18.9 36.9 44.4 49.5
27.4 40.5 47.7 52.2
28.7 41.8 49.4 53.1
5.2. Effect of blockage on flow and heat transfer from a single sphere While the effect of the blockage ratio on the drag on a single sphere has been numerically studied by a few workers [14,28,31], very little information is available on the role of blockage on the recirculation length and separation angle. Table 5 shows the functional dependence of these two parameters on the blockage ratio and the Reynolds number. As expected, for a fixed value of the blockage ratio, both the recirculation length
The values predicted in Table 3 are based on the assumption of (μb / μ0)1 / 4 ⋍ 1. This simplification is reasonable since in the present study the fluid properties have been assumed to be independent of temperature. Bearing this in mind, the present numerical values are seen to be in excellent agreement with the predictions of Eq. (7). Finally, the present solution methodology has also been benchmarked for the case of finite wall effects on a single sphere. Wham et al. [31] have studied the wall effects on a single rigid sphere falling at the axis of a tube. Table 4 shows a comparison between the present values of the drag coefficient and that of Wham et al. [31] for β = 3.2 over a range of the particle Reynolds number 2 ≤ Re ≤ 50. Once again, the two results are seen to be in good agreement, albeit the maximum difference is of the order of 5%. Such differences in numerical results are not uncommon and are often ascribed to the differences inherent in mesh, solution procedures, type of solver, etc. Based on the aforementioned comparisons, it is perhaps fair to conclude that the new results reported herein are reliable to within 2% or so. Table 4 Drag on a confined sphere (β = 3.2) Re
2 5 10 20 50
Values of CD Present value
[31]
35.0 12.0 7.5 5.0 3.0
36.08 12.28 7.33 4.75 3.04
Fig. 2. Effect of blockage ratio and Prandtl number on Nusselt number for a single sphere.
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Table 6 Effects of Reynolds number and inter-sphere spacing on drag coefficient for three aligned spheres (β = 10) Re
Value of CD s=4
s=2
1st sphere 2nd sphere 3rd sphere 1st sphere 2nd sphere 3rd sphere 1 30.42 10 4.39 50 1.58 100 1.08
28.57 3.35 1.00 0.63
29.27 3.22 0.92 0.57
26.89 4.08 1.47 1.01
20.49 2.46 0.69 0.40
24.23 2.55 0.75 0.47
and the angle of separation increase with the increasing Reynolds number. Also, for a fixed value of the Reynolds number, both these parameters also show a positive dependence on the blockage ratio. Furthermore, the values of lw and θs at β = 10 are within 5–6% of the corresponding values for an unconfined sphere. Broadly, the confining walls seem to have a stabilizing effect on the flow and thus the wake formation is somewhat delayed.
Fig. 4. Effect of Reynolds number on streamline patterns for β = 10 (nearly unconfined flow) and s = 2. (a) Re = 10, (b) Re = 50 and (c) Re = 100.
Fig. 3. Effect of Reynolds number on streamline patterns for β = 10 (nearly unconfined flow) and s = 4. (a) Re = 10, (b) Re = 50 and (c) Re = 100.
Fig. 2 shows the effect of blockage ratio on the average Nusselt number for air and water. It is seen that for the extreme value of β = 2 used here, there is about 15% enhancement in the value of the Nusselt number for a sphere. This is in stark contrast to the fact that the drag on a sphere for β = 2 at low Reynolds number could be up to 4 times the value for the unconfined sphere. However, the influence of blockage gradually diminishes with a progressive increase in the value of the Reynolds number. Therefore, the average heat transfer characteristics are much less sensitive to the blockage ratio than the flow characteristics. This finding is also consistent at least qualitatively with the experimental findings of Sastry et al. [15] who reported an enhancement of 25% to 100% in the value of the Nusselt number over the Reynolds number range of 7300 to 45,000 with water. In concluding this section, it is thus appropriate to reiterate here that the blockage influences the flow characteristics (drag, recirculation length, angle of separation) much more than the forced convection heat transfer characteristics for a single sphere placed at the axis of a tube.
A. Maheshwari et al. / Powder Technology 168 (2006) 74–83 Table 7 Comparison between the present results and that of Ramachandran et al. [17] for air Re
Table 9 Effect of blockage on Nusselt number Re
Value of Nusselt number 1st sphere Present
Value of Nusselt number β=2
2nd sphere
3rd sphere
1st
Literature
Present
Literature
Present
Literature
Separation = 2 dp 1 2.03 10 3.32 50 5.42 100 6.98
2.12 3.37 5.50 7.05
1.832 2.34 3.44 4.23
1.812 2.324 3.39 4.18
1.63 2.05 3.08 3.77
1.629 2.025 2.98 3.74
Separation = 4 dp 1 2.20 10 3.328 50 5.399 100 6.91
2.17 3.275 5.401 6.96
1.936 2.720 4.105 5.092
2.03 2.79 4.18 5.16
1.64 2.53 3.52 4.39
1.633 2.49 3.60 4.42
5.3. Results for the three-sphere system In order to explore the combined effects of blockage and of sphere–sphere interactions, the flow past an in-line array of three spheres has been studied for a range of values of β, Reynolds number and for two values of the center-to-center spacing between the spheres, namely, s = 2 dp and 4 dp. The choice of these values is in part governed by the study of Ramachandran et al. [17] who studied heat transfer from an unconfined in-line array of three spheres. Table 6 shows the individual values of drag of the three spheres for a range of values of the Reynolds number and for the two values of the inter-sphere spacing. For the case of the inter-sphere spacing, s = 4 dp, the drag coefficient of the first sphere is almost identical to that of a single sphere, and the 2nd and 3rd spheres show a decreasing value of drag coefficient except at Re = 1. While for the sphere-to-sphere separation of s = 2 dp, the last sphere shows a higher drag than that of the 2nd sphere. These trends are clearly due to the more significant sphere–sphere interactions in
79
β=4 2nd
3rd
s = 4, Pr = 0.74 5 2.67 10 3.79 20 4.82 50 6.39 100 8.09
0.753 1.86 3.23 4.94 6.20
0.211 0.912 2.19 4.14 5.52
s = 2, Pr = 0.74 5 2.67 10 3.79 20 4.82 50 6.38 100 8.08
0.75 1.82 3.05 4.41 5.44
0.217 0.914 2.13 3.73 4.78
s = 4, Pr = 7.0 1 3.69 1.74 5 6.17 4.74 10 7.58 5.81 20 9.38 7.05 50 12.66 9.27 100 16.54 11.81
0.82 3.94 5.13 6.25 8.19 10.43
s = 2, Pr = 7.0 1 3.69 1.71 5 6.16 4.27 10 7.57 5.20 20 9.37 6.33 50 12.64 8.32 100 16.50 10.61
0.825 3.58 4.51 5.51 7.25 9.28
1st
β=6 2nd
3rd
2.98 3.50 4.22 5.64 7.20
2.34 2.91 3.44 4.31 5.23
1.88 2.58 3.16 3.94 4.71
2.94 3.47 4.15 5.59 7.14
2.10 2.52 2.93 3.66 4.46
3.41 5.06 6.21 7.80 10.85 14.43
3.37 5.01 6.15 7.72 10.76 14.27
1st
2nd
3rd
2.92 3.41 4.13 5.53 7.08
2.48 2.83 3.29 5.14 6.05
2.26 2.64 3.04 3.76 4.53
1.75 2.26 2.66 3.28 3.95
2.85 3.37 4.08 5.47 6.99
2.11 2.41 2.79 3.51 4.30
1.96 2.22 2.53 3.15 3.82
2.84 4.11 4.89 5.86 7.58 9.56
2.50 3.77 4.45 5.30 6.74 8.33
3.26 4.84 5.98 7.57 10.62 14.18
2.78 3.90 4.61 5.52 7.23 9.18
2.59 3.60 4.22 4.98 6.37 7.93
2.50 3.57 4.20 4.98 6.48 8.39
2.23 3.23 3.79 4.48 5.72 7.16
3.20 4.76 5.89 7.46 10.49 13.95
2.39 3.30 3.87 4.64 6.14 8.02
2.22 3.04 3.53 4.17 5.42 6.84
the latter case than that for the larger inter-sphere separation, s = 4 dp. The physical reasons for this behaviour are clearly brought out by the streamline patterns shown in Figs. 3 and 4 for the three values of the Reynolds number. For the inter-sphere separation of 4 dp, one can clearly see that even though no flow
Table 8 Effect of blockage ratio on drag coefficient Re
Value of drag coefficient β=2
β=4
β=6
1st
2nd
3rd
1st
2nd
3rd
1st
2nd
3rd
s=4 1 5 10 20 50 100
143.17 28.79 14.60 7.64 3.63 2.33
143.76 28.90 14.65 7.65 3.55 2.13
143.45 28.83 14.62 7.63 3.54 2.12
47.72 10.05 5.53 3.27 1.80 1.22
47.69 9.97 5.31 2.88 1.36 0.82
47.73 9.98 5.31 2.88 1.31 0.764
36.63 8.18 4.71 2.89 1.64 1.12
36.22 7.62 4.02 2.23 1.11 0.69
36.42 7.67 4.03 2.17 1.03 0.63
s=2 1 5 10 20 50 100
142.7 28.67 14.53 7.59 3.61 2.32
142.67 28.65 14.48 7.47 3.29 1.85
142.97 28.73 14.53 7.51 3.31 1.85
45.67 9.68 5.33 3.15 1.73 1.17
43.12 8.67 4.36 2.24 1.01 0.579
45.25 9.10 4.62 2.38 1.04 0.613
33.58 7.68 4.45 2.73 1.54 1.05
28.94 5.79 3.00 1.63 0.776 0.446
32.35 6.37 3.22 1.71 0.82 0.509
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separation occurs at Re = 10, but unlike the first sphere, the 2nd and 3rd spheres are exposed to distorted velocity profiles thereby influencing the values of the drag coefficients. This effect is, however, suppressed at Re = 1, owing to the weak advection effects. However, as the Reynolds number is progressively increased to 50 and 100, Fig. 3 clearly shows the increasing size of the wake behind the first sphere, whereas the wakes formed in the rear of the 2nd and 3rd spheres are clearly smaller (but of comparable size) than that for the 1st sphere. The difference in wake sizes is seen to grow with the increasing value of the Reynolds number. This explains for the differences in the value of the drag coefficient for the three spheres as seen in Table 6. Now turning our attention to Fig. 4, these interactions are further accentuated and one can now clearly see the decreasing size of the wake thereby lowering of the drag for each successive sphere. Finally, it is suffice to add here that notwithstanding the significance of wall effects, the trends seen in Table 6 are qualitatively consistent with the experimental observations of Liang et al. [20]. Based on their numerical computations, Ramachandran et al. [17] presented a correlation for heat transfer from an unconfined in-line array consisting of three spheres with equal and unequal spacing of 2 to 10 dp. The present results are compared with their results for air (Pr = 0.74) in Table 7 for the two values of the sphereto-sphere separation. Once again, the two results are seen to be in excellent agreement, the maximum discrepancy being of the order of 1–1.5%. Such a close correspondence inspires confidence in the reliability of the new results obtained in this study.
Tables 8 and 9 show the effect of blockage on drag and heat transfer for the three-sphere configuration studied here for the three values of β and for the two values of the inter-sphere separation. An inspection of Table 8 reveals qualitatively similar trends as that seen for an unconfined array of three spheres. Furthermore, as noted earlier, the increasing blockage, i.e., the decreasing value of β suggests the delayed flow separation and therefore the drag of the three spheres is pretty much identical up to about Re = 20 for β = 2, irrespective of the value of the sphere-to-sphere separation. On the other hand, as the value of β is gradually increased, the three values of drag begin to differ from each other. The effect of blockage on drag is seen to gradually reduce with the increasing Reynolds number. For instance, irrespective of the value of s, at Re = 1, the drag on the first sphere for β = 2 is nearly four times the corresponding value for β = 6; whereas the corresponding ratio of the drag coefficients is only a little more than 2 at Re = 100. Conversely, one must employ a larger domain in the lateral direction to simulate low Reynolds number flows than the high Reynolds number conditions. Figs. 5 and 6 show the corresponding streamline patterns for β = 2 and β = 6 for Re = 50 and 100 and s = 2 and 4. For s = 4 and β = 2, the wake size is almost identical for the three spheres which in turn yields comparable values of drag for each sphere as seen in Table 8 thereby implying little or no sphere–sphere interactions. Similar trends are seen in Fig. 6 for β = 6. An examination of the results shown in Table 9 reveals an intricate functional dependence of the Nusselt number on the
Fig. 5. Effect of sphere-to-sphere spacing and Reynolds number on streamline patterns for β = 2. (a) Re = 50, s = 4; (b) Re = 100, s = 4; (c) Re = 50, s = 2; (d) Re = 100, s = 2.
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Fig. 6. Effect of sphere-to-sphere spacing and Reynolds number on streamline patterns for β = 6. (a) Re = 50, s = 4; (b) Re = 100, s = 4; (c) Re = 50, s = 2; (d) Re = 100, s = 2.
blockage ratio (β), Prandtl (Pr) and Reynolds (Re) numbers and on sphere-to-sphere separation. The key trends can be summarized as follows: (i) For fixed values of β and s, the Nusselt number varies approximately as Pr1 / 2, except at Re = 1, where there is very little convection. Similarly, all else being equal the results show dependence on the Reynolds number as ∼ Re0.32 to Re0.50 , the value of the exponent of 0.5 can be justified based on the laminar boundary layer analysis whereas the smaller values of index compare favourably with Eq. (6) for a single sphere. (ii) For fixed values of the Reynolds and Prandtl numbers, the Nusselt number is maximum for the first sphere and it is reduced progressively for the 2nd and 3rd spheres. This trend is observed for all values of β and s and it stems from the fact that the driving force for heat transfer (ΔT) decreases as one moves from the first sphere to the 2nd and to the 3rd sphere. This effect is particularly acute for β = 2 and s = 2; it, however, gradually diminishes with the increasing value of β and/or s. (iii) Also, the enhancement in the rate of heat transfer due to the sharpening of temperature gradients by
blockage is only realized when there is some advection, i.e., Re N ∼ 5–10 and at high Prandtl numbers when the thermal boundary layer is thin, e.g., see the results for Pr = 7 as opposed to that for air (Pr = 0.74). (iv) The effect of blockage on the Nusselt number is much weaker than that on drag coefficient, akin to that for the case of a single sphere. Furthermore, in view of the fact that the values of the Nusselt number for β = 6 are within 5% of the corresponding values for the unconfined system, shorter computational domain (in the lateral direction) is thus needed for heat transfer calculations than that for the flow field. This finding is also consistent with our previous work on circular cylinders [24,25]. In summary, the effect of blockage on drag is much stronger than that for heat transfer coefficient, both for a single sphere and for the three-sphere array studied here. The drag of the subsequent spheres may decrease and/or increase (with reference to the first sphere) depending upon the values of β, s and Re. This is directly linked to the size of the wake of each sphere.
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6. Concluding remarks The effect of blockage on the flow and heat transfer characteristics of a single sphere and of an in-line array of three equally spaced spheres has been investigated numerically in the range 1 ≤ Re ≤ 100 for two values of Prandtl number, namely, 0.74 and 7 corresponding to air and water. The blockage seems to influence the drag much more strongly than the heat transfer phenomenon. Conversely, it is possible to obtain reliable heat transfer results by using shorter domains than that required for flow calculations. The inter-sphere spacing also exerts an important influence on the drag and Nusselt number values due to the obvious interactions between the wake of the first sphere and the flow over the succeeding spheres. However, while the drag can increase by a factor of 10, the corresponding enhancement in heat transfer is much less than this amount. Nomenclature Ap projected area normal to flow (m2) Cp heat capacity of fluid (J/kg K) CD drag coefficient (−) dp sphere diameter (−) dt tube diameter (m) FD drag force (N) h heat transfer coefficient (W/m2 K) k thermal conductivity of fluid (W/m K) Ld downstream distance (m) Lu upstream distance (m) Nu Nusselt number (−) p pressure (Pa) Pe Peclet number (−) Pr Prandtl number (−) r radial coordinate (m) Re Reynolds number (−) s non-dimensional center-to-center spacing between spheres (−) t time (s) Ts sphere temperature (K) To inlet fluid temperature (K) U uniform velocity at inlet (m/s) V velocity vector (m/s) z axial coordinate (m) Greek Symbols β blockage ratio, = dt/dp (−) μb fluid viscosity at bulk temperature (Pa s) μ0 fluid viscosity at reference temperature (Pa s) ρ fluid density (kg/m3) ∇ del operator (m− 1) References [1] A. Astrom, G. Bark, Heat transfer between fluid and particles in aseptic processing, J. Food Eng. 21 (1994) 97–125. [2] A. Alhamdan, S.K. Sastry, Bulk average heat transfer coefficient of multiple particles flowing in a holding tube, Trans. Inst. Chem. Eng. 76C (1998) 95–101.
[3] P.N. Baptista, F.A.R. Oliveira, J.C. Oliveira, S.K. Sastry, Dimensionless analysis of fluid-to-particle heat transfer coefficients, J. Food Eng. 31 (1997) 199–218. [4] M. Barigou, S. Mankad, P.J. Fryer, Heat transfer in two-phase solid–liquid food flows: a review, Trans. Inst. Chem. Eng. 76C (1998) 3–29. [5] H.S. Ramaswamy, M.S. Zareifard, Evaluation of factors influencing tubeflow fluid-to-particle heat transfer coefficient using a calorimetric technique, J. Food Eng. 45 (2000) 127–138. [6] R.P. Chhabra, Heat and mass transfer in rheologically complex systems, in: D.A. Siginer, D. DeKee, R.P. Chhabra (Eds.), Advances in the Flow and Rheology of Non-Newtonian Fluids, Elsevier, Amsterdam, 1999, Chapter 39. [7] R.P. Chhabra, J.F. Richardson, Non-Newtonian Flow in the Process Industries, Butterworth-Heinemann, Oxford, 1999. [8] R. Clift, J. Grace, M.E. Weber, Bubbles, Drops and Particles, Academic Press, New York, 1978. [9] R.P. Chhabra, Bubbles, Drops and Particles in Non-Newtonian Fluids, second ed., CRC Press, Boca Raton, FL, 2006. [10] R.P. Chhabra, Fluid flow, heat and mass transfer in non-Newtonian fluids, Adv. Heat Transf. 23 (1993) 187–278. [11] M. Kaviany, Principles of Convection Heat Transfer, Springer Verlag, New York, 1994. [12] B. Melissari, S.A. Argyropoulos, Development of a heat transfer dimensionless correlation for spheres immersed in a wide range of Prandtl number fluids, Int. J. Heat Mass Transfer 48 (2005) 4333–4341. [13] S.D. Dhole, R.P. Chhabra, V. Eswaran, A numerical study on the forced convection heat transfer from an isothermal and isoflux sphere in the steady-symmetric regime, Int. J. Heat Mass Transfer 49 (2006) 984–994. [14] R.P. Chhabra, Wall effects on spheres falling axially in cylindrical tubes, in: D. DeKee, R.P. Chhabra (Eds.), Transport Processes in Bubbles, Drops and Particles, 2nd edition, Taylor and Francis, New York, 2002, Chapter 13. [15] S.K. Sastry, M. Lima, J. Brim, T. Brunn, B.F. Heskitt, Liquid-toparticle heat transfer during continuous tube flow: influence of flow rate and particle to tube diameter ratio, J. Food Process Eng. 13 (1990) 239–253. [16] R. Tal, D.N. Lee, W.A. Sirignano, Heat and momentum transfer around a pair of spheres in viscous flow, Int. J. Heat Mass Transfer 27 (1984) 1953–1962. [17] R.S. Ramachandran, C. Kleinstreuer, T.-Y. Wang, Forced convection heat transfer of interacting spheres, Numer. Heat Transf. 15A (1989) 471–487. [18] J.S. Tsai, A.M. Sterling, The application of an embedded grid to solution of heat and momentum transfer for spheres in a linear array, Int. J. Heat Mass Transfer 33 (1990) 2491–2502. [19] Y. Tsuji, Y. Morikawa, K. Terashima, Fluid-dynamic interaction between two spheres, Int. J. Multiph. Flow 8 (1982) 71–82. [20] S.-C. Liang, T. Hong, L.-S. Fan, Effects of particle arrangements on the drag force of a particle in the intermediate flow regime, Int. J. Multiph. Flow 22 (1996) 285–306. [21] G. Kasper, T. Niida, M. Yang, Measurements of viscous drag on cylinders and chains of spheres with aspect ratios between 2 and 50, J. Aerosol Sci. 16 (1985) 535–556. [22] R.P. Chhabra, T. Singh, S. Nandrajog, Drag on chains and agglomerates of spheres in viscous Newtonian and power law fluids, Can. J. Chem. Eng. 73 (1995) 566–571. [23] I. Orlanski, A simple boundary condition for unbounded hyperbolic flows, J. Comput. Phys. 21 (1976) 251–269. [24] J. Chakraborty, N. Verma, R.P. Chhabra, Wall effects in flow past a circular cylinder in a plane channel: a numerical study, Chem. Eng. Process. 43 (2004) 1529–1537. [25] S. Mettu, N. Verma, R.P. Chhabra. Momentum and heat transfer from an asymmetrically confined circular cylinder in a plane channel, Heat Mass Transf., (in press). [26] S. Nitin, R.P. Chhabra, Wall effects in two dimensional axisymmetric flow over a circular disk oriented normal to flow in a cylindrical tube, Can. J. Chem. Eng. 83 (2005) 450–457.
A. Maheshwari et al. / Powder Technology 168 (2006) 74–83 [27] S. Nitin, R.P. Chhabra, Sedimentation of a circular disk in power law fluids, J. Colloid Interface Sci. 295 (2006) 520–527. [28] J.H. Oh, S.J. Lee, A study on the Newtonian fluid flow past a sphere in a tube, Korean J. Chem. Eng. 5 (1988) 190–196. [29] T.A. Johnson, V.C. Patel, Flow past a sphere up to a Reynolds number of 300, J. Fluid Mech. 378 (1999) 19–70.
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[30] S. Whitaker, Forced convection heat transfer correlations for flow in pipes, past flat plates, single cylinders, single spheres, and for flow in packed beds and tube bundles, AIChE J. 18 (1972) 361–371. [31] R.M. Wham, O.A. Basaran, C.H. Byers, Wall effects on flow past solid spheres at finite Reynolds number, Ind. Eng. Chem. Res. 35 (1996) 864–874.
Effect of blockage on drag and heat transfer from a single sphere and an in-line array of three spheres A. Maheshwari a,⁎, R.P. Chhabra a,⁎, G. Biswas b a b
Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 208016, India Received 20 January 2006; received in revised form 15 April 2006; accepted 12 July 2006 Available online 21 August 2006
Abstract The effect of blockage ratio on the steady flow and heat transfer characteristics of incompressible fluid over a sphere and an in-line array of three spheres placed at the axis of a tube has been investigated numerically. The Navier–Stokes and thermal energy equations have been solved numerically using FLUENT for the following ranges of parameters: for a single sphere, 2 ≤ β ≤ 10; 1 ≤ Re ≤ 100; for the three-sphere system, for two values of sphere-to-sphere distance, namely s = 2 and 4. All computations were carried out for two values of the Prandtl number, i.e., 0.74 and 7, corresponding to the flow of air and water respectively. Extensive results on streamline patterns, wake characteristics (angle of separation and recirculation length), drag coefficient and Nusselt number are presented to elucidate the interplay between the blockage and the Reynolds number and their influence on drag and Nusselt number. © 2006 Elsevier B.V. All rights reserved. Keywords: Nusselt number; Wall effects; Sphere; Three spheres; Prandtl number
1. Introduction The flow of fluids past and heat transfer from spherical particles is encountered in a wide variety of industrial settings including during aseptic processing of food particles in water and polymer solutions in a continuous flow system (e.g., see Refs. [1–5], etc.), processing of suspensions in tubular heat exchangers in mineral, chemical and process engineering applications [6,7], etc. Aside from these wide ranging applications, model studies involving single spheres and arrays of spheres are germane to the development of suitable frameworks for the analysis of multi-particle systems as encountered in real life applications in the processing of liquid–solid suspensions. Notwithstanding the significance of the detailed kinematics of the flow, the temperature and the residence time characteristics, it is readily acknowledged that the hydrodynamic drag force and the convective heat transfer coefficient of a sphere on its own and as a part of an array represent the most important design parameters in such systems [6,8–10]. This study is concerned with the flow over a single sphere and over a linear ⁎ Corresponding authors. Tel.: +91 512 259 7393; fax: +91 512 259 0104. E-mail address: [email protected] (R.P. Chhabra). 0032-5910/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2006.07.007
array of three spheres fixed at the axis of a cylindrical tube for a single sphere and that for the array of three spheres. The effect of the blockage ratio is elucidated on drag and heat transfer by varying the ratio of sphere diameter to that of the tube. However, before presenting the present work in detail, it is useful and instructive to present a concise account of the previous pertinent studies available in this field. 2. Previous work It is customary to report the values of drag force and heat transfer coefficient using dimensionless groups. Thus, for instance, the dimensionless drag coefficient is a function of the blockage ratio and the Reynolds number whereas the Nusselt number shows further dependence on the Prandtl number. Furthermore, additional geometric ratios emerge for the system of multiple spheres. Over the years, voluminous body of information has accrued as far as the flow and heat transfer from a single sphere is concerned, albeit most of it relates to an unconfined sphere [8,11–13]. It is well known that the confining walls give rise to higher drag than that for an unconfined sphere [14], and similarly one would intuitively expect somewhat larger values of the heat transfer coefficient for a
A. Maheshwari et al. / Powder Technology 168 (2006) 74–83
confined sphere than that for an unconfined sphere. This is simply due to that fact that the steeper temperature gradient exists in the fluid medium due to the presence of the confining walls, and this assertion is supported by the preliminary experimental results of Sastry et al. [15]. On the other hand, very little is known about the flow and heat transfer from a chain of three aligned spheres placed on the axis of a tube. Tal et al. [16] studied the flow and heat transfer from a pair of spheres in the viscous flow regime up to the Reynolds number of 40. Due to the interactions, they found the drag coefficient and Nusselt number of either spheres to be less than that for an isolated sphere otherwise under identical conditions. Subsequently, this study has been extended to the case of three spheres [17] and to a linear array of eight spheres [18]. However, it needs to be emphasized here that all these and other numerical studies have considered the case of unconfined spheres only, a condition which is nearly impossible to realize in practice owing to a range of practical constraints [19]. In an extensive experimental and numerical study, Liang et al. [20] have studied interparticle interactions by way of measuring drag force on four model configurations (including a two-particle system, an inline array of three spheres, a hexagonal and a cubic assembly) made up of equal-size spheres. In particular, the effects of interparticle spacing and the Reynolds number were investigated on the drag behaviour and flow patterns. Their results entail only very minor wall effects. Finally, it is also appropriate to add here that a few studies are also available on the free settling behaviour of chains of spheres falling in Newtonian and non-Newtonian media [21,22]. Both of these studies attempted to link the drag on a chain of spheres to that of the primary sphere. Based on the aforementioned discussion, it is thus fair to summarize the current state of the art as follows: while for a single sphere, limited results are available on the severity of the blockage
75
as far as the drag is concerned, and virtually no results are available on the effect of blockage on Nusselt number even for a single sphere. Similarly, as far as known to us, no prior results exist on the extent of blockage effect for a three-sphere system. This study aims to fill this void in the existing literature. In particular, the governing equations are solved numerically in the following ranges of conditions; 1 ≤Re b 100; 2 ≤β ≤ 10 and inter-sphere spacing of 2 dp and 4 dp. All computations have been carried out for two values of the Prandtl number, namely, Pr = 0.74 (air) and Pr = 7 (water). 3. Problem statement and formulation Consider the steady and incompressible flow past a sphere or an array of three spheres as shown schematically in Fig. 1(a) and (b) respectively, together with the details of the domain used for the purpose of computations. Owing to the ϕ-symmetry, the flow is two-dimensional and no flow variable shows dependence on ϕ. The governing equations for the flow and heat transfer are that of continuity, Navier–Stokes and the thermal energy as written below: • Continuity equation jd V ¼ 0
ð1Þ
• Navier–Stokes equation q
DV ¼ −jp þ lj2 V Dt
ð2Þ
• Thermal energy equation qCp
DV ¼ kj2 T Dt
ð3Þ
In writing Eq. (3), the thermo-physical properties have been assumed to be independent of temperature and the viscous
Fig. 1. (a) Schematics of flow over a single sphere. (b) Schematic of flow over a row of three spheres.
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Table 1 Comparison of drag coefficient values for an unconfined sphere
4. Numerical solution methodology
Re
Present value
[28]
[29]
2 5 10 20 50 100 200
14.005 6.565 3.967 2.513 1.516 1.058 0.758
14.00 6.60 4.00 2.50 1.50 1.05 0.75
– – 3.9 2.5 1.50 1.06 0.75
dissipation to be negligible thereby leading to the decoupling of the flow and temperature fields. The physically realistic boundary conditions for this flow are outlined below. • At the surface of the sphere, no slip condition is prescribed, i.e., Vz = 0; Vr = 0 and for heat transfer, the sphere is assumed to be isothermal, i.e., T = Ts. • At the inlet, i.e., uniform axial velocity and temperature of the entering fluid, i.e., T=To, Vz =U, Vr =0 at z= 0 are implemented. • In this work, the walls are assumed to move with the same velocity as the inlet fluid velocity, i.e., at r = dt / 2, Vz = U, Vr = 0 and walls are assumed to be adiabatic, i.e., AT Ar ¼ 0. • At the exit plane, while as such there is no unique prescription for the outflow boundary, but the condition imposed here must not influence the conditions upstream of this plane. The commonly used boundary condition according to Orlanski [23], is equivalent to the so-called zero diffusion flux (for all flow variables) option in FLUENT. • In addition to the aforementioned conditions, owing to the symmetry of the flow, the computations have been carried out only in half of the domain, and thereby using the following symmetry conditions at the axis of the tube:
Thus, Eqs. (1)–(3) together with the above-noted boundary conditions provide the theoretical framework for studying the momentum and heat transfer characteristics over a range of the pertinent dimensionless parameters. These equations have been solved here numerically using FLUENT as described in the next section. Once the fully converged velocity and temperature fields are available, these are further processed to infer values of the integral quantities such as the drag coefficient, recirculation length, separation and Nusselt number as defined here: 2FD qU 2 dAp
hdp Nusseltnumber; Nu ¼ k
5. Results and discussion Dimensional analysis of these flows suggests the drag coefficient to be a function of the Reynolds number, blockage ratio (β) and the dimensionless inter-particle distance whereas the Nusselt number shows additional dependence on the Prandtl number. This study endeavours to establish this functional dependence for a single sphere and for a linear array of three spheres. However, prior to presenting the new results on the effect of the blockage ratio on the flow and heat transfer for the single sphere and for the in-line array of three spheres, it is useful to establish the reliability and accuracy of the results by making a few benchmark comparisons, as described in the next section. 5.1. Validation
AVz AT Vr ¼ 0; ¼0 ¼ 0; Ar Ar
Drag coefficient; CD ¼
The momentum and thermal energy equations have been solved numerically using the segregated solver module of FLUENT (version 6) for a range of conditions. Since detailed descriptions of the solution method and mesh generation, etc. are available elsewhere for the flow of Newtonian fluids past a circular cylinder [24,25] and past a circular disk [26,27], only the salient features are included here. Using GAMBIT, a structured grid was created with triangular cells only. The mesh was continually refined up to the point beyond which any further refinement changed the solution by less than 0.5% at the expense of an enormous increase in CPU time. Similarly, extensive experimentation was carried out to ascertain optimal values of the upstream and downstream distances, Lu and Ld respectively. Therefore, as will be seen below, the results presented herein are believed to be free from domain and grid effects. The accuracy and reliability of the numerical results is ascertained by benchmarking them against the available literature values for a few limiting cases.
ð4Þ ð5Þ
The recirculation length lw is defined as the distance from the rear of the sphere surface up to the point of the closed streamline. This is a measure of the size of the wake region.
Reliable results on drag and heat transfer are now available for an unconfined sphere. In this work, the flow past an unconfined sphere was simulated by using Lu = 150 dp; Ld = 250 dp and β = 400 and the values of drag coefficient and recirculation length were calculated for a range of values of the particle Reynolds number. Tables 1 and 2) show the comparisons between the present results with the literature values; excellent correspondence is seen to exist in both tables without any discernable trends. Similarly, limited results were also obtained for heat transfer from an unconfined sphere for two values of the Prandtl number, namely, Pr = 0.74 and 7.0 corresponding to air and water. The prior numerical and experimental results on Nusselt number for a sphere have been collated by Kaviany [11] who Table 2 Comparison of dimensionless recirculation length for an unconfined sphere Re
Present value
[28]
[29]
50 100 150 200
0.856 1.752 2.423 2.846
0.85 1.750 2.45 2.85
0.86 1.76 2.40 2.84
A. Maheshwari et al. / Powder Technology 168 (2006) 74–83
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Table 3 Comparison of mean Nusselt number for an unconfined sphere for Pr = 7
Table 5 Effect of blockage ratio on lw and θs
Re
Present value
Eq. (6)
Eq. (7)
Re
1 5 10 50 70 100
3.103 4.691 5.859 10.545 12.089 14.155
2.999 4.734 5.938 10.517 11.922 13.939
3.002 4.33 5.36 9.94 11.51 13.53
presented the following equation for estimating the value of the average Nusselt number: Nu−1
1=3 Peþ1 Pe dPr
¼ Re0:41
ð6Þ
A comparison between the present values and the predictions of Eq. (6) is presented in Table 3 for Pr = 7 for a range of values of the Reynolds number. Once again, the two values are seen to be within 1.5% of each other. Included in this table are also the predictions of another experimental correlation according to Whitaker [30]: 1=4 l Nu ¼ 2 þ ð0:4Re1=2 þ 0:06Re2=3 ÞPr0:4 b ð7Þ l0
30 50 70 100
θs
lw β=2
β=6
β = 10
β=2
β=6
β = 10
0.12 0.60 1.04 1.60
0.28 0.90 1.20 1.80
0.31 0.91 1.22 1.81
18.9 36.9 44.4 49.5
27.4 40.5 47.7 52.2
28.7 41.8 49.4 53.1
5.2. Effect of blockage on flow and heat transfer from a single sphere While the effect of the blockage ratio on the drag on a single sphere has been numerically studied by a few workers [14,28,31], very little information is available on the role of blockage on the recirculation length and separation angle. Table 5 shows the functional dependence of these two parameters on the blockage ratio and the Reynolds number. As expected, for a fixed value of the blockage ratio, both the recirculation length
The values predicted in Table 3 are based on the assumption of (μb / μ0)1 / 4 ⋍ 1. This simplification is reasonable since in the present study the fluid properties have been assumed to be independent of temperature. Bearing this in mind, the present numerical values are seen to be in excellent agreement with the predictions of Eq. (7). Finally, the present solution methodology has also been benchmarked for the case of finite wall effects on a single sphere. Wham et al. [31] have studied the wall effects on a single rigid sphere falling at the axis of a tube. Table 4 shows a comparison between the present values of the drag coefficient and that of Wham et al. [31] for β = 3.2 over a range of the particle Reynolds number 2 ≤ Re ≤ 50. Once again, the two results are seen to be in good agreement, albeit the maximum difference is of the order of 5%. Such differences in numerical results are not uncommon and are often ascribed to the differences inherent in mesh, solution procedures, type of solver, etc. Based on the aforementioned comparisons, it is perhaps fair to conclude that the new results reported herein are reliable to within 2% or so. Table 4 Drag on a confined sphere (β = 3.2) Re
2 5 10 20 50
Values of CD Present value
[31]
35.0 12.0 7.5 5.0 3.0
36.08 12.28 7.33 4.75 3.04
Fig. 2. Effect of blockage ratio and Prandtl number on Nusselt number for a single sphere.
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Table 6 Effects of Reynolds number and inter-sphere spacing on drag coefficient for three aligned spheres (β = 10) Re
Value of CD s=4
s=2
1st sphere 2nd sphere 3rd sphere 1st sphere 2nd sphere 3rd sphere 1 30.42 10 4.39 50 1.58 100 1.08
28.57 3.35 1.00 0.63
29.27 3.22 0.92 0.57
26.89 4.08 1.47 1.01
20.49 2.46 0.69 0.40
24.23 2.55 0.75 0.47
and the angle of separation increase with the increasing Reynolds number. Also, for a fixed value of the Reynolds number, both these parameters also show a positive dependence on the blockage ratio. Furthermore, the values of lw and θs at β = 10 are within 5–6% of the corresponding values for an unconfined sphere. Broadly, the confining walls seem to have a stabilizing effect on the flow and thus the wake formation is somewhat delayed.
Fig. 4. Effect of Reynolds number on streamline patterns for β = 10 (nearly unconfined flow) and s = 2. (a) Re = 10, (b) Re = 50 and (c) Re = 100.
Fig. 3. Effect of Reynolds number on streamline patterns for β = 10 (nearly unconfined flow) and s = 4. (a) Re = 10, (b) Re = 50 and (c) Re = 100.
Fig. 2 shows the effect of blockage ratio on the average Nusselt number for air and water. It is seen that for the extreme value of β = 2 used here, there is about 15% enhancement in the value of the Nusselt number for a sphere. This is in stark contrast to the fact that the drag on a sphere for β = 2 at low Reynolds number could be up to 4 times the value for the unconfined sphere. However, the influence of blockage gradually diminishes with a progressive increase in the value of the Reynolds number. Therefore, the average heat transfer characteristics are much less sensitive to the blockage ratio than the flow characteristics. This finding is also consistent at least qualitatively with the experimental findings of Sastry et al. [15] who reported an enhancement of 25% to 100% in the value of the Nusselt number over the Reynolds number range of 7300 to 45,000 with water. In concluding this section, it is thus appropriate to reiterate here that the blockage influences the flow characteristics (drag, recirculation length, angle of separation) much more than the forced convection heat transfer characteristics for a single sphere placed at the axis of a tube.
A. Maheshwari et al. / Powder Technology 168 (2006) 74–83 Table 7 Comparison between the present results and that of Ramachandran et al. [17] for air Re
Table 9 Effect of blockage on Nusselt number Re
Value of Nusselt number 1st sphere Present
Value of Nusselt number β=2
2nd sphere
3rd sphere
1st
Literature
Present
Literature
Present
Literature
Separation = 2 dp 1 2.03 10 3.32 50 5.42 100 6.98
2.12 3.37 5.50 7.05
1.832 2.34 3.44 4.23
1.812 2.324 3.39 4.18
1.63 2.05 3.08 3.77
1.629 2.025 2.98 3.74
Separation = 4 dp 1 2.20 10 3.328 50 5.399 100 6.91
2.17 3.275 5.401 6.96
1.936 2.720 4.105 5.092
2.03 2.79 4.18 5.16
1.64 2.53 3.52 4.39
1.633 2.49 3.60 4.42
5.3. Results for the three-sphere system In order to explore the combined effects of blockage and of sphere–sphere interactions, the flow past an in-line array of three spheres has been studied for a range of values of β, Reynolds number and for two values of the center-to-center spacing between the spheres, namely, s = 2 dp and 4 dp. The choice of these values is in part governed by the study of Ramachandran et al. [17] who studied heat transfer from an unconfined in-line array of three spheres. Table 6 shows the individual values of drag of the three spheres for a range of values of the Reynolds number and for the two values of the inter-sphere spacing. For the case of the inter-sphere spacing, s = 4 dp, the drag coefficient of the first sphere is almost identical to that of a single sphere, and the 2nd and 3rd spheres show a decreasing value of drag coefficient except at Re = 1. While for the sphere-to-sphere separation of s = 2 dp, the last sphere shows a higher drag than that of the 2nd sphere. These trends are clearly due to the more significant sphere–sphere interactions in
79
β=4 2nd
3rd
s = 4, Pr = 0.74 5 2.67 10 3.79 20 4.82 50 6.39 100 8.09
0.753 1.86 3.23 4.94 6.20
0.211 0.912 2.19 4.14 5.52
s = 2, Pr = 0.74 5 2.67 10 3.79 20 4.82 50 6.38 100 8.08
0.75 1.82 3.05 4.41 5.44
0.217 0.914 2.13 3.73 4.78
s = 4, Pr = 7.0 1 3.69 1.74 5 6.17 4.74 10 7.58 5.81 20 9.38 7.05 50 12.66 9.27 100 16.54 11.81
0.82 3.94 5.13 6.25 8.19 10.43
s = 2, Pr = 7.0 1 3.69 1.71 5 6.16 4.27 10 7.57 5.20 20 9.37 6.33 50 12.64 8.32 100 16.50 10.61
0.825 3.58 4.51 5.51 7.25 9.28
1st
β=6 2nd
3rd
2.98 3.50 4.22 5.64 7.20
2.34 2.91 3.44 4.31 5.23
1.88 2.58 3.16 3.94 4.71
2.94 3.47 4.15 5.59 7.14
2.10 2.52 2.93 3.66 4.46
3.41 5.06 6.21 7.80 10.85 14.43
3.37 5.01 6.15 7.72 10.76 14.27
1st
2nd
3rd
2.92 3.41 4.13 5.53 7.08
2.48 2.83 3.29 5.14 6.05
2.26 2.64 3.04 3.76 4.53
1.75 2.26 2.66 3.28 3.95
2.85 3.37 4.08 5.47 6.99
2.11 2.41 2.79 3.51 4.30
1.96 2.22 2.53 3.15 3.82
2.84 4.11 4.89 5.86 7.58 9.56
2.50 3.77 4.45 5.30 6.74 8.33
3.26 4.84 5.98 7.57 10.62 14.18
2.78 3.90 4.61 5.52 7.23 9.18
2.59 3.60 4.22 4.98 6.37 7.93
2.50 3.57 4.20 4.98 6.48 8.39
2.23 3.23 3.79 4.48 5.72 7.16
3.20 4.76 5.89 7.46 10.49 13.95
2.39 3.30 3.87 4.64 6.14 8.02
2.22 3.04 3.53 4.17 5.42 6.84
the latter case than that for the larger inter-sphere separation, s = 4 dp. The physical reasons for this behaviour are clearly brought out by the streamline patterns shown in Figs. 3 and 4 for the three values of the Reynolds number. For the inter-sphere separation of 4 dp, one can clearly see that even though no flow
Table 8 Effect of blockage ratio on drag coefficient Re
Value of drag coefficient β=2
β=4
β=6
1st
2nd
3rd
1st
2nd
3rd
1st
2nd
3rd
s=4 1 5 10 20 50 100
143.17 28.79 14.60 7.64 3.63 2.33
143.76 28.90 14.65 7.65 3.55 2.13
143.45 28.83 14.62 7.63 3.54 2.12
47.72 10.05 5.53 3.27 1.80 1.22
47.69 9.97 5.31 2.88 1.36 0.82
47.73 9.98 5.31 2.88 1.31 0.764
36.63 8.18 4.71 2.89 1.64 1.12
36.22 7.62 4.02 2.23 1.11 0.69
36.42 7.67 4.03 2.17 1.03 0.63
s=2 1 5 10 20 50 100
142.7 28.67 14.53 7.59 3.61 2.32
142.67 28.65 14.48 7.47 3.29 1.85
142.97 28.73 14.53 7.51 3.31 1.85
45.67 9.68 5.33 3.15 1.73 1.17
43.12 8.67 4.36 2.24 1.01 0.579
45.25 9.10 4.62 2.38 1.04 0.613
33.58 7.68 4.45 2.73 1.54 1.05
28.94 5.79 3.00 1.63 0.776 0.446
32.35 6.37 3.22 1.71 0.82 0.509
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separation occurs at Re = 10, but unlike the first sphere, the 2nd and 3rd spheres are exposed to distorted velocity profiles thereby influencing the values of the drag coefficients. This effect is, however, suppressed at Re = 1, owing to the weak advection effects. However, as the Reynolds number is progressively increased to 50 and 100, Fig. 3 clearly shows the increasing size of the wake behind the first sphere, whereas the wakes formed in the rear of the 2nd and 3rd spheres are clearly smaller (but of comparable size) than that for the 1st sphere. The difference in wake sizes is seen to grow with the increasing value of the Reynolds number. This explains for the differences in the value of the drag coefficient for the three spheres as seen in Table 6. Now turning our attention to Fig. 4, these interactions are further accentuated and one can now clearly see the decreasing size of the wake thereby lowering of the drag for each successive sphere. Finally, it is suffice to add here that notwithstanding the significance of wall effects, the trends seen in Table 6 are qualitatively consistent with the experimental observations of Liang et al. [20]. Based on their numerical computations, Ramachandran et al. [17] presented a correlation for heat transfer from an unconfined in-line array consisting of three spheres with equal and unequal spacing of 2 to 10 dp. The present results are compared with their results for air (Pr = 0.74) in Table 7 for the two values of the sphereto-sphere separation. Once again, the two results are seen to be in excellent agreement, the maximum discrepancy being of the order of 1–1.5%. Such a close correspondence inspires confidence in the reliability of the new results obtained in this study.
Tables 8 and 9 show the effect of blockage on drag and heat transfer for the three-sphere configuration studied here for the three values of β and for the two values of the inter-sphere separation. An inspection of Table 8 reveals qualitatively similar trends as that seen for an unconfined array of three spheres. Furthermore, as noted earlier, the increasing blockage, i.e., the decreasing value of β suggests the delayed flow separation and therefore the drag of the three spheres is pretty much identical up to about Re = 20 for β = 2, irrespective of the value of the sphere-to-sphere separation. On the other hand, as the value of β is gradually increased, the three values of drag begin to differ from each other. The effect of blockage on drag is seen to gradually reduce with the increasing Reynolds number. For instance, irrespective of the value of s, at Re = 1, the drag on the first sphere for β = 2 is nearly four times the corresponding value for β = 6; whereas the corresponding ratio of the drag coefficients is only a little more than 2 at Re = 100. Conversely, one must employ a larger domain in the lateral direction to simulate low Reynolds number flows than the high Reynolds number conditions. Figs. 5 and 6 show the corresponding streamline patterns for β = 2 and β = 6 for Re = 50 and 100 and s = 2 and 4. For s = 4 and β = 2, the wake size is almost identical for the three spheres which in turn yields comparable values of drag for each sphere as seen in Table 8 thereby implying little or no sphere–sphere interactions. Similar trends are seen in Fig. 6 for β = 6. An examination of the results shown in Table 9 reveals an intricate functional dependence of the Nusselt number on the
Fig. 5. Effect of sphere-to-sphere spacing and Reynolds number on streamline patterns for β = 2. (a) Re = 50, s = 4; (b) Re = 100, s = 4; (c) Re = 50, s = 2; (d) Re = 100, s = 2.
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Fig. 6. Effect of sphere-to-sphere spacing and Reynolds number on streamline patterns for β = 6. (a) Re = 50, s = 4; (b) Re = 100, s = 4; (c) Re = 50, s = 2; (d) Re = 100, s = 2.
blockage ratio (β), Prandtl (Pr) and Reynolds (Re) numbers and on sphere-to-sphere separation. The key trends can be summarized as follows: (i) For fixed values of β and s, the Nusselt number varies approximately as Pr1 / 2, except at Re = 1, where there is very little convection. Similarly, all else being equal the results show dependence on the Reynolds number as ∼ Re0.32 to Re0.50 , the value of the exponent of 0.5 can be justified based on the laminar boundary layer analysis whereas the smaller values of index compare favourably with Eq. (6) for a single sphere. (ii) For fixed values of the Reynolds and Prandtl numbers, the Nusselt number is maximum for the first sphere and it is reduced progressively for the 2nd and 3rd spheres. This trend is observed for all values of β and s and it stems from the fact that the driving force for heat transfer (ΔT) decreases as one moves from the first sphere to the 2nd and to the 3rd sphere. This effect is particularly acute for β = 2 and s = 2; it, however, gradually diminishes with the increasing value of β and/or s. (iii) Also, the enhancement in the rate of heat transfer due to the sharpening of temperature gradients by
blockage is only realized when there is some advection, i.e., Re N ∼ 5–10 and at high Prandtl numbers when the thermal boundary layer is thin, e.g., see the results for Pr = 7 as opposed to that for air (Pr = 0.74). (iv) The effect of blockage on the Nusselt number is much weaker than that on drag coefficient, akin to that for the case of a single sphere. Furthermore, in view of the fact that the values of the Nusselt number for β = 6 are within 5% of the corresponding values for the unconfined system, shorter computational domain (in the lateral direction) is thus needed for heat transfer calculations than that for the flow field. This finding is also consistent with our previous work on circular cylinders [24,25]. In summary, the effect of blockage on drag is much stronger than that for heat transfer coefficient, both for a single sphere and for the three-sphere array studied here. The drag of the subsequent spheres may decrease and/or increase (with reference to the first sphere) depending upon the values of β, s and Re. This is directly linked to the size of the wake of each sphere.
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6. Concluding remarks The effect of blockage on the flow and heat transfer characteristics of a single sphere and of an in-line array of three equally spaced spheres has been investigated numerically in the range 1 ≤ Re ≤ 100 for two values of Prandtl number, namely, 0.74 and 7 corresponding to air and water. The blockage seems to influence the drag much more strongly than the heat transfer phenomenon. Conversely, it is possible to obtain reliable heat transfer results by using shorter domains than that required for flow calculations. The inter-sphere spacing also exerts an important influence on the drag and Nusselt number values due to the obvious interactions between the wake of the first sphere and the flow over the succeeding spheres. However, while the drag can increase by a factor of 10, the corresponding enhancement in heat transfer is much less than this amount. Nomenclature Ap projected area normal to flow (m2) Cp heat capacity of fluid (J/kg K) CD drag coefficient (−) dp sphere diameter (−) dt tube diameter (m) FD drag force (N) h heat transfer coefficient (W/m2 K) k thermal conductivity of fluid (W/m K) Ld downstream distance (m) Lu upstream distance (m) Nu Nusselt number (−) p pressure (Pa) Pe Peclet number (−) Pr Prandtl number (−) r radial coordinate (m) Re Reynolds number (−) s non-dimensional center-to-center spacing between spheres (−) t time (s) Ts sphere temperature (K) To inlet fluid temperature (K) U uniform velocity at inlet (m/s) V velocity vector (m/s) z axial coordinate (m) Greek Symbols β blockage ratio, = dt/dp (−) μb fluid viscosity at bulk temperature (Pa s) μ0 fluid viscosity at reference temperature (Pa s) ρ fluid density (kg/m3) ∇ del operator (m− 1) References [1] A. Astrom, G. Bark, Heat transfer between fluid and particles in aseptic processing, J. Food Eng. 21 (1994) 97–125. [2] A. Alhamdan, S.K. Sastry, Bulk average heat transfer coefficient of multiple particles flowing in a holding tube, Trans. Inst. Chem. Eng. 76C (1998) 95–101.
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