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Heat transfer enhancement in suddenly expanding annular shear-thinning flows
International Journal of Heat and Mass Transfer 121 (2018) 28–36
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Heat transfer enhancement in suddenly expanding annular shear-thinning flows Khaled J. Hammad Department of Engineering, Central Connecticut State University, 1615 Stanley Street, New Britain, CT 06050, USA
a r t i c l e
i n f o
Article history: Received 25 June 2017 Received in revised form 26 December 2017 Accepted 26 December 2017
Keywords: Annular flows CFD Heat transfer enhancement Non-Newtonian flows Reattaching flows Separating flows Shear-thinning flows Sudden pipe expansion
a b s t r a c t Heat transfer enhancement in suddenly expanding annular pipe flows of Newtonian and shear-thinning non-Newtonian fluids is studied within the steady laminar flow regime. Conservation of mass, momentum, and energy equations, along with the power-law constitutive model are numerically solved. The impact of inflow inertia, annular-diameter-ratio, k, power-law index, n, and Prandtl numbers, is reported over the following range of parameters: Re = {50, 100, 150}, k = {0, 0.5, 0.7}; n = {1, 0.8, 0.6}; and Pr = {1, 10, 100}. Heat transfer enhancement downstream of the expansion plane, i.e., Nusselt numbers greater than the downstream fully developed value, Nu/Nufd > 1, is only observed for Pr = 10 and 100. In general, higher Prandtl numbers, power-law index values, and annular-diameter-ratios, result in more significant heat transfer enhancement downstream of the expansion plane. Heat transfer augmentation, for Pr = 10 and 100, increases with the annular-diameter-ratio. For a given annular-diameter-ratio and Reynolds numbers, increasing the Prandtl number from Pr = 10 to Pr = 100, always results in higher peak Nu values, Numax, for both Newtonian and shear-thinning flows. All Numax values are located downstream of the flow reattachment point, in the case of suddenly expanding round pipe flows, i.e., j = 0. However, for suddenly expanding annular pipe flows, i.e., j = 0.5 and 0.7, Numax values appear upstream the flow reattachment point. For Pr = 10 and 100, shear-thinning flows display two local peak Nu/Nufd values, in comparison with one peak value in the case of Newtonian flows. The highest heat transfer enhancement, Numax/ Nufd 5, is observed at j = 0.7, n = 0.6, and Pr = 100. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Slurries, pastes, suspensions of solids in liquids, and emulsions often display a shear-thinning non-Newtonian behavior. Industries where shear-thinning materials are encountered include those dealing composite materials, rubber, pharmaceuticals, biological fluids, plastics, petroleum, soap and detergents, cement, food products, paper pulp, paint, light and heavy chemicals, oil field operations, fermentation processes, plastic rocket propellants, electrorheological fluids, ore processing, printing, and radioactive waste [1–3]. Understanding the thermal phenomenon of an annular pipe flow of a shear-thinning non-Newtonian fluid over an axisymmetric sudden expansion is important to the design of chemical reactors, heat exchangers, mixers, polymer mixing and injection molding devices, and biomedical applications. Most of the performed work so far is related to flow and heat transfer in suddenly expanding round pipe flows of Newtonian [4–8] and nonNewtonian fluids [9–11]. Suddenly expanding round pipe flows E-mail address: [email protected] https://doi.org/10.1016/j.ijheatmasstransfer.2017.12.134 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
are characterized by a corner recirculation region in the case of Newtonian [11–27] and shear-thinning non-Newtonian fluids [28–35]. However, for yield stress non-Newtonian fluids, corner flow recirculation may or may not exist, depending on the combination of yield stress and flow inertia [36–39]. In the case of annular pipe flows over an axisymmetric sudden expansion, the sharp fluid exit causes flow separation. As the flow proceeds further downstream of the expansion plane, it entrains the surrounding fluid, which induces a slight flow of the surrounding fluid toward the expanding flow. This results in the creation of a large corner recirculation region, a counter rotating central recirculation region, a wall reattachment point, and a stagnation point along the centerline, as shown in Fig. 1 for a fully developed annular pipe inflow. The evolution of the expanding flow is generally characterized by radial spreading, centerline velocity development, and flow redevelopment and transition to a fully developed pipe flow further downstream. Evolution of the expanding annular pipe flow and characteristics of the resulting recirculation regions depend on the inflow velocity profile, inertia, and the fluid’s rheology.
K.J. Hammad / International Journal of Heat and Mass Transfer 121 (2018) 28–36
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Nomenclature BR ER d; D Do Dh ei k K Lr Lx n Nu P Pr r R Re T u ub x
annular blockage ratio, BR ¼ ðd=DÞ ¼ j2 expansion ratio, ER ¼ Do =D ¼ 2 inner and outer nozzle diameters downstream pipe diameter, Do ¼ 2D hydraulic diameter of annular nozzle, Dh ¼ D d relative recirculation intensity, ratio of the maximum amount of backflow in the recirculation region to the inlet flow rate jwmin =wmax j thermal conductivity of fluid consistency index reattachment length, axial extent of outer recirculation region axial extent of the computational domain Power-law index Nusselt number based on bulk temperature, @h=@rjrw =hb non-dimensional pressure, P =qu2b Prandtl number, Kðub =Dh Þn1 =qa non-dimensional radial distance, r =R outer nozzle radius, R ¼ D=2 Reynolds number, qDnh ub2n =K temperature non-dimensional streamwise u =ub R D=2 velocity, 2 inflow bulk velocity, 8 d=2 u r dr =ðD2 d Þ non-dimensional streamwise distance, x =R 2
Greek letters thermal diffusivity wmax maximum value of stream function wmin minimum value of stream function c_ ij rate of deformation tensor, @ui =@xj þ @uj =@xi c_ II second invariant of rate of deformation tensor, c_ ij c_ ij
a
j
h
leff q sij
annular pipe inner-to-outer diameter ratio, j ¼ d=D non-dimensional temperature, ðT T w Þ=ðT i T w Þ non-dimensional effective viscosity,leff =Kðub =Dh Þn1 density stress tensor element
Subscripts/Superscripts ⁄ dimensional quantities b bulk properties c centerline properties cen central recirculation region properties cor corner recirculation region properties d, u downstream and upstream properties i inflow, i.e. properties at x ¼ 0 fd fully developed properties w wall properties
Fig. 1. Schematic of a suddenly expanding annular shear-thinning flow.
Studies focusing on annular flows over an axisymmetric sudden expansion are scarce. Sheen et al. [40] experimentally investigated the flow patterns arising from a concentric annular flow of a Newtonian fluid over an axisymmetric sudden expansion using flow visualization and laser-Doppler anemometry (LDA). The observed flow patterns, downstream the sudden expansion, were classified according to the characteristics of the central and corner recirculation regions. They were open annular flow, closed annular flow, vortex shedding, and stable central flow. The resulting flow pattern, depended on the value of the Reynolds number and whether the Reynolds number was increased or decreased over the Re = 165–5800 range. Further, they reported flow bifurcation for Re >
230, and transition from laminar to turbulent flow when the Reynolds number was about 1600. Del Taglia et al. [41] numerically investigated the spontaneous break of symmetry for annular incompressible Newtonian flows in the steady laminar flow regime. Transition to asymmetry was found to depend on the combination of the blockage ratio, BR, and the Reynolds number, Re. BR is defined as the annular innerto-outer-area-ratio, i.e., BR ¼ j2 , where j is the annular pipe inner-to-outer diameter ratio, i.e., j ¼ d=D. The critical Reynolds number needed to transition from steady symmetric to steady asymmetric laminar flow was found to decrease with increasing the blockage ratio. For BR = 0.5 (or j ¼ 0:707), and 0.7 (or
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K.J. Hammad / International Journal of Heat and Mass Transfer 121 (2018) 28–36
j ¼ 0:837), transition from steady symmetric to steady asymmetric flow took place at about Re = 190 and 85, respectively. Hammad [42] investigated annular viscoplastic non-Newtonian flows and reported on impact of yield stress presence on the flow characteristics. Presence of flow recirculation was found to depend on the combination of yield stress and flow inertia. Nonrecirculating expanding flows were found to exist for sufficiently large yield stress and low incoming flow inertia. Recirculating expanding annular flows were present only for low yield stress and high inflow inertia. A parametric numerical study is conducted to quantify the impact of shear-thinning rheology, inflow inertia, the annular diameter ratio, and thermos-physical properties on heat transfer enhancement from a fully developed shear-thinning annular pipe flow over an axisymmetric sudden expansion within the steady laminar flow regime. Shear-thinning non-Newtonian fluids are highly viscous and most of their flows are typically laminar. The power-law constitutive equation is used to model the rheology of interest
The power-law constitutive equation is used to model shearthinning behavior. The general three-dimensional constitutive equation relating the imposed stresses to the flow kinematics for a power-law non-Newtonian fluid is [1]:
n1 1 2 c_ II c_ ij 2
ð1Þ
Here K and n are the consistency index and power-law index, respectively, properties of the fluid, c_ ij ¼ @ui =@xj þ @uj =@xi is the rate-of-deformation tensor, and c_ II is the second invariant of the tensor c_ ij , given by c_ II ¼ c_ ij c_ ij . The power-law model is simple yet accounts for Newtonian (n ¼ 1), shear-thinning (n < 1) and shear-thickening (n > 1) effects. Materials behaving in the abovedescribed manner include slurries, pastes, plastics, electrorheological fluids, suspensions of solids in liquids, and emulsions. The apparent or effective viscosity displayed by a power-law fluid is:
leff ðc_ II Þ ¼ K
n1 1 2 c_ II 2
ð2Þ
Based on Eq. (1), the non-dimensional form of the continuity and momentum conservation equations governing the laminar and steady flow of an incompressible power-law fluid in a cylindrical geometry are [39]:
1 @ðr v Þ @u þ ¼0 r @r @x u
ð3Þ
@u @u @p þv ¼ @x @r @x 1 @ @u 1 @ @ v @u þ 2leff leff r þ þ Re @x @x r @r @x @r ð4Þ
u
@v @v @p þv ¼ @r @x @r þ
( ) 1 1 @ @v @ @ v @u v 2leff r 2leff leff þ þ Re r @r @x @r @x @r ðrÞ2 ð5Þ
@h @h 1 1 @ @h @ @h ¼ r þ u þv @x @r RePr r @r @r @x @x
leff ¼ leff =Kðub =Dh Þn1 is a non-dimensional effective vis-
cosity expressed as follows:
leff ðc_ II Þ ¼ K
1 c_ II 2
n1 2
ð7Þ
In cylindrical coordinates 12 c_ II is given by:
( ) 2 2 v 2 @u2 1 @v @ v @u þ c_ II ¼ 2 þ þ þ 2 @x @r r @x @r
ð8Þ
The axisymmetric velocity, pressure, and temperature fields obtained via numerically solving the non-dimensional form of the governing equation, i.e. Eqs. (3)–(6), along with the powerlaw constitutive equation, i.e. Eq. (7), are a function of Pr, Re, and n. The Prandtl and Reynolds number used here are:
Pr ¼
Kðub =Dh Þn1
qa
;
Re ¼
qDnh u2n b
ð9Þ
K
Other non-dimensional variables include:
2. Governing equations
sij ¼ K
Here,
ð6Þ
x¼
x ; R
r¼
r ; R
u¼
u ; ub
v¼
v ub
;
p¼
p ; qu2b
h¼
T Tw Ti Tw ð10Þ
3. Results and discussion The flow geometry and the coordinate system are illustrated in Fig. 2. The boundary conditions u ¼ 0 and v ¼ 0 were imposed at all walls. A fully-developed annular inflow velocity profile was used, i.e., u ¼ uðrÞ and v ¼ 0. The inflow temperature was h ¼ 1. hw ¼ 1 and hw ¼ 0 were imposed at x ¼ 0 and x > 0, respectively. Along the centerline, i.e., axis of symmetry, v ¼ @u=@r ¼ @p=@r ¼ 0 and @h=@r ¼ 0. The outflow boundary conditions, i.e. v ¼ @u=@x ¼ 0 and @ðh=hb Þ=@r ¼ 0, were applied sufficiently far downstream the expansion plane to ensure that the computed results were independent of the location of application. A 300 100, was employed in all computations. Furthermore, a 150 50 grid led to results identical to those obtained using the 300 100 grid. Vertical grid spacing was varied to achieve grid clustering near the centerline and the shear-layer of the incoming flow. The horizontal grid spacing was also varied to produce grid clustering in the vicinity of the inflow annular flow, i.e. at x = 0. Eqs. (3)–(6) are solved numerically utilizing a staggered, variable grid-size, finite-differences scheme. Fully second-order accurate finite-difference approximations are used for the derivatives appearing in the governing equations. Centered differences are used in all cases with the exception of the convective streamwise derivatives, which are upwinded, using first order approximations. The finite-difference form of the governing partial differential equations and detailed solution methodology can be found in [43,44]. The resulting system of algebraic equations is solved via an iterative marching technique in which the linearized equations are solved simultaneously along lines in the radial direction using a block-tridiagonal matrix inversion technique, i.e., Thomas algorithm. The convergence parameter employed is the magnitude of the maximum residual in the difference equations. The discretization is made in such a way that the continuity equation is always satisfied at any stage of the solution procedure. Therefore, convergence is checked for the momentum and energy equations and the iterative procedure is terminated when the residual becomes less than 106. The finite-difference scheme used here [44] has been tested and validated using planar laser sheet
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Fig. 2. Computational domain and boundary conditions.
visualizations [35] and particle image velocimetry measurements [36]. In the core and flow reattachment regions of the flow, the rates of deformation attain very low values which lead to singularities and/or large effective viscosities. Singularities and very large values of effective viscosities create convergence problems because the coefficient matrix becomes very ‘‘stiff” due to large differences in the magnitude of its elements. In order to avoid such problems, when the value of the non-dimensional second invariant of rate of deformation tensor, c_ II ¼ c_ ij c_ ij , drops below a certain preset level, 107, it is ‘‘frozen” at that very low value, i.e., 107, thus eliminating singularities and warranting convergence. The same approach has been successfully adopted in the past [43,44]. Note that the smaller the selected preset c_ II value the higher the nondimensional effective viscosity will be in low rates of deformation rates regions for shear-thinning fluids, thus the closer the obtained flow behavior to that of a true power-law fluid. The flow and thermal fields of annular shear-thinning flows over an axisymmetric sudden expansion, as depicted in Fig. 2, are influenced by the inflow velocity profile, the annular diameter ratio, j = d/D, the power-law index, n, the Reynolds number, Re, and the Prandtl number, Pr. The results to be presented are obtained from a parametric study for a fully developed annular pipe flow over the following range of variables: j = {0, 0.5, 0.7}; n = {1, 0.8, 0.6}; Re = {50, 100, 150}; Pr = {1, 10, 100}. The selected Reynolds numbers, Re = 50, 100, and 150, ensure steady and laminar flow conditions throughout the whole flow field [41]. The shear-thinning index allows for the investigation of a non-Newtonian rheology effects on the flow and thermal structures. The selected Prandtl numbers allow for investigating the impact of the thermo-physical properties over a wide range. All fluid properties are assumed constant.
Fig. 3. Streamlines for a Newtonian fluid and Re = 50: (a) j = 0, (b) j = 0.5.
3.1. Streamlines The flow structure induced by the expanding annular pipe flow can be illustrated by inspecting the flow streamlines presented in Figs. 3 and 4, at Re = 50. Impact of the annular diameter ratio, j = 0 and 0.5, on the evolution of the annular flow in the immediate vicinity of the expansion plane is clearly demonstrated in Figs. 3 and 4 for Newtonian, n = 1, and shear-thinning non-Newtonian fluids, n = 0.6, respectively.
Fig. 4. Streamlines for a shear-thinning fluid, n = 0.6 and Re = 50: (a) j = 0, (b) j = 0.5.
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Table 1 Corner relative recirculation intensities and reattachment lengths for Newtonian and shear-thinning fluids at Re = 50, 100, and 150.
j
n
ei Re ¼ 50
Lr R
ei Re ¼ 100
Lr R
ei Re ¼ 150
Lr R
0 0 0 0.5 0.5 0.5 0.7 0.7 0.7
1 0.8 0.6 1 0.8 0.6 1 0.8 0.6
0.1005 0.0802 0.0553 0.1921 0.1689 0.1333 0.3123 0.2776 0.2251
4.728 4.886 4.761 4.579 4.507 4.384 6.413 5.547 4.783
0.1195 0.0964 0.0696 0.2288 0.1974 0.1555 0.3519 0.3114 0.2526
9.215 9.540 9.292 10.073 9.765 9.357 13.479 12.830 10.820
0.1263 0.1021 0.0747 0.2381 0.2039 0.1605 0.3594 0.3135 0.2543
13.730 14.228 13.852 16.163 15.474 14.590 25.494 21.247 17.003
Fig. 5. Reattachment length variation with Reynolds number for a Newtonian fluid and j = 0. A comparison between computed and available experimental data [12,20].
Fig. 7. Wall shear-stress at Re = 50: (a) Newtonian. (b) Shear-thinning, n = 0.6.
Fig. 6. Relative recirculation intensity variation with Reynolds number for a Newtonian fluid and j = 0. A comparison between computed and available experimental data [20].
For a round pipe inflow, i.e., j = 0, the shear layer at the outer edge of the jet spreads radially outwards with the propagating downstream flow and creates a large corner recirculation region, as seen in Figs. 3(a) and 4(a). In the case of annular pipe inflows, i.e., j = 0.5, flow separation and entrainment creates a small central and a much larger corner recirculation regions, downstream the plane of expansion, as shown in Figs. 3(b) and 4(b). The dashed lines indicate counter-clockwise rotation within the outer recirculation region. The closed solid lines indicate clockwise rotation within the inner recirculation region. Fig. 8. Wall shear-stress at Re = 100: (a) Newtonian. (b) Shear-thinning, n = 0.6.
3.2. Reattachment length and relative recirculation intensity Reattachment lengths and corner relative recirculation intensities are listed in Table 1 for all investigated Re, j , and n values. Relative recirculation intensities, defined here as the ratio of the maximum amount of backflow in a recirculation region to the inlet flow rate. Relative recirculation intensity of the corner region, i.e., ei , increase with the Reynolds number and annular-diameter-ratio, for both Newtonian and shear-thinning fluids.
Table 1 demonstrates the impact of inflow inertia, shearthinning rheology, and annular diameter ratio on the extent and intensity of the corner recirculation region. Increasing the powerlaw index, i.e., n, enhances recirculation in the corner region. Shear-thinning has a dramatic impact on the recirculation intensity of the corner region only. In general, the most intense recirculation is associated with the highest power-law index value,
K.J. Hammad / International Journal of Heat and Mass Transfer 121 (2018) 28–36
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Fig. 9. Isotherms for a shear-thinning fluid, n = 0.6, Re = 50, and Pr = 1: (a) j = 0 and (b) j = 0.5.
Fig. 10. Isotherms for a shear-thinning fluid, n = 0.6, Re = 50, and Pr = 100: (a) j = 0 and (b) j = 0.5.
Fig. 11. Newtonian Nusselt curves at Re = 50.
Fig. 12. Shear-thinning Nusselt curves at Re = 50.
i.e., n = 1, and the largest annular diameter ratio, i.e., j = 0.7. The relative recirculation intensity of the corner region diminishes when the shear thinning index is lowered from n = 1 to n = 0.6 for all annular diameter ratios. In general, the extent and intensity of recirculation in the corner region, increases with inflow inertia.
Some of the presently computed results are compared to the flow visualization results of Macagno and hung [12] and the PIV results of Hammad et al. [20]. The comparison, shown in Fig. 5, displays the effect of Re on the length of reattachment, Lr, for a Newtonian fluid and j = 0. It can be seen that the reattachment point
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K.J. Hammad / International Journal of Heat and Mass Transfer 121 (2018) 28–36
increases linearly with Re. In addition, a very favorable agreement is obtained between the present numerical predictions and the experimental results. Further, some of the presently computed relative recirculation intensity results are compared to the PIV data obtained by Hammad et al. [20]. The comparison, shown in Fig. 6, displays the effect of the Reynolds number on the relative recirculation intensity, ei, for a Newtonian fluid and j = 0. Increasing Reynolds numbers lead to stronger corner eddies. However, this dependence is non-linear becoming weaker at higher Reynolds numbers. The relative recirculation intensity seems to approach an asymptotic value of approximately 0.14 at the high end of the shown Reynolds numbers. In addition, the agreement between the present numerical predictions and the experimental results is very strong, confirming the validity of the current computational approach and results. 3.3. Wall shear stress Influence of inflow inertia, shear-thinning, and annular diameter ratio on the wall shear stress distribution is demonstrated in Figs. 7 and 8, for Newtonian and shear-thinning fluids at Re = {50, 100} and j = {0, 0.5}. The wall shear stress curves normalized by the downstream fully developed one, i.e., sw =sw;fd , clearly illustrate the dramatic impact of Re, j, and shear-thinning on the evolution characteristics of the suddenly expanding annular flow field. Corner recirculation regions are characterized by sw =sw;fd < 0. Axial location corresponding to sw =sw;fd ¼ 0, marks the reattachment length, i.e., Lr =R. Lr =R also marks the axial extent of the corner recirculation region. In general, higher negative sw =sw;fd values indicate a more intense recirculation within the corner region. Corner recirculation intensity, as evidenced from sw =sw;fd , is higher in the case of Newtonian fluids for j = 0 and 0.5. In addition more intense corner recirculation is associated with j = 0.5 for both Newtonian and shear-thinning non-Newtonian fluids. This confirms the earlier finding, based on Figs. 3 and 4. Larger negative sw =sw;fd , are obtained at Re = 100, in comparison with Re = 50, indicating an increase in the intensity of recirculation in the corner region with inflow inertia for Newtonian and shear-thinning fluids, and all investigate annular diameter ratios.
used to present Nu=Nufd vs. x, as shown in Figs. 11 and 12, at Re = 50, and Figs. 13 and 14, at Re = 100, for Newtonian and shearthinning rheologies. In the present study, Nufd does not change with Re and Pr, i.e., it only varies with j and n. Heat transfer enhancement downstream of the expansion plane, i.e., Nu=Nufd > 1 , is observed only for Pr = 10 and 100. In the case of Pr = 1, a monotonic increase in Nu=Nufd to the fully developed value is observed. The observed peak Nu=Nufd values corresponding to Pr = 10 and 100, for Newtonian and shear-thinning flows, are always located downstream of the flow reattachment point, in the case of suddenly expanding round pipe flows, i.e., j = 0, as shown in Figs. 11– 14. However, in the case of suddenly expanding annular pipe flows, i.e., j = 0.5, the peak Nu=Nufd values appear upstream the flow reattachment point, as shown in Figs. 11–14. Further, shearthinning non-Newtonian flows display two local peak Nu=Nufd values, for Pr = 10 and 100, in comparison with only one peak value in the case of Newtonian flows.
3.6. Heat transfer enhancement The peak Nu=Nufd values, i.e., Numax =Nufd , and the corresponding axial location, i.e., x@Numax , were extracted and listed in Tables 2 and 3 for Pr = 10 and 100, respectively. Heat transfer augmentation for Pr = 10 and 100, is more dramatic for suddenly expanding annular flows, in comparison with the one observed for a suddenly expanding pipe flow. For a given annular diameter ratio and Reynolds numbers, increasing the Prandtl number from Pr = 10 to Pr = 100, always results in higher Numax =Nufd values, for both Newtonian and shear-thinning non-Newtonian flows. Higher Prandtl and Reynolds
3.4. Isotherms The thermal structure induced by the suddenly expanding annular shear-thinning non-Newtonian flow can be illustrated by a close inspection of the flow field isotherms corresponding to h = {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}, as presented in Figs. 9 and 10, at Re = 50, for Pr = 1 and 100. The h = 0.1 isotherms are close to the walls, while those of h = 0.9 are close to the centerline. Figs. 9 and 10 demonstrate the impacts of Prandtl number and j on the temperature field at Re = 50. Increasing the Prandtl number leads to distortions in the isotherms inside the recirculation region and higher gradients in the thermal boundary layer downstream of the expansion plane for j = {0, 0.5}. The higher temperature gradients near the wall in the case of Pr = 100, will result in higher wall heat transfer rates, in comparison with those corresponding to Pr = 1.
Fig. 13. Newtonian Nusselt curves at Re = 100.
3.5. Wall heat transfer rates The observed impact of Re, Pr, and j on the thermal structure in the present geometry dramatically influences heat transfer characteristics from the suddenly expanding annular shear-thinning flow to the surrounding walls. To illustrate this influence, the computed Nusselt numbers, Nu, are normalized with Nufd , i.e., the fully developed one prevailing far downstream of the expansion plane, and
Fig. 14. Shear-thinning Nusselt curves at Re = 100.
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K.J. Hammad / International Journal of Heat and Mass Transfer 121 (2018) 28–36 Table 2 Heat transfer enhancement for Newtonian and shear-thinning fluids at Pr = 10.
j 0 0 0 0.5 0.5 0.5 0.7 0.7 0.7
n
1 0.8 0.6 1 0.8 0.6 1 0.8 0.6
Numax Nufd
Numax Nufd
Numax Nufd
Re ¼ 50
x@ Numax
Re ¼ 100
x@ Numax
Re ¼ 150
x@ Numax
1.603 1.506 1.428 1.988 1.911 1.773 1.931 2.005 2.007
6.67 8.65 10.60 4.17 3.75 3.50 5.05 4.26 3.62
1.624 1.521 1.441 1.934 1.893 1.798 1.808 1.894 1.943
12.93 16.53 21.07 8.56 7.88 7.50 10.08 9.67 8.40
1.621 1.519 1.443 1.860 1.832 1.761 1.679 1.768 1.872
19.22 24.85 31.90 13.47 12.37 11.79 19.51 16.22 13.24
Table 3 Heat transfer enhancement for Newtonian and shear-thinning fluids at Pr = 100.
j 0 0 0 0.5 0.5 0.5 0.7 0.7 0.7
n
1 0.8 0.6 1 0.8 0.6 1 0.8 0.6
Numax Nufd
Re ¼ 100
x@ Numax
Re ¼ 150
x@ Numax
3.298 3.063 2.898 4.374 4.356 4.085 4.422 4.735 4.806
6.19 8.32 10.33 3.29 3.20 3.13 3.78 3.60 3.26
3.389 3.117 2.929 4.530 4.499 4.304 4.542 4.841 4.971
11.78 16.26 20.24 6.29 6.76 6.88 6.84 7.87 7.53
3.404 3.105 2.925 4.518 4.470 4.273 4.418 4.684 4.905
16.26 23.94 30.32 9.20 10.32 10.45 11.80 12.23 11.75
4. Concluding remarks The impact of inflow inertia, the annular diameter ratio, powerlaw index, and thermo-physical properties on the thermal characteristics of a fully developed annular shear-thinning flow over an axisymmetric sudden expansion was studied within the steady laminar flow regime. Heat transfer enhancement downstream of the expansion plane, i.e., Nu=Nufd > 1 , is only observed for Pr > 1. For suddenly expanding round pipe flows, i.e., j = 0, the peak Nu=Nufd values, i.e., Numax =Nufd , at Pr = 10 and 100, are always located downstream of the flow reattachment point. However, in the case of suddenly expanding annular pipe flows, i.e., j = 0.5 and 0.7, Numax =Nufd values are located upstream of the flow reattachment point. Higher Reynolds and Prandtl numbers, power-law index values, and annular diameter ratios, in general, result in a more substantial heat transfer augmentation downstream of the expansion plane.
None declared.
Numax Nufd
x@ Numax
numbers, in general, result in more dramatic heat transfer augmentation downstream of the expansion plane. A careful comparison between examination of x@Numax listed in Tables 2 and 3, with Lr =R values listed in Table 1 confirms that x@Numax > Lr =R that all Numax =Nufd values, for Newtonian and shear-thinning flows, are always located downstream of the flow reattachment point, i.e., x@Numax > Lr =R , in the case of suddenly expanding round pipe flows, i.e., j = 0. However, in the case of suddenly expanding annular pipe flows, i.e., j = 0.5 and 0.7, all Numax =Nufd values appear upstream the flow reattachment point, i.e. x@Numax < Lr =R.
Conflict of interest
Numax Nufd
Re ¼ 50
Acknowledgment The author acknowledges the support received through a CSUAAUP University Research Grant.
References [1] R.B. Bird, G.C. Dai, B.J. Yarusso, The rheology and flow of viscoplastic materials, Rev. Chem. Eng. 1 (1) (1983) 1–70. [2] R.P. Chhabra, J.F. Richardson, Non-Newtonian Flow and Applied Rheology, Engineering Applications, second ed., Butterworth-Heinemann/IChemE, 2008. [3] A. Shekarriz, K.J. Hammad, M.R. Powell, Evaluation of Scaling Correlations for Mobilization of Double-Shell Tank Waste, Report No. PNNL-11737, Prepared for the U.S. Department of Energy by the Pacific Northwest National Laboratory, Richland, Washington, 1997. [4] D.F. Fletcher, S.J. Maskell, M.A. Patrick, Heat and mass transfer computations for laminar flow in an axisymmetric sudden expansion, Comp. Fluids 13 (2) (1985) 207–221. [5] P. Ma, X. Li, D.N. Ku, Heat and mass transfer in a separated flow region for high Prandtl and Schmidt numbers under pulsatile conditions, Int. J. Heat Mass Transf. 37 (17) (1994) 2723–2736. [6] T. Ota, A survey of heat transfer in separated and reattached flows, Appl. Mech. Rev. 53 (8) (2000) 219–235. [7] R. Kumar, S. Saurav, E.V. Titov, D.A. Levin, R.F. Long, W.C. Neely, P. Setlow, Thermo-structural studies of spores subjected to high temperature gas environments, Int. J. Heat Mass Transf. 54 (4) (2011) 755–765. [8] S.N. Kazi, E. Sadeghinezhad, H. Togun, Heat Transfer to Separation Flow in Heat Exchangers, INTECH Open Access Publisher, 2012. [9] G.C. Vradis, K.J. Hammad, Heat transfer in flows of non-Newtonian Bingham fluids through axisymmetric sudden expansions and contractions, Numer. Heat Transf., Part A: Appl. 28 (3) (1995) 339–353. [10] K.J. Hammad, Effect of hydrodynamic conditions on heat transfer in a complex viscoplastic flow field, Int. J. Heat Mass Transf. 43 (6) (2000) 945–962. [11] K.J. Hammad, Inertial thermal convection in a suddenly expanding viscoplastic flow field, Int. J. Heat Mass Transf. 106 (2017) 829–840. [12] E.O. Macagno, T.K. Hung, Computational and experimental study of a captive annular eddy, J. Fluid Mech. 28 (01) (1967) 43–64. [13] L.H. Back, E.J. Roschke, Shear-layer flow regimes and wave instabilities and reattachment lengths downstream of an abrupt circular channel expansion, J. Appl. Mech. 39 (3) (1972) 677–681. [14] R.G. Teyssandiert, M.P. Wilson, An analysis of flow through sudden enlargements in pipes, J. Fluid Mech. 64 (01) (1974) 85–95. [15] D.J. Latorneli, A. Pollard, Some observations on the evolution of shear layer instabilities in laminar flow, Phys. Fluids 29 (9) (1986) 2828–2835.
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[16] P.S. Scott, F.A. Mirza, J. Vlachopoulos, A finite element analysis of laminar flows through planar and axisymmetric abrupt expansions, Comp. Fluids 14 (4) (1986) 423–432. [17] D. Badekas, D.D. Knight, Eddy correlations for laminar axisymmetric sudden expansion flows, ASME J. Fluids Eng. 114 (1) (1992) 119–121. [18] W.J. Devenport, E.P. Sutton, An experimental study of two flows through an axisymmetric sudden expansion, Exp. Fluids 14 (6) (1993) 423–432. [19] P.J. Oliveira, F.T. Pinho, A. Schulte, A general correlation for the local loss coefficient in Newtonian axisymmetric sudden expansions, Int. J. Heat Fluid Flow 19 (6) (1998) 655–660. [20] K.J. Hammad, M.V. Otugen, E.B. Arik, A PIV study of the laminar axisymmetric sudden expansion flow, Exp. Fluids 26 (3) (1999) 266–272. [21] C.D. Cantwell, D. Barkley, H.M. Blackburn, Transient growth analysis of flow through a sudden expansion in a circular pipe, Phys. Fluids 22 (3) (2010) 034101. [22] E. Sanmiguel-Rojas, C. Del Pino, C. Gutiérrez-Montes, Global mode analysis of a pipe flow through a 1:2 axisymmetric sudden expansion, Phys. Fluids 22 (7) (2010) 071702. [23] E. Sanmiguel-Rojasa, T. Mullin, Finite-amplitude solutions in the flow through a sudden expansion in a circular pipe, J. Fluid Mech. 691 (2011) 201–213. _ Dag˘tekin, M. Ünsal, Numerical analysis of axisymmetric and planar sudden [24] I. expansion flows for laminar regime, Int. J. Numer. Meth. Fluids 65 (9) (2011) 1133–1144. [25] S. Khodaparast, N. Borhani, J.R. Thome, Sudden expansions in circular microchannels: flow dynamics and pressure drop, Microfluid. Nanofluid. 17 (3) (2014) 561–572. [26] V. Varade, A. Agrawal, A.M. Pradeep, Behaviour of rarefied gas flow near the junction of a suddenly expanding tube, J. Fluid Mech. 739 (2014) 363–391. [27] M. Shusser, A. Ramus, O. Gendelman, Flow in a curved pipe with a sudden expansion, ASME J. Fluids Eng. 138 (2) (2016) 021203. [28] A.L. Halmos, D.V. Boger, A. Cabelli, The behavior of a power-law fluid flowing through a sudden expansion: part I. A numerical solution, AIChE J. 21 (3) (1975) 540–549. [29] A.L. Halmos, D.V. Boger, The behavior of a power-law fluid flowing through a sudden expansion. Part II. Experimental verification, AIChE J. 21 (3) (1975) 550–553. [30] B. Pak, Y.I. Cho, S.U. Choi, Separation and reattachment of non-Newtonian fluid flows in a sudden expansion pipe, J. Non-Newton. Fluid Mech. 37 (2) (1990) 175–199.
[31] F.T. Pinho, P.J. Oliveira, J.P. Miranda, Pressure losses in the laminar flow of shear-thinning power-law fluids across a sudden axisymmetric expansion, Int. J. Heat Fluid Flow 24 (5) (2003) 747–761. [32] S.J. Heath, J.A. Olson, K.R. Buckley, S. Lapi, T.J. Ruth, D.M. Martinez, Visualization of the flow of a fiber suspension through a sudden expansion using PET, AIChE J. 53 (2) (2007) 327–334. [33] D. Nag, A. Datta, Variation of the recirculation length of Newtonian and nonNewtonian power-law fluids in laminar flow through a suddenly expanded axisymmetric geometry, ASME J. Fluids Eng. 129 (2) (2007) 245–250. [34] F. Giannetti, P. Luchini, L. Marino, Stability and sensitivity analysis of nonNewtonian flow through an axisymmetric expansion, in: J. Phys.: Conf. Ser., IOP Publishing, 2011, p. 032015. [35] K.J. Hammad, F. Wang, M.V. Ötügen, G.C. Vradis, Suddenly expanding axisymmetric flow of a yield stress fluid, Album Visual. 14 (1997) 17–18. [36] K.J. Hammad, M.V. Ötügen, G.C. Vradis, E.B. Arik, Laminar flow of a nonlinear viscoplastic fluid through an axisymmetric sudden expansion, ASME J. Fluids Eng. 121 (2) (1999) 488–496. [37] P. Jay, A. Magnin, J.M. Piau, Viscoplastic fluid flow through a sudden axisymmetric expansion, AIChE J. 47 (10) (2001) 2155–2166. [38] K.J. Hammad, Suddenly expanding recirculating and non-recirculating viscoplastic non-Newtonian flows, J. Visual. 18 (4) (2015) 655–667. [39] K.J. Hammad, The flow behavior of a biofluid in a separated and reattached flow region, ASME J. Fluids Eng. 137 (6) (2015), https://doi.org/10.1115/ 1.4029727. [40] H.J. Sheen, W.J. Chen, J.S. Wu, Flow patterns for an annular flow over an axisymmetric sudden expansion, J. Fluid Mech. 350 (1997) 177–188. [41] C. Del Taglia, A. Moser, L. Blum, Spontaneous break of symmetry in unconfined laminar annular jets, ASME J. Fluids Eng. 131 (8) (2009), 081202(1-8). [42] K.J. Hammad, Visualization of an annular viscoplastic jet, J. Visual. 17 (1) (2014) 5–16. [43] K.J. Hammad, Hemorheology and the flow behavior in a separated flow region, in: Proceedings of ASME. 56314; Volume 7A: Fluids Engineering Systems and Technologies, V07AT08A012, November 15, 2013, IMECE2013-62548, 2013, doi:10.1115/IMECE2013-62548. [44] G.C. Vradis, K.J. Hammad, Strongly coupled block-implicit solution technique for non-Newtonian convective heat transfer problems, Numer. Heat Transf. 33 (1) (1998) 79–97.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Heat transfer enhancement in suddenly expanding annular shear-thinning flows Khaled J. Hammad Department of Engineering, Central Connecticut State University, 1615 Stanley Street, New Britain, CT 06050, USA
a r t i c l e
i n f o
Article history: Received 25 June 2017 Received in revised form 26 December 2017 Accepted 26 December 2017
Keywords: Annular flows CFD Heat transfer enhancement Non-Newtonian flows Reattaching flows Separating flows Shear-thinning flows Sudden pipe expansion
a b s t r a c t Heat transfer enhancement in suddenly expanding annular pipe flows of Newtonian and shear-thinning non-Newtonian fluids is studied within the steady laminar flow regime. Conservation of mass, momentum, and energy equations, along with the power-law constitutive model are numerically solved. The impact of inflow inertia, annular-diameter-ratio, k, power-law index, n, and Prandtl numbers, is reported over the following range of parameters: Re = {50, 100, 150}, k = {0, 0.5, 0.7}; n = {1, 0.8, 0.6}; and Pr = {1, 10, 100}. Heat transfer enhancement downstream of the expansion plane, i.e., Nusselt numbers greater than the downstream fully developed value, Nu/Nufd > 1, is only observed for Pr = 10 and 100. In general, higher Prandtl numbers, power-law index values, and annular-diameter-ratios, result in more significant heat transfer enhancement downstream of the expansion plane. Heat transfer augmentation, for Pr = 10 and 100, increases with the annular-diameter-ratio. For a given annular-diameter-ratio and Reynolds numbers, increasing the Prandtl number from Pr = 10 to Pr = 100, always results in higher peak Nu values, Numax, for both Newtonian and shear-thinning flows. All Numax values are located downstream of the flow reattachment point, in the case of suddenly expanding round pipe flows, i.e., j = 0. However, for suddenly expanding annular pipe flows, i.e., j = 0.5 and 0.7, Numax values appear upstream the flow reattachment point. For Pr = 10 and 100, shear-thinning flows display two local peak Nu/Nufd values, in comparison with one peak value in the case of Newtonian flows. The highest heat transfer enhancement, Numax/ Nufd 5, is observed at j = 0.7, n = 0.6, and Pr = 100. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Slurries, pastes, suspensions of solids in liquids, and emulsions often display a shear-thinning non-Newtonian behavior. Industries where shear-thinning materials are encountered include those dealing composite materials, rubber, pharmaceuticals, biological fluids, plastics, petroleum, soap and detergents, cement, food products, paper pulp, paint, light and heavy chemicals, oil field operations, fermentation processes, plastic rocket propellants, electrorheological fluids, ore processing, printing, and radioactive waste [1–3]. Understanding the thermal phenomenon of an annular pipe flow of a shear-thinning non-Newtonian fluid over an axisymmetric sudden expansion is important to the design of chemical reactors, heat exchangers, mixers, polymer mixing and injection molding devices, and biomedical applications. Most of the performed work so far is related to flow and heat transfer in suddenly expanding round pipe flows of Newtonian [4–8] and nonNewtonian fluids [9–11]. Suddenly expanding round pipe flows E-mail address: [email protected] https://doi.org/10.1016/j.ijheatmasstransfer.2017.12.134 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
are characterized by a corner recirculation region in the case of Newtonian [11–27] and shear-thinning non-Newtonian fluids [28–35]. However, for yield stress non-Newtonian fluids, corner flow recirculation may or may not exist, depending on the combination of yield stress and flow inertia [36–39]. In the case of annular pipe flows over an axisymmetric sudden expansion, the sharp fluid exit causes flow separation. As the flow proceeds further downstream of the expansion plane, it entrains the surrounding fluid, which induces a slight flow of the surrounding fluid toward the expanding flow. This results in the creation of a large corner recirculation region, a counter rotating central recirculation region, a wall reattachment point, and a stagnation point along the centerline, as shown in Fig. 1 for a fully developed annular pipe inflow. The evolution of the expanding flow is generally characterized by radial spreading, centerline velocity development, and flow redevelopment and transition to a fully developed pipe flow further downstream. Evolution of the expanding annular pipe flow and characteristics of the resulting recirculation regions depend on the inflow velocity profile, inertia, and the fluid’s rheology.
K.J. Hammad / International Journal of Heat and Mass Transfer 121 (2018) 28–36
29
Nomenclature BR ER d; D Do Dh ei k K Lr Lx n Nu P Pr r R Re T u ub x
annular blockage ratio, BR ¼ ðd=DÞ ¼ j2 expansion ratio, ER ¼ Do =D ¼ 2 inner and outer nozzle diameters downstream pipe diameter, Do ¼ 2D hydraulic diameter of annular nozzle, Dh ¼ D d relative recirculation intensity, ratio of the maximum amount of backflow in the recirculation region to the inlet flow rate jwmin =wmax j thermal conductivity of fluid consistency index reattachment length, axial extent of outer recirculation region axial extent of the computational domain Power-law index Nusselt number based on bulk temperature, @h=@rjrw =hb non-dimensional pressure, P =qu2b Prandtl number, Kðub =Dh Þn1 =qa non-dimensional radial distance, r =R outer nozzle radius, R ¼ D=2 Reynolds number, qDnh ub2n =K temperature non-dimensional streamwise u =ub R D=2 velocity, 2 inflow bulk velocity, 8 d=2 u r dr =ðD2 d Þ non-dimensional streamwise distance, x =R 2
Greek letters thermal diffusivity wmax maximum value of stream function wmin minimum value of stream function c_ ij rate of deformation tensor, @ui =@xj þ @uj =@xi c_ II second invariant of rate of deformation tensor, c_ ij c_ ij
a
j
h
leff q sij
annular pipe inner-to-outer diameter ratio, j ¼ d=D non-dimensional temperature, ðT T w Þ=ðT i T w Þ non-dimensional effective viscosity,leff =Kðub =Dh Þn1 density stress tensor element
Subscripts/Superscripts ⁄ dimensional quantities b bulk properties c centerline properties cen central recirculation region properties cor corner recirculation region properties d, u downstream and upstream properties i inflow, i.e. properties at x ¼ 0 fd fully developed properties w wall properties
Fig. 1. Schematic of a suddenly expanding annular shear-thinning flow.
Studies focusing on annular flows over an axisymmetric sudden expansion are scarce. Sheen et al. [40] experimentally investigated the flow patterns arising from a concentric annular flow of a Newtonian fluid over an axisymmetric sudden expansion using flow visualization and laser-Doppler anemometry (LDA). The observed flow patterns, downstream the sudden expansion, were classified according to the characteristics of the central and corner recirculation regions. They were open annular flow, closed annular flow, vortex shedding, and stable central flow. The resulting flow pattern, depended on the value of the Reynolds number and whether the Reynolds number was increased or decreased over the Re = 165–5800 range. Further, they reported flow bifurcation for Re >
230, and transition from laminar to turbulent flow when the Reynolds number was about 1600. Del Taglia et al. [41] numerically investigated the spontaneous break of symmetry for annular incompressible Newtonian flows in the steady laminar flow regime. Transition to asymmetry was found to depend on the combination of the blockage ratio, BR, and the Reynolds number, Re. BR is defined as the annular innerto-outer-area-ratio, i.e., BR ¼ j2 , where j is the annular pipe inner-to-outer diameter ratio, i.e., j ¼ d=D. The critical Reynolds number needed to transition from steady symmetric to steady asymmetric laminar flow was found to decrease with increasing the blockage ratio. For BR = 0.5 (or j ¼ 0:707), and 0.7 (or
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j ¼ 0:837), transition from steady symmetric to steady asymmetric flow took place at about Re = 190 and 85, respectively. Hammad [42] investigated annular viscoplastic non-Newtonian flows and reported on impact of yield stress presence on the flow characteristics. Presence of flow recirculation was found to depend on the combination of yield stress and flow inertia. Nonrecirculating expanding flows were found to exist for sufficiently large yield stress and low incoming flow inertia. Recirculating expanding annular flows were present only for low yield stress and high inflow inertia. A parametric numerical study is conducted to quantify the impact of shear-thinning rheology, inflow inertia, the annular diameter ratio, and thermos-physical properties on heat transfer enhancement from a fully developed shear-thinning annular pipe flow over an axisymmetric sudden expansion within the steady laminar flow regime. Shear-thinning non-Newtonian fluids are highly viscous and most of their flows are typically laminar. The power-law constitutive equation is used to model the rheology of interest
The power-law constitutive equation is used to model shearthinning behavior. The general three-dimensional constitutive equation relating the imposed stresses to the flow kinematics for a power-law non-Newtonian fluid is [1]:
n1 1 2 c_ II c_ ij 2
ð1Þ
Here K and n are the consistency index and power-law index, respectively, properties of the fluid, c_ ij ¼ @ui =@xj þ @uj =@xi is the rate-of-deformation tensor, and c_ II is the second invariant of the tensor c_ ij , given by c_ II ¼ c_ ij c_ ij . The power-law model is simple yet accounts for Newtonian (n ¼ 1), shear-thinning (n < 1) and shear-thickening (n > 1) effects. Materials behaving in the abovedescribed manner include slurries, pastes, plastics, electrorheological fluids, suspensions of solids in liquids, and emulsions. The apparent or effective viscosity displayed by a power-law fluid is:
leff ðc_ II Þ ¼ K
n1 1 2 c_ II 2
ð2Þ
Based on Eq. (1), the non-dimensional form of the continuity and momentum conservation equations governing the laminar and steady flow of an incompressible power-law fluid in a cylindrical geometry are [39]:
1 @ðr v Þ @u þ ¼0 r @r @x u
ð3Þ
@u @u @p þv ¼ @x @r @x 1 @ @u 1 @ @ v @u þ 2leff leff r þ þ Re @x @x r @r @x @r ð4Þ
u
@v @v @p þv ¼ @r @x @r þ
( ) 1 1 @ @v @ @ v @u v 2leff r 2leff leff þ þ Re r @r @x @r @x @r ðrÞ2 ð5Þ
@h @h 1 1 @ @h @ @h ¼ r þ u þv @x @r RePr r @r @r @x @x
leff ¼ leff =Kðub =Dh Þn1 is a non-dimensional effective vis-
cosity expressed as follows:
leff ðc_ II Þ ¼ K
1 c_ II 2
n1 2
ð7Þ
In cylindrical coordinates 12 c_ II is given by:
( ) 2 2 v 2 @u2 1 @v @ v @u þ c_ II ¼ 2 þ þ þ 2 @x @r r @x @r
ð8Þ
The axisymmetric velocity, pressure, and temperature fields obtained via numerically solving the non-dimensional form of the governing equation, i.e. Eqs. (3)–(6), along with the powerlaw constitutive equation, i.e. Eq. (7), are a function of Pr, Re, and n. The Prandtl and Reynolds number used here are:
Pr ¼
Kðub =Dh Þn1
qa
;
Re ¼
qDnh u2n b
ð9Þ
K
Other non-dimensional variables include:
2. Governing equations
sij ¼ K
Here,
ð6Þ
x¼
x ; R
r¼
r ; R
u¼
u ; ub
v¼
v ub
;
p¼
p ; qu2b
h¼
T Tw Ti Tw ð10Þ
3. Results and discussion The flow geometry and the coordinate system are illustrated in Fig. 2. The boundary conditions u ¼ 0 and v ¼ 0 were imposed at all walls. A fully-developed annular inflow velocity profile was used, i.e., u ¼ uðrÞ and v ¼ 0. The inflow temperature was h ¼ 1. hw ¼ 1 and hw ¼ 0 were imposed at x ¼ 0 and x > 0, respectively. Along the centerline, i.e., axis of symmetry, v ¼ @u=@r ¼ @p=@r ¼ 0 and @h=@r ¼ 0. The outflow boundary conditions, i.e. v ¼ @u=@x ¼ 0 and @ðh=hb Þ=@r ¼ 0, were applied sufficiently far downstream the expansion plane to ensure that the computed results were independent of the location of application. A 300 100, was employed in all computations. Furthermore, a 150 50 grid led to results identical to those obtained using the 300 100 grid. Vertical grid spacing was varied to achieve grid clustering near the centerline and the shear-layer of the incoming flow. The horizontal grid spacing was also varied to produce grid clustering in the vicinity of the inflow annular flow, i.e. at x = 0. Eqs. (3)–(6) are solved numerically utilizing a staggered, variable grid-size, finite-differences scheme. Fully second-order accurate finite-difference approximations are used for the derivatives appearing in the governing equations. Centered differences are used in all cases with the exception of the convective streamwise derivatives, which are upwinded, using first order approximations. The finite-difference form of the governing partial differential equations and detailed solution methodology can be found in [43,44]. The resulting system of algebraic equations is solved via an iterative marching technique in which the linearized equations are solved simultaneously along lines in the radial direction using a block-tridiagonal matrix inversion technique, i.e., Thomas algorithm. The convergence parameter employed is the magnitude of the maximum residual in the difference equations. The discretization is made in such a way that the continuity equation is always satisfied at any stage of the solution procedure. Therefore, convergence is checked for the momentum and energy equations and the iterative procedure is terminated when the residual becomes less than 106. The finite-difference scheme used here [44] has been tested and validated using planar laser sheet
K.J. Hammad / International Journal of Heat and Mass Transfer 121 (2018) 28–36
31
Fig. 2. Computational domain and boundary conditions.
visualizations [35] and particle image velocimetry measurements [36]. In the core and flow reattachment regions of the flow, the rates of deformation attain very low values which lead to singularities and/or large effective viscosities. Singularities and very large values of effective viscosities create convergence problems because the coefficient matrix becomes very ‘‘stiff” due to large differences in the magnitude of its elements. In order to avoid such problems, when the value of the non-dimensional second invariant of rate of deformation tensor, c_ II ¼ c_ ij c_ ij , drops below a certain preset level, 107, it is ‘‘frozen” at that very low value, i.e., 107, thus eliminating singularities and warranting convergence. The same approach has been successfully adopted in the past [43,44]. Note that the smaller the selected preset c_ II value the higher the nondimensional effective viscosity will be in low rates of deformation rates regions for shear-thinning fluids, thus the closer the obtained flow behavior to that of a true power-law fluid. The flow and thermal fields of annular shear-thinning flows over an axisymmetric sudden expansion, as depicted in Fig. 2, are influenced by the inflow velocity profile, the annular diameter ratio, j = d/D, the power-law index, n, the Reynolds number, Re, and the Prandtl number, Pr. The results to be presented are obtained from a parametric study for a fully developed annular pipe flow over the following range of variables: j = {0, 0.5, 0.7}; n = {1, 0.8, 0.6}; Re = {50, 100, 150}; Pr = {1, 10, 100}. The selected Reynolds numbers, Re = 50, 100, and 150, ensure steady and laminar flow conditions throughout the whole flow field [41]. The shear-thinning index allows for the investigation of a non-Newtonian rheology effects on the flow and thermal structures. The selected Prandtl numbers allow for investigating the impact of the thermo-physical properties over a wide range. All fluid properties are assumed constant.
Fig. 3. Streamlines for a Newtonian fluid and Re = 50: (a) j = 0, (b) j = 0.5.
3.1. Streamlines The flow structure induced by the expanding annular pipe flow can be illustrated by inspecting the flow streamlines presented in Figs. 3 and 4, at Re = 50. Impact of the annular diameter ratio, j = 0 and 0.5, on the evolution of the annular flow in the immediate vicinity of the expansion plane is clearly demonstrated in Figs. 3 and 4 for Newtonian, n = 1, and shear-thinning non-Newtonian fluids, n = 0.6, respectively.
Fig. 4. Streamlines for a shear-thinning fluid, n = 0.6 and Re = 50: (a) j = 0, (b) j = 0.5.
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Table 1 Corner relative recirculation intensities and reattachment lengths for Newtonian and shear-thinning fluids at Re = 50, 100, and 150.
j
n
ei Re ¼ 50
Lr R
ei Re ¼ 100
Lr R
ei Re ¼ 150
Lr R
0 0 0 0.5 0.5 0.5 0.7 0.7 0.7
1 0.8 0.6 1 0.8 0.6 1 0.8 0.6
0.1005 0.0802 0.0553 0.1921 0.1689 0.1333 0.3123 0.2776 0.2251
4.728 4.886 4.761 4.579 4.507 4.384 6.413 5.547 4.783
0.1195 0.0964 0.0696 0.2288 0.1974 0.1555 0.3519 0.3114 0.2526
9.215 9.540 9.292 10.073 9.765 9.357 13.479 12.830 10.820
0.1263 0.1021 0.0747 0.2381 0.2039 0.1605 0.3594 0.3135 0.2543
13.730 14.228 13.852 16.163 15.474 14.590 25.494 21.247 17.003
Fig. 5. Reattachment length variation with Reynolds number for a Newtonian fluid and j = 0. A comparison between computed and available experimental data [12,20].
Fig. 7. Wall shear-stress at Re = 50: (a) Newtonian. (b) Shear-thinning, n = 0.6.
Fig. 6. Relative recirculation intensity variation with Reynolds number for a Newtonian fluid and j = 0. A comparison between computed and available experimental data [20].
For a round pipe inflow, i.e., j = 0, the shear layer at the outer edge of the jet spreads radially outwards with the propagating downstream flow and creates a large corner recirculation region, as seen in Figs. 3(a) and 4(a). In the case of annular pipe inflows, i.e., j = 0.5, flow separation and entrainment creates a small central and a much larger corner recirculation regions, downstream the plane of expansion, as shown in Figs. 3(b) and 4(b). The dashed lines indicate counter-clockwise rotation within the outer recirculation region. The closed solid lines indicate clockwise rotation within the inner recirculation region. Fig. 8. Wall shear-stress at Re = 100: (a) Newtonian. (b) Shear-thinning, n = 0.6.
3.2. Reattachment length and relative recirculation intensity Reattachment lengths and corner relative recirculation intensities are listed in Table 1 for all investigated Re, j , and n values. Relative recirculation intensities, defined here as the ratio of the maximum amount of backflow in a recirculation region to the inlet flow rate. Relative recirculation intensity of the corner region, i.e., ei , increase with the Reynolds number and annular-diameter-ratio, for both Newtonian and shear-thinning fluids.
Table 1 demonstrates the impact of inflow inertia, shearthinning rheology, and annular diameter ratio on the extent and intensity of the corner recirculation region. Increasing the powerlaw index, i.e., n, enhances recirculation in the corner region. Shear-thinning has a dramatic impact on the recirculation intensity of the corner region only. In general, the most intense recirculation is associated with the highest power-law index value,
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Fig. 9. Isotherms for a shear-thinning fluid, n = 0.6, Re = 50, and Pr = 1: (a) j = 0 and (b) j = 0.5.
Fig. 10. Isotherms for a shear-thinning fluid, n = 0.6, Re = 50, and Pr = 100: (a) j = 0 and (b) j = 0.5.
Fig. 11. Newtonian Nusselt curves at Re = 50.
Fig. 12. Shear-thinning Nusselt curves at Re = 50.
i.e., n = 1, and the largest annular diameter ratio, i.e., j = 0.7. The relative recirculation intensity of the corner region diminishes when the shear thinning index is lowered from n = 1 to n = 0.6 for all annular diameter ratios. In general, the extent and intensity of recirculation in the corner region, increases with inflow inertia.
Some of the presently computed results are compared to the flow visualization results of Macagno and hung [12] and the PIV results of Hammad et al. [20]. The comparison, shown in Fig. 5, displays the effect of Re on the length of reattachment, Lr, for a Newtonian fluid and j = 0. It can be seen that the reattachment point
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K.J. Hammad / International Journal of Heat and Mass Transfer 121 (2018) 28–36
increases linearly with Re. In addition, a very favorable agreement is obtained between the present numerical predictions and the experimental results. Further, some of the presently computed relative recirculation intensity results are compared to the PIV data obtained by Hammad et al. [20]. The comparison, shown in Fig. 6, displays the effect of the Reynolds number on the relative recirculation intensity, ei, for a Newtonian fluid and j = 0. Increasing Reynolds numbers lead to stronger corner eddies. However, this dependence is non-linear becoming weaker at higher Reynolds numbers. The relative recirculation intensity seems to approach an asymptotic value of approximately 0.14 at the high end of the shown Reynolds numbers. In addition, the agreement between the present numerical predictions and the experimental results is very strong, confirming the validity of the current computational approach and results. 3.3. Wall shear stress Influence of inflow inertia, shear-thinning, and annular diameter ratio on the wall shear stress distribution is demonstrated in Figs. 7 and 8, for Newtonian and shear-thinning fluids at Re = {50, 100} and j = {0, 0.5}. The wall shear stress curves normalized by the downstream fully developed one, i.e., sw =sw;fd , clearly illustrate the dramatic impact of Re, j, and shear-thinning on the evolution characteristics of the suddenly expanding annular flow field. Corner recirculation regions are characterized by sw =sw;fd < 0. Axial location corresponding to sw =sw;fd ¼ 0, marks the reattachment length, i.e., Lr =R. Lr =R also marks the axial extent of the corner recirculation region. In general, higher negative sw =sw;fd values indicate a more intense recirculation within the corner region. Corner recirculation intensity, as evidenced from sw =sw;fd , is higher in the case of Newtonian fluids for j = 0 and 0.5. In addition more intense corner recirculation is associated with j = 0.5 for both Newtonian and shear-thinning non-Newtonian fluids. This confirms the earlier finding, based on Figs. 3 and 4. Larger negative sw =sw;fd , are obtained at Re = 100, in comparison with Re = 50, indicating an increase in the intensity of recirculation in the corner region with inflow inertia for Newtonian and shear-thinning fluids, and all investigate annular diameter ratios.
used to present Nu=Nufd vs. x, as shown in Figs. 11 and 12, at Re = 50, and Figs. 13 and 14, at Re = 100, for Newtonian and shearthinning rheologies. In the present study, Nufd does not change with Re and Pr, i.e., it only varies with j and n. Heat transfer enhancement downstream of the expansion plane, i.e., Nu=Nufd > 1 , is observed only for Pr = 10 and 100. In the case of Pr = 1, a monotonic increase in Nu=Nufd to the fully developed value is observed. The observed peak Nu=Nufd values corresponding to Pr = 10 and 100, for Newtonian and shear-thinning flows, are always located downstream of the flow reattachment point, in the case of suddenly expanding round pipe flows, i.e., j = 0, as shown in Figs. 11– 14. However, in the case of suddenly expanding annular pipe flows, i.e., j = 0.5, the peak Nu=Nufd values appear upstream the flow reattachment point, as shown in Figs. 11–14. Further, shearthinning non-Newtonian flows display two local peak Nu=Nufd values, for Pr = 10 and 100, in comparison with only one peak value in the case of Newtonian flows.
3.6. Heat transfer enhancement The peak Nu=Nufd values, i.e., Numax =Nufd , and the corresponding axial location, i.e., x@Numax , were extracted and listed in Tables 2 and 3 for Pr = 10 and 100, respectively. Heat transfer augmentation for Pr = 10 and 100, is more dramatic for suddenly expanding annular flows, in comparison with the one observed for a suddenly expanding pipe flow. For a given annular diameter ratio and Reynolds numbers, increasing the Prandtl number from Pr = 10 to Pr = 100, always results in higher Numax =Nufd values, for both Newtonian and shear-thinning non-Newtonian flows. Higher Prandtl and Reynolds
3.4. Isotherms The thermal structure induced by the suddenly expanding annular shear-thinning non-Newtonian flow can be illustrated by a close inspection of the flow field isotherms corresponding to h = {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}, as presented in Figs. 9 and 10, at Re = 50, for Pr = 1 and 100. The h = 0.1 isotherms are close to the walls, while those of h = 0.9 are close to the centerline. Figs. 9 and 10 demonstrate the impacts of Prandtl number and j on the temperature field at Re = 50. Increasing the Prandtl number leads to distortions in the isotherms inside the recirculation region and higher gradients in the thermal boundary layer downstream of the expansion plane for j = {0, 0.5}. The higher temperature gradients near the wall in the case of Pr = 100, will result in higher wall heat transfer rates, in comparison with those corresponding to Pr = 1.
Fig. 13. Newtonian Nusselt curves at Re = 100.
3.5. Wall heat transfer rates The observed impact of Re, Pr, and j on the thermal structure in the present geometry dramatically influences heat transfer characteristics from the suddenly expanding annular shear-thinning flow to the surrounding walls. To illustrate this influence, the computed Nusselt numbers, Nu, are normalized with Nufd , i.e., the fully developed one prevailing far downstream of the expansion plane, and
Fig. 14. Shear-thinning Nusselt curves at Re = 100.
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K.J. Hammad / International Journal of Heat and Mass Transfer 121 (2018) 28–36 Table 2 Heat transfer enhancement for Newtonian and shear-thinning fluids at Pr = 10.
j 0 0 0 0.5 0.5 0.5 0.7 0.7 0.7
n
1 0.8 0.6 1 0.8 0.6 1 0.8 0.6
Numax Nufd
Numax Nufd
Numax Nufd
Re ¼ 50
x@ Numax
Re ¼ 100
x@ Numax
Re ¼ 150
x@ Numax
1.603 1.506 1.428 1.988 1.911 1.773 1.931 2.005 2.007
6.67 8.65 10.60 4.17 3.75 3.50 5.05 4.26 3.62
1.624 1.521 1.441 1.934 1.893 1.798 1.808 1.894 1.943
12.93 16.53 21.07 8.56 7.88 7.50 10.08 9.67 8.40
1.621 1.519 1.443 1.860 1.832 1.761 1.679 1.768 1.872
19.22 24.85 31.90 13.47 12.37 11.79 19.51 16.22 13.24
Table 3 Heat transfer enhancement for Newtonian and shear-thinning fluids at Pr = 100.
j 0 0 0 0.5 0.5 0.5 0.7 0.7 0.7
n
1 0.8 0.6 1 0.8 0.6 1 0.8 0.6
Numax Nufd
Re ¼ 100
x@ Numax
Re ¼ 150
x@ Numax
3.298 3.063 2.898 4.374 4.356 4.085 4.422 4.735 4.806
6.19 8.32 10.33 3.29 3.20 3.13 3.78 3.60 3.26
3.389 3.117 2.929 4.530 4.499 4.304 4.542 4.841 4.971
11.78 16.26 20.24 6.29 6.76 6.88 6.84 7.87 7.53
3.404 3.105 2.925 4.518 4.470 4.273 4.418 4.684 4.905
16.26 23.94 30.32 9.20 10.32 10.45 11.80 12.23 11.75
4. Concluding remarks The impact of inflow inertia, the annular diameter ratio, powerlaw index, and thermo-physical properties on the thermal characteristics of a fully developed annular shear-thinning flow over an axisymmetric sudden expansion was studied within the steady laminar flow regime. Heat transfer enhancement downstream of the expansion plane, i.e., Nu=Nufd > 1 , is only observed for Pr > 1. For suddenly expanding round pipe flows, i.e., j = 0, the peak Nu=Nufd values, i.e., Numax =Nufd , at Pr = 10 and 100, are always located downstream of the flow reattachment point. However, in the case of suddenly expanding annular pipe flows, i.e., j = 0.5 and 0.7, Numax =Nufd values are located upstream of the flow reattachment point. Higher Reynolds and Prandtl numbers, power-law index values, and annular diameter ratios, in general, result in a more substantial heat transfer augmentation downstream of the expansion plane.
None declared.
Numax Nufd
x@ Numax
numbers, in general, result in more dramatic heat transfer augmentation downstream of the expansion plane. A careful comparison between examination of x@Numax listed in Tables 2 and 3, with Lr =R values listed in Table 1 confirms that x@Numax > Lr =R that all Numax =Nufd values, for Newtonian and shear-thinning flows, are always located downstream of the flow reattachment point, i.e., x@Numax > Lr =R , in the case of suddenly expanding round pipe flows, i.e., j = 0. However, in the case of suddenly expanding annular pipe flows, i.e., j = 0.5 and 0.7, all Numax =Nufd values appear upstream the flow reattachment point, i.e. x@Numax < Lr =R.
Conflict of interest
Numax Nufd
Re ¼ 50
Acknowledgment The author acknowledges the support received through a CSUAAUP University Research Grant.
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