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Heat transfer for a Giesekus fluid in a rotating concentric annulus
Applied Thermal Engineering 122 (2017) 118–125
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Heat transfer for a Giesekus fluid in a rotating concentric annulus Marco Lorenzini a,⇑, Irene Daprà b, Giambattista Scarpi b a b
DIN, Dipartimento di Ingegneria Industriale, Alma Mater Studiorum Università di Bologna, Campus Forlì, Via Fontanelle 40, 47121 Forlì, FC, Italy DICAM, Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali, Alma Mater Studiorum Università di Bologna, Viale Risorgimento 2, 40136 Bologna, Bo, Italy
a r t i c l e
i n f o
Article history: Received 25 November 2016 Accepted 3 May 2017 Available online 5 May 2017 Keywords: Annular flow Non-Newtonian fluid Viscous dissipation Convective heat transfer
a b s t r a c t Annular flow of a viscoelastic fluid described by the Giesekus model has received some attention over the years, both concerning the fluid mechanical and thermal aspects, yet no investigation has been carried out using the analytical solution for the velocity profile in the energy equation to determine temperature distribution and heat transfer characteristics. Moreover, viscous dissipation, when accounted for, is usually defined by a formulation of the Brinkman number which proves inconsistent when applied to nonNewtonian fluids. In this work the purely tangential flow of a Giesekus fluid in an annulus with rotating inner wall subject to thermal boundary conditions of the first kind (imposed temperature) at the walls is investigated employing the analytical solution for the velocity profile. Viscous dissipation is accounted for by first deriving a consistent Brinkman number which is easily related to the one usually employed in previous studies and the temperature profiles are obtained for different values of the Deborah number and the non-dimensional mobility factor. Results for the Nusselt numbers at the inner and outer wall are also discussed. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Annular geometries are a recurring feature in many engineering applications, including heat transfer equipment, e.g. tube-in-tube heat exchanger, machining (drilling), manufacturing (food or polymer extrusion), machine elements (journal bearings), oil industry (well drilling) and viscometry. The fluid confined between the two cylindrical surfaces may flow tangentially, axially, or with a combination of both modes and is often non-Newtonian (e.g. emulsions). The latter feature significantly complicates the study of the flow and heat transfer characteristics of the phenomenon, owing to the non-linearity in the constitutive equation. Therefore, the Newtonian model is often inappropriate, whilst the Giesekus model [1] is suitable for shear-thinning fluids. In the case of microchannels, cases where non-Newtonian fluids are employed abound: amongst applications where annular geometries are involved, micro-drilling can be quoted, [2], together with micro-extruders and micro-mixers, [3]. A review on convective heat transfer in stationary and rotating annuli for Newtonian fluids has been carried out by Childs and Long [4] in the mid-90s, whilst later on Coelho and Pinho [5] analysed viscous dissipation in annuli with axial flow subject to prescribed heat flux or surface temperature boundary conditions. ⇑ Corresponding author. E-mail address: [email protected] (M. Lorenzini). http://dx.doi.org/10.1016/j.applthermaleng.2017.05.013 1359-4311/Ó 2017 Elsevier Ltd. All rights reserved.
Fang et al. [6] studied numerically the forced convection of nonNewtonian fluids in concentric and eccentric annuli in the case of negligible viscous dissipation. Still concerning non-Newtonian fluids, axial flow in pipes and ducts with viscous dissipation was investigated by Pinho and Oliveira [7] for a viscoelastic fluid using a simplified Phan-Tien-Tanner model for constant heat flux and by Coelho et al. [8] for constant wall temperature. The same Authors also extended the study to the thermal entry region of a hydrodinamically fully developed flow under the same conditions in [9] and employed a different fluid model to study the same problem in [10]. Concerning purely tangential flow, Khellaf and Lauriat [11] studied the thermal and fluid dynamic behaviour of a Carreau fluid in the case of rotating inner surface and fixed outer surface, whilst Naimi et al. [12] turned their attention to a power law fluid analysing the Taylor-Couette convective vortices. In more recent years, Jouyandeh et al. [13] investigated the heat transfer characteristics for the tangential flow of a Giesekus fluid between two counter-rotating cylinders; their study accounted for viscous dissipation and employed a simplified velocity profile. The same group [14] also analysed the axial annular flow for a fixed duct using the same fluid model and extending thermal conditions to the case of imposed heat flux at the walls. The axial annular flow of a Giesekus fluid was also studied by Mostafaiyan et al. [15], who proposed an approximation to estimate radial normal stress, thus obtaining an analytic expression for the velocity and pressure profiles, whose validity was checked by comparison with the exact solution in a parametric study.
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For several technical fluids exhibiting viscoelastic characteristics, the Giesekus model suitably describes their behaviour. The model employs three physical parameters in the constitutive equation, namely viscosity, mobility factor and relaxation time [1,16]. The experimental determination of the non-linear mobility factor was carried out by Calin et al. [17] and the rheological characterisation of a high-density polyethylene employing this model is documented by Debbaut and Burhin [18]. The Giesekus model has been applied in a number of works, some directly concerned with annular geometries [19,20], other with heat transfer, usually including viscous dissipation [21,22]. The conclusion one might draw from the above discussion is that all aspects of heat transfer and fluid flow in annular geometries for viscoelastic fluids described by the Giesekus model have been investigated. In fact, this is not the case: for the purely tangential flow, an approximate solution is often employed, as discussed by Daprà and Scarpi [23], who in turn obtained an analytical solution for the case of purely tangential annular flow [24] and this directly affects the results concerning heat transfer. Also, viscous dissipation is often described by a formulation of the Brinkman number which works quite well for Newtonian fluids, [25,26] but is inconsistent for those exhibiting a viscoelastic behaviour, as discussed by Coelho and co-workers [27,28]. In this work the purely tangential flow of a Giesekus fluid in an annulus with rotating inner wall subject to first-order (imposed temperature) thermal boundary conditions at the wall is investigated using the analytical solution for the velocity profile. Viscous dissipation is accounted for by first deriving a consistent Brinkman number which is easily related to the one usually employed in previous studies and the temperature profiles are obtained for different values of the Deborah number and the non-dimensional mobility factor. Results for the Nusselt numbers at the inner and outer wall are also discussed. 2. Problem definition and governing equations The geometry analysed in this study is shown in Fig. 1. A concentric annulus of infinite length is considered, such as that formed by two coaxial cylinders. The outer surface is fixed and has a radius Ro , whilst the inner one, which has a radius Ri , rotates with an v angular velocity Xi ¼ Rh;ii . The outer surface of the annulus is kept at a uniform temperature T o , whereas the inner surface is at temperature T i , which is also uniform; moreover, T o > T i .
A Giesekus fluid fills the annulus and moves with a steady-state, laminar and purely tangential flow, which is induced by the rotation of the inner cylinder relative to the outer one. The constitutive equation of the fluid can be written as:
" # h i ka @s r s þ kr ss þ t rs kr s rt þ ðrtÞT s þ @t l h i ¼ l0 rt þ ðrtÞT
ð1Þ
where t is the velocity vector, s is the stress tensor, t is time, l0 is the zero-shear rate viscosity of the polymer, kr is the stress relaxation time, and a is the non-dimensional mobility parameter ð0 6 a 6 1Þ, which accounts for the degree of anisotropy in the polymer. If a ¼ 0 the fluid exhibits an isotropic mobility, and the rheological equation leads to the upper convective Maxwell model (UCM); at the other end of the range, a ¼ 1 corresponds to the most anisotropic mobility. Supposing a purely tangential flow and using cylindrical coordinates, the velocity is limited to its tangential component, th , which automatically verifies the continuity equation. The radial and tangential components of the momentum equation become
2 @p 1 @ shh t ðr srr Þ þ ¼q h @r r @r r r
ð2Þ
@ 2 r srh ¼ 0 @r
ð3Þ
where p is the pressure, r is the radial coordinate and q is the density. Under the assumption that thermophysical properties and model parameters are unaffected by changes in temperature, the tangential velocity distribution in the radial direction th ðrÞ for an isothermal flow still holds. Owing to the nature of the flows in industrial applications such as journal bearings, polymer extruders or chemical/mechanical mixers, fluid viscosity or velocity gradients may be high and viscous dissipation become a relevant issue, causing an alteration in the radial temperature profile that would normally be encountered when its influence is negligible. If viscous dissipation is accounted for, the energy equation for the abovementioned case becomes
k d dT s : rt ¼ 0 r r dr dr
ð4Þ
where k is the thermal conductivity of the fluid and TðrÞ is the tem s : rt gives the contribution
perature; the tensor scalar product
of viscous dissipation to the temperature profile:
s : rt ¼ srh
dth th dr r
ð5Þ
so that Eq. (4) can be written as
d dT r dth th srh ¼0 r dr dr k dr r
ð6Þ
The following non-dimensional quantities are then introduced:
Fig. 1. Sketch of the annulus.
r ¼ Rro
i T ¼ TTT o T i
th ¼ tth;ih
shr ¼ ls0hrtdh
j ¼ RRoi
De ¼ krdthi
i
p ¼ lpdthi 0
where d ¼ Ro Ri . The Deborah number represents the ratio of the polymer relaxation time, kr to the characteristic time of the phenomenon,here represented by the ratio of the thickness of the annulus, d, to the tangential velocity of the rotating wall: the lower De, the more Newtonian-like the fluid behaves. Conversely, for high values of De the shear-thinning behaviour dominates.
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Fig. 2. Velocity profiles for
j ¼ 0:5; a ¼ 0:4.
With
Eqs. (2), (3) and (6) become
ð7Þ
d 2 r srh ¼ 0 dr
pffiffiffi f1 ¼ fð1Þb ¼ si j2 De a pffiffiffiffiffiffiffiffiffiffiffiffi c ¼ 1a
ð8Þ
The no-slip condition at the inner wall in non-dimensional form yields
l0 t2hi 2 1 d th d dT s r ¼0 r 1 j dr r kðT o T i Þ hr dr dr
ð9Þ
dp d s t ¼ Re þ r srr dr dr hh r
2 h r
where Re ¼ qthi d=l0 is the rotational Reynolds number. The tangential stress srh can be related to the tangential stress at the inner wall si by solving Eq. (8)
srh ¼
j2 si
ð10Þ
r 2
The shear rate c_ can thus be expressed as
c_ ¼ ð1 jÞr where
f ¼ fðr Þ ¼
d dr
v
h r
h i 2 2aj2 si ð2a 1Þf þ r ¼ 2 ð2a 1Þr 2 þ f
ð11Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r 4 4j4 a2 De2 si :
Using the no-slip condition at the outer wall th ð1Þ ¼ 0 and integrating Eq. (11) between r and 1, the velocity profile is obtained h ðr Þ
t
( " # pffiffiffi r bcð2a 1Þðf f1 Þ ¼ 2c a arctanh 2 2Deðj 1Þ ð2a 1Þ2 b c2 f1 f 3 2 pffiffiffi 4 2b a f1 r f pffiffiffi 5 2c a arctanh þð2a 1Þ arctan 4 2 4 4 4b a r þ 1 r 39 2 2 = bc 1 r f f 4 5 þ ð2a 1Þbrpffiffiffi 2 1 2 2 2 4 2 2 ; 4 ð j 1 ÞDe a b c2 b c r b c r 3 2 2 2 b þ c2 r ð2a 1Þ 1 r 5 ð12Þ þ 2 2 b c2 b c2 r 4
th ðr Þ r¼j ¼ 1
ð13Þ ð14Þ
ð15Þ
Eq. (15) can be solved numerically to obtain si , which is then used in Eq. (12) to calculate th ðr Þ. The non-dimensional tangential velocity profile is plotted in Fig. 2 for j ¼ 0:5; a ¼ 0:4 and several values of De. It is to be remarked how the velocity gradients become steeper and steeper but more and more confined to the vicinity of the rotating wall as the Deborah number increases, which marks the transition from a Newtonian to a non-Newtonian behaviour in the fluid. The velocity profiles for lower values of j (wider gap between the walls) increase the gradients at the rotating wall and flatten quickly to almost zero velocity as De increases, whilst when j is higher (smaller gaps) the velocity profile becomes close to linear and only bulges slightly inwards at the middle for high values of De [24]. The Poiseuille number Po ¼ f Re, with f the Fanning friction factor for the Giesekus fluid is compared to that for a Newtonian fluid, PoN , under the same conditions: their ratio Po=PoN represents the ratio between the torque required to rotate the inner cylinder for the two fluids: it is thus an invaluable tool for engineering calculations and has been evaluated for the present case. Its behaviour depends on the physical parameters involved, a and De. For all values of j, an increase in the mobility factor and/or fluid elasticity leads to a decrease of the friction factor ratio (which is always below unity) the faster, the smaller j is. Fig. 3 shows the Po=PoN as a function of De for the radius ratio j ¼ 0:5 and several values of a ð0:001; 0:01; 0:1; 0:2; 0:4; 0:6; 0:8Þ. As either De or a increase, Po=PoN decreases, the faster, the larger a is and the smaller j is. (e.g. for De ¼ 5; Po=PoN P 0:9 is achieved for a 6 0:0015 if j ¼ 0:9 and for a 6 0:000015 if j ¼ 0:1). For De ! 0, all curves show a horizontal tangent. It can be demonstrated that when a >¼ 0:5 there exists a maximum Deborah number which cannot be exceeded and whose
M. Lorenzini et al. / Applied Thermal Engineering 122 (2017) 118–125
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Fig. 3. Friction factor ratio versus Deborah number, for different values of a; j ¼ 0:5.
value can be evaluated analytically. Above this De, the problem has no valid solution: this shows in Fig. 3 for a ¼ 0:6 and a ¼ 0:8. The Brinkman number, which for this case has been defined e.g. by [13] as def
Br 0 :¼
l0 t2hi
ð16Þ
kðT o T i Þ
shall be termed static Brinkman number; owing to the fields of application mentioned earlier, the resulting values of Br are sometimes much higher than those normally occurring in heat transfer applications with a Newtonian fluid [29,30]. Such a definition is expedient because it contains only quantities which are normally needed for determining the velocity and temperature fields, and are thus known a priori, but it is inconsistent when applied to fluid with velocity-dependent viscosity [28], such as a Giesekus fluid. Coelho and Pinho [27,28] discussed the matter at length and developed a general formulation which actually applies the physical definition of Br as the ratio of the dissipated mechanical work to heat transfer rate and yields the traditional form for the pipe flow of a Newtonian fluid. In the present case, the dissipated power is needed to keep the inner cylinder rotating at uniform angular velocity t Xi ¼ Rh;ii and the heat transfer rate is that of a plane slit of height d under a temperature difference DT ¼ T o T i : the temperature profile in the annulus is indeed very close to linear except for low values of j. It can be demonstrated that under the conditions above the Brinkman number becomes
Br ¼
lt
2 0 hi
kðT o T i Þ
i
i
s ¼ Br0 s
ð17Þ
Introducing the static Brinkman number the energy equation, Eq. (9), becomes
d Br 0 d th 2 dT shr r ¼0 r 1 j dr r dr dr Substituting Eq. (10) in Eq. (18) yields
ð18Þ
d j2 d th dT Br 0 si ¼0 r 1 j dr r dr dr
ð19Þ
It has to be noticed that the Brinkman number was not expressed explicitly in Eq. (19), with the form Br0 si being used instead: this is done on purpose, as its computation would imply the knowledge of the velocity field, from which si could be obtained; Br0 , on the contrary, can easily be obtained from the known data, moreover, the temperature profiles become directly comparable to those of other Authors, e.g. [13]. In the following the values of Br0 shall be imposed as a known parameter and the resulting values of Br shall be given as results of the calculations for the different cases studied. With v h ðr Þ and si known, the temperature distribution can be computed from Eq. (19), provided suitable boundary conditions are given. In this work conditions of the first kind (known value of temperature) are imposed at the boundaries, i.e. TðRi Þ ¼ T i and TðRo Þ ¼ T o so that the non-dimensional forms become
T ðjÞ ¼ 0
ð20Þ
T ð1Þ ¼ 1
ð21Þ
To compute the heat transfer between the fluid and the walls from the macroscopic data the fluid bulk temperature T b and the Nusselt number must be known. The bulk or mixing-cup temperature can be normalised in the same way as the temperature T was. The non-dimensional bulk temperature becomes
T b ¼
Tb Ti To Ti
ð22Þ
From the definition of bulk temperature it is easy to prove that for thermophysical properties independent of temperature T b is obtained from the non-dimensional temperature and velocity fields as
R1 T b ¼
j
v h ðr ÞT ðr Þr dr R1 j v h ðr Þr dr
ð23Þ
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Fig. 4. Non-dimensional temperature profiles.
The Nusselt number expresses the ratio of the conductive to convective thermal resistance, and for the case of the annular geometry is defined based on the hydraulic diameter Dh ¼ 2Ro ð1 jÞ, such
hi ¼
ð24Þ
i
Tb Ti
De=0.0 De=0.2 De=0.5 De=1.0 De=2.0 De=3.0 De=5.0
4 3.5 3 * *
kdT dr r¼r
4.5
T (r )
that Nu ¼ hDk h , where h is the convective heat transfer coefficient. Since two surfaces with different radii of curvature bound the fluid domain, two Nusselt numbers are given, one for the inner and one for the outer wall respectively. The corresponding convective heat transfer coefficients are computed differently, namely
5
2.5 2 1.5
for the inner cylinder and
ho ¼
kdT dr r¼r o
1
ð25Þ
To Tb
Br =0 0
for the outer one. It can be shown that the expressions of the Nusselt numbers for the inner and outer cylinders in terms of nondimensional quantities are dT
Nui ¼ 2ð1 jÞ dr Nuo ¼ 2ð1 jÞ
0.5
r ¼j
T b dT dr
1
r ¼1 T b
0 0.1
0.2
0.3
0.4
0.5
0.6
r
0.7
0.8
0.9
1
*
Fig. 5. Influence of the Deborah number on T ðr Þ for
j ¼ 0:1; a ¼ 0:01; Br 0 ¼ 10.
ð26Þ ð27Þ
3. Results and discussion Daprà and Scarpi [24] obtained the velocity profiles for the annular tangential flow of a Giesekus fluid and discussed extensively the limitations to be imposed on the Deborah number and on the mobility factor in order for a solution to exist, demonstrating that when a 6 0:5De may take any non-negative value, whereas it is limited when a > 0:5. In this work three different cases characterised by the ratio of the inner-to-outer radius ðj ¼ 0:1; j ¼ 0:5; j ¼ 0:9Þ are studied. The temperature profiles were computed for several values of the static Brinkman number (Br 0 ¼ 1; 2; 5; 7; 10 and Br 0 ¼ 0 corresponding to no viscous dissipation), Deborah numbers and mobility parameters. The latter two quantities are chosen in accordance to the limits defined by Daprà and Scarpi [24] as summarised above, but always in the range 0 6 De 6 5. The corresponding Brinkman numbers are computed as are the Nusselt numbers. The temperature profile is determined by numerically solving Eq. (19) after determining the corresponding value of si . Computations are carried out via a commercial code, as are all operations necessary to determine the bulk temperature and Nusselt numbers. Computational time for each run is negligible.
The non-dimensional temperature profiles versus the nondimensional radial coordinate are plotted in Fig. 4 for three different values of the inner-to-outer radius ratio, j ¼ 0:1 Fig. 4a, j ¼ 0:5 Fig. 4b and j ¼ 0:9 Fig. 4c respectively for static Brinkman numbers ranging from 0 to 10 in the case of a ¼ 0:1 and De ¼ 0:2. The general trend is the same for all cases, with the increase of viscous dissipation modifying the initial temperature distribution, which is close to linear. When the static Brinkman number is high enough, the temperature in some region within the fluid exceeds that of the outer wall, and it is to be remarked that in this case the maximum temperature shifts towards lower values of r as Br 0 increases, independently of j. This is due to the higher local velocity gradients at the inner wall, which make viscous dissipation more significant, although this effect is partially offset by the lower temperature at the same location. When velocity gradients dominate, as is the case for j ¼ 0:1 this causes a strong distortion in the temperature profile which would set in the absence of viscous dissipation, conversely, when local velocity gradients are smaller, deviations in the temperature profile are also more contained. It is interesting to investigate the effects of the Deborah number on the temperature profiles, which are shown in Fig. 5 for j ¼ 0:1 and Br 0 ¼ 10, where the plot for De ¼ 0 corresponds to the behaviour of a Newtonian fluid. The increase of the Deborah number mitigates the effects of viscous dissipation, flattening the temperature profile until, for high enough values of De it becomes similar to that for Br 0 ¼ 0, which
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M. Lorenzini et al. / Applied Thermal Engineering 122 (2017) 118–125
4
De=0.0 De=0.2 De=0.5 De=1.0 De=2.0 De=3.0 De=5.0
3.5 3
T*(r*)
2.5 2 1.5 1 0.5 0 0.5
Br0=0
0.6
0.7
0.8
0.9
1
r* Fig. 6. Influence of the Deborah number on T ðr Þ for
j ¼ 0:5; a ¼ 0:01; Br0 ¼ 10.
is plotted as a dashed line in the figure. This tenfold reduction in the maximum temperature can be explained in the light of the velocity profile: as the Deborah number increases, the velocity profile becomes flat except for the region closest to the rotating inner wall. This also results in the peak temperature occurring at distances closer to the inner wall as De increases. As a consequence, velocity gradients are confined to a very limited area and viscous dissipation gives a contribution to temperature which gradually becomes negligible, as convection and diffusion mechanism effectively manage the thermal surge. This has a practical significance, namely that Giesekus fluids can withstand much higher rotational velocities than Newtonian fluids without suffering from an overheating, which might otherwise become critical and lead to thermal failure either of the fluid or of the walls. The effect is also present at lower Br 0 , all other conditions being the same, and the trend is similar also for higher values of j, although with lower intensity. As can be seen in Fig. 6, the change in temperature profile is far less marked as De increases, although even for De ¼ 2:0 there is still a significant deviation from the temperature distribution in the absence of viscous dissipation. Again, the velocity distribution is responsible for this behaviour. For larger values of j the trend is similar, but the gain from the increase of De becomes smaller and smaller and finally negligible. The influence of a is also investigated. As is evident from Fig. 7 the increase in the mobility parameter acts in the same way as the increase in the Deborah number, decreasing the maximum temperature and shifting it towards the inner wall: already for a ¼ 0:1 the temperature peak has been more than halved from the corresponding value for a Newtonian fluid under the same thermal and flow conditions. When a ¼ 0:4 the temperature profile is flat for all practical purposes in the interval 0:3 6 r 6 1; in this stretch the fluid behaves as if isothermal and all the heat generated by viscous dissipation flows through the inner wall. For larger values of j the influence of a quickly wanes, as can be seen in Fig. 8, the plot De ¼ 0 again corresponding to a Newtonian fluid: even for j ¼ 0:5 the increase of a up to values of a ¼ 0:4 has a marginal effect on the decrease of temperature: the smaller gaps between the cylinders make the role of anisotropy of little significance. Finally, the Nusselt numbers are analysed. As mentioned earlier, two different values are given, one for each of the surfaces bounding the fluid. The trend of the Nusselt number for the outer surface
Fig. 7. Influence of a on T ðr Þ for
j ¼ 0:1; De ¼ 0:2; Br0 ¼ 10.
Fig. 8. Influence of a on T ðr Þ for
j ¼ 0:5; De ¼ 0:2; Br0 ¼ 10.
Nuo is shown in Fig. 9, for a ¼ 0:1 and De ¼ 0:2 as a function of the static and actual Brinkman numbers, Br 0 and Br respectively. It appears clearly that in both cases Nuo initially decreases as viscous dissipation increases, becomes zero then goes steeply to negative values and inverts it sign becoming positive, dropping again quickly to an asymptotical, positive value. Different values of a and De give similar trends.The seemingly puzzling behaviour is readily explained if one considers the expression of Nuo , Eq. (27) and the plots of the temperature profiles, Fig. 4b. In the absence of viscous dissipation, the normal derivative of the nondimensional temperature is negative, T b < T ðr ¼ 1Þ and the value of Nu0 is positive. As viscous dissipation increases, the temperature in the fluid increases locally and its profile changes its curvature, so that the gradient at the wall changes its sign and becomes positive (heat now flows from the fluid into the wall, rather than vice versa), whilst the bulk temperature is still lower than the temperature at the outer surface but gradually approaches unity (corresponding to the condition T b < T ðr ¼ 1Þ): Nuo is now negative and goes rapidly to minus infinity. If the Brinkman number increases further, the curvature of the temperature profile remains unchanged
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cous dissipation was investigated for different values of the outer-to-inner radius ratio, of Deborah number and of the nondimensional mobility parameter. The main results are summarised below.
150
Nuo,Br
0
Nuo,Br
100
Nu o
50
0
5
10
15
20
25
-50
-100
-150
Br0, Br Fig. 9. Nusselt number at the outer wall as a function of Br 0 ; Br.
50 45 40
Nu i
35 30
De=0.0 De=0.2 De=0.5 De=1.0 De=3.0 De=5.0
25 20 15 10 10-3
10-2
10-1
Br
100
101
102
Fig. 10. Nusselt number at the inner wall as a function of Br for different De.
as the sign of Nuo , which decreases owing to the increase of the difference 1 T b . Different values of a and De give similar trends. The behaviour of the Nusselt number for the inner surface Nui is shown in Fig. 10 as a function of Br and for several values of De on a semilogarithmic plot for convenience. Again, the trend can be explained recalling the definition of Nui , Eq. (26) and the temperature profile, Fig. 4b. Now, the temperature gradient at the wall never changes it sign, nor the temperature profile its type of curvature. Moreover, the change in the value of the wall temperature gradient (this is the rotating surface, hence where velocity gradients are steepest) is faster than that of the value of the bulk temperature, which changes linearly with the Brinkman number, and this makes the Nusselt number increase. It can be shown that for higher values of Br the values of Nui tend asymptotically to a finite value: the data are not plotted because at such high values for Br the assumption of temperature-independent thermo-physical properties fails. 4. Conclusions In this work the purely tangential flow of a Giesekus fluid in an annulus with rotating inner wall subject to set temperature thermal boundary conditions at the walls and in the presence of vis-
– The form of the Brinkman number suitable for the problem studied was obtained and related to that commonly used for Newtonian fluid, which was termed static Brinkman number. – For increasing values of the Deborah number the velocity gradients – on which viscous dissipation depends – become more and more concentrated towards the rotating wall. – The temperature profiles show that viscous dissipation distorts the velocity profile and makes local temperature higher than that at the outer wall, reversing the heat flow there. – Temperature gradients at the walls are higher for low j, and more pronounced at the inner wall, viscous dissipation being higher at that location. – High values of the Deborah number strongly dampen the maximum value of temperature and the temperature profile in general, thus allowing far higher Brinkman numbers to be reached without danger of thermal breakdown. – The increase of the non-dimensional mobility parameter acts in the same way as the Deborah number when j is low, becoming of little influence already at j ¼ 0:5. – The Nusselt number for the inner wall shows an asymptotic behaviour as the Brinkman number increases, owing to the boundary condition at Ro . As the temperature profile changes and bulk temperature increases, the temperature gradient changes sign, Nuo takes quickly increasingly negative values and the difference between outer wall and bulk temperature approaches zero; as T b increases, Nuo approaches a finite value for increasing Br. – The Nusselt number at the inner wall increases steadily with Br, never changing its sign as the inner walls always acts as a heat sink for the fluid. The increase of the Deborah number brings forth an increase in Nui for a fixed value of Br. As the Giesekus model has proven itself capable of describing the behaviour of shear thinning fluids, there are several areas of research within microchannels where it could be employed. Starting from channel geometry, the influence on pressure drop and heat transfer of smoothing the cross-section corners could be investigated, as already done for Newtonian fluids, [31], or on branched structures employed in micromixers and evaporators (e.g. [32,33]). Also, the high temperature gradients at the wall make conjugate heat transfer (i.e. convection in the fluid and conduction in the solid wall) worth of investigation, especially for the changes in axial bulk temperature profiles that it induces, which is well documented for Newtonian flows, [34,35]. Finally, concerning micro-effects, electro-osmotic devices often employ nonNewtonian fluids, [36,37], and the Giesekus model could again be used to predict the heat transfer characteristics of electroosmotic flows [38] and the thermofluidic behaviour of electro osmotic pumps, see [39,40].
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Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Heat transfer for a Giesekus fluid in a rotating concentric annulus Marco Lorenzini a,⇑, Irene Daprà b, Giambattista Scarpi b a b
DIN, Dipartimento di Ingegneria Industriale, Alma Mater Studiorum Università di Bologna, Campus Forlì, Via Fontanelle 40, 47121 Forlì, FC, Italy DICAM, Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali, Alma Mater Studiorum Università di Bologna, Viale Risorgimento 2, 40136 Bologna, Bo, Italy
a r t i c l e
i n f o
Article history: Received 25 November 2016 Accepted 3 May 2017 Available online 5 May 2017 Keywords: Annular flow Non-Newtonian fluid Viscous dissipation Convective heat transfer
a b s t r a c t Annular flow of a viscoelastic fluid described by the Giesekus model has received some attention over the years, both concerning the fluid mechanical and thermal aspects, yet no investigation has been carried out using the analytical solution for the velocity profile in the energy equation to determine temperature distribution and heat transfer characteristics. Moreover, viscous dissipation, when accounted for, is usually defined by a formulation of the Brinkman number which proves inconsistent when applied to nonNewtonian fluids. In this work the purely tangential flow of a Giesekus fluid in an annulus with rotating inner wall subject to thermal boundary conditions of the first kind (imposed temperature) at the walls is investigated employing the analytical solution for the velocity profile. Viscous dissipation is accounted for by first deriving a consistent Brinkman number which is easily related to the one usually employed in previous studies and the temperature profiles are obtained for different values of the Deborah number and the non-dimensional mobility factor. Results for the Nusselt numbers at the inner and outer wall are also discussed. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Annular geometries are a recurring feature in many engineering applications, including heat transfer equipment, e.g. tube-in-tube heat exchanger, machining (drilling), manufacturing (food or polymer extrusion), machine elements (journal bearings), oil industry (well drilling) and viscometry. The fluid confined between the two cylindrical surfaces may flow tangentially, axially, or with a combination of both modes and is often non-Newtonian (e.g. emulsions). The latter feature significantly complicates the study of the flow and heat transfer characteristics of the phenomenon, owing to the non-linearity in the constitutive equation. Therefore, the Newtonian model is often inappropriate, whilst the Giesekus model [1] is suitable for shear-thinning fluids. In the case of microchannels, cases where non-Newtonian fluids are employed abound: amongst applications where annular geometries are involved, micro-drilling can be quoted, [2], together with micro-extruders and micro-mixers, [3]. A review on convective heat transfer in stationary and rotating annuli for Newtonian fluids has been carried out by Childs and Long [4] in the mid-90s, whilst later on Coelho and Pinho [5] analysed viscous dissipation in annuli with axial flow subject to prescribed heat flux or surface temperature boundary conditions. ⇑ Corresponding author. E-mail address: [email protected] (M. Lorenzini). http://dx.doi.org/10.1016/j.applthermaleng.2017.05.013 1359-4311/Ó 2017 Elsevier Ltd. All rights reserved.
Fang et al. [6] studied numerically the forced convection of nonNewtonian fluids in concentric and eccentric annuli in the case of negligible viscous dissipation. Still concerning non-Newtonian fluids, axial flow in pipes and ducts with viscous dissipation was investigated by Pinho and Oliveira [7] for a viscoelastic fluid using a simplified Phan-Tien-Tanner model for constant heat flux and by Coelho et al. [8] for constant wall temperature. The same Authors also extended the study to the thermal entry region of a hydrodinamically fully developed flow under the same conditions in [9] and employed a different fluid model to study the same problem in [10]. Concerning purely tangential flow, Khellaf and Lauriat [11] studied the thermal and fluid dynamic behaviour of a Carreau fluid in the case of rotating inner surface and fixed outer surface, whilst Naimi et al. [12] turned their attention to a power law fluid analysing the Taylor-Couette convective vortices. In more recent years, Jouyandeh et al. [13] investigated the heat transfer characteristics for the tangential flow of a Giesekus fluid between two counter-rotating cylinders; their study accounted for viscous dissipation and employed a simplified velocity profile. The same group [14] also analysed the axial annular flow for a fixed duct using the same fluid model and extending thermal conditions to the case of imposed heat flux at the walls. The axial annular flow of a Giesekus fluid was also studied by Mostafaiyan et al. [15], who proposed an approximation to estimate radial normal stress, thus obtaining an analytic expression for the velocity and pressure profiles, whose validity was checked by comparison with the exact solution in a parametric study.
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For several technical fluids exhibiting viscoelastic characteristics, the Giesekus model suitably describes their behaviour. The model employs three physical parameters in the constitutive equation, namely viscosity, mobility factor and relaxation time [1,16]. The experimental determination of the non-linear mobility factor was carried out by Calin et al. [17] and the rheological characterisation of a high-density polyethylene employing this model is documented by Debbaut and Burhin [18]. The Giesekus model has been applied in a number of works, some directly concerned with annular geometries [19,20], other with heat transfer, usually including viscous dissipation [21,22]. The conclusion one might draw from the above discussion is that all aspects of heat transfer and fluid flow in annular geometries for viscoelastic fluids described by the Giesekus model have been investigated. In fact, this is not the case: for the purely tangential flow, an approximate solution is often employed, as discussed by Daprà and Scarpi [23], who in turn obtained an analytical solution for the case of purely tangential annular flow [24] and this directly affects the results concerning heat transfer. Also, viscous dissipation is often described by a formulation of the Brinkman number which works quite well for Newtonian fluids, [25,26] but is inconsistent for those exhibiting a viscoelastic behaviour, as discussed by Coelho and co-workers [27,28]. In this work the purely tangential flow of a Giesekus fluid in an annulus with rotating inner wall subject to first-order (imposed temperature) thermal boundary conditions at the wall is investigated using the analytical solution for the velocity profile. Viscous dissipation is accounted for by first deriving a consistent Brinkman number which is easily related to the one usually employed in previous studies and the temperature profiles are obtained for different values of the Deborah number and the non-dimensional mobility factor. Results for the Nusselt numbers at the inner and outer wall are also discussed. 2. Problem definition and governing equations The geometry analysed in this study is shown in Fig. 1. A concentric annulus of infinite length is considered, such as that formed by two coaxial cylinders. The outer surface is fixed and has a radius Ro , whilst the inner one, which has a radius Ri , rotates with an v angular velocity Xi ¼ Rh;ii . The outer surface of the annulus is kept at a uniform temperature T o , whereas the inner surface is at temperature T i , which is also uniform; moreover, T o > T i .
A Giesekus fluid fills the annulus and moves with a steady-state, laminar and purely tangential flow, which is induced by the rotation of the inner cylinder relative to the outer one. The constitutive equation of the fluid can be written as:
" # h i ka @s r s þ kr ss þ t rs kr s rt þ ðrtÞT s þ @t l h i ¼ l0 rt þ ðrtÞT
ð1Þ
where t is the velocity vector, s is the stress tensor, t is time, l0 is the zero-shear rate viscosity of the polymer, kr is the stress relaxation time, and a is the non-dimensional mobility parameter ð0 6 a 6 1Þ, which accounts for the degree of anisotropy in the polymer. If a ¼ 0 the fluid exhibits an isotropic mobility, and the rheological equation leads to the upper convective Maxwell model (UCM); at the other end of the range, a ¼ 1 corresponds to the most anisotropic mobility. Supposing a purely tangential flow and using cylindrical coordinates, the velocity is limited to its tangential component, th , which automatically verifies the continuity equation. The radial and tangential components of the momentum equation become
2 @p 1 @ shh t ðr srr Þ þ ¼q h @r r @r r r
ð2Þ
@ 2 r srh ¼ 0 @r
ð3Þ
where p is the pressure, r is the radial coordinate and q is the density. Under the assumption that thermophysical properties and model parameters are unaffected by changes in temperature, the tangential velocity distribution in the radial direction th ðrÞ for an isothermal flow still holds. Owing to the nature of the flows in industrial applications such as journal bearings, polymer extruders or chemical/mechanical mixers, fluid viscosity or velocity gradients may be high and viscous dissipation become a relevant issue, causing an alteration in the radial temperature profile that would normally be encountered when its influence is negligible. If viscous dissipation is accounted for, the energy equation for the abovementioned case becomes
k d dT s : rt ¼ 0 r r dr dr
ð4Þ
where k is the thermal conductivity of the fluid and TðrÞ is the tem s : rt gives the contribution
perature; the tensor scalar product
of viscous dissipation to the temperature profile:
s : rt ¼ srh
dth th dr r
ð5Þ
so that Eq. (4) can be written as
d dT r dth th srh ¼0 r dr dr k dr r
ð6Þ
The following non-dimensional quantities are then introduced:
Fig. 1. Sketch of the annulus.
r ¼ Rro
i T ¼ TTT o T i
th ¼ tth;ih
shr ¼ ls0hrtdh
j ¼ RRoi
De ¼ krdthi
i
p ¼ lpdthi 0
where d ¼ Ro Ri . The Deborah number represents the ratio of the polymer relaxation time, kr to the characteristic time of the phenomenon,here represented by the ratio of the thickness of the annulus, d, to the tangential velocity of the rotating wall: the lower De, the more Newtonian-like the fluid behaves. Conversely, for high values of De the shear-thinning behaviour dominates.
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Fig. 2. Velocity profiles for
j ¼ 0:5; a ¼ 0:4.
With
Eqs. (2), (3) and (6) become
ð7Þ
d 2 r srh ¼ 0 dr
pffiffiffi f1 ¼ fð1Þb ¼ si j2 De a pffiffiffiffiffiffiffiffiffiffiffiffi c ¼ 1a
ð8Þ
The no-slip condition at the inner wall in non-dimensional form yields
l0 t2hi 2 1 d th d dT s r ¼0 r 1 j dr r kðT o T i Þ hr dr dr
ð9Þ
dp d s t ¼ Re þ r srr dr dr hh r
2 h r
where Re ¼ qthi d=l0 is the rotational Reynolds number. The tangential stress srh can be related to the tangential stress at the inner wall si by solving Eq. (8)
srh ¼
j2 si
ð10Þ
r 2
The shear rate c_ can thus be expressed as
c_ ¼ ð1 jÞr where
f ¼ fðr Þ ¼
d dr
v
h r
h i 2 2aj2 si ð2a 1Þf þ r ¼ 2 ð2a 1Þr 2 þ f
ð11Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r 4 4j4 a2 De2 si :
Using the no-slip condition at the outer wall th ð1Þ ¼ 0 and integrating Eq. (11) between r and 1, the velocity profile is obtained h ðr Þ
t
( " # pffiffiffi r bcð2a 1Þðf f1 Þ ¼ 2c a arctanh 2 2Deðj 1Þ ð2a 1Þ2 b c2 f1 f 3 2 pffiffiffi 4 2b a f1 r f pffiffiffi 5 2c a arctanh þð2a 1Þ arctan 4 2 4 4 4b a r þ 1 r 39 2 2 = bc 1 r f f 4 5 þ ð2a 1Þbrpffiffiffi 2 1 2 2 2 4 2 2 ; 4 ð j 1 ÞDe a b c2 b c r b c r 3 2 2 2 b þ c2 r ð2a 1Þ 1 r 5 ð12Þ þ 2 2 b c2 b c2 r 4
th ðr Þ r¼j ¼ 1
ð13Þ ð14Þ
ð15Þ
Eq. (15) can be solved numerically to obtain si , which is then used in Eq. (12) to calculate th ðr Þ. The non-dimensional tangential velocity profile is plotted in Fig. 2 for j ¼ 0:5; a ¼ 0:4 and several values of De. It is to be remarked how the velocity gradients become steeper and steeper but more and more confined to the vicinity of the rotating wall as the Deborah number increases, which marks the transition from a Newtonian to a non-Newtonian behaviour in the fluid. The velocity profiles for lower values of j (wider gap between the walls) increase the gradients at the rotating wall and flatten quickly to almost zero velocity as De increases, whilst when j is higher (smaller gaps) the velocity profile becomes close to linear and only bulges slightly inwards at the middle for high values of De [24]. The Poiseuille number Po ¼ f Re, with f the Fanning friction factor for the Giesekus fluid is compared to that for a Newtonian fluid, PoN , under the same conditions: their ratio Po=PoN represents the ratio between the torque required to rotate the inner cylinder for the two fluids: it is thus an invaluable tool for engineering calculations and has been evaluated for the present case. Its behaviour depends on the physical parameters involved, a and De. For all values of j, an increase in the mobility factor and/or fluid elasticity leads to a decrease of the friction factor ratio (which is always below unity) the faster, the smaller j is. Fig. 3 shows the Po=PoN as a function of De for the radius ratio j ¼ 0:5 and several values of a ð0:001; 0:01; 0:1; 0:2; 0:4; 0:6; 0:8Þ. As either De or a increase, Po=PoN decreases, the faster, the larger a is and the smaller j is. (e.g. for De ¼ 5; Po=PoN P 0:9 is achieved for a 6 0:0015 if j ¼ 0:9 and for a 6 0:000015 if j ¼ 0:1). For De ! 0, all curves show a horizontal tangent. It can be demonstrated that when a >¼ 0:5 there exists a maximum Deborah number which cannot be exceeded and whose
M. Lorenzini et al. / Applied Thermal Engineering 122 (2017) 118–125
121
Fig. 3. Friction factor ratio versus Deborah number, for different values of a; j ¼ 0:5.
value can be evaluated analytically. Above this De, the problem has no valid solution: this shows in Fig. 3 for a ¼ 0:6 and a ¼ 0:8. The Brinkman number, which for this case has been defined e.g. by [13] as def
Br 0 :¼
l0 t2hi
ð16Þ
kðT o T i Þ
shall be termed static Brinkman number; owing to the fields of application mentioned earlier, the resulting values of Br are sometimes much higher than those normally occurring in heat transfer applications with a Newtonian fluid [29,30]. Such a definition is expedient because it contains only quantities which are normally needed for determining the velocity and temperature fields, and are thus known a priori, but it is inconsistent when applied to fluid with velocity-dependent viscosity [28], such as a Giesekus fluid. Coelho and Pinho [27,28] discussed the matter at length and developed a general formulation which actually applies the physical definition of Br as the ratio of the dissipated mechanical work to heat transfer rate and yields the traditional form for the pipe flow of a Newtonian fluid. In the present case, the dissipated power is needed to keep the inner cylinder rotating at uniform angular velocity t Xi ¼ Rh;ii and the heat transfer rate is that of a plane slit of height d under a temperature difference DT ¼ T o T i : the temperature profile in the annulus is indeed very close to linear except for low values of j. It can be demonstrated that under the conditions above the Brinkman number becomes
Br ¼
lt
2 0 hi
kðT o T i Þ
i
i
s ¼ Br0 s
ð17Þ
Introducing the static Brinkman number the energy equation, Eq. (9), becomes
d Br 0 d th 2 dT shr r ¼0 r 1 j dr r dr dr Substituting Eq. (10) in Eq. (18) yields
ð18Þ
d j2 d th dT Br 0 si ¼0 r 1 j dr r dr dr
ð19Þ
It has to be noticed that the Brinkman number was not expressed explicitly in Eq. (19), with the form Br0 si being used instead: this is done on purpose, as its computation would imply the knowledge of the velocity field, from which si could be obtained; Br0 , on the contrary, can easily be obtained from the known data, moreover, the temperature profiles become directly comparable to those of other Authors, e.g. [13]. In the following the values of Br0 shall be imposed as a known parameter and the resulting values of Br shall be given as results of the calculations for the different cases studied. With v h ðr Þ and si known, the temperature distribution can be computed from Eq. (19), provided suitable boundary conditions are given. In this work conditions of the first kind (known value of temperature) are imposed at the boundaries, i.e. TðRi Þ ¼ T i and TðRo Þ ¼ T o so that the non-dimensional forms become
T ðjÞ ¼ 0
ð20Þ
T ð1Þ ¼ 1
ð21Þ
To compute the heat transfer between the fluid and the walls from the macroscopic data the fluid bulk temperature T b and the Nusselt number must be known. The bulk or mixing-cup temperature can be normalised in the same way as the temperature T was. The non-dimensional bulk temperature becomes
T b ¼
Tb Ti To Ti
ð22Þ
From the definition of bulk temperature it is easy to prove that for thermophysical properties independent of temperature T b is obtained from the non-dimensional temperature and velocity fields as
R1 T b ¼
j
v h ðr ÞT ðr Þr dr R1 j v h ðr Þr dr
ð23Þ
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Fig. 4. Non-dimensional temperature profiles.
The Nusselt number expresses the ratio of the conductive to convective thermal resistance, and for the case of the annular geometry is defined based on the hydraulic diameter Dh ¼ 2Ro ð1 jÞ, such
hi ¼
ð24Þ
i
Tb Ti
De=0.0 De=0.2 De=0.5 De=1.0 De=2.0 De=3.0 De=5.0
4 3.5 3 * *
kdT dr r¼r
4.5
T (r )
that Nu ¼ hDk h , where h is the convective heat transfer coefficient. Since two surfaces with different radii of curvature bound the fluid domain, two Nusselt numbers are given, one for the inner and one for the outer wall respectively. The corresponding convective heat transfer coefficients are computed differently, namely
5
2.5 2 1.5
for the inner cylinder and
ho ¼
kdT dr r¼r o
1
ð25Þ
To Tb
Br =0 0
for the outer one. It can be shown that the expressions of the Nusselt numbers for the inner and outer cylinders in terms of nondimensional quantities are dT
Nui ¼ 2ð1 jÞ dr Nuo ¼ 2ð1 jÞ
0.5
r ¼j
T b dT dr
1
r ¼1 T b
0 0.1
0.2
0.3
0.4
0.5
0.6
r
0.7
0.8
0.9
1
*
Fig. 5. Influence of the Deborah number on T ðr Þ for
j ¼ 0:1; a ¼ 0:01; Br 0 ¼ 10.
ð26Þ ð27Þ
3. Results and discussion Daprà and Scarpi [24] obtained the velocity profiles for the annular tangential flow of a Giesekus fluid and discussed extensively the limitations to be imposed on the Deborah number and on the mobility factor in order for a solution to exist, demonstrating that when a 6 0:5De may take any non-negative value, whereas it is limited when a > 0:5. In this work three different cases characterised by the ratio of the inner-to-outer radius ðj ¼ 0:1; j ¼ 0:5; j ¼ 0:9Þ are studied. The temperature profiles were computed for several values of the static Brinkman number (Br 0 ¼ 1; 2; 5; 7; 10 and Br 0 ¼ 0 corresponding to no viscous dissipation), Deborah numbers and mobility parameters. The latter two quantities are chosen in accordance to the limits defined by Daprà and Scarpi [24] as summarised above, but always in the range 0 6 De 6 5. The corresponding Brinkman numbers are computed as are the Nusselt numbers. The temperature profile is determined by numerically solving Eq. (19) after determining the corresponding value of si . Computations are carried out via a commercial code, as are all operations necessary to determine the bulk temperature and Nusselt numbers. Computational time for each run is negligible.
The non-dimensional temperature profiles versus the nondimensional radial coordinate are plotted in Fig. 4 for three different values of the inner-to-outer radius ratio, j ¼ 0:1 Fig. 4a, j ¼ 0:5 Fig. 4b and j ¼ 0:9 Fig. 4c respectively for static Brinkman numbers ranging from 0 to 10 in the case of a ¼ 0:1 and De ¼ 0:2. The general trend is the same for all cases, with the increase of viscous dissipation modifying the initial temperature distribution, which is close to linear. When the static Brinkman number is high enough, the temperature in some region within the fluid exceeds that of the outer wall, and it is to be remarked that in this case the maximum temperature shifts towards lower values of r as Br 0 increases, independently of j. This is due to the higher local velocity gradients at the inner wall, which make viscous dissipation more significant, although this effect is partially offset by the lower temperature at the same location. When velocity gradients dominate, as is the case for j ¼ 0:1 this causes a strong distortion in the temperature profile which would set in the absence of viscous dissipation, conversely, when local velocity gradients are smaller, deviations in the temperature profile are also more contained. It is interesting to investigate the effects of the Deborah number on the temperature profiles, which are shown in Fig. 5 for j ¼ 0:1 and Br 0 ¼ 10, where the plot for De ¼ 0 corresponds to the behaviour of a Newtonian fluid. The increase of the Deborah number mitigates the effects of viscous dissipation, flattening the temperature profile until, for high enough values of De it becomes similar to that for Br 0 ¼ 0, which
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4
De=0.0 De=0.2 De=0.5 De=1.0 De=2.0 De=3.0 De=5.0
3.5 3
T*(r*)
2.5 2 1.5 1 0.5 0 0.5
Br0=0
0.6
0.7
0.8
0.9
1
r* Fig. 6. Influence of the Deborah number on T ðr Þ for
j ¼ 0:5; a ¼ 0:01; Br0 ¼ 10.
is plotted as a dashed line in the figure. This tenfold reduction in the maximum temperature can be explained in the light of the velocity profile: as the Deborah number increases, the velocity profile becomes flat except for the region closest to the rotating inner wall. This also results in the peak temperature occurring at distances closer to the inner wall as De increases. As a consequence, velocity gradients are confined to a very limited area and viscous dissipation gives a contribution to temperature which gradually becomes negligible, as convection and diffusion mechanism effectively manage the thermal surge. This has a practical significance, namely that Giesekus fluids can withstand much higher rotational velocities than Newtonian fluids without suffering from an overheating, which might otherwise become critical and lead to thermal failure either of the fluid or of the walls. The effect is also present at lower Br 0 , all other conditions being the same, and the trend is similar also for higher values of j, although with lower intensity. As can be seen in Fig. 6, the change in temperature profile is far less marked as De increases, although even for De ¼ 2:0 there is still a significant deviation from the temperature distribution in the absence of viscous dissipation. Again, the velocity distribution is responsible for this behaviour. For larger values of j the trend is similar, but the gain from the increase of De becomes smaller and smaller and finally negligible. The influence of a is also investigated. As is evident from Fig. 7 the increase in the mobility parameter acts in the same way as the increase in the Deborah number, decreasing the maximum temperature and shifting it towards the inner wall: already for a ¼ 0:1 the temperature peak has been more than halved from the corresponding value for a Newtonian fluid under the same thermal and flow conditions. When a ¼ 0:4 the temperature profile is flat for all practical purposes in the interval 0:3 6 r 6 1; in this stretch the fluid behaves as if isothermal and all the heat generated by viscous dissipation flows through the inner wall. For larger values of j the influence of a quickly wanes, as can be seen in Fig. 8, the plot De ¼ 0 again corresponding to a Newtonian fluid: even for j ¼ 0:5 the increase of a up to values of a ¼ 0:4 has a marginal effect on the decrease of temperature: the smaller gaps between the cylinders make the role of anisotropy of little significance. Finally, the Nusselt numbers are analysed. As mentioned earlier, two different values are given, one for each of the surfaces bounding the fluid. The trend of the Nusselt number for the outer surface
Fig. 7. Influence of a on T ðr Þ for
j ¼ 0:1; De ¼ 0:2; Br0 ¼ 10.
Fig. 8. Influence of a on T ðr Þ for
j ¼ 0:5; De ¼ 0:2; Br0 ¼ 10.
Nuo is shown in Fig. 9, for a ¼ 0:1 and De ¼ 0:2 as a function of the static and actual Brinkman numbers, Br 0 and Br respectively. It appears clearly that in both cases Nuo initially decreases as viscous dissipation increases, becomes zero then goes steeply to negative values and inverts it sign becoming positive, dropping again quickly to an asymptotical, positive value. Different values of a and De give similar trends.The seemingly puzzling behaviour is readily explained if one considers the expression of Nuo , Eq. (27) and the plots of the temperature profiles, Fig. 4b. In the absence of viscous dissipation, the normal derivative of the nondimensional temperature is negative, T b < T ðr ¼ 1Þ and the value of Nu0 is positive. As viscous dissipation increases, the temperature in the fluid increases locally and its profile changes its curvature, so that the gradient at the wall changes its sign and becomes positive (heat now flows from the fluid into the wall, rather than vice versa), whilst the bulk temperature is still lower than the temperature at the outer surface but gradually approaches unity (corresponding to the condition T b < T ðr ¼ 1Þ): Nuo is now negative and goes rapidly to minus infinity. If the Brinkman number increases further, the curvature of the temperature profile remains unchanged
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cous dissipation was investigated for different values of the outer-to-inner radius ratio, of Deborah number and of the nondimensional mobility parameter. The main results are summarised below.
150
Nuo,Br
0
Nuo,Br
100
Nu o
50
0
5
10
15
20
25
-50
-100
-150
Br0, Br Fig. 9. Nusselt number at the outer wall as a function of Br 0 ; Br.
50 45 40
Nu i
35 30
De=0.0 De=0.2 De=0.5 De=1.0 De=3.0 De=5.0
25 20 15 10 10-3
10-2
10-1
Br
100
101
102
Fig. 10. Nusselt number at the inner wall as a function of Br for different De.
as the sign of Nuo , which decreases owing to the increase of the difference 1 T b . Different values of a and De give similar trends. The behaviour of the Nusselt number for the inner surface Nui is shown in Fig. 10 as a function of Br and for several values of De on a semilogarithmic plot for convenience. Again, the trend can be explained recalling the definition of Nui , Eq. (26) and the temperature profile, Fig. 4b. Now, the temperature gradient at the wall never changes it sign, nor the temperature profile its type of curvature. Moreover, the change in the value of the wall temperature gradient (this is the rotating surface, hence where velocity gradients are steepest) is faster than that of the value of the bulk temperature, which changes linearly with the Brinkman number, and this makes the Nusselt number increase. It can be shown that for higher values of Br the values of Nui tend asymptotically to a finite value: the data are not plotted because at such high values for Br the assumption of temperature-independent thermo-physical properties fails. 4. Conclusions In this work the purely tangential flow of a Giesekus fluid in an annulus with rotating inner wall subject to set temperature thermal boundary conditions at the walls and in the presence of vis-
– The form of the Brinkman number suitable for the problem studied was obtained and related to that commonly used for Newtonian fluid, which was termed static Brinkman number. – For increasing values of the Deborah number the velocity gradients – on which viscous dissipation depends – become more and more concentrated towards the rotating wall. – The temperature profiles show that viscous dissipation distorts the velocity profile and makes local temperature higher than that at the outer wall, reversing the heat flow there. – Temperature gradients at the walls are higher for low j, and more pronounced at the inner wall, viscous dissipation being higher at that location. – High values of the Deborah number strongly dampen the maximum value of temperature and the temperature profile in general, thus allowing far higher Brinkman numbers to be reached without danger of thermal breakdown. – The increase of the non-dimensional mobility parameter acts in the same way as the Deborah number when j is low, becoming of little influence already at j ¼ 0:5. – The Nusselt number for the inner wall shows an asymptotic behaviour as the Brinkman number increases, owing to the boundary condition at Ro . As the temperature profile changes and bulk temperature increases, the temperature gradient changes sign, Nuo takes quickly increasingly negative values and the difference between outer wall and bulk temperature approaches zero; as T b increases, Nuo approaches a finite value for increasing Br. – The Nusselt number at the inner wall increases steadily with Br, never changing its sign as the inner walls always acts as a heat sink for the fluid. The increase of the Deborah number brings forth an increase in Nui for a fixed value of Br. As the Giesekus model has proven itself capable of describing the behaviour of shear thinning fluids, there are several areas of research within microchannels where it could be employed. Starting from channel geometry, the influence on pressure drop and heat transfer of smoothing the cross-section corners could be investigated, as already done for Newtonian fluids, [31], or on branched structures employed in micromixers and evaporators (e.g. [32,33]). Also, the high temperature gradients at the wall make conjugate heat transfer (i.e. convection in the fluid and conduction in the solid wall) worth of investigation, especially for the changes in axial bulk temperature profiles that it induces, which is well documented for Newtonian flows, [34,35]. Finally, concerning micro-effects, electro-osmotic devices often employ nonNewtonian fluids, [36,37], and the Giesekus model could again be used to predict the heat transfer characteristics of electroosmotic flows [38] and the thermofluidic behaviour of electro osmotic pumps, see [39,40].
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