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Transient laminar heat transfer simulations in periodic zigzag channels
International Journal of Heat and Mass Transfer 71 (2014) 758–768
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Transient laminar heat transfer simulations in periodic zigzag channels Zhanying Zheng, David F. Fletcher ⇑, Brian S. Haynes School of Chemical and Biomolecular Engineering, The University of Sydney, NSW 2006, Australia
a r t i c l e
i n f o
Article history: Received 8 July 2013 Received in revised form 12 November 2013 Accepted 24 December 2013 Available online 25 January 2014 Keywords: Heat transfer enhancement Nusselt number Friction factor Tortuous passage Microchannel CFD
a b s t r a c t At sufficiently high Reynolds numbers, laminar flows in tortuous channels become unsteady. A computational fluid dynamics methodology is developed to study transient, laminar flow and heat transfer in a periodic zigzag channel with a semi-circular cross-section. The computational domain consists of seven repeating zigzag units with smoothly joined inlet and outlet sections. Reynolds numbers ranging from 400 to 800 and Prandtl numbers ranging from 0.7 to 20 are examined for constant wall heat flux and constant temperature thermal boundary conditions. Simulation results show that the flow reaches a ‘‘developed’’ state after around three units, where the local velocities fluctuate with time but give well defined average heat transfer rates and pressure loss. The power spectra of the velocity at monitor points located periodically along the channel also become very similar. Significant heat transfer enhancement is observed in the transient regime studied, which is accompanied by a modest pressure-drop penalty, both of which increase with increasing Reynolds number. Vortex structures are visualized at different simulation times and Reynolds numbers and it is found that with increasing Reynolds number, vortices with smaller length-scale are generated, which contribute largely to the enhancement of heat transfer. The results for different Prandtl numbers show that the heat transfer enhancement is proportional to Pr1/3. The Nusselt number for the T boundary condition is found to be always higher than that for the H2 boundary condition. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Due to the great potential of achieving smaller, lighter-weight and lower-cost devices, the interest in the miniaturisation of various equipment can be seen in many industrial applications, such as electronics cooling, solar heat collectors, micro-chemical reactors and compact heat exchangers. These involve the use of micro or mini-scaled channels with the characteristic length ranging from hundreds of microns to a few millimetres. Compared with conventional-sized channels, the Reynolds number in these systems is generally too low to support turbulence, so various tortuous geometries are often used, in which the thermal boundary layer is constantly disrupted, giving rise to heat transfer enhancement relative to flow in straight pipes. The enhancement of heat transfer in tortuous passages is largely attributed to the establishment of secondary flows. Dean [1] studied flow in a coiled pipe with constant curvature theoretically and observed secondary flow patterns having a vortices pair structure in the cross-sectional plane, which are now called Dean vortices. Similar flow patterns can be found in various other channels with tortuous features, which promote cross-stream mixing ⇑ Corresponding author. Address: Chiu Building J14, The University of Sydney, NSW 2006, Australia. Tel.: +61 2 9351 4147; fax: +61 2 9351 3455. E-mail address: david.fl[email protected] (D.F. Fletcher). 0017-9310/$ - see front matter Ó 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.12.056
without incurring large increases in pressure drop, leading to relatively more uniform temperature profiles in the channel cross-section than a straight channel. Cheng and Akiyama [2] presented flow and heat transfer results in curved rectangular channels and observed the vortex pair structure in the secondary flow field. Kalb and Seader [3] studied numerically the fully-developed heat transfer of viscous flow in curved circular tubes and quantified the effect of Dean vortices on the pressure loss and heat transfer. Wang and Liu [4] further reported a variety of secondary flow structures for different flow configurations in slightly curved microchannels of square cross-section and their results showed that the Nusselt number increased significantly, although this was accompanied by a moderate increase in the friction factor. Apart from the curved channels, Manglik and co-workers [5–7] performed a series of computational studies for flow in planar wavy channels and found enhancement of heat transfer and increase of pressure drop resulting from the lateral and helical vortices. Another effect, the so-called chaotic advection, also impacts the flow and heat transfer in tortuous passages significantly. This term was first proposed by Aref [8] in his study of a ‘‘blinking vortex model’’ for describing a special flow regime in which the fluid particle trajectories became chaotic and fluid mixing was achieved very efficiently. The study was conducted for a two-dimensional, unsteady flow system and it was then proved theoretically that chaos could also be generated in a three-dimensional, steady flow.
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Nomenclature A AX c^p d dh e f h k Lz _ m Nu n Pr p Q q_ w Rc S S s s0 T t
area (m2) flow cross-sectional area (m2) mean specific heat capacity at constant pressure (J kg1 K1) channel diameter (m) hydraulic diameter (m) internal energy (J kg1) Fanning friction factor heat transfer coefficient (W m2 K1) or enthalpy (J kg1) thermal conductivity (W m1 K1) half-unit length (m) mass flow rate (kg s1) Nusselt number channel unit number Prandtl number pressure (Pa) volumetric flow rate (m3 s1) wall heat flux (W m2) radius of curvature (m) normalized axial location shear strain rate tensor (s1) channel axial location (m) channel axial path length in one unit (m) temperature (K) time (s)
Liu et al. [9] presented a three-dimensional serpentine microchannel design with a ‘‘C-shaped’’ repeating unit as a means of implementing chaotic advection to passively enhance fluid mixing – it was found that the flow in this channel could enhance fluid mixing 16-fold relative to a straight channel, even for a Reynolds number as low as 70. Stroock et al. [10] proposed bas-relief structures on the floor of the channel, and using this method the length of the channel required for flow mixing was reduced significantly relative to a simple smooth channel. The benefit of chaotic advection in achieving heat transfer enhancement has also been discussed. Acharya et al. [11] examined chaotic mixing as a means of enhancing the convective heat transfer in a modified helical coil and found it led not only to better mixing but also to an increased heat transfer rate. Peerhossaini et al. [12] conducted experiments on flow in a twisted curved channel and found chaotic advection was evident. A heat exchanger coil designed on the basis of the existence of chaotic structures showed 15–18% higher efficiency compared with helically coiled tube heat exchangers. Lasbet et al. [13] studied the heat transfer in a three-dimensional channel similar to that of Liu et al. [9] and found the convective heat transfer coefficient for water flow in this channel was about six times that of the equivalent straight channel for a Reynolds number of 200. In addition to the above studies for various three-dimensional geometries, heat transfer in planar-based geometries have drawn considerable interest recently, mainly because of the planar nature of various heat transfer devices, for example, plate type heat exchangers where very complex three-dimensional channels are impractical. However, Yamagishi et al. [14] indicated that planar channels did not generate chaotic advection as readily as twisted geometries. Aref [15] also pointed out that the fluid particle trajectories were still quite regular in a flat square-wave channel while some chaotic advection was evident in a three-dimensional serpentine channel at a Reynolds number of 50. Schönfeld and Hardt [16] then proposed a new concept by which chaotic advection could be induced by a simple meandering channel, relying on the
tR U u um V Vch y
fluid residence time (s) normalized velocity magnitude velocity vector (m s1) mean flow velocity magnitude (m s1) volume (m3) volume of the zigzag channel (m3) distance from an arbitrary axial plane to the channel flat wall (m)
Greek symbols h channel bend angle (°) H non-dimensional temperature ðT w T m ÞÞ l dynamic viscosity (Pa s) q fluid density (kg m3) s normalized simulation time Ds normalized time step X vorticity vector (s1)
ðH ¼ ðT T m Þ=
Subscripts m mean S value at the normalized axial location S. unit unit-based value w value for wall x, y value at the point (x, y)
switching between different flow patterns along the flow direction. Jiang et al. [17] investigated the flow in this planar-based channel and found the mixing performance is enhanced substantially relative to a straight channel. Sui et al. [18] gave another example using simple periodic wavy channels with rectangular cross-sections and found the generation of secondary flow (Dean vortices) and chaotic advection. We have previously studied steady fully-developed laminar flow and heat transfer in periodic channels with a variety of repeating unit paths, including serpentine [19–22], trapezoidal [23,24] and sinusoidal [25] channels of various geometrical configurations and cross-sections. It was found that Dean vortices that formed at the bends promoted fluid mixing transverse to the main flow direction and this led to significant heat transfer enhancement with relatively small pressure-drop penalty. A ‘‘wrapping’’ method was used in these studies to obtain the fully-developed flow in a single unit and a spatially periodic flow pattern was observed up to a Reynolds number of around 200, with the exact value depending on the geometry. With a further increase of the Reynolds number, the steady, laminar flow pattern became more complex and pronounced reverse flow and the disappearance of the spatial periodicity of the flow was evident, which meant that the wrapping method could no longer be used. A different computational methodology was developed [26] which uses a computational domain containing sufficient repeating units with smoothly joined inlet and outlet sections. Random fluctuations of unit-based pressure-drop and Nusselt number from one repeating unit to the next arose in a zigzag channel with a square cross-section for Reynolds number greater than 200 due to the pronounced chaotic particle trajectories. It was found that the fluctuations of unit-based values also occurred in zigzag channels with semi-circular cross-section over a large range of geometrical parameters [27]. With further increase of the Reynolds number, transient flow is expected to appear, however, studies in this particular flow regime
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are still very limited. Recently, Sui et al. [28] performed simulations of the transient flow and heat transfer in periodic wavy channels for Reynolds numbers ranging from 200 to 400 and calculated the heat transfer enhancement and pressure-drop penalty. They assumed the existence of spatial periodicity regardless of the Reynolds number, although it has been shown that this periodicity breaks down at high Reynolds numbers due to chaotic advection [26,27]. Therefore, a methodology for transient flow and heat transfer in tortuous passages allowing the study of transient, spatially non-periodic flow patterns is required. In this paper, a computational fluid dynamics methodology to study transient flow and heat transfer in repeating tortuous passages is proposed and applied to a semi-circular zigzag channel over an extended Reynolds number range. Fig. 1 shows a schematic of a unit cell of the zigzag channel under consideration. The geometry is fully characterised by the half-unit length, Lz; the diameter of the semi-circular cross-section, d; the radius of curvature of the bends, Rc; and the channel bend angle, h. After non-dimensionalisation, the geometry can be described by only three parameters, which are Lz/d, Rc/d and h. 2. Computational method 2.1. Conservation equations The fluid in the zigzag channel is assumed to be incompressible and Newtonian. The viscous energy dissipation is neglected. The fluid properties, such as density, dynamic viscosity, heat capacity and thermal conductivity are assumed to be constant. Therefore, the three-dimensional, time-dependent laminar flow in the zigzag channel can be described by the following conservation equations for mass, momentum and energy Continuity:
r ðquÞ ¼ 0
ð1Þ
Momentum:
@ðquÞ þ r ðqu uÞ ¼ rp þ r ðlðru þ ruT ÞÞ @t
ð2Þ
Energy:
@ðqeÞ þ r ðquhÞ ¼ r ðkrTÞ @t
ð3Þ
where q is the density, l is the dynamic viscosity and k is the thermal conductivity. u denotes the velocity vector; p, e, h and T denote the pressure, internal energy, enthalpy and temperature, respectively.
Fig. 2. Schematic of the computational domain at: (a) the inlet and (b) the outlet.
2.2. Computational domain and boundary conditions The computational domain consists of seven repeating zigzag units (n = 7) together with smoothly joined inlet and outlet sections, which is similar to that used in our previous studies [26,27]. A schematic of the computational domain is shown in Fig. 2. At the channel wall, a no-slip boundary condition is assumed. Two types of thermal boundary conditions are used in this study, the constant wall heat flux (H2) and constant wall temperature (T) boundary conditions. At the channel inlet, a plug flow velocity profile is applied and the flow rate is constant over time. At the channel outlet, a prescribed area-averaged pressure boundary condition is applied and its value is independent of time. 2.3. Solution methods and convergence criteria The conservation equations are solved using the finite-volume based computational fluid dynamics (CFD) code ANSYS CFX 14.5. A modified Rhie-Chow algorithm is used to link the pressure and velocity fields, which are solved via a coupled solver. The resulting algebraic equations are solved via an Algebraic Multigrid method. A second-order bounded differencing scheme is used for the convective terms and a second-order backward Euler scheme is used for the temporal discretization. During the time marching, the solution at each time step is regarded to have reached a converged state once all the scaled residuals fall below 106 and the global imbalances, representing overall conservation, fall below 103. As the simulations assume constant transport and thermal properties, an efficient means of solving the equations for multiple thermal solutions was devised. Using the additional transport equations that can be included in a simulation in ANSYS CFX it was possible to solve transport equations for energy conservation for multiple Prandtl numbers and different thermal boundary conditions simultaneously. This reduces the computational cost significantly and also means that the comparison of the different thermal conditions is performed for the identical flow field. 3. Results and discussion This computational methodology is used to investigate the transient flow and heat transfer in the semi-circular cross-section zigzag channel. For ease of presentation and comparison normalised variables were introduced. The definition of the Reynolds number is the same as that used in [26,27]. The normalized velocity magnitude is defined as
Fig. 1. Schematic of a repeating zigzag unit with a semi-circular cross-section; the geometric parameters of Rc/d = 0.51, Lz/d = 1.75 and h = 45° are set; the dashed line S represents the axial path of the passage and the total length of s for one single unit is s0. The symbol y denotes the distance from an arbitrary plane located parallel to the channel flat wall.
U¼
juj um
ð4Þ
where u is the fluid velocity vector and um is the mean flow velocity, calculated as Q/AX.
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The normalized simulation time is defined as
s¼
t tR
ð5Þ
where tR is the mean fluid residence time, defined as
tR ¼
V ch um AX
ð6Þ
where Vch is the volume of the channel and AX is the flow cross-sectional area. The channel axial location is also normalized, as follows
S¼
s so
ð7Þ
where s is the axial location and s0 is the axial path length for one unit. Therefore, S = 0 corresponds to the inlet plane of the first unit and S = 7 to the exit plane of the last unit. The peripherally-averaged heat transfer coefficient at a given S and s is defined as
hðS; sÞ ¼
qw ðS; sÞ ðT w ðS; sÞ T m ðS; sÞÞ
ð8Þ
w ðS; sÞ and T w ðS; sÞ are the peripherally-averaged wall heat where q flux and wall temperature at the given S and s, respectively; Tm(S, s) is the bulk mean fluid temperature at the given S and s, which is defined as
T m ðS; sÞ ¼
1 _ ^cp m
Z
qux;y ðS; sÞ^cp T x;y ðS; sÞdA
ð9Þ
_ is the mass flow rate through the channel, q and ^cp are the where m density and heat capacity of the fluid, respectively, and ux,y(S, s) and Tx,y(S, s) denote the magnitude of the component of the local velocity normal to the integration area and temperature at the point (x, y) of the cross-sectional plane at the given S and s. For the steady-state flow simulations, a unit-based heat transfer coefficient can be obtained by integrating h(S) along the axial path within a complete unit as given below
hunit ¼
Z
iþ1
hðSÞdS i ¼ 0; 1; 2; 3; 4; 5; 6
ð10Þ
Fig. 3. Ratio of the simplified definition of Nusselt number given in Eq. (11) for the H2 and Eq. (12) for the T boundary conditions to that from the formally correct definition given in Eq. (10) for transient simulations with Pr = 6.13. The error bars indicate the standard deviation among different units and times for each case.
The consequences of using the above definitions were tested using transient cases for 400 < Re < 800, Pr = 6.13 and several different times. Ratios of the Nusselt number calculated using the definitions in Eqs. (11) and (12) to those in Eq. (10) were calculated for different units and times for each case. The average values are shown in Fig. 3, together with error bars representing the standard deviation. Insignificant differences between the simplified and formally correct definition can be observed for the H2 boundary condition cases. For the T boundary condition cases, Eq. (12) tends to overestimate the unit-averaged value slightly, although the relative difference never exceeds 4%. The unit-based Fanning friction factor at a given s can be defined as
funit ðsÞ ¼
2 dh Dpunit ðsÞ s0
qu2m 4
ð14Þ
i
which in ANSYS CFX requires results to be written to a file and then be post-processed. However, this is not practical for transient simulations, because the above process would have to be employed for each time step. Simplified expressions are therefore used. For the H2 boundary condition, the unit-averaged heat transfer coefficient is defined as
hunit ðsÞ ¼
q_ w ðT w;unit ðsÞ T m;unit ðsÞÞ
ð11Þ
where q_ w is the wall heat flux, T w;unit is the unit-averaged wall temperature at the given s and Tm,unit(s) is the unit-averaged fluid temperature at the given s, which is defined as the average value of the unit inlet and outlet mean fluid temperatures at the given s. For the T boundary condition, the definition of the unit-averaged heat transfer coefficient is
hunit ðsÞ ¼
qw;unit ðsÞ DT lm ðsÞ
ð12Þ
where q_ w;unit ðsÞ is the area-averaged wall heat flux of the entire unit at the given s and DTlm(s) is the log-mean of the unit inlet and outlet wall-fluid temperature differences at the given s. Based on the above expressions, a unit-based Nusselt number for the H2 or T boundary condition at a given s can be defined as
Nuunit ðsÞ ¼
hunit ðsÞdh k
ð13Þ
where Dpunit(s) is the pressure-drop along the unit. 3.1. Grid and time step independence studies A swept hexahedral mesh with 3960 cross-sectional and 240 axial elements for each repeating zigzag unit has been shown previously to provide grid independent solutions up to a Reynolds number of 300 and Prandtl number of 20 [27]. To ensure the grid independence remains in the transient regime at higher Reynolds numbers, a test simulation using a refined cross-sectional mesh (6400 cross-sectional elements) was performed at Re = 800 and Pr = 20 for the H2 boundary condition. It was found that the averaged unit-based Nusselt number changed by 0.2%. Using a refined axial mesh (500 axial elements), the averaged unit-based Nusselt number was found to change by 0.9%. These results confirm the suitability of the earlier mesh for the simulation of transient flow and this mesh has therefore been employed in the current work. Simulations were run with three different time steps, Ds = 103, 4 10 and 2 105, for Re = 600. Fig. 4 shows a comparison of the normalized velocities at a monitor point on the cross-sectional plane of S = 6 as a function of the normalized time using different time steps. With Ds decreased from 104 to 2 105, almost no difference can be found in the velocity magnitudes, indicating that a normalised time step of 104 is sufficiently small to resolve the transient nature of the flow correctly. This value was used for all reported simulations.
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Fig. 4. Normalized velocity magnitudes as a function of the normalized time at a monitor point on the cross-sectional plane of S = 6 and Re = 600 for different time step sizes.
located at the centres of the unit exit planes at S = 1 to 7, respectively. Simulations were run for a sufficient period of time to collect meaningful statistics and data were only extracted after a normalized time of 1.5 (1.5 times the residence time), in order to eliminate any residual effects of the initial conditions. Normalized velocity magnitudes at these points were recorded at each time step, as shown in Fig. 5 for Re = 600. At the first and second points, the velocity magnitudes are found to be constant, indicating the flow pattern remains steady in the first two zigzag units. The flow then develops to a transient state in the third unit, as shown by the temporally varying velocity magnitude at the third monitor point. Transient flow persists downstream but the temporal variation, apparently periodic at S = 3, becomes completely irregular at S = 4 and beyond, as shown in Fig. 5(b). To further understand these transient results, power spectrum analysis was employed on the velocity signals at the points on S = 3 to 7 using the Fast Fourier Transform (FFT) capability of ANSYS CFD Post using the Hanning window. Results are shown in Fig. 6, where the frequency range plotted has a cut-off that is approximately equal to the sample Nyquist frequency, ensuring that no aliasing is present in the plotted data. For the power spectrum at S = 3, the peak at around 200 Hz clearly corresponds to the regular velocity fluctuations apparent in Fig. 5(a); the power density falls off sharply at higher frequencies and is always much lower than corresponding power densities at the later monitor points. Starting from the fourth point, the power spectra become continuous (broadband) and apparently do not change further. These results indicate that the flow has converted from a steady pattern at the channel inlet, to a ‘‘developed’’ transient state after around three zigzag units at Re = 600. The special behaviour in the third unit can be regarded as an intermediate state between the steady and a ‘‘developed’’ transient flow pattern. In Fig. 6, a dashed line representing the 5/3 law of Kolmogorov-Obukhov [29] for turbulent flow is also presented for comparison. (Note the location of the line is arbitrary.) The transient flow spectra in this study do not follow the turbulent law, as would be expected given the relatively low Reynolds number. In fact the decay of the spectrum is faster than the 5/3 law as we do not have an inertial region where viscous effects are negligible, instead they are important across most of the spectrum and result in the rapid destruction of small vortices.
Fig. 5. Normalized velocity magnitude as a function of the normalized time at the monitor points located on the unit exit planes of S = 1 to 3 (a) and 4 to 7 (b) for Re = 600.
3.2. Transient behaviour To investigate the transient flow behaviour, seven monitor points were placed evenly along the computational domain,
Fig. 6. Power spectra of velocity magnitude at the third to seventh monitor point for Re = 600. The dashed straight line has a slope of 5/3 and is shown only for comparison purposes.
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Fig. 7. Power spectra of the velocity magnitudes at the monitor point at S = 6 for different Reynolds numbers.
The temporal variations of velocity magnitudes have also been monitored at other Reynolds numbers. Their spectra at the monitor point at S = 6, together with the 5/3 law, are shown in Fig. 7. It is interesting to note that the spectra for three different Reynolds number are almost identical but as the Reynolds number increases the curve is shifted to the right. This shift results from the increase in resolved small-scale structure that is present with increasing Reynolds number due to reduced viscous damping. Overall, the above analysis indicates that beyond a certain Reynolds number (around Re = 400 for this geometry), a flow regime with spatially and temporally chaotic flow patterns appears in the repeating zigzag channel and in this regime, only a few repeating units are required for the flow to achieve a reasonably stationary state. Statistical properties of the flow are invariant in this flow regime, regardless of the Reynolds number and axial location, which belongs to neither conventional laminar nor turbulent flow.
For the unit-based Fanning friction factor, the mean value is defined as
3.3. Heat transfer enhancement and pressure-drop penalty To investigate the heat transfer and pressure-drop performance of the spatially and temporally chaotic flow, the unit-based Nusselt number and Fanning friction factor in this flow regime were monitored at each time step. The results for Re = 600, Pr = 6.13 using the H2 boundary condition are shown in Fig. 8. For each of the units within the ‘‘developed’’ region, the unit-based Nusselt number and friction factor fluctuate over time and their values for a certain unit and simulation time seem to be completely random. However, time-averaged values tend to be very close to each other from one unit to another. To quantify the overall heat transfer and pressure-drop characteristics in each of the units, these time-dependent values are averaged over time. For the unit-based Nusselt number, its mean value is defined as
Nuunit ¼
Z s2 1 Nuunit ðsÞds s2 s1 s1
Fig. 8. The unit-based Nusselt number (a) and Fanning friction factor (b) as functions of the normalized time for Re = 600, Pr = 6.13 and the H2 boundary condition.
ð15Þ
where s1 and s2 are the start and end points of the integration: here, s1 = 1.5 and s2 = 1.8 are used. A longer time interval with s2 = 2.5 was tested for the case of Re = 600 and no significant change was observed, indicating the current time interval is sufficient for achieving time-independent values.
f unit ¼
Z s2 1 f ðsÞds s2 s1 s1 unit
ð16Þ
In Eqs. (15) and (16), their standard deviations over the integration range can be used to describe the amplitudes of the time-dependent fluctuations. Fig. 9 shows the mean unit-based Nusselt numbers and friction factors for different zigzag units at Re = 400, 600 and 800, Pr = 6.13 and the H2 boundary condition. It can be observed that at each of the Reynolds numbers, Nuunit and f unit for different units become very close to each other after the fourth unit, with the relative differences all below 5%. These results confirm that in this transient chaotic flow regime, the unit-based heat transfer and pressuredrop performance become almost identical from one unit to another, so the overall performance of the entire channel is able to be evaluated based on values in a single unit. Henceforth, in this study, the data for the sixth unit are used to evaluate the overall performance for this geometry. Compared with the equivalent straight channel, the overall heat transfer enhancement (eNu) and pressure-drop penalty (ef) can be defined as follows
eNu ¼
Nuunit ; Nust
ef ¼
f unit fst
ð17Þ
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simulations, the heat transfer enhancement and pressure-drop penalty over a further extended Reynolds number range are obtained, as shown in Fig. 10. For Re < 200, the flow field is in the steady periodic state, where the heat transfer enhancement and pressure-drop penalty are identical from one unit to the next; in the approximate range of 200 < Re < 340, the flow field is still steady, but the flow periodicity is no longer seen and both eNu and ef fluctuate between units with the mean values and standard deviations (error bars) shown in Fig. 10; at around Re > 340, transient behaviour is present and this causes eNu and ef to fluctuate temporally with time-averaged values between units remaining very similar and the relatively small standard deviations shown in Fig. 10. Over the entire flow regime under investigation, both eNu and ef always increase with increasing Reynolds number, with eNu greater than ef at the studied Prandtl number of 6.13. This indicates that with the flow evolving into the transient state, the heat transfer performance in the zigzag channel can be improved continuously, although this is accompanied by a moderate pressure-drop increase. 3.4. Flow pattern and vortex structures
Fig. 9. Nuunit (a) and f unit (b) for different zigzag units at Re = 400, 600 and 800, Pr = 6.13 using the H2 boundary condition.
12
eNu
10
Steady
eNu , ef
8
ef 6
Spatially Periodic
4
2
0
Chaotic
0
200
400
600
800
Re Fig. 10. Heat transfer enhancement and pressure-drop penalty as functions of the Reynolds number in the zigzag channel for a Prandtl number of 6.13 and an H2 boundary condition. The symbols N, s and . denote data in the steady and spatially-periodic, steady chaotic and transient regimes, respectively. The error bars for steady flow data indicate the standard deviations of values for different units, and those for transient flow data indicate the standard deviations over the integration time range.
where Nust and fst are the Nusselt number and friction factor for fully-developed flow in straight channels, and Nuunit and f unit are based on a single zigzag unit. Combining the current results with those for the steady flow in [27], which were re-evaluated using the same definition of unit-based Nusselt number as for transient
To further study the transient heat transfer and pressure-drop characteristics of the zigzag channel, flow fields in the fourth to seventh units are presented in Fig. 11. At s = 1.5 (Fig. 11(a)), the flow pattern is found to be very complex with many small vortices located randomly on the displayed plane (y = 0.1d), which can be seen from the curly structure of the instantaneous streamlines (traced through the velocity field at a given time instant). Significant pressure-drop is caused by these vortical structures, although it is also expected that they disrupt the development of the thermal boundary layer, therefore leading to an enhanced heat transfer rate. At s = 1.6 (Fig. 11(b)), it is found that the flow pattern is very different from that at s = 1.5. There are many obvious changes to the flow field, including the movement and destruction of the vortices present initially, as well as the formation of new ones. These time-varying flow features give rise to intensified fluid mixing, leading to enhancement of the heat transfer rate. The cross-sectional flow and temperature fields have also been investigated and the secondary flow vectors and non-dimensional temperature contours on the planes located at S = 5, 5.5, 6 and 6.5 are presented in Fig. 12. Significant secondary flow can be observed with various vortex structures on each of the cross-sectional planes and additionally, at different cross-sectional planes, the secondary flow pattern varies dramatically. These flow features cause efficient cross-sectional fluid mixing and enhance the heat transfer rate significantly, as indicated by the corresponding temperature profiles, which tend to be very uniform. Hunt [30] proposed the Q criterion, the second invariant of the velocity gradient tensor, to detect vortex regions in transient flow. The quantity Q can be expressed as
Q¼
1 ðjXj2 jSj2 Þ 2
ð18Þ
where X and S are the vorticity and shear strain rate tensors, respectively. Values of Q > 0 occur in vortices while, on the vortex surface, Q = 0. To visualize the variation of the vortex strength at different Reynolds numbers, an isosurface of Q = 3 106 s2 (its value is arbitrary and is chosen to highlight the vortex structure) is presented in Fig. 13. It shows that with increasing Reynolds number, the vortex strength is increased significantly which leads to enhanced cross-stream mixing. Additionally, with an increase of the Reynolds number, vortices with smaller length-scales are generated. Compared with the large ones, these small vortices are shorter lived therefore causing the flow to be well mixed, resulting in
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Fig. 11. Flow patterns with surface streamlines for the fourth to seventh units on a plane of y = 0.1 d at s = 1.5 (a) and 1.6 (b) for Re = 600.
Fig. 12. Secondary flow vectors (left) and non-dimensional temperature profiles ðH ¼ ðT T m Þ=ðT w T m Þ (right) on the cross-sectional planes of S = 5, 5.5, 6 and 6.5 at Re = 600.
enhanced heat transfer. This behaviour is consistent with that expected on the basis of the power spectra shown in Fig. 7, which showed a significant increase in power at all frequencies as the Reynolds number increased through the range 400 to 800.
3.5. Effect of the Prandtl number The heat transfer enhancement as a function of the Reynolds number for different Prandtl numbers is shown in Fig. 14. It is clear
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Fig. 13. Isosurface of Q = 3 106 s2 for the fourth to seventh units at Reynolds numbers of (a) 400, (b) 600 and (c) 800.
Fig. 14. Heat transfer enhancements at different Prandtl numbers as functions of the Reynolds number for the zigzag channel. The symbols N, s and . denote data in the spatially periodic, steady and transient regimes, respectively.
Fig. 16. Nusselt numbers as functions of the Reynolds number in the zigzag channel for the H2 and T boundary conditions. The symbols N, s and . denote data in the spatially periodic, steady and transient regimes, respectively.
Fig. 15. Double-logarithmic plot of eNu versus Pr at different Reynolds numbers.
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that no matter whether the flow regime is steady or transient, the Prandtl number has an obvious effect on the heat transfer enhancement. With an increase of the Prandtl number, a greater heat transfer enhancement is achieved at a given Reynolds number. As the thermal boundary layer decreases in thickness with increasing Prandtl number, it is more easily disrupted by the flow, explaining the increased enhancement at higher Prandtl numbers. In the nonperiodic steady flow regime, a proportional dependence between Pr1/3 of the Nusselt number was found previously [27], over the Reynolds number range from 200 to 320. To examine whether this relationship still holds for a higher Reynolds number, ln (eNu) as a function of ln (Pr) is plotted over an extended range of Reynolds number, as shown in Fig. 15. A linear relationship with identical slopes can be observed for 200 < Re < 800, indicating that the Pr1/3 dependence is valid in both the steady and transient flow regimes.
3.6. Effect of the thermal boundary condition The Nusselt number for a straight channel is dependent on the applied thermal boundary condition in the laminar flow regime. It is interesting to examine the dependence of this boundary condition for the zigzag channel. Fig. 16 shows the Nusselt number as a function of the Reynolds number for the H2 and T boundary conditions for a Prandtl number of 6.13. It is found that compared with that for the H2 boundary condition, the Nusselt number for the T boundary condition is always higher across the entire Reynolds number range. In the low Reynolds number laminar regime having a steady, spatially-periodic flow pattern, the ratio is nearly constant at around 1.2. With the flow evolving into the steady, nonperiodic (chaotic) regime, the ratio becomes greater and then decreases with increasing Reynolds number. In the transient flow regime, the difference is generally diminishing with increasing Reynolds number. It is expected that with a further increase of the Reynolds number towards the turbulent regime, the difference between the two curves will disappear.
4. Conclusions In this paper, a computational fluid dynamics based methodology has been developed to study the transient, laminar flow and heat transfer in various periodic tortuous passages. The flow is shown to develop in a long channel with sufficient repeating units. This ‘‘developed’’ flow pattern is reached in only a few units (four in this case). It is characterised by the occurrence of spatially and temporally chaotic flow patterns as well as the similar properties in different units, such as the velocity power spectrum, heat transfer and pressure-drop performances. The methodology has been used to study the flow and heat transfer in a periodic zigzag channel with a semi-circular crosssection. A Reynolds number range of 400 to 800 and Prandtl numbers ranging from 0.7 to 20 have been considered. The numerical results show that in the transient flow regime, the fluid velocity in the zigzag channel always fluctuates randomly, and has a broadband power spectrum. Statistical analysis based on data over a sufficient period of time has shown that compared with a straight channel, significant heat transfer enhancement can be observed in the transient flow regime, which is accompanied by an increase in pressure-drop. Both eNu and ef are higher than those in the steady flow regime and they increase with increasing Reynolds number. Flow pattern and vortex regions have been visualized at different simulation times and Reynolds numbers. With increasing Reynolds number, vortices with smaller length-scale are generated.
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Significant heat transfer enhancement is achieved due to these flow structures. The effect of the Prandtl number on the heat transfer enhancement has been examined for an extended Reynolds number range and it has been found that the Pr1/3 dependence, shown to occur in the steady regime for 200 < Re < 320, remains valid in the transient regime. The effect of the thermal boundary condition has also been investigated, and it has been shown that the Nusselt number for the T boundary condition is always higher than that for the H2 boundary condition (Re < 800). Acknowledgements Zhanying Zheng acknowledges the financial support from Chinese Scholarship Council and the University of Sydney for his Ph.D. study. The work has been supported by the Australian Research Council Grant DP0985453. References [1] W.R. Dean, Note on the notion of fluid in a curved pipe, Philos. Mag. 4 (20) (1927) 208–223. [2] K.C. Cheng, M. Akiyama, Laminar forced convection heat transfer in curved rectangular channels, Int. J. Heat Mass Transfer 13 (3) (1970) 471–490. [3] C.E. Kalb, J.D. Seader, Heat and mass-transfer phenomena for viscous flow in curved circular tubes, Int. J. Heat Mass Transfer 15 (4) (1972) 801–817. [4] L. Wang, F. Liu, Forced convection in slightly curved microchannels, Int. J. Heat Mass Transfer 50 (5–6) (2007) 881–896. [5] R.M. Manglik, J.H. Zhang, A. Muley, Low Reynolds number forced convection in three-dimensional wavy-plate-fin compact channels: fin density effects, Int. J. Heat Mass Transfer 48 (8) (2005) 1439–1449. [6] J.H. Zhang, J. Kundu, R.M. Manglik, Effect of fin waviness and spacing on the lateral vortex structure and laminar heat transfer in wavy-plate-fin cores, Int. J. Heat Mass Transfer 47 (8–9) (2004) 1719–1730. [7] H.M. Metwally, R.M. Manglik, Enhanced heat transfer due to curvatureinduced lateral vortices in laminar flows in sinusoidal corrugated-plate channels, Int. J. Heat Mass Transfer 47 (10–11) (2004) 2283–2292. [8] H. Aref, Stirring by chaotic advection, J. Fluid Mech. 143 (1984) 1–21. [9] R.H. Liu, M.A. Stremler, K.V. Sharp, M.G. Olsen, J.G. Santiago, R.J. Adrian, H. Aref, D.J. Beebe, Passive mixing in a three-dimensional serpentine microchannel, J. Microelectromech. Syst. 9 (2) (2000) 190–197. [10] A.D. Stroock, S.K.W. Dertinger, A. Ajdari, I. Mezic, H.A. Stone, G.M. Whitesides, Chaotic mixer for microchannels, Science 295 (5555) (2002) 647–651. [11] N. Acharya, M. Sen, H.C. Chang, Heat-transfer enhancement in coiled tubes by chaotic mixing, Int. J. Heat Mass Transfer 35 (10) (1992) 2475–2489. [12] H. Peerhossaini, C. Castelain, Y. Leguer, Heat-exchanger design based on chaotic advection, Exp. Therm. Fluid Sci. 7 (4) (1993) 333–344. [13] Y. Lasbet, B. Auvity, C. Castelain, H. Peerhossaini, A chaotic heat-exchanger for PEMFC cooling applications, J. Power Sources 156 (1) (2006) 114–118. [14] A. Yamagishi, T. Inaba, Y. Yamaguchi, Chaotic analysis of mixing enhancement in steady laminar flows through multiple pipe bends, Int. J. Heat Mass Transfer 50 (7–8) (2007) 1238–1247. [15] H. Aref, The development of chaotic advection, Phys. Fluids 14 (4) (2002) 1315–1325. [16] F. Schonfeld, S. Hardt, Simulation of helical flows in microchannels, AIChE J. 50 (4) (2004) 771–778. [17] F. Jiang, K.S. Drese, S. Hardt, M. Kupper, F. Schonfeld, Helical flows and chaotic mixing in curved micro channels, AIChE J. 50 (9) (2004) 2297–2305. [18] Y. Sui, C.J. Teo, P.S. Lee, Y.T. Chew, C. Shu, Fluid flow and heat transfer in wavy microchannels, Int. J. Heat Mass Transfer 53 (13–14) (2010) 2760–2772. [19] N.R. Rosaguti, D.F. Fletcher, B.S. Haynes, Laminar flow and heat transfer in a periodic serpentine channel with semi-circular cross-section, Int. J. Heat Mass Transfer 49 (17–18) (2006) 2912–2923. [20] P.E. Geyer, N.R. Rosaguti, D.F. Fletcher, B.S. Haynes, Thermohydraulics of square-section microchannels following a serpentine path, Microfluid. Nanofluid. 2 (3) (2006) 195–204. [21] P.E. Geyer, N.R. Rosaguti, D.F. Fletcher, B.S. Haynes, Laminar flow and heat transfer in periodic serpentine mini-channels, J. Enhanced Heat Transfer 13 (4) (2006) 309–320. [22] N.R. Rosaguti, D.F. Fletcher, B.S. Haynes, Laminar flow and heat transfer in a periodic serpentine channel, Chem. Eng. Technol. 28 (3) (2005) 353–361. [23] R. Gupta, P.E. Geyer, D.F. Fletcher, B.S. Haynes, Thermohydraulic performance of a periodic trapezoidal channel with a triangular cross-section, Int. J. Heat Mass Transfer 51 (11–12) (2008) 2925–2929. [24] P.E. Geyer, D.F. Fletcher, B.S. Haynes, Laminar flow and heat transfer in a periodic trapezoidal channel with semi-circular cross-section, Int. J. Heat Mass Transfer 50 (17–18) (2007) 3471–3480.
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[25] N.R. Rosaguti, D.F. Fletcher, B.S. Haynes, Low-Reynolds number heat transfer enhancement in sinusoidal channels, Chem. Eng. Sci. 62 (3) (2007) 694–702. [26] Z. Zheng, D.F. Fletcher, B.S. Haynes, Chaotic advection in steady laminar heat transfer simulations: periodic zigzag channels with square cross-sections, Int. J. Heat Mass Transfer 57 (1) (2013) 274–284. [27] Z. Zheng, D.F. Fletcher, B.S. Haynes, Laminar heat transfer simulations for periodic zigzag semicircular channels: chaotic advection and geometric effects, Int. J. Heat Mass Transfer 62 (1) (2013) 391–401.
[28] Y. Sui, C.J. Teo, P.S. Lee, Direct numerical simulation of fluid flow and heat transfer in periodic wavy channels with rectangular cross-sections, Int. J. Heat Mass Transfer 55 (1–3) (2012) 73–88. [29] U. Frisch, Turbulence. The Legacy of A. N. Kolmogorov, Cambridge University Press, 1995. [30] J.C.R. Hunt, Vorticity and vortex dynamics in complex turbulent flows, Trans. Can. Soc. Mech. Eng. 11 (1) (1987) 21–35.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Transient laminar heat transfer simulations in periodic zigzag channels Zhanying Zheng, David F. Fletcher ⇑, Brian S. Haynes School of Chemical and Biomolecular Engineering, The University of Sydney, NSW 2006, Australia
a r t i c l e
i n f o
Article history: Received 8 July 2013 Received in revised form 12 November 2013 Accepted 24 December 2013 Available online 25 January 2014 Keywords: Heat transfer enhancement Nusselt number Friction factor Tortuous passage Microchannel CFD
a b s t r a c t At sufficiently high Reynolds numbers, laminar flows in tortuous channels become unsteady. A computational fluid dynamics methodology is developed to study transient, laminar flow and heat transfer in a periodic zigzag channel with a semi-circular cross-section. The computational domain consists of seven repeating zigzag units with smoothly joined inlet and outlet sections. Reynolds numbers ranging from 400 to 800 and Prandtl numbers ranging from 0.7 to 20 are examined for constant wall heat flux and constant temperature thermal boundary conditions. Simulation results show that the flow reaches a ‘‘developed’’ state after around three units, where the local velocities fluctuate with time but give well defined average heat transfer rates and pressure loss. The power spectra of the velocity at monitor points located periodically along the channel also become very similar. Significant heat transfer enhancement is observed in the transient regime studied, which is accompanied by a modest pressure-drop penalty, both of which increase with increasing Reynolds number. Vortex structures are visualized at different simulation times and Reynolds numbers and it is found that with increasing Reynolds number, vortices with smaller length-scale are generated, which contribute largely to the enhancement of heat transfer. The results for different Prandtl numbers show that the heat transfer enhancement is proportional to Pr1/3. The Nusselt number for the T boundary condition is found to be always higher than that for the H2 boundary condition. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Due to the great potential of achieving smaller, lighter-weight and lower-cost devices, the interest in the miniaturisation of various equipment can be seen in many industrial applications, such as electronics cooling, solar heat collectors, micro-chemical reactors and compact heat exchangers. These involve the use of micro or mini-scaled channels with the characteristic length ranging from hundreds of microns to a few millimetres. Compared with conventional-sized channels, the Reynolds number in these systems is generally too low to support turbulence, so various tortuous geometries are often used, in which the thermal boundary layer is constantly disrupted, giving rise to heat transfer enhancement relative to flow in straight pipes. The enhancement of heat transfer in tortuous passages is largely attributed to the establishment of secondary flows. Dean [1] studied flow in a coiled pipe with constant curvature theoretically and observed secondary flow patterns having a vortices pair structure in the cross-sectional plane, which are now called Dean vortices. Similar flow patterns can be found in various other channels with tortuous features, which promote cross-stream mixing ⇑ Corresponding author. Address: Chiu Building J14, The University of Sydney, NSW 2006, Australia. Tel.: +61 2 9351 4147; fax: +61 2 9351 3455. E-mail address: david.fl[email protected] (D.F. Fletcher). 0017-9310/$ - see front matter Ó 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.12.056
without incurring large increases in pressure drop, leading to relatively more uniform temperature profiles in the channel cross-section than a straight channel. Cheng and Akiyama [2] presented flow and heat transfer results in curved rectangular channels and observed the vortex pair structure in the secondary flow field. Kalb and Seader [3] studied numerically the fully-developed heat transfer of viscous flow in curved circular tubes and quantified the effect of Dean vortices on the pressure loss and heat transfer. Wang and Liu [4] further reported a variety of secondary flow structures for different flow configurations in slightly curved microchannels of square cross-section and their results showed that the Nusselt number increased significantly, although this was accompanied by a moderate increase in the friction factor. Apart from the curved channels, Manglik and co-workers [5–7] performed a series of computational studies for flow in planar wavy channels and found enhancement of heat transfer and increase of pressure drop resulting from the lateral and helical vortices. Another effect, the so-called chaotic advection, also impacts the flow and heat transfer in tortuous passages significantly. This term was first proposed by Aref [8] in his study of a ‘‘blinking vortex model’’ for describing a special flow regime in which the fluid particle trajectories became chaotic and fluid mixing was achieved very efficiently. The study was conducted for a two-dimensional, unsteady flow system and it was then proved theoretically that chaos could also be generated in a three-dimensional, steady flow.
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Nomenclature A AX c^p d dh e f h k Lz _ m Nu n Pr p Q q_ w Rc S S s s0 T t
area (m2) flow cross-sectional area (m2) mean specific heat capacity at constant pressure (J kg1 K1) channel diameter (m) hydraulic diameter (m) internal energy (J kg1) Fanning friction factor heat transfer coefficient (W m2 K1) or enthalpy (J kg1) thermal conductivity (W m1 K1) half-unit length (m) mass flow rate (kg s1) Nusselt number channel unit number Prandtl number pressure (Pa) volumetric flow rate (m3 s1) wall heat flux (W m2) radius of curvature (m) normalized axial location shear strain rate tensor (s1) channel axial location (m) channel axial path length in one unit (m) temperature (K) time (s)
Liu et al. [9] presented a three-dimensional serpentine microchannel design with a ‘‘C-shaped’’ repeating unit as a means of implementing chaotic advection to passively enhance fluid mixing – it was found that the flow in this channel could enhance fluid mixing 16-fold relative to a straight channel, even for a Reynolds number as low as 70. Stroock et al. [10] proposed bas-relief structures on the floor of the channel, and using this method the length of the channel required for flow mixing was reduced significantly relative to a simple smooth channel. The benefit of chaotic advection in achieving heat transfer enhancement has also been discussed. Acharya et al. [11] examined chaotic mixing as a means of enhancing the convective heat transfer in a modified helical coil and found it led not only to better mixing but also to an increased heat transfer rate. Peerhossaini et al. [12] conducted experiments on flow in a twisted curved channel and found chaotic advection was evident. A heat exchanger coil designed on the basis of the existence of chaotic structures showed 15–18% higher efficiency compared with helically coiled tube heat exchangers. Lasbet et al. [13] studied the heat transfer in a three-dimensional channel similar to that of Liu et al. [9] and found the convective heat transfer coefficient for water flow in this channel was about six times that of the equivalent straight channel for a Reynolds number of 200. In addition to the above studies for various three-dimensional geometries, heat transfer in planar-based geometries have drawn considerable interest recently, mainly because of the planar nature of various heat transfer devices, for example, plate type heat exchangers where very complex three-dimensional channels are impractical. However, Yamagishi et al. [14] indicated that planar channels did not generate chaotic advection as readily as twisted geometries. Aref [15] also pointed out that the fluid particle trajectories were still quite regular in a flat square-wave channel while some chaotic advection was evident in a three-dimensional serpentine channel at a Reynolds number of 50. Schönfeld and Hardt [16] then proposed a new concept by which chaotic advection could be induced by a simple meandering channel, relying on the
tR U u um V Vch y
fluid residence time (s) normalized velocity magnitude velocity vector (m s1) mean flow velocity magnitude (m s1) volume (m3) volume of the zigzag channel (m3) distance from an arbitrary axial plane to the channel flat wall (m)
Greek symbols h channel bend angle (°) H non-dimensional temperature ðT w T m ÞÞ l dynamic viscosity (Pa s) q fluid density (kg m3) s normalized simulation time Ds normalized time step X vorticity vector (s1)
ðH ¼ ðT T m Þ=
Subscripts m mean S value at the normalized axial location S. unit unit-based value w value for wall x, y value at the point (x, y)
switching between different flow patterns along the flow direction. Jiang et al. [17] investigated the flow in this planar-based channel and found the mixing performance is enhanced substantially relative to a straight channel. Sui et al. [18] gave another example using simple periodic wavy channels with rectangular cross-sections and found the generation of secondary flow (Dean vortices) and chaotic advection. We have previously studied steady fully-developed laminar flow and heat transfer in periodic channels with a variety of repeating unit paths, including serpentine [19–22], trapezoidal [23,24] and sinusoidal [25] channels of various geometrical configurations and cross-sections. It was found that Dean vortices that formed at the bends promoted fluid mixing transverse to the main flow direction and this led to significant heat transfer enhancement with relatively small pressure-drop penalty. A ‘‘wrapping’’ method was used in these studies to obtain the fully-developed flow in a single unit and a spatially periodic flow pattern was observed up to a Reynolds number of around 200, with the exact value depending on the geometry. With a further increase of the Reynolds number, the steady, laminar flow pattern became more complex and pronounced reverse flow and the disappearance of the spatial periodicity of the flow was evident, which meant that the wrapping method could no longer be used. A different computational methodology was developed [26] which uses a computational domain containing sufficient repeating units with smoothly joined inlet and outlet sections. Random fluctuations of unit-based pressure-drop and Nusselt number from one repeating unit to the next arose in a zigzag channel with a square cross-section for Reynolds number greater than 200 due to the pronounced chaotic particle trajectories. It was found that the fluctuations of unit-based values also occurred in zigzag channels with semi-circular cross-section over a large range of geometrical parameters [27]. With further increase of the Reynolds number, transient flow is expected to appear, however, studies in this particular flow regime
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are still very limited. Recently, Sui et al. [28] performed simulations of the transient flow and heat transfer in periodic wavy channels for Reynolds numbers ranging from 200 to 400 and calculated the heat transfer enhancement and pressure-drop penalty. They assumed the existence of spatial periodicity regardless of the Reynolds number, although it has been shown that this periodicity breaks down at high Reynolds numbers due to chaotic advection [26,27]. Therefore, a methodology for transient flow and heat transfer in tortuous passages allowing the study of transient, spatially non-periodic flow patterns is required. In this paper, a computational fluid dynamics methodology to study transient flow and heat transfer in repeating tortuous passages is proposed and applied to a semi-circular zigzag channel over an extended Reynolds number range. Fig. 1 shows a schematic of a unit cell of the zigzag channel under consideration. The geometry is fully characterised by the half-unit length, Lz; the diameter of the semi-circular cross-section, d; the radius of curvature of the bends, Rc; and the channel bend angle, h. After non-dimensionalisation, the geometry can be described by only three parameters, which are Lz/d, Rc/d and h. 2. Computational method 2.1. Conservation equations The fluid in the zigzag channel is assumed to be incompressible and Newtonian. The viscous energy dissipation is neglected. The fluid properties, such as density, dynamic viscosity, heat capacity and thermal conductivity are assumed to be constant. Therefore, the three-dimensional, time-dependent laminar flow in the zigzag channel can be described by the following conservation equations for mass, momentum and energy Continuity:
r ðquÞ ¼ 0
ð1Þ
Momentum:
@ðquÞ þ r ðqu uÞ ¼ rp þ r ðlðru þ ruT ÞÞ @t
ð2Þ
Energy:
@ðqeÞ þ r ðquhÞ ¼ r ðkrTÞ @t
ð3Þ
where q is the density, l is the dynamic viscosity and k is the thermal conductivity. u denotes the velocity vector; p, e, h and T denote the pressure, internal energy, enthalpy and temperature, respectively.
Fig. 2. Schematic of the computational domain at: (a) the inlet and (b) the outlet.
2.2. Computational domain and boundary conditions The computational domain consists of seven repeating zigzag units (n = 7) together with smoothly joined inlet and outlet sections, which is similar to that used in our previous studies [26,27]. A schematic of the computational domain is shown in Fig. 2. At the channel wall, a no-slip boundary condition is assumed. Two types of thermal boundary conditions are used in this study, the constant wall heat flux (H2) and constant wall temperature (T) boundary conditions. At the channel inlet, a plug flow velocity profile is applied and the flow rate is constant over time. At the channel outlet, a prescribed area-averaged pressure boundary condition is applied and its value is independent of time. 2.3. Solution methods and convergence criteria The conservation equations are solved using the finite-volume based computational fluid dynamics (CFD) code ANSYS CFX 14.5. A modified Rhie-Chow algorithm is used to link the pressure and velocity fields, which are solved via a coupled solver. The resulting algebraic equations are solved via an Algebraic Multigrid method. A second-order bounded differencing scheme is used for the convective terms and a second-order backward Euler scheme is used for the temporal discretization. During the time marching, the solution at each time step is regarded to have reached a converged state once all the scaled residuals fall below 106 and the global imbalances, representing overall conservation, fall below 103. As the simulations assume constant transport and thermal properties, an efficient means of solving the equations for multiple thermal solutions was devised. Using the additional transport equations that can be included in a simulation in ANSYS CFX it was possible to solve transport equations for energy conservation for multiple Prandtl numbers and different thermal boundary conditions simultaneously. This reduces the computational cost significantly and also means that the comparison of the different thermal conditions is performed for the identical flow field. 3. Results and discussion This computational methodology is used to investigate the transient flow and heat transfer in the semi-circular cross-section zigzag channel. For ease of presentation and comparison normalised variables were introduced. The definition of the Reynolds number is the same as that used in [26,27]. The normalized velocity magnitude is defined as
Fig. 1. Schematic of a repeating zigzag unit with a semi-circular cross-section; the geometric parameters of Rc/d = 0.51, Lz/d = 1.75 and h = 45° are set; the dashed line S represents the axial path of the passage and the total length of s for one single unit is s0. The symbol y denotes the distance from an arbitrary plane located parallel to the channel flat wall.
U¼
juj um
ð4Þ
where u is the fluid velocity vector and um is the mean flow velocity, calculated as Q/AX.
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The normalized simulation time is defined as
s¼
t tR
ð5Þ
where tR is the mean fluid residence time, defined as
tR ¼
V ch um AX
ð6Þ
where Vch is the volume of the channel and AX is the flow cross-sectional area. The channel axial location is also normalized, as follows
S¼
s so
ð7Þ
where s is the axial location and s0 is the axial path length for one unit. Therefore, S = 0 corresponds to the inlet plane of the first unit and S = 7 to the exit plane of the last unit. The peripherally-averaged heat transfer coefficient at a given S and s is defined as
hðS; sÞ ¼
qw ðS; sÞ ðT w ðS; sÞ T m ðS; sÞÞ
ð8Þ
w ðS; sÞ and T w ðS; sÞ are the peripherally-averaged wall heat where q flux and wall temperature at the given S and s, respectively; Tm(S, s) is the bulk mean fluid temperature at the given S and s, which is defined as
T m ðS; sÞ ¼
1 _ ^cp m
Z
qux;y ðS; sÞ^cp T x;y ðS; sÞdA
ð9Þ
_ is the mass flow rate through the channel, q and ^cp are the where m density and heat capacity of the fluid, respectively, and ux,y(S, s) and Tx,y(S, s) denote the magnitude of the component of the local velocity normal to the integration area and temperature at the point (x, y) of the cross-sectional plane at the given S and s. For the steady-state flow simulations, a unit-based heat transfer coefficient can be obtained by integrating h(S) along the axial path within a complete unit as given below
hunit ¼
Z
iþ1
hðSÞdS i ¼ 0; 1; 2; 3; 4; 5; 6
ð10Þ
Fig. 3. Ratio of the simplified definition of Nusselt number given in Eq. (11) for the H2 and Eq. (12) for the T boundary conditions to that from the formally correct definition given in Eq. (10) for transient simulations with Pr = 6.13. The error bars indicate the standard deviation among different units and times for each case.
The consequences of using the above definitions were tested using transient cases for 400 < Re < 800, Pr = 6.13 and several different times. Ratios of the Nusselt number calculated using the definitions in Eqs. (11) and (12) to those in Eq. (10) were calculated for different units and times for each case. The average values are shown in Fig. 3, together with error bars representing the standard deviation. Insignificant differences between the simplified and formally correct definition can be observed for the H2 boundary condition cases. For the T boundary condition cases, Eq. (12) tends to overestimate the unit-averaged value slightly, although the relative difference never exceeds 4%. The unit-based Fanning friction factor at a given s can be defined as
funit ðsÞ ¼
2 dh Dpunit ðsÞ s0
qu2m 4
ð14Þ
i
which in ANSYS CFX requires results to be written to a file and then be post-processed. However, this is not practical for transient simulations, because the above process would have to be employed for each time step. Simplified expressions are therefore used. For the H2 boundary condition, the unit-averaged heat transfer coefficient is defined as
hunit ðsÞ ¼
q_ w ðT w;unit ðsÞ T m;unit ðsÞÞ
ð11Þ
where q_ w is the wall heat flux, T w;unit is the unit-averaged wall temperature at the given s and Tm,unit(s) is the unit-averaged fluid temperature at the given s, which is defined as the average value of the unit inlet and outlet mean fluid temperatures at the given s. For the T boundary condition, the definition of the unit-averaged heat transfer coefficient is
hunit ðsÞ ¼
qw;unit ðsÞ DT lm ðsÞ
ð12Þ
where q_ w;unit ðsÞ is the area-averaged wall heat flux of the entire unit at the given s and DTlm(s) is the log-mean of the unit inlet and outlet wall-fluid temperature differences at the given s. Based on the above expressions, a unit-based Nusselt number for the H2 or T boundary condition at a given s can be defined as
Nuunit ðsÞ ¼
hunit ðsÞdh k
ð13Þ
where Dpunit(s) is the pressure-drop along the unit. 3.1. Grid and time step independence studies A swept hexahedral mesh with 3960 cross-sectional and 240 axial elements for each repeating zigzag unit has been shown previously to provide grid independent solutions up to a Reynolds number of 300 and Prandtl number of 20 [27]. To ensure the grid independence remains in the transient regime at higher Reynolds numbers, a test simulation using a refined cross-sectional mesh (6400 cross-sectional elements) was performed at Re = 800 and Pr = 20 for the H2 boundary condition. It was found that the averaged unit-based Nusselt number changed by 0.2%. Using a refined axial mesh (500 axial elements), the averaged unit-based Nusselt number was found to change by 0.9%. These results confirm the suitability of the earlier mesh for the simulation of transient flow and this mesh has therefore been employed in the current work. Simulations were run with three different time steps, Ds = 103, 4 10 and 2 105, for Re = 600. Fig. 4 shows a comparison of the normalized velocities at a monitor point on the cross-sectional plane of S = 6 as a function of the normalized time using different time steps. With Ds decreased from 104 to 2 105, almost no difference can be found in the velocity magnitudes, indicating that a normalised time step of 104 is sufficiently small to resolve the transient nature of the flow correctly. This value was used for all reported simulations.
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Fig. 4. Normalized velocity magnitudes as a function of the normalized time at a monitor point on the cross-sectional plane of S = 6 and Re = 600 for different time step sizes.
located at the centres of the unit exit planes at S = 1 to 7, respectively. Simulations were run for a sufficient period of time to collect meaningful statistics and data were only extracted after a normalized time of 1.5 (1.5 times the residence time), in order to eliminate any residual effects of the initial conditions. Normalized velocity magnitudes at these points were recorded at each time step, as shown in Fig. 5 for Re = 600. At the first and second points, the velocity magnitudes are found to be constant, indicating the flow pattern remains steady in the first two zigzag units. The flow then develops to a transient state in the third unit, as shown by the temporally varying velocity magnitude at the third monitor point. Transient flow persists downstream but the temporal variation, apparently periodic at S = 3, becomes completely irregular at S = 4 and beyond, as shown in Fig. 5(b). To further understand these transient results, power spectrum analysis was employed on the velocity signals at the points on S = 3 to 7 using the Fast Fourier Transform (FFT) capability of ANSYS CFD Post using the Hanning window. Results are shown in Fig. 6, where the frequency range plotted has a cut-off that is approximately equal to the sample Nyquist frequency, ensuring that no aliasing is present in the plotted data. For the power spectrum at S = 3, the peak at around 200 Hz clearly corresponds to the regular velocity fluctuations apparent in Fig. 5(a); the power density falls off sharply at higher frequencies and is always much lower than corresponding power densities at the later monitor points. Starting from the fourth point, the power spectra become continuous (broadband) and apparently do not change further. These results indicate that the flow has converted from a steady pattern at the channel inlet, to a ‘‘developed’’ transient state after around three zigzag units at Re = 600. The special behaviour in the third unit can be regarded as an intermediate state between the steady and a ‘‘developed’’ transient flow pattern. In Fig. 6, a dashed line representing the 5/3 law of Kolmogorov-Obukhov [29] for turbulent flow is also presented for comparison. (Note the location of the line is arbitrary.) The transient flow spectra in this study do not follow the turbulent law, as would be expected given the relatively low Reynolds number. In fact the decay of the spectrum is faster than the 5/3 law as we do not have an inertial region where viscous effects are negligible, instead they are important across most of the spectrum and result in the rapid destruction of small vortices.
Fig. 5. Normalized velocity magnitude as a function of the normalized time at the monitor points located on the unit exit planes of S = 1 to 3 (a) and 4 to 7 (b) for Re = 600.
3.2. Transient behaviour To investigate the transient flow behaviour, seven monitor points were placed evenly along the computational domain,
Fig. 6. Power spectra of velocity magnitude at the third to seventh monitor point for Re = 600. The dashed straight line has a slope of 5/3 and is shown only for comparison purposes.
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Fig. 7. Power spectra of the velocity magnitudes at the monitor point at S = 6 for different Reynolds numbers.
The temporal variations of velocity magnitudes have also been monitored at other Reynolds numbers. Their spectra at the monitor point at S = 6, together with the 5/3 law, are shown in Fig. 7. It is interesting to note that the spectra for three different Reynolds number are almost identical but as the Reynolds number increases the curve is shifted to the right. This shift results from the increase in resolved small-scale structure that is present with increasing Reynolds number due to reduced viscous damping. Overall, the above analysis indicates that beyond a certain Reynolds number (around Re = 400 for this geometry), a flow regime with spatially and temporally chaotic flow patterns appears in the repeating zigzag channel and in this regime, only a few repeating units are required for the flow to achieve a reasonably stationary state. Statistical properties of the flow are invariant in this flow regime, regardless of the Reynolds number and axial location, which belongs to neither conventional laminar nor turbulent flow.
For the unit-based Fanning friction factor, the mean value is defined as
3.3. Heat transfer enhancement and pressure-drop penalty To investigate the heat transfer and pressure-drop performance of the spatially and temporally chaotic flow, the unit-based Nusselt number and Fanning friction factor in this flow regime were monitored at each time step. The results for Re = 600, Pr = 6.13 using the H2 boundary condition are shown in Fig. 8. For each of the units within the ‘‘developed’’ region, the unit-based Nusselt number and friction factor fluctuate over time and their values for a certain unit and simulation time seem to be completely random. However, time-averaged values tend to be very close to each other from one unit to another. To quantify the overall heat transfer and pressure-drop characteristics in each of the units, these time-dependent values are averaged over time. For the unit-based Nusselt number, its mean value is defined as
Nuunit ¼
Z s2 1 Nuunit ðsÞds s2 s1 s1
Fig. 8. The unit-based Nusselt number (a) and Fanning friction factor (b) as functions of the normalized time for Re = 600, Pr = 6.13 and the H2 boundary condition.
ð15Þ
where s1 and s2 are the start and end points of the integration: here, s1 = 1.5 and s2 = 1.8 are used. A longer time interval with s2 = 2.5 was tested for the case of Re = 600 and no significant change was observed, indicating the current time interval is sufficient for achieving time-independent values.
f unit ¼
Z s2 1 f ðsÞds s2 s1 s1 unit
ð16Þ
In Eqs. (15) and (16), their standard deviations over the integration range can be used to describe the amplitudes of the time-dependent fluctuations. Fig. 9 shows the mean unit-based Nusselt numbers and friction factors for different zigzag units at Re = 400, 600 and 800, Pr = 6.13 and the H2 boundary condition. It can be observed that at each of the Reynolds numbers, Nuunit and f unit for different units become very close to each other after the fourth unit, with the relative differences all below 5%. These results confirm that in this transient chaotic flow regime, the unit-based heat transfer and pressuredrop performance become almost identical from one unit to another, so the overall performance of the entire channel is able to be evaluated based on values in a single unit. Henceforth, in this study, the data for the sixth unit are used to evaluate the overall performance for this geometry. Compared with the equivalent straight channel, the overall heat transfer enhancement (eNu) and pressure-drop penalty (ef) can be defined as follows
eNu ¼
Nuunit ; Nust
ef ¼
f unit fst
ð17Þ
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simulations, the heat transfer enhancement and pressure-drop penalty over a further extended Reynolds number range are obtained, as shown in Fig. 10. For Re < 200, the flow field is in the steady periodic state, where the heat transfer enhancement and pressure-drop penalty are identical from one unit to the next; in the approximate range of 200 < Re < 340, the flow field is still steady, but the flow periodicity is no longer seen and both eNu and ef fluctuate between units with the mean values and standard deviations (error bars) shown in Fig. 10; at around Re > 340, transient behaviour is present and this causes eNu and ef to fluctuate temporally with time-averaged values between units remaining very similar and the relatively small standard deviations shown in Fig. 10. Over the entire flow regime under investigation, both eNu and ef always increase with increasing Reynolds number, with eNu greater than ef at the studied Prandtl number of 6.13. This indicates that with the flow evolving into the transient state, the heat transfer performance in the zigzag channel can be improved continuously, although this is accompanied by a moderate pressure-drop increase. 3.4. Flow pattern and vortex structures
Fig. 9. Nuunit (a) and f unit (b) for different zigzag units at Re = 400, 600 and 800, Pr = 6.13 using the H2 boundary condition.
12
eNu
10
Steady
eNu , ef
8
ef 6
Spatially Periodic
4
2
0
Chaotic
0
200
400
600
800
Re Fig. 10. Heat transfer enhancement and pressure-drop penalty as functions of the Reynolds number in the zigzag channel for a Prandtl number of 6.13 and an H2 boundary condition. The symbols N, s and . denote data in the steady and spatially-periodic, steady chaotic and transient regimes, respectively. The error bars for steady flow data indicate the standard deviations of values for different units, and those for transient flow data indicate the standard deviations over the integration time range.
where Nust and fst are the Nusselt number and friction factor for fully-developed flow in straight channels, and Nuunit and f unit are based on a single zigzag unit. Combining the current results with those for the steady flow in [27], which were re-evaluated using the same definition of unit-based Nusselt number as for transient
To further study the transient heat transfer and pressure-drop characteristics of the zigzag channel, flow fields in the fourth to seventh units are presented in Fig. 11. At s = 1.5 (Fig. 11(a)), the flow pattern is found to be very complex with many small vortices located randomly on the displayed plane (y = 0.1d), which can be seen from the curly structure of the instantaneous streamlines (traced through the velocity field at a given time instant). Significant pressure-drop is caused by these vortical structures, although it is also expected that they disrupt the development of the thermal boundary layer, therefore leading to an enhanced heat transfer rate. At s = 1.6 (Fig. 11(b)), it is found that the flow pattern is very different from that at s = 1.5. There are many obvious changes to the flow field, including the movement and destruction of the vortices present initially, as well as the formation of new ones. These time-varying flow features give rise to intensified fluid mixing, leading to enhancement of the heat transfer rate. The cross-sectional flow and temperature fields have also been investigated and the secondary flow vectors and non-dimensional temperature contours on the planes located at S = 5, 5.5, 6 and 6.5 are presented in Fig. 12. Significant secondary flow can be observed with various vortex structures on each of the cross-sectional planes and additionally, at different cross-sectional planes, the secondary flow pattern varies dramatically. These flow features cause efficient cross-sectional fluid mixing and enhance the heat transfer rate significantly, as indicated by the corresponding temperature profiles, which tend to be very uniform. Hunt [30] proposed the Q criterion, the second invariant of the velocity gradient tensor, to detect vortex regions in transient flow. The quantity Q can be expressed as
Q¼
1 ðjXj2 jSj2 Þ 2
ð18Þ
where X and S are the vorticity and shear strain rate tensors, respectively. Values of Q > 0 occur in vortices while, on the vortex surface, Q = 0. To visualize the variation of the vortex strength at different Reynolds numbers, an isosurface of Q = 3 106 s2 (its value is arbitrary and is chosen to highlight the vortex structure) is presented in Fig. 13. It shows that with increasing Reynolds number, the vortex strength is increased significantly which leads to enhanced cross-stream mixing. Additionally, with an increase of the Reynolds number, vortices with smaller length-scales are generated. Compared with the large ones, these small vortices are shorter lived therefore causing the flow to be well mixed, resulting in
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Fig. 11. Flow patterns with surface streamlines for the fourth to seventh units on a plane of y = 0.1 d at s = 1.5 (a) and 1.6 (b) for Re = 600.
Fig. 12. Secondary flow vectors (left) and non-dimensional temperature profiles ðH ¼ ðT T m Þ=ðT w T m Þ (right) on the cross-sectional planes of S = 5, 5.5, 6 and 6.5 at Re = 600.
enhanced heat transfer. This behaviour is consistent with that expected on the basis of the power spectra shown in Fig. 7, which showed a significant increase in power at all frequencies as the Reynolds number increased through the range 400 to 800.
3.5. Effect of the Prandtl number The heat transfer enhancement as a function of the Reynolds number for different Prandtl numbers is shown in Fig. 14. It is clear
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Fig. 13. Isosurface of Q = 3 106 s2 for the fourth to seventh units at Reynolds numbers of (a) 400, (b) 600 and (c) 800.
Fig. 14. Heat transfer enhancements at different Prandtl numbers as functions of the Reynolds number for the zigzag channel. The symbols N, s and . denote data in the spatially periodic, steady and transient regimes, respectively.
Fig. 16. Nusselt numbers as functions of the Reynolds number in the zigzag channel for the H2 and T boundary conditions. The symbols N, s and . denote data in the spatially periodic, steady and transient regimes, respectively.
Fig. 15. Double-logarithmic plot of eNu versus Pr at different Reynolds numbers.
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that no matter whether the flow regime is steady or transient, the Prandtl number has an obvious effect on the heat transfer enhancement. With an increase of the Prandtl number, a greater heat transfer enhancement is achieved at a given Reynolds number. As the thermal boundary layer decreases in thickness with increasing Prandtl number, it is more easily disrupted by the flow, explaining the increased enhancement at higher Prandtl numbers. In the nonperiodic steady flow regime, a proportional dependence between Pr1/3 of the Nusselt number was found previously [27], over the Reynolds number range from 200 to 320. To examine whether this relationship still holds for a higher Reynolds number, ln (eNu) as a function of ln (Pr) is plotted over an extended range of Reynolds number, as shown in Fig. 15. A linear relationship with identical slopes can be observed for 200 < Re < 800, indicating that the Pr1/3 dependence is valid in both the steady and transient flow regimes.
3.6. Effect of the thermal boundary condition The Nusselt number for a straight channel is dependent on the applied thermal boundary condition in the laminar flow regime. It is interesting to examine the dependence of this boundary condition for the zigzag channel. Fig. 16 shows the Nusselt number as a function of the Reynolds number for the H2 and T boundary conditions for a Prandtl number of 6.13. It is found that compared with that for the H2 boundary condition, the Nusselt number for the T boundary condition is always higher across the entire Reynolds number range. In the low Reynolds number laminar regime having a steady, spatially-periodic flow pattern, the ratio is nearly constant at around 1.2. With the flow evolving into the steady, nonperiodic (chaotic) regime, the ratio becomes greater and then decreases with increasing Reynolds number. In the transient flow regime, the difference is generally diminishing with increasing Reynolds number. It is expected that with a further increase of the Reynolds number towards the turbulent regime, the difference between the two curves will disappear.
4. Conclusions In this paper, a computational fluid dynamics based methodology has been developed to study the transient, laminar flow and heat transfer in various periodic tortuous passages. The flow is shown to develop in a long channel with sufficient repeating units. This ‘‘developed’’ flow pattern is reached in only a few units (four in this case). It is characterised by the occurrence of spatially and temporally chaotic flow patterns as well as the similar properties in different units, such as the velocity power spectrum, heat transfer and pressure-drop performances. The methodology has been used to study the flow and heat transfer in a periodic zigzag channel with a semi-circular crosssection. A Reynolds number range of 400 to 800 and Prandtl numbers ranging from 0.7 to 20 have been considered. The numerical results show that in the transient flow regime, the fluid velocity in the zigzag channel always fluctuates randomly, and has a broadband power spectrum. Statistical analysis based on data over a sufficient period of time has shown that compared with a straight channel, significant heat transfer enhancement can be observed in the transient flow regime, which is accompanied by an increase in pressure-drop. Both eNu and ef are higher than those in the steady flow regime and they increase with increasing Reynolds number. Flow pattern and vortex regions have been visualized at different simulation times and Reynolds numbers. With increasing Reynolds number, vortices with smaller length-scale are generated.
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Significant heat transfer enhancement is achieved due to these flow structures. The effect of the Prandtl number on the heat transfer enhancement has been examined for an extended Reynolds number range and it has been found that the Pr1/3 dependence, shown to occur in the steady regime for 200 < Re < 320, remains valid in the transient regime. The effect of the thermal boundary condition has also been investigated, and it has been shown that the Nusselt number for the T boundary condition is always higher than that for the H2 boundary condition (Re < 800). Acknowledgements Zhanying Zheng acknowledges the financial support from Chinese Scholarship Council and the University of Sydney for his Ph.D. study. The work has been supported by the Australian Research Council Grant DP0985453. References [1] W.R. Dean, Note on the notion of fluid in a curved pipe, Philos. Mag. 4 (20) (1927) 208–223. [2] K.C. Cheng, M. Akiyama, Laminar forced convection heat transfer in curved rectangular channels, Int. J. Heat Mass Transfer 13 (3) (1970) 471–490. [3] C.E. Kalb, J.D. Seader, Heat and mass-transfer phenomena for viscous flow in curved circular tubes, Int. J. Heat Mass Transfer 15 (4) (1972) 801–817. [4] L. Wang, F. Liu, Forced convection in slightly curved microchannels, Int. J. Heat Mass Transfer 50 (5–6) (2007) 881–896. [5] R.M. Manglik, J.H. Zhang, A. Muley, Low Reynolds number forced convection in three-dimensional wavy-plate-fin compact channels: fin density effects, Int. J. Heat Mass Transfer 48 (8) (2005) 1439–1449. [6] J.H. Zhang, J. Kundu, R.M. Manglik, Effect of fin waviness and spacing on the lateral vortex structure and laminar heat transfer in wavy-plate-fin cores, Int. J. Heat Mass Transfer 47 (8–9) (2004) 1719–1730. [7] H.M. Metwally, R.M. Manglik, Enhanced heat transfer due to curvatureinduced lateral vortices in laminar flows in sinusoidal corrugated-plate channels, Int. J. Heat Mass Transfer 47 (10–11) (2004) 2283–2292. [8] H. Aref, Stirring by chaotic advection, J. Fluid Mech. 143 (1984) 1–21. [9] R.H. Liu, M.A. Stremler, K.V. Sharp, M.G. Olsen, J.G. Santiago, R.J. Adrian, H. Aref, D.J. Beebe, Passive mixing in a three-dimensional serpentine microchannel, J. Microelectromech. Syst. 9 (2) (2000) 190–197. [10] A.D. Stroock, S.K.W. Dertinger, A. Ajdari, I. Mezic, H.A. Stone, G.M. Whitesides, Chaotic mixer for microchannels, Science 295 (5555) (2002) 647–651. [11] N. Acharya, M. Sen, H.C. Chang, Heat-transfer enhancement in coiled tubes by chaotic mixing, Int. J. Heat Mass Transfer 35 (10) (1992) 2475–2489. [12] H. Peerhossaini, C. Castelain, Y. Leguer, Heat-exchanger design based on chaotic advection, Exp. Therm. Fluid Sci. 7 (4) (1993) 333–344. [13] Y. Lasbet, B. Auvity, C. Castelain, H. Peerhossaini, A chaotic heat-exchanger for PEMFC cooling applications, J. Power Sources 156 (1) (2006) 114–118. [14] A. Yamagishi, T. Inaba, Y. Yamaguchi, Chaotic analysis of mixing enhancement in steady laminar flows through multiple pipe bends, Int. J. Heat Mass Transfer 50 (7–8) (2007) 1238–1247. [15] H. Aref, The development of chaotic advection, Phys. Fluids 14 (4) (2002) 1315–1325. [16] F. Schonfeld, S. Hardt, Simulation of helical flows in microchannels, AIChE J. 50 (4) (2004) 771–778. [17] F. Jiang, K.S. Drese, S. Hardt, M. Kupper, F. Schonfeld, Helical flows and chaotic mixing in curved micro channels, AIChE J. 50 (9) (2004) 2297–2305. [18] Y. Sui, C.J. Teo, P.S. Lee, Y.T. Chew, C. Shu, Fluid flow and heat transfer in wavy microchannels, Int. J. Heat Mass Transfer 53 (13–14) (2010) 2760–2772. [19] N.R. Rosaguti, D.F. Fletcher, B.S. Haynes, Laminar flow and heat transfer in a periodic serpentine channel with semi-circular cross-section, Int. J. Heat Mass Transfer 49 (17–18) (2006) 2912–2923. [20] P.E. Geyer, N.R. Rosaguti, D.F. Fletcher, B.S. Haynes, Thermohydraulics of square-section microchannels following a serpentine path, Microfluid. Nanofluid. 2 (3) (2006) 195–204. [21] P.E. Geyer, N.R. Rosaguti, D.F. Fletcher, B.S. Haynes, Laminar flow and heat transfer in periodic serpentine mini-channels, J. Enhanced Heat Transfer 13 (4) (2006) 309–320. [22] N.R. Rosaguti, D.F. Fletcher, B.S. Haynes, Laminar flow and heat transfer in a periodic serpentine channel, Chem. Eng. Technol. 28 (3) (2005) 353–361. [23] R. Gupta, P.E. Geyer, D.F. Fletcher, B.S. Haynes, Thermohydraulic performance of a periodic trapezoidal channel with a triangular cross-section, Int. J. Heat Mass Transfer 51 (11–12) (2008) 2925–2929. [24] P.E. Geyer, D.F. Fletcher, B.S. Haynes, Laminar flow and heat transfer in a periodic trapezoidal channel with semi-circular cross-section, Int. J. Heat Mass Transfer 50 (17–18) (2007) 3471–3480.
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