January 12-14, 2009, Abu Dhabi, UAE
The Temperature Distribution in Dean Flow: An Analytical Approach
Charbel Habchi a, b, Mahmoud Khaled a,Thierry Lemenand a,Dominique Della Valle a and Hassan Peerhossaini a, *
a Thermofluids, Complex Flows and Energy Research Group, LTN, CNRS UMR 6607, École Polytechnique de l’Université de Nantes, rue Christian Pauc, B.P. 50609, 44306 Nantes, France
b Agence de l’Environnement et de la Maîtrise de l’Énergie (ADEME), 20 avenue du Grésillé, B.P. 90406, 49004 Angers, France
Abstract
Most experimental techniques for measuring the temperature distribution in internal flows are intrusive and disturb the flow and heat transfer process. Numerical simulations of the flow field and heat transfer require the conjugate solution of the Navier-Stokes and energy equations, a highly compute-intensive process. Here an analytical approach is proposed to solve the energy equation in curved pipes that can provide the temperature field in this flow. Since it requires the flow velocity field, the wall temperature, and the temperature at only one point of the flow cross-section, this approach obtains the entire temperature field in the curved-pipe cross section using only one temperature probe in the flow. This technique can be applied to measurement of the temperature field, thus reducing the number of temperature probes necessary to one and minimizing the flow disturbance. For comparison with existing experimental data, here we apply this technique to the diameter of a curved-pipe cross section.
Keywords: Multifunctional heat exchangers, Dean flow, Curved pipe flow, Secondary flow, Analytical solution, Temperature distribution.
Nomenclature a Duct radius, m
d Pipe diameter, m D Thermal diffusivity, m2 s
r Radial coordinate in the pipe cross section
2
2
y
x +
=
R Bend curvature radius, m T Temperature, K
u, v Radial velocities in the pipe cross section, m s-1 W Mean axial velocity, m s-1
x, y Cartesian coordinates z Curvilinear coordinate Greek symbols
θ Curvature angle in the bend plane, rad µ Dynamic viscosity, Pa s
υ Kinematic viscosity, m2 s-1 ρ Density, kg m-3
Subscripts
c Mixer centerline
w Wall
0 Constant values
Exponents
k, i Iteration number Notations
′ First-degree partial derivative
″ Second-degree partial derivative Nondimensional numbers
De Dean number
= a Re
2/( 4 R )
Re Reynolds number
= W d / υ
1. Introduction
The secondary flow arising in curved tubes by the effect of centrifugal forces (usually called Dean cells) takes the shape of a pair of counterrotating roll-cells).
These Dean roll-cells enhance radial mass and heat transfer compared to a straight pipe (Mori and Nakayama [1], David et al. [2]). This property is useful in numerous applications involving the mixing of viscous fluids: in chemical reactions, multifunctional heat exchangers and food-processing applications, especially when the curved pipe segments are mounted in a twisted-pipe configuration to create chaotic advection
January 12-14, 2009, Abu Dhabi, UAE (Acharya et al. [3], Peerhossaini and Le Guer [4, 5],
Castelain et al. [6]).
The local heat transfer process in a helically coiled heat exchanger has been widely studied experimentally, in particular by Acharya et al [3], Mokrani et al. [7] and Chagny et al. [8]. It has been shown that the presence of the Dean instability has a significant effect on heat transfer and extensively modifies the temperature profiles. Dean instability, appearing above a critical Dean number, forms two additional vortices, called Dean vortices, that appear on the concave wall of the curved tube and rotate in the direction opposite to the Dean roll- cells.
Lemenand and Peerhossaini [9] simplified the Navier-Stokes and energy equations and obtained a thermal model to simulate heat transfer in helically coiled and chaotic twisted pipe. The large database obtained from this simplified heat transfer model provides correlations to the Nusselt number Nu.
Naphon and Wongwises [10] reviewed the heat transfer and flow characteristics of single-phase and two- phase flows in curved pipes. They reported numerous relevant correlations for heat transfer coefficients and friction factors in different curved-tube configurations.
This review points to the lack of analytical solutions for the energy equation in curved ducts.
On the one hand, the experimental study of heat transfer in this geometry requires special temperature measurement techniques that, in addition to being difficult to realize (Naphon and Wongwises [10]) and generally intrusive, interact with the flow and affect the measurement results. On the other hand, numerical simulations of the flow field and the heat transfer require the conjugate solution of the Navier-Stokes and energy equations, which is highly demanding in terms of computer memory and capacity.
Analytical solutions could be an appropriate alternative to the above methods and should allow accessible solutions to this difficult problem. The availability of analytical solutions for the velocity profile in a curved pipe (Dean’s solution [11, 12]) led to the
“discovery” of chaotic advection in twisted-pipe flow by Jones et al. [13]. To best of our knowledge, an equivalent analytical solution for the thermal field of the Dean flow has not appeared in the open literature. Therefore, the
“thermal chaotic advection” has so far been deduced from the hydrodynamic field.
Here we propose an analytical resolution of the energy equation in curved pipes that allows computation of the temperature profile in the tube cross section from the wall temperature and the temperature at one point in the field. The main practical interest of this method is in heat transfer experiments, since it permits reduction (to one) of. the number of sensors in the flow section . While this property can be extended to any type of internal flow, the results presented here concern the curved pipe flow, i.e. regular Dean flow.
This method is based on the resolution of the partial differential equations (PDE) for temperature derived from the nonlinear energy equation. The prerequisites are the known velocity components at each point in the flow cross section, which can generally be measured by non- intrusive optical techniques (LDA, PIV) or can be computed.
The paper is organized as follows. In section 2 we formulate the problem. Section 3 describes the mathematical approach to solving the energy equation.
Results and discussion are presented in section 4, and in section 5 we draw some conclusions.
2. Problem formulation
Lemenand and Peerhossaini [9] attempted to calculate the Nusselt number in a helically coiled and chaotic twisted pipe. Since an analytical solution for the energy equation in the curved pipe segments that constituted the helically coiled or twisted pipe coil was not available, the authors used the Reynolds analogy and solved the hydrodynamic problem. The simplified model obtained has ±20% accuracy due to the strong assumptions used.
Analytical resolution of the energy equation provides a more accurate temperature distribution. In fully developed laminar pipe flow, the temperature field can be reduced to a two-dimensional field. The energy differential equation in Eulerian coordinates is then independent of the streamwise coordinate z (Fig. 1) and can be written as
∂ +∂
∂
= ∂
∂ + ∂
∂
∂
2 2 2 2
y T x D T y ) T y , x ( x v ) T y , x (
u (1)
Centrifugal force
Dean roll - cells
x
y x
y
z
Centrifugal force
Dean roll - cells
x
y x
y
z
Fig. 1. Eulerian frame of reference and secondary flow in a curved pipe
The velocity field is provided by the analytical solution given in Habchi et al. [14]. The solution was calculated from the stream function developed by Jones et al. [13] for the Dean [11, 12] asymptotic solution. The velocity field is expressed in the local frame of reference shown in Fig. 1, and is given by:
( ) ( ) ( )( ) [ ] ( )( )
2 22 2 2
2 2
r 3 r 1 y x 6 v
r 1 r 4 r 3 y 6 r 1 u
−
− α
−
=
−
−
−
−
− α
= (2)
where α=Re/144.
Eq. (1) in its general form is a nonlinear differential equation of second order. In the next section we describe a new mathematical approach to solving these types of equations.
3. Analytical approach
The present approach is based on considering that each point in the tube cross section belongs to a virtual disk of radius 1, on which velocity components (u and v) are known. Hence, we assume that the temperature field on a given domain of n points can be obtained from the solution of a system of n PDEs, each of which can be written as:
January 12-14, 2009, Abu Dhabi, UAE
( ) ( ) ( ) ( )
( )
= +
=
∂ +∂
∂
= ∂
∂ + ∂
∂
∂
1 y x for T y , x T
y y , x T x
y , x D T y
y , x v T x
y , x u T
2 2 w
2 2 2 2 k
k (3)
with k=1→n.
In eq. (3) the velocity components are given by eq.
(2) for each point. Therefore, with this approach, the solution of a nonlinear differential equation is reduced to the solution of a system of n homogeneous partial differential equations.
In order to reduce calculation time, given that the velocities u and v vary only among the different n equations, one can solve the system of eq. (3) by adopting a parametric procedure and solving the following equation:
( ) ( ) ( ) ( )
∂ +∂
∂
= ∂
∂ + ∂
∂
∂
2 2 2 2 0
0 y
y , x T x
y , x D T y
y , x v T x
y , x
u T (4)
This approach is applied to obtain the temperature distribution in Dean flow for a given constant wall temperature, which can be taken as a boundary condition forx2+y2=1.
By applying Fourier’s theorem, it is possible to separate the variables by posing:
( ) ( ) ( )
x,y f xgyT = (5)
Hence, the PDE is reduced to two ordinary differential equations
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
20
0 c
y g
y Dg y g
y v g x f
x Df x f
x
u f′ − ′′ =− ′ + ′′ =±
(6) where ±c2 is a constant.
Here we solve eq. (6) on the y axis in order to compare the analytical results to those obtained experimentally by Mokrani et al. [7]. On the y axis the velocity components of the Dean flow are
( ) ( )( )
0 ) y , 0 ( v
1 y 4 y 23 y 7 y , 0
u 4 2 2
=
+ +
− α
= (7)
Eq. (6) can then be written:
( ) ( ) ( )
( ) ( ) ( )
20 c
y g
y Dg 0 f
0 Df 0 f
0
u f′ − ′′ = ′′ =±
(8) As the PDE is of order 2, an additional boundary condition is needed; the temperature in the center of the tube can be used for this purpose. (This choice is arbitrary; another temperature in the flow cross section could be used instead.) Then the boundary conditions are:
=
= ⇒
=
± ⇒
=
c w
T T y
T T y
,
0 0
1
(9)
Solving the temperature distribution on the y axis yields a parametric expression that depends on the velocity field, the fluid temperature at the tube centerT0,c, and the wall temperatureT : w
( ) ( ) ( ) ( ) ( )
( )( )
∑
∞=
+
+ +
π
+ −
+
−
=
0 i
1 i w
c , 0 c , 0 i 0 w
0 i 1i 2
y i 1 cos
T T u
y 1 u T
y
T (10)
The analytical solution developed here provides the temperature distribution on the cross-section diameter from only one known temperature at the pipe center (the other is on the wall), thus reducing the instrumentation required and minimizing the flow disturbance in practical applications.
The temperature distribution depends implicitly on the flow pattern and the thermophysical properties of the fluid.
4. Results and Discussion
4.1. Experimental validation
Analytical results for Reynolds number 97 are compared with the experimental results of Mokrani et al.
[7] in Fig. 3 (to the best of our knowledge the only available experimental profile in the open literature). Two observations can be made:
a) at the center of the Dean roll-cells the measured temperatures are more uniform and higher than the analytical ones. This can be attributed, to a certain extent, to the intrusive mixing caused by the thermocouple probe, which intensifies the heat transfer.
b) the experimental temperature profile is shifted 0.1 unit from the tube center. This displacement cannot be due to the Dean instability, since this instability appears only for higher Reynolds numbers than in the present case. Contrary to what is suggested by Mokrani et al. [7], the shifted profile may be due to the flow disturbance caused by the presence of the thermocouple probe.
-1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 0,0
0,2 0,4 0,6 0,8 1,0
( )
Tw
y T
a / y
y
Experiments [4] Analytical (Re= 97)
Dean roll-cells center
Fig. 3. Comparison of experimental and analytical temperature profiles for Re= 97
4.2. Parametric study
A parametric study is carried out, by varying the Reynolds number and the boundary conditions (wall and center-line temperatures), to investigate the effects of these parameters on the temperature distribution in curved pipe flows.
4.2.1. Reynolds number effect
January 12-14, 2009, Abu Dhabi, UAE Figure 4 plots the experimental temperature profiles
obtained by Mokrani et al. [7] in dimensionless form. In these experiments the wall and the centerline temperatures varied but their ratio stayed fairly constant ( T0,c/Tw =0.95). It is observed that the temperature profiles change little (in the range of measurement uncertainty) when the Reynolds number varies.
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.5
0.6 0.7 0.8 0.9 1.0
a / y
( )
Tw
y T
Experiments [6]:
Re= 63 Re= 97 Re= 190
Fig. 4. Dimensionless experimental profiles adapted from Mokrani et al. [6] for different Reynolds numbers
This phenomenon can be observed in Fig. 5, where the dimensionless analytical temperature profiles are plotted for different Reynolds numbers. In the analytical solutions the ratioT0,c/Twfor each Reynolds number is the same as those in Mokrani et al. [7]. The small difference between the curves is due to the small variation of the T0,c/Tw ratio in the experiments in [7].
The analytical profiles are independent of the flow Reynolds number, as also observed in the experiments.
The similarities between the experimental and analytical results support the idea that the model proposed here captures the essential physics of the problem.
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
a / y
( )
Tw
y T
Reynolds number:
Re= 63 Re= 97 Re= 190
Fig. 5. Dimensionless analytical profiles for different Reynolds numbers
4.2.2. Variation of the boundary conditions
Another parametric study was carried out by varying the boundary conditions, i.e. T0,c andT . Figure 6 plots w the dimensionless temperature profile for fixed Re=97 while T is fixed and the ratiow T0,c/Tw varies. Of course, this variation is somewhat artificial since the T0,c/Twratio
is normally determined by the flow itself unless an arbitrary fixed temperature is imposed at the tube center.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
a / y
( )
Tw
y T
T0, c/Tw= 0.5 0.6 0.7 0.8 0.9 Increasing T0, c/Tw
Fig. 6. Effect of temperature ratio T0,c/Twon temperature profile (Re=97)
At the center of the Dean roll-cells, the dimensionless temperature value is always the same for all center-to-wall-temperature ratios. In fact, at this position heat transfer is due only to the thermal diffusivity;
the radial velocity (which is zero) does not contribute to the heat transfer.
When T0,c/Tw decreases, the temperature profile minimum shifts to the central region of the pipe. Hence the temperature profile is modified when the boundary conditions vary. Therefore, accurate temperature measurements are necessary, both on the wall and on the tube center, if one uses this analytical model to determine the temperature distribution in the tube cross- section by using the wall and center line temperatures as boundary conditions. It should be noted that in the present study, only one thermocouple need be immersed in the flow, so that the flow disturbance is negligible thanks to the small probe dimensions (a few microns).
4.2.3. Velocity-temperature fields coupling
We attempted to derive a polynomial expression for temperature by assuming that the temperature profile is a polynomial similar to the velocity in the tube cross- section, but found that this velocity-temperature analogy is not mathematically valid. However, the analytical velocity and temperature profiles plotted in Fig. 7 show,a clear coupling between the radial velocity and the temperature distribution.
0.0 0.2 0.4 0.6 0.8 1.0
-2 -1 0 1 2
Velocity Temperature
Dimensionless velocity, u(y)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
a / y
( )
Tw
y T
January 12-14, 2009, Abu Dhabi, UAE Fig. 7. Velocity and temperature profiles for T0,c/Tw=0.9
and Re= 97
The dimensionless radial velocity in Fig. 7 is defined by aWu
udimensionless= R (1)
Near the wall region, the temperature increases to a maximum equal to the imposed wall temperature, while the velocity decreases to zero (no slip at the wall). At the Dean roll-cell center, the radial velocity is zero, so that the convective mechanism makes no contribution; the heat transfer is due only to the molecular thermal diffusivity.
In the tube center the velocity increases, enhancing the radial heat transfer by transporting hot fluid particles from near the wall to the pipe center. Therefore, there is overheating at the pipe center and the temperature profile assumes a “W” shape.
5. Conclusions
The analytical resolution of the energy equation, for the flow in a curved pipe with a given constant wall temperature, allows computation of the temperature profile in the tube cross section by using only one known temperature at the pipe center (the other one is on the wall), and knowing the velocity components (measured or computed). This technique can be extended to any type of internal convective heat transfer situation, thus reducing instrumentation and minimizing flow disturbances. The analytical solution shows fair agreement with the experimental results for low Reynolds numbers, where the Dean instability has not yet appeared. In curved-pipe flow, the fluid stays trapped inside Dean roll-cells and causes zones of overheating at the wall and the tube centerline but no convective heating in the center of the Dean roll-cells, where heat is transported only by molecular diffusion.
It is important to recall that the domain of validity of the thermal solution is limited by the validity of the Dean analytical solution for velocity, that is Dn<100.
References
[1] Mori Y., Nakayama W., Study on forced convective heat transfer in curved pipes (1st report, laminar region). Int. J. Heat Mass Tran.. 8 (1965) 67-82.
[2] David A.N., Smith K.A., Merrill E.W., Brian P.L.T., Effect of secondary fluid motion on laminar heat transfer in helically coiled tubes, AIChE J. 17 (5) (1971) 1114-l 122.
[3] Acharya N., Sen M., Cheng H.C. Heat transfer enhancement in coiled tubes by chaotic advection, Int. J. Heat Mass Tran. 35 (1992) 2475-2489.
[4] Peerhossaini H., Le Guer Y., Chaotic motion in the Dean instability flow -- a heat exchanger design, Bull.
Am. Phys. Soc. 35 (1991) 2229.
[5] Peerhossaini H., Le Guer Y., Effect of curvature plane orientation on vortex distortion in curved channel flow, in Andereck C.D., Hayot F., Editors, Ordered and Turbulent Patterns in Taylor-Couette Flow, Plenum Press, New York (1992).
[6] Castelain C., Berger D., Legentilhomme P., Mokrani A., Peerhossaini H., Experimental and numerical characterization of mixing in a spatially chaotic flow by means of residence time distribution measurements, Int. J. Heat Mass Tran. 43 (2000) 3687-3700.
[7] Mokrani A., Castelain C., Peerhossaini H., The effects of chaotic advection on heat transfer, Int. J.
Heat Mass Tran. 40 (1997) 3089-3104.
[8] Chagny C., Castelain C., Peerhossaini H., Chaotic heat transfer for heat exchanger design and comparison with a regular regime for a large range of Reynolds numbers, Appl. Thermal Eng. 20 (2000) 1615-1648.
[9] Lemenand T., Peerhossaini H., A thermal model for prediction of the Nusselt number in a pipe with
chaotic flow, Appl. Thermal Eng. 22 (2002) 1717- 1730.
[10] Naphon P. and Wongwises S., A review of flow and heat transfer characteristics in curved tubes, Renewable and Sustainable Energy Reviews 10 (2006) 463-490.
[11] Dean W.R., Note on the motion of fluid in curved pipe, Phil. Mag. 4 (1927) 208-227.
[12] Dean W.R., The streamline motion of fluid in curved pipe, Phil. Mag. 5 (1928) 673-695.
[13] Jones S.W., Thomas O.M., Aref H., Chaotic advection by laminar flow in a twisted pipe, J. Fluid Mech. 209 (1989) 335-357.
[14] Habchi C., Lemenand T., Della Valle D., Peerhossaini H., Liquid-liquid dispersion in a chaotic advection flow, submitted for publication, Int. J.
Multiphase Flow (2008).
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