International Journal of Heat and Mass Transfer
Numerical studies of simultaneously developing laminar flow and heat transfer in microtubes with thick wall and constant outside wall temperature
Introduction
Conjugate heat transfer in laminar duct flow was widely analyzed in the last century. Since the entrance problem was first proposed by Graetz [1], [2], a broad effort has been made to get information both on the thermal entrance region and on the fully developed region under different kinds of boundary conditions. A comprehensive review of the huge amount of results on the subject is available in the monograph by Shah and London [3]. One of the issues in the study of developing heat transfer in tubes or ducts is the effect of wall heat conduction. This subject is especially important when the heat transfer in microtubes is concerned. The small size in the spanwise direction in microtubes/ducts makes almost all microtubes/ducts be of thick-walled type. And this is the major concern of this paper. In the following only the references related to the effect of tube/duct wall heat conduction are briefly reviewed. Davis and Gill [4] analyzed the effect of axial wall conduction on the Couette flow between parallel plates. Mori et al. [5], [6] investigated the effects of wall conduction on the convective heat transfer between parallel plates and in circular pipes under the boundary conditions of the first and the second kind. Faghri and Sparrow [7] investigated the simultaneous wall and fluid axial conduction in laminar pipe-flow heat transfer and proposed a criterion for judging the importance of the axial heat conduction. Zariffeh et al. [8] studied the combined effects of wall and fluid axial conduction on laminar heat transfer in circular tubes numerically. Campo and Rangel [9] analytically studied the conjugate effect of 1-D wall and fluid axial conduction. Barozzi and Pagliarini [10] investigated flow in thick-walled pipes with two-dimensional conduction analytically. Campo and Shuler [11] did lumped system analyses for the simultaneous wall and fluid axial conduction in laminar pipe flow heat transfer. Bilir [12] numerically analyzed the combined effect of 2-D (radial–axial) wall conduction and fluid conduction for low Peclet number (Pe ⩽ 20) fully developed laminar flow heat transfer in a thick-walled two-regional large circular pipe which has external constant temperature with a step change at a given section. Above-mentioned references were mostly published before the time when the so-called micro-heat transfer was emerging. And the major concerns of those papers are either the effect of one dimensional wall conduction or the combined effect of two-dimensional wall conduction and fluid axial conduction for low Peclet number duct laminar flow heat transfer with not very large ratio of wall thickness over tube diameter. During the last two decades the development of MEMS stimulated a great interest to study flow and heat transfer in microchannels [13], [14].
One particular characteristic of convective heat transfer in mini-micro scale channels is its rather strong multi-dimensional character [14].
With the increasing of the ratio of the wall thickness over the hydraulic diameter, the coupling between the wall and the bulk fluid temperatures becomes more important, and the axial conduction through the tube wall has to be considered very seriously. In [15] numerical simulation was made for a circular thick tube with third kind of outside boundary condition with different thermal conductivity ratio of wall to fluid. Their results show that the larger thermal conductivity ratio leads to a lower Nusselt number in the laminar fully developed region and the very low ratio approximate the constant wall heat flux, which is consistent with the results obtained by Sparrow and Patankar [16]. The numerical results in [15] also show that the fully developed Nusselt number decreases with the decrease in the ratio of outer over inner tube diameters. Maranzana et al. [17] have analyzed the influence of the axial conduction in the tube wall on microchannels and proposed a non-dimensional number (M) to quantify the effect of axial conduction in walls. Zueco et al. [18] analyzed a thick-walled macro-tube with a step change outside temperature by network method. They concluded qualitatively that the effect of wall conduction on heat transfer increases as the Peclet number and thermal conductivity ratio of solid wall over fluid decrease. Tiselj et al. [19] studied the effect of axial conduction on the water heat transfer in multi-microchannels with triangle cross section. Gamrat et al. [20] studied the flow and heat transfer in a micro-channel by both experimental and numerical methods. They concluded that there is no size effect on heat transfer when the channel spacing is reduced from 1.0 mm down to 0.1 mm. The strong reduction in the Nusselt number observed in experiments cannot be explained by axial conduction in the walls or lack of two-dimensionality of the heat flux distribution. Further investigations are needed in order to reduce the gap between the results of some experiments and numerical simulation. Hetsroni et al. [21] studied water heat transfer in a multi-microchannel and found that for the cases studied the viscous dissipation can be ignored. Bavière et al. [22] studied the bias effects on heat transfer measurements in microchannel flows. It is concluded that a bias effect on the solid/fluid interface temperature measurement may account for the apparent scale effects observed in some references. Celata et al. [23] experimentally measured the heat transfer characteristics in micro tube uniformly heated with inner diameter ranging from 50 μm to 528 μm. They found some decrease in heat transfer performance with respect to the conventional value (Nu = 4.36) and it is more marked for decreasing Reynolds number. They attributed this reduction to a heat loss term. Lelea [24] studied the conjugate heat transfer of water in partially heated microchannels by numerical method. Three different tube wall materials were considered, stainless steel (k = 15.9 W/m K), silicon (k = 189 W/m K) and copper (k = 398 W/m K). Two different cases of the partial Joule heating were considered for the tube wall. The local Nu exhibits the usual distribution as for the non-axial conduction case, and the local Nusselt number at fully developed region has the usual value of 4.36. In [25] experimental measurements were conducted for heat transfer in three microtubes with different thickness over diameter ratio and uniformly heated outside. Their results show that with the increase in Reynolds number the local Nusselt number in the downstream region approaches 4.36. Liu et al. [26] experimentally studied the effect of axial wall heat conduction for convective heat transfer in stainless steel microtube with Joule heating. Distilled water and nitrogen gas were used as the working fluids flow through the stainless steel microtube with inner diameter 168 μm and outer diameter 406 μm. The wall temperature field photos of the microtube are acquired by employing an IR camera. Their results show that the axial heat conduction can reduce the convective heat transfer in the stainless steel microtube and the decrement may reach 2% compared to the convective heat transfer when the working fluid is nitrogen gas, however, the decrement can be neglected for distilled water as the working fluid. Weigand and Gassner [27] investigated the effect of wall conduction for the extended Graetz problem for laminar and turbulent channel flows with a thick-walled model numerically. The channel wall only has a small part located in the center being kept at high temperature while the rest at lower temperature. The heat conduction within the solid wall changes the temperature distribution at the interface between the solid and the fluid. The increasing in solid wall thermal conductivity leads to reducing the abrupt temperature change at the interface.
From above review, it can be seen that even though a great number of numerical and experimental studies have been performed on the wall heat conduction effect on the entrance problem in mini/microchannels or tubes, we still have not a very clear picture on what is the effect of the wall axial heat conduction? With what condition the effect of the axial heat conduction can be neglected? And if the axial heat conduction is very severe, what will be the result? Because of the complexity of heat transfer in micro-channel or tube situation, we may first pay attention to a very simple, yet typical and interesting problem in heat transfer theory as follows: fluid with uniform velocity and temperature flowing into a thick-walled tube with a constant temperature of its outside surface. The two ends of the tube are adiabatic. For that case the negligible axial heat conduction leads to uniform temperature of the inner surface, and hence the fully developed Nusselt number should be 3.66. If the axial heat conduction plays a role, then the inner surface of the wall will depart from the uniform wall surface condition. By using this model the above-referenced three problems can be clearly answered. This is the major purpose of the present paper.
In this paper in order to clarify the above-mentioned three questions, the conjugate effect of two-dimensional wall conduction (both radial and axial) and fluid axial conduction is analyzed for simultaneously developing laminar flow and heat transfer in a microtube with the ratio of wall thickness over tube diameter varying in a wide range and a constant outer surface temperature.
It should be noted that even though the problem studied here seems to be quite simple in both governing equation and boundary conditions, the three questions listed above are remained unsolved in the literatures. To obtain their correct answers is not only just for academic purpose, but also is meaningful for development of theory of MEMS design. Even though many commercial softwares are available, readers do not know the code details of implementation of some numerical treatments. And sometimes a minor difference in code implementation may lead to different results. In order to make sure that every our numerical treatments is correctly coded, we use a self-developed code to perform numerical simulation. Such practice was adopted in [24], [28], [29].
Section snippets
Numerical analysis
The physical problem investigated is as follows. A fluid with a temperature tin and velocity uin is going into a circular tube of inner diameter di and wall thickness (ro − ri) with its outside surface temperature being at constant value tw. The tube is assumed to be long enough. The velocity and temperature fields at different ratio of wall thickness over inner diameter and different thermal conductivity ratio of wall over fluid are searched for. Fig. 1 presents the computational domain.
As
Results and analysis
As the validation of our self-coded program, the numerical results of the variation of (fRe)x and Nux versus X/(RiRe) when δ/Ri = 0 are presented in Fig. 2. It is shown that the numerically predicted hydrodynamic and thermal entrance lengths, the fully developed Poiseuile number and Nusselt number coincide with the conventional results (LH/RiRe = 0.1, LT/RiRe Pr = 0.1, (fRe)∞ = 64, Nu∞ = 3.66) very well when the grid points in the fluids at the radial direction are taken as 20. The relative differences
What is the basic function of axial heat conduction?
From above presentation of our numerical results, it is quite clear that the basic function of the wall axial heat conduction for the case studied is to unify the heal flux at the inner surface of the tube. From our numerical study when ksf is larger than 25, the axial heat conduction becomes so strong that influences from other parameters may be neglected. If ksf is less than this value then other parameters, including the ratio of wall thickness over radius, the values of Re and Pr (i.e., the
Conclusions
A comprehensive numerical study of the simultaneously developing forced laminar flow and heat transfer in thick wall microtubes with constant outer wall temperature has been conducted by taking into account the conjugate effect of two-dimensional wall conduction and fluid axial conduction. Heat transfer characteristics are relied on four non-dimensional groups, Re, Pe(=RePr), ksf and δ/Ri. Computations have been conducted in the following parameter ranges: 30 ⩽ Re ⩽ 2280, 1 ⩽ Pr ⩽ 7, 1 ⩽ ksf ⩽ 10,000, and 0
Acknowledgments
This work was supported by the Key Project of the National and Natural Science Foundation of China (50636050) and the Key Project of Fundamental Research in China (G2007CB206902).
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