1 Introduction

Ducts are used in numerous industrial applications such as connecting devices,biochemical reaction chambers,physical particle separation,inkjet print heads,infrared detectors,diode lasers,and cooling computer chips. Industrial techniques produce channels of several cross-sections. In addition to the classical circular cross-sections,non-circular channels such as rectangular,elliptic,and trapezoidal,which have wide practical applications in micro-electromechanical systems (MEMS)^{[1, 2]},are also produced. The third most common microchannel geometry may be the semi-circular profile,which occurs naturally in hard micromachining with isotropic etching^{[3, 4]}. The semi-circular microchannels that are fabricated using hard or soft micromachining have,in fact,semi-elliptic cross-sections instead. In several cases,the axis ratio of such semi-elliptic sections is far from unity.

Several attempts have been made to solve the problem of flow and heat transfer in semicircular ducts. These studies are relevant to our problem since they stand as the limiting case when the axis ratio of a semi-elliptic channel approaches unity. Date^[5] numerically solved the problem of fully-developed,laminar and turbulent,uniform-property flow in a tube containing a twisted-tape. Hong and Bergles^[6] numerically investigated the laminar flow heat transfer in the entrance region of a semi-circular tube with the uniform heat flux. Two different thermal boundary conditions were considered. The first was the axially constant heat flux and uniform wall temperature around the entire semi-circular tube at each axial location,while the second was the axially constant heat flux and uniform wall temperature around the semi-circular portion and an insulated wall at the flat portion. Other relevant studies include those by Shah and London^[7],Manglik and Bergles^[8],and Sparrow and Haji-Sheikh^[9].

Moreover,as the axis ratio tends to zero,a semi-elliptic channel degenerates to an infinitesimally narrow lenticular passage. The degenerate semi-elliptic channel is also a degenerate full elliptic channel. We refer to two important studies that deal with fully developed convection in elliptic channels,namely,those by Tao^[10] and Bhatti^[11],which are compared with the present results. Fully developed forced convection in ducts of semi-elliptic cross section was numerically studied by Velusamy et al.^[12] who used a control volume-based procedure. Both the isothermal and uniform axial heat flux conditions on the duct walls were considered. With the availability of the exact velocity profile in a semi-elliptic channel provided recently by Alassar and Abushoshah^[13] and Wang^[14],we believe that an exact solution for the case of constant axial heat flux,numerically treated by Velusamy et al.^[12],is possible.

In this paper,an exact solution of the problem of fully developed forced convection through semi-elliptic ducts,under constant axial heat flux and peripherally uniform temperature,is obtained. The solution is validated by comparing the local and average Nusselt numbers with the published approximate and asymptotic values.

2 Problem statement and solution

Consider a channel having a semi-elliptic cross-sectional area with the base length 2a,the height b,and the focal distance c ($c=\sqrt{{{a}^{2}}-{{b}^{2}}}$) (see Fig. 1). We assume thermally and hydrodynamically fully developed steady laminar forced convection. We also assume that a constant axial heat flux per unit area q is applied through the wall.The problem is hosted well in the elliptic cylindrical coordinates system (ξ,θ,z),whose transformation equations are

The corresponding scale factors are The curved surface of the channel is identified with a fixed value of the coordinate variable ξ (namely,ξ₀) which is related to the axis ratio b/a by The values of the coordinate variables on the boundary of the channel are shown in Fig. 1.

The governing equations of the problem under consideration are given in the elliptic cylindrical coordinates as

where w,P,μ,α,z,and T are the fluid velocity in the flow direction,the pressure,the viscosity,the thermal diffusivity,the flow direction,and the temperature,respectively.

Under the constant axial heat flux,a simple energy balance over a segment of the duct of length dz shows^[15] that

where T_m is the mean or mixing-cup temperature,ρ is the density of the fluid,w is the mean velocity,A is the cross-sectional area of the channel,is the perimeter,and c_p is the specific heat capacity. The mean temperature is defined as We define a dimensionless velocity u as and a dimensionless temperature as where T_w is the wall temperature. Equations (3) and (4) may be rewritten as The solution of Eq. (8) obtained by Alassar and Abushoshah^[13] may be written as which can be averaged over the channel to give Here,the area of the semi-elliptic channel (πab/2) is half the area of a full ellipse.

Using Eq. (10),Eq. (9) may now be rewritten as

An obvious particular solution corresponding to the first term on the right-hand side of Eq. (12) is $-\frac{1}{24}$ sech⁴ ξ₀ sinh⁴ ξ sin⁴ θ. To determine a particular solution of the second term on the right-hand side,we note that the product (sinh² ξ+sin² θ) sinh((2k−1)ξ) sin((2k−1)θ) can be expanded into sums of terms of the form sinh(iξ) sin(jθ). Observing that to each of these terms,a corresponding particular solution assumes the form sinh(iξ) sin(jθ)/(i² − j²),where i ≠ j. In other words,

Tedious but straight forward,one can show that a complete particular solution φ_p of Eq. (12),therefore,is where

The symmetry of the problem at θ = π/2 and the conditions at θ = 0 and ξ = 0 make it necessary to assume a general solution in the form of

Applying the condition that φ = 0 at ξ = ξ₀,it is straightforward to show that The solution of Eq. (9) is thus complete. Figure 2 shows the dimensionless temperature distribution for the case b/a = 1/2.

The dimensionless local Nusselt number is defined as

where ${{\left. \frac{\partial T}{\partial n} \right|}_{wall}}$ is the normal derivative of the temperature at the wall,and φ_m is the mean dimensionless temperature. The hydraulic diameter D_h is defined as where E(·) is the complete elliptic integral of the second kind. Figure 3 shows the distribution of the local Nusselt number on the bottom flat wall for various axis ratios. As b/a → 0,the local Nusselt number on the bottom flat wall approaches the limiting distribution of the case of a degenerate full ellipse. In this case,it can be shown that Moreover,as b/a → 1,it can be shown that the Nusselt number approaches the semi-circular limit given by where ζ(·) is the Riemann zeta function,and r is the distance along the bottom base from its center.

Figure 4 shows the local Nusselt number distribution on the curved wall. As b/a → 0,the curved and the flat surfaces coincide,and the same local Nusselt number distribution is expected. As b/a → 1,however,the Nusselt number can be shown to follow the distribution

Figure 4 shows that there are two symmetrical meeting points along the elliptic wall with the Nusselt number independent of b/a. These points are at the inflection of the Nusselt number curves and can be shown to occur at $\theta ={{\cos }^{-1}}\left( \mp \frac{1}{\sqrt{3}} \right)$. Figure 3,although not quite visible,also exhibits the same behavior.

Finally,we calculate the average Nusselt number,Nu,by averaging the local Nusselt number over the perimeter of the channel. It is easy to show that Nu may be given by the expression

Figure 5 shows the variation of Nu with the axis ratio. The two limiting values,namely,as the axis ratio tends to zero and as the axis ratio tends to 1,are $\frac{9{{\pi }^{2}}}{17}$ and $\frac{43200{{\left( {{\pi }^{2}}-8 \right)}^{2}}}{{{\left( 2+\pi \right)}^{2}}\left( -328504+21775{{\pi }^{2}}+18900\zeta \left( 3 \right)+66560\ln 4 \right)}$ ,respectively. These two results are usually reported in the literature as the approximate values of 5.225 and 4.088,respectively. The first,$\frac{9{{\pi }^{2}}}{17}$ ,was obtained by Tao^[10] as a limiting value for a degenerate full ellipse. Of course,a degenerate full ellipse is identical to a degenerate semi-ellipse. We obtain the same value $\frac{9{{\pi }^{2}}}{17}$ by observing that as b/a → 0 (or equivalently ξ₀ → 0),the curved top wall coincides with the bottom flat wall. One can then average the local Nusselt number given by Eq. (18) over the curved surface and then take the limit as ξ₀ → 0,i.e.,

where $ds={{h}_{\theta }}d\theta -c\sqrt{{{\sinh }^{2}}{{\xi }_{0}}+{{\sin }^{2}}\theta }d\theta $ is the elemental arc length along the curved wall,and a = c cosh ξ₀.

Table 1 compares the present exact values of Nu with those approximate values obtained numerically by Velusamy et al.^[12] and few others.