Analytical Velocity and Temperature Distributions for Flow in Microchannels of Various Cross-sections

Analytical solutions are presented for velocity and temperature distributions of laminar fully developed flow of Newtonian, constant property fluids in micro/minichannels for a wide variety of cross-sections. The considered geometries include hyper-elliptical channels and regular polygon ducts, which covers several common shapes. The analysis is carried out under the conditions of constant axial wall heat flux with uniform peripheral heat flux at a given cross section. The boundary conditions are applied using a linear least-squares point matching technique to minimize the residual between the actual and the modeled values on the boundary of the channel. Hydrodynamic and thermal characteristics of the flow are derived; these include pressure drop and local and average Nusselt numbers. The proposed results are successfully verified with existing analytical solutions from literature for a variety of channel cross-sections. The present study provides analytical-based compact solutions for velocity and temperature fields that are essential for basic designs, parametric studies, and optimization analyses required for many thermofluidic applications. NOMENCLATURE a = Hyperellipse major axis, m AC = Cross-sectional area, m 2 cp = Specific heat b = Hyperellipse minor axis, m D𝑕 = Hydraulic diameter, 4A/Γc , m f = Fanning friction factor fRe = Poiseuille number k = Conductivity m = Number of sides in regular polygonal ducts Nu = Nusselt number n = Exponent in hyperellipse formula P = Pressure, N/m q = Heat Flux, W/m Re = Reynolds number s = Half the length of the sides in polygonal ducts, m T = Temperature, K T = Dimensionless temperature u = Axial velocity, m/s u = Dimensionless velocity Greek symbols α = Diffusivity Γc = Perimeter, m ε = Cross-sectional aspect ratio, ε = b/a η = Non-dimensional coordinate, η = r/a μ = Viscosity, N. s/m ρ = Density, kg/m Subscript A = Square root of cross-sectional area w = Wall b = Bulk INTRODUCTION Advances in micro fabrication technologies make it possible to make microchannels with various cross-sections in microfluidic devices. The convective flow and heat transfer in these channels, apart from their theoretical interest, are of considerable practical importance due to practical applications including the thermal management of electronic devices. The developments in the microelectromechanical devices naturally require cooling systems that are equally small. Among the novel methods for thermal management of the high heat fluxes found in microelectronic devices, microchannels are the most effective at heat removal [1]. In addition, porous materials can be modeled as networks of microscale conduits; thus, transport properties of porous structures are closely related to the geometry of the considered microchannels. A proper understanding of fluid flow and heat transfer in these microscale systems is therefore essential for their design and operation. Different methods have been used in the literature to analyze the problem of fully developed laminar flow in non-circular channels, such as analogy method, complex variables method, conformal mapping method, finite difference method, and point matching method. The typical difficulty for obtaining an analytical solution for this problem by means of the well known classical techniques resides in the impossibility of the separation of variables. An additional difficulty is due to the non regular two-dimensional characteristic of the cross section. Sparrow and Haji-Sheikh [2] proposed a method of least squares matching of boundary values for fully developed laminar flow in ducts of arbitrary cross section. Tyagi [3] analyzed the steady laminar forced convection problem of heat transfer in fully developed flow of liquids through a certain class of channels including equilateral triangular and elliptic tube, using complex variables technique. Shah and London [4] surveyed the literature on analytical solution and alternate methods to study such transport phenomena and interpret the results for twenty five different geometries. Shah [5] presented a least squares matching technique to analyze fully developed laminar fluid flow and heat transfer in ducts of arbitrary cross section. Abdel-Wahed and Attia evaluated hydrodynamic and thermal characteristics of fully developed laminar flow in an arbitrarily shaped triangular duct using a finite difference technique [6]. Maia et al [7] solved heat transfer problem in thermally developing laminar flow of a non-Newtonian fluid in elliptical ducts by using the generalized integral transform technique. They transformed the axes algebraically from the Cartesian coordinate system to the elliptical coordinate system in order to avoid the irregular shape of the elliptical duct wall. However, this method cannot be used in more complex geometries for which transformation is not possible. Furthermore, none of the aforementioned studies presented closed form relations for the velocity and temperature distributions in such complex geometries. In fact, accurate information on the velocity and temperature fields are particularly important in devising efficient strategies in a host of engineering applications such as microfluidic, lab-on-chip devices, and fuel cell technologies, to name a few. An in-depth knowledge of velocity distribution plays a key role in determining other transport properties of microchannels such as heat and mass transfer coefficients. As such, having a generalized solution for the velocity distribution in microchannels is of great value. Tamayol and Bahrami [9] approximated the velocity distribution of fully developed laminar flow in straight channels of regular polygon and hyperellipse cross-section, using the matching point technique. However, they did not solve the temperature problem. In this study, analytical solutions are presented for velocity and temperature distributions of laminar fully developed flow of Newtonian, constant property fluids in micro/minichannels in both hyperelliptical and polygonal mini/microchannels. The considered geometries include i) hyper-elliptical channels, encompassing concave/convex shapes from star-shaped, rhombic, elliptical , rectangular with round corners, and rectangular, and ii) regular polygon ducts, which covers several common shapes from equilateral triangular, squared, pentagonal, hexagonal, to circular. The proposed solution is presented in a single unique format that covers all the abovementioned cross-sections. In order to find the temperature distribution, the energy equation should be solved. Since there is a convective term in the energy equation, we have to find the velocity distribution by solving the momentum equation. In this paper, we first derive the governing equations and find a general solution for ducts with arbitrary cross sections. Then by applying the constant heat flux boundary condition we find the velocity and temperature distribution for polygonal and hyper-elliptical cross sections.