Fully Developed Forced Convection Through Semicircular Ducts

Abstract

D UCTS of different cross sections are used in a wide range of industrial applications such as connecting devices, biochemical reaction chambers, particle separation, and cooling computer components. In addition to the classical circular ones, channels with noncircular cross sections that are rectangular, elliptic, trapezoidal, and semicircular are also common. Understanding the characteristics of flow and heat transfer through such channels is very important for design purposes. There is a lot of literature on convection through ducts of different cross-sectional geometry. The problems range from steady-state fully developed convection to transient convection at the tube entrance [1– 5]. Because of the mathematical difficulties of dealing with complex geometries, several attempts have been made to deal (as a lump sum) with flow and heat transfer in ducts of arbitrary shape. Among these studies is that by Shahsavari et al. [6] who presented solutions for velocity and temperature distributions of laminar fully developed flow of hyperelliptical and regular polygonal cross sections under constant axial wall heat flux with uniform peripheral heat flux. A similar study using the same solution technique was originally carried out by Sparrow andHaji-Sheikh [7,8] also for fully developed laminar flow in ducts of arbitrary cross section.With the problemboundary conditions, the constants in the separated series solution could only be determined approximately and mainly through a linear least-squares pointmatching technique thatminimizes the residual between the actual and the predicted values on the boundary of the channel. Several attempts have been made to solve the problem of flow and heat transfer in semicircular ducts, among which are those by Date [9], Hong and Bergles [10], Manglik and Bergles [11], El Hasadi [12], Languari and Hooman [13], Geyer et al. [14], and Zheng et al. [15]. Date [9] numerically solved the problem of fully developed, laminar and turbulent, uniform-property flow in a tube containing a twisted tape. Hong andBergles [10] numerically investigated laminar flow heat transfer in the entrance region of a semicircular tube with 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 semicircular tube at each axial location, whereas the second was axially constant heat flux and uniform wall temperature around the semicircular portion and an insulated wall at the flat portion. These two boundary conditions represent the limits of the fin effect of twisted tapes. The asymptotic values (fully developed conditions downstream) are compared to our results. The recent study by El Hasadi [12] deals with laminar mixed convection in the entrance region of horizontal and inclined semicircular ducts. A numerical solution for the fully developed regime under the same conditions was obtained. Languari and Hooman [13] numerically studied slip-flow forced convection in a microduct of a semicircular cross section subject to temperature jump boundary condition. The studies by Geyer et al. [14] and Zheng et al. [15] deal with laminar heat transfer in periodic trapezoidal and zigzag channels, respectively. Both periodic channels have semicircular cross sections. Velusamy et al. [16] used a control volume-based numerical procedure to solve the problem of fully developed laminar flow and heat transfer in ducts of a semi-elliptical cross section. Both an isothermal and a uniform axial heat flux condition on the duct walls were considered. The numerical results for velocity and temperature profiles for a duct aspect ratio of 0.999 are relevant to compare our exact results with. An exact solution of the numerically obtained velocity profile by Velusamy [16] was reached by Alassar and Abushoshah [17]. This exact solution reduces to the required profile in the present study as the aspect ratio approaches unity. In this Note, we present a simple solution of fully developed forced convection through ducts with semicircular cross sections. Channels with semicircular cross sections arewidely applied in highly compact heat exchangers due to the ease of fabrication in a plate structure by chemical etching or other techniques [15]. Furthermore, twisted-tape inserts inside circular tubes are a particularly effective technique for enhancing laminar flow heat transfer in circular tubes. The swirl flow effects diminish as the severity of the twist is decreased. In the limit, the strip is straight and the tube is divided into two circular segment ducts.When the strip is thin, the situation is essentially that of flow in a semicircular duct [11]. In the present work, under axially constant heat flux, the two cases of peripherally uniform heat flux and peripherally uniform temperature are dealt with. The velocity and temperature profiles are used to calculate Poiseuille and Nusselt numbers, which are compared with published approximate values.