Numerical calculation of heat transfer for developing flow for ducts of arbitrary cross-sectional area

In this work, heat transfer coefficients for four types of laminar (HGA, TGA, HTGA and GA) ducts flows were numerically calculated for various boundary conditions in several cross-sectional ducts. To calculate heat transfer in ducts, Navier- Stokes and energy equations must be solved simultaneously. Velocity distributions, pressure drops, friction factors and incremental pressure drop numbers were calculated by solving Navier-Stokes equation for hydrodynamically developing flow (HGA) in various ducts. Also temperature distributions, local and mean Nusselt numbers were obtained transporting the velocity distributions to energy equation. Variable step finite difference method was employed to solve the differential equations. We encountered with a series of difficulties while applying the method to channel region. These difficulties has especially arisen near the walls of the non uniform channel geometries. In these regions integral and derivations caused the difficulties due to different grid spaces. It was necessary to rearrange the derivation and integral formulas if grid points were not equal near walls. This situation doesn't permit to reach a systematic way to solve the equations. Coordinate transformations were applied to channel region for the elimination of the arising difficulties. Channel geometries were transformed to square and then new coordinate system was used. Application of the finite difference method to square shaped channel geometry becomes easy after transformation and, derivation and integration were also calculated easily. As already known that the accurate solutions can be obtained when the grid lines are fine. But computation times will be longer due to increasing arithmetic operations. To prevent this unwanted situation, we developed a method to obtain fine grids near walls. In this method, a number called 'fining scale' which depends on size was described and the number of the grid lines in the channel can be easily adjusted by this fining scale. If fining scale is greater than 1, grid spaces contract while closing the wall. Otherwise, if fining scale is smaller than 1 grid spaces expand. This means none of the grid spaces can be equal. Derivation and integration formulas were derived for-103- variable step sizes. Obtained derivation formulas were written in differential equations and finite differential equations were found. An implicit method was used for stability. Faster convergence can be achieved by using this method. In this method, knowns and unknowns were multiplied by P and (l-f3) respectively and then finite difference equation was rewritten. 3 number varies between 0 and 1. The method is called as Crank-Nicholson method if 3 is 0.5. 3=0.75 was used in this work. Finite difference equation was applied to new grid system and obtained a system of linear equations. Band width of the obtained matrix from the system of linear equations was 9 and the other elements of the matrix was zero. This band matrix was solved using Gauss-Seidel iterative method. Obtained velocities (u, v, w) were stored for each Az in matrixes to use later. Energy equation was solved using the same method as Navier-Stokes. Temperature distributions, mean temperatures and mean and local Nusselt number in the channel were calculated for thermally and hydrodynamically developing flows. Also, the equations were solved for various Pr number such as 0, 0.7, 5, 20 and oo. Pr=oo is obtained when the flow is hydrodynamically developed and thermally developing flow and Pr=0 was obtained when the flow is thermally developing flow (TGA) and w*=l. Velocity distribution must be known to calculate heat transfer for thermally developing flow (TGA). For this reason, velocity distributions and shape factors for fully developed flows was calculated solving Navier-Stokes equations. Nusselt number for thermally developing flows (TGA) in various channels (Rectangular, circular, square, rightangle triangular, equilateral triangular and ellipse) were calculated by the aid of the velocity distribution. Literature survey showed that there is no general equation which can calculate the heat transfer in various cross-sectional ducts. General equations for constant wall temperature and constant heat flux were derived to calculated heat transfer for thermally developing flow (TGA). The results of these equations were compared with available numerical and theoretical results and maximum deviations were between -8% and +8.7% for constant wall temperature and -6.1% and +6.4% for constant heat flux.