A three dimensional numerical study on natural convection heat transfer from short horizontal rectangular fin array M. Mobedi, H. Yu ¨ncu ¨ Abstract Steady state natural convection heat transfer in a longitudinally short rectangular fin array on a horizontal base is numerically investigated. The problem is three dimensional laminar natural convection heat transfer with open boundaries. A finite difference code based on vor- ticity-vector potential approach is developed to solve the governing equations. The steady behaviour of the flow and temperature distributions is obtained from the solution of transient form of the governing equations. Results are compared with the available experimental results in the literature. A good agreement is seen between the numer- ical and reported experimental results. Flow configurations occurring in the channel of the fin arrays with different geometrical parameters, (fin length, fin height and fin spacing) are analysed. Based on the obtained results, two types of flow patterns are observed. The mechanisms of the flows are discussed and flow patterns are plotted to support the discussion. The study is limited to Rayleigh number based on fin spacing, ranging from 120 to 39000. The fin length and fin height are varied from 2 to 20 and 0.25 to 7-fin spacing, respectively. Nomenclature Gr s Grashof number based on fin spacing H fin height, m h heat transfer coefficient, W/(m 2 Æ K) L fin length, m Nu s average Nusselt number based on fin spacing P pressure, Pa Pr Prandtl number Ra s Rayleigh number based on fin spacing S fin spacing, m T temperature, K t time, s ~ V velocity vector ~ V  dimensionless velocity vector X  ; Y  ; Z  dimensionless Cartesian coordinate s dimensionless time ~ X  dimensionless vorticity ~ W  dimensionless vector potential m kinematic viscosity ~ r gradient vector ~ r  dimensionless gradient vector Subscripts n normal component of a vector to a plane s fin spacing, fin surface t tangential component of a vector on a plane ¥ environment 1 Introduction Finned surfaces are widely used in natural convection to increase heat transfer rate. They offer an economical and trouble free solution to the problem of heat transfer en- hancement. Rectangular fin array on a horizontal base is one type of various fin configurations that is used in heat exchanger between two fluids or between a structure and its surroundings. When vertical fins are attached to a horizontal plate, which is at a higher temperature than surroundings, heat transfer area increases by the amount equal to the fin area. However, these fins also increase the friction and resulting heat transfer enhancement may be negligible. Therefore, it is important to perform a study on this type of fin array in order to have a design with con- siderable heat transfer enhancement. A literature survey on horizontal rectangular fin arrays indicated that experimental studies on this type of fin ar- ray are more common due to difficulties in theoretical modeling. The first work has been performed by Starner and McManus [1]. They have measured the average heat transfer coefficient not only in horizontal but also in 45 degree and vertical base positions. The second study found in literature is the one by Harahap and Mc Manus [2]. The flow patterns in two series of horizontal rectangular fin arrays were observed and reported in their work. Jones and Smith [3] investigated the effect of fin height and fin spacing on heat transfer coefficient. They concluded that fin spacing is the main geometrical parameter and it should be chosen as characteristic length. Mannan [4] investigated the effect of all of the geometrical parameters and temperature difference on heat transfer coefficient. He also found that the most important geometrical parameter is the fin spacing, whereas fin length and fin height are probably the next important geometrical parameters. Heat and Mass Transfer 39 (2003) 267–275 DOI 10.1007/s00231-002-0360-5 267 Received: 15 June 1999 Published online: 7 February 2003 Ó Springer-Verlag 2003 M. Mobedi, H. Yu ¨ncu ¨(&) Mechanical Engineering Department, Middle East Technical University, 06531 Ankara, Turkey E-mail: hafit@rorgual.cc.metu.edu.tr