# DESIGN to examine natural convective heat transfer between isothermal

DESIGN OPTIMISATION OF A FLAT FINNED HEAT EXCHANGERINTRODUCTION:The Exponential increase in the power density of microelectronic components, resulting from rapid advances in semi conductor technology, continues to fuel considerable interest in advanced thermal management of electronic equipment.The theory here comes from strong theoretical base of the Kraus single plate fin design and optimization and extends the benefits of “least-material” single fins to the multiple fin arrays that constitute most heat sinks.The heat transfer q_T can be expressed as,q_(T =? n?_fin q_fin+h_base A_b ?_b )Where the total number of fins is n_(fin ), q_fin is the heat transfer rate from a single fin, h_base is the average heat transfer coefficient for the unfinned base area, A_b, and ?_b is the array base-to-ambient temperature difference. Dividing the area base into unit cells around each fin and rewriting the above equation,q_T=n_fin ?(q?_fin ?+h?_base A_(b,f) ?_b)The base area and fin area, available for heat transfer from each fin, can be found usingA_(b,s)=LsA_fin=2(LH+Ht+Lt?2)Where n_fin is given by,n_fin=  W?((s+t))Assuming equal number of interfin spaces, the total plate fin volume is V_fin= n_fin HLTApplying the Murray-Gardner assumption (insulated tip), the heat dissipation capability of a single fin can be expressed asq_fin= Lk_fin t?_b m(tanhmH)In this relation, m is the fin parameter equal tom= ?(?2h?_fin?(k_fin t))?^(1?2)Where k_fin is the thermal conductivity of the fin, t is the thickness of a plate fin, and h_fin is the average fin area heat transfer coefficient. For such a rectangular fin, the fin efficiency is calculated usingn_fin=  tanhmH?mHElenbass was the first to examine natural convective heat transfer between isothermal vertical flat plates and to document the variation of the heat transfer coefficient with plate spacing.For wide spacing’s, the coefficient was found to approach values associated with isolated plates, whereas for closely spaced plates the heat transfer coefficient decreased to values associated with fully developed, laminar flow.?Nu?_L=C??Ra?_L?^(1?4)The heat transfer coefficient from the Nusselt number,h_base= ?(C?Ra?_L?^(1?4))k_f?LWhere k_f is the fluid thermal conductivity, and ?Ra?_L is the Raleigh Number based on the length of the heat sink base, L, and is given by,?Ra?_L=(g? (?_b PrL^3))?V^2 At the University of Waterloo, models for widely spaced plate arrays have resulted from research by Yovanovich and co-workers. This subsequently led to the development of the Meta code, which utilizes correlations based on the square root of the fluid wetted area as the characteristic dimension.For a typical heat sink fin spacing’s, the heat transfer coefficients are intermediate between the isolated plate and fully developed limits. They can be found from correlations for natural convection in parallel, vertical plate channels.Bar-Cohen and Rohsenow extended the pioneering Elenbass correlation to a variety of boundary conditions. When applied to non-isothermal plates, as encountered in rectangular plate fin arrays, this composite Nusselt number correlation takes the form?Nu?_fin=  (h_fin s)?k_f = ?(576?((?n_fin El)?^2 )+2.873?((?n_fin El)?^(1?2) ))?^(1?2)Where Elenbass number (El) is given by,El=  ((g??_b Prs^4))??Lv?^2 And where n_fin is used to relate the Nusselt number to the average temperature of the fin surface, i.e, ?= n_fin ?_b. The above equation represents a smoothly varying Nusselt number relation, where the first term  576?((?n_fin El)?^2 ) represents the closely spaced channel condition, while the second component 2.873?((?n_fin El)?^(1?2) ) characterizes the isolated plate limit.This relation can be used to determine the Nusselt number and heat transfer coefficient for any channel spacing.Heat sinks are often characterized by their thermal resistance (R_hs) R_hs=  ?_b?qTSystem designers find  R_hs the most useful of the heat sink metrics, while its use masks the effect of heat sink area, volume and material choice, on its thermal performance.In an array heat transfer coefficient (h_a), the effect of base area can be described. Referenced to the area and excess temperature of the base as,h_a=  qT?(LW?_b )