~17-9310~92s5.~+0.00 Q 1992 Pergamon Press Ltd The optimal thickness of a wall with convection on one side J. S. LTM and A. BEJAN Department of Met utnicai Enginee~n~ and Materials Science, Duke U~iversjt~~ Durham, NC 27706, U.S.A. and .I. II. KIM Electric Power Researc I Institute, 3412 Hillview Avenue, Palo Alto, CA 94303, U.S.A. Abstract-This paper documents the conjugate heat transfer through a wall with nonuniform thickness, which is lined on one side by a boundary layer. In the first part, variational calculus shows that the total heat transfer rate is rnin~rniz~ P hen the wall thickness decreases in an optimal manner in the direction of flow. The reductions in total her .t transfer rate are significant when the Riot number is smalier than I. In the second part of the study, thz complete problem of a laminar forced convection boundary layer coupled with conduction through a va~i~~ble-thickens wail is solved numeri~liy. Means for calculating the total heat transfer rate are reported graphically. It was again found that the total heat transfer rate decreases when the wall profile is t:.pered so that the wall thickness decreases in the dir&ion of flow. THE PROBLEM AN ~MPORTA~ characte~stic of man:’ conv~t~on heat transfer configurations is that the heat transfer coefficient varies substantially in the flow direction x. For example, in a for~d-convection laminar bound- ary layer over a flat wall h decreases as x-‘I’, while in a natural-convection laminar boimdary layer h decreases as x- ‘I4 When the wall thitt is swept by the . convective flow has a finite thickness and thermal conductivity, its thermal resistance is added to the resistance of the boundary layer. Intuitively, it seems that a larger wall thickness will have its greatest insu- lation effect in that wall section over which the con- vection heat transfer coefIicient is large. Consider the wall with variabh thickness S(x) shown in Fig. 1. On one side of r1.e wall, the heat transfer coefficient is large enough 30 that the tem- perature of that surface is uniforn, To. The other side is exposed to a flow of different temperature (TO-f-AT), across a heat transfer coefficient whose variation along the wall is known h(x). The wall length L is also specified. The local heat flux driven by the overall constant temperature difference AT is 9’ = s L ATdx -. 01 6 h+k w (2) Of interest is the optimal wall thickness distribution 6(x) for which the heat transfer integral g’ is minimum, while the volume of wall material is fixed. The volume (per unit length) constraint can be written as zyxwvutsrqponm L s 6 dx = 6L (constant) (3) 0 in which 8’ (fixed) is the L-averaged thickness of the wall. SOLUTION BY VARIATIONAL CALCULUS The minimization of the integral (2) subject to the integral constraint (3) is equivalent to the minimiz- ation of the aggregate integral Q= -$$J-t As(x) Fdx (4) ?I+?-- w subject to no constraints (see, for example, Bejan [I]). The factor I in the integrand is a Lagrange multiplier. The optimal thickness can be determined by solving the Euler equation By integrating this over the entire length L we obtain zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJ (9 the total heat transfer rate q’, expressed per unit length in the direction perpendicular to the plane of Fig. 1 in which F is shorthand for the integrand of the @ 1673