Analytical solution is obtained for laminar forced convection inside tubes including wall conduction effects in the axial direction, based on a radially lumped wall temperature model, and accounting for external convection. The ideas in the generalized integral transform technique are extended to accommodate for the resulting more involved boundary condition and accurate numerical results obtained for quantities of practical interest such as bulk fluid temperature, lumped wall temperature, and Nusselt number. The effects of external convection and axial conduction along the wall on these heat transfer quantities are then investigated through consideration of typical values for, respectively, Biot number and a wall-to-fluid conjugation parameter. Convergence characteristics of the present approach are also briefly examined.
BiBi* c p DH h(z) Nomenclature = Biot number with respect to fluid, hjrJK f = Biot number with respect to wall, K L*,L Nu(Z) Pe R,r Z, z = specific heat of the fluid = hydraulic diameter of circular tube, = convective heat transfer coefficient (internal) = convective heat transfer coefficient (external) = thermal conductivities of solid andfluid, respectively = thermal conductivities ratio, K f /K s = duct length, dimensional and dimensionless, respectively = local Nusselt number, h(z)D h /K f = Peclet number, uD h /a = radial coordinate, dimensionless and dimensional, respectively = radius of duct wall, internal and external, respectively = inlet and ambient temperatures, respectively z), T s (r, z) = fluid and solid temperature distributions, respectively = average flow velocity = velocity distribution, dimensional and dimensionless, respectively '= axial coordinate, dimensionless and dimensional, respectively = thermal diffusivity of the fluid = conjugation parameter (dimensionless) = aspect ratio of circular duct, r 2 /r l -dimensionless fluid temperature distribution = eigenvalues of Sturm-Liouville problem = eigenfunctions of Sturm-Liouville problem U(R)