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We investigate the influence of the thermal properties of the boundaries inturbulent Rayleigh-Benard convection on analytical bounds on convective heattransport. Using the Doering-Constantin background flow method, wesystematically formulate a bounding principle on the Nusselt-Rayleigh numberrelationship for general mixed thermal boundary conditions of constant Biotnumber \eta which continuously interpolates between the previously studiedfixed temperature ($\eta = 0$) and fixed flux ($\eta = \infty$) cases, andderive explicit asymptotic and rigorous bounds. Introducing a control parameterR as a measure of the driving which is in general different from the usualRayleigh number Ra, we find that for each $\eta > 0$, as R increases the boundon the Nusselt number Nu approaches that for the fixed flux problem.Specifically, for $0 < \eta \leq \infty$ and for sufficiently large R ($R > R_s= O(\eta^{-2})$ for small \eta) the Nusselt number is bounded as $Nu \leqc(\eta) R^{1/3} \leq C Ra^{1/2}$, where C is an \eta-independent constant. Inthe $R \to \infty$ limit, the usual fixed temperature assumption is thus asingular limit of this general bounding problem.

Bounds on Rayleigh-Benard convection with general thermal boundary conditions. Part 1. Fixed Biot number boundaries