Determining boundary heat flux profiles in an enclosure containing solid conducting block

A numerical implementation of estimating boundary heat fluxes in enclosures is proposed in the present work. Particularly, the flow field is dynamically coupled with the heat convection in the fluid and the heat conduction in the solid domain. An iterative conjugate gradient method is applied such that the gradient of the cost function is introduced when the appropriate sensitivity and adjoint problems are defined. In this approach, no a priori information is needed about the unknown function to be determined. Numerical solutions are obtained for the case of a square enclosure centrally-inserted with a solid block and subjected to an unknown heat flux on one side and to known conditions on the remaining sides. Fluid and heat transports are visualized by the streamlines and heatlines respectively, which are evidently affected by the thermal Rayleigh number, solid body size and thermal conductivity of solid phase, and the functional form of the imposed heat flux. The accuracy of the heat flux profile estimations is shown to depend strongly on the thermal Rayleigh number, body size and relative thermal conductivity of the solid material. Effects of functional form of the unknowns, sensors number and position, and measurement errors on the accuracy of estimation are also investigated. The present work is significant for the flow control simultaneously involving the heat conduction and convection.