Derivation and application of the adjoint method for estimation of both spatially and temporally varying convective heat transfer coefficient

In this study, a numerical algorithm is developed to determine the heat transfer coefficient distribution of mixed convection on a vertical plate with a two dimensional inverse method. Cooling procedures of hot plate include forced convection with buoyancy effect is simulated by solving two dimensional, transient heat conduction equation using finite difference method while the Grashof number is varied through natural convection in combination with forced convection resulting in a variation of the Richardson number from 0.1 to 1. The nonlinear inverse heat conduction problem is then implemented to directly predict the time-space-varying convective heat transfer coefficient of cooling in the mixed convection regime. The sum of squared differences between calculated and measured temperature data at thermocouples' locations is the objective functional. The adjoint method is employed to optimize the functional using conjugate gradient method via the solutions of the direct, adjoint and sensitivity sub-problems in a whole time-domain optimization process. The inverse scheme is validated using exact temperature data without noise. Local heat transfer coefficients are then estimated by the adjoint method at four cooling strategies for ten minutes of data acquisition in the presence of noise with standard deviations of sigma = 0.01 degrees C and sigma = 0.1 degrees C. Although results are affected by noisy simulated temperatures, a satisfactory agreement between exact and estimated heat transfer coefficients is achieved. Furthermore, increasing data acquisition time to sixty minutes reveals that the inverse scheme is able to predict 1200 convective heat transfer coefficient components efficiently employing the rapidly convergent adjoint method.