The focus of this work is on developing a nonlinear grey model for the laminar flow regime of carrier fluid across the flat plate collector. The trust region reflective algorithm was used to solve the nonlinear laminar flow problem and handle the sparse matrix. The stream function, dimensionless velocity, gradients of velocity components, carrier fluid temperature, and the absorber plate temperature were estimated for a flat plate collector operated with and without a circular chimney. Similarly, the same technique was adopted to determine a model for heat transfer across the absorber plate. The numerically obtained solution was also compared with the parameters obtained through the instrumental measurement. The relative tolerance was kept at 10−5 for the white and grey box solutions. The developed model was noticed to have a smaller deviation than the conventional analytical model to predict the experimental values. The slip condition was observed for the proposed scheme. The uncertainty in the temperature measurement through the proposed model varies from ±0.53 to ±1.47 K. The minimum absolute percentage error (MAPE) lies in the domain of 0.52−1.67% for the nonlinear grey model, whereas the linear grey model for measuring absorber plate temperature has a percentage error of 0.0011−0.02%.