Scientifically and accurately predicting the distribution of the bubble sizes on the heating surface is a key step to improve the calculation accuracy of the boiling heat transfer model. In the present study, an ingenious nucleate boiling heat transfer model was developed on the basis of random fractal function of the bubble sizes distribution, and renormalization group theory was introduced to solve this distribution function. Compared to the previous correlations, the distribution function of bubbles for various sizes can be obtained by solving this random fractal distribution function with renormalization group method. Furthermore, the process of increasing the fractal dimension from 1 to 2 was first proposed in this paper to match the whole evolution of the heated liquid from natural convection to nucleate boiling, to transition boiling, and finally to film boiling. Therefore, the present model can reveal the nature of nucleate boiling more comprehensively and deeply. Through comparison, it can be found that the image of bubble distribution obtained from the random fractal model was very similar to the experimental photographs statistically, and the predictions heat transfer were in good agreement with the experimental data when the superheat AT is higher than 10 degrees C, and the deviation is less than 20%. (C) 2018 Elsevier Ltd. All rights reserved.