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　ΔT　is　higher　than　10　°C,　and　the　deviation　is　less　than　20%.　©　2018　Elsevier　Ltd