CHE/ME 109 Heat Transfer in Electronics LECTURE 14 – CONVECTION HEAT AND MOMENTUM ANALOGIES

TURBULENT FLOW HEAT TRANSFER REYNOLD’S NUMBER (DIMENSIONLESS) IS USED TO CHARACTERIZE FLOW REGIMES FOR FLAT PLATES (USING THE LENGTH FROM THE ENTRY FOR X) THE TRANSITION FROM LAMINAR TO TURBULENT FLOW IS APPROXIMATELY Re = 5 x 105 FOR FLOW IN PIPES THE TRANSITION OCCURS AT ABOUT Re = 2100

TURBULENT FLOW CHARACTERIZED BY FORMATION OF VORTICES OF FLUID PACKETS - CALLED EDDIES EDDIES ADD TO THE EFFECTIVE DIFFUSION OF HEAT AND MOMENTUM, USING TIME AVERAGED VELOCITIES AND TEMPERATURES http://boojum.as.arizona.edu/~jill/NS102_2006/Lectures/Lecture12/sphere-flow-comparison.jpg

EQUATIONS FOR MOMENTUM & HEAT TRANSFER EDDY AND MOLECULAR TRANSFER COMPONENTS ARE INCLUDED

EDDY AND MOLECULAR TRANSFER EDDY MOTION IS THE PRIMARY MODE OF ENERGY TRANSPORT IN THE TURBULENT CORE AND MOLECULAR DIFFUSION IS NOT SIGNIFICANT EDDY VALUES GO TO ZERO AT THE SURFACE WHERE MOLECULAR DIFFUSION IS THE DOMINANT MECHANISM http://www.propipe.es/images/img_intro.jpg

FUNDAMENTAL CONSERVATION EQUATIONS ARE APPLIED TO DEFINED CONTROL VOLUMES CONTINUITY EQUATION CONSERVATION OF MASS BASED ON BALANCE OVER A CONTROL VOLUME A UNIT DIMENSION IS USED FOR THE z DISTANCE FOR CONSTANT ρ AND STEADY-STATE TWO-DIMENSIONAL FLOWS THE RESULTING EQUATION FOR A DIFFERENTIAL VOLUME

CONSERVATION OF MOMENTUM ANALYZED IN A SIMILAR MANNER WITH A MOMENTUM BALANCE STRESSES INCLUDED IN THE BALANCE ARE: SHEAR STRESS AT THE SURFACE NORMAL STRESS AT THE SURFACE VISCOUS STRESS IN THE FLUID RESULTING BALANCE FOR A SINGLE DIRECTION (x), IS (6-28):

CONSERVATION OF ENERGY THIS IS THE SAME ANALYSIS AS FOR THE MOMENTUM BALANCE, ONLY USING TEMPERATURE FOR THE DRIVING FORCE THE ENERGY TRANSFER IN AND OUT OF THE DIFFERENTIAL ELEMENT IS ASSUMED TO OCCUR BY THERMAL DIFFUSION AND CONVECTION RESULTING BALANCE EQUATION FOR NEGLIGIBLE SHEAR STRESS (6-35)

CONSERVATION OF ENERGY WHEN SHEAR STRESSES ARE NOT NEGLIGIBLE, A VISCOUS DISSIPATION FUNCTION IS INCLUDED: SO THE EXPRESSION BECOMES

FLAT PLATE SOLUTIONS NONDIMENSIONAL EQUATIONS DIMENSIONLESS VARIABLES ARE DEVELOPED TO ALLOW CORRELATIONS THAT CAN BE USED OVER A RANGE OF CONDITIONS THE REYNOLD’S NUMBER IS THE PRIMARY TERM FOR MOMENTUM TRANSFER USING STREAM FUNCTIONS AND BLASIUS DIMENSIONLESS SIMILARITY VARIABLE FOR VELOCITY, THE BOUNDARY LAYER THICKNESS CAN BE DETERMINED: WHERE BY DEFINITION u = 0.99 u∞

FLAT PLATE SOLUTIONS A SIMILAR DEVELOPMENT LEADS TO THE CALCULATION OF LOCAL FRICTION COEFFICIENTS ON THE PLATE (6-54):

HEAT TRANSFER EQUATIONS BASED ON CONSERVATION OF ENERGY DIMENSIONLESS CORRELATIONS BASED ON THE PRANDTL AND NUSSELT NUMBERS A DIMENSIONLESS TEMPERATURE IS INCLUDED WITH THE DIMENSIONLESS VELOCITY EXPRESSIONS: WHICH CAN BE USED TO DETERMINE THE THERMAL BOUNDARY LAYER THICKNESS FOR LAMINAR FLOW OVER PLATES (6-63):

HEAT TRANSFER COEFFICIENT CORRELATIONS FOR THE HEAT TRANSFER COEFFICIENT FOR LAMINAR FLOW OVER PLATES ARE OF THE FORM: http://electronics-cooling.com/articles/2002/2002_february_calccorner.php

COEFFICIENTS OF FRICTION AND CONVECTION THE GENERAL FUNCTIONS FOR PLATES ARE BASED ON THE AVERAGED VALUES OF FRICTION AND HEAT TRANSFER COEFFICIENTS OVER A DISTANCE ON A PLATE FOR FRICTION COEFFICIENTS: FOR HEAT TRANSFER COEFFICIENTS:

MOMENTUM AND HEAT TRANSFER ANALOGIES REYNOLD’S ANALOGY APPLIES WHEN Pr = 1 (6-79): USING THE STANTON NUMBER DEFINITION: THE REYNOLD’S ANALOGY IS EXPRESSED (6-80): .

MODIFIED ANALOGIES MODIFIED REYNOLD’S ANALOGY OR CHILTON-COLBURN ANALOGY (EQN, 6-83):