CHE/ME 109 Heat Transfer in Electronics

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

2. 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

3. 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

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

5. 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

6. 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

7. 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):

8. 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)

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

10. 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∞

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

12. 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):

13. 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

14. 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:

15. 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): .

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