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

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

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:

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):