Download - CHE/ME 109 Heat Transfer in Electronics LECTURE 14 – CONVECTION HEAT AND MOMENTUM ANALOGIES
CHE/ME 109 Heat Transfer in
ElectronicsLECTURE 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):
Hp
xxfx
Lxf j
VC
hCorNuC 3/2,3/1
, Pr2
Pr2
Re