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