AME 60634 Int. Heat Trans.

D. B. Go

External Convection: Laminar Flat PlateFor a constant property, laminar flow a similarity solution exists for the flow field u(y)

local boundary layer thickness

local skin friction coefficient

average skin friction coefficient

Major flow parameters:

AME 60634 Int. Heat Trans.

D. B. Go

External Convection: Laminar Flat PlateFor a constant property, laminar flow a similarity solution exists for the flow field u(y)

local Nusselt number (Pr > 0.6)

local thermal boundary layer thickness

average Nusselt number

uniform surface temperature, Ts

uniform surface heat flux, q”s

Major heat transfer parameters:

local Nusselt number (Pr > 0.6)

average Nusselt number

AME 60634 Int. Heat Trans.

D. B. Go

External Convection: Turbulent Flat Plate

local skin friction coefficient

local Nusselt number(Pr > 0.6)

average skin friction coefficient

average Nusselt number

For xc= 0 or L >> xc (Rex,L >> Rex,c)

uniform surface temperature, Ts

uniform surface heat flux, q”s

average skin friction coefficient

average Nusselt number

assuming xc for Rex,c = 5×105

uniform surface temperature, Ts

uniform surface temperature, Ts

For turbulent flow, only empirical relations exist

Average parameters

AME 60634 Int. Heat Trans.

D. B. Go

External Convection: Starting Length• The effect of an unheated starting length (USL) can be represented

on the local Nusselt number as:

• Parameters a, b, C, & m depend on – thermal boundary condition: uniform surface temperature (UST) or

uniform heat flux (UHF)– flow conditions: laminar or turbulent

where

LAMINAR TURBULENT

a 3/4 3/4 9/10 9/10

b 1/3 1/3 1/9 1/9

C 0.332

0.453 0.0296 0.0308

m 1/2 1/2 4/5 4/5

AME 60634 Int. Heat Trans.

D. B. Go

External Convection: Starting Length• Uniform Surface Temperature (UST)

• Uniform Heat Flux (UHF)

• The Nusselt number (and heat transfer coefficient) are functions of the fluid properties (ν,ρ,α,cp,k) – the effect of variable properties may be considered by evaluating all

properties at the film temperature

– most accurate solutions often require iteration on the film properties

p = 1 (laminar throughout)p = 4 (turbulent throughout)

numerical integration for laminar/turbulent flow

AME 60634 Int. Heat Trans.

D. B. Go

External Convection: Cylinder in Cross Flow• As with flat plate flow, flow conditions determine heat transfer• Flow conditions depend on special features of boundary layer

development, including onset at stagnation point, separation, and onset of turbulence

• Stagnation point: location of zero velocity and maximum pressure– boundary layer development under a favorable pressure gradient

acceleration of the free stream flow

• There is a minimum in the pressure distribution p(x) and toward the rear of the cylinder, the pressure increases. – boundary layer development under an adverse pressure gradient

AME 60634 Int. Heat Trans.

D. B. Go

External Convection: Cylinder in Cross Flow• Separation occurs when the momentum of the free stream flow is

insufficient to overcome the adverse pressure gradient– the velocity gradient reduces to zero– flow reversal occurs accompanied by a downstream wake

• Location of separation depends on boundary layer transitionnote:

AME 60634 Int. Heat Trans.

D. B. Go

External Convection: Cylinder in Cross Flow• Force (FD) imposed by the flow on the cylinder is composed of

two phenomena– friction boundary layer shear stress– form drag (pressure drag) pressure differential due to wake

dragcoefficient

Af is the area projected perpendicular to free stream

AME 60634 Int. Heat Trans.

D. B. Go

External Convection: Cylinder in Cross Flow• Thermal considerations: uniform surface temperature, Ts

– heat transfer a function of the angel of separation θ– empirical correlations are used to determine average Nusselt

numbers

• Hilpert correlation: Pr ≥ 0.6– also suitable for non-circular cylinders

• Churchill and Bernstein: ReDPr > 0.2

ReD C m0.4-4 0.989 0.330

4-40 0.911 0.385

40-4000 0.683 0.466

4000-40,000 0.193 0.618

40,000-400,000 0.027 0.805

AME 60634 Int. Heat Trans.

D. B. Go

External Convection: Sphere in Cross Flow• Similar flow issues as cylinder in cross flow arise

• Thermal considerations: uniform surface temperature, Ts

– heat transfer again defined by empirical correlations

• Whitaker correlation: – 0.71 < Pr < 380– 3.5 < ReD < 7.6×104

evaluate fluid properties at T∞ except for μs which is evaluated at Ts

AME 60634 Int. Heat Trans.

D. B. Go

External Convection: Impinging Jet• Impinging jet consists of a high speed flow impacting a flat surface

– generates large convection coefficients• The flow and heat transfer are affected by a number of factors

– shape/size of jet, velocity of jet, distance from plate, …• Significant hydrodynamic features:

– mixing and velocity profile development in the free jet– stagnation point and zone– velocity profile development in the wall jet

AME 60634 Int. Heat Trans.

D. B. Go

External Convection: Impinging Jet• Local Nusselt number distribution:

• Average Nusselt number based on empirical correlations for single nozzles and arrays of nozzles– function of Reynolds number, Pr, distance along wall (r or x), height of

jet (H)

AME 60634 Int. Heat Trans.

D. B. Go

External Convection: Impinging Jet• Martin correlation: uniform surface temperature, Ts

– single round nozzle

• Martin correlation: uniform surface temperature, Ts

– single slot nozzle

valid for

valid for