External Convection: Laminar Flat Plate For a constant property, laminar flow a similarity solution exists for the flow field u(y) Major flow parameters: local boundary layer thickness local skin friction coefficient average skin friction coefficient

External Convection: Laminar Flat Plate For a constant property, laminar flow a similarity solution exists for the flow field u(y) Major heat transfer parameters: local thermal boundary layer thickness uniform surface temperature, Ts local Nusselt number (Pr > 0.6) average Nusselt number uniform surface heat flux, q”s local Nusselt number (Pr > 0.6) average Nusselt number

External Convection: Turbulent Flat Plate For turbulent flow, only empirical relations exist local skin friction coefficient uniform surface temperature, Ts uniform surface heat flux, q”s local Nusselt number (Pr > 0.6) Average parameters average skin friction coefficient average Nusselt number assuming xc for Rex,c = 5×105 uniform surface temperature, Ts For xc= 0 or L >> xc (Rex,L >> Rex,c) average skin friction coefficient uniform surface temperature, Ts average Nusselt number

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 9/10 b 1/3 1/9 C 0.332 0.453 0.0296 0.0308 m 1/2 4/5

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

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

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 transition note:

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 drag coefficient Af is the area projected perpendicular to free stream

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 m 0.4-4 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

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

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

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)

External Convection: Impinging Jet Martin correlation: uniform surface temperature, Ts single round nozzle single slot nozzle valid for valid for