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LECTURE 2 Fundamentals of Convection

Flow Over Flat Plates

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conv = hAs (Ts T) Watt

Convection Heat Transfer

By Newtons Law of Cooling, convection heat transfer is

proportional to temperature different :

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Q

h = convection heat transfer coefficient (W/m2 C) As = heat transfer surface area (m

2)

Ts = temperature of surface (C)

T = temperature of media far from surface (C)

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Heat transfer by conduction,

cond = kAs (T1T2) / L = kAs T/ L

Heat transfer by convection,

conv = hAs (T1T2) = hAs T

By taking ratio heat transfer of convection to conduction

gives :

conv / cond =

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Q

Q

QQ

Nusselt Number

TS = T1 T = T2

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or

Reynolds Number

Reynolds Number (Re) gives relation between inertial force and viscous force of fluid flow.

Re = inertial force / viscous force

VL / VD /

V = free stream velocity (m/s)

L = characteristic length (m) for flat plate

D = diameter (m) for tube/cylinder

= kinematic viscosity (m2/s)

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Prandtl Number

The thickness of thermal boundary layer increases in the flow direction of liquid.

The development of velocity boundary layer relative to

thermal boundary layer will have strong effect on

convection heat transfer

The relative thickness of the velocity and thermal boundary

layers are described by the dimensionless parameter called :

Prandtl Number (Pr)

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Prandtl Number

Heat diffuses very quick in liquid metals and gaseous

(Pr =1 and less) and very slow in water/oil (Pr > 1).

Liquid metals 0.004 0.03

Gaseous 0.7 1.0

Water 1.7 13.7

Oil 50 100,000

Typical values Pr Number :

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Properties of Air

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Friction Force over Flat Plate

Because of the friction between the layers, exerting a friction force on the plate.

The friction force over the entire surface is determined as

below, where Cf is friction coefficient

Friction Force, Ff = Cf As V 2 / 2 (N)

The friction force is important parameter in heat transfer

studies since its is directly related to the heat transfer

coefficient.

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Laminar :

Turbulent :

21Re

328.1fC

51Re

074.0fC

Re

1742

Re

074.051fC

Laminar & Turbulent

Friction Coefficient over Flat Plate

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Friction and Pressure Drag

When the flat plate is placed normal to the flow direction,

however, the drag force depends on the pressure only and is

independent of the wall shear since the shear stress in this case

acts in the direction normal to flow.

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Friction and Pressure Drag

The drag force FD depends on the density of the fluid, the

upstream velocity, and the size, shape, and orientation of the

body, among other things.

The drag characteristics of a body is represented by the

dimensionless drag coefficient CD defined as

where A is the frontal area that tends to block the flow. The

frontal area of a cylinder of diameter D and length L, for

example, is A = LD.

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Flow Across Different Bodies

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Plate [ AN = L x w ] Sphere [ AN = D ] Cylinder [ AN = L x D ]

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Limitation of Re over Flat Plate

The transition from laminar to turbulent flow depends on

the surface geometry, surface roughness, surface

temperature, velocity and type of fluid used.

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Limitation of Re over Flat Plate

For fluid flow over the flat surface, the limit of laminar and turbulent flows is given as :

Laminar Flow : Re < 5 x 105

Turbulent Flow : Re 5 x 105

Re = 5 x 105

Transition ?

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Nu, Pr and Re Numbers

Generally, heat transfer analysis on forced convection involves three major numbers : Nusselt number (Nu), Prantdl number (Pr) and Reynolds number (Re).

To simplify the numbers :

C, m and n are constant value that depends on geometry

and flow type of the body (range of Re). The value of

m and n are between 0 to 1.

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Nu = C Rem Prn

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Nusselt Number over Flat Plates

Laminar : Re < 5 x 105 315.0 Pr Re 664.0 Nu

k

hL

Turbulent : Re 5 x 105 318.0 Pr Re 037.0 Nu

k

hL

Laminar & Turbulent : Re 5 x 105

318.0 Pr )871 Re 037.0( Nu k

hL

Nu = C Rem Prn

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The flat plate is assumed to be at uniform temperature during the entire flow.

However, when a flat plate is subjected to uniform heat

flux instead of uniform temperature, the Nu is given as :

Laminar :

Turbulent :

315.0 Pr Re 453.0 Nu k

hL

318.0 Pr Re 0308.0 Nu k

hL

Uniform Temperature/Heat Flux

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q

Uniform temperature

Uniform heat flux

Ts

=W/m2

Uniform Temperature/Heat Flux

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Fluid Properties

The fluid temperature in the thermal boundary layer varies from Ts at the surface to T at the outer edge of boundary.

Therefore, by taking the average, the fluid properties are

usually evaluated at film temperature (Tf ) :

The fluid properties are assumed to remain constant

during the entire flow.

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Tf = (Ts + T) / 2

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Methodology for Solving Problem

Find the fluid properties based on Tf

Calculate the Reynolds Number (Re)

Identify the type of flow, based on Re obtained

Determine h from equation of Nu

Use the right equations

= hAs(TsT)

Cf = Laminar or Turbulent

Ff = Cf A V2 / 2

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Ts = 110 C T= 30 C V = 6 m/s

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5 m

1.5 m

Q

Example 1

Air, at 30C flows with a velocity of 6 m/s over a 1.5 m x 5m

flat plate whose temperature is 110 C. Determine the rate of

heat transfer from the plate if the flows parallel to :

a) 5 m long side b) 1.5 m side

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Properties of Air

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Assume air at 1 atm pressure Tf = (Ts+T)/2 = (110 + 30)/2 = 70C Properties of air at 70C k = 0.02881 W/m.C =1.995 x 10-5 m2/s Pr = 0.7177 Surface are of the flat plate As = W x L = 1.5 x 5 = 7.5 m

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Example 1

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(a) 5 m long side (L = 5 m) Re = 1503759, Re 5 x 105 - Turbulent Assume the entire flow is turbulent

Nu = 2899 h = 16.7 W/m2 C = 10020 W b) 1.5 m long side (L = 1.5 m) Re = 451128, Re < 5 x 105 - Laminar Nu = 399 h = 7.67 W/m2 C = 4602 W

Q

Q

Example 1

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Forced Convection

EXERCISE 1

Consider a hot automotive engine, which can be approximated

as a 0.5-m-high, 0.40-m-wide, and 0.8-m-long rectangular

block. The bottom surface of the block is at a temperature of

100C and the ambient air is at 20C.

Determine the rate of heat transfer from the bottom surface of

the engine block by convection as the car travels at a velocity

of 80 km/h.

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Properties of Air

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EXERCISE 2

A 15-cm x 15-cm circuit board dissipating 15 W of power

uniformly is cooled by air, which approaches the circuit

board at 20C with a velocity of 6 m/s. Disregarding any heat

transfer from the back surface of the board, determine the

surface temperature of the electronic component.

Forced Convection

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Properties of Air

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EXERCISE 3

Consider a case of air flows over hot plates with 20 cm x 50 cm

surface area. The velocity of air is 5 m/s, the temperature of

each plate is 120C and the surrounding air temperature is 60C.

Determine the rate of heat transfer.

Forced Convection

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Properties of Air