(17) external forced convection part1

EXTERNAL FORCED CONVECTION

Prabal TalukdarPrabal TalukdarAssociate Professor

Department of Mechanical EngineeringDepartment of Mechanical EngineeringIIT Delhi

E-mail: [email protected]

Fluid flow over solid bodies frequently occurs in practice, and it is responsible for numerous physical phenomena such as the dragis responsible for numerous physical phenomena such as the drag force acting on the automobiles, power lines, trees, and underwater pipelines; the lift developed by airplane wings;

d d ft f i h il d d t ti l i hi hupward draft of rain, snow, hail, and dust particles in highwinds; and the cooling of metal or plastic sheets, steam and hot water pipes, and extruded wires.

P.Talukdar/Mech-IITD 2

What is Drag?What is Drag?Drag is a mechanical force. It is generated by the interaction and contact of a solid ybody with a fluid. For drag to be generated, the solid body must be in contact with the fluid. If there is no fluid, there is no drag Drag is generated by thethere is no drag. Drag is generated by the difference in velocity between the solid object and the fluid

There must be motion between the object and the fluid. If there is no motion, there is no drag. It makes no difference whether the object moves through a static fluid or whether the fluid moves past a static solid object Drag acts in a


static fluid or whether the fluid moves past a static solid object. Drag acts in a direction that opposes the motion. (Lift acts perpendicular to the motion.)

•We can think of drag as friction, and one of the sources of drag is the skin friction between the molecules of the fluid and the solid surface of object.

• Because the skin friction is an interaction between a solid and a gas, the magnitude of the skin friction depends on properties of both solid and gasmagnitude of the skin friction depends on properties of both solid and gas.

• For the solid, a smooth, waxed surface produces less skin friction. • For the gas, the magnitude depends on the viscosity of the air

•This source of drag depends on the shape of the aircraft and is called form drag.

•As air flows around a body, the local velocity and pressure are changed. A varying pressure distribution will produce a force on the body.

•We can determine the magnitude of the force by integrating (or adding up)•We can determine the magnitude of the force by integrating (or adding up) the local pressure times the surface area around the entire body.

•The component of the force that is opposed to the motion is the drag;

P.Talukdar/Mech-IITD 4•The component perpendicular to the motion is the lift.

Factors that affect dragThe ObjectAircraft geometry has a large effect on

Factors that affect drag

the amount of drag generated. As with lift, the drag depends linearly on the size of the object moving through the air The cross-sectional shape of anair. The cross sectional shape of an object determines the form dragcreated by the pressure variation around the object.If hi k f d d iIf we think of drag as aerodynamic friction, then the amount of drag depends on the surface roughness of the object; a smooth, waxed surfacethe object; a smooth, waxed surface will produce less drag than a roughened surface. This effect is called skin friction and is usually i l d d i th d ffi i t


included in the drag coefficient.

Motion of the AirDrag is associated with the movement of the aircraft through the air, so drag will then depend on the velocity of the air. Like lift, drag actually varies with the square of the velocity between the object and the air How the object is inclinedsquare of the velocity between the object and the air. How the object is inclined to the flow will also affect the amount of drag generated. If the object moves through the air at speeds near the speed of sound, shock waves may be formed on the object which create an additional drag component called wave drag. The

ti f th bj t th h th i l b d l t f thmotion of the object through the air also causes boundary layers to form on the object. A boundary layer is a region of very low speed flow near the surface which contributes to the skin friction.Properties of the AirProperties of the AirDrag depends directly on the mass of the flow going past the aircraft. The drag also depends in a complex way on two other properties of the air: its viscosity and its compressibility. These factors affect the wave drag and skin friction

hich are described abo ewhich are described above.We can gather all of this information on the factors that affect drag into a single mathematical equation called the Drag Equation. With the drag equation we can predict how much drag force will be generated by a given body moving at a


p g g y g y ggiven speed through a given fluid.

Friction and Pressure Drag

The drag force is the net force exerted by a fluid on a body in the directionof flow due to the combined effects of wall shear and pressure forces. The part of drag that is due directly to wall shear stress τw is called the skin p g y wfriction drag (or just friction drag) since it is caused by frictional effects, and the part that is due directly to pressure P is called the pressure drag (also called the form drag because of its strong dependence on the form or shape of the body).When the friction and pressure drag coefficients are available, the total dragcoefficient is determined by simply adding them,

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

pressure,Dfriction,DD CCC +=

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

ffi i t C d fi d F


coefficient CD defined as

AV21

FC2

DD

ρ=

When a fluid is forced to flow over a curved surface at sufficiently high velocities, it will y g ,detach itself from the surface of the body. The low-pressure region behind the body where recirculating and back flows occur is called the separation region The larger thecalled the separation region. The larger the separation area is, the larger the pressure drag will be.

The effects of flow separation are felt far downstream in the form of reduced velocity (relative to the upstream velocity). The region of flow trailing the body where theregion of flow trailing the body where the effect of the body on velocity is felt is called the wake.

The separated region comes to an end when the two flow streams reattachThe separated region comes to an end when the two flow streams reattach, but the wake keeps growing behind the body until the fluid in the wake region regains its velocity. The viscous effects are the most significant in the boundary layer, the separated region, and the wake. The flow outside these


regions can be considered to be inviscid.

• Frictional drag comes from friction between the fluid and the surfaces over which it is flowing This friction is associatedsurfaces over which it is flowing. This friction is associated with the development of boundary layers, and it scales with Reynolds number as we have seen above.

• Pressure drag comes from the eddying motions that are set up in the fluid by the passage of the body. This drag is associated with the formation of a wakeassociated with the formation of a wake.

• Formally, both types of drag are due to viscosity (if the body was moving through an inviscid fluid there would be no drag at all), but the distinction is useful because the two types of drag are due to different flow phenomena.

• Frictional drag is important for attached flows (that is there is• Frictional drag is important for attached flows (that is, there is no separation), and it is related to the surface area exposed to the flow. Pressure drag is important for separated flows, and it

l d h l f h b dis related to the cross‐sectional area of the body.


For the flow of an “idealized” fluid with zero viscosity past a body, both the friction drag and pressure drag are zero regardless of the shape of the body.


Two Opposite SituationTwo Opposite Situation

Drag force acting on a flat plate For parallel flow over a flat plate theDrag force acting on a flat plate normal to flow depends on the pressure only and is independent of the wall shear, which acts

For parallel flow over a flat plate, the pressure drag is zero, and thus thedrag coefficient is equal to the frictioncoefficient and the drag force is equal


normal to flow to the friction force

Heat Transfer CoefficientPr),Re,x(fNu x

*1x = Pr),(RefNu L2=&

nmL Pr.Re.CNu =

Pr > 0.6315.0

xx

x Pr.Re332.0k

x.hNu ==Laminar flowk

0.6 ≤ Pr ≤ 605 105 R 107

Turbulent flow 318.0

xx

x Pr.Re0296.0k

x.hNu ==5 x 105 ≤ Rex ≤ 107xx k

P.Talukdar/Mech-IITD

Parallel flow over flat plateParallel flow over flat plateThe transition from laminar to turbulent flow depends on the

surface geometry,surface roughness, upstream velocityupstream velocity, surface temperature, and the type of fluid, among other

things, and is best characterized by the Reynolds number.

The Reynolds number at a distance x from the leading edgedistance x from the leading edge of a flat plateis expressed as

For flow over a flat plate critical Reρ x.VxVRe


For flow over a flat plate, critical Reν

=μ

=Rex

5crcr 10x5xVRe =

μρ

=

Friction Coefficient

x5δ

664.0C Re < 5 x 105L i fl &2

1

x

x,vRe

=δ2

1

x

x,fRe

C = Rex < 5 x 105Laminar flow &

51

x

x,vRe

x382.0=δ

51

x

x,fRe

0592.0C = 5 x 105 ≤ Rex ≤ 107

Turbulent flow &


Average values

21f

R

328.1C =

ReL < 5 x 105

Laminar flow2

LRe

1f074.0C =Turbulent flow

5 x 105 ≤ ReL ≤ 107

In some cases, a flat plate is sufficiently long for the flow to become turbulent, but not long

51

LRe

, genough to disregard the laminar flow region

⎟⎟⎠

⎞⎜⎜⎝

⎛+= ∫∫

L

turbulent_x,f

x

arminla_x,ff

cr

dxCdxCL1C ⎟

⎠⎜⎝ x0 cr

L

The average friction coefficient over the entire plate is determined to be


Taking the critical Reynolds number to be Recr = 5 x 105L5

1

L

f Re1742

Re

074.0C −= 5 x 105 ≤ ReL ≤ 107

Cf for turbulent flowFor laminar flow, the friction coefficient depends on only the Reynolds number, and the surface roughness has no effectthe surface roughness has no effect.

For turbulent flow, however, surface roughness causes the friction coefficient to gincrease severalfold, to the point that in fully turbulent regime the friction coefficientis a function of surface roughness alone, and independent of the Reynoldsnumber

A curve fit of experimental data for the average h b f b

p a da a a agfriction coefficient in this regime is given by Schlichting as

Rough surface, f

5.2

log621891C−

⎟⎞

⎜⎛ ε

=

In the absence of a better relation, this relation can be used for turbulent flow on rough surfaces for Re > l06, especially


turbulent flow f Llog62.189.1C ⎟

⎠⎜⎝

−= surfaces for Re l0 , especially when ε/L > 10-4.

Variation with x

Pr > 0.6315.0

xx

x Pr.Re332.0k

x.hNu ==Laminar flow

0.6 ≤ Pr ≤ 605 105 ≤ R ≤ 107


xx

x Pr.Re0296.0k

x.hNu ==

21

x

x,fRe

664.0C = Rex < 5 x 105

P.Talukdar/Mech-IITD

5 x 105 ≤ Rex ≤ 107xx k

51

x

x,fRe

0592.0C = 5 x 105 ≤ Rex ≤ 107

Average CoefficientAverage Coefficient

Average quantities∫=L

0x,DD dxC

L1C

L1∫=L

0xdxh

L1h

Pr > 0.6315.0

L Pr.Re664.0kL.hNu ==Laminar flow

Rex < 5 x 105

0.6 ≤ Pr ≤ 605 x 105 ≤ Re ≤ 107


L Pr.Re037.0kL.hNu ==

x


5 x 105 ≤ Rex ≤ 107k

h for combined laminar and b l flturbulent flow

In some cases, a flat plate is sufficiently long for the flow to become turbulent, but not long enough to disregard the laminar flow region

⎟⎟⎠

⎞⎜⎜⎝

⎛+= ∫∫

L

turbulent,x

x

arminla,x

cr

dxhdxhL1h

The average h over the entire plate is determined to be

⎟⎠

⎜⎝ x0 cr

L

0 6 ≤ Pr ≤ 60T b l t fl ( ) 1Lh

Taking the critical Reynolds b b 105

0.6 ≤ Pr ≤ 605 x 105 ≤ ReL ≤ 107

Turbulent flow ( ) 318.0

L Pr.871Re037.0kL.hNu −==

number to be Recr = 5 x 105


Liquid metals such as mercury have high thermal conductivities.Liquid metals such as mercury have high thermal conductivities. However, they have very small Prandtl numbers, and thus the thermal boundary layer develops much faster than the velocity boundary layer. Then we can assume the velocity in the thermal boundary layer to be constant at the free stream value and solve the energy equation. It gives

( ) 21

xx

x Pr.Re565.0k

x.hNu == Pr < 0.05

It is desirable to have a single correlation that applies to all fluids, including liquid metals

1

k

Churchill and Ozoe :

( ) 41

32

315.0

xxx

Pr0468.01

Pr.Re3387.0k

x.hNu

⎥⎦⎤

⎢⎣⎡ +

==

Applicable for all Prandtl numbers and is claimed to be accurate to 1%,

Pr ⎥⎦⎢⎣


(17) external forced convection part1

Documents

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drag force

drag p

sources of drag

solid surface of object

static solid object

solid body

static fluid