Convection

Prepared by: Nimesh Gajjar

CONVECTIVE HEAT TRANSFER

Convection heat transfer involves • fluid motion • heat conduction

The fluid motion enhances the heat transfer, since it brings hotter and cooler chunks of fluid into contact, initiating higher rates of conduction at a greater number of sites in fluid. Therefore, the rate of heat transfer through a fluid is much higher by convection than it is by conduction. In fact, the higher the fluid velocity, the higher the rate of heat transfer.

AIR

20°C5 m/s

20°C.

Q

Warmer air rising

AIR.

Q

No convection current

AIR

a) Forced Convection

b) Free Convection

c) Conduction

Convection heat transfer strongly depends on • fluid properties dynamic viscosity , thermal conductivity k, density

and specific heat • fluid velocity V• Geometry and the roughness of the solid surface• Type of fluid flow (such as being laminar or turbulent).

conv s sQ hA T T NEWTON’S LAW OF COOLING

h = Convection heat transfer coefficientAs = Heat transfer surface areaTs = Temperature of the surfaceT= Temperature of the fluid sufficiently far from the surface

LOCAL HEAT FLUX convq

conv l sq h T T

hl is the local convection coefficient

TOTAL HEAT TRANSFER RATE

s

conv conv s

A

Q q dA s

conv s l s

A

Q T T h dA

1

s

l ss A

h h dAA

convQ

Local and total convection transfer (a) Surface of arbitrary shape. (b) Flat plate.

q”

x dxL

As, Ts

U ,T

dAsq”

As, Ts

V ,T

A fluid flowing over a stationary surface comes to a complete stop at the surface because of the no-slip condition.

A similar phenomenon occurs for the temperature. When two bodies at different temperatures are brought into contact, heat transfer occurs until both bodies assume the same temperature at the point of contact.

Therefore, a fluid and a solid surface will have the same temperature at the point of contact. This is known as NO-TEMPERATURE-JUMP CONDITION.

Zero velocity at the surface

Relative velocity of fluid layers

Uniform approach velocity, V

Solid Block

An implication of the no-slip and the no-temperature jump conditions is that heat transfer from the solid surface to the fluid layer adjacent to the surface is by pure conduction, since the fluid layer is motionless,

0conv cond fluid

y

Tq q k

dy

T represents the temperature distribution in the fluid is the temperature gradient at the surface.

0yT y

conv l sq h T T

fluid y

s

k T yh

T T

0

NUSSELT NUMBER

chLNu

k

k is the thermal conductivity of the fluid Lc is the characteristic length

Heat transfer through a fluid layer of thickness L and temperature difference

Heat transfer through the fluid layer will be by convection when the fluid involves some motion and by conduction when the fluid layer is motionless.

convq h T condT

q kL

conv

cond

q hLh TNu

q k T L k

Nusselt number - enhancement of heat transfer through a fluid layer as a result of convection relative to conduction across the same fluid layer. Larger the Nusselt number, the more effective the convection. Nu = 1 for a fluid layer - heat transfer across the layer by pure conduction

Fluid layer

.

Q

T2

T1

ΔT = T2 – T1

Internal and external flowsEXTERNAL FLOW - The flow of an unbounded fluid over a surface such as a plate, a wire, or a pipe INTERNAL FLOW - flow in a pipe or duct, if the fluid is completely bounded by solid surfaces

External flowAir

Internal flowWater

Laminar versus Turbulent FlowSome flows are smooth and orderly while others are rather chaotic. The highly ordered fluid motion characterized by smooth streamlines is called laminar. The flow of high-viscosity fluids such as oils at low velocities is typically laminar. The highly disordered fluid motion that typically occurs at high velocities characterized by velocities fluctuations is called turbulent. The flow of low-viscosity fluids such as air at high velocities is typically turbulent. The flow regime greatly influences the heat transfer rates and the required power for pumping

Dye Streak

Smooth well rounded Entrance

Q = VA

Pipe

Dye

Laminar

Transitional

Turbulent

Osborne Reynolds in 1880’s, discovered that the flow regime depends mainly on the ratio of the inertia forces to viscous forces in the fluid.

The Reynolds number can be viewed as the ratio of the inertia forces to viscous forces acting on a fluid volume element.

c cVL VLInertia forcesRe

Viscous

One, Two and Three Dimensional FlowsV(x, y, z) in cartesian or V(r, , z) in cylindrical coordinates

r z

V (r)

v e lo c ity p ro file(rem a in s u n ch an g ed )

One-dimensional flow in a circular pipe

VELOCITY BOUNDARY LAYER

Development of a boundary layer on a surface is due to the no-slip condition

BOUNDARY LAYER REGION

INVISCID FLOW REGION

U

U

Boundary-layer thickness, δ xcr

Laminar boundary-layer Turbulent boundary-layerTransition

region

Turbulent layer

Laminar layer

Buffer layer

V

y

x

δ

0.99V

Zero velocity at the surface

(No slip condition)

VU V

Relative velocity of fluid layers

Surface Shear Stress

0s

y

u

y

2

2s fV

C

Skin friction coefficient

Friction force over the entire surface 2

2f f sV

F C A

THERMAL BOUNDARY LAYER

Thermal boundary layer on a flat plate (the fluid is hotter than the plate surface)

The thickness of the thermal boundary layer, at any location along the surface is define as the distance from the surface at which the temperature difference T – Ts equals 0.99(T– Ts). For the special case of Ts = 0, we have T = 0.99 at the outer edge of the thermal boundary layer, which is analogous to u = 0.99 for the velocity boundary layer.

Thermal boundary layer

Uniform Temperature T

T

Free Stream

sT

s sT . (T T ) 0 99

t

Shape of the temperature profile in the thermal boundary layer dictates the convection heat transfer between a solid surface and the fluid flowing over it.

In flow over a heated (or cooled) surface, both velocity and thermal boundary layers will develop simultaneously.

Noting that the fluid velocity will have a strong influence on the temperature profile, the development of the velocity boundary layer relative to the thermal boundary layer will have a strong effect on the convection heat transfer.

PRANDTL NUMBERThe relative thickness of the velocity and the thermal boundary layers is

described by the dimensionless parameter Prandtl number, defined as

pCMolecular diffusivity of momentumPr

Molecular diffusivity of heat k

Fluid Pr

Liquid metals 0.004-0.030

Gases 0.7-1.0

Water 1.7-13.7

Light organic fluids 5-50

Oils 50-100,000

Glycerin 2000-100,000

TYPICAL RANGES OF PRANDTL NUMBERS FOR COMMON FLUIDS

51 2 1 2

5 0 664: and 5 10v,x f ,x x

x x

x .La min ar C , Re

Re Re

5 71 5 1 5

0 382 0 05925 10 10v,x f ,x x

x x

. x .Turbulent : and C , Re

Re Re

0

1 20

1 2

0

1 2 1 2

1

20

1 2

1 2

1

1 0 664

0 664

0 664

2 0 664

1 328

L

f f ,x

L

x

L

L

L

C C dxL

.dx

L Re

. Vxdx

L

. V x

L

. VL

L

.

Re

AVERAGE SKIN FRICTION COEFFICIENT

51 2

1 3285 10f

L

.La min ar : C Re

Re

5 71 5

0 0745 10 10f L

L

.Turbulent : C Re

Re

Problem: Engine oil at 60° C flows over the upper surface of a 5-m long flat plate whose temperature is 20° C with a velocity of 2 m/s (Fig 2.12). Determine the total drag force and the rate of heat transfer per unit width of the entire plate.

Known: Engine oil flows over a flat plate.Find: The total drag force and the rate of heat transfer per unit width of the plate are to be determined.Assumptions:The flow is steady and incompressible.The critical Reynolds number is . 55 10crRe

o2 20 60 2 40 Cf sT T T 3

6 2

876kg m 2870

0 144 W m 242 10 m s

Pr

k . C

sT C 20

T C

V m s

Oil

60

2

L= 5 m

A

-6 2

2m s 5m41322

242 10 m sL

VLRe

0 50 5 31 328 1 328 41322 6 533 10..

f LC . Re . .

2323 2

2

876kg m 2m s 1N6 533 10 5 1m

2 2 1kg m sD f s

VF C A .

= 57.23 N

which is less than the critical Reynolds number

0 50 5 1 3 1 30 664 0 664 41322 2870 1918..

LhL

Nu . Re Pr .k

o20.144 W m= 1918 55 2 W m C

5m

Ckh Nu .

L

o o2 255 2 W m C 5 1m 60 20 C =11,049 Ws sQ hA T T .

Note that, heat transfer is always from the higher-temperature medium to the lower-temperature one. In this case, it is from the oil to the plate. The heat transfer rate is per m width of the plate. The heat transfer for the entire plate can be obtained by multiplying the value obtained by the actual width of the plate.

Problem: A long 10-cm diameter steam pipe whose external surface temperature is 110oC passes through some open area that is not protected against the winds (Fig. 2.23). Determine the rate of heat loss from the pipe per unit length of its length when the air is at 1 atm pressure and 10oC and the wind is blowing across the pipe at a velocity of 8 m/s.

W in dT s = 11 0 o CSchematic:

Known: A steam pipe is exposed to windy air.Find: The rate of heat loss from the steam is to be determined.

Assumptions:• Steady operating conditions exist.• Radiation effects are negligible.• Air is an ideal gas.

The properties of air at the average film temperature of

2f sT T T = (110 + 10)/2 =60

-5 2

0 02808W m C = 0.7202

= 1.896 10 m s

k . Pr

45 2

8m s 0 1m4 219 10

1 896 10 m s

.VDRe .

.

4 55 81 2 1 3

1 42 3

1 2 4 51 34 5 84

1 42 3

0 620 3 1

282 0001 0 4

0 62 4 219 10 0 7202 4 219 100 3 1

282 0001 0 4 0 7202

. Re PrhD ReNu .

k ,. Pr

. . . ..

,. .

= 124.44

20.02808 W m C= 124 44 34 94 W m C

0 1m

kh Nu . .

D .

2

2 2

0 1m 1m 0 314m

34 94 W m C 0 314m 110 10 C = 1097.3W

s

s s

A pL DL . .

Q hA T T . .

Important numbers

LUo

L ReInertial force

Viscous forceReynolds number

a

Pr Momentum diffusivity

Thermal diffusivity

Prandtl number

2

3

LTTsg

Gr Buoyancy forces

Viscous forces

k

hLNu

Conduction

Convection Nusselt number

solidk

hLBi thermal internal resistance

surface film resistance

Grashof number

Biot number

Reference book: Fundamentals of Heat and Mass Transfer, Incropera & DeWitt