Heat Transfer 0300 - Forced Convection on a Horizontal Flat Plate

HEAT TRANSFER Prepared by

Mohammad Fazlur Rahman Asst. Professor (AERO)

B. S. Abdur Rahman University

This document contains the basic information regarding the subject matter

Heat Transfer. The effort is made to help the students getting exposure to

the subject as well as understand the basic and fundamental behaviour of

the heat transfer phenomenon. It must be noted that this document in no way

can avoid the use of text books. For the detailed and deep understanding of

the subject matter students must refer the text books. While providing

information the syllabus of the B. S. Abdur Rahman University has been

targeted.

Forced

Convection over a

Flat Plate

Notes on Heat Transfer prepared by Asst. Professor Mohammad Page 1

Contents

Introduction ............................................................................................................................ 2

Drag and Heat Transfer in External Flow ..................................................................... 3

Friction Drag and Pressure Drag ..................................................................................... 4

Parallel Flow over Flat Plate ............................................................................................. 6

Friction Coefficient .......................................................................................................... 7

Heat Transfer Coefficient .............................................................................................. 9

Special Cases ....................................................................................................................... 11

Flat Plate with Unheated Starting Length ............................................................... 11

Uniform Heat Flux .......................................................................................................... 11


Introduction After having studied the forced convection and having solved the equations involved in it, now

is the time to tackle different cases to discuss and see how the same finding can be used for the

various cases.

The forced convection differs from the natural one only in the case of flow as how the flow

takes place and which is the driving force which affects the flow and its behaviour. First of all

it must be noted that so called free convection is not that in which there is not force acting. The

force acting in it to cause the bulk motion to occur is the natural force of buoyancy and hence

it is more appropriate to call it a natural convection rather than free convection. Now whatever

the flow we have studied in our previous level courses like Fluid Mechanics and Aerodynamics,

they all contain flow of fluid and thats all are actually forced flow. So there is a clear analogy

in the governing equations of forced convection and flows in fluid mechanics and

aerodynamics. The only difference is the energy equation which involves thermal energy

instead of kinetic energy and flow energy.

The forced flow convection process can be further divided into two parts 1. External flow and

2. Internal flow. External flows are the cases when fluid is in contact with one of the solid

surface and other surface of the fluid is free and is in contact with free air. In the case of internal

flow, the entire fluid surface is surrounded by the solid surface and this flow is highly affected

by the viscous effect of the fluid. We have a third type also which is not classified as a separate

class though, in this type of flow top and bottom of the fluid boundary is in contact with the

solid surface and flow takes place in between them. It is known as Couette Flow.

Heat Transfer

(Forced Convection over a Flat Plate)


Drag and Heat Transfer in External Flow The main reason to discuss all these flows and their behaviour is to get the appropriate

expressions for the surface friction drag coefficient and the convection heat transfer coefficient.

There are many natural phenomenon in which the external flow over a solid surface takes place,

like rain drops fall, flow passing automobile, power lines, trees and underwater pipelines etc.

Cooling of metal or plastic sheets, steam and hot water pipes and extruded wires and

rectangular fins etc. these surfaces when exposed to external fluid flow, they not only cause

the heat transfer, rather there exists a shear drag on the solid surface owing to the viscous effect

of the fluid. This entire phenomenon is inter related and mutual interaction of thermal boundary

layer and velocity boundary layer takes place. The knowledge of heat transfer and momentum

transfer relation helps us understand such phenomenon and make us solve the problem

normally we encounter.

Since the geometry which we encounter in the normal life is not always a simple one; the

complicacy of the geometry and their interaction with the fluid flow forces us to go the

numerical way. Because analytical solution for such problems are very complicated.

Availability of high speed computers have made our life a bit easy in this regard, and most of

the data collection and analyses is done through numerical experimentations quickly by

solving the governing equations numerically. The time consuming and expensive experimental

testing are done at the final stage of design to collect some real time data and validate the data

collected numerically.

In this regard we discuss and use two types of velocity frequently. The velocity far away from

the solid body is called the free stream velocity and this is the fluid velocity we expect to be

approaching the body encountering the solid body so it is also the upstream velocity. We call

it approach velocity and denote it by . The subscript denotes and reminds that this is the

velocity far away from the surface which remains unaffected by the presence of the solid. The

upstream velocity may vary depending with respect to time but for the sake of convenience and

ease in analysis we take it to be a steady flow velocity which remains constant with time.

Another velocity in the near vicinity of the solid is the local velocity which is affected by the

solid and its value depends upon the geometry of the solid apart from other influencing

parameters. The local fluid velocity ranges from zero at the surface to the free stream velocity

far away from the surface.


Friction Drag and Pressure Drag When the flow over a body takes place, there exists a drag on the body which tries to pull the

body in the direction of fluid flow. This force is called Drag force. By the mechanism of its

generation this is of two types. 1. Friction drag and 2. Pressure drag. They both act in the same

direction (in the direction of flow) but their origination is different so they have are drag and

anybody in the fluid flow faces a drag which is a combination of these two drags.

Friction drag is clear by name that it is the result of surface friction. This generates due to the

realness of the fluid which and this property of the fluid is called viscosity. This always works

tangentially to the surface. This depends upon the roughness of the surface.

Pressure drag is also clear by its name that it is coming into picture due to the pressure

difference between the front portion and back portion of the body. It is also the integral sum of

the pressure over the surface in the direction of fluid flow. It exists due to the presence of the

wake in the flow leaving the body. By a proper design we can minimize the wake but it can

never be removed completely.

If a flat plate is laid along the flow direction, it will produce very small pressure drag but

comparatively large amount of friction drag due to existence of very thin wake but large surface

area involved. On the other hand if the plate is laid perpendicular to the flow direction, a large

wake will be present there and a huge amount of the pressure

drag will be felt and friction drag also called surface friction

drag will be very small. In the terms of aerodynamics, drag

on any slender body is mostly surface friction drag while

drag on any bluff body is mostly pressure drag. None the

less, at any time or in any case total drag on anybody in the

fluid motion will be given by the sum of these two drags.

Sometimes pressure drag is also referred to as form drag.

= +

For the friction drag, it is viscosity which causes the drag force

to come into being, but for the pressure drag too, it is the

viscosity which becomes the reason for its existence. In the

ideal case when the fluid has no viscosity, both types of drag

are missing no matter what is the shape of the body.


We normally deal with the coefficients of the forces. So the drag force for the pressure and


Parallel Flow over Flat Plate Let us consider upon a normal flat plate having a flow over it. The boundary layer is formed

over it due to the realness of the fluid. Initially there is a laminar boundary layer and then very

soon the flow becomes turbulent following a thin transition region. This happens due the fact

that it is the distance of the location from the leading edge which plays the role of a

characteristic length in the Reynolds number expression. The transition from laminar to

turbulent takes place at a location where Reynolds number reaches its critical value for

transition.

The transition of the flow from laminar to turbulent flow depends upon the surface geometry,

surface roughness, upstream velocity, surface temperature, and the type of fluid among some

more other things. It is best characterised by the Reynolds number which is expressed as below:

=

=

So the value of Reynolds number rises as the location under consideration goes away from the

leading edge. The transition of the flow from laminar to turbulent takes place at about =

1 105 but becomes fully developed turbulent flow typically around = 3 106. In

engineering a general value accepted for the critical Reynolds number is 5 105 which gives

turbulent flow in any case. Actually under controlled condition the flow can be maintained to

be laminar up to a maximum value of Reynolds number equal to 5 105.

=

=


Friction Coefficient

We have already derived the expressions for the boundary layer flow over a flat plate. We can

use them here for finding the friction coefficient.

For the laminar flow:

, =4.91

1 2

and , =0.664

1 2

< 5 105

Same for the turbulent boundary layer will be:

, =0.38

1 5

and , =0.059

1 5

5 105 < < 107

The boundary layer thickness is directly proportional and friction coefficient is inversely

proportional to 1 2 . Therefore at the leading edge where = 0 boundary layer thickness will

be zero while friction coefficient is supposedly infinite. Friction coefficient thereafter decreases

by a factor of 1 2 in the flow direction. The local friction coefficient are higher in turbulent

flow than in laminar flow owing to the intense mixing that occurs in turbulent boundary layer.

In the transition region the friction coefficient increases till it becomes highest in the fully

grown turbulent region. Thereafter it starts decreasing by a factor of 1 5 in the flow direction.

The average friction coefficient over the entire plate is determined by integrating the local

friction coefficient over the entire length of the plate.

For laminar region:

=1

,

0

=1

0.664

1 2

0

=0.664

(

)

1 2

()1 2

0

=0.664

(

)

1 2 1 2

1

2

|

0

= 1.33 (

)

1 2

=.

For Turbulent region:

=1

,

0

=1

0.059

1 5

0


=0.059

(

)

1 5

()1 5

0

=0.059

(

)

1 5 4 5

4

5

|

0

= 0.074 (

)

1 5

=.

So if the flow is entirely laminar or entirely turbulent then we can find the friction coefficient

as per the expressions obtained above. Sometimes the plate is very long and in this case though

both the types of flow will exist on the surface, laminar flow friction coefficient can be ignored

and only turbulent flow can be estimated to find the total friction coefficient. If the plate is long

enough to have a turbulent region but not long enough to ignore the laminar region then the

average friction coefficient can be found using the concept given below:

=1

( ,

0

+ ,

)

The transition region is small enough to be included in the turbulent region. Now again taking

the critical Reynolds number to be = 5 105 and then substituting the critical length in

the above expression and integrating it over the entire plate after the critical length will give

that:

=.

5 105 10

7

The constants in this relation will be different for the different critical Reynolds numbers. Also,

the surfaces are assumed to be smooth, and the free stream to be turbulent free. For laminar

flow, the friction coefficient depends on only the Reynolds number, and the surface roughness

has no effect. For turbulent flow, however, surface roughness causes the friction coefficient to

increase several fold, to the point that in fully turbulent region the friction coefficient is a

function of surface roughness alone, and independent of the Reynolds number. That is the

case of pipe flow.


A curve fit of experimental data for the average friction coefficient in this regime is given by

Schilchting as

Rough surface, turbulent:

(. .

)

.

where is the surface roughness, and L is the length of the plate in the flow direction. In the

absence of a better relation, the relation above can be used for turbulent flow on rough surfaces

for > 106, especially when > 104

Heat Transfer Coefficient

We have already determined the Nusselt number for a location for the laminar flow over a

flat plate:

For Laminar Boundary Layer

=

= 0.332

0.5 1 3 (for > 0.6, 5 10

5)

For turbulent boundary layer

=

= 0.0296

0.8 1 3 (for 0.6 < > 60, 5 10

5 107)

(Note: Remember we have no solution for the flow with Prandtl number less than 0.6. For the Prandtl number

equal to 1 we get a self-similar flow in which velocity boundary layer coincides with the thermal boundary layer.)

For the laminar flow heat transfer coefficient is

proportional to 0.5 and thus to 0.5. Therefore it

is infinite at the leading edge i.e. = 0 and

decreases by a factor of 0.5 in the flow direction.

The variation of boundary layer thickness and

the friction and heat transfer coefficients along an

isothermal flat plate are shown in the figure. The

local friction and heat transfer coefficient are

higher in turbulent flow than they are in laminar

flow. Also, reaches its highest values when the flow becomes fully turbulent, and then

decreases by a factor of 0.2 in the flow direction.


The average Nusselt number over the entire plate is determined by integrating the expression

for the entire length of the plate and then dividing it by the length itself. By doing so we find

the below equations.

For Laminar Boundary Layer

=

= 0.664

0.5 1 3 (for > 0.6, 5 10

5)

For turbulent boundary layer

=

= 0.037

0.8 1 3 (for 0.6 < > 60, 5 10

5 107)

The equation for the laminar boundary layer gives the heat transfer coefficient for the entire

plate when the flow is laminar over the entire plate. The second relation gives the average heat

transfer coefficient for the entire plate only when the flow is turbulent over the entire plate.

Sometimes when the laminar region is too small relative to the turbulent flow region, the

turbulent flow equation can be applied to the entire plate and accepting a little compromise in

the accuracy.

In some other cases when the plate is sufficiently long for the flow to become turbulent, but

not long enough to disregard the laminar flow region, the average heat transfer coefficient over

the entire plate must be estimated using both the

relation over the appropriate regions.

=1

( , + ,

0

)

Again taking the critical Reynolds number to

be = 5 105 and performing the integration

the average Nusselt number over the entire plate will

be given by:

=

= (0.037

0.8 871)1/3

for ( 0.6 < > 60,

5 105 107)

In the above case it has been assumed flow over the plate is partly laminar initially and partly

turbulent afterward. Above relation depends upon the critical Reynolds number and for

different value of critical Reynolds number it will be different.


Liquid metals such as mercury which have high thermal conductivities, and are commonly used

in the applications that require high heat transfer rates. However they have 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:

= 0.565()1/2

< 0.05

Above relation has a limitation of the Prandtl number and will change accordingly as the

Prandtl number changes. It is however desirable to have a single correlation that applies to all

fluids, including liquid metals. By curve-fitting existing data, Churchill and Ozoe (1973)

proposed the following relation which is applicable for all Prandtl numbers and is claimed to

be accurate to 1%.

=

=

. /

/

[ + (. )/]/

These relations have been obtained for the case of isothermal surfaces, but could also be used

for approximately for the case of non-isothermal surfaces by assuming the surface temperature

to be constant at some average value. Also, the surfaces are assumed to be smooth, and the free

stream to be turbulent free. The effect of variable properties can be accounted for by evaluating

all properties at the film temperature.

Special Cases

Flat Plate with Unheated Starting Length

Uniform Heat Flux

Documents

Heat Transfer 0300 - Forced Convection on a Horizontal Flat Plate