Chapter 4 - Convection Heat Transfer Updated

CHAPTER 4

Convection Heat Transfer

Outlines

To examine the methods of calculating convection

heat transfer (particularly, the ways of predicting the

value of convection heat transfer coefficient, h)

Convection heat transfer requires an energy balance

along with an analysis of the fluid dynamics of the

problem concern. So that. The discussion will

consider;

simple relations of fluid dynamics

Boundary layer analysis

Important for basic

understanding of

convection heat

transfer

The region of flow that develops from the leading edge of the plate in which

the effects of viscosity boundary layer (The y position where the

boundary layer ends and the velocity become 0.99 of the free-stream value)

Initially the boundary layer development is laminar but at some

critical distance from the leading edge (depending on the flow field & fluid

properties), small disturbance in the flow begin to become amplified, and

a transition process takes place until the flow become turbulent.

occurs when where

Flat

plate

Tube Laminar flow

Turbulence flow

For transition between

laminar & turbulence

d = Tube diameter

For transition between

laminar & turbulence

Re. no. (Tube)

Other Form:

Define the mass velocity as

So that, mass flow rate

The Reynolds no. may also written

Classification of Fluid Flows

Inviscid Flow

Although no real fluid is inviscid, in some instance the fluid may be treated

as such, and useful to present some of the equation that apply in these

circumstances (the flow at a sufficiently large distance from the plate will

behave as a nonviscous flow system).

or

Where,

Fundamentally a Dynamic Equation

(The Bernoulli equation)

To solve convection heat transfer

coefficient, h we have to:

1) Identify the type of fluid involve to

get the fluid properties

2) State the process

1) Type of Fluid

# Specify the equation of state of fluid to calculate pressure drop in compressible

flow.

An ideal gas

e = Internal Energy

i = Entalpy

Gas

Where:

Where:

Air

2) State the Process

Example: Reversible Adiabatic Flow through a nozzle

The relation involved which is relating the

properties at some points in the flow to the

Mach no. & stagnation properties

Where:

a: Local velocity of sound

For an ideal gas: For air behaving as an ideal gas:

Laminar Boundary Layer on Flat Plate

From the analytical analysis by making a force and momentum balance on the

element yield the momentum equation for the laminar boundary layer with

constant properties.

Can be solved for many

boundary conditions.

For development in this chapter, we

shall satisfied with an approximate

analysis that furnishes an easier

solution without a loss in physical

understanding of process involved.

Laminar Boundary Layer on Flat Plate Consider the boundary layer flow system as shown:

The free-stream velocity outside the boundary

layer is u∞ and the boundary layer thickness is .

We wish to make a momentum-and-force balance

on the control volume bounded by the plane 1, 2,

A-A and the solid wall.

The boundary layer thickness,

Mass flow rate Where:

So that

Example 1: Mass Flow & Boundary-Layer

Thickness

Air at 27 °C and 1 atm flows over a flat plate at a speed of 2 m/s. Calculate the boundary-layer thickness at distances of 20 cm and 40 cm from the leading edge of the plate. Calculate the mass flow that enters the boundary layer between x=20 cm and x= 40 cm. The viscosity of air at 27 ° C is 1.85 x 10-5 kg/ m . s. Assume unit depth in the z direction.

J.P Holman

Solution:

J.P Holman

Dimensionless Numbers in Heat transfer

Dimensionless numbers are tools of heat transfer. They are not actually necessary to solve convection problems but the make it easier.

Key dimensionless numbers used in heat transfer are:

1. Reynolds number (Re)

2. Nusselt number (Nu)

3. Prandtl number (Pr)

4. Grashof number (Gr)

5. Rayleigh number (Ra)

Reynolds Number

The flow regime depends on the ratio of the inertia forces to viscous forces in the

fluid. This ratio is called Reynolds no.

Where.

V = upstream velocity (equivalent to the free-stream velocity for a flat plate)

Lc = characteristic length of the geometries

= kinematic viscosity of the fluid (units: m2/s)

# The critical Reynolds no. The Reynolds no. at which the flow become turbulent

# The value of the critical Reynolds no. is different for different geometries and flow

conditions. For flow over a flat plate, the general value of the critical Reynolds no is

Where is the distance from the

leading edge of the plate at which

transition from laminar to turbulent flow

occurs

Nusselt Number

Represent the enhancement of

heat transfer through a fluid

layer as a result of convection

relative to conduction across

the same fluid layer.

Nu large the more effective the convection

Nu = 1 heat transfer across the layer by pure conduction

Prandtl Number

The relative thickness of the velocity & the thermal boundary layer is best described

by the Prandtl no. (dimensionless parameter).

The Prandtl numbers of fluids range from less than 0.01 for liquid metals to

more than 100,000 for heavy oil.

(Can be found in tables)

Grashof Number

where

This dimensionless number is mainly for natural convection.

It is the ratio of buoyancy forces to the viscous forces. The Grashof

number plays the same role that the Reynolds number plays for forced

convection.

When buoyancy forces overcome viscous forces, the flow starts to become

turbulent.

Rayleigh Number

dimensionless number associated with buoyancy driven flow (also known as free convection or natural convection). When the Rayleigh number is below the critical value for that fluid, heat transfer is primarily in the form of conduction; when it exceeds the critical value, heat transfer is primarily in the form of convection.

This number is just the product of the Grashof Number and the Prandtl number

In general,

Nu = f(Re, Pr, Gr)

The Thermal Boundary layer - Flat Plate

Exist when temperature gradient are present in the flow.

If the fluid properties were constant throughout the flow, an appreciable

variation between the wall and free stream condition which is film temp. Tf define as:

Used Tf to get the fluid properties

from properties fluids table

The Thermal Boundary layer - Flat Plate Consider the system shown

Tw: The temp. of the wall

T∞ : The temp. of the fluid outside the thermal boundary layer

: Thickness of the thermal boundary layer

For the plate heated

over its entire length

Basic: convection/conduction

hx: Heat transfer coefficient in term of the

distance from the leading edge of the plate

0.6 < Pr < 50

Used the average heat transfer coefficient

Case 1

Rex Pr > 100


For the plate heated

starts at

Case 2

or

Where:


Constant Heat Flux

To find the distribution of the plate surface temp.

and

From these equation, can

be produce equation

So that

The energy Equation of the boundary layer (heat flow, q)

& the thermal boundary layer (heat flux, qw)

To determine heat flow, q and heat flux, q w

From energy equation of the

boundary layer

From energy equation of the

thermal boundary layer

Heat flow, q

1) Film temp.

2) The properties of fluid at Tf such as kinematic viscosity, thermal conduction

coefficient, heat capacity, Prandtle no.

There is an appreciable variation between wall & free

stream condition, so that, it is recommended that the

properties be evaluated at film temp.

Determine:

3) Rex at x=xL

4) Nusselt No. 0.6 < Pr < 50

Rex Pr > 100

Then

5) Heat transfer coefficient, h

6) The average heat transfer coefficient,

7) Heat Flow,

Heat flux, qw

Example 2: Isothermal Flat Plate Heated

Over Entire Length

Air at 27 °C and 1 atm flows over a flat plate at a speed of 2 m/s. Calculate the boundary-layer thickness at distances of 20 cm and 40 cm from the leading edge of the plate. Calculate the mass flow that enters the boundary layer between x=20 cm and x= 40 cm. The viscosity of air at 27 ° C is 1.85 x 10-5 kg/ m . s. Assume unit depth in the z direction and the plate is heated over its entire length to a temp. of 60 ° C. Calculate the heat transferred in

(a) The first 20 cm of the plate and

(b) The first 40 cm of the plate.

J.P Holman

Solution:

J.P Holman

Example 3: Flat Plate with Constant Heat

Flux

A 1.0 KW heater is constructed of a glass plate with an electrically

conducting film that produces a constant heat flux. The plate is 60 cm

by 60 cm and placed in an airstream at 27 °C, 1 atm with u∞ = 5 m/s.

Calculate

(a) The average temp. different along the plate and

(b) The temperature difference at the trailing edge.

Example 4: Oil Flow Over Heated Flat Plate

Engine oil at 20 °C is forced over a 20-cm-square plate at a velocity of

1.2 m/s. The plate is heated to a uniform temp. of 60 °C. Calculate

the heat lost by the plate.

J.P Holman

Solution:

J.P Holman

The relation between fluid friction & heat

transfer

The shear stress at the wall be

expressed in term of a friction

coefficient, Cf

Where:

The relation between fluid friction

and heat transfer for Laminar flow

on a flat plate

This is important relation between

friction & heat transfer is the drag force

(D) which is depends on the average

shear stress. The average of shear

stress is a friction coeffiecient Cfx

Drag Force, D = (shear stress) (Area)

Example 5: Drag Force on a Flat Plate

Air at 27 °C and 1 atm flows over a flat plate at a

speed of 2 m/s. Calculate the boundary-layer

thickness at distances of 20 cm and 40 cm from the

leading edge of the plate. Calculate the mass flow

that enters the boundary layer between x=20 cm

and x= 40 cm. The viscosity of air at 27 ° C is 1.85 x

10-5 kg/ m . s. Assume unit depth in the z direction

and the plate is heated over its entire length to a

temp. of 60 ° C. Compute the drag forced exerted

on the first 40 cm of the plate using the analogy

between fluid friction and heat transfer

Turbulent-boundary-layer heat transfer (q)- Flat Plate

1) Determine either the flow is turbulent

region or not. Check based on Re. no.

Turbulent Region

2) Heat transfer (q) from the plate is:

Where:

Turbulent-boundary-layer thickness ( )- Flat Plate

The boundary layer thickness measured when 500000>Re<10000000

The boundary layer is fully turbulent

from the ledge of the plate

The boundary layer follows a laminar

growth pattern up to Rcrit= 5 x 105 and

turbulent growth thereafter

Air at 20◦C and 1 atm flows over a flat plate at 35 m/s. The plate is 75 cm

long and is maintained at 60◦C. Assuming unit depth in the z direction,

i. Calculate the heat transfer from the plate

ii. Calculate the turbulent-boundary-layer thickness at the end of the plate

if it is develop.

(a) from the leading edge of the plate

(b) from the transition point at Recrit.

Example 6: Turbulent Heat Transfer &

boundary Layer Thickness – Flat Plate

i.

ii.

Heat Transfer in Laminar Tube Flow Consider the tube flow system

Aim to calculate the heat transfer

under developed flow condition

(Laminar Flow)

Consider the fluid element derive to get the

velocity and temp. distribution

Heat Transfer in Laminar Tube Flow From the analysis, the analytical solution give;

The velocity at the center of the tube

The velocity

distribution

The temperature

distribution

The Bulk Temperature In tube flow, convection heat transfer coefficient, h

defined by:

Where, Tw : The wall temp. Tb : Bulk temp.

From the analysis, the analytical solution give;

The bulk temp The wall temp

Heat transfer coefficient Heat transfer coefficient in term of the Nusselt No.

Turbulent Flow in a Tube

Velocity profile for turbulent flow in a tube

To determine heat transfer analytically should know the temp.

distribution in the flow

# to obtain temp. distribution, the analysis must take into consideration

the effect of the turbulent eddies in the transfer of heat and momentum)

Turbulent Flow in a Tube

From the analysis, the analytical solution give;

Relates the heat transfer rate to the friction loss in tube flow

Where,

Heat transfer coefficient in term of the Nusselt No.

or

Pr ≈ 1.0

Relation for turbulent heat transfer in smooth tube

# from this analytical solution, shows that h higher than those observed in experiment Pr 2/3

SUMMARY OF

EQUATIONS

Chapter 4 - Convection Heat Transfer Updated

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