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MMU FET EME 4016 HEAT TRANSFER

CONDUCTION 1

INTRODUCTION TO HEAT TRANSFER

1.

Study of heat energy transferbetween material bodies as a result of temperature difference.

2.

Emphasis on the local rate of heat energy flow, rather than the completed overall condition

usually assumed in engineering thermodynamics. Hence it complements thermodynamics.

3.

The flow of heat could be in the mediumof solid, liquid, gas or in a vacuum.

4.

That heat flow can occur in a fluid explains the intimate relationship between heat transfer and

fluid dynamics.

5.

The trio: thermodynamics, fluid dynamics and heat transfer can be described as the field of

thermo-fluids.

6.

The heat transfer influences the temperature of bodies which can have consequences in

materials scienceand engineering.

7.

Likewise, thermal effects can cause thermal stresses in solid mechanics.

8.

Heat transfer is vital in energy studieswhenever heat energy is involved.

Wide applications

1.

Involved in energyproduction, energy utilization, energy conservation, energy storage, and

alternative energy.

2.

In designing of heat transfer equipment, such as boilers, heaters, evaporators, and refrigerators

which extends shelve lives of food.

3.

In heat removal, such as in combustion chambers, electrical machines, transformers,

switchgears, electronic devices, and air-conditioning units that affect our personal comfortindoor and on the road.

4.

In many high technologydevelopment, such as jet engines, rocket motors, space craft, nuclear

reactors.

5.

In environmentalstudies, thermal pollution is of concern in waste discharges into water and air,

and in cities, the heat island effect.

6.

Increasingly important in biologicalsciences and micro-systems.

Basic modes of heat transfer

The 3 basic modes that we are mainly concerned in this course are conduction, convectionand

radiation. Each mode has a basic heat rate equation.

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Conduction

In conduction, heat is transferred by vibrational energy to neighboring molecules within asolid, without

the molecules changing their mean positions. The basic heat rate law is Fouriers law(1822). Stated in 1-

D:

=

where

qis the heat transfer rate [W]

Ais the area normal to the heat flow [m2]

Tis the temperature in the solid [K] or [oC]

is the temperature gradient along the flow direction of axisx[K/m] or [oC/m], and

kis the thermal conductivity, a property of the solid material. [W/mK] or [W/mo

C]

Note that values of k is tabulated in tables, usually assumed to be constant, but sometimes need to

consider its value changing with temperature T.

The Fourier law is an experimentallaw, and note that it is a linearlaw.

It is useful to use the concept of a heat flux "which is defined as

" = [

].

Then Fouriers law can also be stated as

" =

In our context, the heat energy is denoted by the symbol Q [J]. Q = q = " , where tis thetime.

In simple situations, Fouriers law is often written in finite difference form:

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= /.

In conduction, we also need to know about the heat rate laws of convection and radiation, because at

the boundaryof the solid, there could be convection and radiation.

Convection

In convection, heat is transferred by movement of a bulk of fluid itself, from a surface or wall to the

fluid, and so more and quicker can be transferred. It is a well-known that there are 2 categories; forced

convection or free convection. The basic heat rate law is Newtons law of cooling(1791). This is also an

experimental law.

= where

A is the contact area between the surface and the fluid

is the surface temperature (often also denoted as , the wall temperature)is the fluid temperature flowing over the surface, andhis the convection heat transfer coefficient. [W/K]his complicated and will be studied extensively under convection part of the syllabus. It is also the link

with fluid dynamics. But as far as the conduction part of the syllabus is concerned, his usually taken to

be a known constant, thus making Newtons law simple to be used. In terms of heat flux, the law is

" = .

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Radiation

Heat transfer is by electromagnetic (EM) waves, through a vacuum or any media that allows the

transmission of an EM wave. The basic heat rate law is the Stefan-Boltzmann Law(1879). This is an

experimental law with theoretical justification. For a small black bodyplate at temperature surrounded by an infinitely large environment at , and assuming > ,then

= where A is the surface area of the plate, and is the Stefan-Boltzmann constant, 5.669 108W/ . It is important to note that the temperatures must be in K. In terms of heat flux,

" = Now, this law is highly non-linear in temperature. In certain situations, we can approximate it as a linear

law by following the format of Newtons law, and write it as

= .Here, it can be shown that the radiation heat transfer coefficient = .Overall heat energy balance in a control volume or surface

The heat rate equations can be used in combination to estimate overall energy balance, due to

conduction, convection, or radiation. Below left shows two examples of heat balance on two types of

glass windows.

In a general statement, illustrated on the right, energy stored is equal to the net energy into the volume

plus energy produced inside the body (an energy source or generation).

=

So one, or all, or none of the three basic heat rates can be present in each of .

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THE CONDUCTION EQUATION

To study the conduction in details inside a solid, we derive a general differential equation with the help

of Fouriers law. To be general, we use the 3-D form of Fouriers law, which is the vector equation

= .Consider the energy balance in a solid of volume Vwith surface areaS. To be general, include a heat

source[/3] uniformly distributed in the solid. If u is the specific internal energy [J/3], then, forthe whole solid,

Rate of heat stored = Rate of heat input + Rate of heat generation

= (" )

= (" ) Hence removing the integration results in

= +

Substitute in the above Fouriers law and let = , where is the density, and cthe specific heat[J/kg ] . Note that for a solid in most situations there is hardly any difference between and .

= . . Called the conduction equation. (Text calls it heat equation.)

In Cartesian system, the conduction equation is

=

If , c and k are constant, called constant properties, then the conduction equation becomes

=

It is useful to define a new property of the solid, called the thermal diffusivity , which affects the timedependent or transient behavior in conduction. / . Then the conduction equation takes the form

= /

For steady stateconduction, there is no dependence on time t, and

= 0The operator in 3-D must be expanded in full, in accordance with the coordinate system to be used.

For 1-D, in Cartesian coordinate system, the conduction equation is just = 0.

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Essentially, we solve the conduction equation subject to the boundary conditions, which are: known

temperature, or known heat flow, or obey the heat rate equations. Knowing theT, we can then calculate

the heat transfer q anywhere in the solid, using Fouriers law.

STEADY STATE HEAT TRANSFER IN 1-D AND WITHOUT HEAT SOURCE

This is the simplest situation but the results have many practical applications, as a first level analysis.

Consider the example of finding the heat transfer in an infinite vertical plate or slab of thicknessL, with

thermal conductivity k, and subjected to a constant temperatures and on the left and right side,respectively. Without a heat source, =0. First, sketch a diagram with appropriate coordinates.

The conduction equation is 0 = 0, or = 0.

Solution is = , an integration constant. = , where is a second constant to be found.

Applying the boundary conditions,

BC 1: At = 0, = . Giving = 0 + = BC 2: At = , = . Giving = , or = /Hence the solution to the conduction equation is

=

(a linear temperature profile)

The heat flux through the plate is " = =

=

(as expected from the finitedifference form of Fouriers law)

Thermal circuit method

It turns out that because of the linear T profile, it is faster to solve these problems using the thermal

circuit method. The method draws on the analogy between Fouriers law and Ohms law, and does not

require solving the conduction equation.

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The analogue is between the electric circuit with = and the thermal circuit =.

From the finite difference form, write = =

.

Thus the heat rate qcorresponds to the current flow I.

The thermal driving force corresponds to the voltage potential .

The thermal resistance = corresponds to the electrical resistance R.

Going back to the simple example solved using the conduction equation, if we use thermal circuit, we

can write down the heat transfer directly as = =

.

Multiple thermal resistances

The power of the method is evident when there are multiple layers of solids or called acompositewall.

We represent the heat transfers as a circuit and we can sum up the thermal resistances in the same way

as in electrical circuit theory. The temperatures are the nodes. Thus we have in symbolic form

=

Furthermore we can do the same to Newtons law and S-Bs law.

From Newtons law, = = . So the convection thermal resistance is = .

From S-Bs law, = = . So the radiation thermal resistance is = .

And these thermal resistances are used in exactly the same way as the conduction thermal resistances.

In multiple layer calculations, it is convenient to define an overall heat transfer coefficient U which is

effectively an equivalenth for the combined overall heat transfer in a multilayer system. Thus,

=

Or [/ ]

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Example 1. Convection boundary conditions on both sides of an infinite slab.

Thermal circuit:

Heat rate:

=

= 1

1

= , ,

Example 2

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As the same q is flowing through each layer, we can also find any temperature in the circuit.

= , , =, ,

=,

+

= =

= etc

Example 3

As a rough approximation of 1-D flow, 2-D flow is treated as 1-D by using series and parallel resistances.

Alternatively, summation of parallel resistances can be performed with =

or =

+ .

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Thermal resistance of a cylindrical shell

In cylindrical coordinates, the thermal resistance for a cylindrical shell must be re-derived for

conduction. (Not needed for convection and radiation, as they have the same form as in the plate case,

but only involve a different expression for the areaA.)

We apply 1-D Fouriers law to an elementary cylindrical shell of thickness dr, distanceraway from the

centre, inside the cylindrical shell of length L. Assuming heat is transferred outwards, along the r

direction,

= = 2

Write it asdr = -

dT

Then integrating over the whole shell, dr

= -

dT

Giving = -

Or =

. Therefore the thermal resistance of a cylindrical shell is =

Hence when there are multiple layers of cylindrical shells, we use this form of the thermal resistance.

Thermal resistance for a spherical shell

Similarly, for the spherical shell, we write as

= = 4

Then dr

= -

dT

, giving [] = -

which leads to =

. So =

Overall heat transfer coefficient for cylindrical or spherical shells

As we define , the question arises which A is to be used, because the area is different at adifferent radius. The convention is to use either the extreme inside area = , or the extreme outsidearea = . Then we call the ( ) the overall heat transfer coefficient based on the inside (oroutside) area.

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The concept of a critical thickness

Consider a hot pipe with an external radius covered with aninsulationmaterial of radius , therebygiving an insulation thickness of ( to the pipe. The idea of insulation is to reduce heat loss byusing a material of low thermal conductivity.

If the insulation material has thermal conductivityk, and the convection (heat transfer) coefficient

outside the insulation is h, it can be shown that must be greater than a certain critical value in orderfor the insulator to be effective. If < , the insulation material actually increases the heat loss!

The value of = , the critical insulation radius.

Contact resistance

This is essentially the thermal resistance of a small air gap, with partial contact, when two solid surfaces

are in imperfect contact with each other. We denote the thermal contact resistance as [ /],and treat it in the thermal circuit as just another resistance, between two temperature nodes. Since thegap is small, the difference in temperature will appear as a discontinuity in the temperature distribution.