2 conduction

CONDUCTION HEAT TRANSFER


Heat conduction in a medium, in general, is three-dimensional and time dependent.

And the temperature in a medium varies with position as well as time. That is, T = T(x, y, z, t).

Heat conduction in a medium is said to be steady when the temperature does not vary with time, and unsteady or transient when it does.

BASIC PRINCIPLES IN THERMODYNAMICS

THE FIRST LAW OF THERMODYNAMICS

The first law of thermodynamics, also known as theconservation of energy principle, states that energy can neitherbe created nor destroyed; it can only change forms.

The conservation of energy principle for any system undergoingany process may be expressed as follows: The net change(increase or decrease) in the total energy of the system during aprocess is equal to the difference between the total energyentering and the total energy leaving the system during thatprocess.

Energy balance for any system undergoing any process:

Or in rate form:



In steady operation, the rate of energy transfer to a system isequal to the rate of energy from the system.


In heat transfer analysis, it is more convenient to write a heatbalance and to treat the conversion of nuclear, chemical, andelectrical energies into thermal energy as heat generation.

Energy Conservation

Heat in Heat out + Heat generated =The change of the amount

of energy in the system.

Heat in

Heat in

Heat

generated

Heat in Heat out Heat generatedThe change of the amount of

energy in the system.

Time Time- + =

TimeTime

Rate of

heat in

The rate of change of the amount

of energy in the system.Rate of

heat out

Rate of heat

generated- + =

The principle of energy conservation is used to derive general heat transfer

equations and to solve heat transfer problems.

SURFACE ENERGY BALANCE

A surface contains no volume or mass, and thus no energy.Therefore, a surface can be viewed as a fictitious system whoseenergy content remains constant during a process.

The energy balance for a surface can be expressed as:

This relation is valid for both steady and transient conditions, andthe surface energy balance does not involve heat generationsince a surface does not have a volume.


SURFACE ENERGY BALANCE



General Heat Conduction Equation:Rectangular Coordinate

Consider a small rectangular element of length x, width y, and height z, as shown.

Assume the density of the body is and thespecific heat is C.

An energy balance on this element during asmall time interval t can be expressed as:


General Heat Conduction Equation: Rectangular Coordinate


Noting that the heat transfer areas of the element for heat conduction in the x,y, and z directions are Ax =y z, Ay = x z, and Az = x y, respectively, andtaking the limit as x, y, z and t 0 yields:

where again the property = k/Cp is the thermal diffusivity of the material



In the case of constant thermal conductivity:

Under specified condition, the above equation reduces to:


One-dimensional heat conduction: Plane Wall


General Heat Conduction Equation: Cylindrical Coordinates


One-dimensional heat conduction: Cylinders

For the case of constant thermal

conductivity, the equation above

reduces to:

where again the property = k/Cpis the thermal diffusivity of the material


One-dimensional heat conduction: Cylinders


General heat conduction equation: Spherical Coordinates


One-dimensional heat conduction: Spheres

Under specified conditions:


BOUNDARY AND INITIAL CONDITIONS

The mathematical expressions of the thermal conditions atthe boundaries are called the boundary conditions.

To describe a heat transfer problem completely, twoboundary conditions must be given for each direction ofthe coordinate system along which heat transfer issignificant.


DIRECTION BOUNDARYCONDITIONS

1-D 2

2-D 4

3-D 6



The number of boundary conditions that needs to bespecified in a direction is equal to the order of thedifferential equation in that direction.



Initial condition, usually specified at time t = 0, is a mathematicalexpression for the temperature distribution of the mediuminitially.

Initial condition can be specified in the general form as:

where the function f(x, y, z) represents the temperature distribution throughout the medium at time t = 0.



Boundary conditions most commonly encountered in practice are the following:

Specified temperature

Specified heat flux

Convection boundary conditions

Radiation boundary conditions

Boundary and Initial Conditions

Specified temperature boundary condition

Specified heat flux boundary condition


Insulated boundary

- a well-insulated surface can be modelled as a surface with a specified heat flux of zero


Convection boundary condition

- the convection boundary condition is based on a surface energy balance expressed as:


Radiation boundary condition


Consider the base plate of a 1200-W household iron that has a thicknessof L = 0.5 cm, base area of A = 300 cm2, and thermal conductivity of k=15W/m C. The inner surface of the base plate is subjected to uniformheat flux generated by the resistance heaters inside, and the outersurface loses heat to the surroundings at T = 20C by convection, asshown. Taking the convection heat transfer coefficient to be h = 80 W/m2 C and disregarding heat loss by radiation, obtain an expression for thevariation of temperature in the base plate, and evaluate the temperaturesat the inner and the outer surfaces.


The inner surface of the base plate

Is subjected to uniform heat flux at a

rate of:

The differential equation for this problem

can be expressed as:

B. C.

(a)

Applying the boundary conditions:

Noting that

The application of the second boundary condition gives:


Substituting C1 and solving for C2 we

obtain

Now substituting C1 and

C2 to (a) gives

Modelling and Solving Heater Transfer Problem

T = 50CAt t= 0 sec

Insulated surface (boundary condition)

1. First you have to know the boundary conditions and initial conditions.

2. With those conditions defined, the physical system is determined.

3. If you know the subsequent disturbance to the system, you can fully

predict the behavior of the system.

4. Initial conditions are related to transient problems. For steady state,

initial conditions are irreverent.

Think about it in the real world set up. What conditions do you need to

know to define a physical system so that you can predict the system.

Initial condition

Modelling and Solving Heater Transfer Problem

1. Normally the systems in our world can be modeled using a rectangular, a

cylindrical, and a spherical coordinate because of their geometries.

2. Once the system is modeled, and the initial and boundary conditions are

determined. The rest is solving the mathematical equations.

3. When the geometry of the system or the conditions are more complex,

numerical simulation is required.

Wall made of concrete

The wall thickness is 20cm

AC

AC maintains the temperature

of the room at 25C

Out side temperature is at 36C

What is the temperature profile across the wall thickness?

The problem is a 1-D steady

state problem. 25C 36C


Consider a large plane wall of thickness L = 0.06 m and thermal conductivityk = 1.2 W/m C in space. The wall is covered with white porcelain tiles thathave an emissivity of = 0.85 and a solar absorptivity of = 0.26, as shown.The inner surface of the wall is maintained at T1 = 300 K at all times, whilethe outer surface is exposed to solar radiation that is incident at a rate ofqsolar = 800 W/m2. The outer surface is also losing heat by radiation to deepspace at 0 K. Determine the temperature of the outer surface of the wall andthe rate of heat transfer through the wall when steady operating conditionsare reached.


Consider a steam pipe of length L = 20 m, inner radius r1 = 6 cm, outer radius r2 = 8 cm, and thermal conductivity k = 20 W/m C, as shown. The inner and outer surfaces of the pipe are maintained at average temperatures of T1 = 150C and T2 = 60C, respectively. Obtain a general relation for the temperature distribution inside the pipe under steady conditions, and determine the rate of heat loss from the steam through the pipe.


Consider a spherical container of inner radius r1 = 8 cm, outer radius r2=10cm, and thermal conductivity k = 45 W/m C, as shown. The inner and outer surfaces of the container are maintained at constant temperatures of T1 = 200C and T2 = 80C, respectively, as a result of some chemical reactions occurring inside. Obtain a general relation for the temperature distribution inside the shell under steady conditions, and determine the rate of heat loss from the container.

2 conduction

Documents