application of mathematics in chemical engineering · pdf file03.01.2014 · in...

Heat conduction

through a large plane

wall of thickness x

and area A.

CONDUCTIONConduction: The transfer of energy from the more

energetic particles of a substance to the adjacent less

energetic ones as a result of interactions between the

particles.

In gases and liquids, conduction is due to the

collisions and diffusion of the molecules during their

random motion.

In solids, it is due to the combination of vibrations of

the molecules in a lattice and the energy transport by

free electrons.

The rate of heat conduction through a plane layer is

proportional to the temperature difference across the

layer and the heat transfer area, but is inversely

proportional to the thickness of the layer.

When x → 0 Fourier’s law of heat

conduction

Thermal conductivity, k: A measure of the ability

of a material to conduct heat.

Temperature gradient dT/dx: The slope of the

temperature curve on a T-x diagram.

Heat is conducted in the direction of decreasing

temperature, and the temperature gradient becomes

negative when temperature decreases with

increasing x. The negative sign in the equation

ensures that heat transfer in the positive x direction

is a positive quantity.

The rate of heat conduction

through a solid is directly

proportional to its thermal

conductivity.

In heat conduction

analysis, A represents

the area normal to the

direction of heat

transfer.

Thermal

ConductivityThermal conductivity may

be defined as the amount

of heat conducted per unit

time across unit area and

through unit thickness

when a temperature

difference of unit degree is

maintained across the

bounding surfaces.

The thermal conductivity of

a material is a measure of

the ability of the material to

conduct heat.

A high value for thermal

conductivity indicates that

the material is a good heat

conductor, and a low value

indicates that the material

is a poor heat conductor or

insulator.

A simple experimental setup

to determine the thermal

conductivity of a material.

The range of variation of thermal conductivity of different

classes of materials at room temperature

The range of thermal conductivity of

various materials

Thermal Conductivity

Thermal Conductivities (average values at

normal pressure and temperature) of some

common materials are as under

Material k (W/m K) Material k (W/m K)

Diamond 2300 Brick 0.72

Silver 429 Water (l) 0.613

Copper 401 Wood (oak) 0.17

Gold 317 Helium (g) 0.152

Aluminium 237 Refrigerant -

12

0.072

Iron 80.2 Glass fibre 0.043

Mercury (l) 8.54 Air (g) 0.026

Glass 0.78


The thermal conductivity is a property of

material which depends upon

• Material structure (chemical composition,

physical state and texture)

• Density of the material

• Moisture content

• Pressure and temperature

The mechanisms of heat

conduction in different

phases of a substance.

The thermal conductivity

of a material is due to

flow of free electrons (in

case of metals) and

lattice vibrations waves

(in case of fluids)



Accordingly thermal conductivity of a material is

the outcome of migration of free electrons and

lattice vibrational waves. In metal molecules are

closely packed; molecular activity is rather small

and so thermal conductivity is substantially due

to the flow of free electrons.

In fluids, the free electron movement is

negligible small and therefore thermal

conductivity results primarily from the frequency

of interactions between the lattice atoms.


Further metals are the best conductors while

liquids are generally poor conductors. Probably

the disordered structure of the liquids and so of

the gases is not conductive for transmitting

molecular vibration.

Thermal Conductivity of Solids

Thermal conductivity of solids is made up of two

components

1. Due to flow of free electrons and

2. Due to lattice vibration (atom which are

bound in a periodic arrangement called

lattice)

First effect is known as electronic conduction

and second effect is known as photon

conduction

Metals and alloys

In case of pure metals and alloys

a) There is an abundance of free electrons and

the electronic conduction predominates.

Since free electrons are also responsible for

electrical conduction, it is observed that

good electrical conductors are also good

thermal conductors e.g. copper, silver etc.

Metals and alloys

b) Any effect which inhibits the

flow of free electrons in pure

metals reduces the value of

thermal conductivity.

For example with a rise in

temperature, the lattice

vibration increases and this

offers a resistance to the flow

of electrons and therefore for

pure metals thermal

conductivity decreases as

temperature increases

(aluminum and uranium

being the exception)

Thermal conductivities of

materials vary with

temperature

Metals and alloys

Thermal conductivity of aluminum stays almost

constant within temperature range of 130 0C to

370 0C.

Most of the outer electrons of the uranium

atoms are tied up in covalent bonds and as

such the contribution of free electrons to

conduction process is small. Conduction of heat

within uranium depends mainly on the vibration

of atoms. The vibration tendency increases with

temperature rise and so does the thermal

conductivity of uranium.

Metals and alloys

Variation of thermal conductivity with

temperature for a few metals

Metals and alloys

c) Alloying decreases the

value of thermal

conductivity since the

foreign atoms cause

scattering of free electrons,

thus impending their free

flow through the material.

Pure metals have very

high thermal conductivity.

Impurities or alloying

element reduce the

thermal conductivity

considerably.

Metals and alloys

Thermal conductivity of pure copper near

about room temperature is 401 W/m 0C

while presence of traces of arsenic reduces

the value of thermal conductivity to

142 W/m 0C.

Metals and alloys

d) Thermal conductivity of a metal varies

considerably when it (metal) is heat

treated or mechanically

processed/formed (forging, drawing

and bending).

• Heat treatment and mechanical forming

reduce the value of thermal conductivity

of pure metals.

• For example, thermal conductivity of

hardened steel is lower than that of

annealed state.

Metals and alloys

Variation of thermal

conductivity with temperature

for a few alloys

e) Thermal

conductivity

of alloys

generally

increases as

temperature

increases.

Metals and alloys

f) Since the phenomenon of electron

conduction is responsible for both thermal

conduction and electrical conduction, it is

reasonable to presume that there must be

relation between these two quantities. In

fact, Weidemann-Franz law gives this

relation.

Metals and alloys

The Wiedemann and Franz law (based on

experimental results) regarding thermal and

electrical conductivities of a material states as

follows: “The ratio of the thermal and electrical

conductivities is the same for all metals at the

same temperature and that the ratio is directly

proportional to the absolute temperature of the

metal”.

Metals and alloys

Mathematically

k/σ α T

k/σT = C

Where k =Thermal conductivity of metal at

temperature T (K) (W/ m K)

σ = Electrical conductivity of metal at temperature

T (K) (ohm m)-1 and

C = Constant (for all metals) referred to as Lorenz

number (2.45×10-8 W Ohms/K2)

Metals and alloys

• An important application of Wiedemann and Franz

law is to determine the value of thermal

conductivity of a metal at a desired temperature,

knowing the value of electrical conductivity at the

same temperature. Note that it is easier to

measure experimentally the value of electrical

conductivity than that of thermal conductivity.

• Wiedemann and Franz law conveys that the

materials which are good conductors of electricity

are also conductors of heat.

Non-metallic Solids

a) Non-metallic solids do not conduct heat

(there are no free electrons) as efficiently as

metals and hence the thermal conductivity

values are much lower than those of metals.

For many of the building and insulating

materials (concrete, stone, brick, glass wool,

cork etc.) the thermal conductivity may vary

from sample to sample due to variations in

structure, composition, density and porosity.

Non-metallic Solids

For heat insulating

materials, general range

of values of k are from

0.023 W/m0C to

2.9 W/m 0C.

Thermal conductivity

increases with

temperature for

insulating materials.Variation of thermal conductivity

with temperature for insulating

materials

Non-metallic Solids

b) For porous heat insulating material (brick,

concrete, asbestos, slag etc.), thermal conductivity

depends greatly on density of the material and the

type of gas or liquid filling the voids.

Presence of air filled pores and cavities reduce

thermal conductivity because then the heat has to

be transferred across many air spaces and air is

known to be poor heat conductor.

Non-metallic Solids

Thermal conductivity of porous materials also

depends on the moisture content in the material; k

of a damp material is much higher than that of the

dry material and water taken individually.

For dry brick k = 0.35 W/m-deg

For water k = 0.60 W/m-deg

For damp brick k = 1 W/m-deg

This behavior is may be attributed to

(i) capillary movement of water with in the

pores which results in convection heat

transfer (ii) properties of the absorbed

moisture are different from those of free

moisture.

Non-metallic Solids

Density is another parameter that affects the

thermal conductivity of material; thermal

conductivity increases with density growth.

For example, k of asbestos increases from 0.105

to 0.248 W/m 0C as density increases from 400 to

8000 kg/m3

Non-metallic Solids

c) Thermal conductivity of granular materials

increases with temperature since with

increasing temperature, radiation from the

granules also comes into picture along

with conduction of medium filling the

spaces.

Non-metallic Solids

Materials having a crystalline structure have a high

value of thermal conductivity than the substances in

amorphous form.

For quartz (a solid with crystalline structure)

k = 30.5 W/m-deg at -100 0C

= 10.4 W/m-deg at +100 0C

For pyrex (a substance of amorphous form)

k = 1.02 W/m-deg at 0 0C

= 1.73 W/m-deg at 500 0C

Irregular arrangement of the atoms in case of

amorphous solids inhibits the effectiveness of heat

transfer by molecular impact.

Variation of thermal conductivity of solids

with temperature:

In heat transfer calculations, generally we

assume k to be constant when the

temperature range is small; however the

temperature range if large, it is necessary to

take into account the variation of k with

temperature.

Usually, for solids, a linear variation of thermal

conductivity with temperature can be assumed without

loss of much accuracy.

k(T) = k0 ( 1+βT)

Where

k(T) = thermal conductivity at desired temperature T,

W/m 0C

k0= thermal conductivity at reference temperature at

0 0C, W/m 0C

β = a temperature coefficient, 1/ 0C

T = temperature, 0C

Representative values of k0 and β

It may be noted that the variation is linear.

Variation of thermal conductivity with temperature for a few pure metals

Value of β may be positive or negative.

Generally β is negative for metals (exception

being aluminum, uranium and certain non-

ferrous alloys) and positive for non metals

and insulators (magnesite bricks being

exception) and alloys.

Thermal Conductivity of Liquids

Heat propagation in liquid is considered to be due to

elastic oscillations. As per this hypothesis, the thermal

conductivity of liquids is given by

Where cp = specific heat of liquid at constant pressure

ρ = density of liquid

M = Molecular weight of liquid

A = Constant depending on the velocity of elastic

wave propagation in the liquid; it does not depend on

nature of liquid; but on temperature

Non-metallic liquids

31

34

M

Ack

p


It is noted that the product A.cp is nearly constant. As

temperature rises, density of liquid falls and as per

equation

The value of thermal conductivity also drops for

liquids with constant molecular weights (i.e. for non-

associated or slightly associated liquids)

Notable exceptions are water and glycerin, which are

heavily associated liquids. With rising pressure,

thermal conductivity of liquids increases. For liquid k

value ranges from 0.07 to 0.7 W/m 0C

31

34

M

Ack

p


Thermal conductivity of non-metallic liquids

Liquid metal

Liquid metals like sodium, potassium etc. are used in

high flux applications as in nuclear power plants

where a large amount of heat has to be removed in

small area. Thermal conductivity values of liquid

metals are much higher than those for non-metallic

liquids. For example, liquid sodium at 644 K has

k=72.3 W/ m K; liquid potassium at 700 K has k=39.5

W/m K; and liquid bismuth at 589 K has k=16.4 W/ m

K

Thermal Conductivity of Gases

a) Heat transfer by conduction in gases at ordinary

pressure and temperature is explained by the

Kinetic Theory of gases. Temperature is a

measure of kinetic energy of molecules. Radom

movement and collision of gas molecules

contribute to the transport of kinetic energy, and

therefore to transport of heat. So, the two

quantities that come into picture now are: the

mean molecular velocity V and the mean free path,

I. Mean free path is defined as the mean distance

travelled by a molecule before it collides with.


Thermal conductivity of gases is given by

Where V = mean molecular velocity

l = mean free path (average distance travelled by a

molecule before experiencing collision)

cv = specific heat of gas at constant volume

ρ = density

vVlck3

1


b) As pressure increases, density ρ increases, but

the mean free path l decreases almost by the

same proportion and the product l ρ remains

almost constant, i.e. the thermal conductivity of

gases does not vary much with pressure except at

very low (less than 20 mmHg) or very high (more

than 20,000 bar) pressures.

c) As to the effect of temperature on thermal

conductivity of gases, mean molecular velocity V

depends on temperature as follows,

Where G = Universal gas constant = 8314.2 J/kmol K

M = Molecular weight of gas; T = absolute temperature of Gas K


i.e. mean molecular velocity varies directly as the

square root of absolute temperature and inversely

as the square root of the molecular weight of a

gas. Specific heat cv also increases as

temperature increases. As a result, thermal

conductivity of gases increases as temperature

increases.


d) For the reason stated above, gases with

higher molecular weight have small thermal

conductivity than those with lower molecular

weight.

For example

k for hydrogen (mol wt = 2) = 0.190 W/ m-deg

k for oxygen (mol wt = 32) = 0.0272 W/ m-deg


f) Generally, thermal conductivity values for

gases vary in the range of 0.006 to

0.6 W/m 0C

g) Thermal conductivity of steam and other

imperfect gases depend very much on

pressure unlike that of perfect gases.


Variation of k with

temperature for a few

gases

Variation of k with

temperature for

hydrogen and helium

Insulation systems

Materials with large thermal conductivity are called

thermal conductors and those with small thermal

conductivity are called thermal insulators.

Insulating materials are used for obstructing the

flow of heat between an enclosure and its

surroundings. Insulation is required for high

temperature systems as well as low temperature

systems.

Insulation systems may be classified as

(i) Fibrous (ii) Cellular (iii) Powder (iv) Reflective

Insulation systems

In high temperature systems, any leakage of heat

from boilers, furnaces or piping carrying hot fluids

represents an energy loss.

Similarly in low temperature/cryogenic systems, any

heat leakage into the low temperature region

represents an energy loss.

Insulation systems

Low temperature insulation (cork, rock, wool, glass

wool, cattle hair, slag wool and thermocole etc.) are

used when the enclosure is at a temperature lower

than the ambient temperature and it is desired to

prevent the enclosure from gaining heat.

High temperature insulations (asbestos,

diatomaceous earth, magnesia etc.) are used when

it is desired to prevent an enclosure at a

temperature higher than the ambient from losing

heat to surroundings.

Insulation systems

Super insulators include powders, fibres or multi-

layer materials that have been evacuated of all air.

The low conductivity of insulating materials is due

primarily to air (a poorly conducting gas) that is

contained in the pores rather than the low

conductivity of the solid substance.

Substances under low temperature conditions that

have exceeding high thermal conductivity are

known as super conductors. For example thermal

conductivity of aluminium reaches a value of

20000 W/ m-deg at 10 K and this is over 100

times as large as the value that occurs at room

temperature.

Insulation systems

Common Insulation used in Industry

Thermal Diffusivitycp Specific heat, J/kg · °C: Heat capacity per

unit mass

cp Heat capacity, J/m3·°C: Heat capacity

per unit volume

Thermal diffusivity, m2/s: Represents how

fast heat diffuses through a material

A material that has a high thermal

conductivity or a low heat capacity will

obviously have a large thermal diffusivity.

The larger the thermal diffusivity, the faster

the propagation of heat into the medium.

A small value of thermal diffusivity means

that heat is mostly absorbed by the

material and a small amount of heat is

conducted further.

• General Heat Conduction equation in Cartesian

Coordinates

• General Heat Conduction equation in

Cylindrical Coordinates

• General Heat Conduction equation in Spherical

Coordinates

Topics

Heat Conduction through a plane wall and

composite walls

• Heat Conduction through a plane wall

Case I: Uniform k

Case II: Variable k

• Heat Conduction through a composite walls

• The overall heat transfer coefficient

Topics

Heat Conduction through Hollow and composite

cylinder

• Heat Conduction through a hollow cylinder

Case I: Uniform k

Case II: Variable k

• Heat Conduction through a composite cylinder

Topics

Heat Conduction through Hollow and composite

sphere

• Heat Conduction through a hollow sphere

Case I: Uniform k

Case II: Variable k

• Heat Conduction through a composite sphere

Topics

Critical thickness of Insulation

• Insulation – General Aspects

• Critical thickness of insulation

Heat Transfer from extended surfaces (Fins)

• Introduction

• Heat flow through rectangular fin

Heat dissipation from an infinitely long fin

Heat dissipation from a fin insulated at the

tip

Heat dissipation from a fin losing heat at the

tip

Efficiency and effectiveness of fin

Topics

application of mathematics in chemical engineering · pdf file03.01.2014 · in...

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