Fundamentals of Heat Transfer

Submitted to:

Dr. Lubos Hes

By:

Muhammad Mushtaq Ahmed Mangat

(mushtaq.mangat@gmail.com)

Dec 21, 2009

Fundamentals of Heat Transfer

Introduction

Heat is delineate as energy in thermodynamics and think over in a transitional stage between system

and the surrounding. Every substance has thermal energy, which is equal to the total of kinetic

energy (transitional, rotational or vibration of the particles), potential energy ( associated with

vibrational and electric energy of atoms within molecules or crystal, and energy exist in chemical

bonds and free energy of conduction electrons in metal and temperature is average kinetic energy

of a substance. It is pertinent to note that thermal energy is not the entire energy of the system,

rather it is a part of internal energy of a system. Apparently heat and thermal energy looks same by

keeping its qualitative nature but they are never identical in quantitative terms. Thermal energy can

be enhanced by applying other means, e.g. severe agitation can increase the thermal energy of a

system (Leland, 2009). More explicitly, we can express heat is an energy which is colligate with the

movement of atoms or molecules. Furthermore, these molecules and atoms should have the quality

and power to transmit through solid and fluid media by conduction, through fluid media by

convection, and through empty space by radiation. In addition, heat is a thermal energy and it is

transferred from one body to other or it can be used to perform a work. Thermal energy, which is

loosely defined as energy of a body increases with the increase in temperature. People also prefer to

call heat is as energy of a body (Mooney, 1955).

Heat can be jumbled with temperature in daily life. Notwithstanding, that these two words are

interchangeable. Fact is that there is a clear and distinct and diverse variation between these two.

Heat is the energy which an element possess and by instinct will take or give energy from

surroundings to achieve an equilibrium with the surroundings. It is denoted by Q and its measuring

unit in SI system is Joule. Whereas, temperature is represent level of heat which an element posses

and it is a relative term. The concept of temperature comes from a cold or hotter object.

Temperature is used to quantify the level of hotness or level of hotness of any material. What is

more, heat flows from a hotter body to a less hotter body, when both bodies are kept isolated from

any third body (Mooney, 1955). We can notice and estimate from the temperature about condition

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of a surface; cool or hot. For temperature, symbol is T is used and units are Celsius, Fahrenheit and

Kelvin.

Finally it can be said that the term “heat” is a name fixed for the particular form of

energy crossing the boundary. Nevertheless it is more commonly referred as heat transfer, when we

are taking about thermodynamics. It is used to describe the consistent ability of heat to alter the

energy level of a system (NA, 2007). Finally we conclude that heat is a form of energy and it

transfer due to the gradient in temperature between one element and its surroundings. Heat is

macroscopic property of an object and temperature is a quantitative description and measure of

hotness or coldness of a system and it is a measure of energy which an object posses.

Definition of Heat

James Clerk Maxwell is the first Scottish physicist, who defined heat in a comprehensive way in

his Theory of Heat. Maxwell defined four different aspects of heat:

1. Something which is transferable

2. It is measurable

3. It cannot be treated as a substance

4. It is one form of the energy.

In modern science there is a more clarity about heat and is defined more explicitly. After having a

literature survey, we can put our findings in the following way:

1. Heat is never considered as being stored in a body, rather it exist only in transit from one body

to other body.

2. It is treated as noun during the heat transfer flow

3. It is defined as any spontaneous flow of energy, from one body to other due to temperature

gradient.

4. As energy is transient from high temperature to low temperature

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5. It is an interaction between two closed system without exchange of work.

Kern (1950) describes that the science of thermodynamics deals with the quantitative transmission

and rearrangement of energy as heat in bodies of matter. This definition clarifies the whole process

in a precise way. In this definition , the word “rearrangement” explains many things. The main

difference in work and heat is that in case of work there is a no transfer of mass and work is done

not due to the temperature gradient (Siegel and Howell, 2002).

Heat Measurement and Units of Heat

Temperature of any body is the average of kinetic energy of the molecules present in a substance. It

is measured in Fahrenheit, Celsius and Kelvin degree. It does not recount the total heat available in

the substance. However, from temperature we understand and observe the direction of heat flow

since it is incessantly heat flows from a higher to lower temperature. Nevertheless we use heat

units. These units are measured keeping in view the property of heat. As said heat flows from higher

temperature to lower temperature. Considering this characteristics, there are two common units are

uses; British Thermal Unit (BTU) and Calorie. BTU is defined the amount of heat required to raise

temperature of one pound of water by one degree Fahrenheit. Whereas, calorie is amount of heat

required to raise temperature of one gram of water by one degree Celsius (Mooney, 1955).

BTU is a unit which is used since a long time but currently it is being replaced by SI unit of heat;

joule. One BTU is equal to 1.06 Kilo Joules. There are many areas, where still BTU is used,

particularly, in UK, it is still in use. It is also used to exposit the energy of fuels; BTU of furnace oil,

etc. Joule is defined as amount of energy, when a force of one newton is applied for displacement of

any object of one meter. Whereas, “One newton is the force required to cause a mass of one

kilogram to accelerate at a rate of one meter per second squared in the absence of other force-

producing effects”(http://searchcio-midmarket.techtarget.com).

Latent and Sensible Heat

Latent and Sensible heat have a great deal of applications in the real life. Latent heat is referred as

the amount of heat absorbed or released during a process of sate changing without change in

temperature. It is easy to understand from the melting of ice. Ice changes its phase from a solid to

liquid and for this process a certain amount of heat is required. This amount of heat is called latent

of the substance. To describe, precisely, the term specific latent is used. It is denoted for the energy

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required to convert one Kg or one pound of a substance from solid to liquid or vise-versa, without

changing in temperature. The latent heat for a different mass of the substance can be calculated

using the equation:

Q = mL

Where:

Q is the amount of energy released or absorbed during the change of phase of the substance (in kJ

or in BTU),

m is the mass of the substance (in kg or in lb.), and

L is the specific latent heat for a particular substance (kJ-kgm-1 or in BTU-lbm-1). Water has highest

latent heat of vaporization, it is 2260 (at 100oC) kJ/kg.

Sensible heat is the heat absorbed or transmitted by a substance during a change of temperature

which is not accompanied by a change of state.

Flow of Heat or Heat Transfer

Just in case, when a substance posses different level of thermal energy from it surroundings, heat

transfer phenomenon takes place and it is referable to the difference in the level of thermal energy,

which is indicated by the disparity of temperature. It is also known as heat exchange or transfer of

thermal energy. This transfer always come along from higher to lower temperature and will

continue until both bodies attain the same temperature, it is reckon as thermal equilibrium. Heat

transfer between two bodies, if there is a difference of temperature, can never be stopped, however,

it can be slow downed. There are numerous factors which can affect heat transfer process. In fact

heat transfer is a transition of thermal energy from an element ,which is hotter to other element

which is less hotter. Here element means that a substance which can store energy in many ways. It

does not posses only thermal energy, whereas, it contains other forms of energy but transmitting

will be only of thermal. Rate of transmission depends upon the difference of temperature and size

(Fourier’s law).

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Heat Transfer is the science which deals with the rate of exchange of heat between hot and cold

bodies, which are called source and receiver. Furthermore, heat flow is directly proportion to

potential (temperature gradient and inversely proportional to the resistance (Kern, 1950).

Heat flow ! Potential/resistance

Incropera et al,. (2007) set forth that heat transfer is a thermal energy in transit due to a spatial

temperature difference. Literature is full of theories, ideas, models and mathematical equation

which narrate the heat transfer process. This is basically due to the fact that in real life heat transfer

process has great applications. There are three common ways of heat transfer; conduction,

convection and radiation. Mayan (2002) quotes that some people cerebrate phase change or boiling

as fourth method of heat transfer.

Heat Transfer Through Conduction

In conduction transfer of heat takes place between neighboring molecules due to temperature

gradient and it is always form a higher temperature to a lower temperature till there is an

equilibrium. Heat Transfer can take place in any form of substance, it may be in solid, liquid, gas or

plasma form. However, there will be slightly difference in process. For instance in case of solid,

heat transfer takes place due to vibrations of molecules and free electron, which transfer energy.

However, in case of gases and liquids, it is due to collision and diffusion of the molecules, while

they are in a random motion. For conduction there is a prerequisite of direct contact. Transfer of

energy can be classified broadly in two categories; first, transfer due to elastic as in case of fluids

and second through free electron diffusion. In nutshell, heat transfer takes place either through

vibration against each other or movement of electron from one to other substance. It implies that in

case of solids, where contact is significance, heat transfer will be rapid as compare to liquid or gases

where distance is more.

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Figure: Heat Transfer Through Conduction

Malalasekera (2009) summarizes that heat transfer through conduction is a process which takes

place in stationary medium and energy is transferred from a more energetic material to a low

energetic material. In solids it is due to collision (vibrations) and motion of free electrons and in

liquids and gases it is due to molecular collision and molecular diffusion.

Elements which can transfer energy are called conductors. The property of an elements to pass the

heat is called its thermal conductivity or conductivity constant or conduction coefficient. It varies

across the available matters. Metals are surmount conductor than non-metals. It is primarily due to

presence of metallic bonds instead of covalent bonds, which allow free movement of electrons. This

free movement of electrons is finally responsible to transfer heat. Density of the material has

another factor playing a crucial role. There is a decrease in thermal conductivity with the decrease

in density. It is uncomplicated to infer that gap between (low density) will provide less chances to

transfer heat since there will less chances of contact of electrons present in the matter to approach

each other and come to contact. We can say more precisely that in conduction, energy transfer

across a system boundary is due to a temperature difference by the mechanism of inter-molecular

interactions. However, basic requirement of conduction is availability of matter. Nevertheless there

is no bulk movement of matter for heat transfer.

Transient Conduction vs. Steady-State Conduction

Steady state conduction is the form of conduction which happens when the temperature difference

is constant, so that an equilibration time, the spatial distribution of temperatures in an object does

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not change (for example, a bar may be cold at one end and hot at the other, but the gradient of

temperatures along the bar do not change with time). In short, temperature at a section remains

constant and it deviates linearly along direction of heat transfer.

There also exist situations wherein the temperature drop or raise occurs more drastically, such as

when a hot copper ball is dropped into oil at a low temperature, and the interest is in analyzing the

spatial change of temperature in the object over time. This mode of heat conduction can be referred

to as unsteady mode of conduction or transient conduction.

Source: http://en.wikipedia.org/wiki/Heat_transfer

Law of Heat Conduction (Fourier's law)

Investigation on heat transfer by Jean Baptiste Joseph Fourier (1768 – 1830) resulted in many

useful and much known equations. One of his well recognized equation is called Fourier’s Law.

This law governs the heat transfer through conduction from one substance to other, where there is a

divergent of temperature. Fourier’s law implies that rate of heat transfer through a body is

proportional to the negative gradient in the temperature and the area at right angle. There are two

further possibility to take take this law; first in its integral form, where we take total heat transfer

and second differential form, where we see energy flow at local level.

In simple words, Fourier law of conduction describes that heat transfer from one system to other

system will be proportional to:

1. Area normal to the direction of heat flow

2. The gradient of temperature

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Equation of Fourier’s law is self explanatory and not much complicated. This equation allows to

determination of the conduction of heat flux. For this purpose we should have knowledge of

temperature gradient, thermal conductivity and area of contact. Its most general (vector) form for

multidimensional conduction is:

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Implications of Fourier’s Equation:

1. Heat transfer is in the direction of decreasing temperature (basis for minus sign)

2. Fourier’s Law serves to define the thermal conductivity of the medium

3. Direction of heat transfer is perpendicular to lines of constant temperature (isotherms)

4. Heat flux vector may be resolved into orthogonal components.

Considering area equation can be written in this form:

Q = -k A *dT/dx

Here Q is the rate of heat flow (W or J/s),

is the area (m2),

k is the thermal conductivity [W/(m °C) or W/(m. K)].

Fourier’s law is one of the fundamental equations which describes the heat transfer through

conduction. Above discussion depicts that this law involves:

1. Heat flux

2. Thermal conductivity

3. Temperature gradient

4. Area of contact

Law demonstrates that there is direct connection between the heat flow and the instinct property of

material; thermal conductivity. If we heat water in a copper plate, temperature of copper will

increase more rapidly as compared to temperature of water. It is due the thermal conductivity. In

case of water it is only 0.63 whereas, thermal conductivity of copper is 401 W/m K. Other than

thermal conductivity gradient also plays an important role. Equation tells us that heat flow will be

higher when the gradient will be higher. It will be slowed down with the decrease in gradient.

Notwithstanding that it will continue till there is no difference and that the equilibrium is achieved

in a system. It is important to mention that heat transfer can take place through different ways

(conduction, convection and radiation) at same time.

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This law is used to predict temperature of any system or element it is exposed to a certain source of

the heat. This equation can tell us the melting point or boiling point of any system and how much

energy will be required to have a certain temperature. This equation helps a lot in engineering

works, where people are dealing with heat transfer issues.

Insulations: A Process to Reduce the Heat Flow

In Table 01, there is a list of different elements who owns different thermal conductivity. Variation

is due to the chemical composition and sometime due to physical structure of the materials.

Thermal insulation of a foam made of polyester will give different results as compare to a solid

layer of same quantity of polyester. In case of foam, it is filled with air, which also have a

significant low thermal conductivity and thus increases the thermal insulation. When there is a

need to slow down or stop heat flow from one system to other, a material, which is having low

thermal conductivity is put in between so that there should be desired flow of heat. Very commonly

used material in industry is fiber glass. In textile clothing, selection of clothing is linked with the

ultimate desire to control heat flow. In winter we need clothing, which should provide high

insulation with a material which is having very low thermal conductivity, so that heat of our body

should not go outside, where temperature is low than our body temperature. Nevertheless, in

summer, we need to reduce the impact of hot climate by having different clothing, which are

different in chemical nature and even they have different physical structure.

Source: http://thermo-tutorial.blogspot.com/2007/04/fouriers-law-of-heat-conduction.html

Differential Form of Fourier’s Law

Heat flux, which refers sometimes, as heat flux density or heat flow rate intensity, is a flow of

energy per unit area per unit of time. Its measuring unit is [W!m-2]. In addition it has direction and

magnitude, which makes it vectorial quantity. To measure heat flux at a certain point, when area

becomes infinitesimally small, one can take the limiting case in such case. By having differential of

Fourier’s law, we find the following equation:

Where (including the SI units)

is the local heat flux, [W!m"2]

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is the material's conductivity, [W!m"1!K"1],

is the temperature gradient, [K!m"1].

It is important to note that in most of the cases thermal conductivity k is taken as constant but fact

is that there is change, of course minor change in the thermal conductivity with the change in

temperature. For simplicity it is taken constant in most of the cases (Lienhard IV and Lienhard V,

2008).

Assumption of Fourier’s Law

Fourier’s law is based on two fundamental assumptions; one is temperature and other is heat flow.

Both are quantitative figures but assorted in qualitative perspective. Temperature demonstrates the

random movement of the elements of variant nature, whereas, heat flow is an additive event. It

leads to the conclusion that Fourier’s law justify a relationship between two contrary regions having

divergent temperature. It can be said more specifically that it governs the interaction between two

dynamic elements (Kawaguchi et al., 2005).

Contextual relationship between two different phases will be resulted in changed phases. This is

because heat is acting as intervening force. There is also an intrinsic affinity between two different

elements. Since whole process is based on the disparity between the temperature and it will be

resulted as fast as possible (Matsuno and Swenson, 1999, as cited by Kawaguchi et al., 2005).

One can find that energy quantum1 as a coherent enclosure of a standing material is robust and is

not complete. If it is not the situation, then there is no chance of transfer of heat. This implies that

there is an underlying dynamics of variation in the participating contexts. This is only possible if

there is a different of temperature between two different phases present in a different context which

put together (Kawaguchi et al., 2005).

When a contact between quantum and environment establishes, transformation of quantum as

fastest possible rate takes place. This transformation does not stop till there is an equilibrium with

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1 A discrete quantity of energy proportional in magnitude to the frequency of the radiation it represents.

the environment. Regarding the framework of Fourier’s law there is a likelihood that that energy

quantum may play a role of a heat engine, responsible for the transformation, which covers

incoming and outgoing heat (Kawaguchi et al., 2005).

Thermal Conductivity (Conduction Coefficient)

Thermal conductivity is one of the underlying characteristics of materials. Selection of material for

heat transfer depends upon the thermal conductivity. It is denoted by k. It is calculated by

measuring amount of heat Q, which is transmitted through material, having thickness of L in one

direction (from higher temperature to low temperature), of A surface area in time t due to the

temperature difference #T (Lienhard IV and Lienhard V, 2008).

Table 01 Thermal Conductivity of Some Common Materials

Source: http://nptel.iitm.ac.in/courses/Webcourse-contents/IISc-BANG/Heat%20and%20Mass

%20Transfer/pdf/M1/Student_Slides_M1.pdf

Flow of Heat Through Convection

In convection heat transfer is takes place between a solid and flowing fluid having a temperature

gradient. It is also a sort of conveyor where heat is transported from one area to other. There are two

sorts of phenomenon which takes place at a time. Diffusion near the surface, where fluid velocity is

low and bulk motion of molecules away from the surface (Malalasekera, 2009). Frank and David

(1990) mark that convection is one of the major modes of heat transfer and mass transfer. In fluid,

convective heat and mass transfer take place through both diffusion2 and advection3

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2 The random Brownian motion of individual particles in the fluid

3 In this process matter or heat is transported by the larger-scale motion of currents in the fluid

There are two categories of convection:

1-Forced or assisted convection (fluid flow is induced e.g. fan, pump , etc.)

2-Natural or free convection ( it is caused by buoyancy forces mainly due to the density difference

as result of temperature difference, fluid after coming will hot will rise and will be replaced by a

cool fluid e.g. Boiling, condensation , etc.)

In some cases natural and forced may occur simultaneously and this is called mixed convection.

Newton's Law of Cooling

Convection is also known as Newton’s Law of Cooling. A related principle, Newton's law of

cooling, states that the rate of heat loss of a body is proportional to the difference in temperatures

between the body and its surroundings. The law is

Q = Thermal energy in joules

h = Heat transfer coefficient

A = Surface area of the heat being transferred

T = Temperature of the object's surface and interior (since these are the same in this approximation)

Tenv = Temperature of the environment

#T(t) = T(t) " Tenv is the time-dependent thermal gradient between environment and object

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Table 02 Heat Transfer Medium and Calculation

Source: http://en.wikipedia.org/wiki/Newton%27s_Law_of_Cooling#Newton.27s_law_of_cooling

Heat Transfer Coefficients - Units

• 1 W/m2K = 0.85984 kcal/h m2 oC = 0.1761 Btu/ ft2 h oF

• 1 Btu/ft2 h oF = 5.678 W/m2 K = 4.882 kcal/h m2 oC

• 1 kcal/h m2 oC = 1.163 W/m2K = 0.205 Btu/ ft2 h oF

In general the convective heat transfer coefficient for some common fluids is within the ranges:

Air: 10 - 100 (W/m2K)

Water: 500 - 10,000 (W/m2K)

Convective and Convector Heat Transfer Coefficient

There is a convective heat transfer coefficient used in Newton’s equation for cooling. To determine

this coefficient, we use a idea called the convective heat transfer coefficient. Malalasekera (2009)

have summarised this idea with many examples.

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Fig …..Heat transfer from a solid surface to a flowing fluid.

If q is the heat flux (W/m2 K) from wall to the fluid:

q = h(Tw ! T")

is the convective heat transfer coefficient - units W/m2 K. Here is the temperature of the wall

and is the temperature of the fluid.

The value of the convective heat transfer coefficient depends on:

1. type of the flow – laminar, turbulent

2. geometry of the situation

3. physical properties of the fluid

4. the temperature difference

5. position along the surface of the body

6. whether convection mechanism is forced or free convection

Natural Convection

In natural convection heat transfer is due to the density difference, which is the result of

temperature gradient. There is no external force is used to transfer heat. Fluid surroundings takes

heat and reduces its density and movies away due to gravitational force difference and other fluid

comes in contact which is dense due to low temperature and the phenomenon goes on. The

buoyancy 4, a result of difference in fluid density, is the driving force in convection heat transfer

process.

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4 Ability to float in water

For the acceleration there is a need of a force. It may be gravitational force or any equivalent force,

which could a result of acceleration, centrifugal or coriolis5 force. Thereby, natural convectional

does not appear in a free fall context (Kays; Crawford and Weigand, 2004). There are lot of

application of free convection. For example, air cooling without air fan. It may be on small scale,

computer chips or at very high scale.

Jaluria ( 1980) has reviewed the role of mathematics in natural convection and concludes that

natural convection relies on the Grashof number (Gr), which is a ratio of buoyancy force and

viscous force. The Grashof number (Gr) is a dimensionless number and used in natural convection

for fluid dynamics and heat transfer. The number approximates the ratio between buoyancy and

viscous force. These two force are action on the fluid and are responsible in free convection heat

transfer. This number was developed by Franz Grashof. In nutshell we can say that tendency of

natural convection depends pun Grashof number which has the folioing mathematical form:

For vertical flat plates

For pipes

Where the L and D subscripts indicates the length scale basis for the Grashof Number.

g = acceleration due to Earth's gravity

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5 Deflection of movement when viewing from a rotation

# = volumetric thermal expansion coefficient (equal to approximately 1/T, for ideal fluids, where T

is absolute temperature)

Ts = surface temperature

T" = bulk temperature

L = length

D = diameter

$ = kinematic viscosity

Jaluria ( 1980) further explains that the transition to turbulent flow occurs in the range

108 < GrL < 109 for natural convection from vertical flat plates. It means that at higher Grashof

numbers, the boundary layer is turbulent; at lower Grashof numbers, the boundary layer is laminar.

Rayleigh number is a product of Prandtl and Grashof number and it is a dimensionless number

which indicates the convection problem in heat transfer.

This is also an analogous from of the Grashof number used in cases of natural convection mass

transfer problems.

Where:

And

g = acceleration due to Earth's gravity

Ca,s = concentration of species a at surface

Ca,a = concentration of species a in ambient medium

L = characteristic length

$ = kinematic viscosity

% = fluid density

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Ca = concentration of species a

T = constant temperature

p = constant pressure

It is axiomatic in a fluid that there will be two forces working opposite; upward buoyancy of the

heated fluid and the internal friction, which creates resistance in the movement of the fluid, means

viscosity. Apparently thick solution will provide maximum resistance and in case when there is an

infinity viscosity, heat transfer will take place due to conduction not through convection. Grashof

number indicates the ratio of upward force and force slowing down the movement of fluid.

The relative magnitudes of the Grashof and Reynolds number determine which form of convection

dominates, if forced convection may be neglected, whereas if natural

convection may be neglected. If the ratio is approximately one both forced and natural convection

need to be taken into account (Myron, 2002).

Furthermore, geometry of the hot surface plays a significant role in the convection. There are

possibilities of various correlations for calculation of heat transfer coefficient. For this purpose,

Rayleigh number (Ra) is commonly used.

(Pr is Prandtl number). Myron (2002) concludes that following a general

correlation applies for a variety of geometries:

A general correlation that applies for a variety of geometries is

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The value of f4(Pr) is calculated using the following formula

Nusselt Number

Nusselt number is a ratio between convective to conductive heat transfer ratio across the normal to

the boundary. It was named after Wilhelm Nusselt and is a dimensionless number. For Nusselt

number measurement conductive component is measured nude the same conditions as the heat

convection but with an understanding of hypothetically motionless fluid. If Nusselt number is

closed to unity indicates a sluggish flow or laminar flow. Whereas, a large number shows that more

active convection with turbulent flow and it is between 100-1000 range. Along with that convection

and conduction are parallel to each other and to surface normal of the boundary surface, and are all

perpendicular to the mean fluid flow in the simple case (Incropera and Dewitt, 1997).

Where:

L = characteristic length

kf = thermal conductivity of the fluid

h = convective heat transfer coefficient

The Biot Number (Bi)

Biot number is given after the name of a French physicist, Jean-Baptiste Biot (1774-1862). This

number is used in the calculation of heat transfer in non-steady or transient state. It is an index of

ratio of heat transfer resistance inside and at the surface of a body. This ratio indicates the

signification of variance in temperature in space, while the body heats or cools over time from a

thermal gradient applied to its surface.

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Biot number, if less than one, it means that there is less problem and process is simple or there in

big variation. Wherever, value more than one depicts that there is a huge variation in temperature

with in the object (Incropera and Dewitt, 1997).

The Biot number is defined as:

Where:

h = film coefficient or heat transfer coefficient or convective heat transfer coefficient

LC = characteristic length, which is commonly defined as the volume of the body divided by the

surface area of the body, such that

kb = Thermal conductivity of the body

If Biot number is less than 0.1, it indicates that object is thermally thin and it can be presumed that

there will be a constant heat throughout the object. Greater than 0.1, means that we can put a label

on the substance that it is thermally thick and there is a feasibility that heat will not be constant

throughout the system.

We find an analogous version of the Biot number (usually called the "mass transfer Biot number",

or Bim) is also used in mass diffusion processes:

Where:

hm - film mass transfer coefficient

LC - characteristic length

DAB - mass diffusivity.

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Fourier number

Fourier number is a dimensionless number and it given name after the name of Joseph Fourier. This

number specifies the properties of heat conduction. It is a ratio of heat conduction rate to the rate of

thermal energy storage. It is used in Biot umber to elaborate the transient conduction problems. It is

defined as:

Where:

& is the thermal diffusivity [m2/s]

t is the characteristic time [s]

R is the length through which conduction occurs [m]

For transient mass transfer by diffusion, there is an analogous mass Fourier Number (also denoted

Fo) defined as:

Where:

"D" is the Diffusivity

"t" is the characteristic timescale

"L" is the length scale of interest

(Incropera and Dewitt, 1997)

Schmidt Number (Sc)

Sc is a dimensionless number and it is defined as the ratio of momentum diffusivity (viscosity) and

mass diffusivity. Its main application is to elaborate the properties of a fluid flow, where there is a

chance that simultaneous momentum and mass diffusion convection will take place. It was named

after the German engineer Ernst Heinrich Wilhelm Schmidt (1892-1975). We can also define Sc

number as ratio of shear component for diffusivity to the diffusivity for mass transfer D. It relates

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the relative thickness of the hydrodynamic layer layer and mass-transfer boundaries (Incropera and

Dewitt, 1997).

It is defined as:

Where:

$ is the kinematic viscosity

D is the mass diffusivity.

µ is the dynamic viscosity

% is the density

The heat transfer analog of the Schmidt number is the Prandtl number.

Sherwood Number

The Sherwood number, Sh (also called the mass transfer Nusselt number) is a dimensionless

number used in mass-transfer operation. It represents the ratio of convective to diffusive mass

transport, and is named in honor of Thomas Kilgore Sherwood.

It is defined as follows:

Where

L is a characteristic length (m)

D is mass diffusivity (m2.s-1)

K is the mass transfer coefficient (m.s-1)

It can also be further defined as a function of the Reynolds and Schmidt numbers; for example, for a

sphere it can be expressed as:

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This form is particularly valuable to chemical engineers in situations where the Reynolds number

and Schmidt number are readily available. Since Re and Sc are both dimensionless numbers, the

Sherwood number is also dimensionless. These correlations are the mass-transfer version of an

analogous technique in heat transfer of writing the Nusselt number with the Reynolds number and

Prandtl number. For a correlation for a given geometry (e.g. spheres, plates, cylinders, etc.), a heat

transfer correlation (often more readily available from literature and experimental work, and easier

to determine) for Nusselt number Nu in terms of the Reynolds number (Re) and the Prandtl number

(Pr) can be used as a mass transfer correlation by replacing the Prandtl number with the analogous

dimensionless number for mass transfer, the Schmidt number, and replacing the Nusselt number

with the analogous dimensionless number for mass transfer, the Sherwood number.

Heat Transfer Through Radiation (Thermal Radiation)

Thermal Radiation is a result of the movements of atoms and molecules in a substance. Movement

of charged components (electron and protons) emits electromagnetic radiation, which carries energy

away from the surface of the substance. Radiation keeps on since there is constantly bombardment

on surface of radiation from the surrounding. Radiation depends upon the temperature of the

substance. All bodies at all temperature radiate electromagnetic waves. It is mainly due to the

presence of molecular and atomic agitation, which is present at all temperature. There is a long

range of wave length, it may be in meters and microns. However, thermal radiation is consist of

electro magnetic waves ranging from 0.1 to 10.0 µm, whereas, visible light is between 0.3 to 0.7 µm

(Siegel and Howell, 2002).

Radiant energy is transported by electro magnetic waves or by photon at the speed of light.

Quantum theory explains the radiation process and details of absorbing and emitting of radiation.

Plank’s law explains that maximum amount of energy can be emitted at a given temperature and

given wavelength, in this case emitter is called blackbody (Sparrow and Cess, (1970). Radiation is

described as a process in which one body emits energy and this energy travels through any medium

or through space and finally absorbed by another body. More precisely, we can say that thermal

radiation is a phenomenon, in which surface of a substance radiates its thermal energy as

electromagnetic waves. Very simple example is radiation from a household radiator or electric

radiator. It is generated due to the movement of charged particles within atoms is converted to

electromagnetic radiation. Radiation is the third method used in heat transfer. It is also called

thermal radiation.

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When we rise temperature of a rod, its yellow-orange part is quite visible. It is the part which emit

thermal, mainly due to high temp. Human eye cam see a very limited radiation. Major radiation is

in the infra red range, which is not possible to see with naked eye, however van be viewed with

through infrared Thermography, thermal imaging, thermographic imaging, or thermal video. Al

these are different types of infrared imaging science.

Thermal radiation is quite common in nature. It is most commonly used for room heating, since

ancient times. Roman were also using this technology and they made hypocaust, which were used

to heat rooms even common bathrooms. There are many types of room heaters, like, wall,

underground, overhead panels , etc. These heaters do not contain high temperature. Their normal

temperature is between 25-30 C, whereas, the area is too much and it keeps room hot due to thermal

radiation. Nevertheless, there is also convection process is there due to the movement of air.

There are four main properties that characterize thermal radiation:

1. Radiation occurs at a wide range at any temperature. It is governed by the Planck’s law of

radiation (for idealized materials).

2. There is a gradual change in frequency and color with the change in temperature. In case of a

hot rod, it is red and by further heating it will move to middle of the range and then it looks

white and we call it a white hot

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3. There is a fast growth in frequency as compare to temperature. Its ratio is T4, where T is the

absolute temperature of the body.

4. In thermodynamics thermal radiation in is isotropic and unpolarized.

Following are the main properties that characterize thermal radiation:

1. Thermal radiation, at every point of temperature scale has wide range of frequency. Ratio of

frequency can be calculated by Planck’s law of radiation.

2. There is a change in color with the change in temperature. It starts from red to white.

3. The amount of radiation increases rapidly as temperature increases. (it grows as T4, where T

is the absolute temperature of the body)

4. Thermal radiation in a cavity in thermodynamic equilibrium is isotropic and unpolarized.

5. Black body is used as a reference to describe heat transfer phenomenon.

Black Body

Black body is an idealized object, which absorbs all electromagnetic radiation and does not reflect

any one and the object appears black in color when it is cold. However, black body emits radiation

and it is dependent on temperature. These thermal radiation from a black body is called black-body

radiation.

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At room temperature, black bodies emit mostly infrared wavelengths, but as the temperature

increases past a few hundred degrees Celsius, black bodies start to emit visible wavelengths,

appearing red, orange, yellow, white, and blue with increasing temperature. By the time an object is

white, it is emitting substantial ultraviolet radiation.

The term "black body" was introduced by Gustav Kirchhoff in 1860. If the object is a black body in

thermodynamic equilibrium, the radiation is termed black-body radiation. The emitted wave

frequency of the black body thermal radiation is described by a probability distribution depending

only on temperature, and for a genuine black body in thermodynamic equilibrium is given by

Planck’s law of radiation. Wien's law gives the most likely frequency of the emitted radiation, and

the Stefan–Boltzmann law gives the radiant intensity. There are no strictly exact black bodies in

nature, but graphite is a good approximation, and a closed box with graphite walls at a steady state

gives a good approximation to ideal black body radiation.

Gray Bodies and Emissivity Coefficients

For objects other than ideal blackbodies ('gray bodies') the Stefan-Boltzmann Law can be

expressed as

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q = ' ( T4 A (2)

where

' = emissivity of the object (one for a black body)

For the gray body the incident radiation (also called irradiation) is partly reflected, absorbed or

transmitted.

The emissivity coefficient lies in the range 0 < ' < 1 depending on the type of material and the

temperature of the surface. The emissivity of some common materials

• oxidized Iron at 390 oF (199 oC) > ' = 0.64

• polished Copper at 100 oF (38 oC) > ' = 0.03

Laws Governing Black Body Radiations

Planck's law states that

where

I($,T) d$ is the amount of energy per unit surface area per unit time per unit solid angle emitted in

the frequency range between $ and $ + d$ by a black body at temperature T;

h is the Planck constant;

c is the speed of light in a vacuum;

k is the Boltzmann constant;

$ is frequency of electromagnetic radiation; and

T is the temperature in kelvins.

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Wien's displacement law

Wien's displacement law shows how the spectrum of black body radiation at any temperature is

related to the spectrum at any other temperature. If we know the shape of the spectrum at one

temperature, we can calculate the shape at any other temperature.

A consequence of Wien's displacement law is that the wavelength at which the intensity of the

radiation produced by a black body is at a maximum, &max, it is a function only of the temperature:

Where:

The constant, b, known as Wien's displacement constant, is equal to 2.8977685(51)'10"3 m K.

Stefan–Boltzmann law

This law states that amount of thermal radiation emitted per second per unit area of the surface of a

black body is directly proportional to the fourth power of its absolute temperature. That is

where j*is the total energy radiated per unit area per unit time, T is the temperature in kelvins, and (

= 5.67'10"8 W m"2 K"4 is the Stefan–Boltzmann constant

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This diagram provides a relationship between wavelength and temperature and it is obvious that

total radiated amount vary with temperature. Although this plot shows relatively high temperatures,

the same relationships hold true for any temperature down to absolute zero. Visible light is between

380 to 750 nm. Source: K. Huang (2003)

Color Temperature (C)

Fain red glow 480

Dark red 580

Bright red 730

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Color Temperature (C)

Bright orange 930

Pale yellowish 1100

Yellowish white 1300

White (yellow if seen

from a distance)

Temperature >1400

Source: K. Huang, Statistical Mechanics (2003), p278

The Laws of Thermodynamics

Thermodynamics science is a classical example of axiomatic form. There is an explanation of

certain changes keeping some assumption in view and considering them always true. Fro this

purpose a deductive process has been used. Core characteristics of the axiom is that it does not need

to b proven rather there should be a sufficient evidence to make it acceptable in general (Leland,

2009).

There are certain principles which should be able to explain the changes occurring during heat

transferring process. To explain the fundament changes, two laws (first and second law of

thermodynamics) are quite sufficient but we find two more laws available in literature (Zeroth law).

The Intuitive Perception of the First Law

“Energy can be neither created nor destroyed but only transformed”.

The general energy equation:

Energy In = Energy Out

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or

U2 - U1 = Q - W

Where

U1: internal energy of the system at the beginning

U2: internal energy of the system at the end

Q: net heat flow into the system

W: net work done by the system

Following are the elements of a thermodynamics cycle:

1-Working Substance - medium by which energy is carried through the cycle.

2- Heat Source - supplies thermal energy to the working substance.

3- Heat Receiver - absorbs heat from the working substance.

(Source: www.owlnet.rice.edu/~nava102/presentations/lesson03)

First law of thermodynamics is also called law of conservation of energy and mass. There is a

perception present in nature about the conservation of energy and mass. Even Greek philosophers

believe that matter cannot be destroyed. However, in middle ages there was some confusion raised

from the burning of material, which leads to the convection that material can be destroyed.

Nevertheless, Lavoisier proved the conservation of mass during chemical reactions. As compare to

mass conservation, energy conservation is difficult to prove. However, our intuition tells us that

according to natural justice energy should never be created and not be destroyed. Not withstanding

that transformation is possible and is required to perform various jobs in the real world (Leland,

2009).

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The internal energy of a system of particles, U, is the total of the of the kinetic energy and potential

energy. According to first law, any change in the system will be sum of the energy induct into the

system and work done by the system.

#U = Q + W

Where:

#U: difference in internal energy

Q: energy provided to the system

W: work done by the system

We can conclude after observing the whole system at micro level that Q and W to be positive if

energy flows into the system and for a cyclic process:

(Ui = Uf) " Q = - W.

If, in addition, Q = 0 then W = 0

Leland (2009) summarizes the whole journey from initial idea of mass and energy conservation and

to the final observation about the conservation of mass and energy. Leland gives credit to Count

Rumford, who is the first person who observed the conservation of energy during canon

manufacturing process. However, Joule and Kelvin between 1843 and 1848 are the scientists who

came up with accurate and indisputable observation about the conservation of mass and energy

based on experiments. Conservation of energy and mass is a universal law. Considering the

magnitude of energy changes in thermodynamics, these are considered separately.

Second Law of Thermodynamics

We find traces of second law of thermodynamics in the paper, “Reflection on the Motive power of

fire, of Sadi Carnot” by a French physicist published in 1824. Sadi presented the idea that motive

power work is due to the flow of caloric (heat) from a hot body to cold body. The second law of

thermodynamics, which was formulated by Clausius and Thomson following Carnot's earlier

observation. This law is based on the flow of heat from a hot body to a body having low

temperature. This law is also known as the Law of Increased Entropy. It is an inherent and intrinsic

property of the world, that whenever, there is an imbalance in the energy distribution, it will start

working and will minimize the thermodynamic force, which is in fact gradient of potential. To

describe further, Clausius coined the term entropy. Entropy, which is defined as a thermodynamic

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quantity representing the unavailability of a system's thermal energy for conversion into mechanical

work. It is also called as the degree of disorder or randomness in the system. (Symbol: S). In a

natural world, entropy will always increase.

Source: http://www.entropylaw.com/entropy2ndlaw.html

First law states that energy cannot be generated and cannot be destroyed, whereas, this law

describes that the quality of matter/energy deteriorates gradually over time. It is further explained

that during work, useable energy is used and a portion of useable energy is converted into unusable

energy. There is an increase in the unusable energy, which is denoted as entropy. The second law of

thermodynamics represents an idea which is quite common and experienced by every one. This law

explains the direction, which is unique and works under the action of thermodynamics force. It was

not told to people that in case when no pump is used, gravitational force is responsible of water

flow from a higher point to a lower point. People discovered the earth's gravitational force by

different experiments moreover observed that the rate of transfer increases with an increase in the

difference in elevation. Since its reverse never happened, same is our experience about the flow of

heat from hot item to cold item. We found analogous examples in case of heat transfer. We observe

that rate of heat transfer increases when difference of temperature between two items is high and

keep on decreasing until there is no difference of temperature. This observation leads scientist to

develop a theory that temperature difference is the main reason of heat transfer. To derive this

theory no theoretical knowledge was required. Heat transfer phenomenon was observed and more

factors were added, like, area of contact, direction and formally a idea was completed, which is

called second law of thermodynamics (Leland, 2009).

Entropy is a very important idea in thermodynamics and contains versatile ideas. It is defined and

explained in different ways. It is the name of an extensive property of a system. It changes by

multiplying by temperature and gives thermal energy of a system. In a more simple way, change in

entropy is defined quantity of thermal energy which is transported from a system and divided by

temperature, which is a driving force. Also note that temperature is an intensive property, whereas,

entropy is an extensive property. Heat is also defined as the amount of thermal energy crossing

boundary of a system. This change will also lead to the change in the entropy (Leland, 2009). We

can conclude that entropy is measure of randomness or chaos within a closed system.

Table: 03 Common Thermodynamic Driving forces and Their Associated Displacements

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Source: Leland (2009)

http://www.allaboutscience.org/second-law-of-thermodynamics.htm

Third Law of Thermodynamics

It is a statistical law of nature related to entropy and deals with absolute zero temperature.

Commonly its expressed as, “As a system approaches absolute zero, all processes cease and the

entropy of the system approaches a minimum value”. It is not necessary to have minimum value

zero, although it is almost zero in a perfect and pure crystal form (Kittel and Kroemer, 1980).

Kittel and Kroemer (1980) have put forward that this law was developed by the chemist Walther

Nernst, during the years 1906-1912, and is thus sometimes referred to as Nernst's theorem or

Nernst's postulate. The third law of thermodynamics states that the entropy of a system at absolute

zero is a well-defined constant. This is because a system at zero temperature exists in its ground

state, so that its entropy is determined only by the degeneracy of the ground state; or, it states that

"it is impossible by any procedure, no matter how idealised, to reduce any system to the absolute

zero of temperature in a finite number of operations."

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In most simple words, according to third law of thermodynamics, that entropy of most pure

substances approaches zero as the absolute temperature approaches to zero. This provides a point,

which could be absolute entropy. This law further explains that at 0 K no solid solutions should

exist. Phases in equilibrium at 0 K should either be pure elements or atomically ordered phases

(Abriata, and Laughlin, 2004)

The Zeroth Law of Thermodynamics

It is a generalization about the thermal equilibrium. It can be stated as:

"If A and C are each in thermal equilibrium with B, A is also in thermal equilibrium with C."

Thermal equilibrium is a state when there is no change in temperature over time. If A, B and C are

distinct bodies and they are in equilibrium, then we can perceive Euclidean relation and reflexive

relation. Such relation which are both Reflexive and Euclidean are called equivalence relations and

can be expressed in the following equation (Reif, 1965):

if T(A) = T(B)

and T(B) = T(C)

then T(A) = T(C).

Mass Transfer

Mass transfer is process in which mass transfer from a high concentration to low concentration, or

from a high pressure to low pressure. It involves molecular and convective transport of atoms and

molecules within a physical system. It may be fluid flow or separation of different materials.

Evaporation of water, diffusion of impurities in a water solution, distillation , etc. Main force which

is responsible for the mass transfer is the concentration. It is the random motion of molecules,

which hold pressure and move into an area in surrounding where pressure of random molecules is

less.

Analogy Among Molecular Transport, Mass and Heat Transfer

There is a similarity among molecular transport, heat transfer and mass transfer. Three basic

equation dealing with these are:

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1. Newton equation for momentum

2. Fourier’s law

3. Fick’s law for mass transfer

Many people have done much effort to quantify the analogy among these. One of popular effort is

Reynolds analogy. Reynolds assumes that turbulent diffusivities are all same. Chilton and Colburn

J-factor analogy is one of the most commonly quoted in literature. This analogy is based on laminar

and turbulent data. This is an experimental data and van be used to satisfy the results. Ambrosian et

al. (2007) reviewed the work of different writes and finally conclude that for a better correlation of

experimental data related to friction factors or heat transfer analogy between mass, molecular

diffusion and heat transfer plays a significant role. They further explains that Reynolds’ proposal is

a starting point to discuss the analogy between mass and heat transfer. However, there are many

people, who worked on this topic and gave their significant contribution in the body of knowledge

related to this analogy.

References:

Abriata, J. P.; Laughlin, D. E. (2004). The Third Law of Thermodynamics and low temperature

phase stability. Progress in Materials Science 49 (3–4): 367–387.

Ambrosian,W.; Forgone, N.; Manfredini ,A. and Oriole, F. (2007). On various forms of the heat and

mass transfer analogy: Discussion and application to condensation experiments.Nuclear

Engineering and Design 236 (2006) 1013–1027

Engineering Tool Box (2009). Radiation Heat Transfer. Retrieved from http://

www.engineeringtoolbox.com/radiation-heat-transfer-d_431.html Dec1, 2009.

Incropera, Frank P. and Dewitt, David P. (1990). Fundamentals of Heat and Mass Transfer (3rd

ed.). John Wiley & Sons.

Incropera, Frank P.; Dewitt, David P. (1997). Fundamentals of Heat and Mass Transfer (4th

Edition ed.). Wiley. p. 493.

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Incropera, Frank P.; Dewitt, David P.; Bergman, Theodore L. And Lavine, nAdrienne L. (2007).

Introduction to Heat Transfer 5th Edition. John and Wiley Sons.

Incropera, Frank P.; Dewitt, David P. (1990), Fundamentals of Heat and Mass Transfer (3rd ed.),

John Wiley & Sons.

Jaluria, Yogesh. (1980). Natural Convection Heat and Mass Transfer. New York: Pergamon Press.

Jos Uffink, J. van Dis, S. Muijs. Grondslagen van de Thermische en Statistische Fysica. Utrecht

University

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