HEAT TRANSFER

UNIT - 1
Introduction
However a temperature difference exists within a system or when two systems at different temperatures
are brought into contact, energy is transferred. The process by which the energy transport takes place is
known as heat transfer. Heat cannot be measured or observed directly, but the effect it produces is
amenable to observation and measurement.
Difference between heat and temperature
In describing heat transfer problems, we often make the mistake of interchangeably using the terms heat
and temperature. Actually, there is a distinct difference between the two. Temperature is a measure of the
amount of energy possessed by the molecules of a substance. It is a relative measure of how hot or cold a
substance is and can be used to predict the direction of heat transfer. The usual symbol for temperature is
T. The scales for measuring temperature in SI units are the Celsius and Kelvin temperature scales. On the
other hand, heat is energy in transit. The transfer of energy as heat occurs at the molecular level as a result
of a temperature difference. The usual symbol for heat is Q. Common units for measuring heat are the
Joule and calorie in the SI system.
Difference between thermodynamics and heat transfer

Thermodynamics tells us:

• How much heat is transferred (δQ)
• How much work is done (δW)
• Final state of the system
Heat transfer tells us:
• How (with what modes) δQ is transferred
• At what rate δQ is transferred
• Temperature distribution inside the body

Modes of heat transfer

Conduction :Heat conduction is a mechanism of heat transfer from a region of high temperature to a
region of low temperature within a medium or between different medium in direct physical contact.
Examples: Heating a Rod.

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Convection:It is a process of heat transfer that will occur between a solid surface and a fluid medium
when they are at different temperatures. It is possible only in the presence of fluid medium.
Example: Cooling of Hot Plate by air

Radiation: The heat transfer from one body to another without any transmitting medium. It isan
electromagnetic wave phenomenon.
Example: Radiation sun to earth.
Basic laws of heat transfer governing conduction
Basic law of governing conduction:This law is also known as Fourier’s law of conduction.
The rate of heat conduction is proportional to the area measured normal to the direction of heat flow and
to the temperature gradient in that direction

dt
Q 𝖺 −A
dx
dt
Q = −KA
dx

Where,A – Area in m2

dt
–Temperaturegradient, K /m
dx

K–Thermal conductivity, W/mk

Basic law of governing convection: Thislaw is also known as Newton’s law of convection.
An energy transfer across a system boundary due to a temperature difference by the combined mechanisms
of intermolecular interactions and bulk transport. Convection needs fluid matter.

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Newton’s Law of Cooling:

𝑞 = ℎ𝐴𝑠ΔT
Where:
q = heat flow from surface, a scalar, (W)
h = heat transfer coefficient (which is not a thermodynamic property ofthe material, but may depend on
geometry of surface, flowcharacteristics, thermodynamic properties of the fluid, etc. (W/m 2 K)
𝐴𝑠 = Surface area from which convection is occurring. (m 2)
ΔT = Ts − T∞ = Temperature Difference between surface and coolant. (K)

Basic law of governing radiation: This law is also known as SteffanBoltzman law.
According to the SteffanBoltzman law the radiation energy emitted by a body is proportional to the
fourth power of its absolute temperature and its surface area.

𝑞 = 𝜀𝜎𝐴(𝑇𝑠4 − 𝑇𝑠𝑢𝑟4)

Where:
ε = Surface Emissivity
𝜎 = Steffan Boltzman constant
A= Surface Area
Ts = Absolute temperature of surface. (K)
Tsur = Absolute temperature of surroundings. (K)

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Thermal conductivity: Thermal conductivity is a thermodynamic property of a material “the amount of energy
conducted through a body of unit area and unit thickness in unit time when the difference in temperature
between faces causing heat flow is unit temperature difference”.
Derivation of general three dimensional conduction equation in Cartesian coordinate
Consider a small rectangular element of sides dx, dy and dz as shown in figure.The energy balance of this
rectangular element is obtained from first law of thermodynamics
Consider the differential control element shown below. Heat is assumed to flow through the element in the
positive directions as shown by the 6 heat vectors.

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Discussion on 3-D conduction in cylindrical and spherical coordinate systems

Cylindrical coordinate system:

The 3-Dimensional conduction equation in cylindrical co-ordinates is given by,

Spherical coordinate systems:

The 3-Dimensional conduction equation in cylindrical co-ordinates is given by,

In each equation the dependent variable, T, is a function of 4 independent variables, (x,y,z,τ);(r,θ,z,τ);

(r,φ,θ,τ) and is a 2nd order, partial differential equation. The solution of suchequations will normally require a
numerical solution. For the present, we shall simply look atthe simplifications that can be made to the equations
to describe specific problems.
 Steady State: Steady state solutions imply that the system conditions are not changing with time.
Thus ∂T / ∂τ = 0.

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 One dimensional: If heat is flowing in only one coordinate direction, then it follows that there is
notemperature gradient in the other two directions. Thus the two partials associated with these
directionsare equal to zero.
 Two dimensional: If heat is flowing in only two coordinate directions, then it follows that there is
notemperature gradient in the third direction. Thus the partial derivative associated with this third
directionis equal to zero.
 No Sources: If there are no heat sources within the system then the term, q=0.
Note that the equation is 2nd order in each coordinate direction so that integration will resultin 2 constants of
integration. To evaluate these constants two additional equations must bewritten for each coordinate direction based
on the physical conditions of the problem. Suchequations are termed “boundary conditions’.
Boundary and Initial Conditions:
 The objective of deriving the heat diffusion equation is to determine the temperature distribution
within the conducting body.
 We have set up a differential equation, with T as the dependent variable. The solution will give
us T(x,y,z). Solution depends on boundary conditions (BC) and initial conditions (IC).
 How many BC’s and IC’s?
 Heat equation is second order in spatial coordinate. Hence, 2 BC’s needed for each
coordinate.
o 1D problem: 2 BC in x-direction
o 2D problem: 2 BC in x-direction, 2 in y-direction
o 3D problem: 2 in x-dir., 2 in y-dir., and 2 in z-dir.
 Heat equation is first order in time. Hence one IC needed.

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Heat Diffusion Equation for a One Dimensional System:

Consider the system shown above. The top, bottom, front and back of the cube are insulated, so that heat
canbe conducted through the cube only in the x direction. The internal heat generation per unit volume is
q&(W/m3).
Consider the heat flow through an arbitrary differential element of the cube.

From the 1st Law we write for the element:

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One Dimensional Steady State Heat Conduction: The

plane wall:

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Thermal resistance (electrical analogy):

Physical systems are said to be analogous if that obey the same mathematical equation. The above
relations can be put into the form of Ohm’s law:
V=IRelec

Using this terminology it is common to speak of a thermal resistance:

ΔT = qRtherm

A thermal resistance may also be associated with heat transfer by convection at a surface. From
Newton’s law of cooling,
𝑞 = ℎ𝐴(𝑇𝑆 − 𝑇 ) ∞

The thermal resistance for convection is then

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Applying thermal resistance concept to the plane wall, the equivalent thermal circuit for the plane
wall with convection boundary conditions is shown in the figure below

Composite walls:
Thermal Resistances in Series:
Consider three blocks, A, B and C, as shown. They are insulated on top, bottom, front and back. Since the
energy will flow first through block A and then through blocks B and C, we say that these blocks are
thermally in a series arrangement.

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The steady state heat flow rate through the walls is given by:

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The following assumptions are made with regard to the above thermal resistance model:
1) Face between B and C is insulated.
2) Uniform temperature at any face normal to X.

1-D radial conduction through a cylinder:

One frequently encountered problem is that of heat flow through the walls of a pipe or through the
insulation placed around a pipe. Consider the cylinder shown. The pipe is either insulated on the ends or
is of sufficient length, L, that heat losses through the ends are negligible. Assume no heat sources within
the wall of the tube. If T1>T2, heat will flow outward, radially, from the inside radius, R1, to the outside
radius, R2. The process will be described by the Fourier Law.

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Composite cylindrical walls:

Critical Insulation Thickness:

Objective: decrease q, increase RTotal

Vary ro; as ro increases, first term increases, second term decreases.
This is a maximum – minimum problem. The point of extreme can be found by setting

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1-D radial conduction in a sphere:

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Summary of Electrical Analogy:

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