Heat transfer – lecture 1

Heat transfer – lecture 1
LITERATURE
Bejan A., Kraus A.D.: Heat Transfer Handbook. John Wiley & Sons, Inc., New
Jersey 2003
Baehr H.D., Stephan K.: Heat and Mass Transfer. Springer, Berlin 2006
NOMENCLATURE
Roman Letter Symbols
A, (F, S) cross-sectional area, m 2
c specific heat, W/m · K
d differential, diameter, m
e specific energy, J/kg · K
h heat transfer coefficient, W/m 2 · K, specific enthalpy, J/kg
k thermal conductivity, W/m · K
L path length, m
q heat flow, W
q’’ heat flux, W/m 2
R thermal resistance, K/W
r radius, m
S surface area, m 2
T temperature, K
t time, s
V volume, m 3
Vˆ velocity, m/s
Greek Letter Symbols
β coefficient of volumetric expansion, m -1
∆ change, dimensionless
δ thickness, m
µ dynamic viscosity, N/m · s
ν kinematic viscosity, m 2 /s
ρ density, kg/m 3
Roman Letter Subscripts
cd conduction
cv convection
f fluid
s surface condition
w wall condition
1

Introduction
Thermal sciences generally deal with four basic thermal transport modes:
conduction, convection, phase change, and radiation.
The process by which heat diffuses through a solid or a stationary fluid is
termed heat conduction. Situations in which heat transfer from a wetted surface is
assisted by the motion of the fluid give rise to heat convection, and when the fluid
undergoes a liquid–solid or liquid–vapor state transformation at or very near the
wetted surface, attention is focused on this phase-change heat transfer. The
exchange of heat between surfaces, or between a surface and a surrounding fluid, by
long-wavelength electromagnetic radiation is termed thermal heat radiation.
During this short course I describe briefly these modes of heat transfer, with
emphasis on an important parameter known as the thermal resistance to heat
transfer. The calculation methods presented here will be relatively simple to give you
just basis for the further study at this faculty, as for example heat or ventilation
installations design.
The different types of heat transfer
In thermodynamics, heat is defined as the energy that crosses the boundary of
a system when this energy transport occurs due to a temperature difference between
the system and its surroundings. The second law of thermodynamics states that heat
always flows over the boundary of the system in the direction of falling temperature.
However, thermodynamics does not state how the heat transferred depends on this
temperature driving force, or how fast or intensive this irreversible process is. It is the
task of the science of heat transfer to clarify the laws of this process.
Three modes of heat transfer can be distinguished: conduction, convection,
and radiation. The following course deal with their basic laws and phenomenological
description of heat transfer processes, using the thermodynamic concepts of
temperature, heat, heat flow and heat flux. In contrast to thermodynamics, which
mainly deals with homogeneous systems, the so-called phases, heat transfer is a
continuum theory which deals with fields extended in space and also dependent on
time.
This has consequences for the concept of heat, which in thermodynamics is
defined as energy which crosses the system boundary. In heat transfer one speaks
of a heat flow also within the body. This contradiction with thermodynamic
terminology can be resolved by considering that in a continuum theory the mass and
volume elements of the body are taken to be small systems, between which energy
can be transferred as heat. Therefore, when one speaks of heat flow within a solid
body or fluid, or of the heat flux vector field in conjunction with the temperature field,
the thermodynamic theory is not violated.
As in thermodynamics, the thermodynamic temperature T is used in heat
transfer. However with the exception of radiative heat transfer the zero point of the
thermodynamic temperature scale is not needed, usually only temperature
differences are important. For this reason a thermodynamic temperature with an
adjusted zero point, an example being the Celsius temperature, is used.
Heat conduction
Heat conduction is the transfer of energy between neighbouring molecules in a
substance due to a temperature gradient. In metals also the free electrons transfer
energy. In solids which do not transmit radiation, heat conduction is the only process
2

for energy transfer. In gases and liquids heat conduction is superimposed by an
energy transport due to convection and radiation.
The mechanism of heat conduction in solids and fluids is difficult to understand
theoretically. We do not need to look closely at this theory; it is principally used in the
calculation of thermal conductivity, a material property. We will limit ourselves to the
phenomenological discussion of heat conduction, using the thermodynamic
quantities of temperature, heat flow and heat flux, which are sufficient to deal with
most technically interesting conduction problems.
One-Dimensional Conduction
Thermal diffusion through solids is governed by Fourier’s law (Jean Baptiste
Fourier (1768–1830) was Professor for Analysis at the Ecole Polytechnique in Paris
and from 1807 a member of the French Academy of Science. His most important
work “Th´eorie analytique de la chaleur” appeared in 1822. It is the first
comprehensive mathematical theory of conduction and cointains the “Fourier Series”
for solving boundary value problems in transient heat conduction.), the basic law for
the conduction of heat, from 1822. The minus sign in this equation is accounting for
the 2nd law of thermodynamics: heat flows in the direction of falling temperature.
Fourier’s law in one-dimensional form is expressible as:
where q is the heat current, k the thermal conductivity of the medium, A the
crosssectional area for heat flow, and dT/dx the temperature gradient, which,
because it is negative, requires insertion of the minus sign to assure a positive heat
flow q. The temperature difference resulting from the steady-state diffusion of heat is
thus related to the thermal conductivity of the material, the cross-sectional area A,
and the path length L, according to
The form of above equation, where k and A are presumed constant, suggests
that in a way that is analogous to Ohm’s law governing electrical current flow through
a resistance, it is possible to define a conduction thermal resistance as
3

Convective Heat Transfer
Heat Transfer Coefficient Convective thermal transport from a surface to a
fluid in motion can be related to the heat transfer coefficient h, the surface-to-fluid
temperature difference, and the “wetted” surface area S in the form
The differences between convection to a rapidly moving fluid, a slowly flowing
fluid, or a stagnant fluid, as well as variations in the convective heat transfer rate
among various fluids, are reflected in the values of h. For a particular geometry and
flow regime, h may be found from available empirical correlations and/or theoretical
relations. Use of above equation makes it possible to define the convective thermal
resistance as
4

Natural Convection
In natural convection, fluid motion is induced by density differences resulting
from temperature gradients in the fluid. The heat transfer coefficient for this regime
can be related to the buoyancy and the thermal properties of the fluid.
Forced Convection
Here fluid motion in tubes, pipes or channels with the equivalent diameter de is
forced by an external machine such as fun or pump.
Coordinate systems
Heat transfer and fluid flow analyses of objects of various sizes and shapes
and their corresponding flow fields are facilitated by working in a coordinate system
that provides a good fit to the flow geometry.
Most practically important application, is the conduction of heat independent of
time, so called steady conduction, in a flat plate or in a hollow cylinder. The
assumption is made that heat flows in only one direction, perpendicular to the plate
surface, and radially in the cylinder and sphere. The temperature field is then only
dependent on one geometrical coordinate. This is known as one-dimensional heat
conduction.
5