TRANSIENT HEAT
CONDUCTION
"With the new day
comes new strength
and new thoughts.
Either you run the day
or the day runs you."
Eleanor Roosevelt
LUMPED SYSTEM ANALYSIS
Interior temperature of some bodies remains
essentially uniform at all times during a heat
transfer process.
The temperature of such bodies can be taken to
be a function of time only, T (t).
A small copper ball can be
Heat transfer analysis that utilizes this modeled as a lumped system, but
a roast beef cannot.
idealization is known as lumped system
analysis.
3
Integrating with
T = Ti at t = 0 T
= T(t) at t = t
The geometry and
parameters involved in the
lumped system analysis.
time constant
4
• This equation enables us to
determine the temperature T(t)
of a body at time t, or
alternatively, the time t
required for the temperature
to reach a specified value T(t).
• The temperature of a body
approaches the ambient
temperature T exponentially.
• The temperature of the body
changes rapidly at the
beginning, but rather slowly
later on. A large value of b
The temperature of a lumped system indicates that the body
approaches the environment temperature as approaches the environment
time gets larger. temperature in a short time
5
The rate of convection heat transfer
between the body and its environment at
time t
The total amount of heat transfer between
the body and the surrounding medium
over the time interval t = 0 to t
The maximum heat transfer between the
body and its surroundings
Heat transfer to or from a body reaches its maximum value
when the body reaches the environment temperature. 6
Criteria for Lumped System Analysis
• Characteristic length
• Biot number
Lumped system analysis is applicable if
When Bi 0.1, the temperatures
The Biot number can be viewed as the within the body relative to the
ratio of the convection at the surface to surroundings (i.e., T −T) remain
conduction within the body. within 5 percent of each other.
7
Small bodies with high
thermal conductivities
and low convection
coefficients are most
likely to satisfy the
criterion for lumped
system analysis.
When the convection coefficient h is
high and k is low, large temperature
differences occur between the inner and
outer regions of a large solid.
passenger traffic to an island.
8
The temperature of a gas stream is to be measured by a thermocouple whose
junction can be approximated as a 1-mm-diameter sphere, as shown. The
properties of the junction are k 35 W/m · °C, 8500 kg/m3, and Cp 320 J/kg ·
°C, and the convection heat transfer coefficient between the junction and the
gas is h 210 W/m2 · °C. Determine how long it will take for the thermocouple
to read 99 percent of the initial temperature difference.
In order to read 99 percent of the initial temperature difference Ti T between the
junction and the gas, we must have
A person is found dead at 5 PM in a room whose temperature is 20°C. The
temperature of the body is measured to be 25°C when found, and the heat
transfer coefficient is estimated to be h 8 W/m2 · °C. Modeling the body as
a 30-cm-diameter, 1.70-m-long cylinder, estimate the time of death of that
person
TRANSIENT HEAT CONDUCTION IN LARGE PLANE WALLS,
LONG CYLINDERS, AND SPHERES WITH SPATIAL EFFECTS
consider the variation of temperature with time and position in 1-D problems such as
those associated with a large plane wall, a long cylinder, and a sphere.
Transient temperature profiles
in a plane wall exposed to
Schematic of the simple geometries in convection from its surfaces
which heat transfer is 1-D. for Ti >T.
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Nondimensionalized One-Dimensional Transient Conduction
Problem
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Fourier number
Bi = hLc/k
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Nondimensionalization reduces the number of independent variables in one-
dimensional transient conduction problems from 8 to 3, offering great convenience
in the presentation of results.
Exact Solution of 1-D Transient Conduction Problem
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Approximate Analytical and Graphical Solutions
The terms in the series solutions converge rapidly with increasing time, and for >0.2, keeping the first
term and neglecting all the remaining terms in the series results in an error under 2 percent.
Solution with one-term approximation
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(a) Midplane temperature
Transient temperature and heat transfer charts (Heisler and Grober
charts) for a plane wall of thickness 2L initially at a uniform temperature
Ti subjected to convection from both sides to an environment at
temperature T with a convection coefficient of h.
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(b) Temperature distribution
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(c) Heat transfer
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An ordinary egg can be approximated as a 5-cm-diameter sphere.
The egg is initially at a uniform temperature of 5°C and is
dropped into boiling water at 95°C. Taking the convection heat
transfer coefficient to be h = 1200 W/m2·K, determine how long it
will take for the center of the egg to reach 70°C.
Check for Bi??
