Transient Conduction

1
Chapter 5

• Transient Conduction

2
Transient Conduction

• Many heat transfer problems are time dependent
• Changes in operating conditions in a system cause
  temperature variation with time, as well as
  location within a solid, until a new steady state
  (thermal equilibrium) is obtained.
• In this chapter we will develop procedures for
  determining the time dependence of the
  temperature distribution
• Real problems may include finite and
  semi-infinite solids, or complex geometries, as
  well as two and three dimensional conduction
• Solution techniques involve the lumped
  capacitance method, exact and approximate
  solutions, and finite difference methods.
• We will focus on the Lumped Capacitance Method,
  which can be used for solids within which
  temperature gradients are negligible (Sections
  5.1-5.3)

3
Lumped Capacitance Method

• Consider a solid that is initially at a uniform
  temperature, Ti , and at t0 is quenched by
  immersion in a cool liquid, of lower temperature
• The temperature of the solid will decrease for
  time tgt0, due to convection heat transfer at the
  solid-liquid interface, until it reaches

T
t0
4
Lumped Capacitance Method

• If the thermal conductivity of the solid is very
  high, resistance to conduction within the solid
  will be small compared to resistance to heat
  transfer between solid and surroundings.
• Temperature gradients within the solid will be
  negligible, i.e. the temperature of the solid is
  spatially uniform at any instant.

T
5
Lumped Capacitance Method

• Starting from an overall energy balance on the
  solid

where
(5.1)

• Lets define a thermal time constant

Rt is the resistance to convection heat
transfer Ct is the lumped thermal capacitance of
the solid
(5.2)
6
Transient Temperature Response

• Based on eq. (5.1) the temperature difference
  between solid and fluid decays exponentially.

7
Transient Temperature Response
From eq. (5.1) the time required for the solid
to reach a temperature T is
(5.3)
The total energy transfer, Q, occurring up to
some time t is
(5.4)
8
Validity of Lumped Capacitance Method

• Surface energy balance

Ts,1
qcond
qconv
Ts,2
(5.5)
T?
9
Validity of Lumped Capacitance Method
(Rearranging 5.5)

• What is the relative magnitude of DT solid versus
  DT solid/liquid for the lumped capacitance method
  to be valid?

10
Biot and Fourier Numbers

• The lumped capacitance method is valid when

where the characteristic length LcV/AsVolume
of solid/surface area
We can also define a dimensionless time, the
Fourier number
where
Eq. (5.1) becomes
(5.6)
11
True or False?

• A hot solid will cool down faster when it is
  cooled by forced convection in water rather than
  in air.
• For the same solid, the lumped capacitance method
  is likely more applicable when it is being cooled
  by forced convection in air than in water.
• The lumped capacitance method is likely more
  applicable for cooling of a hot solid made of
  aluminum (k237 W/m.K) than copper (k400 W/m.K)
• The transient response is accelerated by a
  decrease in the specific heat of the solid.
• The physical meaning of the Biot number is that
  it represents the relative magnitude of
  resistance due to conduction and resistance due
  to convection.

12
Example (Problem 5.7 Textbook)

• The heat transfer coefficient for air flowing
  over a sphere is to be determined by observing
  the temperature-time history of a sphere
  fabricated from pure copper. The sphere, which is
  12.7 mm in diameter, is at 66C before it is
  inserted into an air stream having a temperature
  of 27C. A thermocouple on the outer surface of
  the sphere indicates 55C, 69 s after the sphere
  is inserted in the air stream.
• Calculate the heat transfer coefficient, assuming
  that the sphere behaves as a spacewise isothermal
  object. Is your assumption reasonable?

13
What if?

• What happens to the rate of cooling if h
  increases?
• What happens to the rate of cooling if the
  diameter of the sphere increases?
• What happens if we have a huge sphere?

14
General Lumped Capacitance Analysis

• In the general case we may have convection,
  radiation, internal energy generation and an
  applied heat flux. The energy balance becomes

Tsur
qrad
qs
T?, h
qconv

• Numerical solutions are generally required
• Simplified solutions exist for no imposed heat
  flux or generation (see equations (5.19, 5.25)
  textbook).

As(c,r)
As,h
15
Example 5.2

• Calculation of the steady state temperature of
  the thermocouple junction.
• How much time is needed for the temperature to
  increase from 25C to within 1C from its steady
  state value?

16
Example 5.2
17
Example (5.33)

• Microwave ovens operate by rapidly aligning and
  reversing water molecules within the food,
  resulting in volumetric energy generation.
• Consider a frozen 1-kg spherical piece of ground
  beef at an initial temperature of Ti-20C.
  Determine how long it will take the beef to reach
  a uniform temperature of T0C, with all the
  water in the form of ice. Assume that 3 of the
  oven power (P1kW total) is absorbed by the food.
• After all the ice is converted to liquid,
  determine how long it will take to heat the beef
  to Tf80C, if 95 of the oven power is absorbed.

18
Other transient problems

• When the lumped capacitance analysis is not
  valid, we must solve the partial differential
  equations analytically or numerically
• Exact and approximate solutions may be used
• Tabulated values of coefficients used in the
  solutions of these equations are available
• Transient temperature distributions for commonly
  encountered problems involving semi-infinite
  solids can be found in the literature

19
Summary

• The lumped capacitance analysis can be used when
  the temperature of the solid is spatially uniform
  at any instant during a transient process
• Temperature gradients within the solid are
  negligible
• Resistance to conduction within the solid is
  small compared to the resistance to heat transfer
  between the solid and the surroundings
• The Biot number must be less 0.1 for the lumped
  capacitance analysis to be valid.
• Transient conduction problems are characterized
  by the Biot and the Fourier numbers.