•In this chapter, we consider the variation of temperature with time as well as position in one- and multidimensional systems.

•We start this chapter with the analysis of lumped systems in which the temperature of a solid varies with time but remains uniform throughout the solid at any time.

LUMPED SYSTEM ANALYSIS

•some bodies are observed to behave like a lump”•whose interior temperature remains essentiallyuniform at all times during a heat transfer process.•which provides great simplification in certain classes of heat transfer problems

b is a positive quantity The reciprocal of b hastime unit (usually s), and is called the time constant.

The temperature of a body approaches the ambient temperature exponentially. The temperature changes rapidly at the beginning, but rather slowly later on.A large value of b indicates that the body will approach the environment temperature in a short time.Note that b is proportional to the surface area, but inversely proportional to the mass and the specific heat of the body.it takes longer to heat or cool a larger mass, especiallywhen it has a large specific heat.

Criteria for Lumped System Analysis

characteristic length

Biot number

•lumped system analysis is exact when Bi = 0 and approximate when Bi > 0.

•smaller the Bi number, the more accurate the lumped system analysis.

•How much accuracy?

•20 percent uncertainty in the convection heat transfer coefficient h in most cases is considered“normal” and “expected.”

•It is generally accepted that lumped system analysis is applicable if Bi ≤ 0.1

the larger the thermal conductivity, the smaller the temperature gradient.

variation of temperature withtime and position in one-dimensional problems such asthose associated with a largeplane wall, a long cylinder, and asphere.

In order to reduce the number of parameters, we nondimensionalize the problem by defining thefollowing dimensionless quantities:

enables us to present the temperature in terms of only: X, Bi, and This makes it practical to present the solution in graphical form.

The one-dimensional transient heat conductionproblem, can be solved exactly for any of thethree geometries, but the solution involvesinfinite series, which are difficult to deal with.However, the terms in the solutions convergerapidly with increasing time, and for > 0.2

it is very convenient to express the solution using this oneterm approximation, given as