Chapter 4TRANSIENT HEAT CONDUCTION

Heat Transfer

ObjectivesAssess when the spatial variation of temperature isnegligible, and temperature varies nearly uniformly withtime, making the simplified lumped system analysisapplicable.

Obtain analytical solutions for transient one-dimensionalconduction problems in rectangular, cylindrical, andspherical geometries using the method of separation ofvariables, and understand why a one-term solution is usually a reasonable approximation.

Solve the transient conduction problem in large mediumsusing the similarity variable, and predict the variation oftemperature with time and distance from the exposedsurface.

Construct solutions for multi-dimensional transient

UMPED SYSTEM ANALYSISor temperature of some

es remains essentially rm at all times during a transfer process.

temperature of such es can be taken to be a ion of time only, T(t). transfer analysis that

es this idealization is wn as lumped system ysis.

A small copper ball can be modeled as a

lumped system but

grating withTi at t = 0T(t) at t = t

The geometry and parameters involved in the lumped system analysis.

Time t t

temperature of a lumped system

• 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 indicates that the body approaches the environment temperature in a short time.

Heat transfer to or from a body reaches its

maximum value when the

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 surroundingmedium over the time interval t = 0 to t

The maximum heat transfer between the body and its surroundings

eria for Lumped System Analysis

Lumped system analysis is applicable if

When Bi 0.1, the temperatures within the body relative to the surroundings (i.e., T −T) remain within 5 percent of each other.

Characteristic length

Biot number

Small bodies with high thermal conductivities and low convectioncoefficients 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.

Heat Transfer in Lumped Systems

ANSIENT HEAT CONDUCTION IN LARGE PLANE LLS, LONG CYLINDERS, AND SPHERES WITH

ATIAL EFFECTSl consider the variation of temperature me and position in one-dimensional ms such as those associated with a large wall, a long cylinder, and a sphere.

Schematic of the i l t i i

Transient temperature profiles in aplane wall exposed to convection

from its surfaces for Ti >T.

dimensionalized One-Dimensional Transientduction Problem

Nondimensionalization reduces the number of independent variables in one-dimensional transient

ct Solution of One-Dimensional Transient duction Problem

The analytical solutions of transient conduction problems typically involve infinite series, and thus the evaluation of an infinite number of terms to determine the temperature at a specified location and time.

proximate Analytical and Graphical Solutions

ution with one-term approximation

terms in the series solutions converge rapidly with increasing time, for > 0.2, keeping the first term and neglecting all the remaining

ms in the series results in an error under 2 percent.

a) Midplane temperature

ient temperature and heat transfer chartser and GrÖber charts) for a plane wall of thickness

c) Heat transfer

he dimensionless temperatures anywhere in a plane wall, linder, and sphere are related to the center temperature by

f f f

e Fourier number is a easure of heat

onducted through a body ative to heat stored.arge value of the urier number indicates

ster propagation of heat ough a body.

Fourier number at time t can be viewed as the

ratio of the rate of heat

e physical significance of the Fourier number

in

ANSIENT HEAT CONDUCTION IN SEMI-INITE SOLIDS

matic of a semi-infinite body.

Semi-infinite solid: An idealized body that has a single plane surface and extends to infinity in all directions.The earth can be considered to be a semi-infinite medium in determining the variation of temperature near its surface. A thick wall can be modeled as a semi-infinite medium if all we are interested in is the variation of temperature in the region near one of the surfaces, and the other surface is too far to have any impact on the region of interest during the time of observation

hort periods of time, most bodies d l d i i fi it lid

Transformation of variables i th d i ti f th

ytical solution for the case of constant temperature Ts on the surface

Error function

unction is a standardmatical function, just like thend cosine functions, whose varies between 0 and 1.

Analytical solutions for different boundary conditions on the surface

Dimensionlesstemperature distribution for transient conductionin a semi-infinite solid whose surface is maintained at a constanttemperature Ts.

act of Two Semi-Infinite Solidstwo large bodies A and B, initially at

m temperatures TA,i and TB,i are ht into contact, they instantly achieve rature equality at the contact e.wo bodies are of the same material, ntact surface temperature is the etic average, Ts = (TA,I + TB,i)/2.

bodies are of different materials, the e temperature Ts will be different he arithmetic average.

Contact of two semi-infinite solids ofdifferent initial temperatures.

The interface temperature of two bodies brought into contact is dominated by the body with the larger kcp.

ANSIENT HEAT CONDUCTION INLTIDIMENSIONAL SYSTEMSng a superposition approach called the product solution, the transient perature charts and solutions can be used to construct solutions for the two-

mensional and three-dimensional transient heat conduction problems ountered in geometries such as a short cylinder, a long rectangular bar, a tangular prism, or a semi-infinite rectangular bar, provided that all surfaces of solid are subjected to convection to the same fluid at temperature T, with the

me heat transfer coefficient h, and the body involves no heat generation.e solution in such multidimensional geometries can be expressed as the oduct of the solutions for the one-dimensional geometries whose intersection he multidimensional geometry.

The temperature in a shortcylinder exposed to convection from all surfaces varies in both the radial and axial directions and thus

A short cylinder of radius ro and height a is the intersection of a longcylinder of radius ro and a plane wall of thickness a

olution for a multidimensional geometry is the product of the solutions for the imensional geometries whose intersection is the multidimensional body.olution for the two-dimensional short cylinder of height a and radius ro is to the product of the nondimensionalized solutions for the one-dimensionalwall of thickness a and the long cylinder of radius ro.

ng solid bar of rectangular profile b is the intersection of two plane

he transient heat transfer for a two-dimensional eometry formed by the intersection of two one-mensional geometries 1 and 2 is

ransient heat transfer for a three-dimensional body ormed by the intersection of three one-dimensional odies 1, 2, and 3 is

tidimensional solutions expressed as products of one-dimensional tions for bodies that are initially at a uniform temperature Ti and osed to convection from all surfaces to a medium at T

imensional solutions expressed as products of one-dimensional ons for bodies that are initially at a uniform temperature Ti and exposed vection from all surfaces to a medium at T

ummaryLumped System Analysis Criteria for Lumped System Analysis

Some Remarks on Heat Transfer in Lumped Systems

Transient Heat Conduction in Large Plane Walls, Long Cylinders, and Spheres with Spatial Effects Nondimensionalized One-Dimensional Transient Conduction

Problem

Exact Solution of One-Dimensional Transient Conduction Problem

Approximate Analytical and Graphical Solutions

Transient Heat Conduction in Semi-Infinite Solids Contact of Two Semi-Infinite Solids

Transient Heat Conduction in Multidimensional Systems