chapter 4 · 2018-01-28 · ansient heat conduction in semi-inite solids matic of a semi-infinite...
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Chapter 4TRANSIENT HEAT CONDUCTION
Heat Transfer
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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
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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
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grating withTi at t = 0T(t) at t = t
The geometry and parameters involved in the lumped system analysis.
Time t t
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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.
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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
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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
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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
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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.
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dimensionalized One-Dimensional Transientduction Problem
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Nondimensionalization reduces the number of independent variables in one-dimensional transient
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ct Solution of One-Dimensional Transient duction Problem
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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.
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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.
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a) Midplane temperature
ient temperature and heat transfer chartser and GrÖber charts) for a plane wall of thickness
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c) Heat transfer
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he dimensionless temperatures anywhere in a plane wall, linder, and sphere are related to the center temperature by
f f f
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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
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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
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Transformation of variables i th d i ti f th
ytical solution for the case of constant temperature Ts on the surface
Error function
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unction is a standardmatical function, just like thend cosine functions, whose varies between 0 and 1.
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Analytical solutions for different boundary conditions on the surface
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Dimensionlesstemperature distribution for transient conductionin a semi-infinite solid whose surface is maintained at a constanttemperature Ts.
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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.
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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
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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.
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ng solid bar of rectangular profile b is the intersection of two plane
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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
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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
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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
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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