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ﻢﯿﻧاﻮﺧ ﯽﻣ ﻞﺼﻓ ﻦﯾا رد ﻪﭽﻧآ :
• Lumped Systems Analysis
• Transient Heat Conduction in Large Plane Walls, Long Cylinders, and Spheres
• Transient Heat Conduction in Semi-Infinite Solids
• Transient Heat Conduction in Multidimensional Systems
variation of temperature with time as well as position in 1D and
multidimensional systems:T(x,y,z,t)
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
دوﺪﺤﻣ ﻢﺘﺴﯿﺳ
• energy balance of the solid for the time interval dt:
Lumped System Analysis
temperature remains uniform within the body at all times and changes
with time only,T=T(t)
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
دوﺪﺤﻣ ﻢﺘﺴﯿﺳ
= & = −
=
@ = 0: =
−
− = =
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
دوﺪﺤﻣ ﻢﺘﺴﯿﺳ
Temp. 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 will
approach the environment
temperature in a short time.
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
دوﺪﺤﻣ ﻢﺘﺴﯿﺳ
• rate of convection heat transfer:
• total amount of heat transfer:
= − ( )
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
دوﺪﺤﻣ ﻢﺘﺴﯿﺳ
• characteristic length: (just for a lumped system)
• And a Biot Number:
• small bodies with high thermal conductivity are good candidates for lumped system analysis
Criteria for Lumped System Analysis
=
=
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
دوﺪﺤﻣ ﻢﺘﺴﯿﺳ
• lumped system analysis is exact when Bi=0 and approximate when Bi>0.
– conduction resistance is zero
• It is generally accepted that lumped system analysis is applicable if:
the smaller the Bi number, the more accurate the lumped system analysis
≤ .
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
دوﺪﺤﻣ ﻢﺘﺴﯿﺳ
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
دوﺪﺤﻣ ﻢﺘﺴﯿﺳ
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
دوﺪﺤﻣ ﻢﺘﺴﯿﺳ
• Heat Conduction
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
يﺪﻌﺑ ﮏﯾ
• Heat transfer takes place between these bodies and their environments by convection with a uniform and constant heat transfer coefficient h.
• There is geometrical and thermal symmetry:
Transient Heat Conduction in Large Plane Walls, Long Cylinders, & Spheres
1D variation of temperature with time and position
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
يﺪﻌﺑ ﮏﯾ
at t=0:
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
يﺪﻌﺑ ﮏﯾ
Transient temperature profiles in a plane wall exposed to convection:
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
يﺪﻌﺑ ﮏﯾ
• Solution involves the parameters:
The formulation of the problems for the determination of the 1D transient temperature distribution T(x,t) in a wall results in a PDE, which can be solved using advanced mathematical techniques.
, , , , , , , For a cylinder or sphere:
Replacing x by r and L by the outer radius r
oارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
يﺪﻌﺑ ﮏﯾ
In order to reduce the number of parameters, we nondimensionalize the problem by defining the following dimensionless quantities:
= ; = [ ℃]
a large value of the Fourier number indicates
faster propagation of heat through a body
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
يﺪﻌﺑ ﮏﯾ
where the constants A & are functions of the Bi number only, and their
values are listed in Table 4–1 against the Bi number for all three geometries.
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
يﺪﻌﺑ ﮏﯾ
• Table 4-1:
: = : =
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
يﺪﻌﺑ ﮏﯾ
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
يﺪﻌﺑ ﮏﯾ
For those who prefer reading charts to interpolating, the relations above are plotted and the one-term approximation solutions are presented in graphical form, known as the transient temperature charts.
, : Sometimes requires interpolation
Heisler charts presented by M. P. Heisler in 1947
supplemented in 1961 by H. Gröber
• 3 charts for each geometry:
Temperature at the center of the geometry at a given time t:
(a) Midplane temperature (from M. P. Heisler)
> .
Temperature at other locations at the same time in terms of T
0(b) Temperature distribution (from M. P. Heisler)
> .
Total amount of heat transfer up to the time t (c) Heat transfer (from H. Gröber et al.)
> .
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
يﺪﻌﺑ ﮏﯾ
• Same charts for cylinder & sphere
• Maximum amount of heat that a body can gain (or lose): (t → ∞ )
• The ratio Q/Qmax plotted in figures 4–13c, 4–14c, and 4–15c.
• body is initially at a uniform temperature,
• T & h of the medium surrounding the body are constant & uniform,
• and there is no energy generation in the body.
Remember:
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
يﺪﻌﺑ ﮏﯾ
Brass plate: ﺞﻧﺮﺑ ﺲﻨﺟ زا يا ﻪﺤﻔﺻ
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
يﺪﻌﺑ ﮏﯾ
Assumptions:
• Heat conduction in the plate is 1D since the plate is large relative to its thickness
• thermal symmetry about the center plane
• Fourier number is > 0.2 so that the one-term approximate
solutions are applicable
Heisler charts
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
يﺪﻌﺑ ﮏﯾ
• Biot number in this case is Bi=1/45.8=0.022, which is much less than 0.1. Therefore, we expect the lumped system analysis to be applicable.
• This is also evident from = 0.99 which indicates that the
temperatures at the center and the surface of the plate relative to
the surrounding temperature are within 1% of each other.
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
يﺪﻌﺑ ﮏﯾ
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
ﺖﯾﺎﻬﻧ ﯽﺑ ﻪﻤﯿﻧ يﺪﻌﺑ ﮏﯾ
• A semi-infinite solid is an idealized body that has a single plane surface and extends to infinity in all directions.
• This idealized body is used to indicate that the temperature change in the part of the body in which we are interested (the region close to the surface)
Transient Heat Conduction in In Semi-infinite Solids
vertical axis correspond to x=0, and represent the surface temperature
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
ﺖﯾﺎﻬﻧ ﯽﺑ ﻪﻤﯿﻧ يﺪﻌﺑ ﮏﯾ
• Exact solution of the transient 1D heat conduction problem in a semi-
infinite medium that is initially at a uniform temperature of Ti and is
suddenly subjected to convection at time t=0:
ارﺬﮔ تراﺮﺣ لﺎﻘﺘﻧا :
ﺖﯾﺎﻬﻧ ﯽﺑ ﻪﻤﯿﻧ يﺪﻌﺑ ﮏﯾ
• Special case: ℎ → ∞
– Surface Temp (Ts)=Fluid Temp (T∞)
