Unformatted text preview: T
Ti T
2
A1e 1 J0( 1r/ro),
T(r, t) T
Ti T
2 sin( 1r /ro)
,
A1e 1
1r /ro 0.2 0.2 0.2 where the constants A1 and 1 are functions of the Bi number
only, and their values are listed in Table 4–1 against the Bi
number for all three geometries. The error involved in oneterm solutions is less than 2 percent when
0.2.
Using the oneterm solutions, the fractional heat transfers in
different geometries are expressed as
Plane wall:
Cylinder:
Sphere: Q
Qmax wall
Q
Qmax cyl Q
Qmax sin 1 0, wall 1 2 1 3 1
1 J1( 1)
0, cyl 1 cos 1 0, sph 1 3
1 sph erfc
erfc x
2 exp t h x
2 t T(x, t) Ti
Ts Ti erfc x
2 (Ts t constant) Using a clever superposition principle called the product solution these charts can also be used to construct solutions for
the twodimensional transient heat conduction problems encountered in geometries such as a short cylinder, a long rectangular bar, or a semiinfinite cylinder or plate, and even
threedimensional problems associated with geometries such
as a rectangular prism or a semiinfinite rectangular bar, provided that all surfaces of the solid are subjected to convection
to the same fluid at temperature T , with the same convection
heat transfer coefficient h, and the body involves no heat
generation. The solution in such multidimensional geometries
can be expressed as the product of the solutions for the
onedimensional geometries whose intersection is the multidimensional geometry.
The total heat transfer to or from a multidimensional geometry can also be determined by using the onedimensional values. The transient heat transfer for a twodimensional geometry
formed by the intersection of two onedimensional geometries
1 and 2 is 1 sin The analytic solution for onedimensional transient heat
conduction in a semiinfinite solid subjected to convection is
given by
T(x, t) Ti
T
Ti where the quantity erfc ( ) is the complementary error function. For the special case of h → , the surface temperature Ts
becomes equal to the fluid temperature T , and the above equation reduces to hx
k
t h2 t
k2 k Q
Qmax total, 2D Q
Qmax Q
Qmax 1 12 Q
Qmax 1 Transient heat transfer for a threedimensional body formed by
the intersection of three onedimensional bodies 1, 2, and 3 is
given by
Q
Qmax total, 3D Q
Qmax 1
Q
Qmax 3 Q
Q
1Qmax 2
Qmax 1
Q
Q
11Qmax 1
Qmax 2 REFERENCES AND SUGGESTED READING
1. ASHRAE. Handbook of Fundamentals. SI version.
Atlanta, GA: American Society of Heating, Refrigerating,
and AirConditioning Engineers, Inc., 1993.
2. ASHRAE. Handbook of Fundamentals. SI version.
Atlanta, GA: American Society of Heating, Refrigerating,
and AirConditioning Engineers, Inc., 1994.
3. H. S. Carslaw and J. C. Jaeger. Conduction of Heat in
Solids. 2nd ed. London: Oxford University Press, 1959. 4. H. Gröber, S. Erk, and U. Grigull. Fundamentals of Heat
Transfer. New York: McGrawHill, 1961.
5. M. P. Heisler. “Temperature Charts for Induction and
Constant Temperature Heating.” ASME Transactions 69
(1947), pp. 227–36.
6. H. Hillman. Kitchen Science. Mount Vernon, NY:
Consumers Union, 1981.
7. F. P. Incropera and D. P. DeWitt. Introduction to Heat
Transfer. 4th ed. New York: John Wiley & Sons, 2002. cen58933_ch04.qxd 9/10/2002 9:13 AM Page 252 252
HEAT TRANSFER 8. L. S. Langston. “Heat Transfer from Multidimensional
Objects Using OneDimensional Solutions for Heat
Loss.” International Journal of Heat and Mass Transfer
25 (1982), pp. 149–50.
9. M. N. Özisik, Heat Transfer—A Basic Approach. New
,
York: McGrawHill, 1985. 10. P. J. Schneider. Conduction Heat Transfer. Reading, MA:
AddisonWesley, 1955.
11. L. van der Berg and C. P. Lentz. “Factors Affecting
Freezing Rate and Appearance of Eviscerated Poultry
Frozen in Air.” Food Technology 12 (1958). PROBLEMS*
Lumped System Analysis
4–1C What is lumped system analysis? When is it
applicable?
4–2C Consider heat transfer between two identical hot solid
bodies and the air surrounding them. The first solid is being
cooled by a fan while the second one is allowed to cool naturally. For which solid is the lumped system analysis more
likely to be applicable? Why?
4–3C Consider heat transfer between two identical hot solid
bodies and their environments. The first solid is dropped in a
large container filled with water, while the second one is allowed to cool naturally in the air. For which solid is the lumped
system analysis more likely to be applicable? Why?
4–4C Consider a hot baked potato on a plate. The temperature of the potato is observed to drop by 4°C during the first
minute. Will the temperature drop during the second minute be
less than, equal to, or more than 4°C? Why?
Cool
air Hot
baked
potato FIGURE P4–4C
4–5C Consider a potato being baked in an oven that is maintained at a constant temperature. The temperature of the potato
is observed to rise by 5°C during the first minute. Will the temperature rise during the second minute be less than, equal to, or
more than 5°C? Why?
4–6C What is the physical significance of the Biot number?
Is the Biot number mor...
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 Spring '10
 Ghaz
 Heat Transfer, TI

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