Unformatted text preview: onal geometry.
Consider a short cylinder of height a and radius ro initially at a uniform tem0, the
perature Ti. There is no heat generation in the cylinder. At time t
cylinder is subjected to convection from all surfaces to a medium at temperature T with a heat transfer coefficient h. The temperature within the cylinder will change with x as well as r and time t since heat transfer will occur
from the top and bottom of the cylinder as well as its side surfaces. That is,
T
T(r, x, t) and thus this is a twodimensional transient heat conduction
problem. When the properties are assumed to be constant, it can be shown that
the solution of this twodimensional problem can be expressed as
T(r, x, t) T
Ti T short
cylinder T(x, t) T
Ti T Plane wall FIGURE 4–27
A long solid bar of rectangular
profile a b is the intersection
of two plane walls of
thicknesses a and b. t) cyl(r, a T(r, t) T
Ti T infinite
cylinder (425) That is, the solution for the twodimensional short cylinder of height a and
radius ro is equal to the product of the nondimensionalized solutions for the
onedimensional plane wall of thickness a and the long cylinder of radius ro,
which are the two geometries whose intersection is the short cylinder, as
shown in Fig. 4–26. We generalize this as follows: the solution for a multidimensional geometry is the product of the solutions of the onedimensional
geometries whose intersection is the multidimensional body.
For convenience, the onedimensional solutions are denoted by
wall(x, b plane
wall t) semiinf(x, t) T(x, t) T
Ti T
T(r, t) T
Ti T
T(x, t) T
Ti T plane
wall
infinite
cylinder (426) semiinfinite
solid For example, the solution for a long solid bar whose cross section is an a b
rectangle is the intersection of the two infinite plane walls of thicknesses
a and b, as shown in Fig. 4–27, and thus the transient temperature distribution
for this rectangular bar can be expressed as
T(x, y, t) T
Ti T rectangular
bar wall(x, t) wall(y, t) (427) The proper forms of the product solutions for some other geometries are given
in Table 4–4. It is important to note that the xcoordinate is measured from the
surface in a semiinfinite solid, and from the midplane in a plane wall. The radial distance r is always measured from the centerline.
Note that the solution of a twodimensional problem involves the product of
two onedimensional solutions, whereas the solution of a threedimensional
problem involves the product of three onedimensional solutions.
A modified form of the product solution can also be used to determine
the total transient heat transfer to or from a multidimensional geometry by
using the onedimensional values, as shown by L. S. Langston in 1982. The cen58933_ch04.qxd 9/10/2002 9:13 AM Page 233 233
CHAPTER 4 TABLE 4–4
Multidimensional solutions expressed as products of onedimensional solutions for bodies that are initially at a
uniform temperature Ti and exposed to convection from all surfaces to a medium at T x 0 ro r r x
θ (r, t) = θcyl(r, t)
Infinite cylinder r
θ (x,r, t) = θcyl (r, t) θwall (x,t)
Short cylinder θ (x,r, t) = θcyl (r, t) θsemiinf (x, t)
Semiinfinite cylinder y
x
x y
z θ (x, t) = θsemiinf (x, t)
Semiinfinite medium θ (x,y,t) = θsemiinf (x, t) θsemiinf (y, t)
Quarterinfinite medium x
θ (x, y, z, t) =
θsemiinf (x, t) θsemiinf (y, t) θsemiinf (z,t)
Corner region of a large medium 2L
2L
y
0 x Lx y
z θ (x, t) = θwall(x, t)
Infinite plate (or plane wall) θ (x, y, t) = θwall (x, t) θsemiinf (y, t)
Semiinfinite plate x
θ (x,y,z, t) =
θwall (x, t) θsemiinf ( y, t) θsemiinf (z , t)
Quarterinfinite plate y
x
z y
x z y x
θ (x, y, t) = θwall (x, t) θwall ( y, t)
Infinite rectangular bar θ (x,y,z, t) =
θwall (x, t) θwall (y, t) θsemiinf (z , t)
Semiinfinite rectangular bar θ (x,y,z, t) =
θwall (x, t) θwall (y, t) θwall (z , t)
Rectangular parallelepiped cen58933_ch04.qxd 9/10/2002 9:13 AM Page 234 234
HEAT TRANSFER transient heat transfer for a twodimensional geometry formed by the intersection of two onedimensional geometries 1 and 2 is
Q
Qmax Q
Qmax total, 2D Q
Qmax 1 12 Q
Qmax (428)
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 Q
Qmax 1 Q
Qmax 3 12 Q
1Qmax Q
Qmax
1 1 1 Q
Qmax (429)
2 The use of the product solution in transient two and threedimensional heat
conduction problems is illustrated in the following examples.
EXAMPLE 4–7 Cooling of a Short Brass Cylinder A short brass cylinder of diameter D 10 cm and height H 12 cm is initially
at a uniform temperature Ti
120°C. The cylinder is now placed in atmospheric air at 25°C, where heat transfer takes place by convection, with a heat
transfer coefficient of h
60 W/m2 · °C. Calculate the temperature at (a) the
center of the cylinder and (b) the center of the top surface of the cylinder
15 min after the start of the cooling. SOLUTION A short cylinder is allowed to cool in atmospheric air. The temperatures at the centers of the cylinder and the top surface are to be determined.
Assumptions 1...
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 Spring '10
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