0.3 Boundary Value Problems 37
Findp(x)in terms ofa,b,andK(constant). Hint: The differential equa-
tion can be solved by integration.15.In a nuclear fuel rod, nuclear reaction constantly generates heat. If we treat
a rod as a one-dimensional object, the temperatureu(x)in the rod might
satisfy the boundary value problem
d^2 u
dx^2+g
κ=hC
κA(u−T), 0 <x<a,
u( 0 )=T, u(a)=T.Here,gis the heat generation rate or power density, and the terms on the
right-hand side represent heat transfer by convection to a surrounding
medium, usually pressurized water. Findu(x).16.Sketch the solution of Exercise 15 and determine the maximum temper-
ature encountered. Typical values for the parameters areg=300 W/cm^3 ,
T= 325 ◦C, κ= 0 .01 cal/cm s◦C, a= 2 .9m,C/A= 4 /cm, h=
0 .035 cal/cm^2 s◦C. It will be useful to know that 1 W= 0 .239 cal/s.
17.An assembly of nuclear fuel rods is housed in a pressure vessel shaped
roughly like a cylinder with flat or hemispherical ends. The temperature
in the thick steel wall of the vessel affects its strength and thus must be
studied for design and safety. Treating the vessel as a long cylinder (that
is, ignoring the effects of the ends), it is easy to derive this differential
equation in cylindrical coordinates for the temperatureu(r)in the wall:
1
r
d
dr(rdu)
dr= 0 , a<r<b,whereaandbare the inner and outer radii, respectively. The boundary
conditions both involve convection, with hot pressurized water at the in-
ner radius and with air at the outer radius:−κu′(a)=h 0(
Tw−u(a))
,
κu′(b)=h 1(
Ta−u(b))
.
Findu(r)in terms of the parameters, carefully checking the dimensions.18.If a beam of uniform cross section is simply supported at its ends and
carries a distributed loadw(x)along its length, then the displacementu(x)
of its centerline satisfies the boundary value problem
d^4 u
dx^4 =
w(x)
EI ,^0 <x<a,
u( 0 )= 0 , u′′( 0 )= 0 , u(a)= 0 , u′′(a)= 0.