Page 424 - Bird R.B. Transport phenomena

406 Chapter 12 Temperature Distributions with More Than One Independent Variable

12D.7. Asymptotic solution for small z for laminar tube flow with constant heat flux. Fill in the
missing steps between Eq. 12.2-23 and Eq. 12.2-24. Insertion of the expression for ф into Eq.
12.2-23 gives

в = J x LI (3) x
J

Why do we introduce the symbols J and ~x> Next, exchange the order of integration to get
@ = ^9X Г U L f * Jexpf-pW*) \dx (12D.7-2)
X [Л\з) x J
J J
Then perform the integration over x and obtain

в ( г A ) = e x 3 ) 3 3
>' Щ-1 P(~^ ~ Ц J " ^ exp(-^ ) ^ - ^ x exp(-^ ) rf^J I (12D.7-3)
o
1
а
Then use the definitions T(a) = f% 1 ~^е~ &\ and T(a, x) = /J F~ e~ dt for the complete and in-
[
A
complete gamma functions.
12D.8. Forced conduction heat transfer from a flat plate (thermal boundary layer extends beyond
the momentum boundary layer). Show that the result analogous to Eq. 12.4-14 for A > 1 is 13

3
H. Schlichting, Boundary-Layer Theory, 7th edition, McGraw-Hill, New York (1979), p. 306.