Boundary Lyer Theory

306 XII. Tllcrrnal hor~ndary layers in laminar flow
I = U ( ) 2 -- 2 11"1- ?i4] == IJ (x) F(7) , (12.92~~)
,.
I htr vt:loc.it,y tlist,riln~t,ion st.il)~~lnl.ctl Ilcrc c:orrcspontls to t,lw I'ohlhauscn assumpti
or^ in oqn. (10.23) :~.t~cl t,I~t: liwm of t.l~e t.cmpcr:~t.~~rc clistribt~tion func:tion is so
srlrc:l.ctl as tm cnsurc: itltwl,ic::tl vclocity :m(l t(:n~~~~ra.l,urr (listril~~~tion~ for BT ---- 0,
as rcrquircrtl 11.y t,hc Jtc~ynolcls analogy for n flnt pI:~t.c at P = I, cqn. (12.64). 011
sul~stitat~ing cqns. (12.!)2:1, 1)) into cqn. (12.91), wo obhin
Performing the intlicatcd integrations, we obtain
2
II(A)= -A-
15
? /In"+
140 ' A4
180
for A < 1
and
3 3 1
~I(A)= lo---+
10 A
2 1
15 A2
3 1 1 1 orA>,
. +- - -- f
140 A' 180 A6
Some numerical valucs of the function [[(A), calculated by W. Dienemann[ll],
havc been listed in Table 12.4.
Tablo 12.4. Nurnoricnl vnlwa of tilo function H(A)
The integration of cqn. (12.93) yields
'I'lle vclority l~oundary-layer thickness (J can bc evaluated with thc aid of cqn. (10.37)
wl~cw it is rcrncml~crt:cl from cqn. (10.24)t thnt ?~/d, = 316/37. Thus
I
I .
t Iptw IIir ankv of~in~plidy tlic rdalci~lalion is ba.wd throughout on the flat-plate relations ( A = 0).
I
Upon dividing eqn. (12.95) by eqn. (12.96), we obtain
U~~UII.(I~
4 1 0
A2. H(A) = i-4 - (12.97)
z
H JUQI x
O
Since H (A) is a known function, Table 12.4, the preceding equation can bc used
to doterrnine A (x). The calculation is best perrornicd by uurccssivc npproximntio~w,
starting with the initial assumption that A -= rot~sl~. Jltwrc wc oMain
The resulting value of A is now int,rvtlucctl into thc 1t:ft.-11:~ntl sitlc of ac111. (12.!)7)
thus leading to an improved value of A. In general, two steps in the itcration nrc
found to be sufficient.
The local rate of heat transfer becomes
and hence the local Nusselt number referred to a characteristic length I is
The steps to be taken to evaluate the thcrmnl boundary layer, and in partficular,
to determine the variation of the Nusselt number along a body of preseribcd shape
are thus the following ones:
1. evaluate A (2) from eqns. (12.97) and (12.978)
2. evaluate d (x) from eqn. (12.96)
3. steps 1 and 2 give dT(x); finally, the local Nusselt number follows from
eqn. (12.98).
Flat plate at zero incidence: The preccding approximato method will now be
compared with the exact solution in tthc case of a flnt plnto at zero incidcncc. Insert-
ing U (z) = U, into eqn. (12.97), wc obtain
The expression A = P-ID constitutes an approximation to the solntion of this equation
which is in error by no more than 5 per ccnt. as compared with the exact solution.
The boundary-layer thickness from cqn. (12.06) is
I