Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
300<br />
XI1. Thrrrnnl borrndary layers in lan~inar flow<br />
The cooling action of a stream of fluitl on a wall is considcm1)ly rctluccrl because<br />
of t,hc hrat gencmtrd by friction. In thr nbsrncc of frirtionnl heat, heat will flow from<br />
the platc to thc fluid (q>0) as long as T, > II', but in actual fact,, if frictional<br />
hcat is prcsont, a flow of hcat persists only if T, > T,, eqn. (12.81). Taking into<br />
acronnt thc valuc tlcducctl for T,, we obtain the condition that heat flows from wall<br />
to fluitl (nplwr sign) or in tho reverse tlircction (lower sign), if<br />
A numcricnl cxamplc may serve to illust~mtc the signifirancc of cqn. (12.82): 111<br />
a stream of air flowing at TJm = 200 m/scc, P = 0.7, cp = 1.006 k.J/kg dcg wr<br />
obtain 1/ P 1JW2/2 c, = IF tlcg C. The wall will begin Lo be cooled whcn<br />
If tltc tenipcrat,urc difference bctwren wall and stream is snlallcr than this value<br />
the wall will pick up a port,ion of thc hcat generated by friction. In particular this<br />
is tho case whcn thc tcmpcraturc of the wall and stream arc equal.<br />
An equation for thc rate of hcat transferred from n flat platc at zcro incitIcnce<br />
but with variable material properties was derived by H. Schuh [110]. The tempe-<br />
rature field on a platc placed in a stream with a linear temperature distribution w54<br />
studicd in ref. [128].<br />
2. Additional sinlilar sol~~tions of the equations for thermal boundary laycrs.<br />
In the casc of a flat platc at zcro incidence, the velocity and the temperature profiles<br />
t~lrnctl out to bc similar among themselves. This means that the distributions at<br />
tliffcrcnt clistanccs z along thc platc coulcl hc mn.tlc congruent by a sniLahlc stretching<br />
in the y-direction. Since it is lcnown that there cxist velocity boundary layers other<br />
i.han those on a flat platc for which this is true (e. g. the wedge profiles discussed<br />
in Chap. IX), it, appcnrs useful to stucly the possibility of tho cxistcncc of additional<br />
similar solutions of the energy equation. This problem was investigated in detail<br />
in ref. 11351. At the prcscnt time, we sha,ll start with the class of velocity boundary<br />
leycrs on wedges and will awnme that t,hc cxtcrnal flow is of the form U(z) = tc1 x"'.<br />
111 an analogous manner, we stipulate that tho wall-tcmpcraturc distribution also<br />
x". Walls<br />
sa1,isfies n power law, say one of thc form T,(x) - Y', = = TI<br />
of constant t,rmpcrat,ure are inclutlctl as thc casc n == 0, and t11c valuc 12. = (1 --711,)/2<br />
corrr~ponds to :I, crn~stc~nl, hrat flux q. 1nt.rotlucing tlic sitnilarit,y variable<br />
wr ol~tnin tlic f'ami1i:ir rquations (0.8~1) for the bdocity u = iJ(.z) . /'(q), or<br />
g. Thcrmal boundary layers in forced flow<br />
ant1 thc solution must satisfy thc boundary contlitions<br />
17~0: a:=]; 17=03: O-:()<br />
lIcrc E =. 71,2/c,, '/I1 rrprcwnts the nppropriaLc form of 1I1r I':ckorl. t~ttrnl)cr li)r illc<br />
prol~lcm.<br />
It is clear from cqn. (12.84) that its right-ltantl sitlc vanisltcs in tltc al)scncc<br />
of frictional heat and that all solutions arc thcn of tlrc similar typc. IIowcvcr, if<br />
frictional hcat is includcd, similar solutions arc rcstrictctl to tltat combinatio~~ of<br />
pnramctms for which thc right-hand sido becomes intlcpcntlcnt of z. 'rl~is occ:urs<br />
whcn 2 ~n - 7t = 0 , tltat is, whcn thcrc cxists a firm cor~pling 1)ctwcon tltc vclocity<br />
distribution in thc cxtcrnal flow ant1 tho tcmperatarc tlistril~ution along t.110 w:ill.<br />
According to this result., thc casc of a co~~st~ant tcmpcmt~ure lratls to similar solut,ions<br />
only on a flat plate (1i1=1t =0). 011 thc olhcr hnntl, if tho contlil,ion 2 111, -- 11. - 0<br />
is satisfied, thcn for every pair of values of m and P thcrc cxists one tlcfinitc valur:<br />
E, for which there is no flow of ltcat (O'(0) = 0). Jn this rasc, the tcmpc:mt,~~rc<br />
distribution along the wa.11, once again lcnown as tfhc atliabalic wall-tcmparat,urc:<br />
distribution T,, is given by<br />
Numcricnl valucs for thc function b(m,P) havc bcnn romput,rtl by 1% A. llrun 171.<br />
In the particular case whcn m = 0, the numerical valucs of l'ablc 12.2 arc rccovcrcd.<br />
Wlicn thc cffcct of dissipative hcat is ncglcctcd, wc obtain the simpler cquation<br />
whose solutions for different valucs of thc parameters m, n, and P have bccn<br />
published by a number of authors [79, 121, 32, 33, 89, 1401. E.1E.G. Eclrcrt [I91<br />
has dcmonst,ratcd that for n = 0, the local Nusselt number is given by thc equation<br />
Eerc<br />
ax U (x) . 1:<br />
N =-=-<br />
% k v---y O' (0) = - id, 0' (0) . ( 12.88)<br />
The function F(m, P) is seen plotted in Fig. 12.14 on thc basis of the numcriral<br />
data provided by 11. 1,. Evans [33]. Jn addition, thc asymptotes for very small<br />
and vary large l'randtl numbers, cqns. (12.57) arid (12.01 n), rrspcctivrly, have also
Change language