35 



The power available for heating is related to 

 the thermal efficiency of the power plant as 

 follows : 



th 



^D j '^ef f = ^D n 



- 1 



th 



eff 



(10) 



where rieff is the effectiveness of transmitting 

 the reject heat to the water in the desired manner. 



If one considers heating only the laminar por- 

 tion of the hull then the power required to accom- 

 plish such heating is 



P = p u 

 H " « 



:AT J" 



tr 



c, „ wdx 



hil 



(11) 



where c is the specific heat of water, cj^jj is the 

 laminar Stanton number for the heated boundary 

 layer at AT = T,,-T . 



Applying the available heating power P^ to the 

 laminar portion of the flow, (P^ = P^) , after some 

 simplification yields 



1 + 



L 



■' c ^ wdx 

 X ft 

 tr 



X 



tr 



fi 



wdx 



=, (12) 



The left side is the ratio of overall friction drag 

 to the laminar friction drag and is configuration 

 dependent. The right side depends on the dimension- 

 less ratio cAT/Uco and on the bracketed parameter 

 in the denominator related to the amount of reject 

 heat that can be transferred to the boundary layer. 

 The bracketed parameter in the numerator is a 

 Reynolds analogy factor which is configuration de- 

 pendent. In order to close the calculation, a 

 relation is needed between AT and transition 

 Reynolds number Re^^ which is also dependent on 

 configuration. 



Example - The Flat Plate 



In order to quantitatively evaluate the prospective 

 drag reduction due to heating, it is necessary to 



choose a particular configuration. The flat plate 

 is chosen because of its great simplicity and be- 

 cause some information on transition with surface 

 heating is available. The results should be repre- 

 sentative of what can be obtained for slender shapes 

 having pressure gradients that are not too large. 

 For a flat plate (w = const) 



/ ^'^c dx = 1.328 ^tr 



'fS. 



X c dx = 0.074 

 tr ft 



(13) 



'fee 



X. 



tr 



tr 



1/5 1/5 

 Rey Re ' 

 L x^ 

 tr 



'fl -2/3 



~ — Pr 



(14) 



Thus for the case of the flat plate, Eq. (12) be- 

 comes 



,„ 4/5 „ 4/5 ^ 

 /Re Re ' 

 L - x^ 

 tr 



1 + 



0.074 



1.328 \ 1/2 

 Re 



CAT 



Pr 



-2/3 (15) 



u D 



F ^ th 



in 



eff 



The left side of equation (15) is the ratio of 



overall friction drag to laminar friction drag for 



a flat plate. 



The variation of transition Reynolds number 



Rev with overheat AT depends on the choice of 



tr 

 transition criterion. A criterion that has been 



shown to give plausible trends is the e^ criterion 

 of Smith and Gamberoni (1956) and Van Ingen (1956) . 

 For low speed flows, these authors correlated tran- 

 sition Reynolds number over plates, wings, and 

 bodies with the amplitude ratio using linear sta- 

 bility theory of the most unstable frequency from 

 its neutral point to the transition point. They 

 found that the transition Reynolds number Rex^„ as 

 predicted by assuming an amplification factor of 

 e^ was seldom in error by more than 20%. Wazzan 

 et al. (1970) have calculated and presented such a 

 curve for heated flat plates in water a portion of 

 which is shown in Figure 2 . Although not quite 

 shown on the figure, Rextj. reaches a maximum value 

 of about 260 x 10^ at an overheat of about 43°C. 

 The most recent data of Barker (1978) taken in a 

 constant-diameter pipe are shown on this figure as 

 well. Barker obtains a considerable increase of 

 transition Reynolds number with heating in the 

 entrance flow boundary layers and his data attests 



loV 



tr 



DC 



3 



Q 



_] 



o 



2 



o 



z 

 < 



and by Reynolds analogy 



WALL OVERHEAT, AT, C 



FIGURE 2. Variation of transition Reynolds number for 

 a flat plate with uniform wall overheat according to 

 an "e transition criterion, T^ = 60°F. 



