The Effect of Additives on Fluid Friction 



This difference in feasible concentrations is simply because the unit of additive 

 in pipe flow is used over and over again, until the end of the pipe, while in a ve- 

 hicle boundary layer, the unit of additive is effective only on a certain wetted 

 surface area for a certain time, before it is discarded into the wake. 



This key difference can be seen more clearly by comparing fully estab- 

 lished pipe flow and high Reynolds number flat-plate boundary- layer flow. A 

 useful measure of performance is 



Additive Effectiveness (A.E.) = ^°^^'' ^^^^^ 



Additive Weight Flow Rate 



with units of lip - hr/kg. 



In the following it is assumed that speeds are kept fixed and that only pipe 

 length and plate length are varied. 



For pipe flow, assume that an additive weight concentration per unit volume, 

 C, produces a percent pressure-drop or friction reduction, R, for fully estab- 

 lished turbulent flow in a pipe of diameter, D, for a throughput of Q with the 

 mean flow velocity u= 4Q/ttB^ . The pumping power saved is RP^ where P^ is 

 the power required for C = . Thus 



RL(P/L) RL 



where P^ /L is the pumping power per unit length for C = . Thus if polymer 

 degradation is negligible, A.E. will increase indefinitely with L. 



For boundary- layer flow, one can assume for a first approximation that the 

 local percent skin friction reduction will require about the same mean concen- 

 tration across the turbulent boundary layer thickness, b , as in pipe flow for 

 S = D/2 and freestream speed U^, = U. The friction reduction factor, R, will be 

 assumed to be determined by c as in the typical pipe flow results given earlier. 



Because the additive concentration in the turbulent boundary layer will be 

 continually reduced by mixing as the boundary layer thickens, more additive will 

 have to be injected at intervals along the plate length, or else the concentration 

 will have to be very large near the leading edge. In either case, the total addi- 

 tive supply rate per unit width will be C(& - h*) u^, where i* is the boundary- 

 layer displacement thickness. If 6^ is the momentum thickness for C = 0, then 

 the thrust power saved per unit width is 



Re (p/2) L' ^ . 



Thus for a flat plate 



^r,A■.■ vff .■ Re^(p/2)\]J 



Additive Er rectiveness = r • 



C ( S - S ) U,^ 



957 



