_ q1 



Since tan oC Q = - cot u S , of + <x £ » r^ and sin ot (t + £ ) = cos cs6 (t- 9). 



Substituting this relationship in equation (25) and dividing through by cos otE 

 cos oi (t - Q) gives 



s r* 



1 _ Bo^ULA tanOtff +_oi_LA tan o(.£ tan oC (t - 0) = 



waR2 ga 



The first two members do not vary with time and are equal. The third member 

 then expresses the error resulting from the assumption that the flow at each ' 

 instant equals the rate of steady flow under the same head. Since tan c< 

 (t - 9)— »-ooas oc (t - 9) — >- 'ii/2 s the error is certainly not negligible 

 over some portion of the cycle. Taking the ratio of the acceleration and 

 viscosity terras given by the first approximation. 



_^_f_LA tanoc £ tan oc (t - 9) e x R^ 

 gf = 8V 



=^ oc £ 



waR'^ 



tan oc (t-9) 



An approximate method of determining a limit for oc such that the 

 acceleration tarm will not appreciably affect the results is to hold this 

 ratio below a value n over a certain percentage of the cycle, p. Thus 



^^-Ip tan oi (t - 9) < nj oC (t - 9) < p (TT (26) 



8 7/ 2 



For example, to hold the ratio of inertia forces to viscous forces below 

 Ool over 75 per cent of the cycle, 



2o^U 



<X= 0.1 X 8 X V 



2.U4. r2 ~ ■ 



Ifith y = 1.0 x 10"5 ft.2/sec. and R = 0.0112 ft. (1/8" pipe), oc = 0.033 

 radians per second. As these figures for oC , R, and')' are approximately 

 those obtained in the experiments reported here, the conditions were such 

 that the inertia forces probably account for some of the discrepancy between 

 the theoretical and experimental results. 



The preceding analysis indicates that in long lines the effect of inertia 

 will have a very great effect on the results before the flow becomes turbulent 

 in the usual sense. Referring back to the Reynolds number (Equation 5a) 



2AR 



^ 



, 2V„R , 



2A 

 a 



R 

 V 



dH' 





V 



dt 





^ 2ARaHQ 



sin 



ex. 



eat 



a >> 



oC Ho sinc<.e|Gos oC (t - 9)]Vt, 

 a V 



30 



(27) 



