88 BULLETIN 376, U. S. DEPARTMENT OF AGETCULTUEE. 



Gardner S. Williams, formerly of Cornell and Michigan Universities, as representing 

 the best average value. They inclined rather toward varying the exponent of V within 

 the range of about 1.75 to 1.95 and they also varied the coefficient m for different kinds 

 and conditions of pipes from about 0.30 to 0.50. 



The exponent 1.25 was determined by Schoder from a plotting of all known observa- 

 tions on pipes covering the entire field from very small drawn-brass pipes to very 

 large tuberculated riveted-steel pipes, and is therefore not based on estimate nor 

 opinion, but on actual facts. ' The winter's observations on wood-stave pipes developed 

 a value of 1.26, which he has since reduced in the formula for practice to 1.25 to con- 

 form to the generally accepted figure. 



In the matter of proposing a new formula, the writer lives in a glass house and can 

 therefore not afford to throw stones. However, there were extenuating circumstances 

 in this case, which need not be discussed here, that ultimately led to a rather wide 

 acceptance of the writer's formula, although he originally offered it with reservations 

 because there were a number of points not satisfactorily explained, and experiments 

 on very large pipes were notably lacking. Thanks to the author's careful experiments 

 the deficiency has now been supplied to a considerable extent and we are closer to the 

 ultimate solution. 



It is interesting to note that the author has arrived at the same figure for exponent of 

 V, namely 1.8, that developed from the writer's experiments. Unfortunately, he 

 had not time to check the author's analysis on this point, nor to make an independent 

 analysis of the data in this paper, but the evidence in support of this average figure 

 may now be fairly considered as having been greatly extended. The evidence in 

 support of 1.25 as the exponent of d is in the writer's opinion equally as good if not 

 better, since its derivation was based on observations on all kinds and sizes of pipes. ^ 

 If we could, therefore, now agree on these two figures and throw the variation iii 

 formula for different classes of pipe into the factor m, a much longer step will have been 

 made toward the general acceptance by engineers and courts of an exponential formula 

 for flow of water in pipes. 



It must be conceded that additional experiments that may be made in the future, 

 especially in the field of diameters between 60 and 160, may have a marked effect on 

 the exponent of d and we will then no doubt be confronted by another formula. More- 

 over, as has already been pointed out, the personal equation of the analyst, especially 

 in the application of weights and methods of reasoning, must be taken into account 

 and another person with the same data as a basis would no doubt arrive at somewhat 

 different conclusions than those given in the present paper. No doubt this factor has 

 been eliminated to the greatest possible degree in the present paper, but complete 



1 Author' s footnote (the italics are his).— A study of original sources does not bear out this statement. In 

 Trans. Amer. Soc. Oiv. Engin., vol. 51, Messrs. Saph and Schoder, on page 305, state: "First of all, the line 

 best fitting the points for the writers' brass pipes has been drawn," which line, as they say on page 306, 

 " forms the lower limit of the zone in which all the plotted points lie. This is another way of stating that 

 these pipes represent the extreme of smoothness and ideal conditions." The exponent of d for this line 

 was found to be 1.25. (This line is A, fig. 7.) The values of m for all kinds of pipes were then platted and 

 Saph and Schoder state (id., p. 308): "It will be seen that a line parallel to the one already drawn wUl 

 represent fairly the other limit" of the zone. (This line is shown as B, fig. 7.) Thus it will be seen that this 

 exponent was derived from a study of smooth brass pipes and then applied to all kinds of pipes by drawing a 

 line parallel, i. e., with the same slope, hence indicating the same exponent of d, or 1.25. In discussing the 

 exponent as derived by Saph and Schoder, Mr. Allen Hazen states (id., p. 321), "It is, unfortunately, a 

 fact that probably none of the large pipe has as smooth surfaces as the brass pipe used by the authors (Saph 

 and Schoder) and some allowances must be made for this fact in the comparisons, and, while the line drawn 

 as a general average would be a matter of individual judgment, the writer (Hazen) would hardly give it 

 an inclination greater than that corresponding to an exponent of 1.20. If the matter were carried further the 

 exponent would probably be lower." It is interesting to again note that the exponent of d is 1.17 in the 

 formula derived from a study of these same pipes of all kinds by Williams and Hazen, and that this is the 

 same exponent found in this study of the flow in wood-stave pipes. (Line E, fig. 7. This question is 

 further discussed on page 94.) 



