Darcy (1803-1858).

H.Darcy was a French engineer and his experiments in 1845 to measure the friction loss in cast iron pipes are among the earliest recorded. He worked with water flowing in cast iron pipes having diameters ranging from about 10 mm to about 500 mm with a test length of 100 m. It is not his results that matter now, although they are still used, but the way in which he chose to present them.

 

At the time people thought about the problem of resistance to motion of water flowing in a pipe or in a channel as being one problem for which a common approach might be successful. This supposition would have been appropriate as piped water supplies did not initially provide water under pressure with taps to control the flow. Instead wooden pipes were laid with a fall and on "water days" water would be allowed to flow into a part of the system and to fill the cisterns of the customers. No provision was made to prevent the cisterns overflowing and when the supply was turned off the pipes drained. The system of pipes produced the same result as physically carrying water in containers to fill the cisterns and the water was simply stored until required. Many of the pipes would have run only partially full, like a channel, for some of the time and full at other times.

 

 The resistance to flow, , whether in a channel or a pipe, was considered to act at the surface in a manner comparable to friction between solids and this resistance was expressed as  where  was the area of the wetted surface,  was the velocity derived from the flow divided by the area of cross-section of flow,  was an index and  was a coefficient. For a given pipe the wetted area would depend on whether the pipe was full. In either case the wetted area is equal to the wetted perimeter, , times the length, . It transpired that, in many applications the ratio of the area of cross-section, , to the wetted perimeter appeared in empirical expressions. The ratio  was given a name, the mean hydraulic depth and denoted  and seen to be the link between pipe and channel flow. This name arose because, if the water were to be thought of as standing on the wetted surface when laid out on a horizontal plane, the depth would be equal to .

 

By Darcy's time wooden pipes were giving way to cast iron pipes which could operate under pressure without excessive leakage and the resistance to flow in these pipes, which always ran full, was becoming an object of interest to those installing pipe systems.

 

In pipes that run full, the wetted area is  where  is the perimeter of the pipe and  the length of the pipe under consideration and, as the force causing flow is  where  is the cross-sectional area of the pipe and  is the pressure drop in length , we can write:

                                         , from which:

                                          .

 

Darcy appears to have decided that the best way to present his results was to convert  to head and to put  equal to 2. Noting that the head lost to friction in the pipe, , equals :

                                                 = .

Now,  for a circular pipe is , and if a new coefficient  is introduced, we get:

                                                       .

Darcy observed that, other things being equal,  appeared to decrease as the pipe diameter increased and allowed for this by putting :-

                                                  .

He gave values of  of about 0.005 for new pipes and about 0.01 for old and encrusted pipes and these figures are still in widespread use for first approximations to friction loss.

 

He could not have foreseen that the best decision was to omit the  term and let  vary and it is the expression :-

                                                     that is now called the Darcy expression. It was the decision to put the expression in terms of the kinetic head of the flowing water as used in the energy equation that ensured its survival.