1. This slide rule was made by George Kent, a company founded in 1883 which made water meters. Its meters are still sold under the name of ABB-Kent.
2. The rule is designed to calculate flow in pipes.
3. The rule does not carry an explanation of which formula is used for the calculation.
4. It's use very simple. By placing the gauge mark for the appropriate material against the hydraulic gradient, the flow or velocity can be read against the pipe diameter on the appropriate scale.
5. The rule is made of cardboard and does not need a cursor.
6. I have two identical rules, obtained at the same time. This one has the word "Dud" written on it and some numbers hand written against one of the scales. The other rule does not. I give below my explanation as to why someone wrote those words on the rule.

Summary of instructions

Around 1768 Chézy, a Frenchman, developed a theory for hydraulic flow. He argued that in a river the water was exchanging potential energy, proportional to height, to kinetic energy, proportional to velocity squared. Since, at a given moment in time, the river flowed at a constant velocity he reasoned that the potential energy was lost by turbulence and friction. He further reasoned that this energy was directly proportional to the area and inversely to the wetted surface. The ratio of area to wetted surface is call the hydraulic radius. This reasoning gave rise to the Chézy equation:

V = C Ö(RS)

where:
   V = velocity, ft/s
   R = hydraulic radius, ft
   S = gradient
   C = Chézy coefficient.

Over time it came to be realised that this formula was not exact; that the value of C was not constant. One of the more successful attempts to improve on this was due to Kutter (1869) whose formula was:

where:
   n = Coefficient of roughness.

Other scientists took an entirely different approach as developed empirical equations. One of these is the Hazen-Williams formula:

V = 0.115 C d^0.63 S^0.54

where:
   V = velocity in ft/s
   d = diameter in inches
   C = roughness coefficient (not the same as the Chézy C above.)

Let's look at a sample calculation, say a pipe 6" diameter with a pressure gradient of 1 : 1000.

Hazen-Williams formula. A typical value of  C for new cast iron is 125. This gives:
   V = 0.115 C 6^0.63 (1/1000)^0.54 = 1.066
The sectional area, A, is:
   A = p d² /4  = p (0.5)² /4 = 0.196
So the flow, Q, is:
   Q = V A = 1.066 * 0.196 = .208 cusecs = 4.67 thousands of gallons per hour   (the units on Kent's rule)

Using the Kutter formula with n=0.010 gives a flow of  4.92 thousand gph; with n = 0.011 the flow is 4.32. As we can see a small change in roughness coefficient, well within the error estimate for an experienced engineer, leads to a significant change in calculated flow.

I also have another hydraulic slide rule by Mear's which uses the Colebrook-White formula. Using this rule and the same variables gives a flow of 5.5 litres/s, which is equivalent to 4.35. In this case the user does not have to select a roughness. There is a series of gauge marks for different pipe materials.

In this case, as with the Mear's rule, there is no need to estimate a coefficient of roughness. In case it only necessary to set the mark for New C.I. (cast iron) against the gradient. This shows the velocity of flow (lower scale, second image) as 1.05 ft/s and the flow as .55 thousand gph. The velocity corresponds well with the value from the Hazen-Williams above, 1.066 ft/s. The Mear's rule gives a velocity of .3 m/s which is equivalent to 0.98 ft/s, again a very similar figure. It therefore appears that whilst the rule gives the correct velocity.

The flow however is given as .55 thousand GPH., a factor of at least too low.

To check the calculation of flow from velocity the rule was set to 24" diameter (i.e. 1ft radius) and a velocity of 1 ft/s. In this case the flow in ft³/s is 3.14 p .  To convert it to thousands of GPH:
Flow thousand GPH = Flow ft³/s * 6.24 * 3600/1000 = 70.5.
Using the rule gives a value of 3.1.

Clearly this is an error - no wonder the owner marked "Dud"   on the rule. But, using the hand written marks on the rule gives a value of 11.7. This is not right either! Given the range of errors it would seem the Kent rule is a factor of ten out.

Not content with identifying the errors with this rule, the owner set about designing his own rules. Two of which are shown below.

The first has a single slide. It is designed to cover the same type of calculations as the Kent rule above. In its favour is that fact that the formula used, Kutter's,  is given explicitly and he also gives the roughness used for the calculation. On the other hand whereas the Kent rule can be used for different pipe materials, this rule cannot. You can also see that the designer copyrighted the rule in his own name, Ronald W.North.

The rule is made of cardboard on ply-wood. The rule is well made with the cardboard being recessed into the ply-wood.

I also have another rule, for house building in relation to plot size, apparently owned by the same person who also designed rules for this purpose. This rule can seen here. As far as I know neither of the rules designed by Ronald W. North was ever produced so it is quite a fitting memorial to him that his rules now should be available for inspection all over the world via the internet.

Front view

Detail - front left

Detail - formula

This rule is made of cardboard on hardboard. Logarithmic graph paper has been used for the sales. Having two slides this rules is more complicated than either of the others but does not seem to have a lot more functionality although it can handle different values of C, the Chézy roughness coefficient.

Front view

Detail - front left