Catalogue of Mear's slide rules

Make Mear's, United Kingdom
Model Water flow calculator

1. This rule is designed for flow in pipes on one side and for flow in open channels on the other side.
2. This rule is quite large - 19.5 cm (7.7 inches) diameter.
3. Although the rule was designed for hydraulics engineers, and must have had help from them, some of the terms used are slightly unusual. For example the use of the phrase "rational formula" for pipe flow (it is more normally applied to urban runoff problems).
4. Both faces of the rule have a solid outer disc, a thinner rotating disc and a cursor with, in effect, multiple gauge points.
5. As the rule is large and as I give detailed instructions below, I have not shown detailed images in the first part of this page.
6. The both discs of the rule are actually a uniform white colour but, maybe because of different material for the outer disc, in the scanned image the outer disc appears much darker.

Front view
9918-mears-hyd-01.jpg (64585 bytes)
Back view
9918-mears-hyd-02.jpg (387890 bytes)
Manufacturing date c 1970
Length 7.7" diameter.
Material Plastic
Scales Special
Cursor Plastic - multi line

Summary of instructions


Firstly it should be noted that this calculator is a double-sided instrument, the two sides being entirely independent.

One side determines the flow of water through pipes and ducts of circular cross section and constructed in a variety of materials, i.e., metal pipes, concrete ducts. coated pipes etc. The other side deals with the flow in large open channels of any cross section, rectangular, round, etc.

Design Basis - Pipes, Ducts & Sewers

This side of the calculator solves the rational formula for the flow of fluids in pipes, incorporating coefficients of friction in accordance with the Colebrook-White equation and pipe roughness factors.

The rational formula as applied to water flow is

P = 3.25 x 107 Q2 f L / D5

    P = pressure loss bar
    D = bore of pipe, mm.
    Q = flow, litre/s
    f = coefficient of friction
    L  = length of pipe, m.

The Colebrook-White equation is :-

    1 / f = - 4 log (Ks/3.7d + 1.255/Rf)

    Ks = absolute roughness of pipe in mm. (This approximates to the dimensions of the particles or granularity of the material causing roughness.)
    R = Reynolds Number. (A dimensionless number related to viscosity and the boundary layer).

The variable coefficient of friction is built into the calculator and needs no separate determination and a Pipe Material scale makes appropriate allowances for roughness.

The Rational formula combined with the Colebrook-White equation for the coefficient of friction on which this model is based takes account of pipe roughness, viscosity, and Reynolds Number and has been practically universally adopted as the most logical and accurate for problems of fluid flow in pipelines. The answers have been checked against over 100 well authenticated practical tests on actual pipelines from small metal tubes to long concrete tunnels up to 6 m dia., and give an accuracy appreciably better than the usual Hazen-Williams formula. The pressure loss obtained is for pipes in new condition and allowance should be made for deterioration with age due to corrosion or incrustation according to experience with the type of water being handled. This will result in an increased pressure loss or a reduction in the flow.

Scales are provided, coloured green, for determination of velocity.

Flow in Ducts Running Partly Full

The flow through any pipe, duct or open channel running only partly full can be read off directly against the Proportional Depth scale on the top quadrant. Thus if the depth of water in the pipe is only one quarter of the diameter, the flow is read off immediately opposite .25 on this scale.

It will be seen that paradoxically there is a slight increase in the flow with a pipe only say 95% full, this being due to the appreciable reduction in the length of the wetted perimeter causing a corresponding reduction in the frictional losses as the water leaves the top of the pipe.

With partly full pipes the flow is determined by the slope. Thus if the slope is 1 in 600 it will be appreciated that there is a loss of pressure head of 1m in every 600m of length. To determine the flow therefore the pressure loss is set at 1m and the length at 600m.


9918-mears-hyd-03.jpg (110126 bytes)Find the flow of water through a smooth concrete pipe 500mm bore, l00 m long with a pressure loss of 100 millibars (1.02 metres head).

1. Set 500mm bore to l00 m long.
2. Set "Smooth Concrete" to 100 millibar Pressure Loss.
3. Read opposite outer arrow:

Flow = 507 litre/s



9918-mears-hyd-03.jpg (110126 bytes)or alternatively if the pipe is only running 30% full read off against .30 on the Proportional Depth scale:

Flow = 102 litre/s

It is interesting to note that the maximum flow does not occur when the pipe is running full but when it is about 0.95 full. THis is because the sectional area is cloe to maximum but the drag caused by the surface of the pipe is proportionally less.


9918-mears-hyd-05.jpg (52582 bytes)From Velocity Scales: Set green arrow to 507 litre/s and read opposite 500 mm bore of pipe

Velocity = 2.55 in/s

Design Basis - Open Channels

To deal with open channels of non-circular section the other side of the calculator is arranged to solve the Manning Formula which is:

Q = A1.66 S0.5 / n P.66


    Q = flow in m3/s
    A = area of channel in m2
    n = Kutters 'n'
    P = wetted perimeter of channel in metres.
    S = slope of channel

It is necessary to obtain the area of the channel occupied by the water and also the corresponding length of the wetted perimeter and to use these values in the calculation.

The Manning Formula will also deal with the flow in circular channels running partly full but it will give low values for the smaller pipes and steeper slopes and it is preferable to use the other side of the calculator


Find the flow in a 3 in square rough concrete channel with a slope of 1 in 1000 running half full.

Area = 3 x 1.5 = 4.5 m2 .(This is fairly straightforward as if the channel is half full the weater depth will be 1.5m).
Wetted Perimeter = 1.5 + 3 + 1.5 = 6 m. (The wetted perimeter is the length around the channel with water touching it. In this case water is touching the whole of the bottom (3m) and both sides for the depth of water (1.5 at each side).)
    9918-mears-hyd-06.jpg (101962 bytes)

1. Set 4.5m2 Area to 6m Wetted Perimeter.
2. Set "Rough Concrete" arrow to 1000.
3. Read: Flow = 7.36 m3/s








From Velocity Scales: Set green arrow to 7.36 m3/s and read opposite 4.5 m2 area. Velocity = 1.64 m/s

9918-mears-hyd-07.jpg (62028 bytes)

Allowance for Bends and Elbows

If the pipe has a number of bends or elbows the following additions to the length should be made for each:
Pipe Diameter (mm) Length to Add (metres)
Welded Elbow Long Radius Bend Globe Valve Gate Valve
13 .3 .15 5.5 .15
25 .6 .25 9 .25
50 .9 .5 16 .4
100 2 .9 32 .75
200 3 1.5 64 1.5
300 4.5 2   2
500 7.5 3.5   3.6
1000 15 6.5   7

Screwed elbow and bends are double the above figures.

Hot Water

The volume flow of water at 65C in steel pipes is approximately 3% greater than at 10C for the same pressure loss.