The lead that you gave has led me to a pleasant time examining Bridge End Mill and what is left of the water management arrangements upstream and downstream of Settle Bridge. The weir just above the bridge I believe was constructed for the sole benefit of Bridge End Mill (there are the remains of another weir below the bridge for the next mill down river), which means that a great deal of water power - up to the full flow of the Ribble - was potentially available.
I did not know where the mill had been, but in the process of delivering a load of cardboard to the Settle Swimming Pool recycling container I bumped into Chris who was helping to stack the contents, and was told that it was the next building but one upstream of Settle Bridge.
The mill wheel which still survives is breast-shot, not undershot. At 15rpm and with a water feed of about 1.6 cubic metres per second, I have calculated (see below) that the potential power was 28kW with a realised output after various losses I would guess at perhaps 75% of that figure.
The formula you quote after conversion to SI units is:
F = 1.8384 [ L – 0.2 H ] H 3/2
where
L = width of the weir in metres
H = depth of water at the weir edge in metres
F = flow rate, m 3 / sec
A Google search for the Francis formula brings up numerous hits, and one of them quotes a "modified Francis weir formula" where the initial constant is 0.415 and a multiplier √ (2×G) is added at the end. The two versions are identical if G is taken as the standard value 9.80665ms –2. (Incidentally, there is no need to bother about dimensions so long as all inputs are in SI units - avoiding of course deprecated or superannuated units such as litre, bar or calorie - when the output will also be in SI units.) Chris told me that he had found the formula in one of his books, where it stipulated that the measurement of H should be taken 2 metres upstream of the weir edge. The website quoted, however, gives the (normally much lesser) value of 3H upstream, which seems to me more reasonable since in many cases the weir sill would be less than 2m wide, and 2m upstream could be in deep water.
I do not know whether the figures you quoted ”Say L = 10' and H = 0.2' ” and “about a 2 metre drop” were meant to relate to this particular wheel (which as I have already pointed out is not ”undershot” but “breastshot”). However, the weir formula is not relevant to estimating the potential power of the mill, because the wheel would be fed from a sluice deeper than the weir crest. The formula can be used instead to estimate the overall potential power of the river. For that we need the depth of water over the weir: would the Settle-based members of the U3A Maths group be prepared to wade or cycle along the weir to measure the depth, at both 3H and 2m upstream? The width of the river between its walls level with the Bridge End car park is about 27m, and the total length of the weir (which I could not measure directly) is probably about 35m.
I had to pay several visits to investigate the mill site, on account of hostile weather and not always having suitable measuring equipment with me. All measurements were made in metres, but since the design would presumably have been in Imperial units I show rounded foot values with the actual metre measurement in brackets.
I believe that the original intake structures have been extensively modified since the wheel went out of use in order to reduce the risk of the building being flooded, with the channel being made narrower and more shallow and with a small new steel sluice gate. Originally, I believe the sluice would have been 12ft (3.6m) wide, the same as the present width of the headrace lower down towards the wheel.
The iron waterwheel is still in place in front of the mill building, made up of two 12ft (3.6m) diameter sections joined side by side, each carrying iron buckets 8ft wide (2.4m) of which a number survive. The buckets are of a sophisticated design, very similar to the buckets on the Little Salked waterwheel, and their arrangement in cross-section forms a volute like a slice through an ammonite or a nautilus shell. The wheel was fed at the level of its axis, so that the fall was about half a diameter or 6ft (1.8m), with the water doing work for one quarter of a revolution. The circuference is divided evenly into 7 segments by spokes, and there are 5 buckets per segment, so that there are 35 buckets in all. (Why 7 spokes? I think that there must be an engineering reason: a minimum number may be needed, and the spokes may be required not to be opposite. Can any of our brilliant engineers help?)
It is hard to calculate the capacity of the buckets because of their curved cross-section and an uncertain filling height. I estimate a height of 1ft (0.30m), a mean width of 6in (0.15m) and a filling height of 0.25m, which taken with a length of 2.4m suggests that that they may have been designed for a normal fill of 20 gallons (0.09 cubic metres or 90L = 90kg approximately). With 2 buckets in line and a fall of 1.8m, the energy potentially available from each pair of buckets as they fall through a quarter of a revolution (from the filling to the emptying points) is 2×1.8×90×G, or approximately 3.2kJ. If each quarter-revolution takes 1 second, so that the rotational speed is 0.25Hz or 15rpm, the power generated from each pair of buckets is 3.2kW. With a total of 35/4 pairs of buckets for each quarter of the wheel, the total power becomes 3.2×35/4 = 28.0kW.
The corresponding volume of water consumed is 2×0.09×35/4 = 1.58 cubic metres per second. With a sluice 3.6m (12ft) wide and a mean flow rate of say 2m per second, the depth of water as it passed through the sluice would be 0.22m (9in).
The wheel is obviously quite massive, and would have the momentum to cope with substantial short-term overloads. Wood-working on a modest scale does not need all that much power, for example a pole lathe powered only by one of its turner's feet manages pretty well! For what relevance it may have, the power required to drive the pair of 4ft diameter Little Salked millstones at one revolution per second while milling wheat I estimated at slightly under 3kW.
I intend to revisit Bridge End Mill and to knock on the door to see whether they have any more information such as drawings or old photographs.