Optimum speed

It's been cold here with temperatures well below freezing. It's a good job that flowing water takes a lot of cold to stop it flowing:

But cold is not what I want to write about in this Diary entry, - rather it is the matter of optimum speed.  
A pelton wheel best converts energy from the force of the jets hitting its cups to rotational energy when it is turning at a particular speed.  That speed will be quite specific to the pelton wheel itself, - size being a major determinant, but also specific to the site at which it is working, - where the net (dynamic) head is a major determinant.

The relationship between rotational speed and power output is commonly presented in text-books in the form of this graph:

 ...which I used in a Diary entry a year ago (see here) to illustrate how changing the speed (by packing off the rotor) had an effect on power output.  As can be gathered from the graph, power as a proportion of maximum power is greatest when speed as a proportion of optimum speed is 1, - i.e. maximum power when speed is at optimum speed.

The difficulty this graph creates for people like me, who want to run their Powerspout so it produces as many kWh's as possible, is the question of what the optimum speed for their particular site is.

To try to answer this question, I have for the past year been measuring speed and efficiency at each and every flow condition at which the turbine has operated.  To date, I have accumulated 29 pairs of speed/efficiency figures, which is enough data to plot a reasonably presentable graph:

Now plotting efficiency against speed is not, of course, the same as plotting [power / optimum power] against speed. Moreover, the efficiency I have used is 'whole system efficiency' rather than 'turbine wheel efficiency', - and 'whole system efficiency' has a lot of variables external to the pelton wheel itself which might corrupt a relationship with speed.

So it is unsurprising that my plot doesn't look exactly like the text-book graph, - it is much 'flatter' in its 'peak'. But I would suggest, nevertheless, that efficiency is a reasonable surrogate for [power/optimum power].  After all, efficiency ought to be highest when speed is at its optimum.

When the graph was first displayed in my computer's spreadsheet, I have to admit to being rather pleased at seeing that its flattish top seems to lie at a shaft speed of around 1050 rpm.

A year ago when I wrote about knowing what my optimum speed might be, I guessed it was between 950 and 1000 rpm.  This latest investigation nudges that guess to a slightly faster speed, - and perhaps gives the new figure an aura of greater respectability by having been arrived at by 'research'.

Having been snowed-in where I live for the past 5 days and with the roads not looking like being clear yet awhile, bringing together this data has, at least, given me something with which to be occupied. Keeping away 'cabin fever' is terribly important !