Tom blok Photo collection by Th.Schiöler Applied mathematics concerning water-lifting 3: The test tube
The test tube
Gul streg
The buckets along the rim of the water-lifting wheel should be slender as a test tube…why?

If you have worked in a laboratory, you have tried to drink bear of a test tube. The first test will be an unsuccessful attempt, as most of the bear will not hit the mouth.

By the next experiment, the test tube will be inclined slowly and the bear will reach the mouth.


Vandløftningshjul med "reagensglas"

Nomenclature

d diameter of the bucket m
l length of the bucket m
the inclination of the bucket radians
V the geometrical volume m3
½V the volume below the diagonal m3
ratio of slenderness l/d number
formel angular velocity of the wheel 1/s
formel the flow m3/s
A the area of the water surface m2
A´ the projection of the same area m2

Vipning af "reagensglasset"

Let us first look at a state of guess. If the water reaches the centre of the wheel, the bucket will be filled with water. This corresponds to an angel zero.

The problem is! how to find the flow from the test tube as a function of the angular velocity and the angel. That is:

formel

From the drawing, we can see that the surface has an elliptical form. The area is:

formel

note:

formel = formel

It is well known that the flow Q in a tube is equal to the cross-sectional area multiplied with the velocity of the water. Less obvious is it, to see that the flow from the test tube is equal to the elliptical area multiplied by the distance to the centre of gravity to the same area:

formel

This is the velocity of the centre of gravity.

formel

Hence the flow :

formel
where:

formel

formel
formel
where:
formel

This formula – about the flow Q – is valid until the surface A’ reach the diagonal of the tube. Remember that: formel.

The flow Q will increase tremendous as the angel augment. If the angular velocity increases the flow will increase as well.

We will now control the formula for Q by calculating the volume, which is to be found below the diagonal.

The total volume leaving the test tube can be calculated by integration

formler

The total volume of the test tube is V.


 Next chapter: The concept of the theoretical efficiency