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 Engineering Muslim Engineers Al-Jazari

VI. Mathematical Analysis

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1. Analysis of the Total Head of Water Jet

The purpose of this analysis is to find the total head of water. A set of detailed calculation and explanation can be found in Appendix 11. The formulas in Appendix 11 used for finding this force is simplified and derived as shown below. These formulas are also programmed into Microsoft Excel programme to calculate the total head for variable conditions.

The Total Head of Water Jet:
HT = HW + HAB – [(tanθ)(rscoop)]
Where
HW = head of water surface to bottom of pool
HAB = half the height of the lower chamber
Θ = angle of strike

The above variables are to be inputed into the Microsoft Excel programme for an estimation of the total head of water jet.

2. Analysis of the System of the Third Water Raising Device

The purpose of this analysis is to find the resulting force capable of raising water. A set of detailed calculation and explanation can be found in Appendix 12. The formulas in Appendix 12 used for finding this force is simplified and derived as shown below. These formulas are also programmed into the Microsoft Excel programme to calculate the force for variable conditions.

The force of the water jet:
Fj = 2 g HT ρ π(rj)2
The force transfer from the jet to the bucket:
FT = 2 g HT ρ π(rj)2 (1 – V*)(1 – cos θ)
The force applied on gear A:
FA = {(1 – μ) [(1 – μ)FT rscoop]} / rA
The Force applied on Gear B
FB = (1 – μ)FA

The Force applied on Gear C
FC = [(1 – μ) (FB rB)] / rC
The Force applied on Gear B
FD = (1 – μ) FC
The Force applied on Sindi Wheel
Fsindi = [(1 – μ) (FDrD)] / rsindi

The Weight of Water Raised:
mwater = Fsindi / g
where
HT = total head of water jet
ρ = density of water
rj = radius of jet outlet
V* = scaled bucket speed
= deflection angle for the bucket
= coefficient of friction
rA = radius of Gear A
rB = radius of Gear B
rC = radius of Gear C
rD = radius of Gear D
rsindi = radius of sindi wheel

The above variables are to be inputted into the Microsoft Excel programme for an estimation of the force capable of raising water and the weight of water raised.

3. Assumptions and Considerations

For the mathematical analysis of the system, several assumption and considerations were taken for the most optimal conditions. The more significant ones are the deflection angle for the bucket, θ, and the maximum weight of water being raised.

3.1. Deflection Angle for the Bucket

The typical deflection angle for the bucket is taken at the most optimal condition when it is at the horizontal position, where θ 1 is equal to θ 2 (see Fig. 27). However the deflection angle will vary as the bucket rotates, at this point, θ 1 is not equal to θ 2 (see Fig. 28). Although a more accurate force might be obtained by calculating this difference, it is noted that the difference in θ will be minute and that using the typical value of 165° is more appropriate.

 Large image Large image Figure 27: The bucket in horizontal position during rotation. Figure 28: The bucket deflected at angle during rotation.

3.2. Maximum Weight of Water Being Raised

Once the force capable of raising water is found, the maximum weight of water being raised can be found by Fsindi = mwaterg, on the assumption that the weight of the copper jars are in equilibrium or symmetry is maintained during the cycle, m1r1 = m2r2 (see Fig. 29). This is attributed to the fact that the weight of the copper jars would counter-balance, thence would be equilibrium.

However, it would not be true when the copper jars are in motion as there would be a difference in moment, m1r1 ≠ m2r2 (see Fig. 30). An opposing torque, TO, will therefore be applied against the torque produced by the force applied on the Sindi Wheel, Tsindi. Therefore the torque applied to raised water, Tw = Tsindi – To.

Although a more accurate value of the weight of water might be obtained by calculating this difference, it is noted that the difference will be minute and that using the formula Fsindi = mwaterg, is more appropriate.

 Large image Large image Figure 29: The copper jars in symmetry. Figure 30: The copper jars during rotation.

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by: Salim T. S. Al-Hassani and Colin Ong Pang Kiat, Wed 23 April, 2008