Free Report On Hydrostatic: Laboratory Report

When a surface is immerged in a fluid, such as water, hydrostatic forces develop on the contacting surface due to the fluid. Knowledge about these forces is of great importance in hydraulic systems, such as in the design of storage tanks, ships, and dams (Kudela 1).
1. Relevant Theory
For stationary fluids (fluids at rest) it is known that the force must be perpendicular to the surface as a reaction to the weight of the object, and because there are no shear stresses present. It is also known that the pressure suffers a linear variation with the depth if the fluid is incompressible. For example, for a horizontal surface, such as the bottom of a liquid-filled tank that can be seen at in the Fig. 1, the intensity of the resultant force is given by: FR = pA. (Kudela 1)
where
p is the uniform pressure on the bottom and A is the area of the bottom.
It is important to say that if atmospheric pressure acts on both sides of the bottom the only force affecting the resultant force on the bottom is the weight of the liquid in the tank. This happens, because the pressure is constantly and uniformly distributed along the bottom surface, therefore, the resultant force acts through the centroid of the area. However, it can be more interesting to analyse cases where the surface contacting with the fluid has an inclination in relation to the horizontal. It is important for the generalization of the theory, resulting in some of the formulas used in this work. For these more general cases where there is an inclined submerged plane the determination of the resultant force that acts on the contacting surface is more complex (Kudela 1). Consider the general surface shown in Fig. 2 (Kudela 2).
The liquid has an effect on the plane area shown as a wall section. The force on the plane surface is due to the pressure p = ρgh = γh acting along the area is given by:
In the present experiment the forces and pressures which are produced by a solid boundary due to fluids are calculated. To achieve the aim of calculating the forces and pressures that are exerted on the contacting solid surfaces, the concept of centre of pressure is significantly important as it allows determining where the hydrostatic forces are acting on the plane. This permits us to calculate the moments of the forces that are actually influencing the pressure of the system.
2. Values used on calculations:
B=0.075 m

kN/m3

H=0.164
W (fully submerged plane) = 0.4 kg
W (partially submerged plane)=0.05 kg
θ = 00
L=0.2 m
g=9.81 m/s2

Data of the problem:

Average angle θ = 5 0
R2 = 0.2 m
R1 = 0.1 m
Average weight (partial submerged plane) W = 0.152 kg
Average height (partial submerged plane) H = 116.7 m
Average weight (full submerged plane) W = 0.4417 kg
Average height (fully submerged plane) H = 0.086 m
dL = 0.0005 m
dW = 0.0005 kg
dθ = 10
dB = 0.0005 m
dR = 0.0005 m
dH = 0.0005 m
3. Sample calculation
3.1 Case 1: Partially submerged plane

Theoretical moment

= 9810*0.075*cos0*0.233-9810*0.075*0.22*0.1642+9810*0.075*sec02*0.16436 = 0.0896 Nm

Experimental moment

Mexp=W*g*Lcosθ*10-3=0.05*9.81*0.2cos0*7=0.6867 Nm

Force

= R2+hsecθ2*cosθ-h=0.2+0.164sec02*cos0-0.164=0.018 m
A = B*R2-hsecθ = 0.075*0.2-0.164 sec 0=0.0027 m2
= 9810*0.018*0.0027= 0.4768 N

Length

= B(R2-hsecθ)12=0.075*(0.2-0.164 sec0)312=2.916*10-7 kgm2
= 0.2cos0+0.0000002916*cos00.018*0.0027=0.206 m
3.2 Case 2: Fully submerged plane

Theoretical Moment

= 9810*0.075*cos03*0.23-0.13-9810*0.0752*0.22-0.12*0.085=0.7787 Nm

Force

= =0.2+0.12*cos0-0.085=0.065 m
A = = 0.075*0.2-0.1=0.0075 m2
= 9810*0.065*0.0075= 4.7824 N

Length

= = 0.075*(0.2-0.1)312=6.25*10-6 kgm2
= 0.2cos0+0.00000625*cos00.065*0.0075=0.200 m

Experimental moment

Mexp=W*g*Lcosθ*10-3=0.4*9.81*0.2*cos0*7=5.4936 Nm
4. Statement of uncertainty
The experimental moment is the formula used to determine the uncertainty within the experiment. This is needed because the formula has conditions, and there is no adjustment in the formula when switching between the partially submerged and fully submerged plane.
4.1 Case 1: Partial submerged plane

The uncertainty calculations for the partial submerged plane are shown next.

σ Mexp = [(68.67Lcosθ*dW)2 + (68.67Wcosθ*dL)2 + (-68.67WLsinθ*dθ)2]1/2
=[(68.67*0.2{cos5}*0.0005)2+(68.67*0.152{cos5}*0.0005)2+(-68.67*0.152*0.2{sin5}*1)2]1/2 = 0.1821 Nm
% σ Mexp = (0.1821/4.93)*100 = 3.6937 %
4.2 Case 2: Fully submerged plane

The uncertainty calculations for the fully submerged plane are shown next.

σ Mexp = [(68.67Lcosθ*dW)2 + (68.67Wcosθ*dL)2 + (-68.67WLsinθ*dθ)2]1/2
= [(68.67*0.2{cos5}*0.0005)2 + (68.67*0.4417{cos5}*0.0005)2 + (-68.67*0.4417*0.2{sin5}*1)2]1/2 = 0.5290 Nm
% σ Mexp = (0.5290/4.93)*100 = 10.73 %
5. Results discussion and conclusions
Some errors in the collect of the experimental data (shown in the tables 1 and 2 of the Annex 1) are inherent to the experimental apparatus. In fact analytical data taken from the tank ruler is known to be unlikely exact. However, it is not expected that is error is very significant on the accuracy of the results. For the partial submerged plane, the obtained error was 3.6937%, which can be considered a reasonably good result. For the fully submerged plane, the obtained error was 10.73%. It is not expected the error for the fully submerged plane to be higher, because all the surface is in contact with the fluid, while on the partially submerged case it is more difficult to put the exact value of the surface area in contact with the fluid. The experimental data looks well taken: weight is quite higher for the fully submerged case, due to the fact that the force exerted by the fluid is much higher. As conclusions of this work, it is possible to say that there are evidences that the analytical equations used in this work might be more accurate for the fully submerged plane case. Further analysis is needed to confirm this, and introduce a newer term to adjust the equation for the fully submerged plane case. The experimental seems to be gathered correctly, as there are no incoherencies on the values, except the possibility that the intrinsic errors of the apparatus might have been stronger in the fully submerged plane that on the partial submerged plane.

References

Kudela, Henryk, Hydrostatic Force on a Plane Surface, n.d. Web. 26 Jan. 2015. < http://
fluid.itcmp.pwr.wroc.pl/~znmp/dydaktyka/fundam_FM/Lecture5.pdf >
ANNEX 1-Experimental data
Table 2-Experimental data for the partially submerged case