Hydrostatic force on a plane surface experiment

Figure 3-1. Hydrostatic Pressure Apparatus

HYDROSTATIC PRESSURE

OBJECTIVE

The purpose of this experiment is to investigate the pressure acting on a submerged surface and to determine the position of the center of pressure experimentally.

INTRODUCTION

The study of pressure forces acting on plane submerged surfaces is a fundamental topic in the subject of hydrostatics. It involves assessment of the value of the net thrust and the concept of center of pressure; both of which are important in the design of innumerable items of hydraulic equipment and civil engineering projects.

The hydraulic model that we will use for this experiment is the Hydrostatic Pressure Apparatus. Fig. 3-1 shows a sketch of the apparatus. This device is designed to allow for measurement of the hydrostatic force and center of pressure on the submerged plane surface. The apparatus consist of a torroid attached to a balance beam with an inner and outer radii that both pass through the pivot point.

The balance beam incorporates a balance pan, an adjustable counterbalance and an indicator which shows when the beam is horizontal. The clear acrylic tank may be leveled by adjusting the screwed feet. Correct alignment is indicated by a circular spirit level mounted on the base of the tank. Water is admitted to the top of the tank by a flexible tube and may be drained through a drain valve in the base. The water level is indicated on a scale.

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Figure 3-2. Schematic views of the Hydrostatic Pressure Apparatus

Theory

In a still body of fluid, the only force that exist is the pressure force that acts normal to any submerged surface. Since the inner and out radius of the torroid body pass through the pivot point, pressure forces acting on the curved surface of the object do not contribute to the net moment about the pivot. Therefore, the only forces contributing to the moment about the pivot are due to the counter-weight, 𝑊 , the balance pan and any added mass, 𝑊, and the force due to the hydrostatic pressure, F, acting on the vertical surface, 𝐴, where:

𝐴 = (𝑟 − 𝑟 )𝑏 if ℎ < 𝑟 (face fully submerged)

𝐴 = (𝑟 − ℎ)𝑏 if ℎ > 𝑟 (face partially submerged)

Figure 3-2 shows a schematic of the apparatus with the forces acting on the torroid body when there is water in the tank.

If the counter weight is set such that the balance beam is horizontal when the torroid is out of the water and no weight is placed on the balance pan, as in Figure 3-3, then,

𝑊 𝐿 = 𝑊 𝐿 + 𝑊 𝐿

(3.1)

Any moment created about the pivot when the torroid is submerged is then due to the hydrostatic force, F, acting on the vertical surface. If mass is added to the balance pan so that the beam again is horizontal, then the moment balance becomes,

𝐹(𝑦 + ℎ) = 𝑊𝐿

(3.2)

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Figure 3-3. Moment balance when torroid is exposed to the atmosphere

Where 𝑊 = 𝑚𝑔 and 𝑚 is the added mass.

One of the goals of the laboratory is to experimentally find 𝑦 . To do this we use the theoretical hydrostatic force

𝐹 = 𝑃 𝐴 = 𝜌𝑔𝑦𝐴

(3.3)

where 𝑃 is the hydrostatic pressure at the centroid of the submerged surface and 𝑦 is the distance from the free surface to the centroid of the submerged portion of the submerged face (note that the location of the centroid will change depending on flow depth).

Using this definition for F and Eq. 3.2 we can write an equation for 𝑦 in terms of measurable quantities,

𝜌𝑔𝑦𝐴(𝑦 + ℎ) = 𝑊𝐿

(3.4)

𝑦 = 𝑚𝐿

𝜌𝑦𝐴 − ℎ

(3.5)

Denote 𝑦 determined with Eq. 3.5 as 𝑦 . or 𝑦 “experimental.”

On the other hand, the distance from the free surface to the center of pressure, 𝑦 , is theoretically found as follows,

𝑦 = 𝑦 + 𝐼

𝑦𝐴

(3.6)

where 𝐼 is the moment of inertia of the submerged surface about the x-axis (passing through the centroid), and 𝐴 is the submerged surface area (Potter and Wiggert, 2015). For this lab we will denote 𝑦 determined using Eq. 3.6 as 𝑦 . , i.e. 𝑦 “theoretical.”

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Experimental Procedure

1. Discuss and determine what information will have to be measured and recorded in order to obtain F and yp. Remember that you will be collecting data for both a partially and fully submerged torroid face.

2. Make sure that the torroid is exposed to the atmosphere and that the balance pan is attached to the balance beam. Adjust the counter-balance weight until the balance arm is horizontal as indicated on the datum level indicator.

3. Add all the weights supplied to the balance pan. Fill the tank with water until the balance beam tips and lifts the weight.

4. Drain out a small quantity of water to bring the balance arm back to a horizontal position. Do not level the balance arm by adjustment of the counter balance weight or the datum setting of the balance arm will be lost.

5. Record the water level shown on the scale, the total mass on the balance pan, and record whether or not the face was fully submerged. Fine adjustment of the water level may be achieved by over- filling and slowly draining, using the drain valve.

6. Remove one or more weights from the balance pan and level the balance arm by draining out more water. When the arm is level, record the depth of immersion shown on the scale, the total mass on the balance pan, and note whether or not the face is fully submerged.

7. Repeat the procedure by reducing the weight on the balance pan in increments.

Data

𝑟 = 200 mm b = 74 mm

𝑟 = 100 mm L = 275 mm

y (mm) m (gr) fully submerged?

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Analysis of Data

1. Categorize the data into “Fully Submerged” and “Partially Submerged.” Keep track of these categories throughout the analysis by organizing the data under Fully Submerged and Partially Submerged categories in tables and by using different colors in plots.

2. For each water depth and mass setting, calculate: a) 𝑦 “experimental” using equation 3.5 b) 𝑦 “theoretical” using equation 3.6

3. Create a table comparing 𝑦 . and 𝑦 . as a function of 𝑦.

4. Plot 𝑦 . and 𝑦 . as a function of 𝑦 for both, the fully submerged and partially submerged cases in the same graph.

5. Calculated ∆𝑦 = 𝑦 . − 𝑦 . for each value of 𝑦.

6. Plot ∆𝑦 vs. 𝑦, for both the fully submerged and partially submerged cases in the same graph. The plot created is a visual help to show the differences between the two quantities, 𝑦 . and 𝑦 . .

Results

y

(mm)

m

(gr)

h

(mm)

𝑦

(mm)

𝐴

(mm2)

F

(N)

𝑦 . (mm)

𝐼

(mm4)

𝑦 . (mm)

P ar

tia ll

y Su

bm er

ge d

S ub

m er

ge d

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Additional Calculations

𝐼 =

H would be the submerged height of the plane surface

Discussion Points

1. How does the design of the apparatus enable the resultant force on the vertical surface to be measured?

2. Are there any significant forces being neglected?

3. Is there any difference in the functional variation of force F with respect to y between the partially and fully submerged cases? If so, what are these differences and what causes them? Add graph of F vs. y if necessary for your analysis.

4. Compare the experimentally measured values of 𝑦 with the theoretical ones using the plots created in your analysis and by calculating the percent error

%𝑒𝑟𝑟𝑜𝑟 = 𝑦 . − 𝑦 .

𝑦 . × 100%

Are there any differences? How small is the percent error? If so, what might they be due to?

5. What is the primary trend in ∆𝑦 vs. 𝑦. Is this what you would expect? Why or why not?

b

H x