Hopwood 1
Blair Hopwood Team 3 7 October 2015
Hydrostatic Pressure Abstract:
An Edibon Hydrostatics Pressure System was used in order to collect data used in calculating hydrostatic force and the center of pressure for a rigid body when it was both partially and fully submerged. The mass of the weights added to the system and the height of the water in the tank when the mechanism was level were collected and used in these calculations. All of the data collected followed expected trends, which suggests accuracy in the data as well as validates the equations used. Relative error values between the “y” (theoretical) values estimated mathematically and the water height (h) values measured with the system ranged from .129% to 6.6%, demonstrating that there is not a large amount of variance in the data. The height of the water shares a linear relationship with the mass of the weights, estimated y value, and distance to the center of pressure. There is also a linear relationship between the hydrostatic force on the surface, and standard deviation between the y and h values. Introduction:
An Edibon Hydrostatics Pressure System can be used to study the hydrostatic pressure on a plane surface (Emaco Group, 2000). The validity of pressure equations that summarize hydrostatic pressure relationships can be tested using this tool. Utilizing a quadrant submerged in a tank of water and a counterweight system, both the resultant of the hydrostatic forces acting on the surface as well as the distance to the center of pressure of a submerged plane surface (ℎ!) can be calculated. When additional mass is added to the counterweight the hydrostatic force became unbalanced and causes the balance bridge arm to tilt. Adding water to the tank exerts a moment in order to counterbalance the resultant moment caused by the hydrostatic forces on the quadrant (Gerry, 2002). Pressure increases with both depth of water in the tank as well as the mass on the counterweight.
A height measurement (h) is taken at each weight and this measurement is compared with a theoretical prediction (y). Both h and y increase as water is added to the tank. These values are used in finding the center of pressure (ℎ!) of a submerged body. Figure 1 demonstrates the location of the center of pressure (ℎ!) in relation to the height of the fluid from the body’s centroid.
Hopwood 2
Figure 1: Center of pressure (ℎ!) compared to depth of liquid (h) (Gerry, 2002)
Objectives:
The objective of this research was to measure the effect of depth of water (that is at rest) on the magnitude and location of resultant force exerted on a partially and fully submerged vertical rectangular surface. Additionally this lab was used to confirm the validity of the theoretical equations used to estimate magnitude and location of resultant force exerted on a partially and fully submerged vertical rectangular surface. Materials and Methods:
Weights of different sizes were used in determining if the pressure equations accurately describe the forces on a partially or fully submerged body. At the beginning, no part of the quadrant was submerged; only the lowest edge was in touch with the water. As each weight was added, additional water was put into the tank until the balance bridge arm was again level and a height measurement was taken. This continued until the height of the water reached approximately 100 mm. This same procedure was then repeated, but this time with the quadrant starting submerged to the upper edge of the vertical surface of the quadrant and finishing fully submerged. In both trials, a set of equations was then used to estimate the distances to the center of pressure (ℎ!) and hydrostatic forces (F) on the surfaces from the heights obtained at each weight. The equations vary slightly when being used to describe a fully submerged body compared to those for one that is partially submerged. These equations can be seen below. Equations (1) and (2) are used for partially submerged bodies in the first trial set and (3) and (4) were used when the quadrant was fully submerged in the second set of measurements.
Figure 2: Diagram of quadrant used in laboratory exercise (Fasina, 2015)
Hopwood 3
ℎ! = 𝑎 + 𝑑 − !! (1)
Where h is the measured height of water (mm) a is the measurement corresponding to that seen in Figure 2 (mm) d is the measurement corresponding to that seen in Figure 2 (mm) 𝐹 = !
!𝜌𝑔ℎ!𝑏 (2)
Where h is the measure height of water (mm) 𝜌 Is the density of water (0.00099795 !
!!!) b is the measurement corresponding to that seen in Figure 2 (mm) g is 9810 !!
!!
ℎ! = 𝑎 + !
!+ !!
!"(!!!!) (3)
Where h is the measured height of water (mm) a is the measurement corresponding to that seen in Figure 2 (mm) d is the measurement corresponding to that seen in Figure 2 (mm) 𝐹 = 𝜌𝑔𝑏 ℎ − !
!𝑑 (4)
Where h is the measured height of water (mm) 𝜌 is the density of water ( !
!!!) b is the measurement corresponding to that seen in Figure 2 (mm) d is the measurement corresponding to that seen in Figure 2 (mm) g is 9810 !!
!!
The height measurements (h) that were taken after each weight was added were also used in another calculation. They were used to estimate the y value, the calculated height that the water should be at to counter balance the mass. This measurement allowed for the examination of the accuracy of the equations being used by calculating the relative error by using equation (7). Low relative error values between the h and y values at each weight suggest a relatively high degree of accuracy; i.e., the equations accurately describe the relationship given by equations (5) and (6). In order to utilize these equations to solve for y, the given pressure equations had to be developed. Manipulation of the variables and their order was crucial. Equation (5) characterizes the situation where the quadrant is partially submerged and (6) describes when it is fully submerged. 𝑚 = !"
!!(𝑎 + 𝑑 − !
!)𝑦! (5)
Where 𝜌 is the density of water ( !!!!)
b is the measurement corresponding to that seen in Figure 2 (mm) d is the measurement corresponding to that seen in Figure 2 (mm) a is the measurement corresponding to that seen in Figure 2 (mm)
Hopwood 4
L is the measurement corresponding to that seen in Figure 2 (mm)
𝑚 = !"#!(𝑎 + !
!+ !!
!"(!!!.!!))(𝑦 − 0.5𝑑) (6)
Where 𝜌 is the density of water ( !!!!)
b is the measurement corresponding to that seen in Figure 2 (mm) d is the measurement corresponding to that seen in Figure 2 (mm) a is the measurement corresponding to that seen in Figure 2 (mm) L is the measurement corresponding to that seen in Figure 2 (mm)
Imputing values in for the quadrant dimension variables (a, b, d, and L) and manipulating the partially submerged and fully submerged pressure equations above resulted in the equations (7) and (8). 0.000040852𝑦! − 0.0245𝑦! +𝑚 = 0 (7) 𝑦 = 0.272𝑚 + 44.4 (8) Results and Discussion: Readings were taken after each time weight and water was added to counterbalance the system. The height of the water in the tank increased proportionally with the addition of weight. These height measurements were used in order to calculate y values, or heights in the vertical direction that the water raised each time. Graphs (1) and (2) both demonstrate the linear relationship between mass and the y values generated. In both the partially submerged and fully submerged data sets, there is a positive and linear relationship between the two variables. As more weight is added, more water is needed to counterbalance the extra force being applied and thus create a moment of zero. Force is the product of mass and area. The area of the rigid body is constant for this system, therefore it is expected that as the mass increased the force would also increase proportionally, as it did in the results form this experiment.
The calculated y values and h measurements were compared to one another for each weight interval. Relative error values were relatively small ranging from .129% to 6.6%, demonstrating that there is not a large amount of variance in the two opposing value sets. Having little variation suggests that the equations accurately describe the relationships they are intended to. Water heights (h) are plotted against the y values in Graphs (3) and (4). Both when partially submerged and when fully submerged, the relationship between them is almost perfectly linear. A perfectly linear relationship has an 𝑅! value of 1.0, both of these sets have a coefficient of determination of about 0.99. Having y and h values that are so similar demonstrates that both the data collection and calculation process are valid. Having a coefficient of determination so close to one also helps support the accuracy of the pressure equations. For example, Equation (8) is a manipulation of (6). This form of the equation is presented in linear form, following the format of y = mx+b where m is the slope and b is the y intercept. In this exercise, the x value is the
Hopwood 5
respective weight at that data point. Thus, by y and h being linearly related, the equation is supported in accurately describing the pressure relationship. As mentioned before relative error between the calculated y values and heights using equation (9) below, these values were relatively small in this laboratory exercise. Graphs (9) and (10) plot this variable vs. height when partially submerged and fully submerged. In both these circumstances, there seems to be no apparent trend in the data showing that the difference in the calculated height and the measured heights were random and the variance in the relative error has to do with human error made in the lab. This lab allowed for a lot of human error because someone had to read weather or not the arm was at equilibrium as another person had to read the height by having humans read these measurements there is room for error in the reading thus causing the variance in relative errors. 𝑅𝐸 = 100 ∗ !!!
! (9)
The changes in height of the water were also used to estimate the distances to
the center of pressure (ℎ!) and hydrostatic forces (F) on the surfaces from using the heights obtained using the equations and method referenced above. Graph (7) and Graph (8) show the resultant hydrostatic force (F) on the surface of the quadrant plotted against the height of the water. This relationship is positively linear for both the partially submerged and fully submerged data. It is perfectly linear when fully submerged, having a correlation coefficient, 𝑅!, of 1.0. The F values are larger when fully submerged than when partially submerged and linearly increase with depth from the first to the last reading. This trend is expected because as depth increases hydrostatic pressure increases (Cengel et. al, 2014).
The distance to the center of pressure plotted against the height of the water has a linear, but negative, relationship. This suggests that as water depth increases, the center of pressure is further from the bottom of the tank, this follows the trend that is expected. As volume increases, the resultant hydrostatic force acting on the body also increases. The line of action of this force passes through the centroid and, for shapes like that of the quadrant, the center of pressure lies on the y-‐axis directly below the centroid (Cengel et al, 2014). Therefore, as volume and height increase, the depth of the center of pressure decreases, as shown in Graph (5) and Graph (6). Tables and Graphs:
Graph 1: Mass vs. Estimated y Value when Partially Submerged
Hopwood 6
Graph 2: Mass vs. Estimated y Value when Fully Submerged Graph 3: Estimated y Value vs. Height when Partially Submerged Graph 4: Estimated y Value vs. Height when Fully Submerged
Hopwood 7
Graph 5: Distance to Center of Pressure vs. Height when Partially Submerged Graph 6: Distance to Center of Pressure vs. Height when Fully Submerged Graph 7: Hydrostatic Forces on the Surface vs. Height when Partially Submerged Graph 8: Hydrostatic Forces on the Surface vs. Height when Fully Submerged
Hopwood 8
Graph 9: Relative Error between Estimated y Values and Heights vs. Height when Partially Submerged Graph 10: Relative Error between Estimated y Values and Heights vs. Height when Partially Submerged Conclusion:
Hydrostatic forces (F) and distances to the center of pressure (ℎ!) were calculated based on measured water height values from an Edibon Hydrostatics Pressure System. The force on the surface of the quadrant increased as weight and water were added while the center of pressure, ℎ!, decreased in height. Both relationships were linear when partially submerged, as well as fully submerged. The data collected and calculated supported that the equations provided accurately describe the relationships between variables. As an example, hydrostatic forces on the surface of the quadrant and the water height when fully submerged had a perfectly linear relationship, a correlation coefficient of exactly 1.0, therefore water depth increased proportionally to pressure. This validates the equation used to calculate F. No relationship examined had a 𝑅! values of less than 0.97 meaning all the equations used were accurate and valid. There also was very little variation between the measured heights and the calculated y values, as seen by the lower relative error between them. References: A. S. Humpherys. 1991. Center-‐of-‐Pressure Gates for Irrigation. ASAE Volume 7(2).
Emaco Group. 2000. Hydrostatics Bench.: Edibon, S.A. Available at: http://www.emacogroup.eu/Fluid%20mech-‐%20aerodynamic.pdf. Accessed 6 October 2014.
Hopwood 9
Gerry. 2002. Experiment 3 Hydrostatic Force on a Submerged Surface. Available at: http://web.cecs.pdx.edu/~gerry/class/EAS361/lab/pdf/lab3_hydrostatics.pdf. Accessed 6 October 2014.
K. M. Robinson. 1992. Predicting Stress and Pressure at an Overfall. ASAE Volume 35(2).
Y. A. Cengel, and J. M. Cimbala. 2014. Fluid Mechanics Fundamentals and Applications. 3rd ed. New York, New York: McGraw-‐Hill.