With z is the elevation of any point in the static fluid mass, constant specific weight γ, and is hydrostatic pressure, the value ofp is also called the water pressure measurement in me
OBJECTIVES OF EXPERIMENT
This experiment helps students understand the basic equation or hydrostatics and apply it to some problems associated with compressed fluid in a static state.
THEORY
The state of static and compressed fluid is described by this equation:
In a static fluid mass, the elevation of any point is represented by \( z \), while the constant specific weight is denoted by \( \gamma \) The hydrostatic pressure, \( p \), is measured in meters and is also referred to as water pressure This relationship allows for various practical applications.
An isobaric surface is defined as a surface where the pressure remains constant A common example of an isobaric surface is the atmospheric surface, where points at the same pressure level are found at equal heights.
To calculate the pressure at any point within a static fluid mass, we can use the fundamental equation: \$p = p_0 + \gamma(z - z_0) = p_0 + \gamma h\$, where \$p_0\$ is the pressure at elevation \$z_0\$ and \$\gamma\$ represents the specific weight of the fluid The reference surface, denoted as 0-0, serves as a standard for comparison in these calculations.
By measuring the height , we can determine the pressure h p
3 Calculating the specific gravity of fluid:
If we have p 0, , both p z 0 and z, from (2) we can measure the specific weight of fluid:
The basic equation (1) is valid when the effects of capillarity are ignored However, this equation becomes inaccurate for fluids in small tubes with diameters less than 3 mm For fluids like water, alcohol, or oil, the fluid level in narrower tubes is higher compared to wider ones Conversely, when dealing with mercury, the fluid level is higher in wider tubes than in narrower ones.
APPARATUSES
The manometer features a series of ten tubes, numbered 1 to 10, mounted on a board with a precision ruler calibrated to the millimeter Most tubes have a diameter of 5 mm, while the second group of tubes has a diameter of less than 3 mm.
• Tube 1 and 3 are used as fluid manometers to measure the atmospheric pressure in vessel T
• Group of tubes 2 containing tube 21 ,22 ,23 with diameters 1mm, 2mm, 3mm respectively are used to observe the capillary phenomenon
• 3 U-shaped tubes are couple tubes 4-5, 6-7, 8-9, containing fluids needed to determine its specific gravity
Tube 10 and 3 are used to observe the acupressure surface The atmospheric pressure of tubes 1, 4,and 6, 8 is the atmospheric pressure in vessel T The elevation of fluid level in tubes is
The pressure system consists of a static closed vessel T and an un-static open vessel Ð, which is suspended from block 12 and filled with water By utilizing crank 11, we can adjust the height of vessel Ð, thereby altering the atmospheric pressure within vessel T.
EXPERIMENT STEPS
To ensure accuracy, verify that the standard figure of the millimeter ruler is positioned in a horizontal plane by comparing the water levels in tube 3 and tube 10; both tubes should display identical water levels.
To lift vessel Ð to the required pitch, use crank 11 to ensure that the free surface of vessel D is elevated above that of vessel T by approximately 15 to 20 cm Begin by measuring and recording the water levels in tubes 1 to 10, as well as in group tube 2 Next, lower vessel Ð to the average position, where the height difference is between 5 and 7 cm, and then to the low position, with a height difference of -15 to -20 cm Repeat the measurement process and document the results in the table.
Barometric pressure and temperature of open air during experiment time are:
The specific weight of water is: (using the appendix in lecture note) γ H2O 9765(N/m 3 )
EXPERIMENT RESULTS AND OBSERVATION
1 According to the three elevations of tank D in comparison with tank T, please record the measured values of 9 tubes, and Group Tube 2 in the following Table 1
Table 1a: The measured results (Unit: cm)
Table 1b: The measured results in tube 2
COMPUTATION AND EXPERIMENTAL RESULT PRESENTATION 6
1 In the fluid static experiment set, which water levels of tubes, or tanksare in the same horizontal plane? Why?
Tube 3 and 10 have the same water level in the hydrostatic experiment because both the free surface of tube 3 and 10 are in contact with air so they have atmosphere pressure
Tube 1 and tank T also have same horizontal plane because they contact through the same fluid as well as they have the same pressure on the free surface
2 In the fluid static experiment set, which water levels of tubes arenot followed the law of fluid static (not in the same horizontal plane)?
The water level in tube 2 does not adhere to the principles of fluid statics due to the varying diameters of tubes 21, 22, and 23, all of which are less than 3 cm This small diameter results in capillarity effects, which are influenced by surface tension.
Due to capillarity , water levels of tubes in tube 2 21 , 2 and 23 are different The smaller the diameter of the tube, the higher the water level
3 Compute the absolute and gage pressure of air in tank T and theserelative errors in three measurement cases Write them in Table 2
4 Compute the specific weight of three fluids in U-Tubes 4-5, 6-7, and8-9 and their relative errors in three measurement cases Write them in
1 To compute the pressure of closed air in Tank T:
7 p = p 0 + γ (z − z 0) = p 0 + γh The pressure of the air, in Tank T can be computed as follows:
The specific weight of water, denoted as γ H2O, varies with the temperature of the environment When considering atmospheric pressure as zero (p a = 0), the gage pressure of air in Tank T can be calculated using the formula: pTd = γH2O (L3 − L1).
According to three locations of Tank D, we can compute three values of pressure of air in Tank T
2 To compute the specific weight of fluids in the U-Tubes:
In a U-tube system, the specific weight of a fluid at rest in tubes i and i + 1 can be determined by considering the pressure of air in Tank T, which is given by the equation \( p_T = p_a + \gamma (L_{i+1} - L_i) \) By comparing this equation with a previously established formula, we can derive the specific weight of the fluid within the U-tube setup.
Tubes can be computed as follows:
3 To compute the error in the method of pressure measurement:
Utilizing the theory on the error, from the formula (3), the relative error, δ p from the pressure measurement can be computed as follows:
(6) Therefore, relative error from the gage pressure measurement is the addition of two terms:
• Relative error from the computation of the specific weight of water:
• Relative error from the readings on the level scales:
In general, we use: δγH2 O = 0 12% and ∆L 1 = ∆L 3 = 0.5mm
4 To compute the error in definition of specific weight, γ of fluid in the U-Tubes
From formula (5), we can compute the relative error in the measurement of specific weight in the U-tubes as follows:
Give your comments on: a) The pressure is closed in tank T
During the initial phase of the experiment, the pressure in the closed air of tank T reaches its maximum at 102.41 Pa As tank D is moved downward, the pressure in the closed air of tank T subsequently decreases This change in pressure can be calculated using a specific formula.
The pressure difference between two tanks is given by the equation \( P_T = \gamma (Z_D - Z_T) \), indicating that a larger difference in elevation (\( Z_D - Z_T \)) results in higher pressure Since the fluid in tank D is in contact with the air, its surface can be treated as a free surface Additionally, the variation in water surface elevations between tanks T and D is significant.
The difference between water elevations between two tanks leads to the difference of pressure in the system
Pressure of the free surface in tank D is atmospheric pressure because it is in contact with the open air
Tube 4, 6, 8 is connected with tank T through closed air so they have same pres- sure of closed air and tube 5, 7 ,9 have atmospheric pressure
When the water level in tank D exceeds that of tank T, the gage pressure in tank T becomes positive, while tubes 5, 7, and 9 experience atmospheric pressure Consequently, the water elevations in tubes 4, 6, and 8 are lower than those in tubes 5, 7, and 9, respectively.
When the water level in tank D is lower than in tank T, tank T experiences vacuum pressure Consequently, tubes 4, 6, and 8 are under vacuum pressure, while tubes 5, 7, and 9 maintain atmospheric pressure This difference in pressure affects the water elevations in tube 4.
6, 8 are higher than those of tube 5, 7, 9 respectively
When tank D descends below tank T, a vacuum pressure is created in tank T At a constant temperature, the relationship described by the formula \$PV = \text{constant}\$ indicates that as pressure (P) decreases, volume (V) must increase Consequently, the water elevation in tube 1 experiences a slight decrease Additionally, the water levels in tubes 3, 10, and tank D are equal and significantly drop due to their connection in the same fluid, with the upper section exposed to open air, resulting in a shared isobar.
To investigate energy equation in current, having changeable cross section b THEORY:
The energy of one weight unit of fluid at a section is determined by 3 parts:
With zi, pi, Vi alternately are elevation, pressure and average velocity of the water current at cross-section i, 𝛼is kinetic fixing coefficient, 𝛼 appears in the component of
… Is due to unsteady velocity distribution (by friction in the current) above the section
𝛼 = 1 𝐴 ∬ ( 𝑢 𝑉 ) 3 𝑑𝐴 (3.1) With u, V alternately are point velocity and average velocity above wet cross-section with area A
In the context of internal currents, when analyzing moving layers with an angle of 𝛼 = 2, the tangled motion occurs within the range of 𝛼 values from 1.05 to 1.15 To achieve greater accuracy, it is essential to understand the distribution rule for the velocity \( u \) in section A and apply formula (3.1) for calculations Typically, for fluctuating flow, we simplify the process by using \( \alpha = 1 \).
So at section (i) the energy of a liquid weight is equal to:
𝐻 𝑖 = 𝑎 2𝑔 𝑖 𝑉 𝑖 2 + 𝑝 𝛾 𝑖 + 𝑧𝑖 With Hi is the total water column (m)
Consider the flow is stable from section 1-1 to section 2-2, it can be described as an energy equation:
(3.2) With ℎ 𝑓1−2 is the energy loss of the flow from section 1-1 to section 2-2
If neglecting the energy loss, equation (3.1)is:
Equation (3.3) show us the derive of potential energy z + p/ and kinetic energy
V 2 /2g of the flow From section with small area to section have big area, the kinetic energy decrease and the potential energy increase
The total energy of a water column is the sum of its potential energy, represented as \( z + \frac{p}{\gamma} \), and its kinetic energy, given by \( \frac{\alpha V^2}{2g} \).
APPARATUS
• Water from tank (6) is pumped into tank (1) flow into glass channel through valve (2) (to change discharge)
• The horizontal section of the glass channel (3) is rectangle, in which bottom width Bxmm
• Broad crested weir (4) has a trapezoidal cross – – section, with the sides angle is
The water level downstream of the broad crested weir is regulated by valve (5) at the channel's end, allowing water to flow into tank (6) via the rectangular overflow.
• Point gauge and Vernier (7) are mounted on the glass channel (3) to determine channel bottom level and water level in the glass channel.
EXPERIMENT STEPS
The section's location in the canal is determined from points 1 to 6, moving from upstream to downstream, with the midpoint located at a distance of 40 cm along the canal The positions of the sections in the canal are illustrated in the accompanying image, and the distances between each section are specified.
2- Measure the depth of the bottom of the cannel Zđi correspond with section 1 to 6 Record the measurement to the table 1
3- Use the valve (5) to adjust the flow and water level in the cannel so that the water level in the downstream is higher than the upstream
To ensure a stable flow, use needle (7) to measure the height of the free surface \( z_i \) from section 1 to section 6, and record these measurements in Table 1 Maintain the flow conditions and adjust the upstream water level using the valve as needed.
In this instance, the downstream flow is lower than the upstream flow, as indicated by the black mark in the channel Once the flow stabilizes, proceed to measure as outlined in step 4 and document the results in Table 1.
INTRODUCTION
The average velocity at section I equal to:
With Ai is the area of section A = Bh i i
The width of cannel B = 78mm;
The water level from the bottom of the cannel: hi = |Z Z |; Zđi– i i and Zđiare the water surface elevation and the depth of the cannel in each section
- Check all the system to make sure it work safely
- Before bumping the water, check the water level in the cannel to avoid the water spill out
2 Calculate the water column velocity
The average water column velocity at section i:
The energy lost between section i and j are determined by using Bernoulli equation (3.1) between 2 sections:
With z and z are the elevation of water level from section I to j; i j 𝑝 𝑖
We also have = 1 and the average water column velocity is calculated like (3.5)
Apply (3.6) to calculate energy loss of the flow from section 1 to 2 (hf1-2), from section 2 to 3 (hf2-3), from section 3 to 4 (h3-4), from section 4 to 5 (hf4-5), from section
5 to 6 (hf5-6) Determine for both cases
4 Determine the change of water surface
If neglecting the energy loss of the flow from section 1 to I randomly (i=26), and if we assume that the flow of all section are stable or hardly changed:
In the experiment, a standard plane is established at the bottom of the channel, which is considered a horizontal plane The initial height is denoted as \( z_i = h_i \) before the step, and after the step, it is represented as \( z_i = h_i + a \) The equation governing the system is given by \( z_1 + \alpha 2g V_1^2 z_i + \alpha 2g V_i^2 \).
Use Vi from (3.4), z from (3.7) to place at equation 3.8: i
If i before and after the step
In equations (3.9) and (3.10), knowing the value on the left side allows us to determine \( h_i \), which results from a third-order equation We can solve this equation using the trial-and-error method to find \( h_i \).
To determine the value of \( h_i \) using the trial-and-error method, start with an initial estimate that may be either larger or smaller than the measured \( h \) Substitute this value into the right side of equations (3.9) or (3.10) If the result on the right side exceeds the left side, reduce \( h_i \) and recalculate until both sides are equal, at which point \( h_i \) can be accepted as the correct value.
5 Draw the water surface line in the cannel
In the diagram, the bottom of the channel and the water surface line are illustrated, allowing for calculations based on equation (3.10) and the variable \( h_i \) The results indicate that the water level downstream is higher than that upstream.
PREPARATION
1 How to measure water level and channel bottom coordinates?
Ans: A measuring needle (7) is used to measure the water level and the bottom coordinate
2 How to adjust the water level in the glass channel? How many water levels do experiment with downstream do?
Ans: A valve (5) is used to adjust the water level in the channel There are two water level modes in the downstream:
+ The water level in the downstream is higher than that on the ladder
+ The water level in the downstream is lower than that on the ladder
3 How many forms of energy loss on this experiment?
Ans: There is one main kind of energy loss which is losses along the wall.
RESULTS
The measurement of the glass channel's bottom (Zđ) and the water surface (Zi) at various water levels is documented in Table 1.
Table 1: The coordinates of bottom and free surface
Distance from section I to section i+1, cm
Accrual dis tance from section 1 to section I + 1, cm
1 Calculate the current velocity and water column at each section based one equation (3.4) and (3.5) Calculate for 2 attempt Write the result into table
2 With 2 attempt, calculate the energy lost among each section based one equation (3.6)
3 Based on the water level at section 1 Calculate the water column h, at sections I with equation (3.9) and (3.10) by testing method Calculate for the first attempt Write the results into table 2
4 In figure 1, draw the bottom of the channel: a Based on the results measure z in Table 3, draw on Figure 1 a i
“measuring” waterline (draw for the first water level) b Based on the results calculate h by testing method in Table 3, drawing on i
Figure 1 an “ideal” waterline (drawing for the first water level) c Discussing two “measuring” waterlines and “ideal” waterlines
Both graphs exhibit a similar trend, showing a decrease in section 2 and an increase in section 4, attributed to the variation in wetted area between sections 2-3 and 4-5 Additionally, the ideal graph demonstrates a significantly greater difference in section 4-5 compared to the measured graph, indicating energy loss.
5 Discussing the water level between section 5 and section 6?
Ans : The water level between two section is nearly equal
6 Please comment, compare and explain the energy loss calculations power between the sections in Table 2
Energy loss between sections is primarily attributed to friction force, which is influenced by the flow velocity and the distance between the two sections Consequently, the head losses \( hf_{2-3} \) and \( hf_{4-5} \) are notably higher than those in other sections.
Table 2: The results of calculated velocity and energy losses
Average Velocity Vi, cm/s Average Velocity head hv1, cm Sec
Table 3: The results of calculated water level Z by testing method 1
Calculated by the test method
Accrual distance from section I to I+1, cm 20 38.2 41.8 60 80
EXPERIMENT 3D: MEASUREMENT OF VOLUMETRIC FLOW RATE
Comparison of flow measurement devices in a duct:
EQUIPMENT SET – UP
The fan inlet features a 149 mm diameter duct equipped with pressure tapings for simultaneous static pressure measurement at four sections These tapings connect to a bank of pressurized manometer tubes Additionally, two flow measurement devices are utilized: a 65 mm orifice plate and a 65 mm to 149 mm diameter venturi nozzle.
In which: 1 Orifice plat 2 Ventury nozzle 3 Fans and electric motors 4 Inverter 5 Measuring tubes 6 Pressure gauges 7 Silicon tubes 1,2,3,4: Order number of the measuring tubes.
THEORY
The volume flow rate at the orifice plate and venturi nozzle in the pipe is determined by formula as follows:
The pressure difference (\( \Delta p \)) from the inlet to the throat is measured using a manometer filled with a liquid of density \( \rho_1 \) This pressure difference can be represented in terms of the manometric head differential (\( \Delta h \)).
The expandability factor (ε) is crucial for understanding gas flows, particularly in scenarios involving significant pressure reductions It is incorporated into the code to account for density changes in these flows In the case of liquid flows and gas flows with moderate pressure variations at the meter, the expandability factor is set to 1.00.
The discharge coefficients of the orifice plat and the venturi nozzle can be determined by empirical formula
When determining Q from ∆p, it is necessary to estimate a value initially as C Re cannot be calculated until Q is known From an initial estimate of (example = 1C C ),
Q can be calculated and thus Re found The value of C can then be corrected and new values of Q and Re cure calculated For the venturi nozzle:
STEPS
Check water level in the tank, it should be at the level of half of tank Open the valve at the outlet of fan and turn on the motor
Take the readings of 4 manometer levels, and the reading values at two electronic pressure measurers (give the different values of gage pressure between two sections in
KPa) In order to decrease or increase the volume flow rate, the valve is closed or opened partly according to the rotating velocity of the motor decrease or increase
Repeat for three valve settings (three rotating velocities: 800 rpm, 600 rpm and 400 rpm), and write the readings of manometer levels and the values of electronic pressure measurers
1 Derive the formula (1) in case of C = 1
The Work-Energy equation written between cross-section 1 in the approach fluid flow and cross-section 2 in the constricted area of flow is shown below:
The pipe is on horizontal plane so Z 1 = Z 2 and we apply the continuity equation velocity Therefore, we get:
Proof To proof the formula (1) we do as below:
• Q ideal is the ideal flow rate through the meter (neglecting viscosity and other friction effects)
• A 2 is the constricted cross-sectional area perpendicular to flow
• P 1 is the approach pressure in the pipe
• P 2 is the pressure in the meter,
The volumetric flow rate, referred to as Q ideal, is derived from an equation that does not account for frictional losses However, in real-world applications, friction losses and other nonideal factors are always present, necessitating the inclusion of a discharge coefficient in the equation for C.
Q but in this question, it assume that C = 1, So is not necessary in this formula C
In addition, it is essential to incorporate the expandability factor, as outlined in the code, to account for the impact of density changes in gas flows during significant pressure reductions For liquid flows and gas flows experiencing moderate pressure variations at the meter, the value of ε is approximately 1.
Finally, we get the formula:
2 Derive the formula (2): We call M,N as the free surface of tube 1 and 2 respectively
The pipe is on horizontal plane so Z 1 = Z 2
Applying the fluid statics, we can get
Because tube 1 and tube 2 are connected with each other through water,so we can apply fluid statics p M = p N + ρ1 g h∆ Finally, we can get the formula:
∆p = p 1 − p 2 = (ρ1 − ρ)g h ∆ Where: a ρ1 = 1000kg/m 3 : Water density b ρ = 1.226kg/m 3 : Flow density
3 Determine the volumetric flow rate in three experiments by using orifice Flat
- First, we will find the value ∆h = h 2 − h 1 then we will use the formula below to find
- Then, we find the difference of pressure measured by pressure gauge and measuring tube by applying following formula:
- To calculate C value in orifice plate, we will use iteration method:
+ Step 1: We assume = 1 C to calculate Q by the function (1):
+ Step 2: Calculate Reynolds number by the function (4)
+ Step 3: Calculate value by function C (3)
+ Step 4: After getting value of , we will replace = 1 with this new value of and C C C calculate Q again by formula (1)
+ Step 5: We keep doing this iteration method until: 5%
After getting the suitable values of C and Q we write them down to the table
(rpm) h 1 (cm) h 2 (cm) (m) (Pa) gauge (Pressure gauge -
4 Determine the volumetric flow rate in three experiments by using venturi nozzle
First, we will find the value ∆h = h 4 − h 3 then we will use the formula below to find
Then, we find the difference of pressure measured by pressure gauge and measuring tube by applying following formula:
We calculate the value of C in the venturi nozzle by using the formula (5):
Finally, we find the value of by using the formula Q (1):
Record the value of C and Q to the table below
(rpm) h 3 (cm) h 4 (cm) (m) (Pa) gauge (Pressure gauge -
5 Compare the computed results between the using of orifice plate and venturi nozzle, give the conclusions
There is always a relative difference in the results between the using of orifice plat and venturi nozzle
When calculating the Q value, using venturi nozzle is better It is because:
The diverging section of a venturi nozzle plays a crucial role in recovering lost pressure that occurs during the measurement of the average velocity or flow rate of the fluid.
The orifice plate experiences greater pressure loss than a venturi meter due to its lack of dedicated components for pressure recovery As the flow passes through the orifice plate, it abruptly expands, causing disturbances in the flow and resulting in unstable measurements.
In conclusion, venturi nozzle is quite more accurate than the orifice plate because it has lower relative errors in its calculation, making it a more secure and reliable measurement instrument
EXPERIEMENT 5A: FRICTION LOSS IN PIPE
- To investigate the variation of friction head along a circular pipe with the mean flow velocity in the pipe
- To investigate the friction factor against Reynolds number and roughness
A centrifugal pump supplies water through a pipe with a diameter of 10.64 cm, which is connected to four test sections spaced 3 meters apart and linked to a bank of pressurized manometer tubes The water flows from the pipe into a concrete channel, where a vee-notch is installed at the end to measure the flow rate, which corresponds to the flow rate in the pipe The water level above the vee-notch is measured using a point gauge vernier mounted on a small tank that is open to the channel The flow rate over the vee-notch is calculated using a specific formula.
Where z is water levelin channel, ZCR is the elevation of the crest of Veenotch, ZCR = 27.8 cm
The flow rate over the Vee-notch is controlled by a pump's valve, while an ammeter in the electric box indicates the motor's current intensity, which correlates with the flow rate in the pipe Pressure differences between the test sections are determined by measuring the water levels in the manometer tubes.
Considering flow at two sections i,j in a pipe, Bernouilli’s equation may be written as:
STEPS OF EXPERIMENT
Before conducting tests, ensure that valves (2) and (4) and locks (9) are securely closed Additionally, lightly turn the spindle of the pump and motor to verify that they move freely without resistance.
To operate the electric box, press the POWER ON and RUN/STOP buttons, then activate the pump by pressing the ON button Fully open valve (4) by turning it counterclockwise until it cannot be turned further It is crucial to ensure that valve (4) is completely open; otherwise, opening valve (2) may cause the manometer tubes to break due to high pressure Gradually open valve (2) and monitor the current intensity until it reaches the desired test value, then proceed to open clocks (9).
When preparing to shut down, it is crucial to first close valve (2) before pressing the OFF button on the RUN/STOP and the power OFF button Immediately after, close valve (4) to retain water in the pile.
Open the locks (9) of the manometer tubes at the section (1) and (2)
To adjust the flow rate levels, modify valve (2) to correspond with the current settings of 21A < I < 26A The first flow rate aligns with I = 25A, the second with I = 23.5A, and the third with I = 22A.
Wait for the water level in the channel to stabilize (the water level in point gauge
(5) is constant), read the following values:
- Water level in manometer tube (1)
- Water level in manometer tube (2)
- Elevation Z at point gauge and vernier (5)
The measurement results are recorded in Table 1 of the report b- The second measurement:
• Adjust the valve (2) to change the five discharge levels corresponding to the current I = 21.5A to 19.5A
• Continue to open the locks of manometer tubes (3), (4)
• For each discharge level, wait for the water level in the channel to stabilize, taking the following measurements:
- Reading water level from the manometer tub (1) to the manometer tube (4)
- Elevation Z at the point gauge (5) The results are recorded in Table 1 of the report
MEASUREMENTS
REPORT
1 Select any three levels of flow rate at the second measurement (I