FUNDAMENTAL THEORY
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
With u, V alternately are point velocity and average velocity above wet cross-section with area A
In fluid dynamics, when analyzing the movement of layers with an internal current, the parameter 𝛼 is set to 2, while during tangled motion, it ranges from 1.05 to 1.15 To achieve greater accuracy, it is essential to understand the distribution rule of the velocity u in section A and apply formula (3.1) accordingly Typically, for fluctuating flow scenarios, a value of 𝛼 equal to 1 is utilized to simplify calculations.
So at section (i) the energy of a liquid weight is equal to:
𝐻𝑖 = 𝑎 𝑖 2𝑔 𝑉 𝑖 2 + 𝑝 𝛾 𝑖 + 𝑧 𝑖 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 defined as the sum of its potential energy, represented by z + p/γ, and its kinetic energy, expressed as αV²/2g This combination reflects the overall energy present in the flow of water.
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 a valve located at the channel's end Subsequently, the water flows into the tank through a 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
To determine the section's location in the canal, assess its position on a scale from 1 to 6, moving from upstream to downstream, with the midpoint located 40 cm along the canal The section's position is illustrated in the accompanying image, and the distance between sections is noted.
𝐿1−2= 𝐿2−3 = 𝐿4−5= 𝐿5−6= 20𝑐𝑚; 𝐿3−4= 3.7𝑐𝑚 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 stable flow, use a needle (7) to measure the height of the free surface (zi) from section 1 to section 6, and record the measurements in Table 1 Additionally, maintain the flow and adjust the upstream water level using a valve.
Ensure that the downstream flow is lower than the upstream, 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:
Q = 0.25 L/s 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:
To conduct the experiment, select a standard plane at the bottom of the canal, which is defined as a horizontal plane For the initial condition, set zi equal to hi before the step, and adjust zi to hi plus a after the step, as indicated in equation (3.8) Utilize Vi from equation (3.4) and z from equation (3.7) to substitute into equation (3.8).
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 hi, which results from a third-order equation We can solve this equation using the trial-and-error method to find the value of hi.
To determine the accurate value of \( h_i \) using the trial-and-error method, start by substituting an initial value of \( h_i \) (either larger or smaller than the measured \( h \)) into equations (3.9) or (3.10) If the result on the right side exceeds the left side, reduce the value of \( h_i \) and recalculate Continue this process of adjustment and comparison until the results on both sides of the equation are equal, at which point \( h_i \) can be confirmed.
5 Draw the water surface line in the cannel
In the diagram illustrating the canal, the water surface line and the bottom of the canal are depicted, allowing for calculations based on equation (3.10) and the variable hi The findings indicate that the water level in the downstream section is higher than that in the upstream section, highlighting the flow dynamics within the canal system.
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 followed by an increase in section 4, attributed to the variations in wetted area between sections 2-3 and 4-5 Additionally, the ideal graph indicates a significantly greater difference in section 4-5 compared to the measured graph, highlighting 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 primarily occurs due to friction forces, which are influenced by the flow velocity and the distance between the two sections As a result, the head losses hf2-3 and hf4-5 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 h
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 (∆p) from the inlet to the throat is measured using a manometer filled with liquid of density (ρ1) This pressure difference can be represented in terms of the manometric head differential (∆h).
The expandability factor (ε) is a crucial component detailed in the code, enabling the analysis of density changes in gas flows, particularly during significant pressure reductions For liquid flows and gas flows experiencing moderate pressure variations at the meter, the expandability factor is set to ε = 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: h
• 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 a specific equation but does not account for frictional losses In real-world applications, friction losses and other nonideal factors are inevitable, 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
Additionally, it is essential to incorporate the factor of expandability, as outlined in the code, which accounts for density changes in gas flows during significant pressure reductions In the case of liquid flows and gas flows with moderate pressure variations at the meter, the expandability factor is approximately equal to 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
Z M − Z N = −∆h 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) h
+ 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 Velocity Tube
(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: h
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, which can be impacted by the measurement of average velocity or flow rate of the fluid.
The orifice plate experiences greater pressure loss compared to a venturi meter due to its lack of components designed for pressure recovery As the flow passes through the orifice plate, it abruptly expands, causing disturbances in the flow and resulting in unstable outcomes.
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 h
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 equipped with four test sections spaced 3 meters apart, connected 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 recorded using a point gauge vernier mounted on a small tank that is open to the channel The flow rate over the vee-notch is then 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 displays the motor's current intensity, which correlates with the flow rate in the pipe Additionally, the pressure difference between the test sections is measured by observing the water level in the manometer tubes.
Considering flow at two sections i,j in a pipe, Bernouilli’s equation may be written as: h
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 check for any resistance; smooth movement indicates proper functioning.
To operate the electric box, press the POWER ON and RUN/STOP buttons, followed by the ON button to start the pump Ensure that valve (4) is fully opened by turning it counterclockwise until it stops moving It is crucial to fully open valve (4) to prevent damage to the manometer tubes from high pressure when opening valve (2) 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, first close valve (2) and then press 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 between 21A and 26A The first flow rate is set for a current of 25A, the second for 23.5A, and the third for 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: h
• 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
The water temperature t o H2O = 30 C o The water kinematic viscosity v = 0.8x10 -6 m 2 /s
MEASUREMENTS
REPORT
1 Select any three levels of flow rate at the second measurement (I