Pressure Transducer Types and Principles PDF

The inclined tube manometer is also used to measure small changes in pressure. It is essentially a U-tube manometer with one leg inclined at an angle Ө, typically from 10 to 30 degrees relative to the horizontal. *B e:A force acting in a direction parallel to a surface or to a planar cr...

The inclined tube manometer is also used to measure small changes in pressure. It is essentially a U-tube manometer with one leg inclined at an angle Ө, typically from 10 to 30 degrees relative to the horizontal. *B e:A force acting in a direction parallel to a surface or to a planar cross section of a body, as for example the pressure of air along the front of an airplane wing. Shear forces often result in shear strain. Resistance to such forces in a fluid is linked to its viscosity. Also called shearing force *se: When an object is placed in a liquid environment, its buoyancy is equivalent to the weight of the liquid it displaces. This is why some low-density objects float in water, for instance, and why other, more dense, items may still sink but will be easier to lift or move while below the surface General categories for pressure transducers are absolute, gauge, vacuum, and differential. These categories reflect the application and reference pressure used. - Absolute transducers have a sealed reference cavity held at a pressure of absolute zero, enabling absolute pressure measurements/signals. - Gauge transducers have the reference cavity open to atmospheric pressure and are intended to measure above or below atmospheric pressure or both. - Differential transducers measure the difference between two applied pressures. Vacuum transducers are a special form of absolute transducer for low-pressure measurements. Pressure transducers are subject to some or all of the following elemental errors: - Resolution - zero shift error - linearity error - sensitivity error - hysteresis - noise -drift due to environmental temperature changes. *Electrical transducers are also subject to loading error between the transducer output and its indicating device *Loading errors increase the transducer nonlinearity Bourdon Tube The Bourdon tube is a curved metal tube having an elliptical cross section that mechanically deforms under pressure. In practice, one end of the tube is held fixed and the input pressure applied internally. Operation: A pressure difference between the outside of the tube and the inside of the tube brings about tube deformation and a deflection of the tube free end. This action of the tube under pressure can be likened to the action of a deflated balloon that is subsequently inflated. The magnitude of the deflection of the tube end is proportional to the magnitude of the pressure difference. Several variations exist, such as the C shape (Fig. 9.9), the spiral, and the twisted tube. The exterior of the tube is usually open to atmosphere (hence, the origin of the term ‘‘gauge’’ pressure referring to pressure referenced to atmospheric pressure), but in some variations the tube may be placed within a sealed housing and the tube exterior exposed to some other reference pressure, allowing for absolute and for differential designs. The Bourdon tube mechanical dial gauge is a commonly used pressure transducer. A typical design is shown in Figure 9.10, in which the secondary element is a mechanical linkage that converts the tube displacement into a rotation of a pointer. Designs exist for low or high pressures, including vacuum pressures, and selections span a wide choice in range. The best Bourdon tube gauges have instrument uncertainties as low as 0.1% of the full-scale deflection of the gauge, with values of 0.5% to 2% more common. But the attractiveness of this device is that it is simple, portable, and robust, lasting for years of use. Bellows and Capsule Elements A bellows sensing element is a thin-walled, flexible metal tube formed into deep convolutions and sealed at one end (Fig. 9.9). One end is held fixed and pressure is applied internally. A difference between the internal and external pressures causes the bellows to change in length.  Construction & operation: The bellows is housed within a chamber that can be sealed and evacuated for absolute measurements, vented through a reference pressure port for differential measurements, or opened to atmosphere for gauge pressure measurements. - A similar design, the capsule sensing element, is also a thin-walled, flexible metal tube whose length changes with pressure, but its shape tends to be wider in diameter and shorter in length (Fig. 9.9). - A mechanical linkage is used to convert the translational displacement of the bellows or capsule sensors into a measurable form. A common transducer is the sliding arm potentiometer (voltagedivider) found in the potentiometric pressure transducer shown in Figure 9.11, or connected to indicating needle. - Another type uses a linear variable displacement transducer (LVDT; see Chapter 12) to measure the bellows or capsule displacement. The LVDT design has a high sensitivity and is commonly found in pressure transducers rated for low pressures and for small pressure ranges, such as zero to several hundred mm Hg absolute, gauge, or differential.  Rise time (tr) is the time required to reach at final value by a under damped time response signal during its first cycle of oscillation.  Pressure measurement 1- Hydrostatic Pressure in a Stationary Fluid: In a fluid that is at rest (stationary), the only pressure present is hydrostatic pressure.  Hydrostatic pressure is the pressure exerted by the fluid at a given point, and it acts equally in all directions. Pressure in a Moving Fluid: If the fluid is set in motion, the pressure still exists, but its magnitude may change. Even when the fluid is moving, hydrostatic pressure continues to act, though its distribution might differ depending on the fluid's motion. Effect on a Thin Disc in the Fluid: Consider a very thin disc placed in the fluid, moving along with it. The static pressure ( Ps) will act on both faces of the disc. The illustration shows that regardless of the fluid's motion, the pressure acts equally on both sides of the thin disc. A stationary disc in a fluid is rotated. 1. Rotation of the Disc:  If the stationary disc is turned by 90°, its orientation changes so that its front face is now perpendicular to the flow direction. 2. Pressure Components: Static Pressure (Ps): The disc still experiences static pressure on its front face, similar to the case when it was aligned with the fluid flow. Dynamic Pressure (Pd): In addition to static pressure, the disc also experiences an additional pressure due to the fluid’s motion, known as dynamic pressure. Dynamic pressure is a result of the fluid's velocity and is given by the formula: 9.7 pressure measurements in moving fluids Pressure measurements in moving fluids, specifically focusing on the flow over a bluff body (a body with a blunt shape). Flow Characteristics:  The upstream flow (before the bluff body) is assumed to be uniform and steady, with negligible energy losses. This means the fluid’s velocity and pressure remain consistent before interacting with the body. Streamline A: As the fluid moves along streamline A, it approaches the bluff body with an initial velocity U1 at point 1. As the flow reaches point 2, directly in front of the body, the fluid slows down and eventually stops. Point 2 is known as the stagnation point because the velocity U2 at this point becomes zero. Above streamline A, the flow moves over the top of the bluff body, while below streamline A, it moves under the body. Streamline B: Along streamline B, which is deflected around the body, the velocity at point 3 is denoted as U3. Due to the conservation of mass and the shape of the body, the velocity U4 at point 4 is greater than U3 because the fluid is constricted and has to move faster around the body. Conservation of Energy: The energy conservation between points 1 and 2 and between points 3 and 4 is expressed using Bernoulli’s equation  Stagnation Point: At this point, U2=0 (velocity is zero). Stagnation Pressure: Here, P2 > P1 by the dynamic pressure, Dynamic pressure represents the kinetic energy per unit mass of the flow. Energy Conservation: If there are no energy losses (e.g., no heat transfer), the kinetic energy converts entirely into pressure at P2. Total Pressure: Known as stagnation or total pressure (Pt). It can be measured by stopping the flow isentropically (without entropy change). Static Pressures: Pressures at points 1, 3, and 4 are static pressures. This is the pressure a fluid particle feels while moving with the flow. Freestream Static Pressure and Velocity: Static pressure and velocity at points 1 and 3 are called freestream static pressure and freestream velocity, respectively.  Pressure Relationship: Since U4 > U3 , P4 < P3. Total pressure at any point is the sum of static pressure and dynamic pressure. Measuring total pressure: Impact Probe: Measures total pressure by aligning a small hole with the flow so that the flow slows to rest at the hole. Pressure Transfer: The pressure sensed at the hole is sent to a pressure-sensing device (like a manometer or transducer). Alignment Sensitivity: o Probes in Figure 9.18a and 9.18b: Slight misalignment (up to ±7 degrees) has a minimal effect (within 1% error). o Kiel Probe (Figure 9.18c): Uses a shroud to guide the flow to the impact port, reducing sensitivity to misalignment up to ±40 degrees static pressure measurement: Measurement Method: o Wall Taps: Small, burr-free holes drilled perpendicular to the flow in ducted flows. ▪ Diameter: 1% to 10% of the pipe diameter (smaller preferred).  ▪ A hose or tube connects the tap to a pressure gauge or transducer. ▪ The tap must be perpendicular and free from burrs. o Static Pressure Probe: Inserted into the flow to measure local static pressure. ▪ Design: Streamlined to minimize flow disturbance. ▪ Size: Frontal area should be ≤ 5% of the pipe flow area. ▪ Port Location: Should be well downstream of the probe’s leading edge to allow for flow realignment. Prandtl Tube: o Design: Includes eight holes around the probe, positioned 8-16 probe diameters downstream and 16 probe diameters upstream of the support stem. ▪ Purpose: Minimize static pressure error caused by the probe’s leading edge and stem. ▪ Pressure Measurement: A transducer or manometer is connected to the probe stem. o Correction Formula: p=C0 pi , where 0.99 < C0 < 0.995 and pi is the indicated pressure. Where: Ci: The amplitude of the i-th sine wave component, representing the magnitude of the fluctuating (dynamic) component. ωi : The angular frequency of the i-th sine wave component, indicating how fast the component oscillates. ϕi : The phase angle of the i-th sine wave component, determining the initial angle (offset) at time t=0 e.g:  Pitot-static pressure probe: Used to measure dynamic pressure, similar in appearance to an improved Prandtl static pressure probe. Structure: Contains an interior pressure tube attached to an impact port at the leading edge. Has two internal cavities/holes: one exposed to total pressure and the other to static pressure. These pressures are measured using a differential pressure transducer to indicate pv (dynamic pressure). Misalignment Sensitivity: Insensitive to misalignment within a yaw angle range of ±15 degrees. Can be rotated to align with mean flow direction by maximizing the signal. Lower Velocity Limit: Viscous effects in the pressure port entry regions limit its use at low velocities. Viscous effects are not an issue if the Reynolds number based on probe radius Rer=Ur/ ν is greater than 500 (where ν is the kinematic viscosity). Corrections for Low Rer : For 10 < Rer < 500, correct dynamic pressure using pv=Cv pi, where Cv=1 + 4/Rer. At Rer10, the uncertainty in measured dynamic pressure can be as high as 40%. At Rer ≥ 500, uncertainty decreases to about 1%  ✓ Misalignment in the context of a pitot-static pressure probe refers to the angular difference between the probe's orientation and the actual flow direction of the fluid. Ideally, the probe should be perfectly aligned with the flow to measure accurate pressure values. However, if the probe is tilted or rotated away from the flow direction, it is considered misaligned. ✓ Ya angle: The yaw angle indicates the horizontal deviation of the probe from the flow direction. o A yaw angle of 0° means the probe is perfectly aligned with the flow direction. o A positive or negative yaw angle indicates that the probe is rotated to the left or right of the flow direction. Compressibility Effects: In high-speed gas flows, compressibility near the probe's leading edge affects the pitot-static pressure probe's governing equations. For a perfect gas, energy balance along a streamline can be written as:  where U is velocity, cp is specific heat at constant pressure, Tt is total temperature, and Tx is temperature at point x. Isentropic Process: The relationship between temperature and pressure is: Mach Number: Mach number M relates local velocity to the speed of sound: M= U / a, where is the speed of sound, and R is the gas constant and for pressure Relationship in Compressible Flow: For compressible flows, the relationship between total pressure pt and static pressure px at any point x is: Low Mach Number Limit: Above Equation simplifies to a basic dynamic pressure equation when M (Mach number) is less than 1. For M ≤ 0.3M, the flow is considered incompressible. Errors in dynamic pressure estimates become significant when M > 0.3. For supersonic flow (M >1), local velocity must be calculated using the Rayleigh relation.  Thermal Anemometry Thermal anemometry Measures fluid velocity by relating heat transfer from a sensor to fluid velocity. Working Principle: 1- Heat transfer Q from a warm sensor to a cooler fluid is proportional to the temperature difference and thermal conductance hA. 2- Thermal conductance increases with fluid velocity, forming the basis of thermal anemometry. Sensor: - Typically a metallic RTD element in a Wheatstone bridge circuit. - Heated by passing a current through it. - Sensor temperature Ts inferred from resistance Rs using Rs=R0[1+α(Ts−T0)]. - Relationship between heat transfer and velocity: Q˙=I2R=A+BUn Where A, B, and n are calibration constants. Types of Sensors: Hot-wire: Tungsten/platinum wire used in nonconducting fluids. Hot-film: Thin film of platinum/gold used in conducting or nonconducting fluids. Operating Modes: Constant Current: Fixed current; resistance varies with cooling velocity. Constant Resistance: Resistance is maintained constant; velocity inferred from power required to maintain temperature.  Velocity measurement Advantages: High sensitivity at low velocities. Multiple sensors measure different velocity components. Can measure dynamic velocities due to high-frequency response. Frequency Limits: Upper limit set by Strouhal frequency f≈0.22U/d Lower Velocity Limit: Given by Red ≥ Gr1/3, ensuring inertial forces dominate over buoyant forces. Lower limit for hot-wire sensors in air is ~0.6 m/s. Flow of velocity can be determined by two types: - Constant current (CC) - Constant resistance (CR)/ constant temperature fs: scattered light at a frequency; fi : the frequency of the incident laser beam; fD: the Doppler shift frequency.  Conclusion: Doppler Anemometry (LDA) 1. What is the Doppler Effect: o Frequency of waves (sound/light) shifts based on the relative motion between the source and observer. Higher frequency if approaching, lower if receding. o Used in astrophysics to measure velocities via frequency shifts (e.g., redshift, blueshift). What does the DA measure: o Measures local fluid velocity using the Doppler effect. o By using emission source (laser) and observer are stationary; small particles in the fluid cause the Doppler shift. Laser Doppler Anemometer (LDA):  o Uses a laser beam as the source. o Measures time-dependent velocity at a specific point in the flow. o A particle passing through the laser beam scatters light; the observed frequency shift is the Doppler shift. Dual-Beam Mode: o The laser beam is split into two coherent beams focused on a point. o The mixing of beams (optical heterodyne) separates the Doppler frequency from the incident frequency. o The velocity is directly related to the Doppler shift. Advantages of LDA: o Measures velocity without probe blockage. o Effective in environments with density/temperature fluctuations or where physical sensors are unsuitable. o High accuracy with good signal-to-noise ratio (SNR) and particle seeding.  o Conclusion: o Particle Image Velocimetry (PIV) Key Points: Purpose: o Measures full-field instantaneous velocities in a planar cross-section of fluid flow by tracking particle displacement over time. Components: o Coherent light source (laser beam), optics, CCD-camera, and signal interrogation software. Basic Operation: o Particles in the flow are illuminated by short, repetitive laser flashes. o The distance particles travel between flashes indicates their velocity. Typical Setup: o A pulsed laser beam is shaped into a 2-D sheet using a cylindrical lens. o The laser illuminates the flow field, and the camera records the illuminated field. o Laser flashes and camera shutter are synchronized for precise image capture. Data Processing: o Each camera flash produces an image frame, which is divided into small interrogation areas. o Cross-correlation between corresponding areas in two consecutive images determines particle displacement. o Velocity vector maps of the flow field are created by repeating this process across the entire area. Applications: o Used in both gases and liquids. o 3D velocity information can be obtained using two cameras. o Resolution and measurable flow speed depend on several factors, including l aser flash width, camera settings, and image magnification. Limitations: o Maximum flow speed measurable is limited by the size of the interrogation area. o Errors can occur if particles are too small relative to the camera's pixel size. Main comparison between the methods: Method Best Suited For Advantages Disadvantages Pitot-Static Pressure - Fluids of constant Simple and inexpensive - Misalignment errors Methods density - No calibration required - Susceptible to - Mean velocity - Minimal blockage in particulate blockage measurement large ducts Thermal Anemometer - Clean fluids - High resolution - Complex interpretation - Constant temperature - Less fragile in hot-film in dynamic flows and density form - Directionally - Dynamic velocity - Inexpensive industrial ambiguous measurement systems Laser Doppler - Hostile, combusting, or - No probe blockage - Expensive Anemometer (LDA) dynamic flows - High frequency - Requires optical access - Point velocity response and scattering particles measurement - Good temporal resolution Particle Image - Full-field velocity - Provides instantaneous - Expensive Velocimetry (PIV) measurement flow snapshot - Requires optical access - Hostile and combusting - Excellent for and scattering particles flows visualizing flow - Limited by camera structures frame rate 4o  Conclusion: Mass Flow Rate: Depends on fluid density, average velocity, and conduit cross-sectional area. For a circular pipe, the mass flow rate is given by: eq.10.1 The area of the pipe is A=π. r2 To measure mass flow rate directly, the device must be sensitive to the area-averaged mass flux per unit volume (ρU) or to the fluid mass passing through per unit time.  Units: kg/s, lbm/s, etc. Volume Flow Rate: Depends on the area-averaged velocity over a cross section of flow: eq.10.2 To measure volume flow rate, the device must be sensitive to the average velocity (U) or to the fluid volume passing through per unit time. Units: m³/s, ft³/s, etc. Difference Between Mass and Volume Flow Rate: Mass flow rate requires sensitivity to the product of density and velocity, or to mass rate. Volume flow rate requires sensitivity only to average velocity, or to volume rate. When density is constant, mass flow rate can be inferred by multiplying the measured volume flow rate by the density. In cases where density changes or isn't well-known, assumptions may not be accurate enough for precise measurements. Flow Character: Flow in a pipe or duct can be laminar, turbulent, or transitional. Characterized by the Reynolds number (Re): eq.10.3 Below 2000: The flow is generally considered laminar. Between 2000 and 4000: The flow is in the transitional phase, where it can fluctuate between laminar and turbulent. Above 4000: The flow is generally considered turbulent. For non-circular conduits, the hydraulic diameter is used in place of diameter d1 Note : For a conduit with a non-circular cross-section, the hydraulic diameter Dh is given by: Dh = (4×Flow Area) / Wetted Perimeter Where: Flow Area: The cross-sectional area through which the fluid flows. Wetted Perimeter: The perimeter of the conduit that is in contact with the fluid. Example: For a rectangular duct with width W and height H: Flow Area A=W×H Wetted Perimeter P=2×(W+H) So, the hydraulic diameter Dh is: Dh=4×(W×H) / 2×(W+H) Conclusion: The above section describes how to determine the volume flow rate in a conduit by measuring the velocity at multiple points across its cross-section. The key idea is to accurately estimate the velocity profile to calculate the volume flow rate using the following procedure: Conclusion: The volume flow rate Q can be determined using the area-averaged velocity across the cross-section, as given by: Q=∫A. u. dA= =U. A Where: u is the velocity at a specific point in the cross-section. A is the cross-sectional area. U is the average velocity. Procedure for Velocity Measurement: 1. Velocity Profile Measurement: o Measure the velocity at multiple discrete positions n along a cross-section of the conduit. This process may be repeated at m circumferential locations around the cross-section, spaced 360∘/m degrees apart. 2. Cross-Sectional Traverses: o Use a velocity probe to traverse along each cross-section, taking velocity readings at each designated position. The positions are typically chosen to either evenly cover the area or according to engineering standards. 3. Area Division: o The flow area can be divided into smaller equal areas. Velocity measurements are taken at the centroid of each smaller area, with the measured velocity assigned to that entire area. 4. Average Flow Rate Calculation: o For each cross-section, the average velocity is calculated using the measured velocities. The flow rate is then calculated by multiplying the average velocity by the cross-sectional area..Application: This method is often used for one-time verification or calibration of flow rates in systems like ventilation setups, where continuous monitoring is not required ressure differential meters are devices used to measure flow rates by analyzing the pressure drop between two points along the flow path. Here's a breakdown of their operating principle and characteristics: Conclusion: Pressure Drop and Flow Rate Relationship: The flow rate Q through a conduit is related to the pressure drop Δp=p1−p2 between two points along the flow path. Q ∝ (Δp)n Where: n=1 for laminar flow between the pressure measurement points. n=1/2 for fully turbulent flow. Area Reduction and Pressure Drop: A deliberate reduction in flow area between the pressure measurement points (locations 1 and 2) creates a measurable pressure drop Δp. This area reduction causes an increase in flow velocity due to the conservation of mass, leading to a decrease in pressure, commonly known as the Bernoulli effect. Additionally, flow energy losses contribute to the overall pressure drop. Obstruction meters are devices used to measure fluid flow rates by creating a pressure drop across a flow obstruction. Common Types of Obstruction Meters 1. Orifice Plate: o A flat plate with a central hole, placed perpendicular to the flow. o Causes a significant pressure drop and is the most widely used type of obstruction meter. 2. Venturi Meter: o Consists of a converging section that narrows to a throat, followed by a diverging section. o Minimizes energy loss due to gradual area changes, resulting in lower pressure drop compared to the orifice plate. 3. Flow Nozzle: o Similar to the venturi meter but with a shorter throat and no diverging section. o Provides a middle ground between the orifice plate and venturi in terms of pressure drop and accuracy. Energy Balance: The operating principle of obstruction meters relies on the relationship between flow rate and pressure drop. Using the Bernoulli equation under assumptions like steady, incompressible, and one-dimensional flow, the energy balance between two points along the flow path gives:  Incompressible Flow Rate: Substituting the continuity equation into the energy balance and rearranging provides the volume flow rate for incompressible flow: Where: Cf: Friction coefficient Cc: Contraction coefficient A0: Throat area Discharge Coefficient: For practical use, the discharge coefficient C is introduced to account for non-idealities: Where: E: Velocity of approach factor K0: Flow coefficient Note that  Compressibility Effects Adiabatic Expansion Factor (Ɣ): For compressible flows, the adiabatic expansion factor Ɣ is introduced to account for compressibility: An orifice meter is a common flow measurement device that consists of a thin, flat plate with a central hole (orifice) installed perpendicular to the flow within a pipe. Orifice Plate: The orifice meter uses a circular plate with a hole in the center, smaller than the pipe diameter. The orifice plate is typically positioned concentrically within the pipe, creating a flow restriction.  Installation: The plate is housed between two flanges in the pipe, which simplifies installation and allows for easy interchangeability of plates with different orifice diameters (d0). This flexibility enables adjustments to be made by swapping out plates with different β ratios (the ratio of orifice diameter to pipe diameter). Flow Area Reduction: The orifice causes a reduction in flow area, which in turn causes a pressure drop across the plate due to increased velocity at the orifice (as per Bernoulli's principle). Flow Coefficient: The discharge coefficient C is used in the flow rate equation to account for real-world factors such as friction and turbulence. The flow coefficient K0 is defined as: K0= C.E Where E is the velocity of approach factor, accounting for the change in velocity as the fluid approaches the orifice. Volume Flow Rate: The volume flow rate for an incompressible fluid is given by: QI=K0.A.(2.Δp/ρ)1/2 Here: A0 is the area of the orifice. Δp is the pressure drop across the orifice. ρ is the fluid density. Pressure Taps: The placement of pressure taps is crucial for accurate measurements. Standard locations include: o Flange taps: 1 inch upstream and 1 inch downstream from the orifice. o d and d/2 taps: 1 pipe diameter upstream and half a diameter downstream. o Vena contracta taps: Located at the point where the fluid velocity is highest (throat ) and pressure is lowest downstream of the orifice. Uncertainties and Accuracy Discharge Coefficient Uncertainty: The relative systematic uncertainty in the discharge coefficient is typically around 0.6% for β values between 0.2 and 0.6, and increases for higher β values. Expansion Factor Uncertainty: The systematic uncertainty in the expansion factor, Ɣ, is about 4×(Δp/p1) % of Ɣ. Overall Uncertainty: The overall uncertainty in flow rate estimation using an orifice meter is generally between 1% and 3%, depending on the β ratio and Reynolds number. Higher β values tend to provide more accurate measurements. https://www.youtube.com/watch?v=o5_5D6Iqe60& ab_channel=TechnicalPiping The venturi meter is a type of flow measurement device that uses a smooth converging section followed by a narrow throat and a diverging section to measure the flow rate of a fluid in a pipeline. Here's a breakdown of its components, operation, and key characteristics: Structure : 1. Converging Section: o The venturi meter has a conical contraction section with an angle of approximately 21 degrees (±1 degree). o This section gradually narrows the pipe's diameter, leading the fluid into the throat of the meter. 2. Throat: o The narrowest part of the venturi meter where the fluid velocity is the highest and pressure is the lowest.  o Pressure taps are located just before the upstream contraction and at the throat to measure the pressure differential. 3. Diverging Section: o After the throat, the meter expands with either a 15-degree or 7-degree divergent angle. o This section gradually restores the pipe to its original diameter, allowing the pressure to recover partially. Key Characteristics  Discharge Coefficient: The discharge coefficient C for venturi meters is quite stable across different flow conditions. It varies slightly depending on whether the meter is cast or precision-machined: o For cast units, C≈0.984 with a systematic uncertainty of 0.7%. o For machined units, C≈0.995 with a systematic uncertainty of 1%. o For pipe diameters larger than 7.6 cm (3 inches) o coefficient is reliable within the Reynolds number range of 2×105 ≤ Red1 ≤ 2×106, 0:4 ≤ β ≤ 0:75 Expansion Factor Ɣ: The expansion factor, which accounts for compressibility effects, has a systematic uncertainty of approximately where β is the (d0/d1). Operating Range: Flow Nozzle Flow nozzles are devices used to measure the flow rate of fluids in pipelines. They are similar to venturi meters but are more compact and less expensive. Key Characteristics: 1. Design and Structure: o A flow nozzle consists of a gradual contraction from the pipe's inside diameter to a narrower throat, which is less abrupt than in an orifice plate but not as smooth as in a venturi meter. 2. Pressure taps are often located either: 1. One pipe diameter upstream of the nozzle inlet and at the nozzle throat (using wall or throat taps).  2. At one pipe diameter upstream and one-half diameter downstream of the upstream nozzle face (using d and d/2 wall taps). Performance and Uncertainties: 1. Discharge Coefficient: o The discharge coefficient C for flow nozzles typically has a systematic uncertainty of about 2% at a 95% confidence level. 2. Expansion Factor: o The systematic uncertainty in the expansion factor Ɣ is about 3. Pressure Loss: o The permanent pressure loss for a flow nozzle is greater than that of a venturi meter but significantly less than that of an orifice plate for the same pressure drop. Sonic Nozzles Sonic nozzles are specialized flow meters used primarily for compressible gases. They operate based on the principle that when the gas flow rate through the nozzle reaches a critical condition, the gas velocity at the throat equals the speed of sound, leading to "choked" flow. Key Points: 1. Choked Flow Condition: o At choked flow, the mass flow rate through the nozzle throat is maximized and cannot be increased further, even with a higher pressure drop across the nozzle. o The critical pressure ratio at which choked flow occurs is given by (p0p1)critical=(2k+1)kk−1\left(\frac{p_0}{p_1}\right)_{\text{critical}} = \left(\frac{2}{k+1}\right)^{\frac{k}{k-1}}(p1p0)critical=(k+12)k−1k, where p0p_0p0 is the throat pressure, p1p_1p1 is the upstream pressure, and kkk is the specific heat ratio. 2. Mass Flow Rate: o The maximum mass flow rate at the throat under choked conditions for an ideal gas is expressed as: m˙max= o The actual flow rate differs from the ideal by no more than 3%, and when calibrations are not available, a discharge coefficient C=0.98±2%C = 0.98 \pm 2\%C=0.98±2% is used. 3. Applications: o Sonic nozzles are valuable for metering and regulating gas flow, especially as local calibration standards due to their precise flow control capability. o They are suitable for various applications but require accommodation for large pressure drops and system pressure losses. Sonic Nozzles Sonic nozzles are specialized flow meters used primarily for compressible gases. They operate based on the principle that when the gas flow rate through the nozzle reaches a critical condition, the gas velocity at the throat equals the speed of sound, leading to "choked" flow. Key Points: Choked Flow Condition: o At choked flow, the mass flow rate through the nozzle throat is maximized and cannot be increased further, even with a higher pressure drop across the nozzle. o The critical pressure ratio at which choked flow occurs is given by Where, p0 is the throat pressure, p1 is the upstream pressure, the Ɣ is the specific heat ratio. Mass Flow Rate:  The maximum mass flow rate at the throat under choked conditions for an ideal gas is expressed as: The actual flow rate differs from the ideal by no more than 3%, and when calibrations are not available, a discharge coefficient C=0.98±2% is used. Applications: o Sonic nozzles are valuable for metering and regulating gas flow, especially as local calibration standards due to their precise flow control capability. o They are suitable for various applications but require accommodation for large pressure drops and system pressure losses. Flow Meter Placement Proper placement of flow meters in a piping system is crucial to ensure accurate measurements. The following points highlight the importance and considerations for placement: 1. Straight Pipe Lengths: o Upstream and Downstream: Providing sufficient straight pipe lengths before and after the flow meter is essential for allowing the flow to develop properly. This ensures that any swirl or turbulence introduced by upstream disturbances (e.g., bends, valves, or fittings) dissipates and the velocity profile becomes symmetric. o Flow Development: Proper flow development leads to accurate pressure readings and minimizes systematic errors in the measurement. 2. Downstream Pressure Recovery: o Sufficient straight lengths downstream of the flow meter are also important to allow for pressure recovery. This helps reduce potential errors caused by pressure fluctuations and ensures that the flow returns to a more stable state after passing through the meter. 3. Installation Effects: o Elbows and Bends: Particular attention must be given to the placement of flow meters relative to elbows and bends in the piping. Out-of-plane double elbow turns are especially challenging, as they can introduce significant disturbances that affect the accuracy of the flow meter.  o Best Practices: Figure 10.13 provides a guideline for the recommended straight lengths and placement to minimize installation effects. However, it's important to note that these are general guidelines, and actual installation effects can vary based on the specific configuration and conditions of the piping system. Key Elements of the Diagram: 1. Configurations and Pipe Layouts: o The left side of the diagram shows various configurations where an orifice or flow nozzle is installed after different types of upstream disturbances (such as bends, elbows, and other fittings). o The right side shows configurations where the flow meter is placed after more complex upstream disturbances like tube turns, long-radius bends, and straightening vanes. 2. Straight Pipe Lengths (A, A', B, C): o A, A', B, C represent the required lengths of straight pipe, expressed as multiples of the pipe diameter, D. These lengths are necessary upstream (A, A') and downstream (B, C) of the flow meter to allow the flow to stabilize. o Upstream lengths (A, A'): Required before the meter to dissipate any disturbances caused by fittings, bends, or other components.  o Downstream lengths (B, C): Required after the meter to allow the flow to recover from any disruptions caused by the meter itself. 3. Diameter Ratio (β): o The X axis represents the diameter ratio β (defined as the ratio of the orifice or nozzle throat diameter to the pipe diameter). This ratio affects the required lengths of straight pipe. o As β increases, the required straight lengths also increase. 4. Curves on the Graph: o Curve A: Represents the required upstream straight length for the most sensitive installation condition, such as elbows or tube turns. o Curve B: Represents the downstream straight length for typical installations. o Curve C: Shows additional required straight lengths when using flow conditioning devices like straightening vanes.  Vortex flow meters measure fluid velocity using a principle of operation referred to as the von Kármán effect, which states that when flow passes by a bluff body, a repeating pattern of swirling vortices is generated. In a Vortex flow meter, an obstruction in the flow path, often referred to as a shedder bar, serves as the bluff body. The shedder bar causes process fluid to separate and form areas of alternating differential pressure known as vortices around the back side of the shedder bar. Relationship between stress and strain, which is fundamental to understanding how materials behave under load. Stress and Strain Relationship: 1. Stress: o Definition: Stress (σa ) is defined as the internal force (FN) per unit area (Ac) acting within a material. Mathematically, it's expressed as: σa=FN. Ac o Context: For a rod with cross-sectional area Ac subjected to a force FN along its axis, the normal stress is the force per unit area perpendicular to Ac. 2. Strain: o Definition: Strain (ϵa) is the measure of deformation representing the displacement between particles in the material body relative to a reference length. It’s calculated as: ϵa=ΔL/L ΔL is the change in length due to the applied load, and L is the original length of the material. o Units: Strain is a dimensionless quantity, often expressed in units of (μϵ), which is 10 −6 times the strain.  3. Stress-Strain Diagram: o Importance: Stress-strain diagrams help visualize the relationship between stress and strain for materials under load. o Behavior of Mild Steel: For materials like mild steel (which is ductile), the stress-strain relationship is linear up to a certain point, known as the elastic limit. Within this elastic region, the material will return to its original shape when the load is removed. 4. Elastic Region and Hooke's Law: o Elastic Region: The range of stress over which the material exhibits a linear stress-strain relationship. o Hooke's Law: Within the elastic region, stress and strain are proportional. This relationship is described by Hooke's Law: σa=Em. ϵa where Em is the modulus of elasticity (Young's modulus) of the material, a measure of the material's stiffness. oApplication: Hooke's Law applies only in the elastic region, meaning that once the stress exceeds this range, the material may not return to its original shape, leading to permanent deformation. 5. Material Behavior Beyond the Elastic Region: o Ductile vs. Brittle: Materials respond differently when the stress exceeds the elastic limit. Ductile materials (like steel) will deform plastically, while brittle materials may fracture without significant deformation. lateral strains in the context of material deformation under axial loads, the concept of Poisson's ratio, and the extension of stress-strain relationships to multidimensional cases. Here's a breakdown: Lateral Strains 1. Lateral Deformation:  When a rod is subjected to an axial load (either tension or compression), not only does its length change, but its cross-sectional area changes as well. o Tension: If the rod is stretched, its length increases, leading to a decrease in the cross-sectional area. o Compression: Conversely, if the rod is compressed, its length decreases, resulting in an increase in the cross-sectional area. 2. Lateral Strain: o Definition: Lateral strain is the ratio of the change in the cross-sectional dimension (e.g., diameter for a circular rod) to the original dimension. o Lateral strain is crucial because it describes how the material deforms perpendicular to the applied load. 3. Poisson’s Ratio: o Definition: Poisson’s ratio (νP ) is a material property that defines the ratio of lateral strain (ϵL) to axial strain (ϵa): o This ratio is a measure of how much a material will contract or expand laterally when stretched or compressed axially. o For most materials, Poisson's ratio is a positive number less than 0.5, indicating that the material will contract laterally when stretched longitudinally. Multidimensional Stress-Strain Relationships 1. Generalization to Two-Dimensional Loading: o Biaxial Stress: When a material is subjected to tensile loads in two perpendicular directions, it experiences normal stresses in both the x and y directions (σx and σy). o Stress-Strain Relations:.  Where, τ is specifically mentioned in the context of a generalized relationship between stress and strain in a biaxial or three-dimensional state of stress. G: This is the shear modulus, or modulus of rigidity, which is a material property that describes how a material deforms under shear stress. It's a measure of the material's stiffness when subjected to shear forces. γxy : This is the shear strain, which represents the angular distortion that occurs in the material due to the applied shear stress. It measures the deformation of the material in response to the shear force Elastic Region (Linear Portion): The initial portion of the curve is linear, indicating that the material behaves elastically. In this region, stress is directly proportional to strain, following Hooke's Law (σ=Eϵ), where E is the modulus of elasticity. If the load is removed while the material is within this region, the material will return to its original shape. Yield Point:  The point where the curve begins to deviate from linearity is known as the yield point. Beyond this point, the material undergoes plastic deformation, meaning it will not return to its original shape when the load is removed. 2. Three-Dimensional Stress-Strain: o Extension to 3D: In real-world applications, materials are often subject to complex loading conditions in three dimensions. The relationship between stress and strain in 3D is more complex but follows similar principles. o Surface Strain Measurements: Strain measurements are typically taken at the surface of an engineering component. These surface strains provide information about the stress state at the surface, which can be used to infer internal stresses.