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Research

Pipe Flowrate Measurement: Problem Solving

A spray tank system is engineered to uniformly distribute a pest-control liquid across plants by using a pressurized mechanism. The tank, pressurized to 150 kPa, holds the pesticide at a height of 0.80 meters. Liquid flows from the tank through a 1.9 meter pipe with a diameter of 0.015 meters, angled at 0.698 radians, ultimately reaching a 0.007 meter nozzle that sprays the pesticide. Accurate calculation of the system&#39;s flow rate is crucial to ensure uniform application, and this is achieved by first determining the flow velocity within the connecting pipe.
The design requires an understanding of fluid mechanics principles. At the outset, the velocity of the fluid in the tank at Point 1 is zero, with the pressure set to 150 kPa. Point 2, located at the nozzle, is assumed to be open to the atmosphere, with a pressure of zero. The height difference between Points 1 and 2, calculated using the 0.698-radian pipe angle and pipe length, determines the elevation change component in the Bernoulli equation. This height difference is found by multiplying the pipe length by the sine of the inclination angle.
To connect the parameters at Points 1 and 2, a modified Bernoulli equation is used, accounting for energy losses due to friction and the nozzle&#39;s loss coefficient. The equation expresses the pressure, velocity, and elevation terms for both points as follows:
<img alt="Equation 1" src="/files/ftp_upload/18085/18085_PT_equation_1.svg" style="text-align: center; height: 30px;" />
Here, p represents pressure, &#961; the fluid density, V the velocity, g the gravitational acceleration, z1 the height of the pesticide inside the tank, and z2&#160; the nozzle height above the bottom of the tank. With values substituted for these terms, the flow velocity in the pipe at Point 1 is solved.
After calculating the pipe&#39;s flow velocity, the flow rate, Q, is determined by multiplying this velocity by the cross-sectional area A of the pipe:
<img alt="Equation 2" src="/files/ftp_upload/18085/18085_PT_equation_2.svg" style="text-align: center; height: 11px;" />
Where A is:
<img alt="Equation 3" src="/files/ftp_upload/18085/18085_PT_equation_3.svg" style="text-align: center; height: 25px;" />
This flow rate calculation ensures that the pesticide is delivered at a consistent and effective rate, optimizing pest control application.

Consider a spray tank system designed to evenly apply pest-control liquid to plants.
The system includes a pressurized tank at 150 kPa filled with ...

Pipe Flowrate Measurement

In pipe flow measurement, orifice, nozzle, and Venturi meters are commonly used to determine fluid flowrates by constricting the flow area, which increases fluid velocity and reduces pressure. This pressure difference, governed by Bernoulli&#39;s principle and adjusted for real-world conditions, is essential for calculating flowrate. Each meter type is suited to specific applications based on accuracy, efficiency, and compatibility with various flow conditions.
The orifice meter is a simple, cost-effective device consisting of a flat plate with a central hole that creates a measurable pressure drop. This device is popular in applications like water treatment, oil and gas processing, and general industrial use where moderate accuracy is acceptable. However, due to head loss from turbulent flow and area contraction, a discharge coefficient is used to correct the idealized flow equation. The flowrate through an orifice meter is proportional to the square root of the pressure difference, accounting for fluid density, orifice area, and the ratio of the hole diameter to the pipe diameter.
The nozzle meter improves on the orifice meter by using a contoured design that reduces flow separation and turbulence, resulting in a smoother flow and increased measurement accuracy. It is often used for high-velocity flows, such as in steam or gas pipelines, chemical processing, and power plants. With a higher discharge coefficient than the orifice meter, the nozzle meter provides improved precision with less energy loss.
The Venturi meter offers the highest accuracy among the three types. It is ideal for applications requiring minimal head loss and reliable measurements, such as water supply systems, natural gas pipelines, and wastewater management. Its smooth contraction and gradual expansion reduce flow separation, conserving energy and enabling accurate flowrate calculations even in low-pressure systems. The discharge coefficient for a Venturi meter, which depends on factors like the ratio of the throat to pipe diameter and flow characteristics, supports exact flow measurement but requires a more complex installation and is generally more costly.
Each meter type offers unique benefits, from essential flow monitoring to precise flow control, chosen based on the application&#39;s accuracy needs and specific conditions.

Flow measurement devices&#8212;such as orifice, nozzle, and Venturimeters&#8212;operate by narrowing the flow area. This increases fluid velocity and ...

Design Example: Designing a Residential Plumbing System

The design of residential plumbing systems requires carefully evaluating water demand, flow rates, and pressure dynamics to ensure both efficiency and reliability. The nature of water flow within pipes is defined by its Reynolds number, which classifies flow as either laminar (smooth) or turbulent.
<img alt="Equation 1" src="/files/ftp_upload/18086/18086_Equation_1.svg" style="text-align: center; height: 42px;" />
This dimensionless value depends on factors such as water velocity and pipe. Residential plumbing systems are designed to meet specific water demand and flow rate requirements based on occupancy and fixture needs.
Determining the nature of water flow &#8212; laminar or turbulent &#8212; requires calculating the Reynolds number, which depends on variables like pipe diameter and water velocity. In residential settings, flow is often turbulent, leading to higher energy loss due to friction, which impacts system efficiency. Energy losses, or head losses, arise as water moves through pipes. These losses are classified as either major or minor. Major losses from friction along pipe walls are calculated using the Darcy-Weisbach equation.
<img alt="Equation 2" src="/files/ftp_upload/18086/18086_Equation_2.svg" style="text-align: center; height: 42px;" />
The Darcy-Weisbach equation requires a friction factor, which is obtained from the Moody chart. This chart accounts for pipe roughness and flow characteristics, which significantly affect turbulent flow friction levels. Minor losses, in contrast, are caused by fittings, bends, and valves, each of which contributes resistance to flow. Each fitting is assigned a loss coefficient that quantifies its resistance and is factored into the total head loss.
<img alt="Equation 3" src="/files/ftp_upload/18086/18086_Equation_3.svg" style="text-align: center; height: 48px;" />
By adding major and minor losses, the overall head loss can be determined and managed so as to maintain adequate pressure, avoiding potential leaks or damage. Flow measurement devices like Venturi or orifice meters ensure the plumbing system provides the necessary flow rates. These meters measure flow rates by detecting pressure differences across constrictions in the pipe, confirming that water flow meets design specifications and household needs.

The residential plumbing system is designed by assessing the building&#39;s water demand and flow rate needs based on occupancy and fixture requirements.
...

Single Pipe Systems

In pipe flow analysis, problems are typically categorized into three types &#8212; Type I, Type II, and Type III &#8212; based on the known parameters and the desired outcome. Each type of problem addresses specific engineering requirements using fluid properties, pipe characteristics, and operational conditions.
In a Type I problem, fluid properties (density and viscosity), pipe characteristics (including diameter, length, and surface roughness), and the flow rate or average velocity are known. The objective is calculating the pressure drop or head loss needed to maintain the specified flow rate. For instance, in a heating system circulating hot water at 0.32 liters per second, knowing the system&#39;s fluid and pipe specifications allows engineers to determine the required pressure to achieve steady flow. Type I problems are common in designing HVAC systems, where maintaining consistent flow is essential for effective temperature control.
A Type II problem provides the applied pressure, fluid properties, and pipe details, to calculate the resulting flow rate. This type of analysis is used in applications such as garden irrigation, where engineers need to determine the water flow rate in a sprinkler system operating at a specified pressure, such as 276 kilopascals. Knowing the pipe dimensions and roughness, the flow rate reaching each sprinkler can be calculated to ensure adequate water distribution.
In a Type III problem, the known parameters include the pressure drop and desired flow rate, to determine the optimal pipe diameter to support efficient flow. This scenario is typical in cooling systems or industrial piping design, where engineers must size the pipe to achieve a flow rate of 0.19 liters per second, balancing efficiency with minimal energy loss. Choosing an appropriate pipe diameter in Type III problems ensures the required flow rate while minimizing pressure losses and pump energy demands.
By systematically categorizing pipe flow problems, engineers can streamline analysis and ensure optimal design and performance for fluid transport systems in heating, irrigation, and industrial applications.

In pipe flow problems, scenarios are grouped by known information and desired calculation and are typically classified as Type I, II, or III.
In a Type I ...

Minor Losses in Pipes

In pipe systems, minor losses refer to energy losses arising from components such as valves, bends, fittings, expansions, and other features that disrupt the steady flow of fluid. These disturbances cause energy dissipation through turbulence and resistance, which engineers quantify to manage system efficiency effectively.
Valves play a significant role in generating minor losses by obstructing or redirecting the fluid flow. When a valve is closed or partially closed, it restricts the flow path, creating additional resistance and inducing turbulence. The resulting energy dissipation depends on the valve&#39;s geometry, position, and flow path design. To quantify this effect, engineers use a loss coefficient, K, which scales with the square of the fluid velocity V as follows:
<img alt="Equation 1" src="/files/ftp_upload/18081/18081_Equation_1.svg" style="text-align: center; height: 30px;" />
where:
<ol style="list-style-type: circle;">
	<li>hLminor is the head loss due to the component,</li>
	<li>KL is the loss coefficient specific to the component geometry,</li>
	<li>V is the fluid velocity</li>
	<li>g is the acceleration due to gravity.</li>
</ol>
For ease of calculation, engineers often represent minor losses from fittings and valves as an equivalent length of straight pipe that would produce the same head loss.
At the entry point of a pipe, sharp edges or abrupt changes in flow area can cause the fluid to separate from the wall, creating turbulence and energy dissipation through viscous effects. This entrance loss contributes to the overall energy loss, depending on the pipe geometry and the flow velocity. When fluid exits a pipe, the kinetic energy disperses into the surrounding environment, creating an exit loss characterized by a unity loss coefficient, as all the kinetic energy is lost from the system.
Sudden expansions in the pipe create high-velocity jets that decelerate as they expand, dissipating energy through viscous effects. These expansions lead to additional energy losses as the fluid experiences abrupt pressure and velocity changes. Bends in the pipe also contribute to head loss, primarily due to flow separation and swirling motions induced by centripetal forces as the fluid changes direction. The frictional resistance over the length of the bend adds further to the overall head loss, with the degree of loss depending on the bend angle and radius.
Understanding these minor losses allows engineers to design pipe systems that manage energy dissipation effectively, optimizing flow efficiency and minimizing pressure drops across the system.

Minor losses in pipe systems arise from components like valves, bends, and fittings that disrupt fluid flow, causing energy dissipation through turbulence ...

Major Losses in Pipes

When a fluid flows through a pipe, it experiences energy losses due to frictional resistance along the pipe walls, known as major losses. These energy losses result in a pressure drop, which varies based on the flow conditions &#8212; whether laminar or turbulent &#8212; and the specific physical properties of the fluid and pipe.
Fluid flow can be classified as laminar or turbulent, primarily based on the Reynolds number. This dimensionless number reflects the relative influence of inertial to viscous forces in the fluid. In laminar flow (Re &#60; 2000), the fluid flows in parallel layers, or streamlines, with minimal cross-stream mixing. Here, the pressure drop mostly depends on the fluid&#39;s viscosity and is generally lower. In contrast, turbulent flow (Re &#62; 4000) is characterized by chaotic eddies and swirling motions. In this regime, the pressure drop is influenced not only by viscosity but also by the roughness of the pipe wall, as these irregularities disrupt the flow further, increasing energy losses.
The pressure drop in a pipe depends on several factors: fluid properties, flow velocity, pipe characteristics, and dimensionless numbers.
The Darcy-Weisbach equation is the standard approach for quantifying the pressure drop due to frictional losses in pipe flow:
<img alt="Equation 1" src="/files/ftp_upload/18080/18080_Equation_1.svg" style="text-align: center; height: 30px;" />
Where:
<ol style="list-style-type: circle;">
	<li>&#916;P is the pressure drop,</li>
	<li>f is the friction factor,</li>
	<li>L is the length of the pipe,</li>
	<li>D is the pipe diameter,</li>
	<li>&#961; is the fluid density,</li>
	<li>v is the flow velocity.</li>
</ol>
The friction factor f is crucial for calculating pressure drops, especially in turbulent flow. It depends on the Reynolds number and the relative roughness of the pipe. Engineers often refer to the Moody chart, which provides empirical values of friction factors across flow regimes and roughness levels.
The Colebrook equation
<img alt="Equation 2" src="/files/ftp_upload/18080/18080_Equation_2.svg" style="text-align: center; height: 30px;" />
Where:
<ol style="list-style-type: circle;">
	<li>f is the friction factor,</li>
	<li>&#1013;/D is the relative roughness</li>
	<li>Re is the Reynolds number.</li>
</ol>
offers a precise method to calculate f for smooth and moderately rough pipes, but it is implicit and requires iterative solutions. To avoid iteration, approximations like the Haaland equation:
<img alt="Equation 3" src="/files/ftp_upload/18080/18080_Equation_3.svg" style="text-align: center; height: 30px;" />
Where:
<ol style="list-style-type: circle;">
	<li>f is the friction factor,</li>
	<li>&#1013;/D is the relative roughness</li>
	<li>Re is the Reynolds number.</li>
</ol>
are commonly used in practice, providing reasonably accurate friction factor values without extensive calculation. Understanding these dynamics enables engineers to design pipe systems that manage flow efficiently, compensating for potential pressure losses over time as pipe roughness increases.

Consider &#160;a fluid flowing through a pipe; it encounters frictional resistance, causing energy losses known as major losses, resulting in a pressure ...

Multiple Pipe Systems

Multipipe systems consist of complex configurations of interconnected pipes designed to transport fluids efficiently across intricate networks. They are essential in engineering applications requiring precise control over flow distribution, pressure, and head loss. They are categorized into series, parallel, loop, and network configurations, each distinguished by unique flow characteristics and applications.
Series Configuration
In a series configuration, fluid flows sequentially from one pipe segment to the next, maintaining a constant flow rate throughout. This setup means each segment experiences the same flow rate, although head losses accumulate progressively across each section. The total head loss from an entry point A to an exit point B is the cumulative sum of head losses in each pipe, governed by the Darcy-Weisbach equation for head loss or similar empirical relationships. This configuration suits applications requiring a consistent flow rate along a fixed path, such as in specific water treatment processes.
Parallel Configuration
Parallel multipipe systems divide fluid flow across multiple paths, enabling a higher overall flow rate and reducing resistance across the network. Here, the total flow rate is the sum of the individual flow rates in each branch. However, a critical feature of parallel systems is that all paths experience the same pressure drop. Consequently, fluid particles in each path encounter identical head losses, regardless of the path taken. This arrangement is effective in systems needing efficient fluid distribution, such as in heating and cooling systems, where multiple paths ensure temperature control across large areas.
Loop and Network Configurations
In loop systems, the flow rate within one pipe is balanced by the combined flow rates of the other connected pipes, enabling fluid redistribution based on demand changes or localized resistance variations, such as in urban water distribution systems. Network configurations represent the most complex setup, involving numerous interconnected pipes with multiple inlets and outlets,such as in oil and gas pipeline networks. This flexibility allows for multiple flow directions, enhancing transport efficiency and adaptability to dynamic system demands.
Applications of Multipipe Systems
Multipipe systems are indispensable in several engineering sectors. Municipal water supply networks often use network configurations to ensure consistent pressure and reliable delivery across urban areas. In deepwater oil and gas pipelines, multipipe systems help manage the challenging pressures and flow conditions of underwater environments. Additionally, industrial cooling systems benefit from the controlled fluid distribution provided by parallel configurations, which improve heat dissipation and maintain optimal temperatures in manufacturing processes.

Multiple pipe systems consist of interconnected pipes that regulate fluid flow across complex networks.
They can be arranged in series, parallel, loop, or ...

Turbulent Flow

Turbulent flow is characterized by unpredictable fluctuations in velocity and pressure, which result in a chaotic fluid movement distinct from the orderly patterns of laminar flow. While laminar flow is governed by smooth, parallel layers with minimal mixing, turbulent flow exhibits highly irregular, three-dimensional patterns. This behavior arises due to instabilities in the fluid&#39;s velocity profile, and amplifies as the flow velocity increases. Minor disturbances, known as turbulent spots, appear once the flow reaches a critical speed. When the Reynolds number &#8212; a dimensionless quantity expressing the ratio of inertial to viscous forces in a fluid &#8212; exceeds a threshold of around 4,000, these spots coalesce, generating fully developed turbulence.
Structure of Turbulent Flow in Pipes
In pipe systems, turbulent flow comprises three distinct regions: the viscous sublayer, the buffer layer, and the outer layer. The viscous sublayer forms close to the pipe wall, a thin boundary where viscous forces dominate, and the flow behaves similarly to laminar flow. Here, velocity follows a near-linear profile influenced by fluid viscosity, providing a stable region within the otherwise chaotic turbulent flow.
Above the viscous sublayer lies the buffer layer, a transitional zone where viscous and turbulent forces contribute to fluid motion. This region is marked by a logarithmic velocity profile that conforms to the logarithmic law of the wall, representing a balance between viscous and turbulent interactions. Further from the wall, turbulent eddies drive fluid movement in the outer layer, promoting rapid mixing and random motion across the flow.
Engineering Applications of Turbulent Flow
The chaotic nature of turbulent flow makes it advantageous in numerous engineering applications where mixing and energy dissipation are crucial. For instance, wastewater treatment systems rely on turbulent flow to enhance the mixing of chemicals, promoting efficient contaminant removal. In hydraulic engineering, turbulent flow dissipates energy in spillways, preventing structural damage by controlling water velocity. The turbulent mixing process also reduces river sediment buildup, maintaining navigable waterways and protecting aquatic ecosystems.

Turbulent flow is a fluid movement marked by chaotic shifts in pressure and velocity.
Unlike laminar flow, turbulent flow displays complex, irregular ...

Turbulent Flow: Problem Solving

Carbonation is a process used to dissolve carbon dioxide gas in a liquid, commonly used in the production of carbonated beverages. Achieving efficient carbonation requires careful control of temperature, pressure, and flow conditions. By adjusting these parameters, carbonation efficiency can be maximized, producing a higher concentration of CO2&#160;in the liquid.
Temperature is a key factor in CO2 solubility. In this case, the CO2 gas and the liquid are cooled to 20&#176;C. Lower temperatures enhance the solubility of CO2, enabling a greater volume of gas to dissolve within the liquid. At this temperature, the CO2&#160;gas is pressurized to approximately 550 kPa. Elevated pressure increases the partial pressure of CO2&#160;over the liquid, further driving dissolution.
CO2 is pumped through a connecting pipe between two tanks, where the flow characteristics are crucial for maintaining steady carbonation. The high Reynolds number, defined as greater than 4000, indicates that the flow is turbulent rather than laminar. Turbulent flow improves gas-liquid contact by promoting mixing, which enhances CO2&#160;dissolution in the liquid. The Reynolds number (Re) is calculated using the equation:
<img alt="Equation 1" src="/files/ftp_upload/18079/18079_Equation_1.svg" style="text-align: center; height: 30px;" />
where:
<ol style="list-style-type: circle;">
	<li>&#961; is the density of CO2,</li>
	<li>v is the velocity of flow,</li>
	<li>D is the pipe diameter,</li>
	<li>&#956; is the dynamic viscosity of CO2.</li>
</ol>
The flow rate of CO2&#160;in this system is controlled by the pipe&#39;s cross-sectional area and the velocity of the gas. To maintain turbulent conditions, the flow velocity v is adjusted based on the desired Reynolds number, which allows the required pipe diameter to be calculated.
The Reynolds number equation can be rearranged to determine the pipe diameter for the required turbulent flow. The&#160;dynamic viscosity &#956; of CO2&#160;at 20&#176;C are obtained using physical property tables. The optimal pipe diameter can be calculated by substituting these values along with the target Reynolds number and flow velocity into the equation. This configuration ensures efficient CO2&#160;transport through the system, optimizing the liquid&#39;s carbonation conditions.

Consider a liquid that needs to be carbonated. Both the liquid and the carbon dioxide gas are cooled. The temperature is selected as 20 degrees Celsius to ...

General Characteristics of Pipe Flow II

When fluid enters a pipe, it first passes through the entrance region, where the velocity profile adjusts due to viscous effects. In this region, a boundary layer forms along the pipe walls and grows until it fully occupies the pipe&#39;s cross-section. Once the boundary layer merges, the flow becomes fully developed, with a steady velocity profile that remains consistent along the pipe&#39;s length.
The distance to reach a fully developed flow is called the entrance length and depends on the flow regime (laminar or turbulent) and the Reynolds number. Laminar flow typically has a shorter entrance length due to its smooth, gradual profile development, whereas turbulent flow requires a longer entrance length due to enhanced mixing and chaotic motion.
A pressure difference along the pipe&#39;s length drives fully developed flow in a constant-diameter pipe. In horizontal pipes, this pressure difference overcomes viscous forces, ensuring steady motion. The entrance region has a higher pressure gradient to account for acceleration as the velocity profile evolves. Once fully developed, the pressure gradient stabilizes, reflecting the energy needed to overcome viscosity.
Shear stress behaves differently in laminar and turbulent flows. In laminar flow, shear stress arises from molecular momentum transfer, resulting in smooth, orderly layers. Turbulent flow stems from momentum exchange between more significant, irregularly moving fluid particles, making it a macroscopic effect. These differences influence pressure distribution and flow characteristics.
For inclined pipes, gravity modifies the pressure gradient. Depending on the pipe&#39;s orientation, gravity can assist or resist flow, altering the force balance required for steady motion. The viscosity, pressure, and gravity interplay determine fluid behavior within the pipe.

As fluid enters a pipe, it first flows through an entrance region where the velocity profile adjusts, forming a boundary layer along the walls. This ...

Laminar Flow

Laminar flow represents a smooth, orderly fluid motion where particles move along parallel paths, resulting in minimal mixing between layers. Streamlined particle paths characterize this flow regime and occur under conditions where viscous forces dominate over inertial forces. The distinction between laminar, transitional, and turbulent flow is primarily determined by the Reynolds number, a dimensionless quantity calculated as:
<img alt="Equation 1" src="/files/ftp_upload/18076/18076_Equation 1.svg" style="text-align: center; height: 30px;" />
where &#961; is the fluid density, v is the flow velocity, D is the pipe diameter, and &#956; is the dynamic viscosity. A Reynolds number below approximately 2100 indicates laminar flow, while values above 4000 denote turbulent flow. Transitional flow occurs between these ranges as the fluid shifts from laminar to turbulent.
Velocity Profile and Entrance Length
In a pipe, fluid initially entering develops a velocity profile over a length known as the entrance length. During this phase, the flow transitions from a uniform profile at the inlet to a fully developed parabolic profile, where the velocity is highest at the center and gradually decreases toward the pipe walls due to viscous forces. The entrance length is a function of the Reynolds number and pipe characteristics, typically shorter in turbulent flow and longer in laminar flow. Once fully developed, the parabolic velocity profile remains stable unless disrupted by changes in pipe geometry, allowing consistent analysis of flow behavior along the pipe&#39;s length.
Energy Loss and Hagen-Poiseuille Law
Laminar flow is advantageous in minimizing energy loss, as the pressure drop along the pipe is dominated by viscous forces rather than turbulence. The relationship between pressure drop and flow rate in a fully developed laminar flow is defined by the Hagen-Poiseuille equation:
<img alt="Equation 2" src="/files/ftp_upload/18076/18076_Equation 2.svg" style="text-align: center; height: 30px;" />
where &#916;P is the pressure drop, &#956; is the dynamic viscosity, L is the pipe length, Q is the volumetric flow rate, and R is the pipe radius. This equation shows that the pressure drop in laminar flow is directly proportional to flow rate and fluid viscosity and inversely proportional to the fourth power of the pipe radius, highlighting the significant impact of pipe diameter on energy efficiency.
Applications of Laminar Flow
Laminar flow conditions are essential in applications that require smooth, predictable fluid motion with minimal mixing. Wastewater treatment systems, for example, benefit from laminar flow to avoid excessive energy loss and facilitate controlled fluid handling. In canal designs for irrigation, laminar flow promotes even distribution without turbulent disturbances, enhancing water delivery efficiency. Hydropower systems can also leverage laminar flow for more efficient energy conversion, particularly in sections where consistent, low-energy flow is needed.

Laminar flow refers to the smooth, orderly motion of fluid particles in which each particle follows parallel paths with minimal mixing between layers.
The ...

Laminar Flow: Problem Solving

Laminar flow occurs when a fluid moves smoothly in parallel layers with minimal mixing and turbulence. In fluid mechanics, ensuring laminar flow within a pipe is essential for precise control of flow characteristics, especially in engineering applications. The key factor in determining whether flow remains laminar is the Reynolds number, a dimensionless quantity that depends on the fluid&#39;s velocity, density, viscosity, and the pipe&#39;s diameter. A Reynolds number of 2100 or lower indicates laminar flow, while higher values lead to turbulence.
For air flowing through a pipe at standard atmospheric conditions and a given mass flow rate, the velocity of the fluid must be controlled to maintain laminar conditions. Since velocity depends on the flow rate and the cross-sectional area of the pipe, the pipe diameter plays a crucial role. By substituting the velocity expression into the Reynolds number formula and solving for the diameter, it is possible to determine the minimum required pipe size.
Using standard values for air density and viscosity at 300 Kelvin, the volumetric flow rate can be computed. This, in turn, allows for the calculation of the necessary pipe diameter. To ensure a Reynolds number of 2100 or lower, the minimum allowable diameter for the pipe is found to be approximately 0.46 meters. This calculation ensures that the flow remains smooth and predictable, avoiding turbulence that could disrupt performance.

Consider air flowing through the pipe at 300 Kelvin at standard atmospheric pressure, with a mass flowrate of 0.0163 kilograms per second. If the flow is ...

General Characteristics of Pipe Flow I

Pipe flow refers to the movement of fluids within fully enclosed conduits, typically cylindrical in shape, such as water pipes or hydraulic hoses. These conduits are designed to withstand high-pressure gradients that drive fluid movement, contrasting with open-channel flows, where gravity is the primary driving force. Rectangular conduits, like air conditioning and heating ducts, generally operate at lower pressures and are less suited for high-pressure applications.
The classification of fluid flow in pipes was pioneered by Osborne Reynolds through his foundational dye injection experiments. Reynolds identified three flow regimes &#8212; laminar, transitional, and turbulent &#8212; based on fluid velocity. At low velocities, the dye maintained a smooth streakline, indicating laminar flow, where fluid particles follow orderly, parallel paths. As velocity increased, the dye began to oscillate, marking the transitional regime, and at higher velocities, it dispersed unpredictably in the turbulent regime, where random, chaotic motion dominates.
These observations led to the development of the Reynolds number, a dimensionless quantity that indicates the balance of inertial and viscous forces in the flow. The Reynolds number is calculated based on fluid velocity, density, viscosity, and pipe diameter and serves as an indicator of flow type: laminar for values below 2100, turbulent for values above 4000, and transitional in the range between. In laminar flow, velocity remains steady over time, while turbulent flow exhibits unsteady, fluctuating velocities with random components perpendicular to the pipe axis. By calculating the Reynolds number, engineers can predict flow behavior in pipe systems, aiding in the design and optimization of pipelines, hydraulic systems, and fluid transport networks across various industries.

Pipe flow describes fluid movement through enclosed, fully-filled conduits, typically cylindrical pipes like water pipes or hydraulic hoses, engineered to ...

Design Example: Creating a Hydraulic Model of a Dam Spillway

Scaled hydraulic models of dam spillways provide a practical way to replicate and study the intricate flow dynamics of these structures. Often built to a 1:15 ratio, these models allow for observing critical water behavior, such as velocity distribution, flow patterns, and energy dissipation.
<img alt="Equation 1" src="/files/ftp_upload/18074/18074_Equation_1.svg" style="text-align: center; height: 38px;" />
Through this controlled setup, researchers can closely examine how water flows over the spillway, supporting a detailed analysis of its performance. A fundamental principle behind these models is using Froude number similarity, which preserves the proper balance between gravitational and inertial forces.
<img alt="Equation 2" src="/files/ftp_upload/18074/18074_Equation_2.svg" style="text-align: center; height: 38px;" />
This similarity is essential, as it ensures that free-surface flow characteristics, including wave formation, turbulence, and flow separation, behave as they would in the full-sized structure. By achieving this accurate representation, scaled models provide valuable insights into how water interacts with spillway features, helping to pinpoint areas where the design might benefit from adjustments to enhance safety or flow efficiency.
In the scaled model, adjustments are also applied to discharge rates, flow velocities, and timing to reflect real-world conditions without requiring extensive water volumes or prolonged test durations. The discharge rate, for example, is reduced by the scale factor to maintain manageable flow rates, enabling safe experimentation.
<img alt="Equation 3" src="/files/ftp_upload/18074/18074_Equation_3.svg" style="text-align: center; height: 36px;" />
Additionally, time is compressed so that events unfolding over an entire day in the full-scale structure can be examined in just a few hours within the model.
<img alt="Equation 4" src="/files/ftp_upload/18074/18074_Equation_4.svg" style="text-align: center; height: 48px;" />
This efficient scaling supports rapid testing and analysis of different flow conditions and potential structural responses. Overall, scaled modeling of spillways is crucial for assessing designs, identifying potential vulnerabilities, and confirming that the full-scale structure will perform reliably under various water flow conditions.

A hydraulic model of a dam spillway is often created by scaling down the design, typically to a 1:15 ratio, to replicate key flow characteristics such as ...

Dimensionless Groups in Fluid Mechanics

Dimensionless groups in fluid mechanics provide simplified ratios that help analyze fluid behavior without relying on specific units. The Reynolds number (Re), which represents the ratio of inertial to viscous forces, distinguishes between laminar and turbulent flows, making it essential in the design of pipelines and aerodynamic surfaces. The Froude number (Fr), the ratio of inertial to gravitational forces, is particularly useful in predicting wave formation and hydraulic jumps in open-channel flows and coastal engineering.
The Euler number (Eu) indicates the ratio of pressure forces to inertial forces and is used to calculate pressure drops in systems such as pipelines and across valves. The Cauchy number (Ca), representing the ratio of inertial to elastic forces, is critical in compressible fluid flow analysis and the study of shockwaves. The Mach number (Ma) compares the flow speed to the speed of sound, making it crucial for studies in supersonic flight and aeroelasticity.
The Strouhal number (St), which compares oscillatory frequency to inertial forces, finds applications in resonance studies, particularly vortex shedding around structures like bridge piers. Lastly, the Weber number (We), showing the ratio of inertial to surface tension forces, is vital in analyzing droplet formation, spray dynamics, inkjet printing, and fuel atomization processes. These groups enable the comparison of fluid behaviors across different scales and systems, supporting a broad range of fluid dynamics applications.

The dimensionless groups in fluid mechanics are ratios of physical quantities used to describe and analyze fluid behavior independently of units of ...

Correlation of Experimental Data

Dimensional analysis simplifies complex physical problems and guides experimental investigations, but it does not provide complete solutions. It identifies the dimensionless groups that influence a phenomenon, but experimental data is needed to establish the specific relationships and validate theoretical predictions.
For example, a spherical particle moving through a viscous fluid experiences drag. Dimensional analysis shows that the drag force depends on the particle&#39;s diameter, velocity, and the fluid&#39;s viscosity. These variables are grouped into dimensionless terms to form a relationship, but experimental data is needed to determine the exact nature of this relationship.
Researchers measure drag force while varying velocity to correlate the dimensionless groups, such as the Reynolds number. This correlation allows the relationship between drag and velocity to be expressed as a constant. Multiple tests verify this constant, confirming the dependency of drag on velocity.
If drag depends only on diameter, velocity, and viscosity, the relationship should be universal for any spherical particle in any fluid under consistent conditions. Multiple tests with different particles and fluids verify this universality, reducing the need for further experiments, as the derived constant remains valid under the specified conditions.

A key use of dimensional analysis is to efficiently manage, interpret, and correlate experimental data.
Dimensional analysis alone cannot fully solve a ...

Typical Model Studies

Fluid mechanics model studies often utilize scaled-down systems to predict fluid behavior in full-scale environments, such as river flows, dam spillways, and structures interacting with open surfaces. Maintaining Froude number similarity in river models is crucial, as it replicates surface flow features like wave patterns and velocities.
<img alt="Equation 1" src="/files/ftp_upload/18073/18073_Equation_1.svg" style="text-align: center; height: 48px;" />
Practical constraints, however, can introduce geometric distortions, impacting how accurately these models represent the real system. Adjustments, such as increased model roughness, help bridge this gap, aligning flow behaviors between the model and the actual river environment.
For hydraulic structures like spillways, the focus shifts to achieving high Reynolds numbers to maintain dynamic similarity. This approach emphasizes balancing inertial and gravitational forces to replicate the prototype&#39;s behavior closely.
<img alt="Equation 2" src="/files/ftp_upload/18073/18073_Equation_2.svg" style="text-align: center; height: 45px;" />
Larger models are generally preferred, as they capture critical force interactions more effectively in full-scale designs. Although exact similarity across all scales may not be feasible due to physical and financial limitations, these larger models still allow close approximations of essential conditions.
Verification tests and prototype comparisons are key for refining model accuracy. Adjusting variables and calibrating outcomes ensure that the model reliably mirrors actual fluid flow behaviors. This predictive capability supports planning and design adaptations, making these models indispensable tools for assessing performance before moving to full-scale implementation.

Typical model studies analyze scaled-down systems to predict fluid behavior, focusing on interactions with free surfaces and flow around immersed bodies.
...

Modeling and Similitude

Scaled modeling is a fundamental technique in engineering, enabling the study of large and complex systems by creating smaller, manageable replicas that recreate critical characteristics of the original. In hydrology and civil infrastructure, for example, scaled models of dams help analyze water flow, turbulence, and pressure. This method allows for accurate predictions of real-world behavior within a controlled environment, significantly reducing the cost and time involved in full-scale testing, which is often impractical. These models provide valuable insights into system performance under various conditions by ensuring that essential design elements match.
For the model to yield valid insights, it must maintain similitude, meaning the replica accurately reflects the prototype&#39;s behavior through geometric, kinematic, and dynamic similarity. Geometric similarity ensures the scaled model retains the same proportional dimensions as the actual structure, while kinematic similarity requires that motion characteristics match between model and prototype. Dynamic similarity further demands that forces acting within the model accurately replicate those in the real system, accounting for any scaling effects.
In fluid dynamics, the Froude number is a key dimensionless parameter that matches scaled models to their full-sized counterparts, especially for applications like ship design. The Froude number is defined as:
<img alt="Equation 1" src="/files/ftp_upload/18072/18072_Equation_1.svg" style="text-align: center; height: 60px;" />
where V represents velocity, g is the gravitational acceleration, and L is a characteristic length, often the ship&#39;s length. By equating the Froude numbers for both the model and the full-scale ships, the gravitational and inertial forces align, allowing the scaled model to accurately predict behaviors such as maneuverability, stability, and wave formation.

A scaled model of a dam is tested in a controlled environment to observe how it will manage water flow, turbulence, and pressure.
This setup predicts the ...

Determination of Pi Terms

The Buckingham Pi theorem is a valuable method in dimensional analysis, reducing complex relationships between variables into dimensionless terms. Relevant variables in analyzing the lift force on an airplane wing include lift force, air density, wing area, aircraft velocity, and air viscosity. Expressing each variable in terms of fundamental dimensions &#8212; mass, length, and time &#8212; provides a consistent foundation for constructing these dimensionless terms.
The theorem indicates that the number of Pi terms equals the total variables minus the number of fundamental dimensions. In this example, five variables and three dimensions result in two Pi terms. Air density, velocity, and wing area are selected as repeating variables, as they independently cover all dimensions. These repeating variables combine with lift force and viscosity to form dimensionless Pi terms.
<img alt="Equation 1" src="/files/ftp_upload/18069/18069_Equation_1.svg" style="text-align: center; height: 100px;" />
The first Pi term expresses the efficiency of lift generation, representing how lift depends on density, velocity, and area. The second Pi term accounts for the relative impact of viscosity, revealing how viscous forces interact with the wing in relation to inertia.
Each Pi term is verified as dimensionless, allowing complex aerodynamic behavior to be represented in simplified expressions. This approach enables the prediction and optimization of lift under various conditions, capturing the essential dynamics without needing to examine each factor individually.

To determine Pi terms using the Buckingham Pi theorem, consider the example of analyzing lift force on an airplane wing.
Relevant variables here include ...

Dimensional Analysis

Dimensional analysis is a valuable technique in fluid mechanics for simplifying complex problems by reducing them into dimensionless groups. These groups capture the essential relationships between the variables involved, allowing researchers and engineers to analyze fluid flow without dealing with each variable individually. This approach reduces the number of independent variables, allowing for easier analysis and better understanding of physical phenomena.
In fluid mechanics, dimensional analysis is used to understand the behavior of fluids in systems such as pipelines, channels, and other structures common in civil engineering. Consider the example of fluid flow through a long, smooth-walled, horizontal circular pipe, which is typical in the design of water distribution systems or sewage networks. The pressure drop per unit length along the pipe depends on variables such as pipe diameter, fluid density, fluid velocity, and viscosity. According to dimensional analysis, the pressure drop can be expressed as a function of these variables:
<img alt="Equation 1" src="/files/ftp_upload/18067/18067_Equation_1.svg" style="text-align: center; height: 12px;" />
This formulation involves five independent variables, which can complicate the analysis. To simplify this, we can use dimensional analysis to reduce the number of variables into two dimensionless groups, thus streamlining the problem. Two important dimensionless groups that emerge from this analysis are the Reynolds number, which characterizes the flow regime, and the friction factor, which correlates with the pressure drop.
<img alt="Equation 2" src="/files/ftp_upload/18067/18067_Equation_2.svg" style="text-align: center; height: 30px;" />
Engineers use the dimensionless groups to generate a universal curve that applies to any smooth-walled pipe and incompressible Newtonian fluid. This curve would allow them to predict the pressure drop for different pipe sizes and fluids without performing numerous experiments for each possible combination.
<img alt="Figure 1" src="/files/ftp_upload/18067/18067_PT_Figure.svg" style="text-align: center; height: 250px;" />
This approach minimizes the time and cost required for experimentation and allows for more straightforward design decisions. Whether designing water distribution networks or analyzing pressure losses in irrigation systems, dimensional analysis is critical for any civil engineer.

Dimensional analysis helps simplify fluid flow problems by using dimensionless groups, reducing complex variables into simpler terms that are easier to ...

The Buckingham Pi Theorem

The Buckingham Pi theorem provides a structured method to simplify fluid dynamics problems by reducing complex systems of variables to dimensionless terms.
<img alt="Equation 1" src="/files/ftp_upload/18068/18068_Equation_1.svg" style="text-align: center; height: 34px;" />
In analyzing flow through a pipe, variables that define the system include the pipe&#39;s diameter, the fluid&#39;s velocity, its density, and viscosity. These can be combined into a single dimensionless term, the Reynolds number (Re), which captures their essential relationship.
<img alt="Equation 2" src="/files/ftp_upload/18068/18068_Equation_2.svg" style="text-align: center; height: 38px;" />
The first step in applying the Buckingham Pi theorem is identifying the fundamental dimensions of each variable. In fluid mechanics, these are typically mass, length, and time. Here, the diameter represents length, velocity combines length and time, density encompasses mass and volume, and viscosity incorporates mass, length, and time.
<img alt="Equation 3" src="/files/ftp_upload/18068/18068_Equation_3.svg" style="text-align: center; height: 140px;" />
The theorem suggests that the number of independent dimensionless groups, known as Pi terms, is the difference in the number of variables k and the number of fundamental dimensions r. Only one dimensionless term is needed with four variables and three dimensions, simplifying analysis.
<img alt="Equation 4" src="/files/ftp_upload/18068/18068_Equation_4.svg" style="text-align: center; height: 32px;" />
This single term, the Reynolds number, compares inertial forces to viscous forces in the fluid. Establishing this ratio reveals whether the flow is likely to be smooth and orderly, known as laminar flow, or chaotic and irregular, known as turbulent flow. Lower values indicate laminar flow, where viscous forces dominate, and the fluid moves in parallel layers. Higher values reflect turbulent flow, where inertial forces lead to chaotic motion. The Buckingham Pi theorem&#39;s application in this context reduces complex interactions to a single term, making it easier to generalize flow behavior under varying conditions.

Consider the flow through a pipe, where key variables include the pipe&#39;s diameter D, fluid velocity V, density &#961;, and viscosity &#956;.
To ...

Navier&#8211;Stokes Equations

For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting tangential stress to velocity differences between layers. Normal and shear stresses define the internal forces within a fluid that influence its movement.
The Navier-Stokes equations are derived by incorporating these stress relationships into differential equations of motion.
<img alt="Equation 1" src="/files/ftp_upload/18062/18062_Equation_1.svg" style="text-align: center; height: 106px;" />
These equations balance inertial, pressure, and gravitational forces in a viscous fluid and are fundamental to predicting fluid behavior under various conditions. The inertia term in the Navier-Stokes equations accounts for fluid acceleration, representing the fluid&#39;s resistance to changes in velocity. This term is critical in capturing the momentum conservation of a moving fluid, as it demonstrates how fluids resist sudden shifts in speed or direction.
The pressure gradient term within the Navier-Stokes equations drives fluid movement by creating a net force from areas of high pressure to low pressure. Meanwhile, the viscous terms represent internal friction arising from molecular interactions within the fluid. Depending on the velocity gradients present, these viscous forces act to slow down the fluid movement.
Each directional Navier-Stokes equation captures internal forces like viscous stress and external forces like gravity. These equations allow for predicting fluid motion in various conditions, from simple to complex flows. For steady or laminar flow scenarios, where flow is smooth and orderly, the equations become more manageable, enabling more straightforward analysis. This simplification is useful in controlled scenarios like boundary layer flow, where flow near a surface has a structured layer, and in Couette flow, describing fluid between two parallel surfaces moving at different velocities.

In fluid motion under viscous conditions, shear stress is directly proportional to fluid deformation. In incompressible Newtonian fluids, this stress ...

Plane Potential Flows

Plane potential flows simplify fluid motion by assuming the fluid to be irrotational and incompressible. These characteristics allow these flows to be described by a velocity potential function, &#981;, representing the flow speed in a given direction, and a stream function, &#968;, that visualizes the flow path, both governed by Laplace&#39;s equation. These parameters help in estimating flow patterns, velocity distributions, and pressure fields around various hydraulic structures.
Uniform Flow
Uniform flow represents the simplest form of potential flow, with fluid moving in a straight line at a constant velocity. Streamlines are straight, parallel, and evenly spaced, clearly visualizing flow across channels or pipes with constant velocity. This flow type is essential in designing canals, pipelines and drainage systems, where the assumption of constant velocity aids in flow rate estimation and hydraulic design.
Source and Sink Flows
Source and sink flows describe radially symmetric flows originating from or converging to a point. In a source flow, fluid moves radially outward from a central point, while in a sink flow, it converges inward. These flows are modeled using a flow rate parameter to represent the strength of the source or sink. This model is widely used in groundwater hydrology, where source flows represent groundwater recharge, such as that from an injection well, and sink flows represent groundwater extraction in a pumping well.&#160; represent groundwater extraction from a pumping well., and sink flows simulate groundwater recharge .
Vortex Flow
Vortex flow involves fluid circulating a central point and appears in two forms: free and forced. In a free vortex, particles move in circular paths around a center point without external force, creating a whirlpool effect often observed in natural bodies of water. This model is used to design spillways and drainage systems that manage spiraling water flow to prevent clogging or turbulence. In a forced vortex, an external force drives the fluid&#39;s rotation, creating a scenario where the fluid rotates as a rigid body. Forced vortices are used in sedimentation tanks and certain hydraulic devices where uniform rotation aids in sediment separation and fluid mixing.
Flow Nets
In flow nets, streamlines represent the fluid flow path, while equipotential lines mark points of equal hydraulic potential. These lines intersect perpendicularly, creating a grid pattern that aids in identifying pressure distributions and potential seepage paths around structures. Flow nets are constructed to assess seepage in soil or porous media. Seepage rates are estimated by carefully analyzing the spacing and orientation of these lines and identifying potential areas of erosion or structural weakness.

Plane potential flows simplify fluid motion by assuming irrotational and incompressible flow, governed by a velocity potential function and a stream ...

Steady, Laminar Flow Between Parallel Plates

Understanding steady, laminar flow between parallel plates is essential for analyzing and designing flow in narrow rectangular channels, commonly found in various water conveyance and drainage systems. The Navier-Stokes equations govern fluid motion and are generally challenging to solve due to their nonlinearity. However, simplifications are possible in certain cases, like the steady laminar flow between parallel plates. For this scenario, we assume steady, incompressible, laminar flow.
Considering two horizontal plates aligned along the x-axis and spaced a distance &#39;2h&#39; apart, with the y-axis perpendicular to the plates, fluid motion is assumed to be parallel. Here, the continuity equation indicates no change in velocity along the x-axis, and the Navier-Stokes equations reduce as follows:
<img alt="Equation 1" src="/files/ftp_upload/18063/18063_Equation_1.svg" style="text-align: center; height: 30px;" />
The equations in the y and z directions show no pressure gradient along these axes, indicating that pressure varies hydrostatically in the y-direction, allowing integration to yield:
<img alt="Equation 2" src="/files/ftp_upload/18063/18063_Equation_2.svg" style="text-align: center; height: 12px;" />
For parallel plates with a pressure gradient along the x-axis, the equation simplifies:
<img alt="Equation 3" src="/files/ftp_upload/18063/18063_Equation_3.svg" style="text-align: center; height: 30px;" />
By integrating twice and applying boundary conditions (u = 0 at y = +h and y = -h), we obtain:
<img alt="Equation 4" src="/files/ftp_upload/18063/18063_Equation_4.svg" style="text-align: center; height: 30px;" />
This expression reveals that the velocity profile across the gap between the plates is parabolic, with maximum velocity at y = 0, midway between the plates. The volumetric flow rate through a unit width of the plates is obtained by integrating the velocity profile across the gap.
Understanding laminar flow between parallel plates has practical applications in groundwater flow, wastewater treatment, and lubrication. For instance, this model can approximate flow in soil layers, where water moves between soil particles like flow between plates. Knowing the flow rate and velocity distribution helps engineers design efficient drainage systems and predict groundwater movement.

Steady, laminar flow between two parallel plates describes fluid movement in a narrow, confined channel, often used to model simple fluid systems.
For ...

Sample Preparation for Analytical Characterization

Source: Laboratory of Dr. B. Jill Venton - University of Virginia
Sample preparation is the way in which&#160;a sample is treated to prepare&#160;for analysis. Careful sample preparation is critical in analytical chemistry to accurately generate either a standard or unknown sample for a chemical measurement. Errors in analytical chemistry methods are categorized as random or systematic. Random errors are errors due to change and are often due to noise in instrument. Systematic errors are due to investigator or instrumental bias, which introduces an offset in the measured value. Errors in sample preparation are systematic errors, which will propagate&#160;through analysis, causing&#160;uncertainty or inaccuracies through improper calibration curves. Systematic errors can be eliminated through correct sample preparation and proper use of the instrument. Poor sample preparation can also sometimes cause harm to the instrument.

Source: Laboratory of Dr. B. Jill Venton - University of Virginia
Sample preparation is the way in which&#160;a sample is treated to prepare&#160;for ...

Internal Standards

Source: Laboratory of Dr. B. Jill Venton - University of Virginia

The goal of many chemical analyses is a quantitative analysis, where the amount of a substance in a sample is determined. In order to accurately calculate the concentration of an unknown from a sample, careful sample preparation is key. Every time a sample is handled or transferred, some of the sample can be lost. There are strategies however, for minimizing sample loss. There are also strategies for coping with sample loss and still making accurate measurements of concentration.
To minimize sample loss, the ideal is to minimize the number of sample handling and transfer steps. For example, massing a solid sample directly into a flask that a solution will be made in reduces a transfer step. If it&#39;s necessary to transfer from one flask to another and a dilution is being made, then triple rinsing the glassware helps ensure all the sample is transferred. Other strategies are more specific to the sample. For example, samples that adsorb to glass, such as proteins, might better be handled in polypropylene disposable tubes. The tubes&#160;are not hydrophilic, so if a small amount of sample is to be pipetted in water, it is best to have already added the water to the tube, so the sample can be pipetted directly into the solvent. It may be better to concentrate, rather than completely dry a sample, due to losses from insolubilities after rehydration.
Another source of sample loss is through incomplete sample manipulations. For example, if a derivatization procedure is used and the derivatization is incomplete, then the full amount of sample is not observed. Errors such as this are systematic errors, and can be solved by correcting the problem, such as changing the derivatization procedure. Another cause of systematic error in measurements is matrix effects. These can interfere with measurement of certain substances and performing calibrations in the same matrix as the sample can reduce this effect.
Quantitative analysis is typically carried out using either external or internal standards. For external standards, a calibration curve is made by measuring different known concentrations of the analyte of interest. Then, the sample is run separately from the standard. For internal standards, the standard is in the same sample as the analyte of interest, allowing the measurement to be taken simultaneously. Typically, a different species is added called the internal standard and the ratio of the response for that internal standard and the analyte is calculated. The idea is that the ratio of the response, called the response factor, is proportional to their concentration. While the method must be able to distinguish between the analyte of interest and the internal standard, any sample losses that occur after the internal standard is added should be similar for both substances and thus the ratio of the response stays the same. A special case of using internal standards is the method of standard additions, where increasing amounts of the analyte is added to the solution and the original amount of analyte is back-calculated. Internal standards can be used in chromatography, electrochemistry, and spectroscopy.

Internal Standards for Quantitative Analysis

Source: Laboratory of Dr. B. Jill Venton - University of Virginia

The goal of many chemical analyses is a quantitative analysis, where the amount of a ...

Method of Standard Addition

Source: Laboratory of Dr. Paul Bower - Purdue University
The method of standard additions is a quantitative analysis method, which is often used when the sample of interest has multiple components that result in matrix effects, where the additional components may either reduce or enhance the analyte absorbance signal. That results in significant errors in the analysis results.
Standard additions are commonly used to eliminate matrix effects from a measurement, since it is assumed that the matrix affects all of the solutions equally. Additionally, it is used to correct for the chemical phase separations performed in the extraction process.
The method is performed by reading the experimental (in this case fluorescent) intensity of the unknown solution and then by measuring the intensity of the unknown with varying amounts of known standard added. The data are plotted as fluorescence intensity vs. the amount of the standard added (the unknown itself, with no standard added, is plotted ON the y-axis). The least squares line intersects the x-axis at the negative of the concentration of the unknown, as shown in Figure 1.
<img alt="Figure 1" src="/files/ftp_upload/10201/10201fig1.jpg" /> 
Figure 1. Graphic representation of method of standard addition.

Method of Standard Addition to Minimize Matrix Effect

Source: Laboratory of Dr. Paul Bower - Purdue University
The method of standard additions is a quantitative analysis method, which is often used when the ...

Calibration Curves

Source: Laboratory of Dr. B. Jill Venton - University of Virginia
Calibration curves are used to understand the instrumental response to an analyte and predict the concentration in an unknown sample. Generally, a set of standard samples are made at various concentrations with a range than includes the unknown of interest and the instrumental response at each concentration is recorded. For more accuracy and to understand the error, the response at each concentration can be repeated so an error bar is obtained. The data are then fit with a function so that unknown concentrations can be predicted. Typically the response is linear, however, a curve can be made with other functions as long as the function is known. The calibration curve can be used to calculate the limit of detection and limit of quantitation.
When making solutions for a calibration curve, each solution can be made separately. However, that can take a lot of starting material and be time consuming. Another method for making many different concentrations of a solution is to use serial dilutions. With serial dilutions, a concentrated sample is diluted down in a stepwise manner to make lower concentrations. The next sample is made from the previous dilution, and the dilution factor is often kept constant. The advantage is that only one initial solution is needed. The disadvantage is that any errors in solution making&#8212;pipetting, massing, etc.&#8212;get propagated as more solutions are&#160;made. Thus, care must be taken when making the initial solution.

Calibration Curves: Principles and Applications

Source: Laboratory of Dr. B. Jill Venton - University of Virginia
Calibration curves are used to understand the instrumental response to an analyte and ...

Ultraviolet-Visible (UV-Vis) Spectroscopy

Source: Laboratory of Dr. B. Jill Venton - University of Virginia
Ultraviolet-visible (UV-Vis) spectroscopy is one of the most popular analytical techniques because it is very versatile and able to detect nearly every molecule. With UV-Vis spectroscopy, the UV-Vis light is passed through a sample and the transmittance of light by a sample is measured. From the transmittance (T), the absorbance can be calculated as A=-log (T). An absorbance spectrum is obtained that shows the absorbance of a compound at different wavelengths. The amount of absorbance at any wavelength is due to the chemical structure of the molecule.
UV-Vis can be used in a qualitative manner, to identify&#160;functional groups or confirm the identity of a compound by matching the absorbance spectrum. It can also be used in a quantitative manner, as concentration of the analyte is related to the absorbance using Beer&#39;s Law. UV-Vis spectroscopy is used to quantify the amount of DNA or protein in a sample, for water analysis, and as a detector for many types of chromatography. Kinetics of chemical reactions are also measured with UV-Vis spectroscopy by taking repeated UV-Vis measurements over time. UV-Vis measurements are generally taken with a spectrophotometer. UV-Vis is also a very popular detector for other analytical techniques, such as chromatography, because it can detect many compounds.
Typically, UV-Vis is not the most sensitive spectroscopy technique, because not a lot of light is absorbed over a short path length. Other spectroscopy techniques such as fluorescence have higher sensitivity, but they are not as generally applicable, as most molecules are not fluorescent. UV-Vis has a similar sensitivity to other absorbance measurements, such as infrared spectroscopy.

Ultraviolet-Visible UV-Vis Spectroscopy: Principle and Uses

Source: Laboratory of Dr. B. Jill Venton - University of Virginia
Ultraviolet-visible (UV-Vis) spectroscopy is one of the most popular analytical ...

Raman Spectroscopy for Chemical Analysis

Source: Laboratory of Dr. Ryoichi Ishihara &#8212; Delft University of Technology
Raman spectroscopy is a technique for analyzing vibrational and other low frequency modes in a system. In chemistry it is used to identify molecules by their Raman fingerprint. In solid-state physics it is used to characterize materials, and more specifically to investigate their crystal structure or crystallinity. Compared to other techniques for investigating the crystal structure (e.g. transmission electron microscope and x-ray diffraction) Raman micro-spectroscopy is non-destructive, generally requires no sample preparation, and can be performed on small sample volumes.
For performing Raman spectroscopy a monochromatic laser is shone on a sample. If required the sample can be coated by a transparent layer which is not Raman active (e.g., SiO2) or placed in DI water. The electromagnetic radiation (typically in the near infrared, visible, or near ultraviolet range) emitted from the sample is collected, the laser wavelength is filtered out (e.g., by a notch or bandpass filter), and the resulting light is sent through a monochromator (e.g., a grating) to a CCD detector. Using this, the inelastic scattered light, originating from Raman scattering, can be captured and used to construct the Raman spectrum of the sample.
In the case of Raman micro-spectroscopy the light passes through a microscope before reaching the sample, allowing it to be focused on an area as small as 1 &#181;m2. This allows accurate mapping of a sample, or confocal microscopy in order to investigate stacks of layers. Care has to be taken, however, that the small and intense laser spot does not damage the sample.
In this video we will briefly explain the procedure for obtaining a Raman spectra, and an example of a Raman spectrum captured from carbon nanotubes will be given.

Source: Laboratory of Dr. Ryoichi Ishihara &#8212; Delft University of Technology
Raman spectroscopy is a technique for analyzing vibrational and other ...

X-ray Fluorescence (XRF)

Source: Laboratory of Dr. Lydia Finney &#8212; Argonne National Laboratory
X-ray fluorescence is an induced, emitted radiation that can be used to generate spectroscopic information. X-ray fluorescence microscopy is a non-destructive imaging technique that uses the induced fluorescence emission of metals to identify and quantify their spatial distribution.

X-ray Fluorescence XRF: Elemental Analysis by Emitted Fluorescent X-Ray

Source: Laboratory of Dr. Lydia Finney &#8212; Argonne National Laboratory
X-ray fluorescence is an induced, emitted radiation that can be used to ...

Gas Chromatography (GC) with Flame-Ionization Detection

Source: Laboratory of Dr. B. Jill Venton - University of Virginia
Gas chromatography (GC) is used to separate and detect small molecular weight compounds in the gas phase. The sample is either a gas or a liquid that is vaporized in the injection port. Typically, the compounds analyzed are less than 1,000 Da, because it is difficult to vaporize larger compounds. GC is popular for environmental monitoring and industrial applications because it is very reliable and can be run nearly continuously. GC is typically used in applications where small, volatile molecules are detected and with non-aqueous solutions. Liquid chromatography is more popular for measurements in aqueous samples and can be used to study larger molecules, because the molecules do not need to vaporize. GC is favored for nonpolar molecules while LC is more common for separating polar analytes.
The mobile phase for gas chromatography is a carrier gas, typically helium because of its&#160;low molecular weight and being chemically inert. Pressure is applied and the mobile phase moves the analyte through the column. The separation is accomplished using a column coated with a stationary phase. Open tubular capillary columns are the most popular columns and have the stationary phase coated on the walls of the capillary. Stationary phases are often derivatives of polydimethylsiloxane, with 5&#8211;10% of the groups functionalized to tune the separation. Typical functional groups are phenyl, cyanopropyl, or trifluoropropyl groups. Capillary columns are usually 5&#8211;50 m&#160;long. Narrower columns have higher resolution but require higher pressures. Packed columns can also be used where the stationary phase is coated onto beads packed in the column. Packed columns are shorter, 1&#8211;5 m. Open tubular capillaries are generally preferred because they allow higher efficiencies, faster analyses, and have higher capacities.
Flame-ionization detection (FID) is a good general detector for organic compounds in GC that detects the amount of carbon in a sample. After the column, samples are burned in a hot, hydrogen-air flame. Carbon ions are produced by the combustion. While the overall efficiency of the process is low (only 1 in 105 carbon ions produce an ion in the flame) the total amount of ions is directly proportional to the amount of carbon in the sample. Electrodes are used to measure the current from the ions. FID is a destructive detector, as the entire sample is pyrolyzed. FID is unaffected by noncombustible gases and water.

Gas Chromatography GC with Flame-Ionization Detection

Source: Laboratory of Dr. B. Jill Venton - University of Virginia
Gas chromatography (GC) is used to separate and detect small molecular weight compounds ...

High-Performance Liquid Chromatography (HPLC)

Source: Dr. Paul Bower - Purdue University
High-performance liquid chromatography (HPLC) is an important analytical method commonly used to separate and quantify components of liquid samples. In this technique, a solution (first phase) is pumped through a column that contains a packing of small porous particles with a second phase bound to the surface. The different solubilities of the sample components in the two phases cause the components to move through the column with different average velocities, thus creating a separation of these components. The pumped solution is called the mobile phase, while the phase in the column is called the stationary phase.
There are several modes of liquid chromatography, depending upon the type of stationary and/or mobile phase employed. This experiment uses reversed-phase chromatography, where the stationary phase is non-polar, and the mobile phase is polar. The stationary phase to be employed is C18 hydrocarbon groups bonded to 3-&#181;m silica particles, while the mobile phase is an aqueous buffer with a polar organic modifier (acetonitrile) added to vary its eluting strength. In this form, the silica can be used for samples that are water-soluble, providing a broad range of applications. In this experiment, the mixtures of three components frequently found in diet soft drinks (namely caffeine, benzoate, and aspartame) are separated. Seven prepared solutions containing known amounts of the three species are used, and their chromatograms are then recorded.

Operation of High-Performance Liquid Chromatography HPLC

Source: Dr. Paul Bower - Purdue University
High-performance liquid chromatography (HPLC) is an important analytical method commonly used to separate and ...

Ion-Exchange Chromatography

Source: Laboratory of Dr. B. Jill Venton - University of Virginia
Ion-exchange chromatography is a type of chromatography that separates analytes based on charge. A column is used that is filled with a charged stationary phase on a solid support, called an ion-exchange resin. Strong cation-exchange chromatography preferentially separates out cations by using a negatively-charged resin while strong anion-exchange chromatography preferentially selects out anions by using a positively-charged resin. This type of chromatography is popular for sample preparation, for example in the cleanup of proteins or nucleic acid samples.
Ion-exchange chromatography is a two-step process. In the first step, the sample is loaded onto the column in a loading buffer. The binding of the charged sample to the column resin is based on ionic interactions of the resin to attract the sample of the opposite charge. Thus, charged samples of opposite polarity to the resin are strongly bound. Other molecules that are not charged or are of the opposite charge are not bound and are washed through the column. The second step is to elute the analyte that is bound to the resin. This is accomplished with a salt gradient, where the amount of salt in the buffer is slowly increased. Fractions are collected at the end of the column as the elution occurs and the purified sample of interest can be recovered in one of these fractions. Another technique, such as spectroscopy, may be needed to identify which fraction contains the sample. Ion-exchange chromatography is especially useful in protein studies, to isolate proteins of interest that have a specific charge or size, as size can determine the number of interactions with the resin.
Ion-exchange chromatography is a more general separation technique than affinity chromatography, which is also often used in preparing protein samples, where an antibody is attached to a column to bind one specific analyte. A new affinity column must be purchased for each analyte, while the same type of ion-exchange column, often with different eluting conditions, can be used to clean up many proteins of the same charge. Ion-exchange chromatography can also be used in conjunction with other types of chromatography that separate based on other properties. For example, size-exclusion chromatography separates based on size and could be used before ion-exchange chromatography to choose compounds of only a given size.

Ion-Exchange Chromatography: Principle and Protein Separation

Source: Laboratory of Dr. B. Jill Venton - University of Virginia
Ion-exchange chromatography is a type of chromatography that separates analytes based on ...

Capillary Electrophoresis (CE)

Source: Laboratory of Dr. B. Jill Venton - University of Virginia
Capillary electrophoresis (CE) is a separation technique that separates molecules in an electric field according to size and charge. CE is performed in a small glass tube called a capillary that is filled with an electrolyte solution. Analytes are separated due to differences in electrophoretic mobility, which varies with charge, solvent viscosity, and size. Traditional electrophoresis in gels is limited in the amount of voltage that can be applied because Joule heating effects will ruin the gel and the separation. Capillaries have a large surface area-to-volume ratio and thus dissipate heat better. Therefore, the voltages applied for a capillary electrophoresis experiment are quite large, often 10,000&#8211;20,000 V.
Capillary electrophoresis is useful for high-performance separations. Compared to liquid chromatography, CE separations are often faster and more efficient. However, capillary electrophoresis works best to separate charged molecules, which is not a limitation of liquid chromatography. CE has a greater peak capacity than high-performance liquid chromatography (HPLC), meaning the separations are more efficient and more peaks can be detected. The instrumentation can be very simple. However, HPLC is more versatile and many stationary and mobile phases have been developed for different types of molecules.

Capillary Electrophoresis: Principle, Procedure and Applications

Source: Laboratory of Dr. B. Jill Venton - University of Virginia
Capillary electrophoresis (CE) is a separation technique that separates molecules in an ...

Introduction to Mass Spectrometry

Source: Laboratory of Dr. Khuloud Al-Jamal - King&#39;s College London

Mass spectrometry is an analytical chemistry technique that enables the identification of unknown compounds within a sample, the quantification of known materials, the determination of the structure, and chemical properties of different molecules.
A mass spectrometer is composed of an ionization source, an analyzer,&#160;and a detector. The process involves the ionization of chemical compounds to generate ions. When using inductively coupled plasma (ICP), samples containing elements of interest are introduced into argon plasma as aerosol droplets. The plasma dries the aerosol, dissociates the molecules, and then removes an electron from the components to be detected by the mass spectrometer. Other ionization methods such as electrospray ionization (ESI) and matrix assisted laser desorption ionization (MALDI) are used to analyze biological samples. Following the ionization procedure, ions are separated in the mass spectrometer according to their mass-to-charge ratio (m/z), and the relative abundance of each ion type is measured. Finally, the detector commonly consists in an electron multiplier where the collision of ions with a charged anode leads to a cascade of increasing number of electrons, which can be detected by an electrical circuit connected to a computer.
In this video, the procedure of ICP-MS analysis will be described by the detection of 56Fe as an example.

Mass Spectrometer: Principle, Operation, Applications

Source: Laboratory of Dr. Khuloud Al-Jamal - King&#39;s College London

Mass spectrometry is an analytical chemistry technique that enables the ...

Scanning Electron Microscopy (SEM)

Source: Laboratory of Dr. Andrew J. Steckl &#8212; University of Cincinnati
A scanning electron microscope, or SEM, is a powerful microscope that uses electrons to form an image. It allows for imaging of conductive samples at magnifications that cannot be achieved using traditional microscopes. Modern light microscopes can achieve a magnification of ~1,000X, while typical SEM can reach magnifications of more than 30,000X.&#160;Because the SEM doesn&#8217;t use light to create images, the resulting pictures it forms are in black and white.&#160;
Conductive samples are loaded onto the SEM&#8217;s sample stage. Once the sample chamber reaches vacuum, the user will proceed to align the electron gun in the system to the proper location. The electron gun shoots out a beam of high-energy electrons, which travel through a combination of lenses and apertures and eventually hit the sample. As the electron gun continues to shoot electrons at a precise position on the sample, secondary electrons will bounce off of the sample. These secondary electrons are identified by the detector. The signal found from the secondary electrons is amplified and sent to the monitor, creating a 3D image. This video will demonstrate SEM sample preparation, operation, and imaging capabilities.

Scanning Electron Microscopy SEM: Operation, Image Analysis

Source: Laboratory of Dr. Andrew J. Steckl &#8212; University of Cincinnati
A scanning electron microscope, or SEM, is a powerful microscope that uses ...

Electrochemical Measurements of Supported Catalysts Using a Potentiostat/Galvanostat

Source: Laboratory of Dr. Yuriy Rom&#225;n &#8212; Massachusetts Institute of Technology
A potentiostat/galvanostat (often referred to as simply a potentiostat) is an instrument that measures current at an applied potential (potentiostatic operation) or measures potential at an applied current (galvanostatic operation) (Figure 1). It is the most commonly used instrument in the electrochemical characterization of anode and cathode materials for fuel cells, electrolyzers, batteries, and supercapacitors.
Conventionally, these anode and cathode materials are interfaced with a potentiostat via a three-electrode electrochemical cell. The electrode leads from the potentiostat are connected to the reference electrode, the counter electrode (often called the auxiliary electrode), and the working electrode (which contains the test material of interest). The electrochemical cell is then filled with a high ionic strength electrolyte solution, such as an acidic, alkaline, or salt solution. The media for this high ionic strength solution is typically aqueous; however, for applications necessitating higher operating cell potential windows, such as batteries and supercapacitors, non-aqueous media is often used. The cell media is degassed with an inert gas (to prevent unwanted side reactions) or with a test gas (if the test reaction involves a gas at one of the electrodes).
Alternatively, a salt bridge or membrane is employed to maintain ionic contact if the two half cells are to be measured in different electrolytes. In heterogeneous electrocatalysis, this type of &#34;two compartment&#34; cell is often used if the test molecule at the working electrode is also reactive at the counterelectrode. This happens frequently as the counterelectrode typically employed is platinum, which is a highly active catalyst for many reactions. Here, single compartment cells will be used, where all three electrodes are in the same media.
This video will explain the process of polishing a working electrode, preparing a catalyst ink, mounting the catalyst ink onto the working electrode, preparing the electrochemical cell, and then performing electrochemical measurements. The measurements that are performed include: cyclic voltammetry (CV), linear sweep voltammetry (LSV), chronopotentiometry (CP), and chronoamperometry (CA).
<img alt="Figure 1" src="/files/ftp_upload/5698/5698fig1.jpg" /> 
Figure 1. An example of a single compartment electrochemical cell. a.) Teflon cap, b.) glass cell, c.) Pt wire counter electrode, d.) working electrode, e.) Ag/AgCl reference electrode, f.) 0.5 M aqueous sulfuric acid electrolyte solution.

Potentiostat/Galvanostat and Voltammetry

Source: Laboratory of Dr. Yuriy Rom&#225;n &#8212; Massachusetts Institute of Technology
A potentiostat/galvanostat (often referred to as simply a ...

Cyclic Voltammetry (CV)

Source: Laboratory of Dr. Kayla Green &#8212; Texas Christian University
A Cyclic Voltammetry (CV) experiment involves the scan of a range of potential voltages while measuring current. In the CV experiment, the potential of an immersed, stationary electrode is scanned from a predetermined starting potential to a final value (called the switching potential) and then the reverse scan is obtained. This gives a &#39;cyclic&#39; sweep of potentials and the current vs. potential curve derived from the data is called a cyclic voltammogram. The first sweep is called the &#39;forward scan&#39; and the return wave is called the &#39;reverse scan&#39;. The potential extremes are termed the &#39;scan window&#39;. The magnitude of reduction and oxidation currents and the shape of the voltammograms are highly dependent on analyte concentration, scan rates, and experimental conditions. By varying these factors, cyclic voltammetry can yield information regarding the stability of transition metal oxidation state in the complexed form, reversibility of electron transfer reactions, and information regarding reactivity. This video will explain the basic setup for a cyclic voltammetry experiment including analyte preparation and setting up the electrochemical cell. A simple cyclic voltammetry experiment will be presented.

Cyclic Voltammetry CV: Measuring Redox Potentials and Currents

Source: Laboratory of Dr. Kayla Green &#8212; Texas Christian University
A Cyclic Voltammetry (CV) experiment involves the scan of a range of potential ...

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