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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 ...

Synthesis Of A Ti(III) Metallocene Using Schlenk Line Technique

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University&#160;
Inorganic chemists often work with highly air- and water-sensitive compounds. The two most common and practical methods for air-free synthesis utilize either Schlenk lines or gloveboxes. This experiment will demonstrate how to perform simple manipulations on a Schlenk line with a focus on solvent preparation and transfer. Through the synthesis of a reactive Ti(III) metallocene complex, we will demonstrate a new, simple method to degas solvent as well as how to transfer solvent by cannula and by syringe on a Schlenk line.
The synthesis of a Ti(III) metallocene compound 3 is shown in Figure 1.1 Compound 3 is highly reactive with O2, (see oxidation of compound 3 to Ti(IV) metallocene 4 shown in Figure 1). Therefore, it is important to run the synthesis under anaerobic conditions. The synthesis of target compound 3 can be monitored visually and progresses through one additional color change before arriving at the desired product, which is blue in color. If during the experiment there is an observed color change from blue to yellow (or green = blue + yellow), this is an indication that O2 entered the flask and that undesired oxidation of compound 3 to the Ti(IV) analog (compound 4) has occurred.
<img alt="Figure 1" src="/files/ftp_upload/10428/10428fig1v2.jpg" /> 
Figure 1. Synthesis of Ti(III) metallocene compound 3 and it&#39;s reaction with O2.

Synthesis Of A TiIII Metallocene Using Schlenk Line Technique

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University&#160;
Inorganic chemists often work with highly air- and water-sensitive ...

Glovebox and Impurity Sensors

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University&#160;
The glovebox provides a straightforward means to handle air- and moisture-sensitive solids and liquids. The glovebox is what it sounds like: a box with gloves attached to one or more sides, which allows the user to perform manipulations within the glovebox under an inert atmosphere.
For manipulations under inert atmospheres, chemists can choose between Schlenk or high-vacuum techniques and a glovebox. Schlenk and particularly high-vacuum techniques offer a higher degree of control of the atmosphere, and are thus suitable for reactions that are greatly air- and moisture-sensitive. The glovebox, however, provides greater access for manipulations in an inert atmosphere. Weighing out reagents, filtering reactions, preparing samples for spectroscopy, and growing crystals are all examples of routine procedures that are more readily performed in a glovebox versus a Schlenk/vacuum manifold. Advancements in glovebox design have increased its performance, such as running reactions at reduced temperatures and spectroscopy within the glovebox.
This video will demonstrate how to bring items in and out of the glovebox and how to qualitatively ensure a good working environment. Basic manipulations within a glovebox will be demonstrated through the synthesis of sodium benzophenone.

Glovebox-Operating Procedure and Use of Impurity Sensors

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University&#160;
The glovebox provides a straightforward means to handle air- and ...

Purification of Ferrocene by Sublimation

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University&#160;
Sublimation, the direct phase transition of a solid into a gas without first becoming a liquid, takes place at temperatures and pressures lower than that of the compound&#39;s triple point (Figure 1).The process of sublimation can be utilized to purify both organic and inorganic solids. During the purification technique, a solid is heated directly into the gas-phase. All non-volatile impurities are left behind while the vaporized compound is then collected (deposition) as a solid on a cold surface. Here, we will use sublimation to purify ferrocene, an inorganic solid with a triple point temperature of 183 &#176;C.1
<img alt="Figure 1" src="/files/ftp_upload/10429/10429fig1v2.jpg" /> 
Figure 1. Generic phase diagram. The colored lines represent the pressure and temperature requirements for phase transitions. Distillation of a solid will occur at pressures and temperatures above the triple point, represented by the green line in the phase diagram. The blue line represents the temperature and pressure conditions where sublimation occurs.

Sublimation: Concept, Purification of Ferrocene, Applications

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University&#160;
Sublimation, the direct phase transition of a solid into a gas without ...

The Evans Method

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University&#160;
While most organic molecules are diamagnetic, wherein all their electrons are paired up in bonds, many transition metal complexes are paramagnetic, which has ground states with unpaired electrons. Recall Hund&#39;s rule, which states that for orbitals of similar energies, electrons will fill the orbitals to maximize the number of unpaired electrons before pairing up. Transition metals have partially populated d-orbitals whose energies are perturbed to varying extents by coordination of ligands to the metal. Thus, the d-orbitals are similar in energy to one another, but are not all degenerate. This allows for complexes to be diamagnetic, with all electrons paired up, or paramagnetic, with unpaired electrons.
Knowing the number of unpaired electrons in a metal complex can provide clues into the oxidation-state and geometry of the metal complex, as well as into the ligand field (crystal field) strength of the ligands. These properties greatly impact the spectroscopy and reactivity of transition metal complexes, and so are important to understand.
One way to count the number of unpaired electrons is to measure the magnetic susceptibility, &#967;, of the coordination compound. Magnetic susceptibility is the measure of magnetization of a material (or compound) when placed in an applied magnetic field. Paired electrons are slightly repelled by an applied magnetic field, and this repulsion increases linearly as the strength of the magnetic field increases. On the other hand, unpaired electrons are attracted (to a larger extent) to a magnetic field, and the attraction increases linearly with magnetic field strength. Therefore, any compound with unpaired electrons will be attracted to a magnetic field.1
When we measure the magnetic susceptibility, we obtain information on the number of unpaired electrons from the magnetic moment, &#181;. The magnetic susceptibility is related to the magnetic moment, &#181; by Equation 12:
<img alt="Equation 1" src="/files/ftp_upload/10304/10304eq1.jpg" /> (1)
The constant <img alt="Equation 2" src="/files/ftp_upload/10304/10304eq2.jpg" /> = [(3kB)/N&#946;2)], where &#946;= Bohr magneton of the electron (0.93 x 10-20 erg gauss-1), N = Avogadro&#39;s number, and kB = Boltzmann constant 
XM = molar magnetic susceptibility (cm3/mol) 
T = temperature (K) 
&#181; = magnetic moment, measured in units of Bohr magneton, &#181;B = 9.27 x 10-24 JT-1
The magnetic moment for complexes is given by Equation 21:
<img alt="Equation 3" src="/files/ftp_upload/10304/10304eq3.jpg" /> (2)
g = gyromagnetic ratio = 2.00023 &#181;B 
S = spin quantum number = &#8721;ms = [number of unpaired electrons, n]/2 
L = orbital quantum number = &#8721;ml
This equation has both orbital and spin contributions. For first-row transition metal complexes, the orbital contribution is small and hence can be omitted, so the spin-only magnetic moment is given by Equation 3:
<img alt="Equation 4" src="/files/ftp_upload/10304/10304eq4.jpg" /> (3)
The spin-only magnetic moment can thus directly give the number of unpaired electrons. This approximation can also be made for heavier metals, though orbital contributions may be significant for second and third row transition metals. This contribution may be so significant that it inflates the magnetic moment enough that the compound appears to have more unpaired electrons than it does. Therefore, additional characterization may be required for these complexes.
In this experiment, the solution magnetic moment of tris(acetylacetonato)iron(III) (Fe(acac)3) is determined experimentally using Evans method in chloroform.

The Evans Method: Calculating Unpaired Electrons and Magnetic Susceptibility

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University&#160;
While most organic molecules are diamagnetic, wherein all their ...

Single Crystal and Powder X-ray Diffraction

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University&#160;
X-ray crystallography is a technique that uses X-rays to study the structure of molecules. X-ray diffraction (XRD) experiments are routinely carried out with either single-crystal or powdered samples.
Single-crystal XRD:
Single-crystal XRD allows for absolute structure determination. With single-crystal XRD data, the exact atomic positions can be observed, and thus bond lengths and angles can be determined. This technique provides the structure within a single crystal, which does not necessarily represent the bulk of the material. Therefore, additional bulk characterization methods must be utilized to prove the identity and purity of a compound.
Powder XRD:
Unlike single-crystal XRD, powder XRD looks at a large sample of polycrystalline material and therefore is considered a bulk characterization technique. The powder pattern is considered a &#34;fingerprint&#34; for a given material; it provides information about the phase (polymorph) and crystallinity of the material. Typically, powder XRD is used to study minerals, zeolites, metal-organic frameworks (MOFs), and other extended solids. Powder XRD can also be used to establish bulk purity of molecular species.
Previously, we have seen how to grow X-ray quality crystals (see video in Essentials of Organic Chemistry series). Here we will learn the principles behind XRD. We will then collect both single-crystal and powder data on Mo2(ArNC(H)NAr)4, where Ar = p-MeOC6H5.

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University&#160;
X-ray crystallography is a technique that uses X-rays to study the ...

Electron Paramagnetic Resonance (EPR) Spectroscopy

Source: David C. Powers, Tamara M. Powers, Texas A&#38;M
In this video, we will learn the basic principles behind Electron Paramagnetic Resonance (EPR). We will use EPR spectroscopy to study how dibutylhydroxy toluene (BHT) behaves as an antioxidant in the autoxidation of aliphatic aldehydes.

EPR Spectroscopy, Paramagnetic Molecule and Zeeman Effect

Source: David C. Powers, Tamara M. Powers, Texas A&#38;M
In this video, we will learn the basic principles behind Electron Paramagnetic Resonance (EPR). ...

M&#246;ssbauer Spectroscopy

Source: Joshua Wofford,&#160;Tamara M. Powers, Department of Chemistry, Texas A&#38;M University&#160;
M&#246;ssbauer spectroscopy is a bulk characterization technique that examines the nuclear excitation of an atom by gamma rays in the solid state. The resulting M&#246;ssbauer spectrum provides information about the oxidation state, spin state, and electronic environment around the target atom, which, in combination, gives evidence about the electronic structure and ligand arrangement (geometry) of the molecule. In this video, we will learn about the basic principles of M&#246;ssbauer spectroscopy and collect a zero field 57Fe M&#246;ssbauer spectrum of ferrocene.

M&amp;#246;ssbauer Spectroscopy

Source: Joshua Wofford,&#160;Tamara M. Powers, Department of Chemistry, Texas A&#38;M University&#160;
M&#246;ssbauer spectroscopy is a bulk ...

Lewis Acid-Base Interaction in Ph3P-BH3

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University&#160;
One of the goals of chemistry is to use models that account for trends and provide insights into the properties of reactants that contribute to reactivity. Substances have been classified as acids and bases since the time of the ancient Greeks, but the definition of acids and bases has been modified and expanded over the years.1
The ancient Greeks would characterize substances by taste, and defined acids as those that were sour-tasting, such as lemon juice and vinegar. The term &#34;acid&#34; is derived from the Latin term for &#34;sour-tasting.&#34; Bases were characterized by their ability to counteract or neutralize acids. The first bases characterized were those of ashes from a fire, which were mixed with fats to make soap. In fact, the term &#34;alkaline&#34; is derived from the Arabic word for &#34;roasting.&#34; Indeed, it has been known since ancient times that acids and bases can be combined to give a salt and water.
The first widely-used description of an acid is that of the Swedish chemist, Svante Arrhenius, who in 1894 defined acids as substances which dissociate in water to give hydronium ions, and bases as substances which dissociate in water to give hydroxide ions. This definition is thus limited to aqueous acids and necessitates that an acid contribute a proton.2 For example, in water, HCl is an acid, as it dissociates to give the hydronium ion (H3O)+ and the chloride ion. Boron trichloride would not be considered an acid, as in water it hydrolyzes to give B(OH)3 and 3 HCl; the product HCl though is an Arrhenius acid.
In 1923, Johannes Nicolaus Br&#248;nsted and Martin Lowry independently defined acids and bases on their ability to donate and accept hydrogen ions, or protons. Thus came the concept of acid-base conjugate pairs, and the expansion of the definition of acids and bases in solvents other than water. For example, ammonium is an acid, as it can donate a proton and generate ammonia. Ammonia can accept a proton, to give ammonium. Thus, ammonia is the conjugate base of ammonium. This acid-base reaction can occur in water, ammonia, or other solvents.
This video deals with the acid-base definition of the American chemist, Gilbert N. Lewis, who also defined acids and bases in 1923. Indeed, this is the same Lewis from Lewis-dot structures in General Chemistry. His approach focuses not on the ability of acids and bases to donate and accept protons, but rather on their ability to accept and donate electron pairs, respectively. This encompasses the Br&#248;nsted-Lowry definition, as H+ accepts an electron pair from a Br&#248;nsted base during protonation. However, it greatly expands the definition of an acid, now encompassing metal ions and main-group compounds. Here, we compare the 31P NMR of the Lewis acid-base adduct Ph3P-BH3 to free triphenylphosphine.

Lewis Acid-Base Interaction in Ph3P-BH3

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University&#160;
One of the goals of chemistry is to use models that account for trends ...

Structure Of Ferrocene

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University&#160;
In 1951, Kealy and Pauson reported to Nature the synthesis of a new organometallic compound, ferrocene.1&#160;In their original report, Pauson suggested a structure for ferrocene in which the iron is singly bonded (sigma bonds) to one carbon atom of each cyclopentadiene ligand (Figure 1, Structure I).1,2,3&#160;This initial report led to wide-spread interest in the structure of ferrocene, and many leading scientists participated in the structure elucidation of this interesting new molecule. Wilkinson and Woodward were quick to suggest an alternative formulization where the iron is &#34;sandwiched&#34; between two cyclopentadiene ligands, with equal binding to all 10 carbon atoms (Figure 1, Structure II).4&#160;Here, we will synthesize ferrocene and decide, based on experimental data (IR and 1H NMR), which of these structures is observed. In addition, we will study the electrochemistry of ferrocene by collecting a cyclic voltammogram. In the course of this experiment, we introduce the 18-electron rule and discuss valence electron counting for transition metal complexes.
<img alt="Figure 1" src="/files/ftp_upload/10347/10347fig1v2.jpg" /> 
Figure 1. Two proposed structures of ferrocene.

Structure Determination of Ferrocene-an Organometallic Complex

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University&#160;
In 1951, Kealy and Pauson reported to Nature the synthesis of a new ...

Application of Group Theory to IR Spectroscopy

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University
Metal carbonyl complexes are used as metal precursors for the synthesis of organometallic complexes as well as catalysts. Infrared (IR) spectroscopy is one of the most utilized and informative characterization methods of CO containing compounds. Group theory, or the use of mathematics to describe the symmetry of a molecule, provides a method to predict the number of IR active C-O vibrational modes within a molecule. Experimentally observing the number of C-O stretches in the IR is a direct method to establish the geometry and structure of the metal carbonyl complex.
In this video, we will synthesize the molybdenum carbonyl complex Mo(CO)4[P(OPh)3]2, which can exist in the cis- and trans-forms (Figure 1). We will use group theory and IR spectroscopy to determine which isomer is isolated.
<img alt="Figure 1" src="/files/ftp_upload/10442/10442fig1.jpg" /> 
Figure 1. The cis- and trans-isomers of Mo(CO)4[P(OPh)3]2.

Symmetry Elements, Group Theory and IR-Active Vibrations

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University
Metal carbonyl complexes are used as metal precursors for the synthesis of ...

Molecular Orbital (MO) Theory

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University
This protocol serves as a guide in the synthesis of two metal complexes featuring the ligand 1,1&#39;-bis(diphenylphosphino)ferrocene (dppf): M(dppf)Cl2, where M = Ni or Pd. While both of these transition metal complexes are 4-coordinate, they exhibit different geometries at the metal center. Using molecular orbital (MO) theory in conjunction with 1H NMR and Evans method, we will determine the geometry of these two compounds.

Molecular Orbital MO Theory: Predicting Geometry of Transition Metal Complexes

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University
This protocol serves as a guide in the synthesis of two metal complexes ...

Quadruply Metal-Metal Bonded Paddlewheels

Source: Corey Burns, Tamara M. Powers, Department of Chemistry, Texas A&#38;M University
Paddlewheel complexes are a class of compounds comprised of two metal ions (1st, 2nd, or 3rd row transition metals) held in proximity by four bridging ligands (most commonly formamidinates or carboxylates) (Figure 1). Varying the identity of the metal ion and the bridging ligand provides access to large families of paddlewheel complexes. The structure of paddlewheel complexes allows for metal-metal bonding, which plays a vital role in the structure and reactivity of these complexes. Due to the diversity of electronic structures that are available to paddlewheel complexes - and the corresponding differences in M-M bonding displayed by these structures - paddlewheel complexes have found application in diverse areas, such as in homogeneous catalysis and as building blocks for metal-organic frameworks (MOFs). Understanding the electronic structure of the M-M bonds in paddlewheel complexes is critical to understanding their structures and thus to application of these complexes in coordination chemistry and catalysis.
<img alt="Figure 1" src="/files/ftp_upload/10441/10441fig1v2.jpg" /> 
Figure 1. General structure of paddlewheel complexes, where M can be a 1st, 2nd, or 3rd row transition metal.
When two transition metals are held in close proximity the d-orbitals overlap, which can result in the formation of M-M bonds. Overlapping d-orbitals can form three types of bonds - &#963;, &#960;, and &#948; - depending on the symmetry of the orbitals involved. If we assign the molecular z-axis to be coplanar with the M-M bond, a &#963; bond is formed by overlap of the dz2 orbitals and &#960; bonds are formed by overlap of the dxz and dyz orbitals. &#948; bonds are generated by overlap of d-orbitals that have two planar nodes (dxy and dx2&#8211;y2). As a result, all four lobes of the d-orbital overlap and the corresponding &#948; bond has two planar nodes (Figure 2). In theory, with the addition of &#948; bonds, paddlewheel complexes are capable of supporting quintuple bonds, or five bonds between metal atoms.1 In most complexes, the dx2&#8211;y2 forms strong metal-ligand bonds and does not meaningfully contribute to M-M bonding. Thus, quadruple bonds are the maximum bond order in many complexes.
<img alt="Figure 2" src="/files/ftp_upload/10441/10441fig2v2.jpg" /> 
Figure 2. Visual representation of &#963;, &#960;, and &#948; bonding MOs resulting from the linear combination of metal d-orbitals. The dz2 atomic orbitals have the best spatial overlap, followed by the dxz and dyz orbitals. The dxy atomic orbitals have the least amount of spatial overlap.
In this video, we will synthesize the dimolybdenum paddlewheel complex Mo2(ArNC(H)NAr)4, where Ar = p-(MeO)C6H4, which features a quadruple bond. We will characterize the compound by NMR spectroscopy and use X-ray crystallography to study the M-M bond.

Source: Corey Burns, Tamara M. Powers, Department of Chemistry, Texas A&#38;M University
Paddlewheel complexes are a class of compounds comprised of two ...

Dye-sensitized Solar Cells

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University
Today&#39;s modern world requires the use of a large amount of energy. While we harness energy from fossil fuels such as coal and oil, these sources are nonrenewable and thus the supply is limited. To maintain our global lifestyle, we must extract energy from renewable sources. The most promising renewable source, in terms of abundance, is the sun, which provides us with more than enough solar energy to fully fuel our planet many times over.
So how do we extract energy from the sun? Nature was the first to figure it out: photosynthesis is the process whereby plants convert water and carbon dioxide to carbohydrates and oxygen. This process occurs in the leaves of plants, and relies on the chlorophyll pigments that color the leaves green. It is these colored molecules that absorb the energy from sunlight, and this absorbed energy which drives the chemical reactions.
In 1839, Edmond Becquerel, then a 19-year old French physicist experimenting in his father&#39;s lab, created the first photovoltaic cell. He illuminated an acidic solution of silver chloride that was connected to platinum electrodes which generated a voltage and current.1&#160;Many discoveries and advances were made in the late 19th and first half the 20th century, and it was only in 1954 that the first practical solar cell was built by Bell Laboratories. Starting in the 1950s, solar cells were used to power satellites in space.2
Solar cells are electrical devices that utilize light to create a current. This video demonstrates preparation and testing of one such type of cell, the dye-sensitized solar cell (DSSC). First invented at UC Berkeley by Brian O&#39;Regan and Michael Gr&#228;tzel, Gr&#228;tzel pursued this work at the &#201;cole Polytechnique F&#233;d&#233;rale de Lausanne in Switzerland, culminating in the first highly efficient DSSC in 1991.3&#160;These solar cells, like plants, use a dye to help harness energy from the sun.

Dye-sensitized Solar Cells: Principle, Fabrication and Performance

Source: Tamara M. Powers, Department of Chemistry, Texas A&#38;M University
Today&#39;s modern world requires the use of a large amount of energy. While ...

Synthesis of an Oxygen-Carrying Cobalt(II) Complex

Source: Deepika Das, Tamara M. Powers, Department of Chemistry, Texas A&#38;M University
Bioinorganic chemistry is the field of study that investigates the role that metals play in biology. Approximately half of all proteins contain metals and it is estimated that up to one third of all proteins rely on metal-containing active sites to function. Proteins that feature metals, called metalloproteins, play a vital role in a variety of cell functions that are necessary for life. Metalloproteins have intrigued and inspired synthetic inorganic chemists for decades, and many research groups have dedicated their programs to modeling the chemistry of metal-containing active sites in proteins through the study of coordination compounds.
The transport of O2 is a vital process for living organisms. O2-transport metalloproteins are responsible for binding, transporting, and releasing oxygen, which can then be used for life processes such as respiration. The oxygen-carrying cobalt coordination complex, [N,N&#39;-bis(salicylaldehyde)ethylenediimino]cobalt(II) [Co(salen)]2&#160;has been studied extensively to gain understanding about how metal complexes reversibly bind O2.1
In this experiment, we will synthesize [Co(salen)]2 and study its reversible reaction with O2 in the presence of dimethylsulfoxide (DMSO). First, we will quantify the amount of O2 consumed upon exposure of [Co(salen)]2 to DMSO. We will then visually observe the release of O2 from the [Co(salen)]2-O2 adduct by exposing the solid to CHCl3.

Synthesis of an Oxygen-Carrying [Cosalen]2 Complex

Source: Deepika Das, Tamara M. Powers, Department of Chemistry, Texas A&#38;M University
Bioinorganic chemistry is the field of study that investigates ...

Photochemical Initiation Of Radical Polymerization Reactions

Source: David C. Powers, Tamara M. Powers, Texas A&#38;M
In this video, we will carry out the photochemically initiated polymerization of styrene to generate polystyrene, which is an important commodity plastic. We will learn the fundamentals of photochemistry and use simple photochemistry to initiate radical polymerization reactions. Specifically, in this module we will examine the photochemistry of benzoyl peroxide and its role as a photo-initiator of styrene polymerization reactions. In the described experiments, we will investigate the role of wavelength, photon absorption, and excited state structure on the efficiency (measured as quantum yield) of photochemical reactions.

Source: David C. Powers, Tamara M. Powers, Texas A&#38;M
In this video, we will carry out the photochemically initiated polymerization of styrene to ...

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