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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 ...

Testing the Heat Transfer Efficiency of a Finned-tube Heat Exchanger

Source: Michael G. Benton and Kerry M. Dooley, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Heat exchangers transfer heat from one fluid to another fluid. Multiple classes of heat exchangers exist to fill different needs. Some of the most common types are shell and tube exchangers and plate exchangers1. Shell and tube heat exchangers use a system of tubes through which fluid flows1. One set of tubes contains the liquid to be cooled or heated, while the second set contains the liquid that will either absorb heat or transmit it1. Plate heat exchangers use a similar concept, in which plates are closely joined together with a small gap between each for liquid to flow1. The fluid flowing between the plates alternates between hot and cold so that heat will move into or out of the necessary streams1. These exchangers have large surface areas, so they are usually more efficient1.
The goal for this experiment is to test the heat transfer efficiency of a finned-tube heat exchanger (Figure 1) and compare it to the theoretical efficiency of a heat exchanger without fins. The experimental data will be measured for three different flow rates of monoethylene glycol (MEG). Two different water flow rates for each MEG flow rate will be used. Using the Wilson plot method the heat transfer coefficients will be determined from the experimental data. Additionally, the Reynold&#39;s number and the amount of heat transferred will be compared for flow with and without the fins to evaluate heat transfer efficiency.
<img alt="Figure 1" src="/files/ftp_upload/10437/10437fig1.jpg" /> 
Figure 1: Finned-tube Heat Exchanger. 1) MEG outlet temperature 2) water inlet temperature 3) MEG inlet temperature 4) water outlet temperature 5) water meter 6) MEG accumulation sight glass/cylinder.

Heat Transfer Efficiency Test of a Finned-tube Heat Exchanger

Source: Michael G. Benton and Kerry M. Dooley, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Heat exchangers transfer ...

Using a Tray Dryer to Investigate Convective and Conductive Heat Transfer

Source: Michael G. Benton and <a href="https://www.jove.com/author/Kerry+M._Dooley">Kerry M. Dooley</a>, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Dryers are utilized in numerous industrial processes. The function of a dryer is to use heat transfer processes to dry solids. A variety of dryer types exist. Adiabatic dryers use convection and direct contact with gases to dry solids, whereas non-adiabatic dryers use methods other than heated gas contact to dry1, including conduction, radiation, and radio frequency drying1. Dryers can be used for batch processes or they be in continuous use1.
In this experiment, the effects of temperature and air velocity on the drying rate of sand will be determined using a tray dryer. Three different power settings (1000 W, 1500 W and 2500 W) for two different air flow rates will be tested, providing a total of six data sets. From this data, the heat and mass transfer coefficients can be calculated.

Tray Dryer Test to Investigate Convective and Conductive Heat Transfer

Source: Michael G. Benton and Kerry M. Dooley, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Dryers are utilized in ...

Viscosity of Propylene Glycol Solutions

Source: Michael G. Benton and <a href="https://www.jove.com/author/Kerry+M._Dooley">Kerry M. Dooley</a>, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Viscosity is a measure of a fluid&#39;s resistance to flow, and it is a useful parameter in the design of efficient product processing and quality control in a wide range of industries. A variety of viscometers are used to obtain the most accurate readings of experimental materials. The standard method of measuring viscosity is through a glass tube viscometer, which estimates viscosity by measuring the amount of time it takes fluid to flow through a capillary tube made of glass1.
Rotational viscometers operate by applying shearing forces and measuring the time it takes a flowing1. These viscometers make use of the flowing force of the fluid, and they can use either a spring system or a digital encoder system1. Different measuring systems exist as well, with the standard being a cone and plate system, where fluid flows under the cone shape and over the plate, in order to minimize shear stress1. Parallel plate systems use two parallel plates and is ideal for measuring across temperature gradients, allowing a smooth transition1. Couette systems use a cup and filling material, and the fluid flows in between the two1. These systems are best for materials with low viscosity, since this system minimizes shear stress, but the system is also harder to operate routinely due to issues with cleaning and needing larger volumes of fluid1.
In this experiment, a Cannon-Fenske viscometer will be used to measure the viscosities of several propylene glycol solutions to determine the relationship between viscosity and composition.

Measuring Viscosity of Propylene Glycol Solution Using a Glasstube Viscometer

Source: Michael G. Benton and Kerry M. Dooley, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Viscosity is a measure of a ...

Porosimetry of a Silica Alumina Powder

Source: <a href="https://www.jove.com/author/Kerry+M._Dooley">Kerry M. Dooley</a> and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Surface area and pore size distribution are attributes used by adsorbent and catalyst manufacturers and users to ensure quality control and to determine when products are at the end of their useful lives. The surface area of a porous solid is directly related to its adsorption capacity or catalytic activity. The pore size distribution of an adsorbent or catalyst is controlled such that pores are large enough to easily admit molecules of interest, but small enough to provide a high surface area per mass.
Surface area and pore size distribution can be measured by the technique of isothermal nitrogen adsorption/desorption. In this experiment, a nitrogen porosimeter will be used to measure the surface area and pore size distribution of a silica/alumina powder.

Measuring Porosimetry of Silica Alumina Powder Using a Nitrogen Porosimeter

Source: Kerry M. Dooley and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Surface area and pore size ...

Demonstration of the Power Law Model Through Extrusion

Source: <a href="https://www.jove.com/author/Kerry+M._Dooley">Kerry M. Dooley</a> and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Polymer melts are often formed into simple shapes or &#34;extrudates&#34;, such as cylindrical pellets, flat sheets, or pipe, using an extruder.1 Polyolefins are among the most common extrudable polymers. Extrusion involves transporting and melting solid feed, which is sometimes mixed with non-polymeric materials, and the pressure build-up and transport of the melt or mixture. It is applied to thermoplastic polymers, which deform when heated and resume their earlier &#34;no-flow&#34; properties when cooled.
Using a simple lab extruder, the effect of operating conditions on polymer output and pressure drop can be examined and the resulting data can be correlated using the &#34;Power Law&#34; model for flow of polymer melts and solutions. This model is used to scale up the process to more complex extruders. The relationship between operating conditions and the deviations from theoretical displacement behavior (&#34;slippage&#34;) and extrudate shape (&#34;die swell&#34;) can be determined.
In this experiment, a typical thermoplastic polymer, such as a high-density polyethylene (HDPE) copolymer (of ethylene + a longer chain olefin) will be used. The operating temperature for the die and zones depend on the material. The flow rate can be determined by weighing the die output at timed intervals. All other necessary data (screw speed, zone temperatures, pressure entering the die) can be read from the instrument panel.

Power Law Model Through Extrusion, Start-up and Heat-soak

Source: Kerry M. Dooley and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Polymer melts are often ...

Gas Absorber

Source: Michael G. Benton and <a href="https://www.jove.com/author/Kerry+M._Dooley">Kerry M. Dooley</a>, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Gas absorbers are used to remove contaminants from gas streams. Multiple designs are used to accomplish this objective1. A packed bed column uses gas and liquid streams running counter to each other in a column packed with loose packing materials, such as ceramics, metals, and plastics, or structured packing1. The packed bed uses surface area created by the packing to create a maximum amount of efficient contact between the two phases1. The systems are low maintenance and can handle corrosive materials with high mass transfer rates1. Spray columns are another type of absorber, which uses constant direct contact between the two phases, with gas moving up and liquid being sprayed down into the gas flow1. This system only has one stage and poor mass transfer rates, but is very effective for solutes with high liquid solubility1.
The goal of this experiment is to determine how variables including gas flow rate, water flow rate, and carbon dioxide concentration affect the overall mass transfer coefficient in a gas absorber. Understanding how these parameters affect CO2 removal enables contaminant removal to be optimized. The experiment uses a randomly packed water counter-flow gas absorption column. Eight runs with two different gas flow rates, liquid flow rates, and CO2 concentrations were used. During each run, the partial pressures were taken from the bottom, middle, and top of the column unit, and the equilibrium partial pressures were calculated. These pressures were then used to find the mass transfer coefficient, and the mass transfer coefficients were compared to theoretical values.

Gas Absorber, Variable Flow Rate and Mass Transfer Coefficient

Source: Michael G. Benton and Kerry M. Dooley, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Gas absorbers are used to ...

Vapor-liquid Equilibrium

Source: Michael G. Benton and <a href="https://www.jove.com/author/Kerry+M._Dooley">Kerry M. Dooley</a>, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Vapor-liquid equilibrium is paramount in engineering applications such as distillation, environmental modeling, and general process design. Understanding the interactions of components in a mixture is very important in designing, operating and analyzing such separators. The activity coefficient is an excellent tool for relating molecular interactions to mixture composition. Finding the molecular interaction parameters allows future prediction of the activity coefficients for a mixture using a model.
Vapor-liquid equilibrium is a critical factor in common processes in the chemical industry, such as distillation. Distillation is the process of separating liquids by their boiling point. A liquid mixture is fed into a distillation unit or column, then boiled. Vapor-liquid equilibrium data is useful for determining how liquid mixtures will separate. Because the liquids have different boiling points, one liquid will boil into a vapor and rise in the column, while the other will stay as a liquid and drain through the unit. The process is very important in a variety of industries.
In this experiment, the activity coefficients of mixtures of various compositions of methanol, isopropanol, and deionized water will be obtained using a vapor-liquid equilibrium apparatus and gas chromatograph. Additionally, the binary interaction parameters of the system will be determined using Wilson&#39;s equation and the activity coefficients.

Vapor-liquid Equilibrium, Activity Coefficient, Fugacity

Source: Michael G. Benton and Kerry M. Dooley, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Vapor-liquid equilibrium is ...

The Effect of Reflux Ratio on Tray Distillation Efficiency

Source: <a href="https://www.jove.com/author/Kerry+M._Dooley">Kerry M. Dooley</a> and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Tray and packed columns are both commonly used for distillation, absorption, and stripping separation operations.1,2 The goal of this experiment is to distill a mixture of alcohols (methanol, isopropanol) and water in a sieve tray column and examine how closely simple theories of distillation based on equilibrium assumptions are followed. Sieve trays provide maximum interfacial area between the liquid and vapor. A P&#38;ID schematic of the sieve tray (each tray contains holes in a support plate) distillation system can be found in Appendix A.
In this demonstration, the Tray Distillation Unit (TDU) is started in total reflux mode. After a steady reflux drum level is attained, a switch to finite reflux mode is made by adjusting the bottoms, distillate and reflux flow rate controllers as necessary to maintain steady levels in the reflux drum and the reboiler, and to maintain a target reflux ratio RD = L/D. Once steady state is achieved (takes at least 90 min), liquid samples will be taken from the reflux drum, reboiler and on each tray and analyzed in a gas chromatograph. A typical protocol is to investigate the effects of reflux ratio over a wide range. From the sample analyses, the tray efficiencies can be determined for all three components on all six trays assuming constant molar overflow (McCabe-Thiele method). The results can also be simulated using an equilibrium process simulator, if available. These two methods can also be used to determine the overall tray efficiency. Additionally, data reconciliation of the mass balances can be performed to determine if gross measurement errors exist. Any Separations or Unit Operations textbook covers the fundamentals of distillation including basic concepts such as reflux ratio, Murphree efficiencies and the McCabe-Thiele method and diagram.2

Distillation Column Operation and Effects of Reflux Ratio

Source: Kerry M. Dooley and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Tray and packed columns are ...

Efficiency of Liquid-liquid Extraction

Source: <a href="https://www.jove.com/author/Kerry+M._Dooley">Kerry M. Dooley</a> and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Liquid-liquid extraction (LLE) is a separation technique used instead of distillation when either: (a) the relative volatilities of the compounds to be separated are very similar; (b) one or more of the mixture components are temperature sensitive even near ambient conditions; (c) the distillation would require a very low pressure or a very high distillate/feed ratio.1The driving force for mass transfer is the difference in solubility of one material (the solute) in two other immiscible or partially miscible streams (the feed and the solvent). The feed and solvent streams are mixed and then separated, allowing the solute to transfer from the feed to the solvent. Normally, this process is repeated in successive stages using counter-current flow. The solute-rich solvent is called the extract as it leaves, and the solute-depleted feed is the raffinate. When there is a reasonable density difference between the feed and solvent streams, extraction can be accomplished using a vertical column, although in other cases a series of mixing and settling tanks may be used.
In this experiment, the operational goal is to extract isopropanol (IPA, ~10 - 15 wt. %, the solute) from a mixture of C8-to-C10 hydrocarbons using pure water as solvent. A York-Scheibel type (vertical mixers and coalescers, one each per physical stage) extraction column is available. Like most extractors, the overall efficiency (number theoretical stages/physical stages) of this column is quite low, especially in comparison to many distillation columns. The low efficiencies arise from both slow mass transfer (two liquid resistances instead of one as in distillation) and often also from maldistribution of the phases. The effect of agitator speed on both the solute recovery in the extract and the overall column efficiency will be evaluated.

Liquid-liquid Extraction and York-Scheibel Column Operation

Source: Kerry M. Dooley and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Liquid-liquid extraction ...

Liquid Phase Reactor: Sucrose Inversion

Source: <a href="https://www.jove.com/author/Kerry+M._Dooley">Kerry M. Dooley</a> and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Both batch and continuous flow reactors are used in catalytic reactions. Packed beds, which use solid catalysts and a continuous flow, are the most common configuration. In the absence of an extensive recycle stream, such packed bed reactors are typically modeled as &#34;plug flow&#34;. The other most common continuous reactor is a stirred tank, which is assumed to be perfectly mixed.1 One reason for the prevalence of packed bed reactors is that, unlike most stirred tank designs, a large wall area to reactor volume ratio promotes more rapid heat transfer. For almost all reactors, heat must either be added or withdrawn to control the temperature for the desired reaction to take place.
The kinetics of catalytic reactions are often more complex than the simple 1st order, 2nd order, etc. kinetics found in textbooks. The reaction rates can also be affected by rates of mass transfer - reaction cannot take place faster than the rate at which reactants are supplied to the surface or the rate at which products are removed - and heat transfer. For these reasons, experimentation is almost always necessary to determine the reaction kinetics prior to designing large-scale equipment. In this experiment, we explore how to conduct such experiments and how to interpret them by finding a reaction rate expression and an apparent rate constant.
This experiment explores the use of a packed bed reactor to determine the kinetics of sucrose inversion. This reaction is typical of those characterized by a solid catalyst with liquid phase reactants and products.
sucrose&#160;&#8594; glucose (dextrose) + fructose(1)
A packed bed reactor will be operated at different flow rates to control the space time, which is related to residence time and is analogous to elapsed time in a batch reactor. The catalyst, a solid acid, will first be prepped by exchanging protons for any other cations present. Then, the reactor will be heated to the desired temperature (isothermal operation) with the flow of reactants. When the temperature has equilibrated, product sampling will begin. The samples will be analyzed by a polarimeter, which measures optical rotation. The mixture&#39;s optical rotation can be related to the conversion of sucrose, which can then be used in standard kinetics analyses to determine the order of the reaction, with respect to the reactant sucrose, and the apparent rate constant. The effects of fluid mechanics - no axial mixing (plug flow) vs. some axial mixing (stirred tanks in series) - on the kinetics will be analyzed as well.

Reaction Kinetics in Packed Bed Reactor and Sucrose Inversion

Source: Kerry M. Dooley and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Both batch and continuous ...

Crystallization of Salicylic Acid via Chemical Modification

Source: <a href="https://www.jove.com/author/Kerry+M._Dooley">Kerry M. Dooley</a> and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Processing of biochemicals involves unit operations such as crystallization, ultracentrifugation, membrane filtration, and preparative chromatography, all of which have in common the need to separate large from small molecules, or solid from liquid. Of these, crystallization is the most important from a tonnage standpoint. For that reason, it is commonly employed in the pharmaceutical, chemical and food processing industries. Important biochemical examples include chiral separations,1 purification of antibiotics,2 separation of amino acids from precursors,3 and many other pharmaceutical,4-5 food additive,6-7 and agrochemical purifications.8 The control of crystal morphology and size distribution is critical to process economics, as these factors affect the costs of downstream processing operations such as drying, filtration, and solids conveying. For more information about crystallization, consult a specialized textbook or a Unit Operations textbook.9
The crystallizer unit (Figure 1) enables study of: (a) the effects of key parameters, such as supersaturation and cooling/heating rates, on solids content, morphology and crystal size distribution; (b) and the on-line control of crystallization processes. Supersaturation can be controlled by altering conditions such as agitation rate and temperature. The different classifications of crystallization include cooling, evaporative, pH swing and chemical modification. In this experiment,an offline microscope will measure from crystals ranging in size from 10-1000 &#956;m, a typical size range for biologicals.
<img alt="Figure 1" src="/files/ftp_upload/10407/10407fig1.jpg" /> 
Figure 1: P&#38;ID schematic (left)and picture (right)of Crystallizer. <a href="//cloudfront.jove.com/files/ftp_upload/10407/10407fig1large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
This experiment will demonstrate a &#34;chemical modification&#34;,or &#34;pH-swing&#34; crystallization, to generate salicylic acid (SAL) (precursor of aspirin) crystals from the rapid reaction of aqueous solutions of basic sodium salicylate (NaSAL), which are basic, and sulfuric acid (H2SO4) at anywhere from 40 - 80&#176;C.11
Na+SAL + 0.5 H2SO4 SAL (ppt) + Na+ + 0.5 SO42-
The byproduct sodium sulfate remains soluble. The apparatus consists of two feed tanks, three variable speed (peristaltic) pumps, the crystallizer (stirred tank to approximate uniform temperature and concentration, ~5 L), a circulating bath for temperature control, power controller, product tank, and a makeup tank for feed regeneration with NaOH solution (if desired). Samples will be analyzed by a UV-Vis spectrometer for the residual soluble salicylate ion, and the salicylic acid crystal product will be dried and weighed.A pH probe can be used to determine steady-state when reaction conditions are altered.

Source: Kerry M. Dooley and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Processing of biochemicals ...

Single and Two-phase Flow in a Packed Bed Reactor

Source: <a href="https://www.jove.com/author/Kerry+M._Dooley">Kerry M. Dooley</a> and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
The goal of this experiment is to determine the magnitude of maldistribution in typical packed bed reactors in both single phase and two-phase (gas-liquid) flow and evaluate the effects of this maldistribution on pressure drop. The concepts of residence time distribution and dispersion are introduced through the use of tracers, and these concepts are related to physical maldistribution.
Channeling in a single-phase flow can occur along walls or by preferential flow through a larger portion of the bed cross-section. Channeling in two-phase flow can result from even more complex causes, and simple two-phase flow theories seldom predict pressure drops in packed beds. A design goal is always to minimize the extent of channeling by finding the optimal bed and particle diameters for the design flow rates and by packing a bed in a way to minimize settling. It is always important to quantify how much maldistribution might occur and to over-design the unit to account for its occurrence.
The permeameter apparatus measures pressure drop, &#916;P, and the concentration of tracer (dye) exiting horizontal packed beds of armored glass for either water, air, or two-phase flow (Figures 1 and 2). Water enters through a control valve and can be routed through manual valves to any of five beds (48&#34; long, 3&#34; I.D.) with different size glass bead dumped (random) packings. The pressure drop is measured using a pressure transmitter. The water flow is measured by a differential pressure (DP, orifice) transmitter and the air flow by a dry test meter (similar to a home gas meter). The dye sample is injected upstream by an automated sampling valve. The exit concentration of the dye from a bed is measured using a UV-Vis spectrometer. Residence time distributions are calculated from the tests and compared to the predictions of theories on dispersion in packed beds. Two-phase flow will be studied in bed 5, which contains the largest particles.
<img alt="Figure 1" src="/files/ftp_upload/10431/10431fig1.jpg" /> 
Figure 1: Process and instrumentation diagram of the apparatus.
<img alt="Figure 2" src="/files/ftp_upload/10431/10431fig2.jpg" /> 
Figure 2. 3-D rendering of the apparatus. Bed #1 is at the top, bed #5 at the bottom. The water control valve is on the left (red bonnet). The DP transmitter is at the top center (blue).

Source: Kerry M. Dooley and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
The goal of this experiment ...

Kinetics of Addition Polymerization to Polydimethylsiloxane

Source: <a href="https://www.jove.com/author/Kerry+M._Dooley">Kerry M. Dooley</a> and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Polymers are molecules consisting of many repeating monomer units that are chemically bonded into long chains. They exhibit a broad range of physical properties, which are affected by their chemical structure, molecular weight and degree of polymerization. The polymer industry manufactures thousands of raw materials used in a broad variety of commercial products.1,2
The goal of this video is to perform an addition polymerization reaction and then evaluate the resulting product to understand how viscosity can be used to determine polymer molecular weight. Additionally, this experiment will investigate how molecular weight can be related to monomer conversion.

Synthesis of Polydimethylsiloxane via Addition Polymerization

Source: Kerry M. Dooley and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
Polymers are molecules ...

Catalytic Reactor: Hydrogenation of Ethylene

Source: <a href="https://www.jove.com/author/Kerry+M._Dooley">Kerry M. Dooley</a> and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
The hydrogenation of ethylene (C2H4) to ethane (C2H6) has often been studied as a model reduction reaction in characterizing new metal catalysts.1-2 While supported nickel is not the most active metal catalyst for this reaction, it is active enough that reaction can take place at &#60; 200&#176;C.
The reaction typically involves adsorbed, dissociated hydrogen (H2) reacting with adsorbed ethylene. In other words, both hydrogen-atoms and ethylene molecules form bonds with a metal site (here denoted &#34;S&#34;). The strong bonding of ethylene with S weakens the double bond sufficiently to allow hydrogen atoms to add to ethylene, forming ethane, which is not adsorbed.
The purpose of this experiment is, first, to convert raw composition measurements to limiting reactant fractional conversions.3 These conversions can then be used in a plug-flow reactor (PFR) to fit the data to a standard power-law kinetics model by the &#34;Integral Method&#34;.3 A comparison of the experimental orders of reaction for both ethylene and hydrogen with the theoretical orders reveals in this case that the reaction is kinetically controlled rather than mass-transfer controlled.

Heterogeneous Catalytic Reactor and Hydrogenation of Ethylene

Source: Kerry M. Dooley and Michael G. Benton, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
The hydrogenation of ...

Evaluating the Heat Transfer of a Spin-and-Chill

Source: Michael G. Benton and Kerry M. Dooley, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
The Spin-and-Chill uses heat transfer and fluid flow fundamentals to chill beverages from room temperature to 38 &#176;F in as little as 2 min. It would take a refrigerator approximately 240 min and an ice chest approximately 40 min to achieve an equivalent temperature change. This is accomplished Spin and Chill by spinning a can or bottle at up to 500 rpm, which creates little or no foaming.
In this experiment, the efficacy of spinning a cylinder (i.e., soda can) at high speeds to cool a soft drink will be evaluated. Operational parameters, such as rpm and spin time, will be varied to assess their effect on heat transfer, and the heat transfer coefficient will be calculated using a lumped parameter model.

Convective Spin-and-Chill Heat Transfer: Lumped Parameter Model

Source: Michael G. Benton and Kerry M. Dooley, Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA
The Spin-and-Chill uses ...

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