Kara Sears
The power law model is a mathematical relationship used to describe the behaviour of non-Newtonian fluids. It is one of the most common viscosity models used in hydraulic analysis. The model defines the apparent viscosity of a fluid in terms of the shear stress required to shear it at a given rate. The power law model is typically used to model non-Newtonian fluid behaviour in various fluid flow contexts, such as chemical and polymer fluid processing. It can also be used to describe any material that exhibits power-law behaviour, which is a proportional response of stress to the shear rate. The model is simple and widely used, but it only approximately describes the behaviour of real non-Newtonian fluids. To employ this model, users must input values for the Power Law constants, K and n, or provide rheological test data.
| Characteristics | Values |
|---|---|
| Model Type | Non-Newtonian fluids |
| Use | One of the most common viscosity models used in hydraulic analysis |
| Equation | Where τ is the shear stress, du/dy is the shear rate (or velocity gradient), and K and n are the Power Law constants |
| Power Law Constants | K and n |
| Fluid Types | Shear Thinning (pseudoplastic) fluid, Newtonian fluid, and Shear Thickening (dilatant) fluid |
| Power-Law Fluid Definition | A type of generalized Newtonian fluid |
| Power-Law Fluid Behaviour | The shear stress is given by τ and is directly proportional to the shear rate |
| Power-Law Fluid Examples | Water, most aqueous solutions, oils, corn syrup, glycerine, air and other gases |
| Dilatant Fluid Examples | Silly Putty, uncooked paste of cornstarch and water (oobleck) |
| First-Order Fluid | A power-law fluid with exponential dependence of viscosity on temperature |
| Multi-Bubble System | Coupling the power-law model with the VOF model to account for bubble interaction, drag force, and liquid rheological properties |
| Disadvantage | As the shear rate tends to zero, the viscosity tends to infinity |
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What You'll Learn
- The Power Law model is used to find viscosity in non-Newtonian fluids
- It is a popular and simple kinetic model used in HDT of heavy oil and residue
- The model defines viscosity as a function of the exponent
- It is also known as the Ostwald-de Waele power law
- The model is used to describe any material that shows power-law behaviour
The Power Law model is used to find viscosity in non-Newtonian fluids
The Power Law model is a popular viscosity model used in hydraulic analysis to describe non-Newtonian fluids. It is also known as the Ostwald-de Waele power law, named after Wilhelm Ostwald and Armand de Waele. This model is used to describe the relationship between shear stress and shear rate in non-Newtonian fluids.
The Power Law model states that the shear stress required to shear a fluid at a given rate can be described by the equation: τ = K * (du/dy)^n, where τ is the shear stress, du/dy is the shear rate (or velocity gradient), and K and n are the Power Law constants. The apparent viscosity of the fluid, η, is defined by this model as well. When n=1, the fluid is Newtonian, when 0 < n < 1, it is a shear-thinning (pseudoplastic) fluid, and when n > 1, it is a shear-thickening (dilatant) fluid.
The Power Law model is advantageous due to its simplicity, and it is often used to describe non-Newtonian fluid behaviour in various flow processes, such as chemical and polymer fluid processing. It can also be used to describe encapsulant flow without considering the curing effect, making it useful in applications like the flip chip underfill process and microchip encapsulation.
To employ the Power Law model, values for the constants K and n are required, or rheological test data can be used to determine these constants. The model's applicability to highly turbulent flow is dependent on the test data used to determine the Power Law constants. The Power Law model is a valuable tool for understanding and predicting the behaviour of non-Newtonian fluids in a range of engineering and industrial contexts.
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It is a popular and simple kinetic model used in HDT of heavy oil and residue
The power-law model is a popular and simple kinetic model used in the hydrotreating (HDT) of heavy oil and residue. HDT is a crucial process in refineries for upgrading heavy oil and residue into cleaner fuel and valuable products. The power-law model is particularly useful in this context as it provides a simple and accurate way to model the complex kinetics of HDT reactions.
In the power-law model, the reaction rate is given by a power-law equation where the rate is directly proportional to the concentration of the reactants raised to a certain power. This equation takes into account the rate constant, 'k', which is influenced by factors such as the properties and structure of the reactants, their environment (temperature, pressure, etc.), and the activity of catalysts. The reaction order, denoted as 'n', is another important factor in the equation.
The power-law model is well-suited for describing non-Newtonian fluid behaviour in various fluid flow contexts, including chemical and polymer fluid processing. It is particularly valuable when dealing with non-Newtonian fluids, where the viscosity is dependent on the shear rate. The model defines the apparent viscosity, η, of the fluid in relation to the shear stress, shear rate, and the Power Law constants, K and n.
The versatility of the power-law model is evident in its application to a range of HDT kinetic studies. Martinez and Ancheyta, for instance, proposed a power-law rate equation for various reactions, including HDS, HDNi, HDV, and HDAsp, by considering both thermal and catalytic reactions. Their work highlighted the adaptability of the power-law model in capturing the intricacies of different reactions. Furthermore, Al Bazzaz et al. employed a power-law model for HDS, HDNi, HDV, HDAsph, and HDCCR kinetics, assuming a reaction order of 2.0 for all reactions.
The power-law model's popularity in HDT of heavy oil and residue stems from its simplicity and ability to capture the complex kinetics of these reactions. It provides a straightforward framework for understanding the relationships between reaction rates and reactant concentrations, environmental factors, and catalytic activity. This makes it a valuable tool for engineers and scientists working in the petroleum refining industry.
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The model defines viscosity as a function of the exponent
The power law model is a popular and simple kinetic model used to describe non-Newtonian fluid behaviour in various fluid flow scenarios, such as chemical and polymer fluid processing. It is also known as the Ostwald-de Waele power law. This model defines viscosity as a function of the exponent, where the shear stress required to shear a fluid at a given rate can be described by an equation involving the shear stress, shear rate, and the Power Law constants, K and n. The apparent viscosity of the fluid, denoted as η, is defined by this model as: when n=1, it corresponds to a Newtonian fluid; 0 < n < 1 indicates a shear-thinning (pseudoplastic) fluid; and n > 1 represents a shear-thickening (dilatant) fluid.
The exponent, n, determines the nature of the relationship between viscosity and shear rate. For most polymers, the exponent n is less than 1, resulting in a decrease in viscosity as the shear rate increases, known as shear thinning behaviour. Conversely, when the exponent is greater than 1, the viscosity increases with the shear rate, referred to as shear thickening. This behaviour is observed in dilatant or shear-thickening fluids, where the apparent viscosity rises at higher shear rates. An example of a dilatant fluid is a mixture of cornstarch and water, which exhibits significantly increased viscosity when squeezed between the fingers.
The power law model simplifies the description of non-Newtonian fluid behaviour by using the exponent n. However, it only approximates this behaviour, as real fluids have minimum and maximum effective viscosities that depend on molecular-level physical chemistry. The model's advantage lies in its simplicity, but it may not accurately capture the entire flow behaviour of shear-dependent fluids.
To employ the power law model effectively, users must have values for the Power Law constants, K and n, or utilise rheological test data. The constants, K and n, are typically obtained from laboratory measurements and experimental data fitting. The model is widely used in engineering calculations and has applications in hydraulic analysis, particularly for non-Newtonian fluids.
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It is also known as the Ostwald-de Waele power law
The power law model is a mathematical relationship used to model non-Newtonian fluid behaviour in various types of fluid flow. It is also known as the Ostwald-de Waele power law, named after Wilhelm Ostwald and Armand de Waele. This model is useful due to its simplicity, but it only approximately describes the behaviour of a real non-Newtonian fluid.
The power law model is one of the most common viscosity models used in hydraulic analysis. It defines the apparent viscosity of a fluid and is given by an equation where τ is the shear stress, du/dy is the shear rate (or velocity gradient), and K and n are the Power Law constants. The power law model can be used to describe encapsulant flow without considering the curing effect during IC encapsulation. It has been applied in various processes, such as the flip chip underfill process, force-injection encapsulation, and microchip encapsulation.
Power-law fluids can be categorized into three types based on their flow behaviour index: pseudoplastic, Newtonian fluid, and dilatant. A pseudoplastic fluid, also known as a shear-thinning fluid, has a lower apparent viscosity at higher shear rates. Its behaviour is time-independent, and it is often observed in solutions with large polymeric molecules in a solvent with smaller molecules. A common example of a pseudoplastic fluid is styling gel. On the other hand, a dilatant or shear-thickening fluid increases in apparent viscosity at higher shear rates. These fluids are commonly used in viscous couplings in automobiles.
The power law model has two fitting constants, K and n, making it simpler than other models that describe non-Newtonian fluids, such as the Cross model or Carreau-Yasuda model, which have four and five fitting constants, respectively. The constant K is the consistency index, and n is the power law index or flow behaviour index. When n is equal to one, the power law model becomes the Newtonian fluid model, and the consistency index, K, has the unit of viscosity.
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The model is used to describe any material that shows power-law behaviour
The power law model is a popular and simple model used in engineering calculations and fluid dynamics. It is particularly useful for describing non-Newtonian fluids, which are those that do not follow Newtonian fluid behaviour, where the shear stress is directly proportional to the shear rate. Power-law fluids can be further subdivided into three types based on their flow behaviour index: pseudoplastic, Newtonian, and dilatant or shear-thickening fluids.
In the power law model, the shear stress required to shear a fluid at a given rate is described by an equation involving the shear stress, shear rate, and the Power Law constants, K and n. The apparent viscosity of the fluid, denoted by η, is defined within this model. When n=1, the model corresponds to a Newtonian fluid, while 0
The power-law model is advantageous due to its simplicity, but it only approximately describes the behaviour of non-Newtonian fluids. For instance, if n is less than one, the model predicts that viscosity decreases indefinitely as the shear rate increases. However, real fluids have minimum and maximum effective viscosities that depend on their molecular physical chemistry. Therefore, the power law is most accurate for the range of shear rates to which the coefficients are fitted.
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Frequently asked questions
The Power Law model is a mathematical relationship used to describe the behaviour of non-Newtonian fluids. It is one of the most common viscosity models used in hydraulic analysis.
The model uses an equation where τ is the shear stress, du/dy is the shear rate, and K and n are the Power Law constants. The apparent viscosity of the fluid, η, is defined by this equation.
The Power Law model is limited by the test data used to determine the Power Law constants, K and n. It also only approximately describes the behaviour of real non-Newtonian fluids. For example, the model predicts that viscosity will decrease as the shear rate increases indefinitely when n is less than one, which is not accurate for real fluids.
