Pressure Drop Calculation Non-Newtonian Fluids Bingham Plastic Laminar Flow
Understanding the flow behavior of non-Newtonian fluids is crucial in various engineering applications, ranging from oil and gas transportation to food processing and pharmaceuticals. These fluids, unlike their Newtonian counterparts, exhibit a non-linear relationship between shear stress and shear rate. This complex behavior necessitates specialized approaches to analyze their flow characteristics, particularly pressure drop within pipelines. This article delves into the calculation of pressure drop for a specific non-Newtonian fluid flowing under laminar conditions through a pipe, considering the fluid's yield stress and plastic viscosity. We will explore the relevant equations and principles, providing a step-by-step guide to solve this practical engineering problem.
Understanding Non-Newtonian Fluid Behavior
Before diving into the calculations, it's essential to grasp the fundamental concepts of non-Newtonian fluid behavior. Unlike Newtonian fluids like water or oil, which exhibit a linear relationship between shear stress and shear rate, non-Newtonian fluids display more complex behaviors. These fluids can be broadly categorized into several types, including:
- Yield stress fluids: These fluids require a certain amount of stress (yield stress) to be applied before they start to flow. Below this stress threshold, they behave like solids. Examples include toothpaste, mayonnaise, and drilling mud.
- Shear-thinning fluids (Pseudoplastic): Their viscosity decreases with increasing shear rate. Paints, blood, and polymer solutions often exhibit this behavior.
- Shear-thickening fluids (Dilatant): Their viscosity increases with increasing shear rate. Cornstarch suspensions and some ceramic slurries are examples.
- Viscoelastic fluids: These fluids exhibit both viscous and elastic properties. Polymer melts and some biological fluids fall into this category.
In this specific case, we are dealing with a fluid characterized by both yield stress and plastic viscosity, indicating that it is a Bingham plastic fluid. Bingham plastic fluids require a finite yield stress to be overcome before flow initiates, and beyond that, they exhibit a linear relationship between shear stress and shear rate, similar to a Newtonian fluid but with an offset. The yield stress (τ₀) represents the minimum stress required for flow, and the plastic viscosity (µp) describes the fluid's resistance to flow once the yield stress is exceeded.
Problem Statement
We are given a non-Newtonian fluid with the following properties:
- Yield stress (τ₀) = 13.51 Nm⁻²
- Plastic viscosity (µp) = 243.5 mNs m⁻² = 0.2435 Ns m⁻²
The fluid flows under laminar conditions through a pipe with a diameter (D) of 65.4 mm (0.0654 m). The center line velocity (v_max) is given as not specified in the original input. To properly solve this, a center line velocity is needed. Let's assume that the center line velocity is 0.5 m/s. We aim to calculate the resulting pressure drop (ΔP) over a certain length (L) of the pipe. For demonstration purposes, let us assume that the pipe length is 10 meters. Pressure drop will depend on the pipe length as well as the fluid parameters.
Methodology: Calculating Pressure Drop for Bingham Plastic Fluids in Laminar Flow
To determine the pressure drop (ΔP) for a Bingham plastic fluid flowing under laminar conditions in a pipe, we can utilize the Buckingham-Reiner equation. This equation is specifically designed for fluids exhibiting yield stress and plastic viscosity. The equation considers the interplay between yield stress, plastic viscosity, pipe dimensions, and flow rate to provide an accurate pressure drop prediction.
The Buckingham-Reiner Equation
The Buckingham-Reiner equation is expressed as follows:
ΔP/L = (4τ₀ / D) + (32µp * v_avg / D²)
Where:
- ΔP is the pressure drop
- L is the length of the pipe
- τ₀ is the yield stress
- D is the pipe diameter
- µp is the plastic viscosity
- v_avg is the average fluid velocity
This equation highlights the two primary contributors to pressure drop in Bingham plastic fluids: the yield stress term (4τ₀ / D) and the viscous term (32µp * v_avg / D²). The yield stress term accounts for the energy required to overcome the fluid's initial resistance to flow, while the viscous term represents the pressure drop due to the fluid's viscosity as it flows through the pipe.
Determining Average Velocity (v_avg)
The Buckingham-Reiner equation requires the average fluid velocity (v_avg), which is related to the center line velocity (v_max) in laminar flow. For a Bingham plastic fluid, the relationship between average velocity and center line velocity is given by:
v_avg = v_max * (1/2)
This equation indicates that the average velocity is half of the center line velocity for laminar flow of a Bingham plastic fluid. This relationship arises from the non-Newtonian velocity profile, which is flatter than the parabolic profile observed in Newtonian fluids. The plug flow region in the center of the pipe contributes to this flatter profile, resulting in a lower average velocity relative to the center line velocity.
Step-by-Step Calculation
Now, let's apply the Buckingham-Reiner equation to calculate the pressure drop for the given scenario. We will follow a step-by-step approach, ensuring clarity and accuracy in our calculations.
1. Calculate the Average Velocity (v_avg)
Given that the center line velocity (v_max) is 0.5 m/s, we can calculate the average velocity using the following formula:
v_avg = v_max * (1/2) = 0.5 m/s * (1/2) = 0.25 m/s
Therefore, the average velocity of the fluid in the pipe is 0.25 m/s.
2. Apply the Buckingham-Reiner Equation
Now that we have the average velocity, we can substitute all the known values into the Buckingham-Reiner equation to calculate the pressure drop per unit length (ΔP/L):
ΔP/L = (4τ₀ / D) + (32µp * v_avg / D²)
Substituting the given values:
ΔP/L = (4 * 13.51 Nm⁻² / 0.0654 m) + (32 * 0.2435 Ns m⁻² * 0.25 m/s / (0.0654 m)²)
3. Calculate the Pressure Drop per Unit Length
Performing the calculations:
ΔP/L = (54.04 Nm⁻² / 0.0654 m) + (1.948 Ns m⁻¹ / 0.004277 m²)
ΔP/L = 826.30 Nm⁻³ + 455.47 Nm⁻³
ΔP/L = 1281.77 Nm⁻³
Therefore, the pressure drop per unit length is 1281.77 Pascals per meter (Pa/m).
4. Calculate the Total Pressure Drop
To determine the total pressure drop (ΔP) over the 10-meter pipe length, we multiply the pressure drop per unit length by the length of the pipe:
ΔP = (ΔP/L) * L
ΔP = 1281.77 Pa/m * 10 m
ΔP = 12817.7 Pa
Therefore, the total pressure drop over the 10-meter pipe length is 12817.7 Pascals, or approximately 12.82 kPa.
Conclusion
In conclusion, we have successfully calculated the pressure drop for a Bingham plastic fluid flowing under laminar conditions through a pipe. By understanding the unique flow behavior of non-Newtonian fluids and applying the Buckingham-Reiner equation, engineers can accurately predict pressure drops in various industrial applications. The step-by-step approach outlined in this article provides a clear and concise methodology for solving similar problems, ensuring efficient and reliable design and operation of fluid transport systems. The key parameters influencing the pressure drop are the yield stress, plastic viscosity, pipe diameter, average velocity, and pipe length. Proper consideration of these factors is essential for optimizing system performance and preventing operational issues.
Key Takeaways
- Non-Newtonian fluids exhibit complex flow behavior, requiring specialized analysis techniques.
- Bingham plastic fluids have both yield stress and plastic viscosity, influencing their flow characteristics.
- The Buckingham-Reiner equation is a crucial tool for calculating pressure drop in Bingham plastic fluids.
- Average velocity is a key parameter related to center line velocity in laminar flow.
- Accurate pressure drop prediction is essential for efficient design and operation of fluid transport systems.
Further Considerations
This analysis assumes laminar flow conditions. In practical applications, it is crucial to verify that the flow remains laminar by calculating the Reynolds number. If the Reynolds number exceeds the critical value for laminar flow, the flow becomes turbulent, and a different approach is required to calculate the pressure drop. Additionally, the temperature dependence of fluid properties, such as viscosity and yield stress, should be considered for accurate predictions in real-world scenarios. Furthermore, pipe roughness can also affect pressure drop, especially in turbulent flow regimes. Therefore, it is essential to account for these factors to ensure reliable and accurate results in engineering design and analysis.
Pressure drop calculation for non-Newtonian fluids in laminar flow is a critical aspect of chemical engineering and fluid mechanics. Understanding how to determine pressure drop in pipes, especially for complex fluids, is essential for designing efficient transport systems. This article provides a detailed guide on calculating pressure drop specifically for Bingham plastic fluids. The key equation used is the Buckingham-Reiner equation, which accounts for both yield stress and plastic viscosity. The plastic viscosity and yield stress are crucial fluid properties that influence flow behavior. By understanding these properties, we can accurately predict pressure drop in pipes. This article covers a practical example, showcasing the step-by-step process of pressure drop calculation, including determining the average velocity and applying the Buckingham-Reiner equation. Laminar flow conditions are assumed in this analysis, which simplifies the calculation. Accurately calculating pressure drop helps engineers optimize system performance and prevent operational issues. This knowledge is vital for industries dealing with non-Newtonian fluids, such as oil and gas, food processing, and pharmaceuticals. The principles of fluid mechanics are applied in this pressure drop calculation, emphasizing the importance of understanding fluid properties and flow regimes. For accurate pressure drop calculation, consider factors like fluid properties, pipe dimensions, and flow conditions. The Buckingham-Reiner equation is a powerful tool for predicting pressure drop in Bingham plastic fluids under laminar flow conditions. Ultimately, the goal of pressure drop calculation is to ensure efficient and reliable fluid transport systems.
