91Ó°ÊÓ

The Reynolds' number (Re) is defined as \(\mathrm{Re}=\rho\left\langle\mathrm{v}_{x}\right\rangle d / \eta,\) where \(\rho\) and \(\eta\) are the fluid density and viscosity, respectively; \(d\) is the diameter of the tube through which the fluid is flowing; and \(\left\langle\mathrm{v}_{x}\right\rangle\) is the average velocity. Laminar flow occurs when \(\operatorname{Re}<2000\), the limit in which the equations for gas viscosity were derived in this chapter. Turbulent flow occurs when \(\mathrm{Re}>2000 .\) For the following species, determine the maximum value of \(\left\langle\mathrm{v}_{x}\right\rangle\) for which laminar flow will occur: a. \(\mathrm{Ne}\) at \(293 \mathrm{K}(\eta=313 \mu \mathrm{P}, \rho=\) that of an ideal gas) through a 2.00 -mm-diameter pipe. b. Liquid water at \(293 \mathrm{K}\left(\eta=0.891 \mathrm{cP}, \rho=0.998 \mathrm{g} \mathrm{mL}^{-1}\right)\) through a 2.00 -mm-diameter pipe.

Step by step solution

01

Write the Reynolds' number equation for laminar flow

Since we need to find the maximum value of average velocity to ensure laminar flow, we will use the definition of Reynolds' number for laminar flow, which is \(Re<2000\). We can set up the equation as follows: \(2000=\rho\langle v_x\rangle d / \eta\)

02

Solve the equation for average velocity

To find the maximum value of average velocity, we will solve the equation from Step 1 for \(\langle v_x\rangle\): \(\langle v_x\rangle = \frac{2000\eta}{\rho d}\) Now we will solve for the average velocity for each species.

03

Calculate the average velocity for Neon gas

For Neon gas, we have: Temperature: \(T = 293\ K\) Viscosity: \(\eta = 313 \times 10^{-6}\ Pa \cdot s\) Density: Since Neon gas is an ideal gas, we can apply the ideal gas law to find the density: \(\rho = \frac{P M}{R T}\) (Note that we are not given the pressure directly, so we cannot calculate the exact value of the density for Neon) Pipe diameter: \(d = 2 \times 10^{-3} m\) We can substitute these values into the equation from Step 2 to get the maximum average velocity for Neon gas to maintain laminar flow: \(\langle v_x\rangle = \frac{2000\left(313 \times 10^{-6}\ Pa \cdot s\right)}{\rho \left(2 \times 10^{-3} m\right)}\) Please note that without the pressure value, we cannot calculate a specific value for the average velocity.

04

Calculate the average velocity for liquid water

For liquid water, we have: Temperature: \(T = 293\ K\) Viscosity: \(\eta = 0.891 \times 10^{-3}\ Pa \cdot s\) Density: \(\rho = 0.998 \times 10^{3}\ kg/m^{3}\) Pipe diameter: \(d = 2 \times 10^{-3} m\) We can substitute these values into the equation from Step 2 to get the maximum average velocity for liquid water to maintain laminar flow: \(\langle v_x\rangle = \frac{2000\left(0.891 \times 10^{-3}\ Pa \cdot s\right)}{\left(0.998 \times 10^{3}\ kg/m^{3}\right) \left(2 \times 10^{-3} m\right)}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Laminar Flow

Laminar flow is a type of fluid motion where the fluid travels smoothly in layers, without mixing between layers. This pattern of flow is characterized by smooth streaming rather than eddies or swirls. An everyday example might include the smooth flow of honey from a spoon. In the context of our problem, Reynolds' number is a key factor in determining if the flow will be laminar. For Reynolds' numbers less than 2000, the flow tends to be laminar.

To understand the conditions for laminar flow, the relationship established through Reynolds' number is crucial. The lower the number, the more likely the flow is laminar. This is important when it comes to applications such as designing pipelines in chemical plants or medical devices where controlled flow rates are crucial.

Turbulent Flow

Contrastingly, turbulent flow is more chaotic and is characterized by eddies and vortices – think of the way whitewater rapids look. In turbulent flow, different layers of fluid mix with each other, which can increase the rate of momentum and mass transfer across the fluid. This can be both beneficial and detrimental, depending on the application. For instance, turbulent flow is advantageous for mixing in industrial processes but can cause damage to structures like pipelines and ship hulls over time.

Knowing when flow transitions from laminar to turbulent is vital in engineering. With Reynolds' number greater than 2000, flow typically becomes turbulent, thereby affecting how substances are mixed, how heat is transferred, and the overall resistance against the flow.

Fluid Dynamics

Fluid dynamics is the study of how fluids (liquids and gases) move and the forces that develop as a result. This is a branch of physics with wide-ranging applications, from predicting weather patterns to designing aircraft. In the exercise given, the fluid dynamics aspect is centered on understanding the flow conditions within a pipe, whether it will remain laminar, or if it will transition to turbulent flow.

The principles of fluid dynamics help in determining flow patterns and designing systems that can effectively handle the fluids in question. Whether it’s to ensure that a gas flows smoothly through a pipeline or to predict how water will behave when it strikes a turbine blade, fluid dynamics form the foundation of analysis.

Viscosity

Viscosity is a measure of a fluid's resistance to deformation. It is often thought of as the 'thickness' or 'stickiness' of a fluid. High-viscosity fluids like syrup or oil flow more slowly than low-viscosity fluids like water or air. Viscosity plays a crucial role in determining Reynolds' number and, consequently, the flow regime of a fluid. The higher the viscosity, the more likely the flow will be laminar, given the same conditions for fluid density and velocity.

In the exercise solution, we calculate with specific viscosities for Neon gas and liquid water to find the maximum average velocity for laminar flow. Understanding viscosity is also necessary when designing machinery and equipment that handle fluids to ensure efficiency and safety.

Fluid Density

Fluid density is the mass per unit volume of a fluid. Denser fluids exert more force and therefore have more momentum for a given velocity than less dense fluids. Fluid density critically affects the Reynolds' number where a higher density can potentially contribute to a lower velocity threshold for the onset of turbulent flow.

In the example we're exploring, the question reflects the need to understand how the density of Neon gas or water affects the flow within a pipeline. It can determine the dimensions of a pipeline in a processing plant or the necessary power for a pump in a hydraulic system. Correctly accounting for fluid density is essential for these calculations and designs.

Ideal Gas Law

The ideal gas law is a fundamental equation that relates the pressure, volume, temperature, and the amount of substance of a gas through the formula PV = nRT, where P is pressure, V is volume, n is the amount in moles, R is the ideal gas constant, and T is the absolute temperature. It assumes the gas particles are small compared to the distances between them and that there are no intermolecular forces affecting them.

In the case of the Neon gas in our exercise, we use the ideal gas law to estimate the density of the gas, which is then used in the calculation of the Reynolds' number to determine the flow regime. This demonstrates how the ideal gas law can be applied in practical situations to relate the physical properties of gases with flow dynamics.