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The Reynolds number is a dimensionless group defined for a fluid flowing in a pipe as \(R e=D u \rho / \mu\) where \(D\) is pipe diameter, \(u\) is fluid velocity, \(\rho\) is fluid density, and \(\mu\) is fluid viscosity. When the value of the Reynolds number is less than about \(2100,\) the flow is laminar- -that is, the fluid flows in smooth streamlines. For Reynolds numbers above \(2100,\) the flow is turbulent, characterized by a great deal of agitation. Liquid methyl ethyl ketone (MEK) flows through a pipe with an inner diameter of 2.067 inches at an average velocity of \(0.48 \mathrm{ft} / \mathrm{s}\). At the fluid temperature of \(20^{\circ} \mathrm{C}\) the density of liquid \(\mathrm{MEK}\) is \(0.805 \mathrm{g} / \mathrm{cm}^{3}\) and the viscosity is 0.43 centipoise \(\left[1 \mathrm{cP}=1.00 \times 10^{-3} \mathrm{kg} /(\mathrm{m} \cdot \mathrm{s})\right]\). Without using a calculator, determine whether the flow is laminar or turbulent. Show your calculations.

Step by step solution

01

Formula substitution

First, substitute all the given parameters into the Reynolds number formula: \(R e = D u \rho / \mu\) where \(D = 2.067\) inches, \(u = 0.48\) ft/s, \(\rho = 0.805\) g/cm鲁, and \(\mu = 0.43\) cP.

02

Convert units

Next, convert all units into consistent ones. Convert inches to metres, f/s to m/s, g/cm鲁 to kg/m鲁 and cP to kg/(m路s). This involves the use of the following conversions: \[1 inch = 0.0254 m\], \[1 ft/s = 0.3048 m/s\], \[1 g/cm鲁 = 1000 kg/m鲁\], and \[1 cP = 0.001 kg/(m路s)\]. After the conversion, the values for \(D\), \(u\), \(\rho \) and \(\mu\) will be 0.0525 m, 0.146 m/s, 805 kg/m鲁 and 0.00043 kg/(m路s) respectively.

03

Calculate Reynolds number

After replacing the units' values, calculate the Reynolds number. However, this exercise asks to determine without using a calculator, if the flow is laminar or turbulent. To do this, compare the result of the multiplication of the parameters in the numerator with the denominator before actually calculating the result. If the numerator is greater than the denominator, the Reynolds number will be greater than 2100, indicating a turbulent flow. If the numerator is smaller, the flow is laminar.

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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 that is smooth and orderly. Picture water flowing through a long, straight tube, where the fluid travels in parallel layers with no disruption between them. Each layer moves smoothly past the adjacent layers.
In a laminar flow, the fluid's velocity is constant at any point within the stream. This means there is little mixing perpendicular to the direction of flow. Laminar flow is typically observed when fluid moves at low speeds and the Reynolds number is below 2100.
This style of flow occurs because the fluid's viscous forces are strong enough to dampen any disturbances that could cause turbulence. It's crucial in applications where precise flow control is needed, such as in biological processes or chemical manufacturing.

Turbulent Flow

Unlike laminar flow, turbulent flow appears chaotic and is highly irregular. This type of flow is characterized by eddies, swirls, and unpredictable fluctuations in velocity and pressure. You can think of it as a river with many rocks, where water splashes and whirls around obstacles.
In a pipe or channel, when the Reynolds number exceeds 2100, the flow typically transitions from laminar to turbulent. Here, inertial forces, which encourage mixing and irregular flow patterns, dominate over viscous forces.
Turbulent flow, though seeming disorderly, is vital in various industrial and natural processes. It enhances mixing, reduces temperature and concentration gradients, and is utilized in a wide array of engineering applications, from aircraft design to HVAC systems.

Unit Conversion

Unit conversion is critical when using formulas like the Reynolds number, which requires consistency in units for accurate results. When handling diverse data, it's common to encounter different unit systems, such as Imperial (inches, feet, pounds) and Metric (meters, kilograms, seconds).
For example, converting the diameter from inches to meters involves multiplying by 0.0254. Similarly, to convert velocity from feet per second to meters per second, you multiply by 0.3048.
Density and viscosity may also require conversion. As a rule of thumb, remember that

• 1 g/cm鲁 = 1000 kg/m鲁
• 1 centipoise (cP) = 0.001 kg/(m路s)

Correct unit conversion ensures the consistency required in engineering and scientific calculations, avoiding errors and ensuring meaningful results.

Fluid Properties

Fluid properties such as density and viscosity play a crucial role in determining the flow behavior of a fluid. Density (\( ho\) ) measures a fluid's mass per unit volume and affects buoyancy and pressure within the flow. In many calculations, it ensures that forces related to weight and gravity are accurately accounted for.
Viscosity (\(u\) ), on the other hand, indicates a fluid's resistance to deform under shear stress. It defines how 'thick' or 'sticky' a fluid is - honey, for instance, has a much higher viscosity than water.
Together, these properties influence the Reynolds number:

• A higher viscosity generally results in lower likelihood of turbulence, favoring laminar flow.
• A higher density combined with higher velocity and diameter will lean towards turbulent flow if other factors remain constant.

Understanding these properties enables engineers to design systems that effectively manage and control fluid systems like pipelines, lubrication, and hydraulic systems.