/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 Viscosity of fluid plays a signi... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 39

Viscosity of fluid plays a significant role in the analyses of many fluid dynamics problems. The viscosity of water can be determined from the following correlation: $$ \mu=c_{1} 10\left(\frac{\varepsilon_{2}}{T-\varepsilon_{1}}\right) $$ where $$ \begin{aligned} \mu &=\text { viscosity }\left(\mathrm{N} / \mathrm{s} \cdot \mathrm{m}^{2}\right) \\ T &=\text { temperature }(\mathrm{K}) \\ c_{1} &=2.414 \times 10^{-5} \\ c_{2} &=247.8(\mathrm{~K}) \\ c_{3} &=140(\mathrm{~K}) \end{aligned} $$ What is the appropriate unit for \(c_{1}\), if the above equation is to be homogeneous in units?

Short Answer

Expert verified
The appropriate unit for \(c_{1}\) for the equation to be homogeneous in units is N.s/m虏.

Step by step solution

01

Identify the Units

First, identify the units on each side of the equation. On the RHS, the first term is \(c_{1}\) whose unit is unknown. So, let's say the unit of \(c_{1}\) is A. The term inside the exponential is dimensionless. Therefore, the complete RHS has the same unit as \(c_{1}\), which is A.
02

Set Equation for Unit Homogeneity

For the equation to be homogeneous in units, the units on both sides must be the same. So, compare the RHS with the LHS. On the LHS, \(\mu\) is the viscosity with its unit as Newton second per square meter (N.s/m虏).
03

Equate the Units

Setting the units equal gives us: A = N.s/m虏. Thus, the unit for \(c_{1}\) must be N.s/m虏 for the equation to be homogeneous in units.

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Key Concepts

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

Viscosity
Viscosity is an essential property of fluids that describes their resistance to deformation or flow. Imagine trying to stir a pot of honey and a pot of water with a spoon. Honey is much thicker and resists your attempts to stir, making it more viscous than water.
Viscosity plays a crucial role in fluid dynamics as it determines how a fluid flows under various forces. It's measured in units of \( \text{N路s/m}^2 \), which can also be expressed in \( \text{Pa路s} \) (pascals路second). This measure helps engineers and scientists understand how fluids will behave under certain conditions, such as in a pipeline or around an aircraft wing.
There are two types of viscosity: dynamic viscosity, which relates to internal fluid friction, and kinematic viscosity, which considers how internal friction interacts with fluid density. Understanding viscosity ensures safety and efficiency in applications like lubrication, hydraulic systems, and in the medical field for the flow of blood.
Dimensional Analysis
Dimensional analysis is a mathematical technique used in fluid dynamics to check the consistency and solve problems involving different physical quantities. It provides a way to understand complex equations by examining the dimensions of the units involved.
This process involves identifying the base dimensions like \( \text{Mass} (M), \text{Length} (L), \) and \( \text{Time} (T) \) in an equation and ensuring they remain consistent. For example, a force has the dimension of mass times acceleration (ML/T虏). By ensuring the dimensions on both sides of an equation match, we can verify that our calculations are heading in the right direction.
Dimensional analysis can also be used to develop dimensionless numbers, like the Reynolds number in fluid dynamics, that help predict flow patterns in different fluid flow situations. This method simplifies the process of comparing similar types of flow, making predictions about fluid behavior more straightforward.
Unit Homogeneity
Unit homogeneity is a principle stating that the equations used in physics and engineering should have consistent units on both sides. This means that when you equate two quantities, they should have the same units for the equation to be valid.
For instance, if you have an equation describing the viscosity of water, such as the one in the problem, both sides of the equation need to be in the same unit of measurement. This ensures that the mathematical relationship is correctly representing the physical phenomenon.
In our equation problem, the left-hand side has units of \( \text{Newton second per square meter (N.s/m虏)} \), representing viscosity. To maintain unit homogeneity, the constant \( c_1 \) on the right-hand side must also carry these units. Checking for unit homogeneity is an essential step in verifying the correctness of any physical equation.

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Most popular questions from this chapter

A machine shop has a rectangular floor shape wit dimensions of \(30 \mathrm{ft}\) by \(50 \mathrm{ft}\). Express the area of the flo in \(\mathrm{ft}^{2}, \mathrm{~m}^{2}\), in \(^{2}\), and \(\mathrm{cm}^{2}\). Show the conversion steps.

For gases under certain conditions, there is a rela ship between the pressure of the gas, its volume, a temperature as given by what is commonly called the ideal gas law. The ideal gas law is given by $$ P V=m R T $$ where \(P=\) absolute pressure of the gas, \(\mathrm{Pa}\) \(V=\) volume of the gas, \(\mathrm{m}^{3}\) \(m=\) mass, \(\mathrm{kg}\) \(R=\) gas constant \(T=\) absolute temperature, Kelvin What is the appropriate unit for \(R\), if the above equation is to be homogeneous in units?

Express the kinetic energy \(\frac{1}{2}(\mathrm{mass})(\mathrm{speed})^{2}\) of car with a mass of \(1200 \mathrm{~kg}\) and moving at speed \(100 \mathrm{~km} / \mathrm{h}\) using SI, \(\mathrm{BG}\), and U.S. Customary unit Show the conversion steps.

In an annealing process-a process wherein materials such as glass and metal are heated to high temperatures and then cooled slowly to toughen them - the following equation may be used to determine the temperature of a thin piece of material after some time \(t\). $$ \frac{T-T_{\text {eavironment }}}{T_{\text {initial }}-T_{\text {environment }}}=\exp \left(-\frac{2 b}{\rho c L} t\right) $$ where $$ \begin{aligned} T &=\text { temperature }\left({ }^{\circ} \mathrm{C}\right) \\ b &=\text { heat transfer coefficient } \\ \rho &=\text { density }\left(\mathrm{kg} / \mathrm{m}^{3}\right) \end{aligned} $$ Those of you who will pursue aerospace, chemical, mechanical, or materials engineering will learn about the underlying concepts that lead to the solution in your heat-transfer class. What is the appropriate unit for h if the preceding equation is to be homogeneous in units? Show all steps of your work.

The head loss due to flow of a fluid inside a pipe is calculated from \(b_{\text {Loss }}=f \frac{L}{D} \frac{V^{2}}{2 g}\), where \(b_{\text {Loos }}(\mathrm{m}), f\) is friction factor, \(L\) is pipe length ( \(\mathrm{m}\) ), \(D\) is pipe diameter \((\mathrm{m}), V\) is average flow velocity \((\mathrm{m} / \mathrm{s})\), and \(g\) is the acceleration due to gravity \(\left(\mathrm{m} / \mathrm{s}^{2}\right)\). What is the appropriate unit for friction factor \(f\) ?
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