91Ó°ÊÓ

Specific gravity is a dimensionless quantity that describes the density of a fluid compared to the density of water. In practical terms, it tells us how heavy or light a fluid is relative to water. A specific gravity of 0.87 means the fluid in the given exercise is 87% the density of water. To find the density \( \rho \) of the fluid, we use the formula \( \rho = \text{SG} \times 1000 \, \text{kg/m}^3 \), where \( \text{SG} \) is the specific gravity and \( 1000 \, \text{kg/m}^3 \) is the approximate density of water. This calculation yields \( 870 \, \text{kg/m}^3 \) for this fluid. Specific gravity is crucial for engineers as it impacts the flow behavior of fluids in pipes and systems.

Kinematic viscosity is the measure of a fluid's internal resistance to flow under the influence of gravity. It's given in units of \( \text{m}^2/\text{s} \) and is used to depict how easily a fluid can flow. It is derived from dynamic viscosity \( \mu \) and density \( \rho \) as \( u = \frac{\mu}{\rho} \). In our exercise, the kinematic viscosity of \( 1.20 \times 10^{-4} \, \text{m}^2/\text{s} \) shows that the fluid flows with moderate ease. Understanding kinematic viscosity is essential for predicting flow patterns, especially in turbulent flow systems such as pipes. It helps in characterizing the fluid's behavior and ensures systems are designed to accommodate the specific needs of different fluids.

A hydraulically smooth pipe assumes no roughness interruptions influence the fluid flow. This means the pipe's internal surface is smooth enough that even in turbulent conditions, the roughness elements are fully submerged within the viscosity's damping layer. In this state, the friction factor depends primarily on the Reynolds number, an indicator of flow regime, rather than on the actual roughness of the pipe. In the exercise, the pipe's consistency allows for easy calculation of the maximum velocities without additional adjustments for friction losses due to rough surfaces. Understanding the conditions of a hydraulically smooth pipe is vital for accurate modeling and calculations, especially when aiming for efficiency in fluid transport systems.

Shear stress in fluid mechanics refers to the force per unit area exerted by a fluid against a surface. It's a significant concept when determining the durability and design requirements of materials handling fluids. In pipes, shear stress directly influences the maximum flow rate and the longevity of the pipe.The exercise specifies a maximum allowable shear stress of 125 Pa, which dictates the limit on flow rate to prevent damage or excessive wear on the pipe's surface. Engineers calculate shear stress to ensure structural integrity while optimizing fluid transport systems. The relationship \( \tau = \frac{1}{4} \rho u^2_f \) is used in our step-by-step solution to solve for the friction velocity, which is key to determining the feasible average velocity and, subsequently, the flow rate.