91Ó°ÊÓ

Standard air flows at a rate of \(500 \mathrm{~L} / \mathrm{s}\) in an existing duct that is \(250 \mathrm{~mm}\) wide and \(500 \mathrm{~mm}\) high. The estimated equivalent sand roughness of the duct material is \(0.1 \mathrm{~mm}\). Ductwork renovations call for replacement of the existing duct by another rectangular duct having the same cross-sectional area and the same length but a different aspect ratio. The aspect ratio is defined as the height divided by the width. Aspect ratios in the range of \(0.1-3\) are being considered for the replacement duct. (a) Plot the ratio of the head loss in the replacement duct to the head loss in the existing duct as a function of the aspect ratio. (b) What aspect ratio in the replacement duct would give the least head loss?

Step by step solution

01

Determine the Cross-sectional Area of the Existing Duct

Given an existing duct with dimensions width \( W = 250 \mathrm{~mm} = 0.25 \mathrm{~m} \) and height \( H = 500 \mathrm{~mm} = 0.5 \mathrm{~m} \), calculate the cross-sectional area \( A \) as \( A = W \times H = 0.25 \times 0.5 = 0.125 \mathrm{~m^2} \).

02

Define Replacement Duct Dimensions

Given that the replacement duct must have the same cross-sectional area of \(0.125 \mathrm{~m^2}\), express the height \( H' \) in terms of the width \( W' \) and aspect ratio \( AR \). Thus, \( H' = AR \times W' \). Solve \( W' \times H' = 0.125 \), substituting \( H' \) to get \( W' = \sqrt{\frac{0.125}{AR}} \).

03

Calculate Hydraulic Diameter

For a rectangular duct, the hydraulic diameter \( D_h \) is given by \( D_h = \frac{2 \times W \times H}{W + H} \). For the replacement duct: \( D_h' = \frac{2 \times W' \times (AR \times W')}{W' + AR \times W'} = \frac{2 \times AR \times (W')^2}{(1 + AR) \times W'} = \frac{2 \times AR \times W'}{1 + AR} \).

04

Assume Similar Flow Conditions

Assume that the flow conditions, including flow rate and length, are constant. The flow resistance is determined by the Darcy-Weisbach equation: \( h_f = f \cdot \frac{L}{D_h} \cdot \frac{V^2}{2g} \). Only the friction factor \( f \) and hydraulic diameter \( D_h \) vary with duct shape. For turbulent flow, express \( f \) typically as a function of Reynolds number and relative roughness using the Colebrook equation.

05

Ratio of Head Losses

Express the ratio of head losses between the replacement and existing duct as \( \frac{h_f'}{h_f} = \frac{f' \cdot \frac{L}{D_h'}}{f \cdot \frac{L}{D_h}} = \frac{f'}{f} \cdot \frac{D_h}{D_h'} \). Solve for all possible aspect ratios \( AR \) given \( 0.1 \leq AR \leq 3.0 \). Assume consistent friction factor approximation for comparable roughness and turbulence dragging.

06

Plot and Analyze

Plot the computed head loss ratio \( \frac{h_f'}{h_f} \) against aspect ratio \( AR \). Analyze the plot to identify the aspect ratio that minimizes the head loss, occurring at the lowest point on the curve.

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Key Concepts

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

Understanding Aspect Ratio

In the evaluation of duct systems, the aspect ratio (AR) is an essential parameter. This is defined as the ratio of the duct's height to its width.
It represents the geometry of the duct cross-section and can significantly influence the fluid dynamics within the duct.
For example, a square duct has an aspect ratio of 1 because its height and width are equal.
Understanding how different aspect ratios affect a duct's performance is crucial in design optimization. When the aspect ratio changes, while maintaining the same cross-sectional area, it impacts the hydraulic diameter, affecting flow characteristics and efficiency.

• AR = Height / Width
• Wide and short duct: Low aspect ratio
• Tall and narrow duct: High aspect ratio

Different aspect ratios will lead to variations in pressure drop and head loss, which are critical for efficient duct design.

Exploring Hydraulic Diameter

The hydraulic diameter is a concept utilized to characterize non-circular conduits, like ducts with rectangular cross sections.
It's a crucial factor in calculating the flow characteristics because it provides a simplified measure of the duct's size.
The hydraulic diameter (D_h ) for a rectangular duct is defined via the formula:\[D_h = \frac{2 \times W \times H}{W + H}\]Replacing the dimensions with new duct configurations involves recalculation to capture changes in duct performance.
However, for the replacement duct, the expression becomes:\[D_h' = \frac{2 \times AR \times W'}{1 + AR}\]The hydraulic diameter assists in determining friction factors and Reynolds numbers, which are fundamental for assessing fluid flow and related losses.

The Concept of Head Loss

Head loss refers to the reduction in the total energy of the fluid as it flows through a duct or pipe.
It involves losses from friction due to the duct's internal surface as well as minor losses from bends, fittings, etc.
In ductwork, minimizing head loss is vital to maintain efficient airflow and reduce operational costs.
Factors affecting head loss in ducts include:

• Length of the duct
• Hydraulic diameter
• Roughness of the duct surface
• Aspect ratio

A critical part of the exercise involves plotting the ratio of head loss in the replacement duct against the original duct as a function of aspect ratio. This plot helps determine which aspect ratio minimizes head loss, thereby guiding the optimal design choice.

Understanding the Darcy-Weisbach Equation

The Darcy-Weisbach equation is a fundamental relationship in fluid dynamics utilized to calculate head loss due to friction in a pipe or duct. It establishes a link between flow characteristics and frictional losses.
The equation is given as:\[ h_f = f \cdot \frac{L}{D_h} \cdot \frac{V^2}{2g}\]where:

• \( h_f \) is the head loss due to friction
• \( f \) is the friction factor
• \( L \) is the length of the duct
• \( D_h \) is the hydraulic diameter
• \( V \) is the velocity of the flow
• \( g \) is the acceleration due to gravity

Using this equation, you can compare various duct configurations by calculating the ratio of their head losses.
The choice of an optimal aspect ratio from this analysis will depend on achieving the minimum value of head loss, ensuring efficient duct operation.