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Linear Approximations

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...

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Related Experiment Video

Updated: Jun 29, 2026

Experimental Protocol to Investigate Particle Aerosolization of a Product Under Abrasion and Under Environmental Weathering

Experimental Protocol to Investigate Particle Aerosolization of a Product Under Abrasion and Under Environmental Weathering

Published on: September 16, 2016

Numerical Simulation Study on Spatial Diffusion Behavior of Non-Point Source Fugitive Dust under Different Enclosure

Jinjun Guo^1,2, Weiqi Lin^1,2, Hao Li³

¹Yellow River Laboratory, Zhengzhou University, Zhengzhou 450001, China.

International Journal of Environmental Research and Public Health

|March 11, 2023

Summary

Enclosures effectively reduce fugitive dust from road construction. Optimal heights of 3-3.5m significantly lower particulate matter concentrations in nearby residential areas, protecting health.

Keywords:

air velocities diffusion behavior enclosure heights gas-solid two-phase flow non-point source dust

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Last Updated: Jun 29, 2026

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Published on: May 22, 2020

Area of Science:

Environmental Engineering
Air Pollution Control
Computational Fluid Dynamics

Background:

Fugitive dust from municipal road construction poses significant health risks to workers and residents.
Ambient air pollution from construction sites is a major environmental concern.

Purpose of the Study:

To analyze the diffusion behavior of fugitive dust under various enclosure heights using a computational model.
To evaluate the effectiveness of enclosures in mitigating dust spread to residential areas.

Main Methods:

Utilized a gas-solid two-phase flow model to simulate dust diffusion dynamics.
Analyzed the impact of different enclosure heights and wind loads on dust dispersion.
Quantified particulate matter concentration reduction in simulated residential zones.

Main Results:

Enclosures demonstrate significant physical blocking and reflux effects, effectively restraining dust diffusion.
An enclosure height of 3-3.5m reduced particulate matter concentration below 40 μg/m³ in most residential areas.
Dust particle diffusion height above enclosures (2-3.5m) was concentrated between 1.5-2m at wind speeds of 1-5 m/s.

Conclusions:

Enclosures are a scientifically validated method for controlling construction dust.
Provides data-driven recommendations for optimal enclosure and sprinkler heights to minimize air pollution impacts.
Offers strategies to improve air quality in residential areas adjacent to construction sites and safeguard public health.