Estimating proportions in petrographic mixing equations by least-squares approximation

If a reaction has a small equilibrium constant, the equilibrium position favors the reactants. In such reactions, a negligible change in concentration may occur if the initial concentrations of reactants are high and the Kc value is small. In such circumstances, the equilibrium concentration is approximately equal to its initial concentration. This estimation can be used to simplify the equilibrium calculations by assuming that some equilibrium concentrations are equal to the initial...

When a fluid flows through a pipe, it experiences energy losses due to frictional resistance along the pipe walls, known as major losses. These energy losses result in a pressure drop, which varies based on the flow conditions — whether laminar or turbulent — and the specific physical properties of the fluid and pipe.
Fluid flow can be classified as laminar or turbulent, primarily based on the Reynolds number. This dimensionless number reflects the relative influence of inertial to viscous...

The alternative coordinate method, also known as the Shoelace Formula, is a technique for determining the area of a traverse using Cartesian coordinates. This method relies on the sequential arrangement of x and y coordinates for each point of the shape, ensuring accuracy and ease of application.In this approach, each corner's x and y coordinates are listed as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator. These coordinates are arranged sequentially...

In warehouse roofing applications, corrugated or curved metal sheets are commonly used to improve structural strength, water drainage, and ventilation efficiency. To accurately estimate material requirements and optimize design parameters, engineers must determine the curved surface area of these sheets. Because the sheet profiles often repeat smoothly along their length, they can be effectively approximated by parabolic curves, enabling the use of numerical integration techniques for area...

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...

The midpoint rule for a double integral provides a practical method for estimating volume over a rectangular region when the surface height varies continuously. In civil engineering, this method is useful for approximating the amount of soil to be moved when planning a road across uneven terrain. The road footprint is represented as a rectangle in the xy-plane. At the same time, the terrain elevation above a flat reference level is described by a continuous height function f(x,y). The objective...