(a) A large storage tank contains a salt solution of...

(a) A large storage tank contains a salt solution of variable density given by $\rho=1050+\mathrm{kh}$ in $\mathrm{kg} / \mathrm{m}^3$, where $\mathrm{k}=50 \mathrm{~kg} / \mathrm{m}^4$, at a depth h metres below the free surface. Calculate the pressure intensity at the bottom of the tank holding 5 m of the solution. (b) A Bourdon type pressure gauge is connected to a hydraulic cylinder activated by a piston of 20 mm diameter. If the gauge balances a total mass of 10 kg placed on the piston, determine the gauge reading in metres of water.

(a) A large storage tank contains a salt solution of variable density given by $\rho=1050+\mathrm{kh}$ in $\mathrm{kg} / \mathrm{m}^3$, where $\mathrm{k}=50 \mathrm{~kg} / \mathrm{m}^4$, at a depth h metres below the free surface. Calculate the pressure intensity at the bottom of the tank holding 5 m of the solution.
(b) A Bourdon type pressure gauge is connected to a hydraulic cylinder activated by a piston of 20 mm diameter. If the gauge balances a total mass of 10 kg placed on the piston, determine the gauge reading in metres of water.

Key Concepts

Pressure Conversion

Pressure conversion involves translating a pressure measurement from one unit to another, such as from pascals to the equivalent height of a fluid column (e.g., metres of water). This is important because different industries and applications might use different units of pressure, and being able to convert between them is crucial for ensuring consistency, safety, and accuracy in design and measurement.

Bourdon Pressure Gauge

A Bourdon pressure gauge is a mechanical device used to measure pressure, typically based on the deformation of a curved tube under pressure. The device converts the pressure-induced deformation into a dial reading, which can be expressed in various units including, in some cases, equivalent heights of a fluid column. Understanding the operational principle of a Bourdon gauge is key for interpreting its measurements in hydraulic systems.

Variable Density

Variable density describes a situation where the density of a fluid is not constant throughout, but changes with depth or another parameter. In the context of hydrostatic pressure, when density varies with depth, the pressure calculation cannot rely on the simple formula for constant density; instead, it requires integrating the density function over the depth. This concept is crucial for accurately modeling scenarios where solutal or thermal gradients cause density variations.

Hydrostatic Pressure

This concept refers to the pressure experienced at a point within a fluid due solely to the weight of the fluid above that point. It is calculated by considering the force exerted by the fluid, which depends on both the fluid's density and the depth. Understanding hydrostatic pressure is essential when dealing with fluids at rest and is widely used in various engineering applications, from designing dams to assessing pipeline systems.

Integration in Pressure Calculations

When the density of a fluid is not constant, calculating the pressure at a certain depth necessitates the use of integration. By summing up the contributions of each differential layer of fluid, each with its own density, integration enables the accurate computation of the cumulative pressure. This application of calculus is fundamental in physics and engineering for dealing with non-uniform systems.

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