{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Manometry\n", "\n", "## Learning outcomes\n", "\n", "* Apply knowledge of hydrostatic pressure to pressure measurement\n", "* Understand the manometer instrument \n", "* Learn how an inclined manometer increases measurement sensitivity\n", "\n", "# Introduction\n", "\n", "Manometers are a class of instruments used to measure fluid pressure which make use of the **Hydrostatic Equation**:\n", "\n", "\\begin{align}\n", "p_2 - p_1 = -\\rho g \\left(z_2 - z_1 \\right)\n", "\\end{align}\n", "\n", "For a fluid of known density in a constant gravity well, a change in elevation $(z_2 - z_1)$ linearly corresponds to a change in pressure.\n", "\n", "Consider the section of pipe shown below. A liquid flows along the length of the pipe (we can assume it is much longer than shown and the flow moves entirely parallel to the pipe) and at some point along its length a hole is drilled and a small open tube is inserted normal to the upper surface. This arrangement is called a **piezometer tube**.\n", "\n", "\"Piezometer\"\n", "\n", "\n", "Shown below is a cross sectional view of the geometry. The arrows indicate the flow direction. The small vertical tube is also shown. \n", "\n", "**What do think the value of $h$ is?** \n", "\n", "Will the flow in the main pipe squirt out of this smaller tube? Will the flow continue along its path and ignore the perpendicular tube completely?\n", "\n", "\"Piezometer\"\n", "\n", "The answer is that for some constant velocity in the main pipe is that the flow will rise up the perpendicular tube some finite $h$ and stop once it's reached an equilibrium.\n", "If we change the velocity in the pipe the level $h$ will change accordingly. Why? In order to increase the velocity in the pipe we have too apply more pressure.\n", "\n", "If we are applying a positive gauge pressure at the pipe inlet and then let the pipe exit to the atmosphere (zero gauge pressure) then there will be a *pressure gradient* along the length of the pipe and the liquid will flow from high to low pressure — at a **constant velocity** (it's a little more complicated than that but more on that later). This result is that the hydrostatic pressure is constant perpendicular to the direction of flow (assuming a sufficiently small diameter). In fact if we place a series of these *'piezometer tubes'* along the length of the pipe we could visualise the pressure gradient along the pipe.\n", "\n", "\"Piezometer\"\n", "\n", "The pressure at each location $n$ can be measured by measuring the local height $h_n$:\n", "\n", "\\begin{align}\n", "p_{abs}(x) = p_{atm} + \\rho g h_{n}\n", "\\end{align}\n", "\n", "or \n", "\n", "\\begin{align}\n", "p_{gauge}(x) = \\rho g h_{n}\n", "\\end{align}\n", "\n", "It is the static pressure that pushes (*piezo* is greek for push) the fluid laterally into the piezometer tube. The velocity of flow moving along the pipe does not directly play a role in the height $h$. In fact in the pipe was instead a closed pressure vessel containing a static fluid as observed in our Bourdon gauge example we would observe the same behaviour. However the Piezometer tube may need to be extremely long for any practical usage!\n", "\n", "## Pressure head\n", "\n", "We can use the height of a column of liquid as a unit of pressure. We see this used in the Mercury Barometer where atmospheric pressure is often recorded in $mmHg$ or $inHg$. Since the height of the liquid column is dependent on the density and acceleration due to gravity (the **specific weight**, $\\gamma = \\rho g$):\n", "\n", "\\begin{align}\n", "\\text{Pressure Head} = \\frac{\\Delta P}{\\gamma}\n", "\\end{align}\n", "\n", "When working with a liquid in a complex flow system such as the pipe network of a chemical plant it is often convenient to work in pressure head." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The U-tube Manometer\n", "\n", "We've seen various arrangements of tubes used to measure fluids so far; from Boyle's *'J-tube'* to the Piezometer tube. Now we will consider another, the *'U-tube Manometer'*\n", "\n", "If we take a U-shaped tube, open to atmosphere at both ends, and partially fill it form one side with a liquid, what happens? The fluid flows down and around and eventually reaches an equilibrium with the free surfaces in each arm perfectly level with one another.\n", "\n", "