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zenterix
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- Homework Statement
- The following is from the book "Thermodynamics for Chemical Engineers" by Kenneth Hall.
- Relevant Equations
- Consider the following figure
Manometers measure differential pressures.
The differential equation that expresses pressure is
$$\frac{\partial P}{\partial z}=-\rho g\tag{1}$$
where ##\rho## is the density of the fluid and ##g## is the local acceleration of gravity.
If the density is constant, the integration of (1) is
$$P_2-P_1=-\rho g\Delta z=-\rho g(z_2-z_1)=-\rho g h\tag{2}$$
$$P_1=P_2+\rho gh\tag{3}$$
This indicates that the hydraulic pressure (gauge pressure) is ##\rho gh##.
First of all, where does (1) come from?
What is an intuitive explanation for the meaning of ##-\rho g##?
##\partial P/\partial z## is the rate of change of pressure relative to position ##z##. Since we have a negative sign on the rhs, it seems that ##z## is being measured from bottom to top.
##\rho g## is thus a rate of change of pressure and so ##\rho g h## is change in pressure.
(3) says that the pressure at position 1 is the sum of the pressure at position 2 plus the term ##\rho gh##.
Note that ##\rho g h## is also the pressure that a vertical column of fluid exerts at its base due to gravity.
$$P=\frac{F}{A}=\frac{mg}{A}=\frac{\rho Vg}{A}=\frac{\rho Ahg}{A}=\rho g h$$
In terms of dimensions, we have ##\frac{[m]}{[L]^3}[a][L]=\frac{[F]}{[L]^2}=[P]##. I am not sure if this is correct in terms of notation. Can I write ##[a]## like this as a dimension?
Next, consider the following snippet
Atmospheric pressure ##P_{atm}## is the pressure caused by the weight of the Earth's atmosphere on an object. We might find this pressure called "barometric" pressure.
Absolute pressure ##P_{abs}## is the total pressure, and absolute pressure of zero is perfect vacuum. All thermodynamic calculations must use absolute pressure.
Gauge pressure or manometric pressure ##P_{man}## is the pressure relative to atmospheric pressure.
$$P_{man}=P_{abs}-P_{atm}\tag{4}$$
$$P_{abs}=P_{atm}+P_{man}\tag{5}$$
I would like to understand what gauge pressure is exactly.
It seems that (3) is in the form of (5) but what allows us to conclude this?
As for the U-tube glass manometer depicted in the picture above, it seems that the opening at 2 is open to the atmosphere and the opening at 1 is subjected to some pressure we would like to measure.
We know the density of the fluid (at the temperature of the fluid being used).
The difference in the pressure at 1 relative to the pressure at 2 is the pressure exerted by the column of fluid on the right with height ##h## (which is the height difference between 1 and 2).
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