L2 slide 18: Is p_c = p_d? (ENGR5002)

L2 slide 18: Is p_c = p_d?From lectureCurrent spec2:27 marks8 min

L2 - Pressure and its measurement, slides 18-19 (Pressure only a function of depth?)

An open tank of water has its free surface at pressure $p_0$ (atmospheric). Several solid partitions of different shapes stand on the floor of the tank, so the water is split into regions that are all connected above the partitions. At the right-hand wall a narrow slot, closed at the bottom and on its left by a solid wall, holds a column of mercury of height $\Delta z_m$ ; the slot is filled with water above the mercury, and that water is continuous with the rest of the tank. The depth of water above the mercury surface is $\Delta z_w$ .

Points $a$ , $b$ and $c$ are in the water, each in a different region between the partitions, all at the same depth $\Delta z_2$ below the free surface. Point $d$ is in the mercury, at the same level as $a$ , $b$ and $c$ (so $\Delta z_w + \Delta z_m = \Delta z_2$ ).

The lecturer asks: is $p_c = p_d$ ?

Formulas you may need

Pressure at depth in one fluid: $p(z) = p(z=0) + \rho g z$ , $z$ measured down from the free surface (on the formula sheet)
Pressure does not vary horizontally in a fluid at rest: $\dfrac{\partial p}{\partial x} = \dfrac{\partial p}{\partial y} = 0$ (on the formula sheet)
Through a stack of fluid layers, add $\rho_i g\,\Delta z_i$ for each layer (learn this)

(a)
Write the pressure at $a$ , $b$ and $c$ and explain why it is the same at all three points even though they are separated by partitions.
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(b)
Is $p_c = p_d$ ? Explain, write an expression for $p_d$ in terms of $p_0$ , $\Delta z_w$ and $\Delta z_m$ , and hence find $p_d - p_c$ .
[3]
(c)
[Added, to put a number on it] Take $\Delta z_m = 0.10$ m, $\rho_{water} = 1000\ \mathrm{kg\,m^{-3}}$ , $\rho_{mercury} = 13\,546\ \mathrm{kg\,m^{-3}}$ (the value on slide 23) and $g = 9.81\ \mathrm{m\,s^{-2}}$ . Calculate $p_d - p_c$ .
$p_{d}$ - $p_{c}$ =
kPa
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