Compressibility of the PTV.

The compressibility is defined as the fractional rate of change of the volume with pressure, $\gamma_{_{PTV}} \equiv \frac{1}{V}\frac{\partial V}{\partial p}$. The volume of water is given by $V=\pi r_i^2 h$ where $r_i$ and $h$ are the inner radius and interior length of the PTV. The compressibility of the cylindrical PTV is

  $$\gamma_{_{PTV}} = 2 \frac{1}{r_i}\frac{dr_i}{dp} + \frac{1}{h}\frac{dh}{dp}~.$$ (5)

Roark (page 638, Case no. 1b) gives the following expressions for the fractional rates of change of inner radius and length with pressure:

  $$\frac{1}{r_i}\frac{dr_i}{dp} = \frac{r_o^2(1+\sigma) + r_i^2(1-... ...\frac{dh}{dp} = \frac{r_i^2(1-2\sigma)}{\kappa \left(r_o^2 - r_i^2 \right)}~.$$ (6)

Substituting the above expressions into (5) yields

  $$\gamma_{_{PTV}} = \frac{2 r_o^2(1+\sigma) + 3 r_i^2 (1-2\sigma)} {\kappa \left ( r_o^2 - r_i^2 \right)}~.$$ (7)

Evaluating (7) using parameters from Table 2 gives the PTV compressibility as

  $$\gamma_{_{PTV}} = \frac{2.6 \cdot (18.5~\mathrm{in})^2 + 1.2 \cd... ...{in})^2 - (12~\mathrm{in})^2 \right]} = 1.8 \cdot 10^{-7}~\mathrm{psi}^{-1}~.$$ (8)

Comparing (8) with water compressibility (see Table 3) shows that the PTV compressibility is more than an order of magnitude less than that of water. The fact that the PTV is incompressible compared to water means that pressure-induced expansions of the PTV can be neglected in all other computations done in this article.