The ballast flow at any time, t:
$$
\frac {d W}{d t} = C _ {F} \rho_ {b} A \sqrt {2 g h}
$$
so that:
$$
\sum_ {t = 0} ^ {3} \Delta W _ {b} = \int_ {0} ^ {\tau} C _ {F} \rho_ {b} A \sqrt {2 g h} d t
$$
where:
C $ F $ is a coefficient of discharge, dependent upon Reynolds number of the flow through the opening
In this equation only $ \sqrt{2}g $ and A are constants (if temperature effect on the opening A is neglected), $ P_{b} $ is dependent upon temperature of the fluid and h is dependent upon the shape of the vessel containing the fluid and time t.
If ballast flow is controlled by atmospheric pressure:
$$
\sum_ {t = a} ^ {\dagger} \Delta W _ {b} = \sum_ {t = a} ^ {r} \frac {d W}{d t} + p > p _ {c}
$$
with a fixed valve opening (open-or-closed valve)
where $ ^{\dagger} \mathrm{p} > \mathrm{p}_{\mathrm{c}} $ represents the time when atmospheric pressure is greater than the pressure of control. Here, again, $ \frac{\mathrm{d} w}{\mathrm{d} t}=C_{F} \rho_{b} A \sqrt{2 g h} $ With ballast flow proportional to D=D
With ballast flow proportional to $ p-p_{c} $
$$
\sum_ {t = 0} ^ {\dagger} \Delta W _ {b} = \sum_ {t = 0} ^ {\dagger} \frac {d \left(\frac {d w}{d t}\right)}{d \Delta p} \left(p - p _ {c}\right) t _ {p > p _ {c}}
$$
where:
$$
\frac {d \left(\frac {d w}{d t}\right)}{d \Delta p}
$$
relationship between rate of flow and pressure difference ( p-p $ _{c} $ ) where $ p > p_{c} $
If we include a rate of pressure change control or a rate of ascent control such that there is no ballast flow if rate of pressure change is less than some value $ - \left( \frac{dp}{dt} \right)_{c} $ or rate of ascent is greater than some value $ \left( \frac{dz}{dt} \right)_{c} $ , we impose the condition for ballast flow in the above two cases that for flow to occur $ p > p_{c} $ , and $ \frac{dp}{dt} > \left( \frac{dp}{dt} \right)_{c} $ or $ \frac{dz}{dt} \left(\frac {d p}{d t}\right) _ {c}
$$