The ballast flow at any time, $t$ :
$$
\frac{dW}{dt} = C_F \rho_b A \sqrt{2gh}
$$
so that:
$$
\sum_{t=0}^{t} \Delta W_b = \int_{0}^{t} C_F \rho_b A \sqrt{2gh} \, dt
$$
where:
$C_F$ is a coefficient of discharge, dependent upon Reynolds number of the flow through the opening
In this equation only $\sqrt{2g}$ and $A$ are constants (if temperature effect on the opening $A$ is neglected), $\rho_b$ 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:
$$
\sum_{t=0}^{t} \Delta W_b = \sum_{t=0}^{t} \frac{dW}{dt} \quad t > p_c \quad \text{, with a fixed valve opening (open-or-closed valve)}
$$
where $t > p_c$ represents the time when atmospheric pressure is greater than the pressure of control. Here, again, $\frac{dW}{dt} = C_F \rho_b A \sqrt{2gh}$
With ballast flow proportional to $p - p_c$ :
$$
\sum_{t=0}^{t} \Delta W_b = \sum_{t=0}^{t} \frac{d\left(\frac{dW}{dt}\right)}{d\Delta p} \left(p - p_c\right) \quad t > p_c
$$
where:
$$
\frac{d\left(\frac{dW}{dt}\right)}{d\Delta p} \quad \text{relationship between rate of flow and pressure difference } (p - p_c) \text{ where } p > p_c
$$
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{dp}{dt}\right)_c
$$
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