The equation for the force due to buoyancy will then become:
$$
F _ {b} = V _ {b} \left[ \frac {P _ {a}}{R _ {a} T _ {a}} - \frac {P _ {a}}{T _ {g}} \left(\frac {x _ {p}}{R _ {p}} + \frac {x _ {a}}{R _ {a}}\right) \right]
$$
If the balloon is of the type that will carry no internal pressure $ p_{a}=p_{g} $ , and we may state that:
$$
F _ {b} = V _ {b} p _ {a} \left[ \frac {1}{R a T a} - \frac {1}{T g} \left(\frac {x _ {p}}{R p} + \frac {x _ {a}}{R a}\right) \right]
$$
Discussions of the contamination of the lifting gas are included under the section on "Diffusion and Leakage of Lifting Gas" of this report.
The force due to the weight of the system $ F_{W}=W $ The weight of the balloon system at any time is a function of the original weight of the system plus the change of weight of the system. This change in the weight of the system is caused by the loss of ballast and the weight of the system at any time ( $ \uparrow $ ):
$$
W _ {t} = W _ {0} - \sum_ {t = 0} ^ {t} \Delta W _ {b}
$$
where:
$$
W _ {0} = \text {t h e o r i g i n a l w e i g h t o f t h e s y s t e m}
$$
$ \sum_{t=0}^{r} \Delta W_{b} = $ the sum of all the losses of ballast from time at which $ W=W_{0} $ until the time $ \dagger $
The value of the term $ \sum_{t=0}^{\infty} \Delta W_{b} $ depends on the type of ballast control. With no ballast:
$$
\sum_ {\mathrm {t} = 0} ^ {+} \Delta W _ {\mathrm {b}} = 0 \quad \mathrm {a n d} \quad W _ {\dagger} = W _ {0}
$$
If a constant ballast flow is used:
$$
\sum_ {t = 0} ^ {1} \Delta W _ {b} = \frac {d W}{d t} t
$$
where:
$ \frac{dW}{dt}=rate of ballast flow $
$ \uparrow $ = elapsed time from $ \uparrow = 0 $ to $ \uparrow = \uparrow $
If a practical fixed opening type or ballast control is used:
where:
$$
\sum_ {t = 0} ^ {+} \Delta W _ {b} = f (t, h, \mu_ {b}, \rho_ {b}, A)
$$
↑ = time
h = head of ballast above opening
$ \mu_{b}= $ viscosity of ballast fluid
$ P_{b} $ density of ballast fluid
A = area of opening