Efflux Time

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Efflux time - It is the time required for draining out the vessel contents. It is obtained by the application of Bernoulli equation by solving unsteady state mass balance equation.

The problem is to find the the efflux time (time needed to empty the vessel contents), for the given experimental setup consisting of Circular tank

(i) with Orifice opening at the bottom

(ii) with an exit pipe extending from the bottom of the tank

Time needed to empty the vessel (t_efflux) can be found theoretically by unsteady state mass balance and steady state energy balance.

Mass Balance:

Rate of mass in - Rate of mass out = rate of change of mass accumulation

If there is no input, then

- rate of mass out = rate of change of mass accumulation

- m_out = dm/dt

m_out = volumetric flow rate x density = A_o v₂ r

Rate of change of mass accumulation = rate of change of volume x density

=r dV/dt

where dV is the change in volume of water for a time interval of dt

Since V = area of tank x height of water = A_T h,

and, dV = A_Tdh

Therefore,

A_o v₂ r = A_T r dh/dt à 1

v₂ is obtained by making energy balance between the section 1 and 2:

p₁ = 0 atm (g)

p₂ = 0 atm (g)

v₁ = 0 (negligible velocity compared to position 2)

Taking reference as position 2, (position 1 and 2 are in a continuous column of fluid)

z₂ = 0

Therefore, Bernoulli equation reduces to

v₂² = 2gz₁

v₂ = Ö (2gz₁)

The height z₂ - z₁ can be taken as h. (water level with respect to position at any time t)

Therefore,

v₂ = Ö (2gh) à 2

Substituting from Equn.2 for v₂ in Equn.1,

Ö (2gh) = (A_T/Ao) dh/dt

Separating the variables,

(A_T/A_o) dh/Ö (2gh) = dt

Integrating between the limits z₁ to z₂ for a time of 0 to t_efflux

t_efflux = 2 A_T [Ö z₁ - Ö z₂] / [A_oÖ (2g)]

To account for the effect of contraction, C_o is introduced; and the is modified as,

t_efflux = 2 A_T [Ö z₁ - Ö z₂] / [C_oA_oÖ (2g)]