minimum volume of sump volume for pumpsmaximum

Question

Minimum volume of sump volume for pumps

Maximum pumping rate = Q_P

Volume of sump = V

Inflow Rate = Q_i

Cycle Time T_c= t₁ + t₂

t₁ = V / Q_P - Q_i

t₂ = V / Q_i

T_c = Minimum Cycle Time

V = Tc / {1/ (Q_P - Q_i) + 1 / Q_i}

For Minimum Volume,

δV / δQ_i = δ/δQ_i[T_c / {1/ (Q_P - Q_i) + 1/ Q_i}]

Minimum Volume occurs when Q_i = 0.5Q_p

Hence, Minimum Volume of sump = T_c / {1/ (Q_P - 0.5Q_p) + 1/0.5Q_p}

            = T_c Q_P / 4