JNTU B.Tech III PROCESS CONTROL INSTRUMENTATION Previous Question Paper (March/April - 2021 - R15/R13)

For a conical tank with height H and base radius R, the radius r at any liquid level h is given by the ratio r/h = R/H, or r = (R/H)h. The volume of liquid in the tank is V = (1/3)πr²h = (πR²/3H²)h³. The mass balance equation is dV/dt = q_in - q_out. Since V is a function of h, we apply the chain rule: (dV/dh)(dh/dt) = q_in - q_out. Differentiating V with respect to h gives dV/dh = π(R²/H²)h². Substituting this into the mass balance yields: π(R²/H²)h²(dh/dt) = q_in - q_out. For a nonlinear process, the outflow q_out is typically proportional to the square root of the head, q_out = C√h, where C is the discharge coefficient. The final nonlinear model is: (dh/dt) = (H²/πR²h²)(q_in - C√h).

Consider two non-interacting tanks in series where the outflow of the first tank, q_1, does not depend on the level of the second tank. For the first tank, the mass balance is A_1(dh_1/dt) = q_in - q_1, where q_1 = h_1/R_1. Taking the Laplace transform: A_1 s H_1(s) = Q_in(s) - H_1(s)/R_1. Rearranging gives H_1(s)/Q_in(s) = R_1/(τ_1 s + 1), where τ_1 = A_1 R_1. For the second tank, the inflow is q_1, so A_2(dh_2/dt) = q_1 - q_2, where q_2 = h_2/R_2. Taking the Laplace transform: A_2 s H_2(s) = Q_1(s) - H_2(s)/R_2. This gives H_2(s)/Q_1(s) = R_2/(τ_2 s + 1), where τ_2 = A_2 R_2. Since the tanks are non-interacting, the overall transfer function is the product of the individual transfer functions: G(s) = H_2(s)/Q_in(s) = [R_1/(τ_1 s + 1)] * [R_2/(τ_2 s + 1)] = K/[(τ_1 s + 1)(τ_2 s + 1)], where K = R_1 R_2.