A stirred-tank blending system can be described by a first-order transfer function between the exit composition x and the inlet composition x₁ (both are mass fractions of solute):

X′(s) / X′_i (s) = K / τs + 1

where K = 0.6 (dimensionless) and τ = 10 min. When the blending system is at steady state (x̅ = 0.3), the dynamic behavior is tested by quickly adding a large amount of a radioactive tracer, thus approximating an impulse function with magnitude 1.5.

(a) Calculate the exit composition response x(t) using Laplace transforms and sketch x(t). Based on this analytical expression, what is the value of x(0)?
(b) Using the Initial Value Theorem of Section 3.4, determine the value of x(0).
(c) If the process is initially at a steady state with x̅ = 0.3, what is the value of x(0)?
(d) Compare your answer for parts (a)–(c) and briefly discuss any differences.