1. Problem statement Consider the liquid-level control system shown in Figure 1. The liquid level is measured and the level transmitter (LT) output is sent to a feedback controller (LC) that controls liquid level by adjusting volumetric flow rate q2. A second inlet flow rate q₁ is the disturbance variable. The level control system shown in Figure 1, implemented with a computer whose inputs and outputs are calibrated in terms of full range (100%). The tank is 1 m in diameter, while the valve on the exit line acts as a linear resistance with R = 6.37 min/m². The level transmitter has a span of 2.0 m and an output range of 0 to 100%. The valve characteristic of the equal percentage control valve is related to the fraction of lift / by the relation f = (30)-1. The air-to-open control valve receives a 3 to 15 psi signal from an I/P transducer, which, in turn, receives a 0 to 100% signal from the computer-implemented controller. When the control valve is fully open (f= 1), the flow rate through the valve is 0.2 m³/min. At the nominal operating condition, the control valve is half-open (1 = 0.5). The valve relation between flow rate & and fraction of lift / can be expressed as: q = 0.2 (30)-1 Assume: 1. The liquid density p and the cross-sectional area of the tank A are constant. 2. The flow-head relation is linear, q3 = h/R. 3. An electronic controller with input and output in % is used (full scale = 100%). 91 h LT hm P LC- I/P 93 Fig 1: Liquid-level control system. Gd U Yu P Ysp Ys E sp Ge G₁₂ G₂ Km 92 Im Gm Fig 2: Standard block diagram of a feedback control system. Y 2. Dynamic Model and transfer functions of the control system 2.1. Process Transfer Functions Derivation. 2.1.1. Dynamic Models Formulation 2.1.2. Degree of Freedom analysis and variables classification 2.1.3. Dynamic Model Linearity. 2.1.4. Dynamic Models in Deviation Variables. 2.1.5. Process Transfer Functions Derivation. 2.2. Instrumentation Transfer Functions Derivation.