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3.18 A continuous, stirred-tank reactor is initially full of water with the inlet and exit volumetric flow rates of water having the same numerical value. At a particular time, an operator shuts off the water flow and adds caustic solution at the same volumetric flow rate q, but with concentration c,. If the liquid volume Vis constant, the dynamic model for this process is + gc = qč, c(0) = 0 where c(f) is the exit concentration. Calculate c(t) and plot it as a function of time. Data: V= 2 m3, q = 0.4 m/min; c, = 50 kg /m 3.19 A liquid storage facility can be modeled by ü + 5ý + y(t) = 8ù + u(t) where y is the liquid level (m) and u is an inlet flow rate (m/s). Both are defined as deviations from the nominal steady-state values. Thus, y = u = 0 at the nominal steady state. Also, the initial values of all the derivatives are zero. a. If u(t) suddenly changes from 0 to 1 m/s at t = 0, determine the liquid level response, y(t). b. If the tank height is 2.5 m, will the tank overflow? c. Based on your results for (b), what is the maximum flow change, umay, that can occur without the tank overflowing? (Hint: Consider the Principle of Superposition.)

3.18 A continuous, stirred-tank reactor is initially full of water with the inlet and exit volumetric flow rates of water having the same numerical value. At a particular time, an operator shuts off the water flow and adds caustic solution at the same volumetric flow rate q, but with concentration c,. If the liquid volume Vis constant, the dynamic model for this process is + gc = qč, c(0) = 0 where c(f) is the exit concentration. Calculate c(t) and plot it as a function of time. Data: V= 2 m3, q = 0.4 m/min; c, = 50 kg /m 3.19 A liquid storage facility can be modeled by ü + 5ý + y(t) = 8ù + u(t) where y is the liquid level (m) and u is an inlet flow rate (m/s). Both are defined as deviations from the nominal steady-state values. Thus, y = u = 0 at the nominal steady state. Also, the initial values of all the derivatives are zero. a. If u(t) suddenly changes from 0 to 1 m/s at t = 0, determine the liquid level response, y(t). b. If the tank height is 2.5 m, will the tank overflow? c. Based on your results for (b), what is the maximum flow change, umay, that can occur without the tank overflowing? (Hint: Consider the Principle of Superposition.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3.18 A continuous, stirred-tank reactor is initially full of water with the inlet and exit volumetric flow rates of water having the same numerical value. At a particular time, an operator shuts off the water flow and adds caustic solution at the same volumetric flow rate q, but with concentration c;. If the liquid volume V is constant, the dynamic model for this process is dc V dt + qc = qči c(0) = 0 where c(t) is the exit concentration. Calculate c(t) and plot it as a function of time. Data: V= 2 m3; q = 0.4 m/min; c; = 50 kg /m3 3.19 A liquid storage facility can be modeled by ÿ + 5ý + y(t) = 8ù + u(t) where y is the liquid level (m) and u is an inlet flow rate (m/s). Both are defined as deviations from the nominal steady-state values. Thus, y = u = 0 at the nominal steady state. Also, the initial values of all the derivatives are zero. a. If u(t) suddenly changes from 0 to 1 m/s at t = 0, determine the liquid level response, y(t). b. If the tank height is 2.5 m, will the tank overflow? c. Based on your results for (b), what is the maximum flow change, umax, that can occur without the tank overflowing? (Hint: Consider the Principle of Superposition.)
Transcribed Image Text:3.18 A continuous, stirred-tank reactor is initially full of water with the inlet and exit volumetric flow rates of water having the same numerical value. At a particular time, an operator shuts off the water flow and adds caustic solution at the same volumetric flow rate q, but with concentration c;. If the liquid volume V is constant, the dynamic model for this process is dc V dt + qc = qči c(0) = 0 where c(t) is the exit concentration. Calculate c(t) and plot it as a function of time. Data: V= 2 m3; q = 0.4 m/min; c; = 50 kg /m3 3.19 A liquid storage facility can be modeled by ÿ + 5ý + y(t) = 8ù + u(t) where y is the liquid level (m) and u is an inlet flow rate (m/s). Both are defined as deviations from the nominal steady-state values. Thus, y = u = 0 at the nominal steady state. Also, the initial values of all the derivatives are zero. a. If u(t) suddenly changes from 0 to 1 m/s at t = 0, determine the liquid level response, y(t). b. If the tank height is 2.5 m, will the tank overflow? c. Based on your results for (b), what is the maximum flow change, umax, that can occur without the tank overflowing? (Hint: Consider the Principle of Superposition.)
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