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Question

A small water reservoir depicted in Fig. 1 has a cylindrical shape with a radius of 0.1m, and its output pipe has a radius of 0.005m. During exploitation, the reservoir water level Y changes in the range between 0.05m and 0.15m, where the maximal volume flow rate is qin = 1.5 * 10^(-4)m^3/s. The nonlinear model for the rate of change of the level is qinS A^2 2A^2gY (1) S^2 = A^2 where S = 0.12Tm^2, A = (0.005)^2T, and g = 9.81kg. (a) If the reservoir level is at the constant level Y* = 0.1, what is the corresponding constant flow rate qin? Explain your result and use the above-provided values to compute qin. (b) If u = qin - qin is a small variation of qin around qin, use qin, Y* from (a), and the above-provided values to find a linear differential equation for the corresponding small variation of y = Y - Y. (c) Apply Laplace transform to the linear differential equation from (b) to obtain the transfer function from u to y and provide the plot of the corresponding magnitude frequency characteristic. The vertical axis of the plot should be the magnitude of the transfer function, and the horizontal axis should be log10(w). (d) Write a MATLAB script that computes an array q,n for an array of Y levels that are in the range [0.05,0.15] (use 30 equally spaced levels from the range). Provide the plot of Y* values (x-axis) vs. qin values (y-axis). (e) For all the 30 pairs (qin,Y*) from (d), provide the magnitude frequency characteristic, and the horizontal axis should be log10(w).

          A small water reservoir depicted in Fig. 1 has a cylindrical shape with a radius of 0.1m, and its output pipe has a radius of 0.005m. During exploitation, the reservoir water level Y changes in the range between 0.05m and 0.15m, where the maximal volume flow rate is qin = 1.5 * 10^(-4)m^3/s. The nonlinear model for the rate of change of the level is qinS A^2 2A^2gY (1) S^2 = A^2 where S = 0.12Tm^2, A = (0.005)^2T, and g = 9.81kg.

(a) If the reservoir level is at the constant level Y* = 0.1, what is the corresponding constant flow rate qin? Explain your result and use the above-provided values to compute qin.

(b) If u = qin - qin is a small variation of qin around qin, use qin, Y* from (a), and the above-provided values to find a linear differential equation for the corresponding small variation of y = Y - Y*.

(c) Apply Laplace transform to the linear differential equation from (b) to obtain the transfer function from u to y and provide the plot of the corresponding magnitude frequency characteristic. The vertical axis of the plot should be the magnitude of the transfer function, and the horizontal axis should be log10(w).

(d) Write a MATLAB script that computes an array q,n for an array of Y* levels that are in the range [0.05,0.15] (use 30 equally spaced levels from the range). Provide the plot of Y* values (x-axis) vs. qin values (y-axis).

(e) For all the 30 pairs (qin,Y*) from (d), provide the magnitude frequency characteristic, and the horizontal axis should be log10(w).

a small water reservoir depicted in fig 1 has a cylindrical shape with the radius of 01m and its output pipe has the radius of 0005m during exploitation the reservoir water level y changes i 94444

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