Question

2. For the following liquid level system, qmi(t) is the mass flow rate input to the system. There is a valve in the pipe, and the flow through the pipe is turbulent, therefore the restrictions are all modeled using the square root relation, 4.-J . a. Develop the nonlinear model for the behavior of the deviation liquid level, h. b. Determine the linearized resistance at Amir = 3 m/s. c. Re-derive the model for the deviation height using your answer to b.

Answer 1

Given Equation am-se o for the given arrangement, flow rate should be constant from above equation, when ple pressure drop across the length with resistance R. Pea P2 P3 these properly are the pressure drop across the length with resistance RI, R., Rz. Here, we get equivalent resistance by Sum of all pressures de equnalent Peg-Pit PetP - ats flow rate is constant, am - am - se On Squaring both the sides, we get PanR Similarly, P. Luke, b anks Therefore Pegadur Ren -) Per-Pi + P2 +P3 air Reg-am R, tam ketam Rz Reg-R + R2 + Ry] B) Assuming to be the initial height of the liquid. Continuty of flow is defined as mean velouties at all cross sections having equal areas are then ecual and if the areas are not equal, the velocities are inverely peoportional to the areas of respective cross section. It is given as

a - A4, = A₂ V2 when a n the volumetric flow rate A is the cross sectional area of flow. & is the mean veelocity. V = f2 ₂ - 0 We know that, Bernoulli's prinuple states that the speed of a moving fluid increases, the pressure within the thad decreases. By Bernoull's equation for ② & ③ P. + 1 2 8 viet Sgh, Pet 8 du 3ghz since P.-P2 = Patn & desav., ALLLA, Now Patm t 123vie Ighi - Paton + 2 l vu + Sghe x = gCh, kn) &.= segh (herhaah) substituting Ve in een , we get AL - - A sign (we have &... db) dh-Aasia di P sh - - Aragt &c since, sro - c at tao, ho ho 55= -An seg & & The Sho-Su - Az sagt