Question

66. In a paint mixing plant, two tanks supply fluids to a mixing cistern. The height, h, of the fluid in the cistern is dependent upon the difference between the input mass flow rate, q, and the output flow rate, qe. A nonlinear differential equation describing this depen- dency is given by (Schiop, 2010 dh A pA where A - cross-sectional area of the cistern, Ac - cross-sectional area of the exit pipe, g - acceleration due to gravity, and ρ-liquid density. a. Linearize the nonlinear equation about the equilibrium point (ho, qo) and find the transfer function relating the output cistern fluid level H(s), to the input mass flow rate, Q(s)

Answer 1

given that - In a point mixing plant two tanks supply thids to a mixing cintain 2011- Gimen differential Equation is The output flow rate is, Te = PAZ Y = (Pag vag) Jh. For Simplicity Consider the Constants ki = Aedag ka pa kz= PAS Jag Ther fore, the equations are h =k, Jh tkar -1) a nd 2a = k3Jh - The functional form of the Equations are fCh, q)= h-k, Jh+kar and h (hq) = &q=&z 5h The relation between equilibrium point (20 ho) is ro = -1 11 Sho 90 = 2g Az Tho rate is water level and input man frow h=hot sh 220t aq The tay lor series expansion about the equilibrium Condition

h = f(h. q) = f(ho, 2o) + df 1 hi ho (h-ho) + of haho 19-20) de 19=20 Where - | if I hoho | | 7-Zs = dk, Jh tkaq I haho đi | 7-io | |--hỏ , gk, Oh + k, A-16 d2 12-20 From the of to, ho relation h=ho - A22 Pg From the expansion of taylor Scries. tha - pa sh + På Az Soc) Dom [ 14 ] Therefore final transfer function in a AHC) el 10 C5) St-2