Consider the system shown at right Assume the resistances R1

Consider the system shown at right. Assume the resistances R_1 and R_2 are constants, and the areas A_1 and A_2 are constant so that the tank capacitances are constant (C_1 and C_2). Consider the inflow rate q_mi to be an input variable and the height of liquid in the second tank, h_2, to be an output variable. Write equations for the flow through each pipe and the changing height in each tank, and use these to obtain the transfer function relating input q_mi to output h_2, that is, derive H_2(s)/Q_mi(s).

Solution

solution:

1)as here are two tank are available hence this system have two variable to govern it output hence we have to write two transfer function one for first tank and second for second tank

2)here for first tank ,as per mass balance equation we have

Qm+Q1=Q1\'

Qm=input flow rate

Q1=volume change of tank

Q1\'=output flow

2)where from capacitance of tank we have

V=C1P1

P1=density*g*h1

we can write further as

dP1/dt=Q1/C1

wherer flow resistance in pi[e is

P1\'=R1*Q1\'

where by KPE

P1+P1\'=Pa

Pa=0=it is not atmospheric pressure but pressure difference across pipe length,here it is zero

P1=-P1\'

hence putting all in first equation we get

dP1/dt=(Q1\'-Qm)/C1

dP1/dt=(P1\'/R1-Qm)/C1

dP1/dt=(-P1/R1-Qm)/C1

2)whaterver flow Q1\' moving out from first tank is adding to second tank hence mass balance equation become

Q1\'+Q2=Q2\'

where capacitance relation is written as

dP2/dt=Q2/C2

dP2/dt=Q2\'-Q1\'/C2

where Q2\'=flow in output pipe suffer from resistance R2,hence

Q2\'=P2\'/R2

as first tank here again

P2\'=-P2

dP2/dt=(-P2/R2-Q1\')/C2

Q1\'=Q1+Qm

hence puting value we get that

dP2/dt=(-P2/R2-Q1-Qm)/C2

P2=density*g*h2

on putting in equation we get

dh2/dt=(-P2/R2-Q1-Qm)/density*g*C2

on differentiating we get that

P2/R2=constant as it is pressure loss

hence we get height in second tank as function of Qm only as follows

d^2h2/d^2t=(1/density*g*c2)Qm\'(t)

hence on applying laplace transform we get that

S^2*h2(s)=(1/density*g*C2)SQm(s)

hence transfer function becomes

h2(s)/Qm(s)=(1/S)(1/density*g*c2)