Let's break down the problem **step-by-step**: --- ## **Step 1: Define Variables** - \( h_1, h2 \): Liquid heights in tanks 1 and 2. - \( A_1, A_2 \): Cross-sectional areas of tanks 1 and 2. - \( q_i \): Inlet flow rate (manipulated, independent of head). - \( q_1 \): Flow from tank 1 to tank 2 (through valve \( R_1 \)). - \( q_2 \): Outflow from tank 1 (through valve \( R_2 \)). - \( w_2 \): Inflow to tank 2 from tank 1 (same as \( q_1 \)). - \( w_3 \): Outflow from tank 1 (same as \( q_2 \)). - \( R_1, R_2 \): Flow resistances for valves. - \( \rho \): Density of fluid (constant). --- ## **Step 2: Flow Relationships** Given: - \( q_1 \) and \( q_2 \) are proportional to the square root of the liquid head (difference). For valves, the flow is also proportional to head difference: - \( q_1 = \frac{h_1 - h_2}{R_1} \) - \( q_2 = \frac{h_1}{R_2} \) --- ## **Step 3: Mass Balance for Each Tank** ### **Tank 1** Rate of change of liquid volume in tank 1: \[ \frac{d}{dt}(A_1 h_1) = \text{Inflow} - \text{Outflow} \] \[ A_1 \frac{dh_1}{dt} = q_i - q_1 - q_2 \] ### **Tank 2** \[ A_2 \frac{dh_2}{dt} = q_1 \] --- ## **Step 4: Substitute Flow Relationships** #### **For Tank 1:** \[ A_1 \frac{dh_1}{dt} = q_i - \frac{h_1 - h_2}{R_1} - \frac{h_1}{R_2} \] #### **For Tank 2:** \[ A_2 \frac{dh_2}{dt} = \frac{h_1 - h_2}{R_1} \] --- ## **Step 5: Final Dynamic Model** ### **State-Space Equations** \[ \boxed{ \begin{align*} A_1 \frac{dh_1}{dt} &= q_i - \frac{h_1 - h_2}{R_1} - \frac{h_1}{R_2} \\ A_2 \frac{dh_2}{dt} &= \frac{h_1 - h_2}{R_1} \end{align*} } \] --- ### **Explanation** - The first equation is the dynamic balance for tank 1: input minus the two outputs. - The second equation is the dynamic balance for tank 2: the only input is from tank 1. --- ## **Summary Table** | State Variable | Differential Equation | |:--------------:|:---------------------| | \( h_1 \) | \( A_1 \frac{dh_1}{dt} = q_i - \frac{h_1 - h_2}{R_1} - \frac{h_1}{R_2} \) | | \( h_2 \) | \( A_2 \frac{dh_2}{dt} = \frac{h_1 - h_2}{R_1} \) | --- ### **These two coupled ODEs form the dynamic model for the given system.**