**Title: Liquid-Level System Analysis****Description:**The diagram displays a liquid-level system consisting of two tanks with cross-sectional areas denoted as \(A_1\) and \(A_2\). A pump connects to Tank 1 via a valve with linear resistance \(R_1\). The pump's inlet is exposed to atmospheric pressure, and the fluid pressure rises by \(\Delta p\) as it traverses the pump. The liquid travels from Tank 1 to Tank 2 through a valve with linear resistance \(R_2\), and exits Tank 2 through another valve with linear resistance \(R_3\) into the atmosphere. It is assumed that the liquid density \(\rho\) remains constant and both tanks are open to atmospheric pressure, denoted \(p_a\).**Diagram Explanation:**- **Tanks:** Two cylindrical tanks, each with cross-sectional areas \(A_1\) and \(A_2\). The liquid inside each tank is depicted with its height represented by \(h_1\) and \(h_2\).- **Pump and Valves:** - A pump increases pressure by \(\Delta p\). - Three valves with resistances \(R_1\), \(R_2\), and \(R_3\) regulate flow between the tanks and into the atmosphere.- **Pressure and Flow:** - Initial pressure \(p_a\) at pump inlet. - Increasing pressure \(\Delta p\) through the pump. - Final output flow denoted by \(q_o\).**Your Tasks:**A. **Derive a differential equation model** for the system behavior in terms of the liquid heights \(h_1\) and \(h_2\).B. **Convert the differential equations** into a second-order matrix form.

To Find:The differential equation model for the system behavior in terms of the liquid heights and The second-order matrix form of the differential equation.Consider the diagram:Calculation:Mass balance for Tank 1:The flow out is due to the valve connecting tank 1 to tank 2:The rate of change of liquid in Tank 1 is given by the difference between the flow in and the flow out:Mass balance for Tank 2:The rate of change of liquid in tank 2 is given by the difference between the flow in from tank 1 and the flow out through the valve connected to tank 2Substitute the values in the expression:Calculation:The second-order matrix can be obtained as:This matrix form allows for a systematic analysis and control design for the liquid-level system.The differential equation model for the system behavior in terms of the liquid heights is obtained.The second-order matrix form of the differential equation is .

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The figure below shows a liquid-level system in which two tanks have cross-sectional areas, A₁ and A2, respec- tively. A pump is connected to tank 1 through a valve of linear resistance R₁. The inlet to the pump is open to atmosphere, and the pressure of the fluid increases by Ap when crossing the pump. The liquid flows from tank 1 to tank 2 through a valve of linear resistance R₂ and leaves tank 2 through a valve of linear resistance R3, exiting at at- mospheric pressure. Assume the density p of the liquid is constant and note that both tanks are open to atmosphere as shown. Pa Your Tasks: Ap Pa A₁ R₂ Pa A₂ R3 -% A. Derive a differential equation model for the system behavior in terms of the liquid heights hi and h₂. B. Put the differential equations into second-order matrix form.

The figure below shows a liquid-level system in which two tanks have cross-sectional areas, A₁ and A2, respec- tively. A pump is connected to tank 1 through a valve of linear resistance R₁. The inlet to the pump is open to atmosphere, and the pressure of the fluid increases by Ap when crossing the pump. The liquid flows from tank 1 to tank 2 through a valve of linear resistance R₂ and leaves tank 2 through a valve of linear resistance R3, exiting at at- mospheric pressure. Assume the density p of the liquid is constant and note that both tanks are open to atmosphere as shown. Pa Your Tasks: Ap Pa A₁ R₂ Pa A₂ R3 -% A. Derive a differential equation model for the system behavior in terms of the liquid heights hi and h₂. B. Put the differential equations into second-order matrix form.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

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Work and Heat Transfer

It is generally related to thermal engineering which tells about the use, creation, conversion and transfer of heat energy between the systems. It is also considered that to transfer heat from one source to another, transfer of masses is also required. There are various mechanisms by which, transfer of heat takes place i.e. thermal conduction, convection, radiation or by change of phase. All these mechanisms have their own specific features which sometimes occur simultaneously within the same system.

Question

**Title: Liquid-Level System Analysis**

**Description:**

The diagram displays a liquid-level system consisting of two tanks with cross-sectional areas denoted as \(A_1\) and \(A_2\). A pump connects to Tank 1 via a valve with linear resistance \(R_1\). The pump's inlet is exposed to atmospheric pressure, and the fluid pressure rises by \(\Delta p\) as it traverses the pump. The liquid travels from Tank 1 to Tank 2 through a valve with linear resistance \(R_2\), and exits Tank 2 through another valve with linear resistance \(R_3\) into the atmosphere. It is assumed that the liquid density \(\rho\) remains constant and both tanks are open to atmospheric pressure, denoted \(p_a\).

**Diagram Explanation:**

- **Tanks:** Two cylindrical tanks, each with cross-sectional areas \(A_1\) and \(A_2\). The liquid inside each tank is depicted with its height represented by \(h_1\) and \(h_2\).
- **Pump and Valves:**
- A pump increases pressure by \(\Delta p\).
- Three valves with resistances \(R_1\), \(R_2\), and \(R_3\) regulate flow between the tanks and into the atmosphere.
- **Pressure and Flow:**
- Initial pressure \(p_a\) at pump inlet.
- Increasing pressure \(\Delta p\) through the pump.
- Final output flow denoted by \(q_o\).

**Your Tasks:**

A. **Derive a differential equation model** for the system behavior in terms of the liquid heights \(h_1\) and \(h_2\).

B. **Convert the differential equations** into a second-order matrix form.

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