Given Solution:

Steam temperature, liquid temperature, steam pressure, liquid pressure, and liquid flow rate are the state variables of the system.
Assuming that the tank has been thoroughly stirred, the mass and energy balances should be performed. The mass balance is required to determine the rate of material accumulation in the tank, whereas the energy balance is required to determine the rate of heat transfer to the liquid.
Heating system with a stirred tank (Fig. II.3) The differential equation characterizes the system. where P is the liquid pressure, P0 the atmospheric pressure, A2 a constant, and K a constant of proportionality.
Integrating the differential equation yields the state model:

Given initial conditions of P = P0, x1 = 0, and t = 0, this equation holds.

a reactor that uses a stirred tank that maintains a constant temperature

Take, for example, the reactor seen in Figure II.4.

Liquid is introduced into the reactor, and the temperature is maintained at all times.

A reactor is being stirred by a permanent-rotation stirrer.

Liquid is pumped into the reactor at regular intervals.

The rate of this exothermic reaction is directly related to the size of the reactor.

The reactor's target temperature TR and the flow rate of liquid QF are the two parameters that define the system.

The temperature TR is always the same and is always the same as the liquid's temperature.

In a nuclear reactor, the liquid is fed in at a constant rate known as the feed rate QF.

Q, the flow rate of the reactor's waste water, is a defining characteristic of the system.

The system is characterized by the temperature TR and the liquid flow rate QF.

Explanation:

State variables of the system.

1. The system's state variables are the liquid static pressure and the steam flow rate.

2.The steam flow rate and liquid static pressure are the input-output relationships.

3. The differential equation that describes the relationship between the steam flow rate and the liquid static pressure is the system's dynamic model.

4. When the system is operating at its nominal operating point, the steady-state operating characteristic is the relationship between the steam flow rate and the liquid static pressure.

5. If the system is operated above its nominal operating point, the system will increase the liquid static pressure in response to a step input in the steam flow rate.

If the system is operated below its nominal operating point, the system's response to a step change in the steam flow rate is to reduce the liquid static pressure.

6. If the system is operated above its nominal operating point, the system will increase the liquid static pressure in response to a step input in the effluent flow rate.

If the system is operated below its nominal operating point, the system will decrease the liquid static pressure in response to a step input in the effluent flow rate.

7. It makes no difference whether the system is run at a high or low static pressure.

8. The system can operate at a static pressure greater than the steam pressure.

9. The benefits of operating the system at a static pressure greater than the steam pressure are that the system can be operated at a higher steam flow rate and is less likely to be overloaded.

The disadvantages of operating the system at a static pressure greater than that of the steam are that the system is more likely to be unstable and damaged if the steam flow rate is too high.

10. The system is stable when the steam flow rate is controlled in a closed loop. When the static pressure is kept constant, the system is stable.

A proportional-integral-derivative (PID) controller is used in the closed-loop feedback control system to regulate the steam flow rate so that the system operates at the nominal operating point.

A proportional-integral-derivative (PID) controller is used in the closed-loop feedback control system to keep the static pressure at its nominal operating point.

balances you should perform

A material balance is the first balance that should be performed.

This will provide the overall composition of the tank as well as the flow rates of the various components.

The following balance to be performed is an energy balance.

This will give the heat input to the tank as well as the heat lost to the environment.

A momentum balance should be performed as the final balance.

This gives the force on the tank walls as well as the liquid flow rates.

Develop the state model that describes the dynamic behavior of the system.

The following state model (in transfer function form) describes the tank heater's dynamic behavior: where

The volume of the tank's contents, V (ft3)

tank level (h) = amount of liquid in storage (ft)

Effluent flow is caused by the static pressure of liquid, denoted by P. (psi)

Q1 = tank effluent discharge rate (gpm)

Steam Flow Rate, U1 (gpm)

Heater with a stirred tank, as shown in Figure II.3
A Case in Point:

Consider the stirred-tank heater shown in Figure II.4. The steam is injected directly in the liquid.
The area of the tank's cross section is denoted by the parameter A1.
Assume that the effluent flow rate is proportional to the liquid static pressure that causes its flow
Create a state model that characterizes the system's dynamic behavior.

The dynamic behavior of the tank heater is described by the following state model (in transfer function form):
where

The volume of the tank's contents, V (ft3)

h = liquid level in tank (ft)

P = liquid static pressure causing effluent flow (psi)

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