Stirred tank heater

Temperature control in a stirred-tank heater is a common example (Fig. 2.9). We will come across it many times in later chapters. For now, we present the basic model equation, and use it as a review of transfer functions. 

Figure 2.9. A continuous flow stirred-tank heater.

A brief review is in order Recall that Laplace transform is a linear operator. The effects of individual inputs can be superimposed to form the output. In other words, an observed output change can be attributed to the individual effects of the inputs. From the stirred-tank heater example in Section 2.8.2 (p. 2-23), we found ... 

It is important to understand that the time constant xp of a process, say, a stirred tank is not the same as the space time x. Review this point with the stirred-tank heater example in Chapter 2. Further, derive the time constant of a continuous flow stirred-tank reactor (CSTR) with a first-order chemical reaction... 

Consider the stirred-tank heater again, this time in a closed-loop (Fig. 5.4). The tank temperature can be affected by variables such as the inlet and jacket temperatures and inlet flow rate. Back in Chapter 2, we derived the transfer functions for the inlet and jacket temperatures. In Laplace transform, the change in temperature is given in Eq. (2-49b) on page 2-25 as... 

We now walk through the stirred-tank heater system once again. This time, we ll take a closer look at the transfer functions and the units (Fig. 5.5). 

With the stirred-tank heater, we know quite well by now that we want to manipulate the heating coil temperature to control the tank temperature. The process function Gp is defined based on this decision. In this simple illustration, the inlet temperature is the only disturbance, and the load function is defined accordingly. From Section 2.8.2 and Eq. (2-49b) on page 2-25, we have the first order process model ... 

The closed-loop characteristic equation of the stirred-tank heater system is hence ... 

When we developed the model for the stirred tank heater, we ignored the dynamics of the heating coil. Provide a slightly more realistic model which takes into consideration the flow rate of condensing steam. 

How we model the stirred tank heater is subject to the actual situation. At a slightly more realistic level, we may assume that heat is provided by condensing steam and that the coil metal is at the same temperature as the condensing steam. The heat balance and the Laplace transform of the tank remains identical to Chapter 2 ... 

Example 4—Selecting physically meaningful inputs to characterize stability. Consider a continuous stirred-tank heater modeled by the following equations, in continuous time ... 

Example 1.1 Controlling the Operation of a Stirred Tank Heater... 

Figure 1.4 Feedforward temperature control for stirred tank heater.

The disturbances F, and T, of the stirred tank heater (Figure 1.1) are easily measured thus they are considered measured disturbances. On the other hand, the feed composition for a distillation column, an extraction unit, reactors, and the like, is not normally measured and consequently is considered an unmeasured disturbance. As we will see later, unmeasured disturbances generate more difficult control problems. 

Figure 3.1 Hardware elements for the feedback control of a stirred tank heater.

Figure 3.1 describes the hardware elements used for the control of the stirred tank heater. 

In the stirred tank heater system shown in Figure 1.1, the flow rate F of the effluent stream is proportional to the square root of the liquid level h in the tank. Show that such a system is self-regulating (i.e., if the inlet flow rate increases or decreases by a unit, the tank will not overflow or empty completely). 

Let us now proceed to develop the state equations for the stirred tank heater. We will apply the conservation principle on the two fundamental quantities the total mass and the total energy. 

Figure 4.5 Dynamic response of a stirred tank heater to a step decrease in inlet flow rate.

Equation (5.1) is the mathematical model of the stirred tank heater with T the state variable, while T, and Ta are the input variables. Let us see how we can develop the corresponding input-output model. 

The stirred tank heater is modeled by two equations containing six variables, thus yielding four degrees of freedom (Example 5.3). This is true if the effluent flow rate F is determined by a pump, valve, and so on. 

Consider again the stirred tank heater, but now under feedback control (Figure 5.5). Control loop 1 maintains the liquid level at a desired value by measuring the level of the liquid and adjusting the value of the effluent flow rate. Therefore, control loop 1 introduces a relationship between F and h. Similarly, control loop 2 maintains the temperature of the liquid at the desired value by manipulating the flow of steam and thus the flow of heat Q. Consequently, control loop 2 introduces a relationship... 

During the reduction in the number of degrees of freedom for a chemical process, care must be exercised not to specify more control objectives than it is possible for the particular system. Thus we can have at most two control objectives for the stirred tank heater. When we attempt to have three control objectives, we are led to an overspecified system with / < 0. 

Consider the stirred tank heater discussed in Example 4.4. 

II.4 Do the same work as in Problem II.3 for the stirred tank heaters system shown in Figure PII.4. For tank 1, the steam is injected directly in the liquid water. Water vapor is produced in the second tank. A i and A 2 are the cross-sectional areas of the two tanks. Assume that the effluent flow rates are proportional to the liquid static pressure that causes their flow. A, is the heat transfer area for the steam coil. 

System A The three-tank system of Figure PII.2 (Problem II.2). System B The two stirred tank heaters system of Figure PII.4 (Problem II.4). 

The mathematical model of the stirred tank heater in terms of deviation variables was developed in Example 5.1 and is given by eq. (5.3) ... 

Example 11.3 The Stirred Tank Heater as a System with Two Interacting Capacities... 

The stirred tank heater of Example 4.4 is characterized by its capacity to store mass and energy. It is easy to show that these two capacities interact when the inlet flowrate changes. Thus, a change in the inlet flowrate affects the liquid level in the tank, which in turn affects the temperature of the liquid. Consequently, the temperature response to an inlet flowrate change exhibits second-order overdamped characteristics. The reader should note that the two capacities do not interact when the inlet temperature changes. Therefore, the temperature response to inlet temperature changes exhibits first-order characteristics. 

What are the basic hardware components of a feedback control loop Identify the hardware elements present in a feedback loop for the temperature control of a stirred tank heater. 

The question that arises is How do we design feedforward controllers The reader may have suspected already that conventional P, PI, or PID controllers will not be appropriate. Let us start with an example, the design of feedforward controllers for a stirred tank heater. 

In Example 4.4 we developed the dynamic mass and energy balances for the stirred tank heater of Figure 1.1. They are given by eqs. (4.4a) and (4.5b). 

For instance, in Example 21.2, for the stirred tank heater we can easily identify the static and dynamic parts of the process transfer functions [see eq. (21.4)] ... 

Figure 21.8 (a) Temperature response of a stirred tank heater under feedforward control alone, and with feedback trimming (b) corresponding block diagram. 

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