# Gerald D. Anderson. Education Technical Specialist

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1 Gerald D. Anderson Education Technical Specialist

3 Selecting the Sensor and Tank The type of sensor employed influences the relationship between controller settings and the tank area required. This interaction is displayed in the form of equations relating the surface area of the liquid to the proportional band setting, the maximum expected throughput, and the rated span of the sensor. The basic formula is given by: In this discussion, the liquid surface area (A) is expressed in ft 2 ; while (PB) is the controller proportional band setting in percent. (1) The constant (K) is determined by the assumed gain margin of 2.25, by the dimensions chosen for the variables and parameters, and by the assumption that the correct valve characteristic has been selected (usually linear for liquid level systems). In the instance of flow measured in gpm and area in ft 2, K equals It is assumed that the valve is sized such that its maximum capacity is equal to the maximum process flow rate. If the valve is oversized, flow rate (Q) will increase; and then either A or PB must increase to satisfy Equation (1). The rated span (RS) of the sensor is the level change, in inches, required to cause a full output change out of a transmitter or sensor. (For example, with a Fisher Type 249 utilizing a 14-inch displacer, RS is 14. M represents the dynamic loop gain of the system evaluated at specific frequency, which is the critical frequency that is, the frequency at which a system cycles. The value of M depends on the type of sensor, transmitter, controller, and positioner employed in the loop. Table 1 lists some typical values of M to be used in Equation (1). Table 1. M Values to be Used in Equation (1) ACTUATOR- POSITIONER TYPE 249 DISPLACEMENT SENSOR WITH DAMPING ORIFICE COMBINATION 2 Ft Equalizing Arm 5 Ft Equalizing Arm / / / / / /80 470/30 470/40 470/60 470/ The reason M changes as the equalizing arm length changes is evident after examining Equation (2). where: W v = g = L 1 = L 2 = A 1 = A 2 = (2) Undamped natural frequency acceleration of gravity Acceleration of gravity Length of equalizing arm Wetted length of displacer Area of equalizing line Area of annular space between displacer and cage wall. This equation was derived in TM-7, The Dynamics of Level and Pressure Control. In general, the shorter L 1 is, the more stable and more responsive the system is. Equation (1) is a guideline derived from a stability analysis of liquid level systems and is based on the assumption that level is within the range of the sensor. Therefore, the equation is not valid when disturbances exceed the sensor range, as might be expected during plant start up or shut down. Of course, the extremes in level which can be tolerated depend upon the process. In one instance, the level control objective may be that the tank never overflow nor run dry. While in another, the level must be kept within inches of the set point to prevent damage to processing equipment, such as heat exchangers. Regardless of the allowable variance in level, a properly designed system accounts for the greatest disturbance that can occur. Controllability Index A relatively simple check can be made to determine whether or not conventional equipment will provide suitable level control. By rearranging Equation (1), the factors related to the process can be separated from those related to control equipment. (3) To provide some significance to the Q/A ratio of Equation (3), the units gpm/ft 2 should be converted in in / sec: (4) Equation (4) indicates that if the process is uncontrolled, if the maximum throughput is Q gallons 3

4 per minute, and if the tank area is A square feet, the level will change Q/A inches each second. Since a great many successful level control systems have a Q/A ratio of less than 4.5 in / sec, Equation (4) provides an index of controllability that proves useful to system designers, equipment suppliers, and process engineers, alike. For instance, the designer can first use Equation (1) to determine a tank size and then check the reasonableness of the size with Equation (4). The equipment supplier can use Equation (4) to properly select the required process equipment, while the process engineer can use Equation (1) to gauge the effect of a change in proportional band. Examples To illustrate use of the guidelines, consider an example. Assume that a tank is to be sized for a system in which maximum expected throughput is 645 gpm. The vessel is to be pressurized at 100 psig. The control valve is to be on the outlet of the tank, with the downstream pressure approximately atmospheric. The liquid being processed has a specific gravity of one. To meet the maximum expected throughput, a 2 cage-style globe valve with a linear flow characteristic is chosen. This valve, with a Cv = 65.3, provides a flow under the assumed conditions of 653 gpm, which is acceptable. A suitable valve positioner/actuator combination for the valve is the /60. Suppose the steady-state level must stay at some point ± one inch. This can be achieved by using a Type 249 with a 14 long displacer and a Type 4181 proportional plus reset controller. Since a 2-foot long liquid equalizing arm between tank and cage will be used, the value of M from table 1 is Arbitrarily choosing a proportional band setting of 50 percent to allow enough adjustment for adequate tuning of the controller, and substituting these values into Equation (1), the necessary liquid surface area can be calculated. (5) Assuming that any baffles, tubing, etc. within the vessel have a cross-sectional area of two square feet, the tank area must be the sum of the liquid surface area and the equipment area. (6) The tank diameter then would be given by Equation (6). (7) Equation (3) now should be used to check this answer by substituting 653 for Q and 4.94 for A. The solution shown in Equation (7) indicates that the equipment and tank size selected should provide stable control. (8) As a second example, a differential pressure sensor is used instead of a displacer. The sensor is calibrated for a span of 25 inches; and assume that M at maximum damping is All other conditions remain as in the first example. Substituting these values into Equation (1) yields the liquid surface area: (9) Again assuming two square feet for equipment within the tank, the tank area would be: (10) And the tank diameter again can be calculated. (11) Utilizing the controllability check provided by Equation (3), the Q/A ratio is found to be far greater than the desired 4.5 in/sec. (12) Although the system would be stable if the tank was extremely long, it still might not be controllable even if the definition of control meant keeping the level between safe and reasonable values. Since the assumed control equipment included a differential pressure sensor with maximum damping, the dynamic response of the control equipment would be relatively slow, thereby intensifying skepticism 4

5 about being able to control the system. This does not mean that differential pressure sensors should not be used for level control. It simply means that for this particular system, a larger surface area than that calculated should be employed. Use of a larger surface area means there will be less of a level change when a change in load alters the net flow. It also allows use of a smaller proportional band setting, as shown by Equation (4). The smaller proportional band means the control valve strokes farther for a given change in level, the result being that the level is maintained closer to set point. Summary A simple mathematical guide to stable liquid level control systems has been described which dictates that control components should be selected so that Equation (1) is satisfied. This guideline is easy to use and yet complete enough to assure confidence in the results obtained. 5

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