5-4. Controller Tuning

The increase in loop period, the decrease in process time constant, and the increase in process gain require a dramatically lower controller gain per Eq. (4-16) for low flow. For an X ratio less than one, a controller gain tuned for quarter amplitude response may have to be decreased by a factor of four to ten for a reduction of 50% in flow. The factor of four corresponds to a more dead time dominant system (e.g., tube side throttled) where the controller gain becomes more dependent upon the open-loop gain than on the dynamics (Ref. 2). Limit cycles develop for excursions below a ratio of one. Some users have found out experimentally that elevation of the controller low output limit helps. The use of an equal percentage trim can cut the gain change in half, because the valve gain becomes proportional to flow. The improvement is greater for a more constant pressure drop across the valve stays, so that the installed characteristic is closer to the inherent trim characteristic. The benefit of an equal percentage characteristic can be achieved more accurately by application of the signal divider, detailed in Eq. (5-4), to the temperature controller output.

where:

The proper solution is adaption of the temperature controller gain and reset settings. Eqs. (5-5) and (5-6) provide approximate rules of thumb for adaption of series (i.e., real) controllers for control of liquid - liquid exchangers with an X ratio of less than one and manipulation of either a valve with a linear installed characteristic or a flow controller set point. The derivative setting (i.e., rate time) doesn't need adaption unless the residence time is much larger than two minutes or a bare element is used, because the setting depends mostly upon the secondary measurement time constant (e.g., thermowell lag).

For tube side throttling:

For shell side throttling:

For tube or shell side throttling:

where:

Most industrial exchangers have a total loop dead time of thirty seconds to two minutes and thus a period of one to eight minutes and a reset setting of 1.0 to 0.1 repeats per minute. The loop dead time is the sum of process dead time, the controller scan time, the dead time from valve dead band, and the portion of the thermowell time constant that becomes dead time. The process dead time is negligible for manipulation of an exchanger bypass (Fig. 5-6). For direct throttling of exchanger low, the dead time is the portion of the residence time and thermal lag across the metal heat transfer surfaces converted to dead time. The metal thermal lag is the product of the metal mass and heat capacity divided by the product of the overall heat transfer coefficient and area. This time constant (TC_a) is usually small for most exchangers (e.g., 0.1 minute). The sum of the sensor and thermowell measurement time constants (TC_m) is larger (e.g., 0.2 to 0.8 minute). Since the derivative time is normally set equal to the sum of these secondary time constants in the loop, temperature controllers for exchangers should have a nominal derivative or rate setting of 0.3 to 0.9 minute that largely depends upon the thermowell material and air gap. For jackets and coils in reactors, the metal lag and other secondary lags are larger, which translates to larger derivative settings for the reactor temperature controllers.

The process gain is inversely proportionally to the manipulated exchanger flow. For large shells, the interaction with the other side can be ignored, and the process gain simplifies to Eq. (4-6), which is used for reactors, fermentors, and vessels. Otherwise, the energy balance on both sides and the effect of the log mean temperature difference should be included. Eqs. (5-7a) and (5-7b) provide a steady solution of the outlet temperature, given the flows and other temperatures. It can be solved for the outlet temperature of choice (i.e., controlled variable) and then differentiated with respect to the manipulated flow to give the process gain. Since the conduction heat transfer resistance is negligible and the heat transfer coefficient changes with 0.8 power of flow, the variation in the heat transfer coefficient can be approximated by Eqs. (5-8a) through (5-8e).

where:

For exchangers where the inner loop is a steam pressure controller, the change in steam pressure corresponds to a change In heat transfer surface temperature. Thus, the process gain is the change in steam temperature divided by the change in steam pressure from steam tables, multiplied by the change in controlled temperature divided by the change in steam temperature at operating conditions.