5-5. Control Errors

The peak error is rather large for fast disturbances due to a time constant to dead time ratio that ranges from poor (e.g., 4.0) to lousy (e.g., 0.4). The larger ratios correspond to moderate flows and temperature control on the shell side where more backmixing occurs. The worse case is a small coolant flow and temperature control on the tube side with large transportation delays from low plug flow.

The integrated error is quite small despite large peak errors because the loop period is relatively small for temperature control. Thus, the use of exchanger temperature as an inner (i.e., slave or secondary) loop of a cascade control system for vessel temperature control is quite effective because the rapid oscillations are effectively filtered by the large process time constant of the liquid volume.

When the peak error or initial transient must be minimized, feedforward control should be used. A rather simple energy balance that equates heat lost from the hot side to heat gain by the cold side yields a solution per Eqs. (5-9a) (coolant flow) and (5-9b) (steam flow) for the required final element flow for a given cold and hot inlet temperature/ feed flow, and set point. Normal operating values are used for those inputs not measured that are relatively constant. For example, if the main upset is feed flow, a measurement of this flow is required, but assumed operating conditions can be used for the inlet temperatures that are not measured. The equations show temperature control of the hot side. For control of a cold side, only the subscripts need to be changed. It is critical that the controlled temperature be the set point rather than the measurement to avoid positive feedback.

where:

The loop set point must be used for the controlled temperature!

The feedforward signal is added to the output of the feedback controller, A bias of 50% is also added so that the temperature controller can make a negative correction as large as the positive correction to the feedforward signal The manipulated flow is best achieved by means of a flow controller and a cascade of exchanger temperature to coolant or steam flow. If the temperature controller output goes directly to a control valve, signal characterization of the installed valve characteristic should be used to convert from desired flow to required valve position. The signal divider for compensation of process gain nonlinearities should be applied to the controller output before the summer, as shown in Fig. 5-8. Of course, the coolant or steam valve should have a positioner. If the manipulated variable is a steam pressure controller or a regulator set point, the substitution of the steam temperature for the hot side temperatures of Eq. (5-7a) and the subsequent solution of the equation for steam temperature will yield a corresponding saturated steam pressure. Here, the objective is to maintain the proper log mean temperature difference or driving force and let the pressure controller or regulator determine the actual steam flow needed. The feedforward loop may not be needed, because a fast inner steam pressure loop or regulator can correct for disturbances before they affect the temperature loop.

Fig. 5-8. The total and feedback output should have separate signal compensation for the installed valve characteristic and process gain, respectively.

If the exchanger outlet temperature can be adjusted, a valve position controller can be used to slowly change the temperature set point to optimize the coolant valve position, as shown in Fig. 5-9. There may be an optimum position that minimizes the fouling of heat transfer surfaces by prevention of low throttle positions but also maintains the process gain and reduces utility usage by the prevention of high throttle positions.

Fig. 5-9. A valve position controller will slowly trim the temperature set point to minimize fouling of heat transfer surfaces at the best possible process gain and utility usage.

For exchanger bypass control (see Fig. 5-6), the blend of the hot and cold streams is achieved by a split of the total flow into an actual exchanger flow and a bypass flow. Since the heat capacities are approximately equal, the ratio of bypassed flow to total flow can be approximated as proportional to the ratio of the biased controller set point to the biased bypass temperature, per Eq. (5-10). The bias is a subtraction of the actual exchanger outlet temperature. Note that the bypass temperature is also the inlet temperature to the exchanger.