Prcess Gain and Lp Gain By nw, it is evident that ne can calculate the sensitivity fr each cmpnent in a cntrlled prcess. Smetimes, this sensitivity is referred t as a gain. The cnfusin is understandable since bth sensitivity and gain are defined as the rati f the change in utput fr a unit change in input. The priry difference between sensitivity and gain is the use f units. When calculating gain, the input and utput signals are expressed in the same units, usually a cntrl signal, s the final value f gain is unit-less. Hwever, sensitivity usually invlves expressing the same rati in terms f engineering units. This statement is best demnstrated by determining the sensitivity and gain f a given cntrller. Cnsider a cntrl lp that cntrls the utlet temperature n the shell side f a shell and tube heat exchanger by nipulating the flw rate f ht water n the tube side. The sensr is a 100 t 200 F RTD with an utput f 4 t 20. The cntrller sends a 4 t 20 signal t an electrnic valve actuatr. Assume a thrttling range f 25 F. The cntrller sensitivity is then: Output CS 0.64 Input F F 25 Nte that the input signal is expressed in terms f an engineering unit while the utput signal is expressed in terms f a cntrl signal. T determine cntrller gain, we must express bth input and utput in terms f the same signal. We can d this by expressing these quantities as cntrl signals. In this case, we will express the input as a current () instead f temperature. This requires we find the sensitivity f the sensr. Cntrller Figure 1 Cntrl f a heat exchanger 0.16 0 ( 200 100) F sensr sens F Once the sensitivity is knwn, the thrttling range f 25 F, which represents cntrller input, can be expressed as: 0.16 25 F 4 m F a
Then cntrller gain is: Output gain Kc 4 Input 4 Sensitivity arund the lp Just as we can express cntrller perfrnce r cmpnent perfrnce in terms f a sensitivity r gain, we can als express the perfrnce f a cntrlled prcess in terms f a gain r sensitivity. This is referred t as prcess gain. The prduct f this prcess gain and the cntrller gain is referred t as lp gain. Let s lk at an example. Depicted belw is a blck diagram f the abve shell and tube cntrl lp. Cnsider the prtin f the lp frm pint 1 clckwise arund the lp t pint 2. The cntrl element is the valve actuatr and cntrl valve. The nipulated variable is the flw rate thrugh the tube-side f the heat exchanger. The prcess is the heat exchange between the shell side fluid and the tube side fluid. Physically, it cnsists f the heat exchanger itself, the fluid flwing thrugh the exchanger, the temperature f the fluids and the flw rates f the fluids. The cntrlled variable is the temperature f the fluid leaving the exchanger n the shell side. The sensr is likely an RTD, thermistr, r thermcuple. The sensr cnnects t a transmitter (a signal cnditiner), which translates the millivltage r resistance signal frm the sensr t a signal capable f being received by the cntrller; usually a vltage, current r pneutic signal. Each f these elements can be assigned sme value f sensitivity, the prduct f which is the prcess gain. Mathetically: Change in prcess utput signal at pint 2 prcess gain K p Change in prcess input signal at pint1 Change in prcess utput signal at pint1 cntrller gain Kc Change in prcess input signal at pint 2 Lp Gain K K K L p c
DISTURBANCE SETPOINT CONTROLLER 1 ACTUATOR CONTROL ELEMENT MANIPULATED VARIABLE PROCESS CONTROLLED VARIABLE 2 TRANSMITTER SENSOR Figure 2 Blck diagram f a shell and tube heat exchanger Let's assume the prcess heat exchanger, a nminal 6,000 Mbtuh unit, has the fllwing design characteristics: Tube side: Flwrate: 0 t 350 gpm Inlet Temp: 240 F Shell side: Flwrate: 350 gpm (nminal) Inlet Temp: 130 F Outlet Temp: 160 F Let us als assume the fllwing cnditins: Actuatr: A 4 t 20 signal mves the actuatr thrugh a full strke f 2.50. This results in a sensitivity f 2.50 in / 16 0.15625 in/ Valve: A linear valve with a 2.50 strke cntrlling a flw frm 0 t 350 gpm. Sensitivity is equal t 350 gpm / 2.5 in 140 gpm/in Prcess: Heat exchanger as defined abve. Assume a sensitivity f 0.1174 F/gpm. Sensr: A 1000 Ω RTD with a resistance f 1146.84 t 1359.70 Ω ver a temperature range f 100 t 200 F. The sensitivity is 2.1286 Ω/ F Transmitter: A 4 t 20 utput signal with an input resistance f 1146.84 t 1359.70 Ω. Transmitter sensitivity equals 0.07516 /Ω Sensitivities are multiplicative. We can determine the sensitivity f the lp, that is, the lp gain, as fllws: K p in gpm F Ω 0.15625 140 0.1174 2.1286 0.07516 0.411 in gpm F Ω This means if the signal t the valve actuatr changes by ne milliamp, (prcess input), yu will see a change f 0.411 milliamps at the utput f the sensr/transmitter at the
ther end. If the cntrller has a gain f 4, then we can determine the lp gain as: KL Kp Kc 0.411 4 1.64 This means that fr a signal change f 1 t the input terminals f a prcess cntrller, the signal will travel arund the lp and be amplified by a factr f 1.64. The cntrller will then see a change at it s input terminals f 1.64 as feedback. It is wrth nting the abve analysis is quite simplified. The priry reasns deal with the nnlinearities f certain cmpnents f the cntrl lp as fllws. The cntrl valve was assumed a linear valve. In general, a linear valve is inapprpriate fr the cntrl f a heat exchanger. In practice, the installed valve is likely an equal percentage valve. A pssible exceptin t this is the cntrl f very high temperature prcess water (400+ F) with a large T acrss the exchanger (100 200 F). Under such circumstances, the heat exchanger exhibits near linear characteristics and a linear valve y be apprpriate. This brings up the secnd pint. A sensitivity was prvided fr the heat exchanger ver a given range. This sensitivity was based upn the premise the utput f the heat exchanger is linear. Again, this is nt the case. Mst heat exchangers exhibit extreme nnlinear characteristics. Anther surce f nnlinearity is the temperature sensr. Temperature sensrs are nt generally linear devices. Hwever, the abve example was based upn the use f an RTD. As discussed during the terial n sensrs and transmitters, an RTD des exhibit nearly linear characteristics. Hwever, if the sensr was a thermcuple r a thermistr, these devices exhibit a greater degree f nnlinearity. As such, it is necessary that a prperly designed transmitter is used t cmpensate fr these nnlinearities. All cntrl lps and cntrlled prcesses exhibit any number f dead times and time delays. Althugh these time elements d nt affect the sensitivity f any given device, it has a great impact n the respnse f the cntrl lp. Variable Lp Gain We cvered the variability f installed valve characteristics and heat exchanger characteristics at the beginning f the quarter. As was discussed at the time, the ideal is t tch the valve t the heat exchanger perfectly. This wuld result in a linear heat exchanger utput relative t valve stem mvement. Hwever, imprperly sized cntrl valves r the necessity t install 'ff-the-shelf' valves that d nt tch the characteristics
f the heat exchanger y result in nnlinearities. Als, it is quite difficult t linearize the utput f a steam cil. The end result might be smething as shwn in Figure 3. 5,250 Heat Exchanger Output (Mbtuh) 2100 720 0 4 4 0% 100% % Valve Open Figure 3 Illustratin f variable prcess gain The graph indicates a valve accepting a 4 t 20 signal t strke full pen t full clsed. The heat exchanger is designed fr an utput f 0 t 5,250 Mbtuh ver this range. The heat utput is nt linear relative t valve psitin. If we lk at tw specific pints, namely 70% valve lift and 15% valve lift, we find very different values f sensitivity (gain). At the lwer range, the sensitivity is abut 2100/4 525 Mbtuh/ while at the upper range, the sensitivity is abut 720/4 180 Mbtuh/. This represents almst a threefld variatin in lp gain if the prcess perated between these extremes. Needless t say, if we have multiple elements in the cntrl lp exhibiting such variatin in sensitivity, the prcess will becme very difficult, if nt impssible, t tune prperly. If the prcess is prperly tuned at that pint where the lp exhibits high lp gain, the cntrl system will be sluggish at the ther extreme. Cnversely, if the lp is tuned t respnd prperly at that pint where the lp exhibits a lwer degree f gain, the lp will likely be ver aggressive, even unstable, at the ppsite end.