# Simulation of Process Conditions for Calibration of Fisher Level Controllers and Transmitters Supplement to 249 Sensor Instruction Manuals

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1 Instruction Manual Supplement 249 Sensors Simulation of Process Conditions for Calibration of Fisher Level Controllers and Transmitters Supplement to 249 Sensor Instruction Manuals Displacer / torque tube sensors are transducers that convert a buoyancy change into a shaft rotation. The change in buoyancy is proportional to the volume of fluid displaced, and the density of the fluid. The change in rotation is proportional to the change in buoyancy, the moment arm of the displacer about the torque tube, and the torque rate. The torque rate itself is a function of the torque tube material, the temperature of the material, the wall thickness, and the length. If the density of the process fluid, process temperature, and torque tube material of the sensor are known, simulation of process conditions may be accomplished by one of the following means (1) : 1. Weight or Force Method: The interface application is the most general case. The level application can be considered an interface with the upper fluid SG 0, and the density application can be considered as a variable SG application with the interface at the top of the displacer. The buoyancy for a given interface level on the displacer is given by: ρ w [ ( SG L - SG U ) F B * V D * SG U H disp * ] (1) Figure 1. Cutaway View of Fisher 249 Displacer Sensor DRIVER ROD F B buoyant force ρ w density of water at 4C, 1 atmosphere Kg/liter ( lb/in 3 ) V D displacer volume H disp height of interface on displacer, normalized to displacer length SG U specific gravity of upper fluid (0.0 for Level) SG L specific gravity of lower fluid LIQUID DISPLACER TORQUE TUBE SUSPENSION ROD W Note that this document does not consider the effects of the thermal expansion of the moment arm, or the thermal expansion of displacer volume.

2 249 Sensors Instruction Manual Supplement Figure 2. Fisher 2500 or 2503 Level Controller Transmitter on Caged 249 Sensor Figure 3. FIELDVUE DLC3010 Digital Level Controller 2500 OR 2503 CONTROLLER/ TRANSMITTER 249 SENSOR W7977 W8334 For the density application, H disp 1.0, SG u lowest expected density, and SG L becomes the independent variable, the actual process density. The net load on the driver rod is then computed from the equation: W net W D - F B (2) W net net load on driver rod W D weight of displacer To simplify equations in the following discussion, let us define a few intermediate terms: The minimum buoyancy, developed when the interface level is at the bottom of the displacer, is given by: FB min ρ w * V D * SG U (3) The change in buoyant force as the normalized interface level rises on the displacer is: ΔF B ρ w * V D * (SG L - SG U ) * H disp (4) The maximum change in buoyancy, developed when the interface level is at the top of the displacer is: (ΔF B ) max ρ w * V D * (SG L - SG U ) (5) 2

3 Instruction Manual Supplement 249 Sensors Temperature Effect As process temperature increases, the torque rate decreases due to the change in modulus of rigidity. This effect can be represented by normalizing the modulus vs. temperature curve for a given material to the room temperature value, and using it as a scale factor on the torque rate. See figure 4 and table 1 or 2. Because we can simulate the rotation of a more compliant torque tube by increasing the load, the weight value may be divided by the same scale factor to simulate the process condition: (6) W net_test W D - F B W net_test net load adjusted to simulate process temperature effect normalized modulus of torque tube material, a function of temperature. Displacer Rise Effect Note that equation 6 simulates the process level on the displacer. The actual level in the cage or vessel will be different, due to the rise of the displacer as the torque tube load is decreased by the increasing buoyancy. On a 14 inch displacer, or on a 249VS with a long driver rod, the displacer rise can become a significant fraction of the span. If the torque tube rate and driver rod length are known, change in rotation can be computed by dividing the torque change by the rate. ΔAngle ΔF B * Driver * ( π /180 ) R amb * (7) ΔAngle resulting change in torque tube angle in radians Driver driver rod length R amb Torque rate (torque per rotation) at ambient temperature 3

4 249 Sensors Instruction Manual Supplement Figure 4. Gnorm : Theoretical Temperature Effect on Torque Rate for Most Commonly Used Materials 1.00 TORQUE RATE REDUCTION (NORMALIZED MODULUS OF RIGIDITY) N05500 N06600 N S TEMPERATURE (C) 1.00 TORQUE RATE REDUCTION (NORMALIZED MODULUS OF RIGIDITY) N05500 N06600 N S TEMPERATURE (F) NOTE: THIS CHART DEPICTS THE REVERSIBLE CHANGE ONLY. THE IRREVERSIBLE DRIFT IS A FUNCTION OF THE NET LOAD, THE ALLOY, AND THE LEVEL OF STRESS EQUALIZATION ACHIEVED IN MANUFACTURING. (THE IRREVERSIBLE EFFECT CAN ONLY BE COMPENSATED BY PERIODIC ZERO TRIM.) N05500 IS AN APPROPRIATE SPRING MATERIAL FOR TEMPERATURES BELOW AMBIENT AND UP TO 232C (450F). ABOVE 260C (500F), THE INDUSTRY DOES NOT RECOMMEND USING IT AS A SPRING MATERIAL. N06600 IS CONSIDERED ACCEPTABLE TO APPROXIMATELY 399C (750F) WITH PROPER STRESS EQUALIZATION. 4

5 Instruction Manual Supplement 249 Sensors Recognizing that the displacer rise is the side opposite this changing angle, in a triangle of which the drive rod is the hypotenuse, the displacer rise can be approximated applying the small angle sine approximation to the result of equation 7. ΔRise disp ( ) ΔF B * Driver 2 R amb * * π 180 The above expression may be factored to produce an ambient temperature displacer rise rate. (9) (π / 180 )* Driver 2 RiseRate amb R amb The net rise is then restated with aid of the temperature correction term Gnorm. (10) ΔRise disp ΔF B * RiseRate amb (8) The digital instrument firmware makes an internal correction for the displacer rise, so we must account for it in our weight calculation to make sure that we get the expected digital value of interface level in the cage. (Analog electronic and pneumatic devices don't have this correction capability, but the accuracy of the calibration would still be improved by accounting for the effect during simulation.) The ratio of the level change on the displacer to the level change in the cage needed to produce it is: (11) ΔH disp L D proc ΔH cage L D( ΔF B ) max * RiseRate amb / L D length of displacer H cage interface level in cage, normalized to displacer length. The above equation is valid only for 0.0 < H disp < 1.0 (since interface level excursions above or below the displacer produce no additional change in buoyancy) The % span error introduced by neglecting the displacer rise effect becomes smaller as the displacer length increases. The displacer rise at the initial condition, (displacer completely submerged in the upper fluid), is given by: Rise 0proc FB min_eff * RiseRate amb FB min_eff [ FB min Max( (W disp W max ), 0.0) ] W max the load that will cause the linkage to contact the lower travel stop. (12) Max( (W disp W max ), 0.0 ) the amount of buoyancy required to lift the displacer off the travel stop so that rise can actually commence. The maximum available shift below the zero rest position at ambient temperature is also limited by this travel stop, so the value of W max will be a function of temperature. To account for this, replace W max by (W max_ambient * ) in the FB min_eff equation. Note that W max_ambient would have to be determined by experiment on the specific physical hardware, but for a first approximation, you could use the maximum un buoyed displacer weight value given in table 5. 5

6 249 Sensors Instruction Manual Supplement Figure 5. Illustration of Displacer Rise Effect Initial Offset Ref Zero Shift Rise i Rise f Final Offset T amb F b 0 T proc F b 0 T proc F b f(sg U ) T proc F b f(sg L ) 6

7 Instruction Manual Supplement 249 Sensors Temperature Induced Zero Shift at Zero Buoyancy If the physical zero reference was established at zero buoyancy and ambient temperature, there is an additional zero shift to take into account. The zero buoyancy position of the bottom of the displacer at process temperature will be lower, because of the reduction in torque rate. ZeroShift proc (13) W D *RiseRate amb *( -1.0 ) Note that the combination of: a. the location of the displacer bottom relative to the external reference at ambient, b. the zero-shift at process temperature, and c. the initial displacer rise at process temperature, will determine the extent of any unobservable region between the external zero reference and the displacer bottom. We must decide what our process variable (PV) calculation is going to use for a zero reference. Since any interface level excursion below the bottom of the displacer cannot change the output, it is convenient to call the displacer bottom zero for the test, and this has been standard procedure in pilot mounting. In the digital level controller, Level Offset is used to adjust the digital output to zero at this condition. Lowest Observable Cage Interface Level If it is desired to line up the calculation with the physical external reference, the Level Offset (and range values) can be adjusted according to the following. (14) H cage0 ZeroShift proc Rise 0proc L D H cage0 highest possible value of cage interface level (normalized to displacer length), relative to zero buoyancy, ambient temperature coupling point, when displacer interface level is 0.0 (bottom of displacer). This is the physical interface level below which a change is unobservable. The range values or alarm values should be set within the observable range of PV to make sure that over and under flow conditions are reported to the control system. 7

8 249 Sensors Instruction Manual Supplement Weight Calculation Procedure To compute the weight required, at room temperature, to simulate a given process condition cage interface level: a. Start with an initial buoyancy based on the SG of the upper fluid, b. subtract it from the displacer weight, and c. correct the result for process temperature. This will give the test weight for the lowest observable process condition. W net_test i W D - F Bmin (15) The change in weight for a process temperature, cage (or vessel) interface level condition, one displacer length higher than the above state, is given by: ( ΔH * ΔW disp / ΔH cage ) proc ( ΔF B ) max f (16) The net weight for the 100% cage process condition is: (17) W net_test W net_test ΔW f i f Other values of ΔW can be computed from: ΔW ΔH cage *( ΔH disp /ΔH cage ) proc * (ΔF B ) max (18) ΔH cage H cage H cage0 Valid for H cage0 < H cage < [1/ (ΔH disp / ΔH cage ) proc ] Remember that it is common to arbitrarily set H cage0 to zero for test purposes when using weights. (For water column calculations in the next section, it is more important to keep track of the initial process condition cage level to simulate the initial buoyancy correctly.) The resultant net weights for the intermediate levels are given by: W net_test W net_test - ΔW i (19) This assumes that the net weights do not violate the maximum or minimum load for the torque tube. Refer to table 5. 8

9 Instruction Manual Supplement 249 Sensors 2. Water Column Method: It is possible to simulate a range of buoyancy adjusted for process temperature effect by using a water column at room temperature. At the ambient SG 1.0 level application, the corrections should all cancel out, leaving H cage desired PV. Cage Water Level Required to Simulate Interface Levels If we have computed an equivalent weight for a given process condition in section 1, the ambient temperature water level on the displacer that will produce the same torque tube rotation is: H disp_eq W D - W net ρ w * V D (20) For high temperatures and high SG, the range of conditions that can be simulated will contract, since we are limited by the actual displacer weight and the nominal density of water. We must next convert this equivalent displacer level into an equivalent cage level using the inverse of the relationship in equation 11, without the temperature compensation. ΔH disp ΔH cage amb L D ρ w * V D * RiseRate amb L D (21) The result is: (22) H cage_eq ( ΔH disp / ΔH cage ) amb * ρ w * V D ( W D -W net ) Process Interface Level Simulated by a Given Cage Water Level We can also write a generic equation for the process condition displacer interface level simulated by a given room temperature displacer water level: First, convert the ambient cage water level to an ambient displacer water level: H disp_eq H cage ΔH cage / ΔH disp amb ( ) (23) Next, define an intermediate variable to compute the apparent SG being simulated at process conditions by the ambient displacer water level: (24) SGapp sim ( 1- ) * W D * H disp_eq ρ w * V D 9

10 249 Sensors Instruction Manual Supplement Now use this apparent SG value to compute the simulated interface level on the displacer: H disp sim SGapp sim -SG U SG L -SG U (25) Finally convert the simulated process conditions displacer interface level to simulated process condition cage interface level, by the equation: H cagesim H disp sim H cage0 ( ΔH disp / ΔH cage ) proc (26) Where H cage0 is either 0.0 or the value computed in equation 14, per the practice being followed for PV reference. 3. Tables of Nominal Values If the calibration is being run per standard practice the values of the parameters for the above equations are generally available for observation in the instrument memory. For analog instruments, a table of nominal values may be consulted to generate good approximations. Table 1. Gnorm for Common Torque Tube Materials Above Room Temperature Gnorm Material C F C F C F C F C F C F C F C F N N N S These values are approximations derived from various metal-alloy industry publications Table 2. Gnorm for 316 SST Below Room Temperature Gnorm Material C F C F C F C F C F S Low temperature data for N05500, N06600, and N10276 not available at time of publication. Table 3 provides the theoretical unloaded rate, and the composite or effective torque rate measured by the digital level controller at the end of the pilot shaft. The physical rotation at the far end of the torque tube may be a bit greater than what these tables would predict, due to some wind up of the pilot shaft. Table 3. Theoretical Room Temperature Torque Rates Family / Wall 249, 249B, 249BF 249BP, 259B, 249P (CL ), 249W HEAVY 1. N05500 is the default material. 2. Appropriate for 2500 controllers only. Composite Rate Material Torque Tube Unloaded Rate (2) W/O Insulator W/Insulator Part Number Nm/deg lbfin/deg Nm/deg lbfin/deg Nm/deg lbfin/deg N05500 (1) 1K4497X N P8662X S K A N K continued- 10

11 Instruction Manual Supplement 249 Sensors Table 3. Theoretical Room Temperature Torque Rates (continued) 249, 249B, 249BF 249BP, 259B, 249P (CL ), 249W STANDARD Family / Wall 249C 249CP 249PT 249VT HEAVY 1. N05500 is the default material. 2. Appropriate for 2500 controllers only. 249B 249, 249BP, 259B, 249P (CL ), 249W THIN Family / Wall 249C 249CP 249PT 249VT STANDARD 1. N05500 is the default material. 2. Appropriate for 2500 controllers only. Family / Wall 249K, 249L, 249N, 249VS, 249P (CL ), STANDARD 1. N05500 is the default material. 2. Appropriate for 2500 controllers only. Family / Wall 249K, 249L, 249N, 249VS, 249P (CL ) THIN 1. N05500 is the default material. 2. Appropriate for 2500 controllers only. Material Torque Tube Part Number Unloaded Rate (2) W/O Insulator Composite Rate W/Insulator Nm/deg lbfin/deg Nm/deg lbfin/deg Nm/deg lbfin/deg N05500 (1) 1K4493X N K A S K A N K A Material Torque Tube Part Number Unloaded Rate (2) Composite Rate W/O Insulator W/Insulator Nm/deg lbfin/deg Nm/deg lbfin/deg Nm/deg lbfin/deg N05500 (1) 1K4495X N K A S K A N K4529X Composite Rate Material Torque Tube Unloaded Rate (2) W/O Insulator W/Insulator Part Number Nm/deg lbfin/deg Nm/deg lbfin/deg Nm/deg lbfin/deg N05500 (1) 1K4499X N K A S K A N K9159X Composite Rate Material Torque Tube Unloaded Rate (2) W/O Insulator W/Insulator Part Number Nm/deg lbfin/deg Nm/deg lbfin/deg Nm/deg lbfin/deg N05500 (1) 1K4501X N P S K

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