1. DYNAMICAL EFFECTS in EXPANDING VOLUME FLOW CALIBRATORS Thomas O. Maginnis, Ph.D. CELERITY GROUP, Unit Instruments Division 2002 NCSL International Workshop and Symposium August 2002

2. Basic Method for Gas Mass Flow Measurement M = mass of gas (collected) n = number of moles of gas M 0 = molecular weight of gas species P = gas pressure T = gas absolute temperature V = volume of gas collected Z = compressibility correction (for non-ideal gases & vapors) R = universal gas constant t = time

3. Volumetric Primary Calibrator V1 V2 P T Difficulties: Determining barometric pressure accurately, needs maintenance, only non-reactive gas, sealing fluid issues, batch device. Strengths: Commonly available, operates at atmospheric pressure.

4. Bell Provers

5. OUTLINE PURPOSE LEVELS OF MODELING SIMPLE RESULTS SUMMARY

6. UNDERSTAND WHAT LIMITS PERFORMANCE flow range of specific instruments / the general technique accuracy of specific instruments / the general technique usable displacement of specific instruments LEARN IF/HOW LIMITATIONS CAN BE OVERCOME any fundamental limitations to range or accuracy? what must be done to improve performance? is technique cost-effective compared to alternatives? IMPROVE PERFORMANCE modify existing instruments design better instruments for wider flow range increase flexibility of existing instruments PURPOSE

7. ELEMENTARY QUASISTATIC assumes zero acceleration of bell or piston displacement velocity set by gas flow rate at constant P, T assumes perfect mass counterbalance BASIC SINGLE DEGREE OF FREEDOM DYNAMIC MODEL includes bell or piston inertia acceleration derived from Newton's laws basic driving force is ΔpA motion of sealing fluid (for bells) ignored TWO DEGREE OF FREEDOM MODELS (for Bell Provers) bell and fluid inertia effects both included coupled non-linear differential equations LEVELS OF MODELING

8. ELEMENTARY QUASISTATIC Fundamental Assumption: Bell or piston coasts upward at constant velocity dX/dt = Q/A where Q is the volume flow of gas into the gas collecting chamber. A is the internal cross-sectional area of the collecting chamber.  Zero net force acting along direction of motion.  near perfect counterbalancing  negligible friction Can then employ measured  X and  t to determine Q

9. How Much Steady Acceleration can be Tolerated?

10. For constant acceleration In terms of Q

11. Numerical Example Want to measure a velocity v = 1 mm/s with a precision of 1 part in 1000 over a displacement of 10 cm. Then we require

12. Oscillatory Acceleration Plus Constant Velocity

13. For Oscillatory Acceleration for precision We require

14. Numerical Example for Superimposed Oscillation Want to measure a velocity v = 1/6 mm/s with a precision of 1 part in 1000 over a displacement of 10 cm. Fill time is 600 s ~ oscillation period. Then we require

16. Variables for Dynamical Equations: Simplified Bell Prover

17. BASIC SINGLE DEGREE OF FREEDOM DYNAMIC MODEL

18. SINGLE DEGREE OF FREEDOM (if no counterweight)

31. Continuing solution past a = g in Systems with Counterweight 1-D equation valid only when a <= g because chain/cord tension cannot be negative Integrate until a = g, determine common positions, velocity at that point. Solve new equations for decoupled motions of bell/piston and counterweight, using those position, velocities as new initial conditions. Continue solution in time until instant when slack in chain/cord is gone Force both velocities to a common value at that instant by conserving linear momentum Continue solution of original equation for re-coupled motion, using new positions, common velocity as starting point, until a=g again. Solve equations for de-coupled motion again slack is once again used up, etc. This process leads to impulsive braking once each cycle that limits oscillation amplitude even without frictional damping.

36. Summary Piston/bell motion often assumed to be at constant velocity Residual acceleration can limit accuracy Inertial effects are important. Simplest dynamical model predicts unstable oscillation until vertical acceleration exceeds g, when model breaks down. Introducing appropriate frictional damping can control the oscillation and smooth out the motion Even without damping, oscillation amplitude limited in counterweighted systems by chain/cord going slack.