# Basics of Analyzer Sample Systems - Parts 1-2

## Here's How to Know Your Process Conditions by Calculating Dead Spaces, System Lag Time and System Pressure Drop, Simplifying a Planned System and Picking the Right Equipment for It

Where:

t = time

V = sample system volume

L = distance from the sample point to the analyzer sensor

P_{a} = absolute pressure

Z = compressibility factor

F_{s} = flow rate under standard conditions

T_{a} = absolute temperature

#### Compressibility Is a Factor for Gases at Higher System Pressures

For liquids, compressibility is negligible and the compressibility factor is Z = 1.0. However, in gas systems operating at more than about 35 to 50 psia, compressibility must be considered. For gases, compressibility changes as a function of pressure and temperature according to the rules of the ideal gas law, as shown in Equation 2

Where:

Z = compressibility factor

P_{a} = absolute pressure

V = volume

n = moles of fluid

R = gas constant

T_{a} = absolute temperature

The compressibility factory Z can be determined from compressibility charts and the associated reduced temperature Tr and reduced pressure P_{r}.

The reduced temperature and pressure are calculated as follows:

T_{r} = T_{a}/T_{c}

P_{r} = P_{a}/P_{c}

Where:

T_{c} = y_{1}T_{c1} + y_{2}T_{c2} + y_{3}T_{c3} … (y_{x} is the mole fraction and Tcx is the critical temperature of component x)

P_{c} = y_{1}P_{c1} + y_{2}P_{c2} + y_{3}P_{c3} … (y_{x} is the mole fraction and Pcx is the critical pressure of component x)

In addition, don't forget that the ideal gas law uses absolute pressures (P_{a}) and temperatures (T_{a}), so calculations must be done in psia or kPa (abs) and degree Rankine (R = F + 460) or degrees Kelvin (K = C + 273.15). Also, by combining and rearranging Equation 2 at two conditions and neglecting n, which remains constant, it is also possible to estimate the effect of pressure or temperature on volume.

Thus,

Where:

Subscript 1 refers to the inlet condition

Subscript 2 refers to the outlet condition.

#### Calculate Sample Flow

If you have a certain size and length of line and want to figure out an appropriate sample flow rate (F_{s}), at standard conditions, rearrange Equation 1 as shown in Equation 3

Once you know the volumetric sample flow rate (F_{s} in liters/min), you can determine the velocity (v in ft/sec) of a stream using Equation 4.

Where:

F_{s} = volumetric sample flow rate (liters/min)

0.1079 = a conversion factor to get the final result into ft/sec

D = internal pipe diameter (inches).

As a general rule of thumb, the sample system velocity should be in the range of 1 to 2 m/s (3 to 6 ft/sec) to ensure that any components in the sample are carried along with the sample proper and do not drop out of solution.

#### System Pressure Drop Depends on Velocity

The pressure drop in the system can be calculated using the sample system velocity calculated in Equation 4. This is not as difficult as it sounds, although it is important. Often the hardest part of the exercise is getting an estimate of the stream properties. The equation for pressure drop per 100 feet of tubing is shown in Equation 5.

Where:

ΔP_{100} = pressure drop per 100 feet of tubing (psi)

f_{d} = Darcy Friction Factor

ρ= density (lb/ft3)

v = velocity (ft/s)

D = pipe diameter (inches)

To calculate the Darcy friction factor (fd) we need to calculate the Reynold's number, as shown in Equation 6.

Where:

R_{e} = Reynolds number

ρ = density

v = velocity

μ = viscosity

If the Reynolds number is less than 4000, the Darcy friction factor is calculated as shown in Equation 7

However, if the Reynolds number is greater than 4000, then A.K. Jaini's non-iterative equation can be used, as shown in Equation 8.

Where:

f = Darcy friction factor

ε = absolute roughness in inches

D = diameter of the pipe in inches

R_{e} = Reynolds number

Meanwhile, the Moody friction factor, also known as the Fanning friction factor, is one quarter (+) the Darcy friction factor calculated in Equation 7 or Equation 8. Make sure you know which friction factor you're using and, if needed, adjust accordingly.

The last step in the pressure drop calculation is to determine the equivalent length of pipe. The equivalent length (L_{e}) is a parameter used to represent the total length of pipe of a single diameter that would be equivalent to the actual pipe with all its fittings and line size changes. Crane Technical Paper 410-C is the standard that is used to obtain these parameters.

The Crane standard uses the concept of "equivalent length" to assign a factor to each type of fitting or change in pipe diameter to a length of straight pipe that would equate to the same pressure drop as the fitting. Each type of pipe change is assigned a "K" factor as a function of a nominal friction factor (f_{t}). The Crane factor (f_{t}) is a function of nominal pipe size. The equivalent length K factor in the Crane manual is empirically determined from experimental data. After the K factors have been determined for all the fittings, they're summed, and this total equivalent length is then added to the actual pipe run length to calculate a total equivalent length. For example, a pipe system with two 90° elbows and plug valve, the calculation would be as follows:

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