Flow Injection Analysis (FIA) is a non-equilibrium technique so there is no mathematical theory than can adequately explain it. In its normal use, FIA is an interaction between chemical reactions and physical dispersion. The FIA manifold is a three dimensional (diameter and length) chemical reactor with an infinite number of small volumes that each have slightly different concentrations of sample and reagent. These concentrations change with time as they travel through the tubing and the rate of change is dependent upon:

1. Flow rate

2. Sample size

3. Tubing Length

4. Tubing Diameter

5. Temperature

6. Viscosity

The closest mathematical theory to explain FIA is the random walk model first proposed by Einstein to explain Brownian motion. The random walk is dependent upon laminar flow and molecular diffusion. A visualization of the random walk is a very drunk person trying to walk a straight line. Each time he staggers off the line there is a chemical reaction. Another example would be to watch what happens when cold milk is poured into hot coffee. The little swirlies that happen are a result of diffusion as the milk slowly mixes with the coffee. The cold milk and coffee are made to react faster by swirling the cup. This is the same as adding loops to a FIA manifold. 

The WAD cyanide by ASTM D6888 system is a FIA system. The sample containing cyanide is injected into a flowing stream of dilute acid solution. The cyanide combines with acid by molecular diffusion and adding a mixing coil speeds up the process. The hydrogen cyanide formed is passed under a hydrophobic membrane and drawn into a receiving stream of sodium hydroxide. 

The total Cyanide ASTM D7511 system is a segmented flow analysis system. The sample containing cyanide is injected into a flowing stream of acid solution that is segmented with air. The air segments eliminate the laminar flow profile and cause reactions to occur in each little packet of sample + reagent. The end over end mixing as these packets travel down the tubing allows equilibrium to occur so that all sample and reagent react. (the biggest limitation to this is not having segments of equal size). The sample + reagent is irradiated by UV light at 312nm. This is sufficient to separate strong metal cyanide bonds recombining cyanide with H+ to form HCN and complexing the metals to form soluble sulfate salts. A reducing agent in the acid counter reacts peroxide radical formation minimizing oxidation of iron or decomposition of thiocyanate. After emerging from the UV reactor, a sulfide complexing reagent combines with the sample stream and the sample passes under a hydrophobic membrane and is drawn into a receiving stream of sodium hydroxide. 

This non-distillation cyanide method was created to eliminate certain interferences that occur during a cyanide distillation that cause negative bias, positive bias, or both. Since the matrices of samples are rarely known, it is impossible to determine if bias will occur from the distillation step. These interferences can be so extreme as to cause cyanide to be not detected when it is really there, or to be detected when it is not. Some examples of these interferences are distillation of thiocyanate in the presence of nitrate, which causes cyanide detections when it is not there, or distillation of cyanide in the presence of thiocyanate alone that causes negative bias. The problem with the examples is significant because thiocyanate and nitrate both are common in environmental samples. Other interferences include oxidizers, sulfite, thiosulfate, elemental sulfur, metallic sulfide complexes, and others. The non-distillation method has been experimentally proven to not be affected by most of these interferents with the exception of a small (<1%) degradation of thiocyanate to cyanide as it passes through the UV reactor. This degradation of thiocyanate limits the time the sample should stay in the UV if thiocyanate is present. In other words, the method should be optimized to maximize cyanide complex recovery and minimize recovery of cyanide from thiocyanate.