Modelling Process G&L Engineering Ltd. 020 8810 5828

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In this modelling example, ammonia (NH3) was introduced into a reformer flue to convert Nitric Oxide (NOx) to Nitrogen (N2) and Water (H2O). NOx is not a naturally occurring element, but it is generally accepted that thermal NOx consists of 90% N2 and 10% NO2. The purpose of this model was to dynamically predict the response and chemical behaviour as a result of injecting ammonia before the catalyst.

The process reaction that occurred was:
4NO + 4NH`3` + O`2`	®	4N`2` + 6H`2`O
6NO`2` + 8NH`3`	®	7N`2` + 12H`2`O

At 100% process load the reformer furnace produced 130ppm of NOx. The flue gas velocity was 46698 m3/h

Volumes of NO and NO`2` produced at 100% load
Volume of NO	130 x 0.9 1000000	x	46698	=	5.464 m3/h
Volume of NO`2`	130 x 0.9 1000000	x	46698	=	0.607 m`3`/h

Converting to kg moles

By applying the gas constant referenced to standard metric conditions, the mass of the above volumes were calculated. Using the component molecular weights the kg mole values were obtained. Substituting these into the chemical equation showed the number of NH3 moles required to convert 1kg mole of NOx.

NO	=	5.464 23.646	=	0.2311	kg_moles/h
NO`2`	=	0.607 23.646	=	0.0257	kg_moles/h

Substituting the kg mole values into the chemical equation, showed the number of NH3 moles required to convert 1kg_mole of NOx

4NO + 4NH + O`2`	º	0.2311NO + 0.2311NH`3`+ 1/4O`2`
6NO`2` + 8NH`3`	º	0.0257NO`2` + 8/6(0.0257)NH`3`

		NOx moles	NH`3` Moles
NO	®	0.2311	0.2311
NO`2`	®	0.0257	0.0343
		------- 0.2568 -------	------- 0.2654 -------

1 mole of NOx required 0.2654 / 0.2568 = 1.033 moles of NH3 to be converted.
Substituting molecular weights and allowing for a 5ppm ammonia slip across the catalyst bed
1kg mole NOx required 0.5813kg NH3 to be converted.

Steady State Equation

Calculating the density of NOx produces the units kg/m3 and knowing the ammonia required to convert 1kg of NOx, and using an 85% NOx reduction across the catalyst, produced the equation: The equation was using process design data at this stage to allow it's validity to be checked on the process.

130 x 0.85 x 1.337 x 46698 x 0.5813 = 4.01 kg/h NH`3` 1000000
NOx reduction	*FGV

*FGV = Flue Gas Volume
Constants from this equation were multiplied together to produce a single constant.

k = 1.337 x 0.5813 = 0.000000772 kg/m3

The equation was rearranged replacing the engineering units with generic labelling to represent our interest. NO analysers not NOx analysers were installed on the process therefore the equation became:

(NOin - NOout) (1.111) x FGV x k = NH3/h

The equation was accurate for steady state conditions, but did not predict the dynamic process behaviour. The equation was converted to a deviation equation.

(NOout - NOin)

NH3
FGV x K 1.111

This equation still assumed an infinite time to complete the reaction, where as the catalyst was sized to promote the reaction time to convert 85% of the NOx as the flue gas passed over the catalyst bed.

1.0
1 - 0.85

6.666

Dynamic Equation

Representing the rate of change from a steady state by

dNO
dt

the deviation equation became:

dNO
DT

kp((NOin - NOout)

NH3
FGV x k

)6.6666

Rearranging to represent the NOout in terms of NOout deviations and NH3 ppm. The process model differential equation became.

Frequency Domain Equation

Having developed the process equation, our interest was to design a control system. The output from a controller is dynamic and can be considered sinusoidal. To obtain conditionally stable control the time delays between the input and the output of the process must be know. Frequency Domain modelling allowed the response of the system to a sinusoidal input to be modelled and time delays to be modelled as phase shifts. The process was essentially linear therefore the relationship between the output and the input at a frequency was represented as the ratio of the steady state output amplitude to the input amplitude.

Xo
Xi

output_amplitude
input_amplitude

with a phase shift of the output relative to the input

A sinusoidal input to the process would result in a differentiated sinusoidal output. Representing the input as an error from the steady state that must lead the output the differential equation can be modified as:

((NOin - NOout)

NH3
FGV x k

)6.6666

sinwt+Ø

This can be represented as E sinwt+Ø. The output dNOout/DT requires differentiating, the sinusoidal becomes:

w NOout sin (wt +

pi
2

)

using the j operator to represent a 90° rotation anticlockwise the above can be substituted by jw. The error from the steady state E is now represented as a phasor containing both amplitude and phase (phasors are represented in bold)

NOout jw = 0.00021968 (E)

The time taken from opening the ammonia valve to measuring a change in NOout was 215 seconds. Modelling in the form of exp(-jTLw) and rearranging to form a transfer function, the transfer function of the process was:

NOout
E

0.00021968 exp(-215jw)
jw

For examples of the control system and calculations associated with this model click on one of the pages below

Control System Model

Calculations for Compensated Indications