Written Report

A “Personalized” linear second order PDE of research topic

Date: April 11, 2014

Present by: Manuel J. Cano.

Professor: Rolando Barrera

INTRODUCTION

Waste generation is an inevitable result of different human activities on the earth. Currently several

kinds of strategies are applied for these undesirable materials in order to prevention (no

generation), re-use, recycling, recovery or final disposal; these are together known as the waste

management hierarchy which goes from the best option, no generation, to the worse, final disposal.

The waste incineration is a kind of thermal treatment for final disposal in which the combustion of

different kind of materials is involved. As a result of the waste incineration process there are a

variety of toxics, pollutants and undesirable compounds generate by the combustion like carbon

monoxide (CO), carbon dioxide (CO2), nitrogen oxides (NOX), hydrochloric acid (HCl), hydrofluoric

acid (HF), dioxins, furans, heavy metals and so on.

Dioxin family represent a group of chemicals known as polychlorinated dibenzo-para-dioxins

(PCDDs) which are one of the most toxics and carcinogenic compounds existing and discovered by

researchers. Exposition to these kinds of compounds may produce several harm to humans or

animals such depressed immune system, reproductive and developmental problems.

Otherwise nitrogen oxides groups nitric oxide (NO) and nitrogen dioxide (NO2); both are produced

during the combustion of waste in the incineration process, these compounds cause several

problems to the environmental such as acid rain, destruction of ozone layer and contribute to the

greenhouse effect and smog formation.

Dioxins, furans and nitrogen oxides are all produced in the incinerator process of the waste and

represent a serious issue for companies, people around these ones and environmental. Currently

simultaneous remotion of dioxins (PCCDs), furans (PCDFs) and nitrogen oxides (NOx) can be

achieved through a selective catalytic reduction over vanadium pentoxide (V2O5) supported on

titanium oxide with ammonia at same operation temperature for reduction of NOx, however these

catalysts is poisoned easily and the use of NH3 as reduction agent lead to a corrosive and toxic

environment.

In this research project is proposed that the simultaneous remotion of dioxins and nitrogen oxides

can be achieved using a catalyst of Pd/Co-H-MOR and sulfated zirconia (SrO42-/ZrO2) and methane as

the reduction agent instead of ammonia.

Selected physical problem and model description.

When a stream of gases has significant amounts of particle matter, dust, coal and so on tends to

obstruct in packed-bed reactor, because this to avoid the problem it is common used monolith

(parallel plate reactor), then the reacting gas mixture flows through channels and between parallels

plates, the reaction takes place on the surface of the plates.

Monoliths catalysts Pd/Co-H-MOR will be used for reduction of the NO with methane in the

experimental phase. The monoliths used for Selective Catalytic Reduction (SCR) have a honeycomb

structure with parallel channels in which the catalyst will be impregnated.

Nitrogen oxides are mainly composed by NO and for the experimental phase it will be used a

simulated mix of NO 90% and NO2 10%. The gases flows through the parallel channels and the

reduction reaction takes placed on the surface of the impregnated monolith. There are involved two

main phenomena mass transfer and chemical reaction.

General reaction is given by:

CH4 + 2NO + O2 N2 + 2H2O+CO2

The overall rate of reaction will be equal to the slowest step in the mechanism: diffusion,

adsorption, surface reaction, desorption and diffusion. This work is focused on the diffusion of the

reactant NO on a particular channel between the bulk fluid and the external surface of the catalyst.

In the model it is needed to have in account the convection along of the channels (“z” direction), the

diffusion (“x” and “y” directions) and the chemical reaction over catalyst surface.

Next a mass balance of NO in the surface of the catalyst is set up departing from a cubic control

volume next to the catalyst as is show on figure 1.

Figure 1: Control volume near the catalyst surface

Definition of variables:

- 𝐶𝑁𝑂: Concentration of NO.

- t: time.

- 𝐷𝑁𝑂𝑔: Diffusion coefficient of NO in the gas mix.

Main assumptions:

- There is not diffusion on “z” direction because the effect of convection is most relevant than

diffusion.

- There are diffusion on “x” and “y” direction because change of the concentration of NO

between the center and the catalytic surface.

- The diffusion coefficient DNOg is constant.

- The monolith is not interconnected; this means that the gas can’t cross the monolith wall.

- There is not reaction in homogeneous phase, reaction only take place in the catalytic surface.

Detailed deduction of the linear second order PDE:

Departing form the general expression for mass balance:

𝐸𝑞. 1 (𝑅𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑖𝑛

) − (𝑅𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑜𝑢𝑡

) + (𝑅𝑎𝑡𝑒 𝑜𝑓

𝑚𝑎𝑠𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛) = (

𝑅𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠𝑎𝑐𝑐𝑢𝑚𝑎𝑙𝑎𝑡𝑖𝑜𝑛

)

𝐸𝑞. 2 (𝑅𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑖𝑛

) − (𝑅𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑜𝑢𝑡

)|𝑥

= (𝐽(𝑥) − 𝐽(𝑥 + ∆𝑥))∆𝑦∆𝑧

𝐸𝑞. 3 (𝑅𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑖𝑛

) − (𝑅𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑜𝑢𝑡

)|𝑦

= (𝐽(𝑦) − 𝐽(𝑦 + ∆𝑦))∆𝑥∆𝑧

There is no diffusion on the “z” direction, only convection:

𝐸𝑞. 4 (𝑅𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑖𝑛

) − (𝑅𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑜𝑢𝑡

)|𝑧

= 𝑢𝑧(𝐶𝑁𝑂(𝑧) − 𝐶𝑁𝑂(𝑧 + ∆𝑧))∆𝑥∆𝑦

𝐸𝑞. 5 (𝑅𝑎𝑡𝑒 𝑜𝑓

𝑚𝑎𝑠𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛) = 0

𝐸𝑞. 6 (𝑅𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠𝑎𝑐𝑐𝑢𝑚𝑎𝑙𝑎𝑡𝑖𝑜𝑛

) =𝑛𝑁𝑂(𝑡 + ∆𝑡) − 𝑛𝑁𝑂(𝑡)

∆𝑡 ∆𝑥∆𝑦∆𝑧

Replacing equations from 2 to 6 on eq. 1 is obtained the next equation:

𝐸𝑞. 7 (𝐽(𝑥) − 𝐽(𝑥 + ∆𝑥))∆𝑦∆𝑧 + (𝐽(𝑦) − 𝐽(𝑦 + ∆𝑦))∆𝑥∆𝑧 + 𝑢𝑧(𝐶𝑁𝑂(𝑧) − 𝐶𝑁𝑂(𝑧 + ∆𝑧))∆𝑥∆𝑦

=𝑛𝑁𝑂(𝑡) − 𝑛𝑁𝑂(𝑡 + ∆𝑡)

∆𝑡 ∆𝑥∆𝑦∆𝑧

On eq. 7 dividing for ∆𝑥∆𝑦∆𝑧 in both sides and taken limits when control volume and time tends to

zero:

𝐸𝑞. 8 −𝜕𝐽

𝜕𝑥−

𝜕𝐽

𝜕𝑦− 𝑢𝑧

𝜕𝐶𝑁𝑂

𝜕𝑧=

𝜕𝐶𝑁𝑂

𝜕𝑡

On eq. 7 dividing for ∆𝑥∆𝑦∆𝑧 in both sides and taken limits when control volume and time tends to

zero:

Now we need to take in account the Fick law:

𝐸𝑞. 9 𝐽𝑁𝑂 = −𝐷𝑁𝑂𝑔

𝜕𝐶𝑁𝑂

𝜕𝑥

𝐸𝑞. 10 𝐽𝑁𝑂 = −𝐷𝑁𝑂𝑔

𝜕𝐶𝑁𝑂

𝜕𝑦

Replacing Eq. 9 and Eq. 10 on Eq.8:

−𝜕

𝜕𝑥(−𝐷𝑁𝑂𝑔

𝜕𝐶𝑁𝑂

𝜕𝑥) −

𝜕

𝜕𝑦(−𝐷𝑁𝑂𝑔

𝜕𝐶𝑁𝑂

𝜕𝑦) − 𝑢𝑧

𝜕𝐶𝑁𝑂

𝜕𝑧=

𝜕𝐶𝑁𝑂

𝜕𝑡

Assuming diffusion coefficient constant this equation becomes in:

𝐸𝑞. 11 𝐷𝑁𝑂𝑔 (𝜕2𝐶𝑁𝑂

𝜕𝑥2+

𝜕2𝐶𝑁𝑂

𝜕𝑦2 ) =𝜕𝐶𝑁𝑂

𝜕𝑡+ 𝑢𝑧(𝑥, 𝑦)

𝜕𝐶𝑁𝑂

𝜕𝑧

Classification of the PDE:

Rearranging equation 11:

𝐸𝑞. 12: 𝐷𝑁𝑂𝑔 (𝜕2𝐶𝑁𝑂

𝜕𝑥2+

𝜕2𝐶𝑁𝑂

𝜕𝑦2 ) − 𝑢𝑧(𝑥, 𝑦)𝜕𝐶𝑁𝑂

𝜕𝑧−

𝜕𝐶𝑁𝑂

𝜕𝑡= 0

Then,

A = 𝐷𝑁𝑂𝑔

B = 0

C = 𝐷𝑁𝑂𝑔

Computing:

B2 – 4AC = (0)2 – 4(𝐷𝑁𝑂𝑔)*( 𝐷𝑁𝑂𝑔)

B2 – 4AC = – 4𝐷2𝑁𝑂𝑔 < 0

Then eq. 12 is an elliptic equation.