Absorption coefficients for СО_{2} and СН_{4} * α*, m

^{3}/m

^{3}of water at

*=0.1 MPa and*В

Т

*=273 К*

_{0}

Open access peer-reviewed chapter

By M. I. Shilyaev and E. M. Khromova

Submitted: February 9th 2012Reviewed: September 5th 2012Published: July 24th 2013

DOI: 10.5772/53094

Downloaded: 2047

The process of complex cleaning of gases, injected into the atmosphere, for instance, by thermal power plants, metallurgical, chemical or other industrial enterprises, from dust and harmful gaseous admixtures by means of their irrigation by wash liquids (water or specially selected water solutions) is considered. This process can be implemented in gas pipes or gas-cleaning apparatuses (direct flow or counter flow jet scrubbers) [1]. The process of gas cleaning from dust and gas admixtures is carried out in the following manner. The fluid dispersed by jets is introduced into the dust-vapor-gas flow in the form of droplets, interacts with it, and under nonisothermal conditions the increased moisture content leads to intensive condensation of liquid vapors on particles, their significant enlargement and efficient absorption of liquid droplets due to collisions of the latter with particles [2]. Simultaneously, the liquid droplets and condensate on particles absorb harmful gas components, dissolving them and removing from the vapor-gas flow.

The authors failed to find the mathematical description of this complex process in literature. From the engineering point of view the importance of development of generalized mathematical models, which reflect properly the interaction of heat and mass transfer with the effects of gas components removal and dust capture by the droplets of irrigating liquid in jet scrubbers and reactors, is undisputable, and it is determined by significant opportunities for optimization of operation conditions and constructions of energy-intensive and large-scale equipment in various industries both in terms of reducing of material and energy costs.

Advertisement## 2. Problem statement, main equations and assumptions

d V → d d τ = R → d + g → − V → d m d d m d d τ ; ![]()

E1с f m d d Т d d τ =- α d π δ d 2 ( Т d - Т ) + ∑ r i d m i d d τ + с δ ρ δ V c π δ d 2 4 η S t k ( T δ - T 0 ) ; E2d m i δ d τ = − β i d π δ d 2 ( ρ i d − ρ i ) ; ![]()

E3d m i δ d τ = − β i δ π δ 2 ( ρ i δ − ρ i ) ; ![]()

E4∂ ρ i ∂ τ + d i v ( ρ i U→ ) = − d m i d d τ n d − d m i δ d τ n δ ; ![]()

E5∂ ρ g ∂ τ + d i v ( ρ g U → ) = 0 ; ![]()

E6∂ ρ δ ∂ τ + d i v ( ρ δ U → ) = ∑ d m i δ d τ n δ − ρ δ V c π δ d 2 4 η Stk n d ; ![]()

E7∂ ρ d ∂ τ + d i v ( ρ d V → d ) = d m d d τ n d ; ![]()

E8с δ m δ d T δ d τ = − α δ π δ 2 ( T δ − T ) + ∑ r i d m i δ d τ ; ![]()

E9ρ d с ( T − T 0 ) d τ = α d π δ d 2 ( T d − T ) n d + α δ π δ 2 ( T δ − T ) n δ ; ![]()

E10d m d d τ = ∑ d m i d d τ + ρ δ V c π δ d 2 4 η S t k ![]()

E11d m δ d τ = ∑ d m i δ d τ ; ![]()

E12∂ ρ p ∂ τ + d i v ( ρ p U → ) = − ρ p V c π δ d 2 4 η S t k n d . ![]()

E13R → d = − ξ ˜ ( V → d - U → ) τ d ; E14ξ ˜ = 1 + 0 , 197 Re d 0 , 63 + 2 , 6 ⋅ 10 − 4 Re d 1 , 38 (0,1 ≤ Re d ≤ 3 ⋅ 10 5 ) [ 5 ] , ![]()

E15τ d = ρ f δ d 2 18 μ , Re d = V c δ d ρ μ , V c = | V → d − U → | ; ![]()

E16η S t k = ( S t k S t k + 0 , 5 ) 2 + 2 , 5 δ δ d , ![]()

E17Stk = τ δ V c δ d , τ δ = ρ f δ δ 2 18 μ ( ρ f δ ≈ ρ f ) , ![]()

E18Nu i d = β i d δ d D i = 2 ( 1 + 0 , 276 Re d 0 , 5 S с i 0 , 33 ) K c i , S с i = μ ρ D i ; E19K с i = 1 + P i d + P i 2 B ; ![]()

E20B = P g + ∑ P i ; ![]()

E21ρ = ρ g + ∑ ρ i ; ![]()

E22ρ g = M g P g R T , ρ i = M i P i R T , ρ i d = M i P i d R T d , ρ i δ = M i P i δ R T δ ; ![]()

E23D i = D i 0 B 0 B ( T T 0 ) 1 , 75 , B 0 = 0.1 M P a , T 0 = 273 K ; ![]()

E24Nu d = α d δ d λ = 2 + 0 , 459 Re d 0 , 5 Pr 0 , 3 , Pr = μ c λ ; ![]()

E25n d = ρ d m d , ![]()

E26n δ = ρ δ m δ ; ![]()

E27α δ = 2 λ δ ( Nu δ = 2 ) , β i δ = 2 D i δ ( N u ′ i δ = 2 ) , ![]()

E28c = ∑ ρ i c i ρ ; ![]()

E29r i = M i − 1 R T 2 d ln m p x , i d T , ![]()

E30P i d = m p x , i x i d , P i δ = m p x , i x i δ ; ![]()

E31x i d , δ = c m i d , δ M i c m i d , δ M i + 1 M d i s , ![]()

E32d c m i d , δ d τ = d m i d , δ d τ 6 π δ d , f 3 ρ f . ![]()

E33d δ f 3 d τ = 6 π 1 ρ f d m v δ d τ , δ d = 6 m d π ρ f 3 ; ![]()

E34δ = 6 m δ π ρ s + δ 0 3 3 , ( ρ s ≈ ρ f ) , ![]()

E35

In the current work we suggest the model for mathematical description of the above process with the following assumptions:

droplets and particles are considered monodispersed with equivalent sizes, equal to mass-median by distributions;

concentrations of droplets of irrigating liquid, dust particles and harmful gas components are low, what allows us to use the Henry’s law for equilibrium of gas components in liquid and gas phases at the interface and assume that the solution in droplet is ideal;

the mean-mass temperature of droplets and temperature of their surfaces are equal because of their small sizes [3];

the typical time of gas component dissolution in droplet is significantly less than the typical time of mass transfer processes, commonly occurring in the apparatus;

the motion velocities of particles with condensate on their surface (“formations”) and vapor-gas flow are equal;

the moisture content in the flow can be high, what requires consideration of the Stefan correction in mass transfer equations for evaporation-condensation process on droplets and “formations”;

we do not take into account the evaporation-condensation correction for the resistance and heat transfer coefficients of droplets and “formations”, it is insignificant and becomes obvious only at the initial stages of the process at high moisture contents [4];

in equation of droplet motion we take into account variability of its mass;

the radiant component in the process of heat transfer is neglected because of low temperatures of droplet, “formations” and flow;

mutual coalescence of droplets and “formations” is not taken into account, and merging of droplets and “formations” due to collision is the basis of condensation-inertial mechanism of dust capture in jet scrubbers [2].

Under the above conditions equations of model system will take the following form:

Motion equation of a mass-median droplet with variable mass

equation of heat transfer between droplet and vapor-gas flow

equation of mass transfer between droplet and the i-^{th} component of vapor-gas flow

equation of mass transfer between “formation” and the i-^{th} component of vapor-gas flow

continuity equation for i-^{th} reacting components, including vapor of liquid

continuity equation for (mass concentration) of non-reacting component of the vapor-gas mixture

continuity equation for (mass concentration) of “formations”

continuity equation for (mass concentration) of droplets

equation of heat transfer between “formation” and vapor-gas flow

equation of convective heat transfer between vapor-gas flow and droplets and “formations”

general rate of droplet mass change due to evaporation-condensation and absorption of removed gas components (droplet collision is assumed unlikely) and “formation” absorption

general rate of “formation” mass change (“formation” collision is assumed unlikely )

continuity equation for (mass concentration) of dry particles

The following closure relationships shall be added to equations (1-13):

for the force of droplet aerodynamic resistance per a unit of droplet mass,

where relative coefficient of droplet resistance is

coefficient of “formation” entrainment according to the empirical formula of Langmuir–Blodgett with Fuchs correction on engagement effect [1]

where is efficient density of “formation’;

mass transfer coefficient of the i-^{th} component with droplets both via evaporation-condensation and absorption-desorption [2]

Stefan correction on increased moisture content

barometric (total) pressure

density of vapor-gas mixture

state equation for gas components and vapor of liquid

diffusion coefficient of the i-^{th} component in non-reacting component of the vapor-gas flow (we assume that its fraction in the flow is predominant)

coefficient of droplet heat transfer according to Drake’s formula

countable concentrations of droplets and “formations”

heat and mass transfer coefficients of “formations”

heat capacity of the vapor-gas mixture

specific heat of gas absorption with the made assumptions [6]

it can be assumed for water vapors that ≈2500 kJ/kg [2-4];

according to Henry’s law for partial saturation pressure at the interface between i-^{th} gas components, the equilibrium condition is [6]

where is a molar part, equal to the number of moles of dissolved gas per the total number of moles in solution,

The equation for mass concentration of dissolved i-^{th} gas component in the kilogram per 1 kg of dissolvent in the droplet and “formation” is written as

Diameters of specific spherical volume of dissolvent for “formation” δ_{f} and droplet δ_{d} are calculated by equations:

“formation” diameter is

where ρ_{s} is solution density, kg/m^{3}.

The reactive force in equation (1) is neglected because of evaporation-condensation and absorption [2]. In equation (2) specific heat capacity is taken constant and equal to specific heat capacity of dissolvent because of low concentrations of absorbed dust and absorbed gases. For small particles and significant amount of condensate on them [2] we will take

As it is shown in [2], in most technically implemented situations it is possible to use a single-dimensional model for calculation of heat and mass transfer in irrigation chambers, what is determined by the vertical position of apparatuses (hollow jet scrubbers HJC); at their horizontal position it is determined by high velocities of cleaned gases, dust particles and droplets (Venturi scrubber VS), when the gravity force, influencing the flow components and causing its 2D character, is low in comparison with the inertia forces.

The calculation scheme of the problem for the vertical construction of apparatus is shown in Fig. 1а). The scheme of interaction between a droplet of washing liquid dispersed by the jets with vapor-gas flow and dust particles is shown in Fig. 1b).

The hollow jet scrubber HJS can have direct-flow and counter-flow construction. In the direct-flow scheme the initial parameters of the vapor-gas flow, irrigating liquid and dust are set on one side (inlet) of apparatus, and the resulting parameters are achieved at the apparatus outlet. In the counter-flow scheme the parameters of vapor-gas flow and dust are set on one side of apparatus, the parameters of irrigating liquid are set on the opposite side (at apparatus outlet). Scheme 1а) is attributed to the counter-flow. From the point of numerical implementation the direct-flow scheme is the Cauchy problem, and the counter-flow scheme is the boundary problem. Let’s perform calculations for the direct-flow scheme according to the known experimental data for generalized volumetric mass transfer coefficients, shown in [6, p. 562], for different gases absorbed on dispersed water. The calculation scheme is shown in Fig. 2 (it is conditional, the construction can differ).

The problem will be solved in the stationary statement. The boundary conditions are set at * x*=0 (

Continuity equations (6) and (8) in stationary single-dimensional case can be reduced to the following, as in [2], analytical dependences:

where

* k*is the number of reacting components, including liquid vapors,

efficiency of dust capture and gas component removal is determined by relationships:

Calculation results on absorption of СО_{2} by water droplets in the direct-low hollow jet scrubber are shown in Fig. 3. There are no any restrictions for calculations by solubility limit (concentration of gas dissolving in the droplet). However, the solubility limits exists as the experimental fact for gases, presented in tables of Hand-books as absorption coefficient α, in volumetric fractions reduced to 0 °С and pressure of 0.1 MPa or in the form of solubility coefficient * q*in mass fractions to solution or dissolvent [8, 9]. Thus, in [9] for

, | 273 | 283 | 293 | 303 | 313 | 323 | 333 | 353 | 373 |

1.713 | 1.194 | 0.878 | 0.665 | 0.530 | 0.436 | 0.359 | … | … | |

^{3} | 55.6 | 41.8 | 33.6 | 27.6 | 23.7 | 21.3 | 19.5 | 17.7 | 17.0 |

It follows from this table that the limit value of СО_{2} concentration in a water droplet is

where _{2} (Table 1).

This value shall limit concentration of СО_{2} dissolved in the droplet. It can be seen in Fig. 3а) that according to Table 1 calculated value of ^{-3} kg of СО_{2}/kg of water. It can be seen in Fig. 3b) that as a result of water vapor condensation and СО_{2} absorption the size of droplet increases insignificantly, less than by 0.05 %, i.e., a small amount of water vapors condenses on the droplet and a small amount of СО_{2} is absorbed by the droplet, Fig. 3а). The calculated amount of the mass of gas component absorbed by liquid droplets in the scrubber is determined by formula, kg/h∙m^{2},

For calculated situation with consideration of partial density of dry air at the inlet ^{3}, gas content _{0}=0.25 m/s and

Let’s compare the obtained result with the value achieved via the empirical volumetric mass transfer coefficient, shown in [6, p. 562] (in our nomenclature):

Here * Q*is irrigation density, m/h,

Let’s write down the value of obtained coefficient per an area unit of apparatus cross-section via coefficient ^{2}:

where * х*=0,

Substituting (46) into (47) and assuming * l*=

where q is irrigation coefficient, m^{3} of water/ м^{3} of vapor-gas flow at apparatus inlet.

Let’s transform formula (48), and finally for calculation we obtain dependence

For the considered situation ^{2}/s [6].

Let’s take the average experimental data of [6] as the calculation working height of absorber =(

since

Calculated (45) and experimental (50) results differ by Δ≈13 %. If we take * Н*=4.3 m, then

It follows from formula (42) for calculated concentration difference at apparatus inlet and outlet that

where calculated value is =

Previous comparison can be made in the relative form, what will prove the validity of calculation of mass transfer coefficient as a measure determining process intensity, on the basis of model in comparison with its experimental expression [6]:

where it is assumed that * Н*=7 m and

Here _{2} concentrations at the inlet and outlet at the example of Fig. 4 is even negative: ^{3}, here =

Extractions per a total volume of apparatus can be presented as, kg/h,

On the other hand

Hence, with consideration of formulas (46) and (53) we will obtain the relationship for volumetric mass transfer coefficients (theoretical and experimental ones)

after elementary reductions in numerator and denominator this corresponds to formula (51). Here * D*is apparatus diameter.

Calculations results for the same situation as in Fig. 3 are shown in Fig. 4, but for the increased moisture content _{0}=0.5 kg/kg of dry air. The theoretical value of absorbed СО_{2} is:

Calculation by formula (48) for height * Н*=4.3 m gives the following

what differs from the theoretical value by 3 %. Here ^{3} by calculation (=7 m

According to comparison, the model agrees well with the experimental data.

In calculations tabular data for water solution of СО

Partial pressures of saturated water vapors on droplet and “formation” surfaces were calculated by formula [2] (the partial pressure of saturated vapors of gas components were not taken into account)

where

= 221.29 10

For hydrogen sulfide for water solution [6] was approximated by dependence:

Calculation results on absorption of hydrogen sulfide on a water droplet from the vapor-gas flow are shown in Fig. 5.

Theoretical value of _{2}S (^{3}) is

Calculation by formula (48) with experimental mass transfer coefficient gives for * Н*=4.3 m

where ^{3}. Difference between results of (57) and (58) is Δ≈8 %. For calculated height * Н*=7 m

If there are no tabular data for (or

Thus, in [8, p. 260-261] there are tabular data for SO_{2} for * l*and

, ºC | 0 | 10 | 20 | 30 | 40 |

79.8 | 56.7 | 39.4 | 27.2 | 18.8 | |

_{s} | 22.8 | 16.2 | 11.3 | 7.8 | 5.41 |

2.8571 | 2.8571 | 2.8680 | 2.8676 | 2.8777 |

According to this Table, ^{3} and it is almost constant value.

First, for this case we calculate * m*:

As a result, the following approximation was obtained by formula (61) for SO_{2}

Knowing _{2} absorption by water:

Following calculation is performed by the general scheme (formulas (31)–(35)).

Results of calculation are shown in Fig. 6 at ventilation of air humidity _{0}=0.02 kg/kg of dry air, _{2} concentration in a droplet is not achieved even for СН_{4}. According to tabular data on absorption coefficient α, m^{3}/m^{3} of water (see Table 1):

_{4}/ kg of water.

According to calculation of extracted SO_{2} for the given case (^{3}):

for * Н*=12 m,

at * Н*=7 m (

Comparison of _{2} proves good agreement between theory and experiment.

According to calculation, Fig. 6c), methane is not absorbed by water. However, even for methane comparison of calculation with experiment yields satisfactory agreement:

at * Н*=4.3 m, Δ=11.75 %. At

We should note that absorber height * Н*in experimental dependence for

For СН_{4} * m*is approximated by dependence

Calculations of combined condensation dust capture and absorption extraction of hydrogen sulfide from the vapor-air flow in direct-flow hollow scrubber are shown in Fig. 7. Calculated parameters are shown below the figure. According to Fig. 7а), even at increased moisture content the size of droplets increases weak due to condensation. Therefore, for similar processes the equation of droplet motion can be calculated with a constant mass.

An increase in the size of “formations” is more significant due to condensation of water vapors on them: for _{0}=0.01 μm it is 2.1, for _{0}=0.1 μm it is 2.3, and for _{0}=1 μ it is 32 and more for the same total concentration of dust at the inlet of 1.72 g/m^{3}. In the first case, particles are not caught, in the second case, about 5.76 % of particles are caught, and in the third case, 100 % of particles are caught at the inlet to the apparatus. For this version of calculation the stable state by concentrations of _{2}S dissolved in droplets and in condensate on “formations” occurs far from the flow escape from the scrubber. For particles water vapor condensation at flow escape from the scrubber has been also competed already (see Fig. 7g). Therefore, in this case the height of absorber above 1.5 m is excessive, and in construction it can be limited by 2 m. According to Figs. 7d) and 7e), concentration of _{2}S dissolved in condensate on the particle and in droplets increases, but it does not exceed the solubility limit (in this case it is about 3.85∙10^{-3} kg of _{2}S/kg of water).

Calculation results on Н_{2}S absorption and condensation capture of dust with different sizes in Venturi scrubber are shown in Fig. 8. As an example the Venturi scrubber with following parameters was chosen for calculations: diameter of Venturi tube mouth _{m}=0.02 m, diffuser length * l*=0.2 m, diffuser opening angle α=6° (α=6–7°,

The mean-mass size of droplets in the tube mouth was calculated by Nukiyama-Tanasava formula [1]:

where _{f} (kg/m^{3}), (Pa∙s),

Velocity * U*was calculated with consideration of diffuser expansion angle [2, 11].

Dependences of droplet size along the diffuser length are presented in Fig. 8а). It can be seen that firstly condensation of water vapors occurs intensively, then this process stops at the length of * х*/

A change in droplet temperature due to convective heat transfer between droplets and vapor-gas flow, thermal effects of water vapor condensation on droplets, and gas dissolution is shown in Fig. 8d). A change in mass concentration of Н_{2}S dissolved in a droplet is shown in Fig. 8c). It is obvious that absorption is almost completed at the length of tube diffuser 1 for this version of calculation. The same circumstance is illustrated by mass concentration of Н_{2}S in “formation” condensate along the diffuser in Fig. 8c). According to the figure, the solubility limit on “formations” and droplets is not achieved as in the hollow jet scrubbers. A change in “formation” size due to water vapor condensate on their surfaces is illustrated in Fig. 8f). It can be seen that firstly water vapors condense very intensively, then at the distance of about * x*/

It is necessary to note that these calculation versions do not meet the conditions of optimal scrubber operation; they only illustrate the character of complex gas cleaning. To determine the optimal regimes, a series of calculation on the basis of suggested model should be carried out and analyzed for the specific industrial conditions.

Let’s turn to comparison of calculation results with the known experimental data. The experimental volumetric mass transfer coefficient is shown in [6] for NH_{3} absorption in the Venturi tube with mouth diameter =0.02 m. There no geometrical and other parameters. This coefficient is presented as

where * q*is irrigation coefficient in l/m

The experimental value of Н_{2}S absorbed in Venturi tube is expressed by formula, kg/h,

where * V*is diffuser volume, Δ

The theoretical value of mass of absorbed gas, kg/h, is

Then

where (see Fig. 9) the volume of truncated cone is

Substituting (67), (71) into (70), at α=6°, * l*=0.2 m,

where for Н_{2}S

The amount of absorbed Н_{2}S for the version of calculation in Fig. 8 (^{3}) is

For the scrubber with =0.1 m,

The experimental values of efficiency of condensation capture of submicron dust are compared with results of model calculation inn [2, 11, 12] at the example of deposition of ash particles from cracking gases under the industrial conditions in hollow jet scrubbers [13]; good agreement is achieved.

Let’s consider this important question in detail as an addition to iss. 2 at the example of water absorption of SO_{2}, comparing calculation and experimental data [6] on volumetric mass transfer coefficient.

It follows from equation (3) that

where, according to formulas (26) and (38)

In (73) and (74), according to calculation results, it is assumed that

Let’s put

where

Lets’ turn dependence (74) to the following form using the theorem about an average for integral:

where ^{2}/s, _{3},

Expressing velocity ^{2}/s (see Fig. 10 b)), ^{2}/s, ^{2}, where ^{2} is obtained via model calculation (see Fig. 11).

We should note that multiplier

Numerical calculation by the model give the value of SO_{2} extraction

The difference is 17 %, what is a sequence of simplifications and averaging in dependence (76).

If we assume average concentration difference in accordance to average experimental height * Н*=7 m

The difference with ^{2} is 4.2 %.

If we take ^{3}, then ^{2}, what differs from result of (77) by similar 4.3 % with accuracy of estimation error. This proves the fact that calculated volumetric mass transfer coefficient agrees empirical dependence (46) of [6].

On the basis of analysis performed the calculated concentration difference should be recommended for practical application as the most appropriate

at determination of the value of extracted gas component by formula (49), thus, it is necessary to measure ^{3}.

The suggested physical-mathematical model of complex heat and mass transfer and condensation-absorption gas cleaning from dust and harmful gaseous components is confirmed by the known experimental data and can be used for engineering calculations and optimization of construction and operation parameters of hollow jet scrubbers of direct and counter flow types. This was proved by its numerical implementation for the specific conditions. Calculations on absorption of some gases (СО_{2}, Н_{2}S, SO_{2}, CH_{4}) on water droplets, dispersed by coarse centrifugal nozzles in hollow direct-flow jet scrubber and pneumatic Venturi scrubber from wet air is shown in the current paper together with calculation of combined absorption-condensation air cleaning from Н_{2}S and various-sized fine dust in these apparatuses at an increased moisture content. The system of model equations is written at some certain conditions for the multicomponent vapor-gas mixture with particle. This makes it possible to use this system for calculation of complex gas cleaning from several harmful gas components and several fractions of dust particles and investigate regularities of this process.

Advertisement## Nomenclature

^{2}∙К

^{3}

^{th} component of vapor-gas flow by concentration difference, m/s

^{3}

^{3}

^{3}

^{3}

^{2}∙К

^{3}

^{3}

^{3}

^{2}/s

^{th} component of the vapor-gas mixture, Pa

^{ths} components of extracted gases, Pa

^{3}

^{3}/s

^{3}/s

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