Absorption coefficients for СО2 and СН4
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) . 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 . 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.
2. Problem statement, main equations and assumptions
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 ;
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 ;
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 .
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 
mass transfer coefficient of the i-th component with droplets both via evaporation-condensation and absorption-desorption 
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 
according to Henry’s law for partial saturation pressure at the interface between i-th gas components, the equilibrium condition is 
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” δf and droplet δd are calculated by equations:
“formation” diameter is
where ρs is solution density, kg/m3.
The reactive force in equation (1) is neglected because of evaporation-condensation and absorption . In equation (2) specific heat capacity
3. Numerical implementation of the model, comparison of calculation results with experimental data
As it is shown in , 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
efficiency of dust capture and gas component removal is determined by relationships:
Calculation results on absorption of СО2 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
It follows from this table that the limit value of СО2 concentration in a water droplet is
where is the table value of absorption coefficient for СО2 (Table 1).
This value shall limit concentration of СО2 dissolved in the droplet. It can be seen in Fig. 3а) that according to Table 1 calculated value of does not reach the solubility limit and it is one order lower. Thus, (, see Fig. 3c)) =2.81∙10-3 kg of СО2/kg of water. It can be seen in Fig. 3b) that as a result of water vapor condensation and СО2 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 СО2 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∙m2,
For calculated situation with consideration of partial density of dry air at the inlet =1.0363 kg/m3, gas content =0.2 kg/kg of dry air, U0=0.25 m/s and =0.022262 according to formula (44) we will obtain
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):
Let’s write down the value of obtained coefficient per an area unit of apparatus cross-section via coefficient in the following form, kg/h∙m2:
where is calculated concentration pressure on the way of gas component obtaining
Substituting (46) into (47) and assuming
where q is irrigation coefficient, m3 of water/ м3 of vapor-gas flow at apparatus inlet.
Let’s transform formula (48), and finally for calculation we obtain dependence
For the considered situation m2/s .
Let’s take the average experimental data of  as the calculation working height of absorber
since and total multiplier in (50) is reduced.
Calculated (45) and experimental (50) results differ by Δ≈13 %. If we take
It follows from formula (42) for calculated concentration difference at apparatus inlet and outlet that
where calculated value is . Thus, for
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 :
where it is assumed that . Thus, for our case at
Here has the meaning of efficient drop of gas concentration, not real, but corresponding to extraction of the gas component due to absorption on a liquid droplet. Real drop of СО2 concentrations at the inlet and outlet at the example of Fig. 4 is even negative: kg/m3, 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
Calculation by formula (48) for height
what differs from the theoretical value by 3 %. Here kg/m3 by calculation (). We should note that even for the increased moisture contents the size of droplets increases slightly due to condensation and absorption (less than by 1 %) (Fig. 4b). For calculated scrubber height
According to comparison, the model agrees well with the experimental data.
In calculations tabular data
Partial pressures of saturated water vapors on droplet and “formation” surfaces were calculated by formula  (the partial pressure of saturated vapors of gas components were not taken into account)
For hydrogen sulfide
Calculation results on absorption of hydrogen sulfide on a water droplet from the vapor-gas flow are shown in Fig. 5.
Theoretical value of for Н2S (kg/m3) is
Calculation by formula (48) with experimental mass transfer coefficient gives for
where , kg/m3. Difference between results of (57) and (58) is Δ≈8 %. For calculated height
3.2. Calculation of absorption by solubility of l and qs
If there are no tabular data for
Thus, in [8, p. 260-261] there are tabular data for SO2 for
According to this Table, =2.8655≈2.9 kg/m3 and it is almost constant value.
First, for this case we calculate
As a result, the following approximation was obtained by formula (61) for SO2
Knowing , we determine specific heat of SO2 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
, kg of СН4/ kg of water.
According to calculation of extracted SO2 for the given case (kg/m3):
Comparison of and for SO2 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:
We should note that absorber height
3.3. Calculation of combined absorption-condensation dust-gas cleaning
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
3.4. Calculation of absorption and condensation dust capture in Venturi scrubber
Calculation results on Н2S 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
The mean-mass size of droplets in the tube mouth was calculated by Nukiyama-Tanasava formula :
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
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 Н2S 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 Н2S 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
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  for NH3 absorption in the Venturi tube with mouth diameter
The experimental value of Н2S absorbed in Venturi tube is expressed by formula, kg/h,
The theoretical value of mass of absorbed gas, kg/h, is
where (see Fig. 9) the volume of truncated cone is
Substituting (67), (71) into (70), at α=6°,
where for Н2S =0.7426, i.e., the difference between the theory and experiment is less than 6 %, what is a good agreement, considering the assumed parameters for normalized Venturi tube.
The amount of absorbed Н2S for the version of calculation in Fig. 8 (=0.072959, =0.7541 kg/m3) is
For the scrubber with
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 ; good agreement is achieved.
4. The choice of the value of calculated concentration difference for the absorbed gas component
Let’s consider this important question in detail as an addition to iss. 2 at the example of water absorption of SO2, comparing calculation and experimental data  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 from (19) to (73) at and from (74), we obtain, proved by estimates,
Lets’ turn dependence (74) to the following form using the theorem about an average for integral:
where is the average value of kinematic viscosity of the vapor-gas flow in the scrubber volume, m2/s, is the average value of diffusion coefficient of methane NH3, is average velocity of droplets on the 0–Н way at motion from scrubber inlet to the outlet, m/s, .
Expressing velocity and in m/h, and assuming m (700 μm), m2/s (see Fig. 10 b)), m2/s, m/s (see Fig. 10 e)), , we obtain for calculation parameters of Fig. 10 kg/h∙m2, where kg/m2 is obtained via model calculation (see Fig. 11).
We should note that multiplier was included into formula (76) as a correction like to was dome for empirical dependence (46).
Numerical calculation by the model give the value of SO2 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
The difference with =83 kg/h∙m2 is 4.2 %.
If we take kg/m3, then =104.5 kg/h∙m2, 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 .
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 and at apparatus inlet and outlet. Calculation of moisture content is shown in Fig. 10 c), at this, it was obtained that m/s, kg/m3.
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 (СО2, Н2S, SO2, CH4) 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 Н2S 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.
vector of droplet velocity
nvector of vapor-gas low velocity
vector of gravity acceleration
droplet mass (variable value due to evaporation-condensation and absorption), kg
specific heat capacity of liquid, J/ kg∙К
mean mass temperature of droplets, К
heat transfer coefficient of droplet, W/m2∙К
mass-median size of droplet, m
temperature of vapor-gas flow, К
specific heat of absorption, evaporation-condensation, J/kg
specific heat capacity of “formation”, J/kg∙К
mass concentration of “formations’ in the vapor-gas flow, kg/m3
module of relative droplet velocity, m/s
coefficient of “formation” capture by droplets
mean mass temperature of “formations”, К
coefficient of droplet mass transfer with the i-th component of vapor-gas flow by concentration difference, m/s
and partial densities (mass concentrations) of saturated vapors of dissolvent and gas components near droplet surface and far from it (in the flow), kg/m3
and calculated concentrations of droplets and “formations” in the flow, 1/m3
partial density of non-reacting gas component, kg/m3
mass concentration of droplets, kg/m3
heat transfer coefficient of “formation”, W/m2∙К
size of “formation”, m
density of vapor-gas flow, kg/m3
specific heat capacity of vapor-gas flow, J/ kg∙К
mass concentration of dry dust particles in the flow, kg/m3
density of liquid (droplets), kg/m3
dynamic viscosity of vapor-gas flow, Pa∙s
molar masses of components of the vapor-gas mixture, kg/kmole
kJ/kmole∙К universal gas constant
diffusion coefficient of mixture component, m2/s
partial pressure of the i-th component of the vapor-gas mixture, Pa
partial saturation pressures of mixture components, calculated by droplet and “formation” temperature, Pa
constants of phase equilibrium of solutions of i-ths components of extracted gases, Pa
mole fractions of gas components dissolved in a droplet and “formation” condensate
mass fractions of gas components in droplet and “formation” dissolvent, kg/kg of dissolvent
molar mass of dissolvent, kg/kmole
rate of “formation” mass change due to evaporation-condensation of liquid, kg/s
initial size of dust particles, m
density of solution on “formation” due to condensation of liquid vapors and absorp-tion of gas components, kg/m3
moisture content, kg of vapors/kg of dry non-reacting component of vapor-gas mixture
gas content, kg of reacting gas component/ kg of dry non-reacting component of vapor-gas mixture
volumetric flow rate of liquid, m3/s
volumetric flow rate of vapor-gas mixture at apparatus inlet, m3/s