# Simultaneous Use of Mass Transfer and Thermodynamics Equations to Estimate the Amount of Removed Greenhouse Gas from the Environment by a Stream of Water

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## Abstract

The main objective of this research is to formulate a novel mathematical model. For modeling the emission of pollutants, the physical principles are captured and simple hypotheses are postulated by the mathematical expressions derived from the stability equations of the mass. The results show the rate of greenhouse gas emissions increases by about 26% when the water flow rate reaches from 1.1 to 1.92 m/s. Also, the obtained data from the model show a very good agreement with the experimental data. Also, the results of this study show the amount of greenhouse gas emissions increases by approximately 2.46% with increasing ambient temperature from 21.4 to 9.43 °C. Finally, the results of this study show it is possible to estimate the degree of gas solubility in water flow with equations of mass transfer and thermodynamics.

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## Abbreviations

A :

bed cross section, m2

A c :

particle surface area in the continuous phase, m2

A b :

particle surface area in the dispersed phase, m2

C c :

gas concentration in continuous phase, $$\raisebox{1ex}{\mathrm{kg}}\!\left/ \!\raisebox{-1ex}{{\mathrm{m}}^3}\right.$$

C d :

gas concentration in dispersed phase, $$\raisebox{1ex}{\mathrm{kg}}\!\left/ \!\raisebox{-1ex}{{\mathrm{m}}^3}\right.$$

C in :

inlet concentration of SO2 into the bed, $$\raisebox{1ex}{\mathrm{kg}}\!\left/ \!\raisebox{-1ex}{{\mathrm{m}}^3}\right.$$

C g :

gas outlet concentration, $$\raisebox{1ex}{\mathrm{kg}}\!\left/ \!\raisebox{-1ex}{{\mathrm{m}}^3}\right.$$

C 0 :

initial concentration of SO2 $$\raisebox{1ex}{\mathrm{kg}}\!\left/ \!\raisebox{-1ex}{{\mathrm{m}}^3}\right.$$

D AB :

SO2 diffusion coefficient in gas, $$\raisebox{1ex}{{\mathrm{m}}^2}\!\left/ \!\raisebox{-1ex}{\sec }\right.$$

D eff :

gas effective diffusion coefficient in continuous phase, $$\raisebox{1ex}{{\mathrm{m}}^2}\!\left/ \!\raisebox{-1ex}{\sec }\right.$$

L :

bed length, m

m acc :

accumulated mass of SO2 in the control volume, $$\raisebox{1ex}{\mathrm{kg}}\!\left/ \!\raisebox{-1ex}{\sec }\right.$$

m bdis. :

amount of mass flow rate of dispersed phase transferred by velocity, $$\raisebox{1ex}{\mathrm{kg}}\!\left/ \!\raisebox{-1ex}{\sec }\right.$$

m mdis. :

dispersed phase molecular amount of mass flow rate, $$\raisebox{1ex}{\mathrm{kg}}\!\left/ \!\raisebox{-1ex}{\sec }\right.$$

m bcon. :

amount of mass flow rate of continuous phase transferred by velocity, $$\raisebox{1ex}{\mathrm{kg}}\!\left/ \!\raisebox{-1ex}{\sec }\right.$$

m bcon. :

continuous phase molecular amount of mass flow rate, $$\raisebox{1ex}{\mathrm{kg}}\!\left/ \!\raisebox{-1ex}{\sec }\right.$$

m r :

mass flow disappeared by reaction, $$\raisebox{1ex}{\mathrm{kg}}\!\left/ \!\raisebox{-1ex}{\sec }\right.$$

n :

number of orifices in the gas distributor

N :

number of bubbles

Q :

gas volumetric flow rate exchanged between phases, $$\raisebox{1ex}{{\mathrm{m}}^3}\!\left/ \!\raisebox{-1ex}{\sec }\right.$$

z :

vertical direction, m

D b :

bubble diameter, m

D b0 :

bubble diameter at the bed entrance, m

D p :

particle diameter, m

E a :

activation energy, $$\raisebox{1ex}{\mathrm{kJ}}\!\left/ \!\raisebox{-1ex}{\mathrm{mol}}\right.$$

F :

dispersed phase volume fraction

g :

Gravitational acceleration, $$\raisebox{1ex}{\mathrm{m}}\!\left/ \!\raisebox{-1ex}{{\sec}^2}\right.$$

k :

overall reaction constant, $$\raisebox{1ex}{\mathrm{m}}\!\left/ \!\raisebox{-1ex}{\sec }\right.$$

k G :

gas mass transfer coefficient from dispersed phase to continuous phase, $$\raisebox{1ex}{\mathrm{m}}\!\left/ \!\raisebox{-1ex}{\sec }\right.$$

k s :

reaction constant, $$\raisebox{1ex}{\mathrm{m}}\!\left/ \!\raisebox{-1ex}{\sec }\right.$$

k s0 :

frequency factor, $$\raisebox{1ex}{\mathrm{m}}\!\left/ \!\raisebox{-1ex}{\sec }\right.$$

r :

solubility rate, $$\raisebox{1ex}{{\mathrm{k}}^{\mathrm{g}}}\!\left/ \!\raisebox{-1ex}{\sec }\right.$$

r u :

diameter of unreacted core, m

r p :

R :

universal gas constant, $$\raisebox{1ex}{\mathrm{kJ}}\!\left/ \!\raisebox{-1ex}{\mathrm{kg}\ \mathrm{K}}\right.$$

t :

time, sec

T :

absolute temperature, K

U :

actual gas velocity, $$\raisebox{1ex}{\mathrm{m}}\!\left/ \!\raisebox{-1ex}{\sec }\right.$$

U mf :

gas minimum fluidization velocity, $$\raisebox{1ex}{\mathrm{m}}\!\left/ \!\raisebox{-1ex}{\sec }\right.$$

V c :

continuous volume fraction in the control volume

V d :

dispersed volume fraction in the control volume

ρ g :

gas density, $$\raisebox{1ex}{\mathrm{kg}}\!\left/ \!\raisebox{-1ex}{{\mathrm{m}}^3}\right.$$

ρ p :

particle density, $$\raisebox{1ex}{\mathrm{kg}}\!\left/ \!\raisebox{-1ex}{{\mathrm{m}}^3}\right.$$

μ :

gas viscosity,$$\raisebox{1ex}{\mathrm{kg}}\!\left/ \!\raisebox{-1ex}{\mathrm{m}\ \mathrm{s}}\right.$$

f :

fugasity in pure state of component

φ̂ :

fugasity coefficient of component

γ :

activity coefficient component

P Sat :

vapor pressure in pure state of component

Γ:

thermodynamic function

c :

concentration of species (i) in phase

K Ga:

overall mass transfer coefficient based on gas phase

Ð :

Stefan-Maxwell diffusivity

P :

pressure

v :

specific molar volume

x :

mole fraction in liquid phase

g E :

excess Gibbs energy

y :

mole fraction in gas phase

N :

bulk motion

J :

diffusion motion

k :

mass transfer coefficient

a :

constant

b :

constant

H :

Henry′s constant

V :

velocity

D :

Fick′s diffusivity

ρ :

density

μ :

viscosity

D :

diffusion factor

l :

length

Sh:

Sherwood number

r :

R :

gas constant

T :

temperature

cosh:

hyperbolic cosine

z :

distance

m :

mass

δ :

thickness

1, 2:

number of component

i and j :

Component

sat:

saturation

liq:

liquid

Vap:

vapor

E:

excess

eq:

equilibrium

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Authors

## Ethics declarations

All the research and presented data in this manuscript have been done by me from beginning to end. In addition, NO other author or body has contributed to the writing or research related to this article. Also, NO funds have been received from any organ or department for this research. I also state that there is NO conflict of interest. I further confirm that according to the subject of the research, there is no need for ethical approvals or tests related to human diseases.

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Farahbod, F. Simultaneous Use of Mass Transfer and Thermodynamics Equations to Estimate the Amount of Removed Greenhouse Gas from the Environment by a Stream of Water. Environ Model Assess 26, 779–785 (2021). https://doi.org/10.1007/s10666-020-09740-8