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.
Similar content being viewed by others
Abbreviations
 A :

bed cross section, m^{2}
 A _{c} :

particle surface area in the continuous phase, m^{2}
 A _{b} :

particle surface area in the dispersed phase, m^{2}
 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 SO_{2} 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 SO_{2} \( \raisebox{1ex}{$\mathrm{kg}$}\!\left/ \!\raisebox{1ex}{${\mathrm{m}}^3$}\right. \)
 D _{AB} :

SO_{2} 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 SO_{2} 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} :

radius of particle, m
 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 _{G}a:

overall mass transfer coefficient based on gas phase
 Ð :

StefanMaxwell 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 :

radius
 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
References
Thomas, J. A., & Veera, G. G. (2018). A microbial desalination process with microalgae biocathode using sodium bicarbonate as an inorganic carbon source. International Biodeterioration & Biodegradation, 130, 91–97.
Fan, H., Su, Z., Wang, P., & Lee, K. Y. (2020). A dynamic mathematical model for oncethrough boilerturbine units with superheated steam temperature. Applied Thermal Engineering, 170, 114912.
Farahbod, F., Mowla, D., Jafari Nasr, M. R., & Soltanieh, M. (2012). Experimental study of forced circulation evaporator in zero discharge desalination process. Desalination, 285, 352–358.
Zou, S., & He, Z. (2018). Efficiently “pumping out” valueadded resources from wastewater by bioelectrochemical systems: a review from energy perspectives. Water Research, 131, 62–73.
Concas, A., Pisu, M., & Cao, G. (2020). Mechanochemical immobilization of heavy metals in contaminated soils: A novel mathematical modeling of experimental outcomes. Journal of Hazardous Materials, 38815, 121731.
Farahbod, F., & Omidvar, M. (2018). Experimental evaluation of collection, thermal, and conductivity efficiency of a solar distiller pond as a free concentration unit in wastewater treatment process. Energy Science & Engineering, 6, 584–594.
Taherizadeh, M., Farahbod, F., & Ilkhani, A. (2019). Experimental evaluation of solar still efficiencies as a basic step in treatment of wastewater. Heat Transfer  Asian Research, 49(Part 1), 236–248.
Ebrahimi, A., Najafpour, G. D., & Yousefi Kebria, D. (2018). Performance of microbial desalination cell for salt removal and energy generation using different catholyte solutions. Desalination, 432, 1–9.
Kokabian, B., Ghimire, U., & Gnaneswar, V. G. (2018). Water deionization with renewable energy production in microalgaemicrobial desalination process. Renewable Energy, 122, 354–361.
Afzali, S., Ghamartale, A., Rezaei, N., & Zendehboudi, S. (2020). Mathematical modeling and simulation of wateralternatinggas (WAG) process by incorporating capillary pressure and hysteresis effects. Fuel, 2631, 116362.
Weber, R., & Mancini, M. (2020). On scaling and mathematical modelling of large scale industrial flames. Journal of the Energy Institute, 93, 43–51.
Jinshah, B. S., & Balasubramanian, K. R. (2020). Thermomathematical model for parabolic trough collector using a complete radiation heat transfer modela new approach. Solar Energy, 197, 58–72.
Aragones, D. G., SanchezRamos, D., & Calvo, G. F. (2020). SURFWET: a biokinetic model for surface flow constructed wetlands. Science of the Total Environment, 72325, 137650.
Farahbod, F. (2019). Investigation of heat transfer equations for evaluation of drinkable water production rate as an efficiency of closed solar desalination pond. International Journal of Ambient Energy, 1, 1–6.
Author information
Authors and Affiliations
Corresponding author
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.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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/s10666020097408
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10666020097408