# Basic Models of Computational Mass Transfer

• Kuo-Tsong Yu
• Xigang Yuan
Chapter
Part of the Heat and Mass Transfer book series (HMT)

## Abstract

The computational mass transfer (CMT) aims to find the concentration profile in process equipment, which is the most important basis for evaluating the process efficiency as well as the effectiveness of an existing mass transfer equipment. This chapter is dedicated to the description of the fundamentals and the recently published models of CMT for obtaining simultaneously the concentration, velocity and temperature distributions. The challenge is the closure of the differential species conservation equation for the mass transfer in a turbulent flow. Two models are presented. The first is a two-equation model termed as $$\overline{{c^{{{\prime }2}} }} - \varepsilon_{{{\text{c}}^{{\prime }} }}$$ model, which is based on the Boussinesq postulate by introducing an isotropic turbulent mass transfer diffusivity. The other is the Reynolds mass flux model, in which the variable covariant term in the equation is modeled and computed directly, and so it is anisotropic and rigorous. Both methods are validated by comparing with experimental data.

## Keywords

Computational mass transfer Reynolds averaging Closure of time-averaged mass transfer equation Two-equation model Turbulent mass transfer diffusivity Reynolds mass flux model

## Nomenclature

[B]

Matrix of inverted Maxwell–Stefan Diffusivities, m−2 s

c

Instantaneous mass concentration of species i, kg m−3; Molar concentration of species i in Sect. 3.4.2, mol s−3

$$c_{\text{t}}$$

Total molar concentration of component i per m3, mol m−3

C

Time-average concentration, bulk concentration, kg m−3 in Table 3.1 mass fraction

C+

Dimensionless concentration

c

Fluctuating concentration, kg m−3

$$\overline{{c^{{{\prime }2}} }}$$

Variance of fluctuating concentration, kg2 m−6

D

Molecular diffusivity, m2 s−1

De

Effective mass diffusivity, m2 s−1

Dt

Isotropic turbulent mass diffusivity, m2 s−1

D

Maxwell-Stefan diffusivity, m2 s−1

$${\mathbf{D}}_{\text{t}}$$

Anisotropic turbulent mass diffusivity, m2 s−1

g

Gravity acceleration, m s−2

[I]

Identity matrix, dimensionless

Jw

Mass flux at wall surface, kg m−2 s−1

k

Fluctuating kinetic energy, m2 s−2; mass transfer coefficient, m s−1

[k]

Matrix of mass transfer coefficients, m s−1

l

Characteristic length, m

[Ni]

Molar mass flux of diffusing species i, mol−2 s−1

[Nt]

Molar mass flux of multicomponent solution, mol−2 s−1

p

Fluctuating pressure, kg m−1 s−2

P

Time-average pressure, kg m−1 s−2

Pe

Peclet number

[R]

Matrix of inverted mass transfer coefficients, m−1 s

rc

Ratio of fluctuating velocity dissipation time and fluctuating concentration dissipation time

S

Source term

Sc

Schmidt number

Sct

Turbulent Schmidt number

t

Time, s

T

Fluctuating temperature, K

$$\overline{{T^{{{\prime }2}} }}$$

Variance of fluctuating temperature, K2

T

Time-average temperature, K

u

Instantaneous velocity of species i, m s−1

u

Fluctuating velocity, m s−1

uτ

Frictional velocity, m s−1

u+

Dimensionless velocity, m s−1

U, V, W

Time-average velocity in three directions, m s−1

[X]

Matrix of correction factor

y+

Dimensionless distance, m

αt

Turbulent thermal diffusivity, m−1 s−1

[β]

Matrix of molar exchange of mass transfer in counter-diffusion due to the difference of latent hear of vaporization between component i and j, dimensionless

δ

Kronecker sign; thickness of fluid film, m

ε

Dissipation rate of turbulent kinetic energy, m2 s−3

εc′

Dissipation rate of concentration variance, kg2 m−6 s−1

εt

Dissipation rate of temperature variance, K2 s−1

μ

Viscosity, kg m−1 s−1

μt

Turbulent viscosity, kg m−1 s−1

νe

Effective turbulent diffusivity, m2 s−1

ρ

Density, kg m−3

[Γ]

Matrix of non-ideality factor (in terms of activity coefficient γ), dimensionless

τμ, τc, τm

Characteristic time scale, s

τw

Near-wall stress, kg m−1 s−2

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