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title: Concentrations, velocities and fluxes
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# Concentrations, velocities and fluxes
Unit 5, Lecture 42

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## Concentrations, Velocities and Fluxes

- Course: Heat and Mass Transfer (DI05019071)
- Unit 5: Mass Transfer
- Lecture 42: Concentrations, Velocities and Fluxes

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Welcome back. Building upon our introduction to mass transfer, today we will formalize the mathematical definitions of concentration, velocity, and flux for chemical mixtures.
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## Lecture Agenda

- 1. Mixture Composition
- 2. Mass Concentration (Density)
- 3. Molar Concentration
- 4. Mass and Mole Fractions
- 5. Mixture Velocities
- 6. Mass and Molar Fluxes

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We will start by looking at mixture composition in terms of mass and moles, then move to how we define the velocity of a mixture containing differently moving components, and finally define fluxes.
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## Mixture Composition: An Overview

- To describe mass transfer mathematically, we must quantify the composition of a mixture.
- A mixture consists of two or more chemical species (e.g., Species A, Species B, Species C, ...).
- The concentration of a species can be expressed in two primary ways:
- 1. On a mass basis (Mass Concentration)
- 2. On a molar basis (Molar Concentration)

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Whenever we analyze a mass transfer problem, our first step is to establish the concentration of the species in the mixture. We can do this either by measuring mass or counting moles.
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## Mass Concentration (Partial Density)

- Mass concentration (ρ_A) of species A is the mass of species A per unit volume of the mixture.
- ρ_A = m_A / V (Unit: kg/m^3)
- The total mass density (ρ) of the mixture is the sum of the mass concentrations of all species:
- ρ = ρ_A + ρ_B + ... + ρ_n = Σ(ρ_i)

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Mass concentration is essentially the partial density of a species. If you sum up the partial densities of all species in the mixture, you get the total density of the mixture.
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## Molar Concentration

- Molar concentration (C_A) of species A is the number of moles of species A per unit volume.
- C_A = n_A / V (Unit: kmol/m^3 or mol/L)
- The total molar concentration (C) of the mixture is the sum of the molar concentrations of all species:
- C = C_A + C_B + ... + C_n = Σ(C_i)
- Relation between mass and molar concentration: ρ_A = C_A * M_A (where M_A is molecular weight).

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Molar concentration is highly useful in chemical reactions and ideal gas laws. Remember that mass concentration and molar concentration are related simply by the molecular weight of the species.
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## Mass and Mole Fractions

- Mass Fraction (m_A): The ratio of the mass concentration of species A to the total density.
- m_A = ρ_A / ρ (Note: Σ(m_i) = 1)
- Mole Fraction (x_A or y_A): The ratio of the molar concentration of species A to the total molar concentration.
- x_A = C_A / C (Note: Σ(x_i) = 1)
- Note: x_A is typically used for liquids/solids, y_A for gases.

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Fractions are dimensionless quantities that describe the relative makeup of the mixture. Whether you use mass fractions or mole fractions, their sum across all species in the mixture is always exactly 1.
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## Mixture Velocity

- In a moving mixture, each species may have its own absolute velocity (v_i).
- The overall velocity of the mixture can be defined in two ways:
- 1. Mass-average velocity (v)
- 2. Molar-average velocity (V)
- This is important because mass transfer involves movement relative to the bulk fluid motion.

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If water is flowing in a pipe and salt is diffusing through the water, the salt molecules and water molecules have different absolute velocities. We need a way to define the 'average' velocity of the mixture itself.
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## Mass and Molar Average Velocities

- **Mass-Average Velocity (v)**:
- v = Σ(ρ_i * v_i) / ρ
- It is the velocity based on the momentum of the mixture.
- **Molar-Average Velocity (V)**:
- V = Σ(C_i * v_i) / C
- It is the velocity based on the molar flow of the mixture.

<!--
The mass-average velocity is what we usually mean by fluid velocity in fluid mechanics. However, in chemical engineering, molar-average velocity is often more convenient.
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## Concept of Diffusion Velocity

- The velocity of a species relative to the bulk mixture velocity is its **diffusion velocity**.
- Mass diffusion velocity of species A = (v_A - v)
- Molar diffusion velocity of species A = (v_A - V)
- Diffusion velocity tells us how fast a species is moving purely due to concentration gradients, independent of the bulk flow.

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If you are walking on a moving walkway, the walkway's speed is the bulk velocity, your walking speed relative to the walkway is your diffusion velocity, and your speed relative to the ground is your absolute velocity.
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## Mass and Molar Fluxes

- **Flux** is the amount of species transferred per unit area per unit time.
- **Absolute Mass Flux (n_A)** = ρ_A * v_A (kg/m^2·s)
- **Mass Diffusion Flux (j_A)** = ρ_A * (v_A - v)
- **Absolute Molar Flux (N_A)** = C_A * v_A (kmol/m^2·s)
- **Molar Diffusion Flux (J_A)** = C_A * (v_A - V)
- Relationship: n_A = j_A + ρ_A * v

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Absolute flux is the total transport of a species. Diffusion flux is the transport strictly due to diffusion. Total flux is the sum of diffusion flux and the flux due to bulk fluid motion.
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## Summary

- Concentration can be expressed by mass (ρ_A) or by moles (C_A).
- Mass fractions (m_A) and mole fractions (x_A) sum to 1.
- A mixture has a mass-average velocity and a molar-average velocity.
- Fluxes can be defined based on absolute velocity or diffusion velocity.
- Total flux = Diffusion flux + Convective (Bulk) flux.

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To summarize, we've established the nomenclature and mathematical definitions needed for mass transfer analysis. These definitions are crucial for formulating Fick's law, which we will do next time.
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## Next Lecture Preview

- Topic: Fick's Law of Diffusion
- Key questions to ponder:
- - How is diffusion flux mathematically related to the concentration gradient?
- - What is the physical significance of the diffusion coefficient (diffusivity)?
- - How does Fick's Law compare to Fourier's Law of heat conduction?

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In the next lecture, we will introduce Fick's Law, the fundamental governing equation for molecular diffusion, connecting the fluxes we defined today to the concentration gradients we discussed previously.
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