Basic Concepts

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Basic Concepts

• Mass transfer: transport of a substance that is involved as a component (constituent, species) in a fluid mixture.

– An example is the transport of salt in saline water.

• Convective mass transfer is analogous to Convective heat transfer

• A fluid mixture of volume V and mass m. Let the subscript i refer to the ith component (component i) of the mixture. The total mass is equal to the sum of the individual masses 𝑚_𝑖.

Multicomponent Flow: Basic Concepts

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Basic Concepts

• Concentration of component i:

• aggregate density of the mixture must be the sum of all the individual concentrations:

• When chemical reactions are of interest it is convenient to work in terms of an alternative description, one involving the concept of mole.

• if there are n moles in a mixture of molar mass M and mass m, then

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Basic Concepts

• mass fraction of component i:

• Similarly the mole fraction of component i is:

• These quantities are related by:

• where, the equivalent molar mass (M) of the mixture is given by:

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Basic Concepts

• For example, if the mixture can be modeled as an ideal gas, then its equation of state is:

• where the gas constant of the mixture (𝑅_𝑚) and the universal gas constant (R) are related by:

• partial pressure Pi of component i:

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Basic Concepts

• the pressure of a mixture of gases at a specified volume and temperature is equal to the sum of the partial pressures of the components.

• To relate 𝐶_𝑖 to 𝑃_𝑖:

• For: mixture in equilibrium, that is, to a fluid batch whose composition, pressure, and temperature do not vary from point to point.

• In a convection study involved with a nonequilibrium mixture, which we view as a patchwork of small equilibrium batches: the equilibrium state of each of these batches is assumed to vary only slightly as one moves from one batch to its neighbors.

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Mass Conservation in a Mixture

• apply the principle of mass conservation to each component in the mixture.

• we use the notation ρ_𝑖 instead of 𝐶_𝑖 for the concentration of component i.

• In the absence of component generation:

• Summing over i,

• the same as

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Mass Conservation in a Mixture

• the mass-averaged velocity:

• diffusion velocity of component i:

• diffusive flux of component i:

diffusion

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Mass Conservation in a Mixture

• Reverting to the notation 𝐶_𝑖 for concentration, and assuming that the mixture is incompressible,

• ^𝜕𝐶

𝜕𝑡: temporal variation: ینامز تارییغت

• 𝑢 ^𝜕𝐶_𝜕𝑥 + 𝑣^𝜕𝐶_𝜕𝑦: local variation (convection, advection): ییاج هب اج ،یناکم تارییغت

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• For a two-component mixture:

• 𝐷₁₂: Mass diffusivity of component 1 into component 2

• diffusivity D, whose unit is ^𝑚

2

𝑠 , has a numerical value which in general depends on the mixture pressure, temperature, and composition

Fick’s law of diffusion

& 𝒋₂ = −𝐷₂₁𝛻𝐶₂

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Mass Conservation in a Mixture

• For a homogeneous situation:

• The analogy between this equation and the corresponding energy equation:

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• So far we have been concerned with the fluid only, but now we consider a porous solid matrix saturated by fluid mixture.

• Recalling the Dupuit-Forchheimer relationship:

Porous medium:

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Mass Conservation in a Mixture

• Some authors invoke tortuosity and produce a more complicated relationship between 𝐷_𝑚 and 𝐷.

• The diffusive mass flux in the porous medium:

• If the mass of the substance whose concentration is C is being generated at a rate 𝑚ሶ ^′′′ per unit volume of the medium:

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Combined Heat and Mass Transfer

• In the most commonly occurring circumstances the transport of heat and mass (e.g., salt) are not directly coupled, and both Eqs. (heat and mass) (which clearly are uncoupled) hold without change.

not directly coupled

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Combined Heat and Mass Transfer

• coupling takes place because the density of the fluid mixture depends on both temperature T and concentration C (and also, in general, on the pressure P).

• For sufficiently small isobaric changes in temperature and concentration:

• volumetric thermal expansion coefficient:

• volumetric concentration expansion coefficient:

double-diffusive convection

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Combined Heat and Mass Transfer

• In some circumstances there is direct coupling. This is when cross-diffusion (Soret and Dufour effects) is not negligible.

Direct coupling

The Soret effect refers to mass flux produced by a temperature gradient

The Dufour effect refers to heat flux produced by a concentration gradient

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Combined Heat and Mass Transfer

is the mass diffusivity

in liquids the Dufour coefficient is an order of magnitude smaller than the Soret effect.

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Effects of a Chemical Reaction

• Application: chemical reactors of porous construction

• Suppose that we have a solution of a reagent whose concentration C is defined as above. If m is the molar mass of the reagent, then its concentration in moles per unit volume of the fluid mixture is:

• rate equation for the reaction:

• The integer power n is the order of the reaction. The rate coefficient k is a function of the absolute temperature T given by the Arrhenius relationship:

𝐶_𝑚 = 𝐶 𝑚

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Effects of a Chemical Reaction

• E is the activation energy of the reaction (energy per mole), R is the

universal gas constant, and A is a constant called the pre-exponential factor.

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• Mass transfer equation:

Mass Transfer Equation

𝑑𝐶_𝑚

𝑑𝑡 = −𝑘 𝐶_𝑚^𝑛 → 𝑑𝐶

𝑚 𝑑𝑡 = −𝑘 𝐶 𝑚

𝑛

→ 𝑑𝐶

𝑑𝑡 = −𝑘𝑚^1−𝑛 𝐶^𝑛

𝜑 𝑑𝐶

𝑑𝑡 = 𝜑 −𝑘𝑚^1−𝑛 𝐶^𝑛 = −φ𝐴 exp − 𝐸

𝑅𝑇 𝑚^1−𝑛𝐶^𝑛

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: ∆𝐻 • هب هک شنکاو یپلاتنا

یازا C هدام

نازیم هب

∆𝐻 𝑚

𝑑𝐶

دنک یم دیلوت یژرنا 𝑑𝑡

.

Energy Equation

∆𝐻𝑑𝐶_𝑚

𝑑𝑡 = ∆𝐻 𝑚

𝑑𝐶 𝑑𝑡

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• These equations are appropriate if the reaction is occurring entirely within the fluid. Now suppose that we have a catalytic reaction taking place only on the solid surface of the porous matrix.

– φ = 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑝𝑜𝑟𝑜𝑠𝑖𝑡𝑦 = 𝑣𝑜𝑙𝑢𝑚𝑒 𝑝𝑜𝑟𝑜𝑠𝑖𝑡𝑦

– and if the reaction rate is proportional to the mass of the solid material

• 𝜑𝐴 → 1 − 𝜑 𝜌

_𝑆

𝐴

^′

𝑤ℎ𝑒𝑟𝑒 𝐴

^′

: 𝑛𝑒𝑤 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 𝑓𝑎𝑐𝑡𝑜𝑟

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Multiphase Flow

• If two or more miscible fluids occupy the void space in a porous medium, then even if they occupy different regions initially they mix because of diffusive and other dispersive effects, leading ultimately to a multicomponent mixture such as what we just have been considering.

immiscible fluids

“two-phase” fluid flow in a porous medium

solid matrix liquid phase (suffix 𝑙) gas phase (suffix 𝑔)

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Multiphase Flow

• a representative elementary volume V occupied by the liquid, gas, and solid, whose interfaces may move with time, so:

• We define the phase average of some quantity 𝜓_𝛼:

• The intrinsic phase average of 𝜓_𝛼 is defined as:

the integration is carried out over only 𝛼 phase

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Multiphase Flow

• Since 𝜓_𝛼 is zero in the other phases,

• Where:

is the fraction of the total volume occupied by the 𝛼 phase.

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Multiphase Flow

• In terms of the porosity:

• We define deviations (from the respective average values, for the 𝛼 phase)

• It can be written as:

• Therefore:

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Three useful theorem

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Multiphase Flow: Conservation of Mass

• The microscopic continuity equation for the liquid phase is:

• which can be integrated over an elementary volume:

• Application of the transport theorem to the first term and the averaging theorem to the second term of this equation and using

Conservation of Mass

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Multiphase Flow: Conservation of Mass

• where 𝐴_𝑙𝑔 and 𝐴_𝑙𝑠 are the liquid-gas and liquid-solid interfaces that move with velocities 𝑤_𝑙𝑔 and 𝑤_𝑙𝑠.

mass transfer due to a change of phase from liquid to gas, and in

general this is nonzero

there is no mass transfer across the liquid-solid interface and =0

Dispersive term: small:

neglected

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Multiphase Flow: Conservation of Mass

• Similarly the macroscopic continuity equations for the gas:

• continuity equations for the solid:

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• The mass gained by change of phase from liquid to gas is equal to the mass lost by change of phase from gas to liquid.

equal in magnitude but opposite in sign

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Multiphase Flow: Conservation of Mass

• Adding the three equations:

• If the volumetric liquid and gas saturation, 𝑆_𝑙 and 𝑆_𝑔, are defined by

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• Final form: Conservation of Mass

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Multiphase Flow: Conservation of Momentum

• The momentum equation for the liquid phase is:

• where 𝑃_𝑙, 𝜏_𝑙 , and f are, respectively, the pressure, the viscous stress tensor, and the body force per unit mass of the liquid.

• If the body force is entirely gravitational,

1. If substitute 𝑓 = 𝑔 = −𝛻Φ into above equation,

2. integrate the resulting equation over an elementary volume, 3. apply the transport theorem to the first term

4. apply averaging theorem to the second, third, and fourth terms, 5. use

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Multiphase Flow: Conservation of Momentum

6. and use mass conservation equation:

7. and replace

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• where density gradients have been assumed to be small compared to the velocity gradients.

1

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• For an isotropic medium, Gray and O’Neill:

• and

• where B and F are constants that depend on the nature of the isotropic medium.

2

3

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Multiphase Flow: Conservation of Momentum

• Substituting Eqs. (2) and (3) into Eq. (1) and neglecting the inertia terms in the square brackets and the term 𝜇𝛻² 𝑽_𝑙 :

• K denotes the intrinsic permeability of the porous medium, as defined for one-phase flow.

• The new quantity 𝑘_𝑠𝑙 is the relative permeability of the porous medium saturated with liquid. It is a dimensionless quantity.

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• Similarly, when inertia terms and the term 𝜇_𝑔𝛻² 𝑽_𝑔 are neglected, the momentum equation for the gas phase is:

• where 𝑘_𝑠𝑔 denotes the relative permeability of the porous medium saturated with gas.

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Multiphase Flow: Conservation of Momentum

• These equations are the Darcy equations for a liquid-gas combination in an isotropic porous medium.

• A similar expression for an anisotropic medium has been developed by Gray and O’Neill. A permeability tensor is involved. They also obtain an expression for flow in an isotropic medium with non-negligible inertial effects.

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Multiphase Flow: Conservation of Energy

• The microscopic energy equation, in terms of enthalpy for the liquid phase, is:

• where ℎ_𝑙 and 𝑘_𝑙 are the enthalpy and thermal conductivity of the liquid.

• In writing this equation, viscous dissipation, thermal radiation, and any internal energy generation are neglected.

1. Integrating this equation over a representative elementary volume 2. applying the transport equations to the first and fourth terms

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Multiphase Flow: Conservation of Energy

3. Applying &

to the second term

4. Applying this eq. to the third term:

5. Applying to the fifth term.

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• where 𝑘_𝑙^∗ is the effective thermal conductivity of the liquid in the presence of the solid matrix.

• This 𝑘_𝑙^∗ is the sum of the stagnant thermal conductivity 𝑘_𝑙^′ (due to molecular diffusion) and the thermal dispersion coefficient 𝑘_𝑙^′′ (due to mechanical dispersion), which in turn are defined by

Energy Eq. for Liquid phase

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Multiphase Flow: Conservation of Energy

• In the energy equation:

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• where q is the conduction heat flux across the interface, and ℎ_𝑙 is defined as the local volume averaged heat transfer coefficient at the liquid-solid interface, which depends on the physical properties of the liquid and its flow rate.

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Energy Eq. for Gas phase

Energy Eq. for Solid phase

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Multiphase Flow: Conservation of Energy

• where 𝑘_𝑔^∗ and 𝑘_𝑠^∗ are defined analogously to 𝑘_𝑙^∗, and similarly for the various Q terms.

• Note that:

• where ℎ_𝑔 is the heat transfer coefficient at the gas-solid interface.

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• The difference between 𝑃_𝑔 and 𝑃_𝑙 is called the capillary pressure. In many circumstances, the capillary pressure can be neglected, so in this case we have:

• Furthermore, we can usually assume local thermodynamic equilibrium and so

• Adding the energy equations for the three phases:

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Multiphase Flow: Conservation of Energy

• effective thermal conductivity of the porous medium saturated with liquid and gas at local thermal equilibrium, with the heat conduction assumed to be in parallel.