Transport Phenomena Continuity Equation Flow From Spherical Aggregate

Transport Phenomena are irreversible processes that arise from the random continuous motion of molecules. The three types of transport phenomena are momentum, mass of a component in a multicomponent mixture, and thermal energy. A complete description of the transport phenomena is embodied in the following two relationships:

• Conservation laws
• Constitutive molecular flux laws

The conservation laws are continuity equations; these equations dictate that the accumulation of a quantity ${\displaystyle \Psi }$ in a control volume equals the transport of ${\displaystyle \Psi }$ into and out of the control volume plus any generation or consumption of ${\displaystyle \Psi }$ in the control volume due to non-conservative mechanisms.

In other words,

${\displaystyle \mathrm {Accumulation} =\mathrm {Inflow} -\mathrm {Outflow} +\mathrm {Generation} -\mathrm {Consumption} }$

The conservation laws can be formulated as either microscopic (differential) balances or as macroscopic (integral) balances.

Microscopic balance

In the microscopic description, the balance equation is formulated on a per volume basis. For instance, if ${\displaystyle {\hat {\psi }}}$ is a transport quantity ${\displaystyle \Psi }$ is on a per volume basis, the differential balance is

${\displaystyle {{\partial \left(\rho {\hat {\psi }}\right)} \over {\partial t}}=-\left({\nabla \cdot \mathbf {\Phi } _{\Psi }}\right)+{\dot {r}}_{_{\Psi }}}$

(1)

where

Macroscopic balance

In the macroscopic description and there are three types mass, momentum and energy, ${\displaystyle \Psi =\iiint \limits _{CV}{\rho {\hat {\psi }}dV}}$ , and the integral balance equation is

${\displaystyle {{d\Psi }/{dt}}=-\iint \limits _{CS}{\left({\mathbf {\Phi } _{\Psi }\cdot {\bf {\hat {n}}}}\right)dA}-\iint \limits _{CS}{\left({{\mathbf {\dot {r}} }_{\Psi }^{s}\cdot {\bf {\hat {n}}}}\right)dA}+\iiint \limits _{CV}{{\dot {r}}_{\Psi }}dV}$

(2)

where

Flux

Flux ${\displaystyle \Phi }$ describes the movement of a property. For this class specifically, we will only be dealing with transport properties of Momentum, Energy and Mass. Flux is a vector that has both magnitude and direction and usually has units of the property per unit area per unit time.

For example,

• Mole Flux has units of moles per unit area per unit time.

${\displaystyle {\frac {moles}{m^{2}\cdot s}}}$

• And Energy Flux has units of energy per unit area per unit time.

Different mechanisms allow for flux to occur.

1. Bulk Flow or Advective Mechanism
2. Molecular or Diffusive / Conductive Mechanism

Advective Flux

Advection is the bulk movement of a property due to a velocity field. That is, the advective movement occurs in the direction of the velocity field.

For example,

After perfume is sprayed in a room with the wind blowing, someone standing in the path of the wind will smell the perfume, because the wind carries the perfume particles along with it until the particles are evenly distributed. This all occurs due to the advective movement of the particles.

Convective Flux

Convective flux ${\displaystyle \Phi _{\Psi }}$ is the movement of a property due to both advective and molecular mechanisms. It has dimensions of "property per unit time per unit area" and has the general form

${\displaystyle \Phi _{\Psi }=\left({\rho {\hat {\psi }}}\right)\mathbf {v} +\mathbf {j} _{\Psi }}$

(3)

where

Constitutive molecular flux laws

The constitutive equations describe how the flux of ${\displaystyle \Psi }$ , or ${\displaystyle {\bf {j}}_{\Psi }}$ is related to a particular gradient (Cartesian, Cylindrical, Spherical). The three most common flux laws are

Diffusivity and the analogy between the three types of molecular transport

In the absence of external bulk flow and non-conservative mechanisms, the microscopic balance reduces to

${\displaystyle {{\partial {\rho {\hat {\psi }}}} \over {\partial t}}=-\left(\nabla \cdot {\bf {j}}_{\Psi }\right)}$

(4)

Substituting the constitutive molecular flux laws into the above equation for momentum, heat, and mass, each equation can be written explicitly as

${\displaystyle {\begin{aligned}{{\partial {\bf {v}}} \over {\partial t}}&={\nu }\nabla ^{2}{\bf {v}}\\[10pt]{{\partial T} \over {\partial t}}&=\alpha \nabla ^{2}T\\[10pt]{{\partial \rho _{A}} \over {\partial t}}&={\mathcal {D}}_{A}\nabla ^{2}\rho _{A}\end{aligned}}}$

(5)

In each case, the coefficient before the Laplacian operator has dimensions of length ² per unit time and is known as the diffusivity. The diffusivities for the three cases are

Hence, the diffusivities give the characteristic rates at which the velocity field, temperature field, and density field spreads in the absence of advective (bulk flow) mechanisms.

Summary of transport relationships

	Momentum	Mass	Thermal Energy
Transport quantity ${\displaystyle \Psi }$	${\displaystyle m{\bf {v}}}$	${\displaystyle m_{A}}$	${\displaystyle mc_{v}T}$ (Internal energy) or ${\displaystyle mc_{p}T}$ (Enthalpy)
Transport quantity per unit volume ${\displaystyle \psi }$	${\displaystyle \rho {\bf {v}}}$	${\displaystyle \rho _{A}}$	${\displaystyle \rho c_{v}T}$ or ${\displaystyle \rho c_{p}T}$
Solution of microscopic balance	Velocity ${\displaystyle {\bf {v}}}$ field	Density ${\displaystyle \rho _{A}}$ or concentration ${\displaystyle c_{A}}$ field	Temperature ${\displaystyle T}$ field
Flux law ${\displaystyle {\bf {j}}_{\Psi }}$	${\displaystyle {\boldsymbol {\tau }}=-\mu \left({\nabla \mathbf {v} +\nabla \mathbf {v} ^{\intercal }}\right)}$	${\displaystyle {\bf {j}}_{A}=-{\mathcal {D}}_{A}\nabla {\rho _{A}}}$	${\displaystyle {\bf {\dot {q}}}=-k\nabla {T}}$
Diffusivity (Length2/time)	${\displaystyle \nu =\mu /\rho }$	${\displaystyle {\mathcal {D}}_{A}}$	${\displaystyle {\alpha }={k/{\rho c_{p}}}}$
Convective transfer coefficient		${\displaystyle k_{A}}$	${\displaystyle h}$
Convective boundary condition		${\displaystyle j_{A}=-k_{A}\Delta \rho _{A}}$	${\displaystyle {\dot {q}}=-h\Delta T}$
Dimensionless convection/diffusion ratio	Reynolds number ${\displaystyle {\rm {Re}}{=}v\ell _{c}/\nu }$	Sherwood number ${\displaystyle {\rm {Sh}}{=}k_{\rho }\ell _{c}/{\mathcal {D}}_{A}}$	Nusselt number ${\displaystyle {\rm {Nu}}{=}h\ell _{c}/k}$
Corresponding macroscopic balance	Mechanical energy balance	Component mass balance	Internal energy balance
Key dimensionless numbers	Friction factor, Reynolds number	Sherwood number, Reynolds number, Schmidt number	Nusselt number, Reynolds number, Grashof number, Prandtl number
Correlation relationship	${\displaystyle f_{D}=f\left({\rm {Re}}\right)}$	${\displaystyle {\rm {Sh}}=f\left({\rm {Re,Sc}}\right)}$	${\displaystyle {\rm {Nu}}=f\left({\rm {Re,Pr}}\right)}$ (forced convection) or ${\displaystyle {\rm {Nu}}=f\left({\rm {Gr,Pr}}\right)}$ (natural convection)
Other transport modes			Radiation

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Source: https://www.ryantoomey.org/wiki/Transport_phenomena

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