1. Using thermodynamics, we can prove that, for all positions r and r’ in a system, equilibrium is given by
p(r) = p(r’)
T(r) = T(r’)
μ(r) = μ(r’)
where p(r) is the pressure, T(r) is the temperature, and μα(r) is the chemical potential of species α at position r. Mathematically, this tells us that the pressure, temperature, and chemical potentials are uniform in a system at equilibrium.
If there is a gradient in any of these quantities, then the system is
out of equilibrium. As a consequence, momentum, energy, and mass will flow through the system to try to bring it to equilibrium.
Most processes that are of practical interest are not in equilibrium and never truly achieve equilibrium. In order to describe these systems, we need to study fluid mechanics, heat transfer, and mass transport, which are also known collectively as non-equilibrium thermodynamics or transport phenomena.

2. NON-EQLBM THERMODYNAMICS
Postulate I
Although system as a whole is not in eqlbm., arbitrary small elements of it are in local thermodynamic eqlbm & have state fns. which depend on state parameters through the same relationships as in the case of eqlbm states in classical eqlbm thermodynamics.

3. NON-EQLBM THERMODYNAMICS
Postulate II
Entropy gen rate
affinities
fluxes
All sorts of transport only take place when a force, called a driving force, is applied. Transport is generally expressed as a flux J, which is defined by the amount of mass, energy, momentum, volume, or charges that are being transported pr. area pr. time.

4. NON-EQLBM THERMODYNAMICS
Purely “resistive” systems
Flux is dependent only on affinity
at any instant
at that instant
System has no “memory”-

5. NON-EQLBM THERMODYNAMICS
Coupled Phenomenon
Since Jk is 0 when affinities are zero,

6. NON-EQLBM THERMODYNAMICS
where
kinetic Coeff
Relationship between affinity & flux from ‘other’ sciences
Postulate III

7. NON-EQLBM THERMODYNAMICS
Postulate IV
Onsager theorem {in the absence of magnetic fields}

8. Analysis of thermo-electric circuits
Addl. Assumption : Thermo electric phenomena can be taken as LINEAR RESISTIVE SYSTEMS
{higher order terms negligible}
Here K = 1,2 corresp to heat flux “Q”, elec flux “e”

9. Analysis of thermo-electric circuits
 Above equations can be written as
Substituting for affinities, the expressions derived earlier, we get

10. Analysis of thermo-electric circuits
We need to find values of the kinetic coeffs. from exptly obtainable data.
Defining electrical conductivity as the elec. flux per unit pot. gradient under isothermal conditions we get from above

12. Substituting the formula of normalvector:
In transport phenomena (heat transfer, mass transfer and fluid dynamics), flux is defined as the rate of flow of a property per unit area, which has the dimensions [quantity]·[time]−1·[area]
j=I/A, where I=dq/dt
In all cases the frequent symbol j, (or J) is used for flux, q for the physical quantity that flows, t for time, and A for area.
As a mathematical concept, flux is represented by the surface integral of a vector field
The surface has to be orientable, i.e. two sides can be distinguished: the surface does not fold back onto itself. Also, the surface has to be actually oriented, i.e. we use a convention as to flowing which way is counted positive; flowing backward is then counted negative.
Substituting the formula of normalvector:

13. Fluxes
Quantity transferred through a given area per unit time
Flux occurs because of a spatial and time-varying gradient of an associated system property.
Flux occurs in opposition to spatial gradient
Flux continues until the gradient is nullified and equilibrium is reached.
Flux continues until external force stop
The most general relation for a flux in the x-direction is:
This relates a flux to a gradient of a property (not specically defined to be anyting at this point)
Flux is a linear response; this treatment assumes a perturbative displacement from equilibrium.

14. Transport fluxes
Eight of the most common forms of flux from the transport phenomena literature are defined as follows:
Momentum flux, the rate of transfer of momentum across a unit area (N·s·m−2·s−1).
(Newton's law of viscosity)
Heat flux, the rate of heat flow across a unit area (J·m−2·s−1).
(Fourier's law of conduction)(This definition of heat flux fits Maxwell's original definition.)
Diffusion flux, the rate of movement of molecules across a unit area (mol·m−2·s−1).
(Fick's law of diffusion)
Volumetric flux, the rate of volume flow across a unit area (m3·m−2·s−1). (Darcy's law of groundwater flow)
Mass flux, the rate of mass flow across a unit area (kg·m−2·s−1). (Either an alternate form of Fick's law that includes the molecular mass, or an alternate form of Darcy's law that includes the density.)
Radiative flux, the amount of energy transferred in the form of photons at a certain distance from the source per unit area per second (J·m−2·s−1). Used in astronomy to determine the magnitude and spectral class of a star. Also acts as a generalization of heat flux, which is equal to the radiative flux when restricted to the infrared spectrum.
Energy flux, the rate of transfer of energy through a unit area (J·m−2·s−1). The radiative flux and heat flux are specific cases of energy flux.
Particle flux, the rate of transfer of particles through a unit area ([number of particles] m−2·s−1)
These fluxes are vectors at each point in space, and have a definite magnitude and direction. Also, one can take the divergence of any of these fluxes to determine the accumulation rate of the quantity in a control volume around a given point in space. For incompressible flow, the divergence of the volume flux is zero.

15. NON-EQLBM THERMODYNAMICS
Heat Flux :
Momentum :
Mass :
Electricity :

17. Concentration / chemical potential
Diffusion
Mass flow process by which species change their position relative to their neighbours
Driven by thermal energy and a gradient
Thermal energy → thermal vibrations → Atomic jumps
Concentration / chemical potential
Gradient
Electric
Magnetic
Stress

18. Matter transport is down the concentration gradient
Fick’s I law
Diffusion coefficient/ diffusivity
No. of atoms crossing area A per unit time
Cross-sectional area
Concentration gradient
Matter transport is down the concentration gradient
Flow direction A
As a first approximation assume D  f(t)

19. Steady-State Diffusion
Diffusion flux does not change with time
Concentration profile:
Concentration (kg/m3) vs. position
Concentration gradient: dC/dx (kg / m4)

20. Steady-State Diffusion
Fick’s first law: J proportion to dC/dx
D=diffusion coefficient
Concentration gradient is ‘driving force’
Minus sign means diffusion is ‘downhill’: toward lower concentrations

21. Fick’s first law

22. Diffusivity (D) → f(A, B, T)
Steady state diffusion C1
D  f(c)
Concentration → C2
D = f(c)
x →

23. Non-steady state J = f(x,t)
D  f(c)
Steady state J  f(x,t)
D = f(c)
Diffusion
D  f(c)
Non-steady state J = f(x,t)
D = f(c)

24. Equations are based on the following physical principles:
Mass is conserved
Newton’s Second Law:
The First Law of thermodynamics: De = dq - dw, for a system.

25. Control Volume Analysis
The governing equations can be obtained in the integral form by choosing a control volume (CV) in the flow field and applying the principles of the conservation of mass, momentum and energy to the CV.

26. Consider a differential volume element dV in the flow field
Consider a differential volume element dV in the flow field. dV is small enough to be considered infinitesimal but large enough to contain a large number of molecules for continuum approach to be valid.
dV may be:
fixed in space with fluid flowing in and out of its surface or,
moving so as to contain the same fluid particles all the time. In this case the boundaries may distort and the volume may change.

27. Substantial derivative
(time rate of change following a moving fluid element)

28. Conservation of mass
Control element u r DX
Mass balance: or

29. Compartmental Analysis
X in X
X out
X is some variable of interest
A change in X = X in – X out
in derivative terms:

30. What is Diffusion?
Definition – the process by which a substance disperses within an ambient medium over time
Modeled using compartmental analysis
Net change of substance at a point =
(inflow rate of substance) – (outflow rate of the substance)

31. Diffusion through a Region Ω
The region may contain sources and sinks.

32. Modeling Diffusion (the nuts and bolts)
u = u(x, y, z, t) = density at a point in the region
f = f(x, y, z, t) = production at a point (a density) in the region due to sources and sinks
J = J(x, y, z, t) = flux density at a point

33. Some Calculus = total mass in region = rate of mass change in region
= total production of mass in region due to sources and sinks

34. The Confusing Quantity of Flux
J = J(x, y, z, t) = flux density at a point (density x velocity)
The units of flux are:
small flux in the negative x direction
large flux in the positive x direction

35. What can we do with Flux?
Mass leaving the region through the boundary will be the component of the flux that is in the direction of the surface normal
= total flux out of the region in the direction of at a point on the boundary
= total flux (accumulation of mass) out of the region through the boundary

36. Putting Together the Pieces
Using compartmental analysis we can write an equation relating all the components we have developed thus far.
rate of substance change = amount produced in the region – amount that escapes the region

37. gradient
Divergence Theorem (Gauss’s Theorem) a Method of modifying the flux integral
The amount of mass that overflows the boundary must be equal to the total change in mass within the boundary.
Total flow across the boundary = cumulative flow in the region omega
The Bathtub Overflow Theorem
Johann Carl Friedrich Gauss
Born: 30 April 1777
Brunswick, Germany
Died: 23 Feb 1855
Göttingen, Germany

38. Simplification Divergence Theorem
Eq2: a = b – total cumulative rate of flow

39. Almost there…
Theorem from Real Analysis
If you integrate over all subregions of “OMEGA” and get zero, the expression integrated must also be equal to zero

40. The Unrefined Diffusion Equation
density change at a point over time = density production at that point per unit time – the divergence (the rate of flow from that point)

41. Fick’s Law Gradient Adolph Fick
During diffusion we assume particles move in the direction of least density. They move down the concentration gradient
In mathematical terms we will assume
Flux density = Diffusion coefficient * the gradient of the concentration, negative means flow is in the direction of least density
Where D is a constant of proportionality called the Diffusion Coefficient
Adolph Fick
Born: 1829
Cassel, Germany
Died: 1879

42. Ockham’s Razor
When faced with a choice between two things, choose the simpler.
If we assume D is constant and substitute the new expression we found for flux using Fick’s Law we can remove the flux density component of our diffusion equation as follows:
William of Ockham
Born: 1288
Ockham, England
Died: 9 April Munich, Bavaria

43. The Laplacian Pierre-Simon Laplace
Born: 23 March 1749 Normandy, France
Died: 5 March 1827
Paris, France

44. The Refined Diffusion Equation
Separate slide for Laplacian explanation, picture

45. Initial and Boundary Conditions
1. Initial conditions (IC) :
The initial state of the primary variables of the system:
For non-horizontal systems:
Where Pref is reference pressure & ρ is fluid densities

46. Multiphase Flow Continuity equation for each fluid phase :
Darcy equation for each phase :
Oil density equation:
roL: the part of oil remaining liquid at the surface roG : the part that is gas at the surface

47. Continuity Equation
 Consider the CV fixed in space. Unlike the earlier case the shape and size of the CV are the same at all times. The conservation of mass can be stated as:
Net rate of outflow of mass from CV through surface S = time rate of decrease of mass inside the CV

48. The net outflow of mass from the CV can be written as
Note that by convention is always pointing outward. Therefore
can be (+) or (-) depending on the directions of the velocity and the surface element.

49. Total mass inside CV
Time rate of increase of mass inside CV (correct this equation)
Conservation of mass can now be used to write the following equation
 See text for other ways of obtaining the same equation.

50. Integral form of the conservation of mass equation thus becomes
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
Integral form of the conservation of mass equation thus becomes

51. Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR

52. An infinitesimally small element fixed in space

53. Net outflow in x-direction
Net outflow in y-direction
Net outflow in z-direction
Net mass flow =

54. Net rate of outflow from CV = time rate of decrease of mass
volume of the element = dx dy dz
mass of the element = r(dx dy dz)
Time rate of mass increase =
Net rate of outflow from CV = time rate of decrease of mass
within CV or

55. Which becomes
The above is the continuity equation valid for unsteady flow
Note that for steady flow and unsteady incompressible flow the
first term is zero.
Figure 2.6 (next slide) shows conservation and non-conservation forms of the continuity equation. Note an error in Figure 2.6: Dp/Dt should be replace with Dr/Dt.

56. Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR

57. Momentum equation

59. Summary
Apply conservation equations to a control volume (CV)
As the CV shrinks to infinitesimal volume, the resulting partial
differential equations are the Navier-Stokes equations
Taylor series can be used to write the variables
The total derivative consists of local time derivative and convective
derivative terms
In incompressible flow, divergence of velocity is a statement of
the conservation of volume
Need surface and body forces to write the momentum equation
Surface forces are: pressure forces, forces due to normal stresses
and forces due to shear stresses
Body forces are due to weight, magnetism and electrostatics
Momentum equation is a vector equation. Can be written in terms
of its components.