Advanced Topics in Heat, Momentum and Mass Transfer

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Advanced Topics in Heat, Momentum and Mass
Transfer

• Lecturer
• Payman Jalali, Docent
• Faculty of Technology
• Dept. Energy Environmental Technology
• Lappeenranta University of Technology

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• How do we use commercial softwares for CFD
  problems?
• After generation of the geometry and grids, one
  has to define boundary conditions. Here are a
  list of different boundary conditions
• Momentum description The boundary conditions
  for momentum equation can be categorized in
  different forms.
• 1) Wall It represents solid boundaries where
  fluid sticks and moves with the wall if there is
  no-slip boundary condition. If there is a
  difference between the velocity of the wall and
  the fluid velocity (slip velocity) it is
  specified in different ways.

2) Velocity inlet The velocity vector over a
boundary or part of it is known.
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3) Symmetry The symmetry boundary condition is
used for other equations and quantities as well.
The normal derivative of different quantities
such as velocity, temperature, concentration etc
is set to zero at the boundary.
4) Pressure inlet The value of pressure over a
boundary or part of the boundary is known.
Velocity will be determined as a result of
calculations. 5) Pressure outlet The value of
pressure over a boundary is known. Velocity will
be then determined from calculations.
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6) Axis The axis boundary condition refers to
axisymmetric problems.

• Energy description The boundary conditions for
  energy equation are listed below.
• 1) Given heat flux This boundary condition
  represents solid boundaries which generate heat
  given by the value of heat flux (constant or via
  a profile).
• 2) Given temperature The value of temperature
  (constant or with a given distribution) is given
  on the boundary.
• 3) Convection heat transfer The value of
  convective heat transfer and the free stream
  temperature is given for the boundary.

4) Convection heat transfer The value of
convective heat transfer and the free stream
temperature is given for the boundary. Radiation
can be also added to the boundary conditions
simultaneously with convective heat transfer.
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• In addition to boundary conditions, the
  selection of model is another part of the
  solution procedure.
• Steady state model or unsteady state?
• Two-dimensional, three-dimensional or
  axisymmetric?
• The solver type (pressure or density based) and
  formulation (explicit or implicit)?
• The viscous model (Inviscid, laminar, k-?
  turbulence model etc)?
• The system model (multiphase flow, species
  transport, radiation, solidification and melting
  etc)
• Material properties and model-dependent
  variables. For example, viscosity, density,
  conductivity are working fluid properties that
  must be given in any case. But, the coefficients
  of the k-? model are only given if this model is
  employed.
• In commercial CFD packages there are possibility
  for user-defined functions (UDF). Users can apply
  any certain rules, functions and conditions in
  the CFD problem which is not available in the
  package.

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Periodicity The solution domain may be subject
to periodic boundary conditions (PBC). According
to the PBC the solution is repeated in
neighboring domains. In this case, the boundary
conditions are not defined on opposite edges, but
we have to define the periodic conditions
separately.
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First example for solving by FLUENT The
following picture shows the geometry of two
connected canals. The geometry is
two-dimensional, so we have canals instead of
pipes. The case of two connected pipes must be
solved using a three-dimensional geometry, and
even not an axisymmetric problem since you can
not find any symmetry axis for the entire system.
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Here are two different grids that can be used
for this problem
Unstructured meshes
Structured meshes
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Boundary conditions
Wall (no-slip, zero heat flux)
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Two models are employed for this problem, 1)
Laminar flow, 2) standard k-? turbulent
model. The energy equation is also solved for
calculating temperature field. The working fluid
is water, so we have single-phase,
single-component flow. Solution procedure -
Initialization As we discussed in the cavity
problem, almost all CFD problems involved with
convective flow are based on iterative schemes.
- Iterations An initial solution must be
produced which will be improved and corrected in
each iteration. After a number of iterations the
solution must converge. In other words, the mean
difference between two consecutive solution
(residual) must reach low enough. This limit is
defined separately for each quantity such as mass
(continuity), velocity components and
temperature. Other quantities can be also defined
whose convergence will determine our stop point
for calculations. - Accuracy Initially, start
with lower accuracy for discretized terms of
pressure, momentum, energy and other possible
variables whose governing equations are solved.
When it is converged then increase the accuracy
to the second order and continue the solution for
another convergence.
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Solution procedure - Under-relaxation The
solution scheme follows consequent iterations so
that the solution improves iteration by
iteration. Under-relaxation factor is defined as
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This figure shows the variation of residuals
through consecutive iterations. For example,
calculation is stopped when continuity residual
is reached to 10-3 while all other residuals have
already converged.
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The preliminary solution for the contours of
velocity magnitude is shown below.
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Similarly, the preliminary solution for
temperature field is shown here.
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Vector field can be visualized too.
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Streamlines (pathlines) are visualized below.
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We can see the distribution of any variable over
a boundary or predefined line. For example,
temperature distribution is shown over the outlet
boundary of the lower duct.
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As another example, the pressure distribution is
plotted along the outer wall of the bend.
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Now, we can increase the order of discretization
for all involved terms from the first order to
the second order and continue iterations.
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The visualization shows that the solution can
improve near the boundary by increasing the order
of discretization. However, it may not be always
true since increasing the discretization degree
and decreasing the mesh size should be usually
done simultaneously.
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In this problem, we see that there is large
gradient of temperature in some region.
Therefore, one could refine meshes in those
regions in order to obtain more accurate
solution. This process is called adaption. It can
be done for different reasons in addition to
gradient-based adaption.
For example, the geometry of the boundaries is
another factor for refining meshes. Also,
near-wall characteristics of turbulent flows is
another factor that can be considered for
adaption. The following figure shows the
distribution of adaption function based on
temperature gradient.
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The following figure shows the new grid after
adaption. Small cells within the elliptical
region have had higher gradient of temperature,
so they have been divided into several cells.
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The following figures display how the adaption
has been done based on the accuracy required for
the temperature field.
Interpolated cell temperatures
Cell temperatures