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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• More topics in fluid mechanics and heat transfer
  solved using CFD
• The presence of radiation heat transfer in the
  problem.
• The effect of fluid flow (convection) on heat
  transfer and flow quantities.
• The presence of chemical reactions along with
  fluid flow and heat transfer, e.g. combustion or
  chemical reactions at surface or in volume.
• Coexistance of multiphases in the system such as
  two-phase flows of liquid water and vapor, or
  melting and solidification.
• The presence of turbulence and its effects on
  mass, momentum and heat transport.

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Radiation Boundary Condition
Consider the plane-wall transient conduction
problem with constant k and with radiation
boundary condition. Find the difference equations
and temperature distribution in the wall.
This example is a one dimensional case in
unsteady state condition with an internal heat
generation and constant k , and at the left side
of the wall the heat is transferred by radiation
mechanism and at the right side of the wall a
constant heat flux exits from the wall. The wall
is initially in a temperature equal to T0 and
undergoes an unsteady process until it reaches to
a steady state condition. We are solving this
example firstly by writing the governing equation
and boundary conditions.
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We use 8 nodes along the thickness of the wall,
at point 1 we have radiation BC and at point 8 we
have constant heat flux BC. So the energy balance
equation at node 1 is
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The latter can be written as
If we use the explicit Euler method in order to
discretize the time derivation, we will get as
follows
By replacing p in this equation, one can write it
as
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For a middle point like i, we can write
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Finally for node 8 which is another boundary
point, we can write the energy balance equation
as follows
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We can see equations I, II and III express the
value of temperature at time step n1 versus the
information on the step n, so in the matrix form
we have
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As we can see, in this approach it is not
necessary to take the inverse of matrix of
coefficients, and temperature vector is
determined by the above explicit expression. Note
that one element of the matrix of coefficients
(corresponding to the first node) depends on the
temperature at stage n and it should be
recalculated at each time stage. It is due to the
nonlinear property of the radiation mechanism of
heat transfer.
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Introduction to the application of CFD in solving
Navier-Stokes equations Conservation equations of
mass, momentum and energy are written as
governing equations of fluid. When Stokes
assumption is applied in the governing equations
(relationship between stresses and strains) the
result is referred as Navier-Stokes equations.
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In Cartesian coordinates, two-dimensional
geometries we have
Source term
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Vorticity-Stream Function Approach Very popular
method for 2D incompressible NS equations. We
change the velocity variable in equations to
vorticity, which is defined as
The stream function in 2D is defined by the
following equations
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These new variables and equations can be combined
with the two momentum equations given above in
which pressure terms can be eliminated. The
result is called the vorticity transport equation
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An additional equation involving vorticity and
stream function can be obtained directly by
combining the definitions of the two quantities
shown above
This equation is called the Poisson equation,
which is also called diffusion equation. We have
shown examples of solving diffusion equation
using finite-difference scheme.
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Using this approach, we have separated the mixed
elliptic-parabolic NS equations into one
parabolic (the vorticity transport equation) and
one elliptic (the Poisson equation) equation.

• These equations are normally solved using a
  time-marching procedure which is described by the
  following steps
• Give initial values for ? and ? at time t0.
• Solve the vorticity transport equation for ? at
  each interior grid point at time t?t.
• Find new ? values at all points by solving the
  Poisson equation using new values of ? at
  interior points from step 2.
• Calculate velocity components from u?y and
  v-?x.
• Determine boundary values of ? using ? and ?
  values at interior points.
• Return to step 2 if the solution is not converged.

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This procedure requires that appropriate
expressions for ? and ? be specified at the
boundaries. The stability and accuracy of the
solution depends on these boundary conditions. At
the wall surface, ? is a constant and usually 0.
To find ? at the surface we use the Taylor
expansion of ? about the wall point (i,1)
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After completing these steps, velocity components
are known. To determine pressure at each grid
point, it is necessary to solve an additional
equation which is the Poisson equation for
pressure. It is obtained after differentiating
momentum equations and adding them, then using
the continuity equation to give
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One of the test problems is the cavity problem as
shown below
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PDE Toolbox of MATLAB
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