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Aula Teórica 9&10. Equação de Evolução. Exemplos. Caso 1D. Equação de evolução. Ou: . Diffusion plus advection. Where is diffusive flux maximum?. Analysis of the evolution equation . For the control volume: What is the sign of ? - PowerPoint PPT Presentation
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Aula Teórica 9&10
Equação de Evolução. Exemplos. Caso 1D
Equação de evolução
iojjj
j SSxxx
utdt
d
iojjj
j SSxxx
utdt
d
Ou:
• Diffusion plus advection
• Where is diffusive flux maximum?
Analysis of the evolution equation
• For the control volume:• What is the sign of ?• What does that mean physically (how does the
advective flux vary with x1)?• What is the relative value of diffusive flux in the lower
and upper faces of the control volume?• How does diffusive flux along x2 contribute to the
concentration inside the control volume? • If the flow is stationary and the material is
conservative what is the relation between advection and diffusion? what is the divergence of the total flux (advection + diffusion)?
iojjj
j SSxc
xxcu
tc
dtdc
x1
x2
11 xcu
Answers• The advective term is negative. The velocity is positive and • In other words, the advective flux decreases with x1, i.e., the quantity entering is bigger
than the quantity leaving (per unit of area).• The diffusive flux in the lower face is about zero. If located exactly over the symmetry line,
the flux will be exactly zero, because the gradient is null • On the upper face the diffusive flux is positive, i.e., the material is transported along the axis
x2 because the concentration is higher inside the control volume than above the control volume.
• The divergence of the vertical diffusive flux contributes to decrease the concentration inside the volume.
• The divergence of the horizontal diffusive flux is negative because the gradient on the left side of the volume is higher than on the right side. Horizontal advection is however the main mechanism to increase the concentration inside the control volume.
• If the flow is stationary the horizontal advection plus the horizontal diffusion balance the vertical diffusion. The divergence of both fluxes would be zero
• If the velocity was increased, the concentration inside the control volume would tend to increase because the quantity entering would increase and thus the quantity leaving would have to increase too.
01
xc
cu1
2xc
dif
0
jj
j xccu
x
Case of concentrationConsider two parallel plates and a property with a parabolic type distribution (blue) with maximum concentration at the center. Assume a stationary flow and a conservative property.a) Draw a control volume and indicate the fluxes (sense and relative magnitude). b) Where would the diffusive flux be maximum? Could this property be a
concentration? What kind of property could it be?c) What is the sign of the material time derivative?d) If the material is conservative (no sink or source), can the problem be
stationary?e) If it was a concentration without adsortion, what would be the final profile?
a) Diffusive flux increase with “r”. Advective flux depends on the longitudinal gradient, not yet known. b) Property gradient is maximum at the boundary. This means that there the diffusive flux is maximum and
consequently the property can pass through the boundary. The property can be a concentration only if there is adsortion at the boundary. It could be a temperature or momentum as well.
c) Property is being lost across the boundary and consequently there is a longitudinal negative gradient. The total derivative is negative (and so is the advective derivative)
d) yes the system could be stationary, but the fluid would loose property along the flow.e) If it is stationary and conservative what would be the sign of the longitudinal gradient component?f) If it was a concentration this profile could not maintain. The gradient should become null at the solid boundary (no
diffusive flux). The final profile would be uniform. Transversal diffusion would homogenize it.
yyxxx x
cxx
cxx
cu
y
Nitrogen Transport in the Baltic Sea• The figure shows the
distribution of nitrate discharged by the river Oder after 4, 8, 12 and 16 years of emission.
• In the region next to the river mouth the concentration becomes constant after 8 years. What does that mean?
• Is advection important in this system?
1D case• Vamos supor o caso de um reservatório com a forma de um
canal rectangular de 10 m de largura e 200 de comprimento, com velocidade nula. A concentração é elevada na zona central e nula na generalidade do canal. O material é conservativo.
• Escreva a equação que rege a evolução da concentração.
iojjj
j SSxc
xxcu
tc
dtdc
• E se existisse decaimento?.
With first order decay
• How will the concentration evolve in each case?
kcxc
xtc
dtdc
t0
t1
t2
t∞• In case of decay, concentration
are lower and tend to zero.
If there is a lateral discharge?
• Concentration will grow because of the discharge, will get homogenized because of diffusion and will decay because of decay.
• When discharge balances total decay, the system will reach equilibrium.
Solution of the problem
• The first case (pure diffusion) has analytical solution, but not the others.
• How to solve the problem numerically?• The Reynolds Theorem:
• The lagrangian formulation:
dAnvdVdtddV
t surfacesistemavc
.
dAncSiSodVdtd
surfacesistema
.)(
The Eulerian formulation )(.. SiSodAncdAnudV
t surfacesurfacevc
• This evolution equation states that “the rate of change inside a control volume balances the fluxes, plus sources minus sinks”.
• In differential form the equation becomes:
ioj
jj
iojjj
j
SSxccu
xtc
SSxc
xxcu
tc
)(.. SiSodAncnudVt surfacevc
Let’s split the channel into elementary volumes
Cell i-1 Cell i Cell i+1
And apply the equation to each of them:
dVolkcdAnccdVt vc
.
If the property can be considered uniform inside each cell and along each surface:
xAVol
kcVolxccA
xccA
tccVol iiiiti
tti
11
Where A is the area of the cross section between the elementary volumes, assumed constant in the academic example.
Numerical solution
• Dividing all the equation by the volume:
xAVol
kcVolxccA
xccA
tccVol iiiiti
tti
11
kcxcc
xcc
tcc iiiiti
tti
21
21
• In this equation we have two variables, , and an extra variable, c, which time we have not defined. Are the actual concentrations and are the concentrations that we want to calculate. The concentration c is the concentration used to calculate the fluxes and decay. What is it?
ttic
tictic
ttic
Flux calculation
• The flux equation gives quantity per unit of time. When we equation as:
• We are computing the amount that crossed the surface during a time period ( ) and dividing it by the length of the time. The most convenient time to allocate to c is the middle of the time interval:
kcxcc
xcc
tcc iiiiti
tti
21
21
t
2
tti
ti ccc
The equation becomes
21
21
21
21
22
222
xcc
xcc
cckxcc
xcc
tcc
tti
tti
tti
tti
ti
tti
ti
ti
ti
ti
ti
tti
222222 21
21
21
21
ti
ti
ti
ti
ti
tti
tti
tti
tti
tti
ti
tti kc
xcc
xcckc
xcc
xcc
tcc
ti
ti
ti
tti
tti
tti c
xtck
xtc
xtc
xtck
xtc
xt
1221212212 22221
222221
2
ti
ti
ti
tti
tti
tti cDckDcDcDckDcD 1111 22
1222
12
This equation has 3 unknowns. The all channel requires the resolution of a system of equations in each time step.
Explicit calculation
• That can be solved explicitly, but has stability problems.
• If we had assumed:ticc
• We would have got: t
iti
ti
tti DcckDDcc 11 21
Implicit calculation
• That has less computation than the “average” approach, but still requires the resolution of a system of equations.
• If we had assumed:tt
icc
• We would have got:
ti
ti
tti
tti cDcckDDc
11 21
Results (D<0.5)D= 0.25
DT= 100
dx 20
difusividate 1
Time k= 0
i-3 i-2 i-1 i i+1 i+2 i+3
0 0 0 0 0 0 1 0 0 0 0 0
100 0 0 0 0.25 0.5 0.25 0 0 0
200 0 0 0.06 0.25 0.38 0.25 0.06 0 0
300 0 0.02 0.09 0.23 0.31 0.23 0.09 0.02 0
400 0 0.03 0.11 0.22 0.27 0.22 0.11 0.03 0
500 0.01 0.03 0.12 0.21 0.25 0.21 0.12 0.04 0.01
The results evolve as expected.
Results (D=0.75)D= 0.75
DT= 300
dx 20
difusividate 1
Time k= 0
i-3 i-2 i-1 i i+1 i+2 i+3
0 0 0 0 0 0 1 0 0 0 0 0
300 0 0 0 0.75 -0.5 0.75 0 0 0
600 0 0 0.56 -0.8 1.38 -0.8 0.56 0 0
900 0 0.42 -0.8 1.83 -1.8 1.83 -0.8 0.42 0
1200 0.32 -0.8 2.11 -2.9 3.65 -2.9 2.11 -0.8 0.32
1500 -0.79 0.71 -3.9 5.77 -6.2 5.77 -3.9 2.24 -0.8
The results are strange. Concentration bigger than initial are obtained. Negative concentrations are also obtained….The method is unstable
Stability
• The method became unstable because the parenthesis becomes negative when D>0.5;
• When the parenthesis changes sign, the effect of ci at time t changes. If it is positive the larger is the initial concentration the larger is the subsequent concentration. If it is negative the larger is the initial concentration the smaller is the subsequent, which is physically impossible.
• This is the limitation of the explicit methods. They are conditionally stable. In pure diffusion problems the D (the diffusion number) must be smaller than 0.5. If other processes exist (e.g. k≠0) D must be samller.
ti
ti
ti
tti DcckDDcc 11 21
Implicit methods
• Implicit (as well as semi-implicit) methods are more difficult to program, but do not have stability limitations allowing larger values for D, meaning that we can combine small grid sizes with large time steps, while in explicit methods the time step is associated to the square of the spatial step.