Transcript Dynamic model

Dynamic plant uptake modeling
Stefan Trapp
Steady-state vs. dynamic models
Steady-state considerations: simple & small data need
However: often emission pattern is non-steady, e.g.:
 non-steady plant growth (logistic)
 pesticide spraying
 application of manure and sewage sludge on
agricultural fields
In these scenarios: steady-state solutions are not valid, and
dynamic simulation is required.
Dynamic input patterns
•
•
Three different types of input, namely
pulse input (pesticide spraying)
•
•
constant input (deposition from air)
irregular input (this we cannot solve --> use numerical integration)
Dynamic models
Designed for
- repeated input
- dynamic growth
- pesticides
- manure or sewage appl.
- 1 year or 10 year
- easy to handle (excel)
Dynamic models
5.0E+00
C(t)
4.0E+00
3.0E+00
2.0E+00
1.0E+00
1.0E-06
0
10
20
30
40
50
60
Real time (d)
C3 Stem
C2 root
C1 soil bioavailable
C4a Leaves
C4 Fruit
● Repeated pulse input from soil or air or constant emission
● Logistic growth of plants (here: summer wheat)
● n variable periods (30 in practice)
Dynamic Model
Differential equation system
In words
soil:
change of mass = + Input - degradation - uptake into plants
roots:
change of mass = + uptake from soil - loss to stem - degradation
stem:
change of mass = + uptake from roots - loss to leaves (fruits) - deg
leaves: change of mass = +uptake from stem ± exchange air - degradation
fruits:
change of mass = +uptake from stem ± exchage air - degradation
Differential equation system
soil
roots
(stem)
leaves
fruits
dCSoil
CSoil
 kdegCSoil 
 Q  input
dt
M Soil K SW
dCR CSoil

 Q / M R  CR / K RW  Q / M R  k R  CR
dt
K SW
dCSt
C
 Q / M St  R  CSt / K StW  Q / M St  k St  CSt
dt
K RW
AL  vdep
dCL
QL
AL  g  1000 L m 3

 CSt 
 CA 
 CL  k L  CL
dt M L  K StW
ML
K LA  M L
dCF
QF
AF  g
AF  g  1000 L m 3

 CSt 
 CA 
 CF  k F  CF
dt
M F  K StW
MF
K FA  M F
Cascade of compartments
Mass balance:
"The change of mass in
tank 2 is what flows out of
tank 1 minus what flows
out of tank 2"
dm1
 k1 m1
dt
dm2
 k1m1  k 2 m2
dt
Differential equation system
1 soil
dC1
 k1C1  b1
dt
2 roots
dC2
 k12C1  k 2 C2  b2
dt
3 stem
dC3
 k 23C2  k 3C3  b3
dt
4a leaves
dC 4
 k 34C3  k 4 C 4  b4
dt
4b fruits
dC 4
 k 34C3  k 4 C 4  b4
dt
The system written in a schematic
way:
Each DE always relates to the DE
before, but not to any other DE
Structure of the multi-cascade crop model
transfer rate constants kij (d-1)
loss rate constant ki (d-1)
constant external input b (mg kg-1 d-1).
Dynamic Model
Same processes, same differential equations,
but formulated as matrix
 k1
 
dC  k12

dt  0
 0
0
0
 k2
0
k 23
 k3
0
k34
1 is soil
2 is roots
3 is stem
4 is leaves or fruits
b is the input vector
0 
 
0 
C  b
0 
 k 4 
k1 loss rate k12 transfer rate
k2 loss rate k23 transfer rate
k3 loss rate k34 transfer rate
k4 loss rate
Cascade
Steady-state solution
Set dC/dt (left hand) to zero. Then
C1 (t  ) 
C2 (t  ) 
I2
k
 12 C1 (t  )
k2 M 2 k2
C3 (t  ) 
etc.
I1
b
 1
k1 M 1 k1
I3
k
 23 C2 (t  )
k3 M 3 k3
Conc. = Input / loss
Analytical solution for pulse input, i.e. C(0) ≠ 0
C1 (t )  C1 (0)  e  k1t
 e  k1t
e  k 2t 
  C2 (0)  e k2t
C2 (t )  k12C1 (0)  

 (k2  k1 ) (k1  k2 ) 


e  k1t
e  k2t
e  k3t
C3 (t )  k12k 23C1 (0)



(
k

k
)(
k

k
)
(
k

k
)(
k

k
)
(
k

k
)(
k

k
)
2
1
2
3
3
1
3
2 
 1 2 1 3
 e k2t
e k3t 

 k23C2 (0)  

 ( k3  k 2 ) ( k 2  k3 ) 
 C3 (0)  e k3t
etc. …
Cascade with constant input
Analytical solution for all t
C1 t  


b1
1  e  k1t  C1 0  e  k1t
k1




C2 t   A  e  k1t  e  k2t  B  1  e  k2t  C2 0  e  k2t






C3 t   D  e  k1t  e  k 3t  E  e  k 2 t  e  k 3t  F  1  e  k 3t  C3 0  e  k 3t
A
C1 0k12 k1  k12b1
k 2  k1 k1
k12b1  k1b2
B
k1k 2
DA
k 23
k3  k1
k C 0   A  B 
E  23 2
k3  k 2
k 23 B  b3
F
k3
That's what you always wanted to know
about math, wasn't it?
Questions?
Principle of superposition
Concentrations are additive
We can thus calculate several subsequent periods with different
values, and the output from one period is the input to the next.
This allows to simulate non-constant conditions.
Our "cascade model" has by default 24 periods to 5 days (= 120
days, i.e. one vegetation period), but this is variable.
Principle of superposition
Figure: Concentrations are additive
Plant growth
Most annual crops show a logistic growth curve
initial growth is exponential
towards ripening, growth slows down
and finally stops
Change of plant mass M [kg]:

dM
M 


 k  M 1 
dt
 Mmax 
k
First-order rate constant
(for exponential growth)
[1/d]
Mmax
Maximum plant mass
[kg]
Plant mass as a function of time
M t  
Mmax
M
 k t
1   max  1  e
 M0

M0
Initial plant mass
[kg]
Plant growth and transpiration
Growth and transpiration of plants are related by
the water use efficiency (kg plant / L water) or the
transpiration coefficient TC (L water / kg plant).
Typical values range between 200 and 1000 L/kg dry weight
Default value for TC is 100 L/kg fresh weight.
dM
Q  TC 
 TC  k  M
dt

M 
1 

 Mmax 
Q
Transpiration
[L/d]
TC
Transpiration coefficient
[L/kg dw]
In our model, transpiration takes place only when plants are growing
Plant growth and transpiration
Data obtained from agricultural handbooks
(summer wheat)
Annual seed plant
data related to 1 m2
- Initial mass 10-4 kg (0.1 g for seeds)
- Growth rate constant k = 0.1 d-1
(doubling time ≈ 1 week)
- Final mass 1 kg
Transpiration coefficient TC = 50 L/kg fw
(water content green plans ≈ 90%)
Plant growth and transpiration
Standard scenario: summer wheat
M t  
Mmax
M
 k t
1   max  1  e
 M0

Q  TC 
Maximum transpiration Qmax is at ½
with
at time
Qmax  1  TC  k  Mmax
4

1 
1

t   ln
k  Mmax M0  1
dM
 TC  k  M
dt
Mmax (inflection point)

M 
1 

M
max 

Plant growth and transpiration
Dynamic model: Default scenario
Annual seed plant
Growth is exponential
for t < 70 d
Absolute growth & transpiration
peak at t = 92 d
Growth almost stops for t > 135 d
Biomass M and transpiration Q
of summer wheat
= phase in which fruit or corn ripe
leaves decay and plants dry out
Reading:
Example simulation for a repeated pesticide
application
0.8
C(t)
0.6
0.4
0.2
0.0
0
5
10
15
20
25
30
35
Time (d)
C3 Stem
C2 root
C4 Fruit
C1 soil
Repeated application of insecticide
by drip irrigation to soil
Comparison to measured data
Model result before calibration
0.25
C Fruit
0.2
0.15
0.1
0.05
0
0
5
10
15
20
Time (d)
Measured
Model
25
30
35
Comparison to measured data
C Fruit
After fit of two parameters (temperature, soil depth)
0.3
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
20
Time (d)
Measured
Model
25
30
35
More reading:
Lucky You - you survived this part.
Questions?