Systems Dynamics, Computer Experiments and Simulation Technologies (Vensim)

Models based on ordinary differential
___equations

The models built basing on the studies of the processes and
phenomena described by ordinary differential equations are
considered in this part.

2

1.1. NEWTON’S HEAT CONDUCTIVITY LAW
STUDY

If the temperature difference between the studied object T, e.g., a
cup of coffee and the environment Ts is not very large, the rate of
temperature change of the object can be considered proportional to
the given temperature difference. This statement can be written in
the form of the differential equations as follows:

f = – r – ( r – T s ) (1.1)

where r is the cooling coefficient, dt is the time discretization step,
the minus sign allows avoiding the unphysical increase in body
temperature at T > Ts. Eq.1.1 is called Newton’s law of thermal
conductivity.

3

1.1. NEWTON’S HEAT CONDUCTIVITY _ LAW
STUDY

Lets consider three models to study it:

• Model 1 – describing the numerical solution of Eq.1.1.

• Model 2, which is a modification of Model 1 that takes into
account the case of instantaneous changes of the body temperature,
e.g. by 10 ° C at a given point in time.

• Model 3, in which the cooling coefficient r is “adjusted” in
accordance with the experimental data.

4

Model 1. The numerical solution of
ordinary ̂differential equations (Eg. 1 . 1 ).

To solve the differential equation 1.1:
dT , A

r – f f – T i )

we need to find dependence T as a function of time t, i.e. T(t).

The influence diagram (Fig. 1.1) is used for the numerical solution
ofEq.1.1: To

Fig. 1.1. Diagram of Newton’s heat equation influence. T0- initial 7(0)
temperature of studying body.

5

Vensim equations for Model 1

(1) “dT/dt”=-r*(T-Ts)
(2) FINAL TIME =60
(3) INITIAL TIME =0
(4) r=0.1
(5) SAVEPER = TIME STEP (1.2)
(6) T= INTEG (“dT/dt”, To)
(7) TIME STEP =1
(8) To=100
(9) Ts=25

6

Graphical solution for Model 1

Fig. 1.2 The solution plot for Equation 1.1. at r=0.1,
Ts=25°C, T(0)=100°C

Tasks To Do. Task 1 The numerical
solution of ordinary – -differential equations-■

Task 1. Base on equation — = — r – ( T — T s) construct a model as

shown in Fig. 1.1. and provide answers to the following questions:

1. What time is required to cool the object from 90°C to 75°C?

2. When the temperature difference between the studied object and
the environment will decrease by 1/2, 1/4, 1/8 from the initial one?

8

Model 2. The acceleration of the cooling
by the addition of the coolant

Let’s assume that in the moment of time time of mix the object cools
down by the value of T of mix instantaneously.

T of mix To

Fig 1.3. Model 2, T of mix is the value the body (coffee) is cooled for,
time of mix is the time during which the coolant is added

9

Model 2. The acceleration of the cooling
by the addition of the coolant.

The graphical solution of the given task is shown in Fig. 1.4.

Fig. 1.4. The graphical solution of the Model 2 at T of
mix =10°C, time of mix=6 and 16 min

10

Vensim Equations for Model 2

Mathematical equations for Model 2 using VenPLE language are
different from those for Model 1 and are described by Eq.1.3:

T= INTEG (“dT/dt”-IF THEN ELSE( time of mix=Time, T of mix, 0), To) T
of mix = 10 (1.3)
time of mix = 1

11

Tasks To Do. Task 2 The numerical
solution of ordinary differential
——equa tmns^wrth changing conditions
——
Task 2. Construct a model basing on Fig.1.3. Perform a modelling and
answer to the following questions:

Let’s assume that cooling by adding a coolant decreases the
temperature by 10°C instantaneously. In what case the temperature
will decrease from 95°C to 75°C faster:

a) if the coolant is added immediately or
b) wait until the temperature decreases to 85°C and add coolant
then?

12

Model 3. The cooling coefficient r is “adjusted” in
accordance with the experimental data_______

Let’s find the parameters for the Model 1 that are in an agreement with
experimental data (Table 1) of the coffee cup cooling at the ambient
temperature of Ts=25°C.

Table 1. Experimental data for the coffee cup cooling.

t, min T°C t, min T°C t, min T°C t, min T°C
0 100 10 65 20 54.1 30 46.1
1 90 11 63.9 21 52.9 31 45.8
2 86.5 12 61.9 22 52 32 45
3 82.5 13 61 23 51 33 44.5
4 79 14 60.1 24 50.2 34 44.1
5 76.5 15 58.9 25 49.4 35 43.9
6 74 16 57.4 26 48.9 36 43.1
7 72 17 55.9 27 48 37 43
8 69.5 18 55 28 47.2 38 42.1
9 67.1 19 54.3 29 47 39

40

42

41.1

Model 3. The diagram of the influence

The diagram of the influence of Model 3 and Plots are shown
in Fig.1.5. and Fig.1.6.

O

table 1

T of mix
To

TT

dTdt

4
Ts time of mix

T-experiment

T

Time (Minute)

T Mcxlell ———————————
T: Model: ——————————–
•T-expericMEt’ : McxJtL? •■tÄ

Fig.1.5. The diagram of
the influence of Model 3

Fig.1.6. Plots of the experimental
and modelled data at r=0.1

14

Model 3. Vensim Equations for Model 3

Mathematical equations using VenPLE language for Model 3
different from Model 1 are described by Eq.1.4:

FINAL TIME = 40
“T-experiment”=tablel(Time)
tablel(
[(0,0)-

(40,100)],(0,100),(1,90),(2,86.5),(3,82.5),(4,79),(5,76.5),(6,74),
(7,72),(8,69.5),(9,67.1),(10,65),(11,63.9),(12,61.9),(13,61),(14,60.1),
(15,58.9),(16,57.4),(17,55.9),(18,55),(19,54.3),(20,54.1),(21,52.9),
(23,51),(24,50.2),(25,49.4),(26,48.9),(27,48),(28,47.2),(29,47),
(30,46.1),(31,45.8),(32,45),(33,44.5),(34,44.1),(35,43.9),(36,43.1),
(37,43),(38,42.1),(39,42),(40,41.1))

How to fill up the tablel see next slide.
15

Model 3. Vensim Equations
______________variable type – Lookup

To use variable tablel we need to choose Type – Lookup:
Edit: tablel

Variable Information
Name tablel

^ype Lookup 7S Sub-Type _d

Units ” ——————— ▼ Check Units

As Graph I

Supplemen

Group Newton 03-2 t Mm Max

Equations

C
[(0.0)-(40.100)],(0,100).(1,90),(2,86.5).(3,82.5).(4,
(11,63,9),(12,61.9),(13,61),(14,60.1),(15,58.9),(16,5
(23,51),(24,50.2),(25,49 4).(26, 48. 9).(27,48).(28,47.
(34.44.1),(35.43.9),(36.43.1),(37,43),(3S,42.1),(39,4

To fill up this table the best way is to choose As Graph method
Edit: tablel

Variable Information
Name |tablel

Type Lookup w Sub-Type J
Units ^ Check Units Supplemen

Group Newton 03-2 7 Min Max

Equations

(
[(0,0) — (40,100)],(0,100).(1,905,(2,86 5),(3,82 . 5),(4,
(11,63.9),(12,61.9),(13,61),(14,60.1),(15.50.9),(16,5
(23,51),(24,50.2),(25,49 4).(26.48.9).(27,48).(28,47.
(34.44.1),(35.43.9),(36.43.1),(37, 43), (38. 42.1), (39,

4

16

Model 3. Vensim Equations
variable type – Lookup

Then we need to fill up Input/Output columns base on Table 1
(Slide 13), where Input corresponds to t, min and Output – T°C:

Graph Lookup – tablel

nput Dutput

m 100
1 90
2 86 5

3 82.5

4 79

5 76.5

6 74

7 72

8 69.5

9 671

10 65

Print

New

i

/
^
/

Y-max:

100 H

Y-min:

F“
I r

X-min:|0

– >:=18 35 y=-3.07 X-max:|4D .1 Reset Scaling |

OK Clear Points Clear All Points
* –

Cur->Ref Clear Reference | Ref->Cur Cancel
«

17

Tasks To Do. Task 3 Adjusting a model in
accordance with the experimental data

Task 3. Find r value that fits the corresponding real process the most
by performing the simulation of the Model 3 Fig.1.7.

Fig.1.7. Diagram of the experimental and modelled data at r=? (find the
value by performing the modelling)

18

NEWTON’S HEAT CONDUCTIVITY LAW
STUDY

Tasks to do
1. You need to create 3 Models based on Vensim software.

Model 1 for Task 1,
Model 2 for Tasks 2 and
Model 3 for Task 3

2. Provide all simulations and get all graphs that you may find the
lecture.

3. Save these graphs as INPUT/OUTPUT objects.
4. Add your Answer for my questions as Comment in Vensim model

files.
5. Upload the model’s files into the Moodle system as one zip

archive with your name and surname as your answers.
P.S.
You can Free Download Vensim PLE version from
https://vensim.com/free-download/

https://vensim.com/free-download/

equations
1.1. NEWTON’S HEAT CONDUCTIVITY LAW STUDY
1.1. NEWTON’S HEAT CONDUCTIVITY _ LAW STUDY

ordinary^ differential equations (Eg.1.1).
Tasks To Do. Task 1 The numerical solution of ordinary ■ – -differential equations-
Tasks To Do. Task 2 The numerical solution of ordinary differential equa tmns^wrth changing conditions
Model 3. The cooling coefficient r is “adjusted” in accordance with the experimental data

Model 3. The diagram of the influence
4

Model 3. Vensim Equations for Model 3
Model 3. Vensim Equations variable type – Lookup
Model 3. Vensim Equations variable type – Lookup
Tasks To Do. Task 3 Adjusting a model in accordance with the experimental data
NEWTON’S HEAT CONDUCTIVITY LAW
Tasks to do