Newton cooling from scratch

We will simulate a basic cooling system follwoing Newton’s law. For that purpose, OpenModelica is a standard solution.

You can refer to https://mbe.modelica.university/ website to learn and play with Modelica PDE solver.

Requirements

  • install OpenModelica on your platform:
    • from OpenModelica website,
    • or using some package manager (eg. for debian/ubuntu): 
      for deb in deb deb-src; do echo "$deb http://build.openmodelica.org/apt `lsb_release -cs` release"; done | sudo tee /etc/apt/sources.list.d/openmodelica.list
wget -q http://build.openmodelica.org/apt/openmodelica.asc -O- | sudo apt-key add -
apt update && apt install openmodelica
  • check that omc command will be recognized in your PATH

Problem setup

We will then work on the ‘NewtonCooling’ example, which solves a basic PDE on temperature, provided in the Funz-Modelica/samples directory:

// @ref http://book.xogeny.com/behavior/equations/physical/
model NewtonCooling "An example of Newton's law of cooling"
  parameter Real T_inf=25 "Ambient temperature";
  parameter Real T0=90 "Initial temperature";
  parameter Real h=0.7 "Convective cooling coefficient";
  parameter Real A=1.0 "Surface area";
  parameter Real m=0.1 "Mass of thermal capacitance";
  parameter Real c_p=1.2 "Specific heat";
  Real T "Temperature";
initial equation
  T = T0 "Specify initial value for T";
equation
  m*c_p*der(T) = h*A*(T_inf-T) "Newton's law of cooling";
end NewtonCooling;

Our engineering goal is to adjust cooling convection in order to control the minimum temperature reached. So, in order to ‘Funzify’ this model, we will just replace the numerical value of the convection coefficient by a parametrized expression:

// @ref http://book.xogeny.com/behavior/equations/physical/
model NewtonCooling "An example of Newton's law of cooling"
  parameter Real T_inf=25 "Ambient temperature";
  parameter Real T0=90 "Initial temperature";
  parameter Real h=$convection "Convective cooling coefficient";
  parameter Real A=1.0 "Surface area";
  parameter Real m=0.1 "Mass of thermal capacitance";
  parameter Real c_p=1.2 "Specific heat";
  Real T "Temperature";
initial equation
  T = T0 "Specify initial value for T";
equation
  m*c_p*der(T) = h*A*(T_inf-T) "Newton's law of cooling";
end NewtonCooling;

… and now play with this ‘functional’ wrapping …

Funz

Install

Install Funz & Modelica plugin:

  • get Funz-Modelica distribution: https://github.com/Funz/plugin-Modelica/releases/… (which also include some basic algorithms like gradient descent optimization and root finding)
  • unzip in current directory: unzip Funz-Modelica.zip
  • move in Funz directory: cd Funz-Modelica

Wake up the 3 Funz ‘daemons’ which will provide calculation services: ./FunzDaemon_start.sh 3

We will use Funz.sh (or Funz.bat) to launch Funz calculations from command-line bash (or cmd.exe).

Basic parametric run

Launching 6 calculations for different convection values (0.5, 0.6, 0.7, 0.8, 0.9, 1.0) is done as follows:

./Funz.sh Run -m Modelica -if samples/NewtonCooling.mo.par -iv convection=0.5,0.6,0.7,0.8,0.9,1.0 -oe "min(T)" -v 0 -pf "convection" "min(T)" "info"

and returns:

| convection | min(T)            | info          |
|------------|-------------------|---------------|
| 0.5        | 25.98653154436357 | Run succeded. |
| 0.6        | 25.43373687652521 | Run succeded. |
| 0.7        | 25.19937564256052 | Run succeded. |
| 0.8        | 25.08828738995784 | Run succeded. |
| 0.9        | 25.04045790290655 | Run succeded. |
| 1.0        | 25.01947293170708 | Run succeded. |

Algorithm-driven root finding

Now we will ask a (quite simple) algorithm to find the convection value leading to min(T) = 25.2 (with relative precision of 0.01 on convection value):

./Funz.sh RunDesign -m Modelica -if samples/NewtonCooling.mo.par -iv convection=[0.5,1.0] -oe "min(T)" -d Brent -do ytarget=25.2 ytol=0.01 -v 0


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