# Cooling Revisited - NewtonCoolingSteadyThenDynamic

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The Modelica By Example target code (using 'initial equation') is:
``````
"Dynamic cooling example with steady state conditions"
type Temperature=Real(unit="K", min=0);
type ConvectionCoefficient=Real(unit="W/(m2.K)", min=0);
type Area=Real(unit="m2", min=0);
type Mass=Real(unit="kg", min=0);
type SpecificHeat=Real(unit="J/(K.kg)", min=0);

parameter ConvectionCoefficient h=0.7 "Convective cooling coefficient";
parameter Area A=1.0 "Surface area";
parameter Mass m=0.1 "Mass of thermal capacitance";
parameter SpecificHeat c_p=1.2 "Specific heat";

Temperature T_inf "Ambient temperature";
Temperature T "Temperature";
initial equation
der(T) = 0 "Steady state initial conditions";
equation
if time<=0.5 then
T_inf = 298.15 "Constant temperature when time<=0.5";
else
T_inf = 298.15-20*(time-0.5) "Otherwise, decreasing";
end if;
m*c_p*der(T) = h*A*(T_inf-T) "Newton's law of cooling";
``````

The exported Modelica code is:

``````
parameter ConvectionCoefficient h(start=0.7,fixed=true);
parameter Area a(start=1.0,fixed=true);
parameter Mass m(start=0.1,fixed=true);
parameter SpecificHeat c_p(start=1.2,fixed=true);
Temperature t_inf;
Temperature t;
TemperatureRate dt(start=0.0,fixed=true);
equation
m*c_p*der(t)=h*a*(t_inf-t);
if time<=0.5 then
t_inf=298.15;
else
t_inf=298.15-20*(time-0.5);
end if;
dt=der(t);
type ConvectionCoefficient=Real(unit="W/(m2.K)");
type Area=Real(unit="m2");
type Mass=Real(unit="kg");
type SpecificHeat=Real(unit="J/(K.kg)");
type Temperature=Real(unit="K");
type TemperatureRate=Real(unit="K/s");
So an additional initialised variable `dt` is used:
``````
Where `dt = der(t)`.