Heat Conduction Model

This example demonstrates the thermal response of two masses connected by a conducting element. The two masses have the same heat capacity but different initial temperatures (T1=100 [°C], T2=0 [°C]). The mass with the higher temperature will cool off, while the mass with the lower temperature heats up. They will each asymptotically approach the calculated temperature TfinalK that results from dividing the total initial energy in the system by the sum of the heat capacities of each element.

using ModelingToolkitStandardLibrary.Thermal, ModelingToolkit, OrdinaryDiffEq, Plots
using ModelingToolkit: t_nounits as t

@mtkmodel HeatConductionModel begin
    @parameters begin
        C1 = 15
        C2 = 15
    end
    @components begin
        mass1 = HeatCapacitor(C = C1, T = 373.15)
        mass2 = HeatCapacitor(C = C2, T = 273.15)
        conduction = ThermalConductor(G = 10)
        Tsensor1 = TemperatureSensor()
        Tsensor2 = TemperatureSensor()
    end
    @equations begin
        connect(mass1.port, conduction.port_a)
        connect(conduction.port_b, mass2.port)
        connect(mass1.port, Tsensor1.port)
        connect(mass2.port, Tsensor2.port)
    end
end

@mtkbuild sys = HeatConductionModel()
prob = ODEProblem(sys, Pair[], (0, 5.0))
sol = solve(prob)

T_final_K = sol[(sys.mass1.T * sys.C1 + sys.mass2.T * sys.C2) / (sys.C1 + sys.C2)]

plot(title = "Thermal Conduction Demonstration")
plot!(sol, idxs = [sys.mass1.T, sys.mass2.T],
    labels = ["Mass 1 Temperature" "Mass 2 Temperature"])
plot!(sol.t, T_final_K, label = "Steady-State Temperature")