Solve PDEs with Events over Regions

Model thermostat-controlled heat generation in a room with three insulated walls and a glass front

Set up the parameters including the region to solve over, the temperature outside as a function of time, the physical constants, and the boundary conditions:

In[1]:=

Ω=Rectangle[{0,0},{3/2,1}];outsideTemp[t_]:=15+10*Sin[2πt/24];kd=0.78;Ld=0.05;Γ=NeumannValue[Ld/kd*(outsideTemp[t]-u[t,x,y]),{x0}];

A heater load is turned up or down at an event, as defined by the following function:

In[2]:=

heaterLoad=26;heater[upQ_,t_,tEvent_]:=If[upQ1,Min[20*Max[(t-tEvent),0],1],1-Min[20*Max[(t-tEvent-1/8),0],1]]*heaterLoad

The PDE models heat diffusion through air while generating heat inside a circle

(x-1/2)

(y-1/2)

≤1/25

and losing heat through the glass window:

In[3]:=

ρ=1.225;Cp=1005.4;With[{heating=heater[a[t],t,eventT[t]]},pde=D[u[t,x,y],t]-ρ*Cp*Laplacian[u[t,x,y],{x,y}]If[(x-1/2)^2+(y-1/2)^2≤(2/10)^2,heating,0]+Γ];

If the thermostat at position

(1.25,1.25)

measures a temperature below/above a trigger, and if the discrete variable

changed, the heater is switched on/off:

In[4]:=

triggerLow=18;triggerHigh=20;events={a[0]1,eventT[0]0,WhenEvent[u[t,1.25,.25]<triggerLow,{eventT[t],a[t]}{If[a[t]0,t,eventT[t]],1}],WhenEvent[u[t,1.25,.25]>triggerHigh,{eventT[t],a[t]}{If[a[t]1,t,eventT[t]],0}]};

Monitor the time integration of the PDE with initial condition equal to the outside temperature:

In[5]:=

eqn={pde,u[0,x,y]outsideTemp[0],events};res=Monitor[NDSolveValue[eqn,{u,a},{t,0,2*24},{x,y}∈Ω,DiscreteVariables{eventT[t],a[t]},EvaluationMonitor(monitor=Row[{"t = ",CForm[t]}])],monitor]

Out[5]=



InterpolatingFunction



Domain: {{0.,48.},{0.,1.5},{0.,1.}}

Output: scalar

Data not in notebook. Store now



,InterpolatingFunction

Domain: {{0.,48.}}

Output: scalar



Visualize the temperature measured at the thermostat, the outside temperature, and the triggers for the heater. A blue background is shown where the heater is on:

In[6]:=

hp=Plot25res2[t],{t,0,2*24},

;ShowPlot{res〚1〛[t,1.25`,0.25`],outsideTemp[t],18,20},{t,0,224},

...

,hp,

...



Out[6]=

	Thermostat
	Outside

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Contributed by: Wolfram Staff

Solve PDEs with Events over Regions

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