#!/usr/bin/env python # -*- coding: utf-8 -*- """ *********************************************************************************** tutorial4.py DAE Tools: pyDAE module, www.daetools.com Copyright (C) Dragan Nikolic *********************************************************************************** DAE Tools is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License version 3 as published by the Free Software Foundation. DAE Tools is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with the DAE Tools software; if not, see <http://www.gnu.org/licenses/>. ************************************************************************************ """ __doc__=""" This tutorial introduces the following concepts: - Discontinuous equations (symmetrical state transition networks: daeIF statements) - Building of Jacobian expressions In this example we model a very simple heat transfer problem where a small piece of copper is at one side exposed to the source of heat and at the other to the surroundings. The lumped heat balance is given by the following equation:: rho * cp * dT/dt - Qin = h * A * (T - Tsurr) where Qin is the power of the heater, h is the heat transfer coefficient, A is the surface area and Tsurr is the temperature of the surrounding air. The process starts at the temperature of the metal of 283K. The metal is allowed to warm up for 200 seconds, when the heat source is removed and the metal cools down slowly to the ambient temperature. This can be modelled using the following symmetrical state transition network: .. code-block:: none IF t < 200 Qin = 1500 W ELSE Qin = 0 W The temperature plot: .. image:: _static/tutorial4-results.png :width: 500px """ import sys from time import localtime, strftime from daetools.pyDAE import*# Standard variable types are defined in variable_types.py from pyUnits import m, kg, s, K, Pa, mol, J, WclassmodTutorial(daeModel):def__init__(self, Name, Parent=None, Description=""): daeModel.__init__(self, Name, Parent, Description) self.m=daeParameter("m", kg, self, "Mass of the copper plate") self.cp=daeParameter("c_p", J/(kg*K), self, "Specific heat capacity of the plate") self.alpha=daeParameter("α", W/((m**2)*K), self, "Heat transfer coefficient") self.A=daeParameter("A", m**2, self, "Area of the plate") self.Tsurr=daeParameter("T_surr", K, self, "Temperature of the surroundings") self.Qin=daeVariable("Q_in", power_t, self, "Power of the heater") self.T=daeVariable("T", temperature_t, self, "Temperature of the plate")defDeclareEquations(self): daeModel.DeclareEquations(self) # If equation expressions are long the computational performance can be improved by # creating and storing the derivative expressions in the equation. # This can be achieved by setting the boolean property BuildJacobianExpressions to True. # Derivative expressions are printed later, once the simulation is initialised. eq=self.CreateEquation("HeatBalance", "Integral heat balance equation") eq.BuildJacobianExpressions=True eq.Residual=self.m()*self.cp()*dt(self.T())-self.Qin()+self.alpha()*self.A()*(self.T()-self.Tsurr()) # Symmetrical STNs in DAE Tools can be created by using IF/ELSE_IF/ELSE/END_IF statements. # These statements are more or less used as normal if/else if/else blocks in all programming languages. # An important rule is that all states MUST contain the SAME NUMBER OF EQUATIONS. # First start with the call to IF( condition ) function from daeModel class. # After that call, write equations that will be active if 'condition' is satisfied. # If there are only two states call the function ELSE() and write equations that will be active # if 'condition' is not satisfied. # If there are more than two states, start a new state by calling the function ELSE_IF (condition2) # and write the equations that will be active if 'condition2' is satisfied. And so on... # Finally call the function END_IF() to finalize the state transition network. # There is an optional argument eventTolerance of functions IF and ELSE_IF. It is used by the solver # to control the process of discovering the discontinuities. # Details about the eventTolerance purpose will be given for the condition Time < 200, given below. # Conditions like Time < 200 will be internally transformed into the following equations: # time - 200 - eventTolerance = 0 # time - 200 = 0 # time - 200 + eventTolerance = 0 # where eventTolerance is used to control how far will solver go after/before discovering a discontinuity. # The default value is 1E-7. Therefore, the above expressions will transform into: # time - 199.9999999 = 0 # time - 200 = 0 # time - 200.0000001 = 0 # For example, if the variable 'time' is increasing from 0 and is approaching the value of 200, # the equation 'Q_on' will be active. As the simulation goes on, the variable 'time' will reach the value # of 199.9999999 and the solver will discover that the expression 'time - 199.9999999' became equal to zero. # Then it will check if the condition 'time < 200' is satisfied. It is, and no state change will occur. # The solver will continue, the variable 'time' will increase to 200 and the solver will discover that # the expression 'time - 200' became equal to zero. It will again check the condition 'time < 200' and # find out that it is not satisfied. Now the state ELSE becomes active, and the solver will use equations # from that state (in this example equation 'Q_off'). # But, if we have 'time > 200' condition instead, we can see that when the variable 'time' reaches 200 # the expression 'time - 200' becomes equal to zero. The solver will check the condition 'time > 200' # and will find out that it is not satisfied and no state change will occur. However, once the variable # 'time' reaches the value of 200.0000001 the expression 'time - 200.0000001' becomes equal to zero. # The solver will check the condition 'time > 200' and will find out that it is satisfied and it will # go to the state ELSE. # In this example, input power of the heater will be 1500 Watts if the time is less than 200. # Once we reach 200 seconds the heater is switched off (power is 0 W) and the system starts to cool down. # Regarding the logical expression, we can use simply use: # Time() < 200 # However, in order to be unit-consistent we may use also Constant() function that takes # as an argument quantity object and returns adouble. This way the unit consistency is always checked. self.IF(Time()<Constant(200*s), eventTolerance=1E-5) eq=self.CreateEquation("Q_on", "The heater is on") eq.Residual=self.Qin()-Constant(1500*W) self.ELSE() eq=self.CreateEquation("Q_off", "The heater is off") eq.Residual=self.Qin() self.END_IF()classsimTutorial(daeSimulation):def__init__(self): daeSimulation.__init__(self) self.m=modTutorial("tutorial4") self.m.Description=__doc__defSetUpParametersAndDomains(self): self.m.cp.SetValue(385*J/(kg*K)) self.m.m.SetValue(1*kg) self.m.alpha.SetValue(200*W/((m**2)*K)) self.m.A.SetValue(0.1*m**2) self.m.Tsurr.SetValue(283*K)defSetUpVariables(self): self.m.T.SetInitialCondition(283*K)defprintJacobianExpressions(simulation): # Print the Jacobian expressions for all equations import pprintforeqinsimulation.m.Equations: print(eq.CanonicalName, ':')foreeiineq.EquationExecutionInfos: print(' %s:'%eei.Name) # dictionary {overall_index : (block_index,derivative_node)}foroi, (bi,node)ineei.JacobianExpressions.items(): print(' %d : %s'%(bi, node))defrun(**kwargs): simulation=simTutorial()returndaeActivity.simulate(simulation, reportingInterval=10, timeHorizon=500,**kwargs)if__name__=="__main__": guiRun=Falseif(len(sys.argv)>1andsys.argv[1]=='console')elseTrue run(guiRun=guiRun)