The chosen Hybrid system is a model aproximation to a temperature control used at ITBA’s Alkaline Electrolyser. Electrolysis is an exothermic process which produces heating of KOH solution. Intending to analyze process efficience as a function of temperature, it is considered a group of temperatures to make tests. With this results, relationships between power consumed, production of gases and their purity will be concluded.

To maintain system temperature , it is considered natural convection with environment temperature of the entire equipment, and dissipated power by a cooling system composed of a countercurrent flow of water and a radiator which also interacts with the environment, . For this approximation it will be considered without the intermediate step of cooling water: .

This system can be considered Hybrid because of the Cooling System Switching. Through state of q, it is shown the activation of .

Then, the proposed model is:

Used parameters are:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Function run %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function run Parameters % initial conditions q_0 = 0; Temp_0 = 20; x0 = [q_0;Temp_0]; % simulation horizon TSPAN=[0 1000]; JSPAN = [0 20]; % rule for jumps % rule = 1 -> priority for jumps % rule = 2 -> priority for flows rule = 1; options = odeset('RelTol',1e-6,'MaxStep',.1); maxStepCoefficient = .1; % set the maximum step length. At each run of the % integrator the option 'MaxStep' is set to % (time length of last integration)*maxStepCoefficient. % Default value = 0.1 % simulate [t x j] = HyEQsolver( @f,@g,@C,@D,x0,TSPAN,JSPAN,rule,options,maxStepCoefficient); % plot solution figure(1) % position clf subplot(2,1,1),plotflows(t,j,x(:,1)) grid on ylabel('q') subplot(2,1,2),plotjumps(t,j,x(:,1)) grid on ylabel('q') figure(2) % velocity clf subplot(2,1,1),plotflows(t,j,x(:,2)) grid on ylabel('Temp') subplot(2,1,2),plotjumps(t,j,x(:,2)) grid on ylabel('Temp') % plot hybrid arc plotHybridArc(t,j,x(:,2)) xlabel('j') ylabel('t') zlabel('Temp') % figure(3) % plot(t,y(:,1));

Parameters (Parameters.m)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Definition of Parameters for Model of Cooling Hysteresis %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% global TempMin TempMax Q Knatconv Kcool TempEnv TempMin = 50; % °C TempMax = 55; % °C Q = 1000; % kW Knatconv = 5; % kW/K Kcool = 50; % kW/K TempEnv = 20; % °C

Flow map (f.m)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Declaration of Flow Map F %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function xdot = f(x) % Parameters global Q % kW global Knatconv % kW/K global Kcool % kW/K global TempEnv % °C % State variables q = x(1); Temp = x(2); % Differential equations xdot = [0 ; (Q + (Knatconv + q * Kcool) * (TempEnv - Temp)) / 1000]; end

Flow set (C.m)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Declaration of Flow Set C %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function inC = C(x) % Out: % 0 if value is outside C % 1 if value is inside C global TempMin global TempMax q = x(1); Temp = x(2); if (((Temp < TempMax) && (q == 0)) || ((Temp > TempMin) && (q == 1))) inC = 1; else inC = 0; end end

Jump map (g.m)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Declaration of Jump Map G %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function xplus = g(x) % State variables q = x(1); Temp = x(2); xplus = [1-q ; Temp]; end

Jump set (D.m)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Declaration of Jump Set D %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function inD = D(x) % Out: % 0 if value is outside D % 1 if value is inside D global TempMin global TempMax q = x(1); Temp = x(2); if (((Temp <= TempMin) && (q == 1)) || ((Temp >= TempMax) && (q == 0))) inD = 1; else inD = 0; end end

Figures 1 and 2 represents temperature and Hybrid arc , as a function of continuous and discrete time, and .