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PSACycle.m
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PSACycle.m
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function [objectives, constraints, a, b, c, d, e, t1, t2, t3, t4, t5] = PSACycle(vars, material, x0, type, N, it_disp)
%Skarstrom: Simulate a 5-Step Modified Skarstrom PSA cycle
% This function is able to simulate a 5-Step Modified Skarstrom PSA cycle
% and provide the state variables and process objectives. Finite Volume
% method is used for calculating the derivatives due to the inherent mass
% and energy conservation due to calculating the flux. An example of the
% FVM set up is shown below with the ordering of the volumes
%
% Diagram of Column divided into sections shown below with corresponding
% designation for inlet and outlet. N is a user defined value below
%
% Inlet Outlet
% -------------------------------\ \----------------------
% 1 | 2 | 3 | 4 | N| N+1 | N+2
% --------------------------------\ \---------------------
%
% Input:
% vars: Process variables which are in order: Length of column [m]
% adsoprtion pressure [Pa], inlet molar flux [mol/s/m^2], time of
% adsorption step [s], light product reflux ratio [-], heavy product
% reflux ratio [-], and intermediate pressure. Not all of these
% variables are used for all cycles, so when the simulated cycle do
% not have the step relating the "vars", assign these to zero. These
% variables are inputed as they are desing variables and can be
% changed to be optimized.
%
% x0 : Initial profile of state variables in the column, it's not mandatory
% N : Number of Finite Volumes, it's not mandatory
%
% it_disp: This tells the program whether to show the cycle number along
% with the CCS values or not. This is automatically set to no to
% allow for the speed up of calculations.
%
% Output:
% The output variables can be customized depend upon of the purpose. Here
% are two cases:
% 1) Optimization: it is necessary to get the "objectives" variable: this
% provides the purity and recovery of CO2. And also the constraints:
% Provides whether a constraint has been violated or not. There are
% some constraints that need to be satisfied. First, the recovery of
% the process must be over 90%. Second, the purity of the final product
% must be greater than the entering stream.
%
% a-e: These are the state variables for the five steps:
% CoCPressurization, Adsorption, Heavy Reflux, CnCDepressurization and
% Light Reflux.
% The matrix is set up where each row contains all the values of the
% state variables at a single moment in time, and each column contains
% the values of a single state variable at a single location throughout
% the step. As shown above in the graph, for each state variable there
% are N volumes for which the value is known at the center of volume.
% In addition, for calculating purity and recovery, the state variable
% values at the two ends of the column are also provided. NOTE: since
% there are no spatial derivatives in the solid loading equations, and
% purity and recovery do not need these values, they are assumed to be
% equal to the volume next to it. The order of the state variables are
% as follows
%
% a-e(:, 1:N+2) are the dimensionless pressure. In order to retrieve the
% true pressure, multiply the dimensionless pressure by the adsorption
% pressure, P_0 or P_H.
% a-e(:, N+3:2*N+4) are the CO2 gas mole fraction
% a-e(:, 2*N+5:3*N+6) are the dimensionless CO2 molar loadings. In order
% to retrieve the true molar loading, multiply the dimensionless molar
% loading by the molar loading scaling factor, q_s
% a-e(:, 3*N+7:4*N+8) are the dimensionless N2 molar loadings. In order
% to retrieve the true molar loading, multiply the dimensionless molar
% loading by the molar loading scaling factor, q_s
% a-e(:, 4*N+9:5*N+10) are the dimensionless column temperature. In order
% to retrieve the true column temperature, multiply the column temperature
% by the feed temperature, T_0
%
% t1-t5 are the dimensionless times for the five steps. In order to
% retrieve the true time, multiply the dimensionless time by the ratio
% of the length to the velocity scaling factor (L/v_0)
%
% 2) Data collection intended to ANNs training: To get an output suitable
% to train Artificial Neural Network of each step of the PSA cycle,
% it is necessay to collect the initial and final states of variables
% ans store it in adequate variables to be used by the ANN toolbox.
% These variables are: a_fin, b_fin, c_fin, d_fin, e_fin, a_in, b_in,
% c_in, d_in, e_in.
% Where fin means final state, and in means initial state.
%
% The following assumptions were made:
% 1) Ideal Gas Law is used to describe the gas phase
% 2) No concentration, pressure, or temperature gradient in the the radial
% or azmuth directions
% 3) Linear Driving Force is used to describe the gas diffusion into the
% adsorbent
% 4) Adsorbent properties, and void fraction are constant throughout the
% column
% 5) Viscosity of the gas is independent of pressure
% 6) There is thermal equilibrium between the adsorbent and the gas phase
% 7) Ergun Equation is used to describe the pressure drop across the bed
% 8) Column operates adiabatically, so any wall energy balance have to be
% used
% 9) An axially dispersed plug flow model is used to represent bulk fluid
% flow
%
%% Check number of inputs
% If no value is given for the iteration variable, it is no. If no value
% is given for N, it is 10. If no value is given for the x, it is empty.
if nargin < 6
it_disp = 'no';
if nargin < 5
N = 10 ;
if nargin < 4
type = 'ProcessEvaluation' ;
end
end
end
%
%% Initialize parameter for the simulation
% Specify the parameters for the simulation
% Initialize objectives and constraints output
switch type
case 'ProcessEvaluation'
constraints = [0, 0, 0];
case 'EconomicEvaluation'
constraints = [0, 0, 0];
otherwise
error('Error. %s is not a recognizable type of operation.' , type) ;
end
objectives = [0, 0] ;
% Input parameters
InputParams = ProcessInputParameters(vars, material, N) ;
Params = InputParams{1} ;
IsothermParams = InputParams{2} ;
Times = InputParams{3} ;
EconomicParams = InputParams{4} ;
% economic_class = InputParams{5} ;
% Retrieve process parameters
N = Params(1) ;
ro_s = Params(4) ;
T_0 = Params(5) ;
epsilon = Params(6) ;
r_p = Params(7) ;
mu = Params(8) ;
R = Params(9) ;
v_0 = Params(10) ;
q_s0 = Params(11)/ro_s ;
P_0 = Params(17) ;
L = Params(18) ;
MW_CO2 = Params(19) ;
MW_N2 = Params(20) ;
y_0 = Params(23) ;
ndot_0 = vars(3) ;
P_l = Params(25) ;
P_inlet = Params(26) ;
alpha = Params(30) ;
beta = Params(31) ;
y_HR = Params(33) ;
T_HR = Params(34) ;
ndot_HR = Params(35) ;
% Call PSA cycle functions
CoCPressurization_fxn = @(t, x) FuncCoCPressurization(t, x, Params, IsothermParams) ;
Adsorption_fxn = @(t, x) FuncAdsorption(t, x, Params, IsothermParams) ;
HeavyReflux_fxn = @(t, x) FuncHeavyReflux(t, x, Params, IsothermParams) ;
CnCDepressurization_fxn = @(t, x) FuncCnCDepressurization(t, x, Params, IsothermParams) ;
% Retrieve times of PSA steps
t_CoCPres = Times(1) ;
t_ads = Times(2) ;
t_HR = Times(6) ;
t_CnCDepres = Times(3) ;
t_LR = Times(4) ;
% Dimensionless times
tau_CoCPres = t_CoCPres*v_0/L ;
tau_ads = t_ads*v_0/L ;
tau_HR = t_HR*v_0/L ;
tau_CnCDepres = t_CnCDepres*v_0/L ;
tau_LR = t_LR*v_0/L ;
% Initialize the column (initial conditions)
if nargin < 2 || isempty(x0)
q = Isotherm(y_0, P_l, 298.15, IsothermParams) ;
x0 = zeros(5*N+10,1) ;
x0(1:N+2) = P_l/P_0 ;
x0(N+3) = y_0 ;
x0(N+4:2*N+4) = y_0 ;
x0(2*N+5:3*N+6) = q(1)/q_s0 ;
x0(3*N+7:4*N+8) = q(2)/q_s0 ;
x0(4*N+9) = 1 ;
x0(4*N+10:5*N+10)= 298.15/T_0 ;
end
% opts1 = odeset( 'RelTol', 1e-6) ;
% opts2 = odeset( 'RelTol', 1e-6) ;
% opts3 = odeset( 'RelTol', 1e-6) ;
% opts4 = odeset( 'RelTol', 1e-6) ;
% opts5 = odeset( 'RelTol', 1e-6) ;
opts1 = odeset( 'JPattern', JacPressurization(N), 'RelTol', 1e-6) ;
opts2 = odeset( 'JPattern', JacAdsorption(N), 'RelTol', 1e-6) ;
opts3 = odeset( 'JPattern', JacAdsorption(N), 'RelTol', 1e-6) ;
opts4 = odeset( 'JPattern', Jac_CnCDepressurization(N), 'RelTol', 1e-6) ;
opts5 = odeset( 'JPattern', Jac_LightReflux(N), 'RelTol', 1e-6) ;
%
%% Begin simulating PSA cycle. This is skipped if the first constraint is violated.
% Run the simulation until the change in the temperature, gas mole fraction
% and CO2 molar loading is less than 0.5%. For the molar loading and the
% mole fraction, the simulation also stops if the absolute change in the
% state variable is less than 5e-5 and 2.5e-4 respectively.
% initialize variables to store the desired data to be collected
a_in = [] ;
b_in = [] ;
c_in = [] ;
d_in = [] ;
e_in = [] ;
a_fin = [] ;
b_fin = [] ;
c_fin = [] ;
d_fin = [] ;
e_fin = [] ;
if constraints(1) == 0
for i=1:700
%disp(['Iteration for CSS condition number: ', num2str(i)])
% Store initial conditions for CoCPressurization step - all iterations
a_in = [a_in; x0'] ;
%% 1. Simulate CoCPressurization step
[t1, a] = ode15s(CoCPressurization_fxn, [0 tau_CoCPres], x0, opts1) ;
% Correct the output (clean up results from simulation)
idx = find(a(:, 1) < a(:, 2)) ; % P_1 < P_2
a(idx ,1) = a(idx, 2) ; % P_1 = P_2
a(idx, N+3) = a(idx, N+4) ; % y_1 = y_2
a(idx, 4*N+9) = a(idx, 4*N+10) ; % T_1 = T_2
a(:, 2*N+5) = a(:, 2*N+6) ; % x1_1 = x1_2
a(:, 3*N+7) = a(:, 3*N+8) ; % x2_1 = x2_2
a(:, 3*N+6) = a(:, 3*N+5) ; % x1_N+2 = x1_N+1
a(:, 4*N+8) = a(:, 4*N+7) ; % x2_N+2 = x2_N+1
a(:, N+3:2*N+4) = max(min(a(:, N+3:2*N+4), 1), 0) ; % 0 <= y => 1
% Store final conditions for CoCPressurization step - all iterations
% and the CO2 and total moles at the Front and End of the column
[totalFront, CO2Front, ~] = StreamCompositionCalculator(t1*L/v_0, a, 'HPEnd') ;
[totalEnd, CO2End, ~] = StreamCompositionCalculator(t1*L/v_0, a, 'LPEnd') ;
a_fin = [a_fin; a(end, :), CO2Front, totalFront, CO2End, totalEnd] ;
% Prepare initial conditions for Adsorption step
x10 = a(end, :)' ; % Final state of previous step is the
% initial state for current step
x10(1) = P_inlet ; % BC z=0 P: P_1 = P_inlet
x10(N+2) = 1 ; % BC z=1 P: P_N+2 = 1
x10(N+3) = y_0 ; % BC z=0 y: y_1 = y_0
x10(2*N+4) = x10(2*N+3) ; % BC z=1 y: y_N+2 = y_N+1
x10(4*N+9) = 1 ; % BC z=0 T: T_1 = 1
x10(5*N+10) = x10(5*N+9) ; % BC z=1 T: T_N+2 = T_N+1
% Store initial conditions for Adsorption step - all iterations
b_in = [b_in; x10'] ;
% Initial conditions of states at first step of the PSA cycle
statesIC = a(1, [2:N+1, N+4:2*N+3, 2*N+6:3*N+5, 3*N+8:4*N+7, 4*N+10:5*N+9]) ;
%
%% 2. Simulate Adsorption step
[t2, b] = ode15s(Adsorption_fxn, [0 tau_ads], x10, opts2) ;
% Correct the output (clean up results from simulation)
idx = find(b(:, N+1) < 1) ; % P_N+1 < 1 = P_N+2
b(idx, N+2) = b(idx, N+1) ; % P_N+2 = P_N+1
b(:, 2*N+5) = b(:, 2*N+6) ; % x1_1 = x1_2
b(:, 3*N+7) = b(:, 3*N+8) ; % x2_1 = x2_2
b(:, 3*N+6) = b(:, 3*N+5) ; % x1_N+2 = x1_N+1
b(:, 4*N+8) = b(:, 4*N+7) ; % x2_N+2 = x2_N+1
b(:, N+3:2*N+4) = max(min(b(:, N+3:2*N+4), 1), 0) ; % 0 <= y => 1
if Params(end) == 0
%b = VelocityCorrection(b, ndot_0, 'HPEnd') ;
b = velocitycleanup(b) ;
end
% Store final conditions for Adsorption step - all iterations
% and the CO2 and total moles at the Front and End of the column
[totalFront, CO2Front, ~] = StreamCompositionCalculator(t2*L/v_0, b, 'HPEnd') ;
[totalEnd, CO2End, TEnd] = StreamCompositionCalculator(t2*L/v_0, b, 'LPEnd') ;
b_fin = [b_fin; b(end, :), CO2Front, totalFront, CO2End, totalEnd] ;
% Add and update necessary parameters for Light Reflux step. These
% are the composition and temperature going out of the adsorption
% (light product end), which are those for the inlet of the LR step
y_LR = CO2End/totalEnd ;
T_LR = TEnd ;
ndot_LR = totalEnd/t_ads ;
Params(27) = y_LR ;
Params(28) = T_LR ;
Params(29) = ndot_LR ;
% Call the function for Light Reflux step with updated parameters
LightReflux_fxn = @(t, x) FuncLightReflux(t, x, Params, IsothermParams) ;
% Prepare initial conditions for Heavy Reflux step
x20 = b(end, :)' ; % Final state of previous step is the
% initial state for current step
x20(1) = P_inlet ; % BC z=0 P: P_1 = P_inlet
x20(1) = x20(2) ;
x20(N+2) = 1 ; % BC z=1 P: P_N+2 = 1
x20(N+3) = y_HR ; % BC z=0 y: y_1 = y_HR
x20(2*N+4) = x20(2*N+3) ; % BC z=1 y: y_N+2 = y_N+1
x20(4*N+9) = T_HR/T_0 ; % BC z=0 T: T_1 = T_HR/T_0
x20(5*N+10) = x20(5*N+9) ; % BC z=1 T: T_N+2 = T_N+1
% Store initial conditions for Heavy Reflux step - all iterations
c_in = [c_in; x20'] ;
%
%% 3. Simulate Heavy Reflux step
[t3, c] = ode15s(HeavyReflux_fxn, [0 tau_HR], x20, opts3) ;
% Correct the output (clean up results from simulation)
idx = find(c(:, N+1) < 1) ; % P_N+1 < 1 = P_N+2
c(idx, N+2) = c(idx, N+1) ; % P_N+2 = P_N+1
c(:, 2*N+5) = c(:, 2*N+6) ; % x1_1 = x1_2
c(:, 3*N+7) = c(:, 3*N+8) ; % x2_1 = x2_2
c(:, 3*N+6) = c(:, 3*N+5) ; % x1_N+2 = x1_N+1
c(:, 4*N+8) = c(:, 4*N+7) ; % x2_N+2 = x2_N+1
c(:, N+3:2*N+4) = max(min(c(:, N+3:2*N+4), 1), 0) ; % 0 <= y => 1
if Params(end) == 0
c = VelocityCorrection(c, ndot_HR, 'HPEnd') ;
%c = velocitycleanup(c) ;
end
% Store final conditions for Heavy Reflux step - all iterations
% and the CO2 and total moles at the Front and End of the column
[totalFront, CO2Front, ~] = StreamCompositionCalculator(t3*L/v_0, c, 'HPEnd') ;
[totalEnd, CO2End, ~] = StreamCompositionCalculator(t3*L/v_0, c, 'LPEnd') ;
c_fin = [c_fin; c(end, :), CO2Front, totalFront, CO2End, totalEnd] ;
% Prepare initial conditions for CoCDepressurization step
x30 = c(end,:)' ; % Final state of previous step is the
% initial state for current step
x30(1) = x30(2) ; % BC z=0 P: P_1 = P_2
x30(N+2) = x30(N+1) ; % BC z=1 P: P_N+2 = P_N+1
x30(N+3) = x30(N+4) ; % BC z=0 y: y_1 = y_2
x30(2*N+4) = x30(2*N+3) ; % BC z=1 y: y_N+2 = y_N+1
x30(4*N+9) = x30(4*N+10) ; % BC z=0 T: T_1 = T_2
x30(5*N+10) = x30(5*N+9) ; % BC z=1 T: T_N+2 = T_N+1
% Store initial conditions for CoCDepressurization step - all iterations
d_in = [d_in; x30'] ;
%
%% 4. Simulate CnCDepressurization step
[t4, d] = ode15s(CnCDepressurization_fxn, [0 tau_CnCDepres], x30, opts4) ;
% correct the output (clean up results from simulation)
idx = find(d(:, 2) < d(:, 1)) ; % P_2 < P_1
d(idx ,1) = d(idx, 2) ; % P_1 = P_2
d(:, 2*N+5) = d(:, 2*N+6) ; % x1_1 = x1_2
d(:, 3*N+7) = d(:, 3*N+8) ; % x2_1 = x2_2
d(:, 3*N+6) = d(:, 3*N+5) ; % x1_N+2 = x1_N+1
d(:, 4*N+8) = d(:, 4*N+7) ; % x2_N+2 = x2_N+1
d(:, N+3:2*N+4) = max(min(d(:, N+3:2*N+4), 1), 0) ; % 0 <= y => 1
% Store final donditions for CnCDepressurization step - all iterations
% and the CO2 and total moles at the Front and End of the column
[totalFront, CO2Front, ~] = StreamCompositionCalculator(t4*L/v_0, d, 'HPEnd') ;
[totalEnd, CO2End, ~] = StreamCompositionCalculator(t4*L/v_0, d, 'LPEnd') ;
d_fin = [d_fin; d(end, :), CO2Front, totalFront, CO2End, totalEnd] ;
% Prepare initial donditions for Light Reflux step
x40 = d(end,:)' ; % Final state of previous step is the
% initial state for durrent step
x40(1) = P_l/P_0 ; % Bd z=0 P: P_1 = P_l/P_0
%x40(N+2) = 2*P_l/P_0 ; % Bd z=1 P: P_N+2 = 2*P_l/P_0 % NOTE: be aware of this alpha here, on the other dode is just 2
x40(N+3) = x40(N+4) ; % Bd z=0 y: y_1 = y_2
x40(2*N+4) = y_LR ; % Bd z=1 y: y_N+2 = y_LR
x40(4*N+9) = x40(4*N+10) ; % Bd z=0 T: T_1 = T_2
x40(5*N+10) = T_LR/T_0 ; % Bd z=1 T: T_N+2 = T_LR/T_0
% Store initial conditions for Light Reflux step - all iterations
e_in = [e_in; x40'] ;
%
%% 5. Simulate Light Reflux step
[t5, e] = ode15s(LightReflux_fxn, [0 tau_LR], x40, opts5) ;
% Correct the output (clean up results from simulation)
idx = find(e(:, 2) < e(:, 1)) ; % P_2 < P_1
e(idx ,1) = e(idx, 2) ; % P_1 = P_2
e(:, 2*N+5) = e(:, 2*N+6) ; % x1_1 = x1_2
e(:, 3*N+7) = e(:, 3*N+8) ; % x2_1 = x2_2
e(:, 3*N+6) = e(:, 3*N+5) ; % x1_N+2 = x1_N+1
e(:, 4*N+8) = e(:, 4*N+7) ; % x2_N+2 = x2_N+1
e(:, N+3:2*N+4) = max(min(e(:, N+3:2*N+4), 1), 0) ; % 0 <= y => 1
e = VelocityCorrection(e, ndot_LR*alpha, 'LPEnd') ;
%e = velocitycleanup(e) ;
% Store final conditions for Light Reflux step - all iterations
% and the CO2 and total moles at the Front and End of the column
[totalFront, CO2Front, TFront] = StreamCompositionCalculator(t5*L/v_0, e, 'HPEnd') ;
[totalEnd, CO2End, ~] = StreamCompositionCalculator(t5*L/v_0, e, 'LPEnd') ;
e_fin = [e_fin; e(end, :), CO2Front, totalFront, CO2End, totalEnd] ;
% Calculate necessary parameters for Heavy Reflux step
y_HR = CO2Front/totalFront ;
T_HR = TFront ;
ndot_HR = totalFront.*beta/t_HR ;
Params(33) = y_HR ;
Params(34) = T_HR ;
Params(35) = ndot_HR ;
HeavyReflux_fxn = @(t, x) FuncHeavyReflux(t, x, Params, IsothermParams) ;
% Prepare initial conditions for CoCPressurization step
x0 = e(end, :)' ; % Final state of previous step is the
% initial state for current step
x0(1) = x0(2) ; % BC z=0 P: P_1 = P_2
x0(N+2) = x0(N+1) ; % BC z=1 P: P_N+2 = P_N+1
x0(N+3) = y_0 ; % BC z=0 y: y_1 = y_0
x0(2*N+4) = x0(2*N+3) ; % BC z=1 y: y_N+2 = y_N+1
x0(4*N+9) = 1 ; % BC z=0 T: T_1 = 1
x0(5*N+10) = x0(5*N+9) ; % BC z=1 T: T_N+2 = T_N+1
% Final conditions of states at lat step of the PSA cycle
statesFC = e(end, [2:N+1, N+4:2*N+3, 2*N+6:3*N+5, 3*N+8:4*N+7, 4*N+10:5*N+9]) ;
%
%% Check CCS condition
% [cyclic_check, cyclic_display] = CCS_Check(a, b, c, d, e, t1, t2, t3, t4, t5) ;
%
% if strcmpi(it_disp, 'yes') == 1
% display([i, cyclic_display]) ;
% end
%
% if cyclic_check == 1
% break
% end
% CCS condition of states
CSS_states = norm(statesIC-statesFC) ;
% Mass balance condition
[~, ~, massBalance] = ProcessEvaluation(a, b, c, d, e, t1, t2, t3, t4, t5) ;
% Check if CCS has been acheived or not
if CSS_states <= 1e-3 && abs(massBalance-1) <= 0.005
break
end
% % Condition to stop if the mass balance is not satisfied but also is
% % not changing between ten consecutive iterations
% mb(i) = massBalance ;
% if i > 15
% if CSS_states <= 1e-3 && abs(massBalance-1) > 0.005
% stateMB = mb(end-15+1:end) ;
% %diffStatesMB = stateMB(end:-1:2)-stateMB(end-1:-1:1) ;
% diffStatesMB = stateMB(end-1:-1:1)-stateMB(end) ;
% normStatesMB = norm(diffStatesMB) ;
% if normStatesMB < 1e-5
% break
% end
% end
% end
%
end
%% Process and Economic evaluation
[purity, recovery, MB] = ProcessEvaluation(a, b, c, d, e, t1, t2, t3, t4, t5) ;
desired_flow = EconomicParams(1) ;
% cycle_time = EconomicParams(3) ;
cycle_time = t_CoCPres + t_ads + t_HR + t_CnCDepres + t_LR ;
% Calculate the amount of flue gas that is fed during the cycle
[n_tot_pres, ~, ~] = StreamCompositionCalculator(t1*L/v_0, a, 'HPEnd') ;
[n_tot_ads, ~, ~] = StreamCompositionCalculator(t2*L/v_0, b, 'HPEnd') ;
gas_fed = n_tot_pres + n_tot_ads ; % mols/m^2
% Calculate the required radius of the column to satisfy the molar flow rate
radius_inner = sqrt((desired_flow.*cycle_time/gas_fed)/pi()) ; % m
r_in = radius_inner ;
% Calculate the energy required during the Pressurization step
E_pres = CompressionEnergy(t1*L/v_0, a, 1e5) ; % kWh
% Calculate the energy required during the Feed step
E_feed = CompressionEnergy(t2*L/v_0, b, 1e5) ; % kWh
% Calculate the energy required during the Heavy Reflux Step
E_HR = CompressionEnergy(t3*L/v_0, c, 1e5) ; % kWh
% Calculate the energy required during the Counter Current Depressurization step
E_bldwn = VacuumEnergy(t4*L/v_0, d, 1e5) ; % kWh
% Calculate the energy required during the Light Reflux Step
E_evac = VacuumEnergy(t5*L/v_0, e, 1e5) ; % kWh
% Calculate the total energy required
energy_per_cycle = E_pres + E_feed + E_HR + E_bldwn + E_evac ; % [kWh per cycle]
% Calculate the CO2 recovered during the cycle [ton CO_2 per cycle and mol/cycle]
[~, n_CO2_CnCD, ~] = StreamCompositionCalculator(t4*L/v_0, d, 'HPEnd') ;
[~, n_CO2_LR, ~] = StreamCompositionCalculator(t5*L/v_0, e, 'HPEnd') ;
CO2_recovered_cycle = (n_CO2_CnCD+(1-beta)*n_CO2_LR)*r_in^2*pi()*MW_CO2/1e3 ;
CO2_recovered_cycle2 = (n_CO2_CnCD+(1-beta)*n_CO2_LR)*r_in^2*pi() ;
%Calculate the productivity of the column and the energy requirments
mass_adsorbent = L*pi()*r_in^2*(1-epsilon)*ro_s ;
productivity = CO2_recovered_cycle2./cycle_time./mass_adsorbent ;
energy_requirments = energy_per_cycle./CO2_recovered_cycle ;
% Compile objectives and constraint violations
con = recovery/MB - 0.9 ;
if con < 0
constraints(2) = abs(con) ;
end
switch type
case 'ProcessEvaluation'
objectives(1) = -purity ;
objectives(2) = -recovery/MB ;
con = purity - y_0 ;
if con < 0
constraints(3) = abs(con) ;
constraints(3) = 0 ;
end
case 'EconomicEvaluation'
objectives(1) = -productivity ;
objectives(2) = energy_requirments ;
con = purity - 0.9 ;
if con < 0
constraints(3) = abs(con) ;
end
end
%
end
%
%% Filter stored data
% Prepare the collected data to store it in the output variables only
% every 5 cycle but guaranteeing that the last cycle (CSS condition)
% is stored
if mod(i, 5) ==0
idx_out = [1, 5:5:i] ;
else
idx_out = [1, 5:5:i, i] ;
end
% a_fin = a_fin(idx_out, :) ;
% b_fin = b_fin(idx_out, :) ;
% c_fin = c_fin(idx_out, :) ;
% d_fin = d_fin(idx_out, :) ;
% e_fin = e_fin(idx_out, :) ;
% a_in = a_in(idx_out, :) ;
% b_in = b_in(idx_out, :) ;
% c_in = c_in(idx_out, :) ;
% d_in = d_in(idx_out, :) ;
% e_in = e_in(idx_out, :) ;
%
%% Complementary Functions
function [n_tot, n_CO2, Temp] = StreamCompositionCalculator(time, state_vars, ProductEnd)
%MoleStreamCalculator: Calculate the composition of streams
% Calculate the composition (moles of CO2 and total moles), and
% temperature of a stream at any end of the column end.
%
% Input:
% time : Dimensional time vector supplied by the ODE solver
% state_vars: Dimensionless state variable matrix supplied for the
% step from the ODE solver
% ProductEnd: End of the column where the composition will be
% calculated. OPTIONS: HPEnd and LPEnd. HPEnd stands for
% heavy product end, which is at the bottom of the column,
% and LPEnd stands for light product end, which is at the
% top of the column.
%
% Output:
% n_tot : Average total number of moles at the desired end of the
% requested step
% n_CO2 : Average number of moles of CO2 at the desired end of the
% requested step
% Temp : Average temperature at the desired end of the requested
% step
%
%% Check number of inputs
% If no value is given for the ProductEnd, this is set up by default
% as HPEnd, which is the heavy product end for any step.
if nargin < 3
ProductEnd = 'HPEnd';
end
%
%%
% Differential section length of the column
dz = L/N ;
% Dimensionalize all variables at the two ends of the columns
% Collect pressure, temperature and mole fraction at the column end
% of interest.
if strcmpi(ProductEnd, 'HPEnd') == 1
P = state_vars(:, 1:2)*P_0 ;
y = state_vars(:, N+3) ;
T = state_vars(:, 4*N+9)*T_0 ;
% Calculate the density of the gas [kg/m^3]
ro_g = (y*MW_CO2 + (1-y)*MW_N2).*P(:, 1)/R./T ;
% calculate concentrations [mol/m^3]
C_tot = P(:, 1)/R./T ;
C_CO2 = C_tot.*y ;
elseif strcmpi(ProductEnd, 'LPEnd') == 1
P = state_vars(:, N+1:N+2)*P_0 ;
y = state_vars(:, 2*N+4) ;
T = state_vars(:, 5*N+10)*T_0 ;
% calculate the density of the gas [kg/m^3]
ro_g = (y*MW_CO2 + (1-y)*MW_N2).*P(:, 2)/R./T ;
% calculate concentrations [mol/m^3]
C_tot = P(:, 2)/R./T ;
C_CO2 = C_tot.*y ;
else
error('Please specify in which end of the column the composition will be calculated. OPTIONS: HPEnd and LPEnd')
end
% Calculate the pressure gradient at the edges [Pa/m]
dPdz = 2*(P(:, 2)-P(:, 1))/dz ;
% calculate superficial velocity using ergun equation [m/s]
viscous_term = 150*mu*(1-epsilon)^2/4/r_p^2/epsilon^3 ;
kinetic_term = (1.75*(1-epsilon)/2/r_p/epsilon^3) * ro_g ;
v = -sign(dPdz).*(-viscous_term+(abs(viscous_term^2+...
4*kinetic_term.*abs(dPdz))).^(.5))/2./kinetic_term ;
% calculate molar fluxes [mol/m^2/s]
ndot_tot = abs(v.*C_tot) ;
ndot_CO2 = abs(v.*C_CO2) ;
% calculate total moles per column area [mol/m^2]
n_tot = trapz(time, ndot_tot) ;
n_CO2 = trapz(time, ndot_CO2) ;
% calculate the average temperature of the gas [K]. Only important
% if the emissions are being used in another step in the cycle (e.g.
% light reflux, light product pressurization)
energy_flux_tot = ndot_tot.*T ;
energy_tot = trapz(time, energy_flux_tot) ;
Temp = energy_tot/n_tot ;
%
end
function [purity, recovery, mass_balance] = ProcessEvaluation(varargin)
%ProcessEvaluation: Calculate the process objectives
% Calculate the purity and the recovery of the heavy product and the
% overall mass balance of the heavy product for the cycle.
%
% Input:
% a-e : The dimensionless state variables of each step for the
% cycle
% t1-t5 : The time steps for each step of the cycle
%
% Output:
% purity : Purity of the heavy product
% recovery : Recovery of the heavy product
% mass_balance: Mass balance of the heavy product (ensure all heavy
% product that enters, leaves. No accumulation)
%
%%
step = cell(nargin/2,1) ;
tau = cell(nargin/2,1) ;
for st = 1:nargin/2
step{st} = varargin{st} ;
tau{st} = varargin{nargin/2+st} ;
end
%
%% Calculate total moles going in and out of column [mols/m^2]
[~, n_CO2_CoCPres_HPEnd, ~] = StreamCompositionCalculator(tau{1}, step{1}, 'HPEnd') ;
[~, n_CO2_ads_HPEnd, ~] = StreamCompositionCalculator(tau{2}, step{2}, 'HPEnd') ;
[~, n_CO2_ads_LPEnd, ~] = StreamCompositionCalculator(tau{2}, step{2}, 'LPEnd') ;
[~, n_CO2_HR_LPEnd, ~] = StreamCompositionCalculator(tau{3}, step{3}, 'LPEnd') ;
[~, n_CO2_HR_HPEnd, ~] = StreamCompositionCalculator(tau{3}, step{3}, 'HPEnd') ;
[n_tot_CnCDepres_HPEnd, n_CO2_CnCDepres_HPEnd, ~] = StreamCompositionCalculator(tau{4}, step{4}, 'HPEnd') ;
[~, n_CO2_LR_LPEnd, ~] = StreamCompositionCalculator(tau{5}, step{5}, 'LPEnd') ;
[n_tot_LR_HPEnd, n_CO2_LR_HPEnd, ~] = StreamCompositionCalculator(tau{5}, step{5}, 'HPEnd') ;
%
%% Calculate Purity, recovery and mass balance of the column
purity = (n_CO2_CnCDepres_HPEnd+(1-beta)*n_CO2_LR_HPEnd)/(n_tot_CnCDepres_HPEnd+(1-beta)*n_tot_LR_HPEnd) ;
recovery = (n_CO2_CnCDepres_HPEnd+(1-beta)*n_CO2_LR_HPEnd)/(n_CO2_CoCPres_HPEnd+n_CO2_ads_HPEnd) ;
mass_balance = (n_CO2_CnCDepres_HPEnd+n_CO2_ads_LPEnd+n_CO2_HR_LPEnd+n_CO2_LR_HPEnd)/...
(n_CO2_CoCPres_HPEnd+n_CO2_ads_HPEnd+n_CO2_HR_HPEnd+n_CO2_LR_LPEnd) ;
%
end
function energy = CompressionEnergy(time, state_vars, Patm)
%CompressionEnergy: Calculate the compression energy from entrance
% Calculate the total energy required by the compressor to increase
% the pressure to the desired value. Intended to be used with PSA
% dimensionless simulations.
%
% Input:
% time : Dimensional time vector supplied by the ODE solver
% state_vars: Dimensionless state variable matrix supplied for the
% step from the ODE solver
% Patm : Atmospheric pressure [Pa]. This value can also be changed
% if the flue gas is at a higher pressure. This value is
% the cutoff for the compressor. Above this value, energy
% is required. Below his value, no energy is required
%
% Output:
% energy : Energy Requirments [kWh]
%
%% Check number of inputs
% Differential section length of the column
dz = L/N ;
% Compresor parameters
adiabatic_index = 1.4 ;
compressor_efficiency = 0.72 ;
% Calculate the pressure gradient at the edges [Pa/m]
P = state_vars(:, 1:2)*P_0 ;
y = state_vars(:, N+3) ;
T = state_vars(:, 4*N+9)*T_0 ;
% Calculate the density of the gas [kg/m^3]
ro_g = (y*MW_CO2 + (1-y)*MW_N2).*P(:, 1)/R./T ;
dPdz = 2*(P(:, 2)-P(:, 1))/dz ;
% calculate superficial velocity using ergun equation [m/s]
viscous_term = 150*mu*(1-epsilon)^2/4/r_p^2/epsilon^3 ;
kinetic_term = (1.75*(1-epsilon)/2/r_p/epsilon^3) * ro_g ;
v = -sign(dPdz).*(-viscous_term+(abs(viscous_term^2+...
4*kinetic_term.*abs(dPdz))).^(.5))/2./kinetic_term ;
% Calculate the compression ratio along with the impact it has on the
% energy requirments
ratio_term = ((P(:, 1)/Patm).^((adiabatic_index-1)/adiabatic_index)-1) ;
ratio_term = max(ratio_term, 0) ;
%Calculate the total energy required by the compressor
integral_term = abs(v.*P(:, 1).*ratio_term) ;
energy = trapz(time, integral_term).*((adiabatic_index)./(adiabatic_index-1))./compressor_efficiency*pi()*r_in.^2 ;
energy = energy/3.6e6 ;
end
function energy = VacuumEnergy(time, state_vars, Patm, ProductEnd)
%VacuumEnergy: Calculate the vacuum energy at both ends of the column
% Calculate the total energy required by the vacuum pump to operate
% at the desired pressure. Intended to be used with PSA dimensionless
% simulations.
%
% Input:
% time : Dimensional time vector supplied by the ODE solver
% state_vars: Dimensionless state variable matrix supplied for the
% step from the ODE solver
% ProductEnd: End of the column where the composition will be
% calculated. OPTIONS: HPEnd and LPEnd. HPEnd stands for
% heavy product end, which is at the bottom of the column,
% and LPEnd stands for light product end, which is at the
% top of the column.
% Patm : Atmospheric pressure [Pa].This value is the cutoff for
% the vacuum. Above this value, energy is not required.
% Below his value, energy is required
%
% Output:
% energy : Energy Requirments [kWh]
%
%% Check number of inputs
% If no value is given for the ProductEnd, this is set up by default
% as HPEnd, which is the heavy product end for any step.
if nargin < 4
ProductEnd = 'HPEnd';
end
%
%%
% Differential section length of the column
dz = L/N ;
% Vacuum parameters
adiabatic_index = 1.4 ;
vacuum_efficiency = 0.72 ;
% Dimensionalize all variables at the two ends of the columns
% Collect pressure, temperature and mole fraction at the column end
% of interest.
if strcmpi(ProductEnd, 'HPEnd') == 1
P = state_vars(:, 1:2)*P_0 ;
y = state_vars(:, N+3) ;
T = state_vars(:, 4*N+9)*T_0 ;
% Calculate the density of the gas [kg/m^3]
ro_g = (y*MW_CO2 + (1-y)*MW_N2).*P(:, 1)/R./T ;
P_out = P(:, 1) ;
elseif strcmpi(ProductEnd, 'LPEnd') == 1
P = state_vars(:, N+1:N+2)*P_0 ;
y = state_vars(:, 2*N+4) ;
T = state_vars(:, 5*N+10)*T_0 ;
% calculate the density of the gas [kg/m^3]
ro_g = (y*MW_CO2 + (1-y)*MW_N2).*P(:, 2)/R./T ;
P_out = P(:, 2) ;
else
error('Please specify in which end of the column the composition will be calculated. OPTIONS: HPEnd and LPEnd')
end
% Calculate the pressure gradient at the edges [Pa/m]
dPdz = 2*(P(:, 2)-P(:, 1))/dz ;
% calculate superficial velocity using ergun equation [m/s]
viscous_term = 150*mu*(1-epsilon)^2/4/r_p^2/epsilon^3 ;
kinetic_term = (1.75*(1-epsilon)/2/r_p/epsilon^3) * ro_g ;
v = -sign(dPdz).*(-viscous_term+(abs(viscous_term^2+...
4*kinetic_term.*abs(dPdz))).^(.5))/2./kinetic_term ;
% Calculate the compression ratio along with the impact it has on the
% energy requirments
ratio_term = ((Patm./P_out).^((adiabatic_index-1)/adiabatic_index)-1) ;
ratio_term = max(ratio_term, 0) ;
integral_term = abs(v.*P_out.*ratio_term) ;
%Calculate the total energy required by the compressor
energy=trapz(time, integral_term).*((adiabatic_index)./(adiabatic_index-1))./vacuum_efficiency*pi()*r_in.^2;
energy = energy/3.6e6 ;
%
end
function x_new = VelocityCorrection(x, n_hr, CorrectionEnd)
%% Check number of inputs
% If no value is given for the CorrectionEnd, this is set up by
% default as HPEnd, which is the heavy product end for any step.
if nargin < 3
CorrectionEnd = 'HPEnd';
end
%
%%
x_new = x ;
% Differential section length of the column
dz = L/N ;
% Dimensionalize all variables at the two ends of the columns
if strcmpi(CorrectionEnd, 'HPEnd') == 1
T = x(:, 4*N+9)*T_0 ;
y = x(:, N+3) ;
P = x(:, 2)*P_0 ;
elseif strcmpi(CorrectionEnd, 'LPEnd') == 1
T = x(:, 5*N+10)*T_0 ;
y = x(:, 2*N+4) ;
P = x(:, N+1)*P_0 ;
else
error('Please specify in which end of the column the velocity correction will be calculated. OPTIONS: HPEnd and LPEnd')
end
MW = MW_N2+(MW_CO2-MW_N2)*y ;
a_1 = 150*mu*(1-epsilon)^2*dz/2/4/r_p^2/epsilon^3/R./T ;
a_2_1 = 1.75*(1-epsilon)/2/r_p/epsilon/epsilon/epsilon*dz/2 ;
a_2 = a_2_1/R./T*n_hr.*MW ;
a_a = a_1+a_2 ;
b_b = P./T/R ;
c_c = -n_hr ;
vh = (-b_b+sqrt(b_b.^2-4.*a_a.*c_c))/2./a_a ;
a_p = a_1.*T*R ;
b_p = a_2_1.*MW/R./T ;
% Correction
if strcmpi(CorrectionEnd, 'HPEnd') == 1
x_new(:, 1) = ((a_p.*vh+P)./(1-b_p.*vh.*vh))./P_0 ;
elseif strcmpi(CorrectionEnd, 'LPEnd') == 1
x_new(:, N+2) = ((a_p.*vh+P)./(1-b_p.*vh.*vh))./P_0 ;
end
%
end
function x_new = velocitycleanup(x)
x_new=x;
numb1=150*mu*(1-epsilon)^2/4/r_p^2/epsilon^2;
ro_gent=x(:,2).*P_0/R/T_0;
numb2_ent=(ro_gent.*(MW_N2+(MW_CO2-MW_N2)*x(:,N+3)).*(1.75*(1-epsilon)/2/r_p/epsilon));
x_new(:,1)=(numb1*v_0+numb2_ent*v_0*v_0)*L/P_0/2/N + x(:,2);
end
%
%% Jacobian patterns functions
function J_pres = JacPressurization(N)
%JacPressurization: Calculates a Jacobian pattern for the step
% This function calculates a jacobian pattern for the pressurization
% step. This is to be used with the ode solver to decrease the
% computational time
%
% Input:
% N : Number of finite volumes used in the column
% Output:
% J_pres: The sparse Jacobian pattern scheme
%
%% Create individual segments
% Four band Jacobian scheme for advection terms
B4 = ones(N+2, 4) ;
A4 = full(spdiags(B4, -2:1, N+2, N+2)) ;
% One band Jacobian scheme for adsorption/desoprtion term
B1 = ones(N+2, 1) ;
A1 = full(spdiags(B1, 0, N+2, N+2)) ;
A1(1, 1) = 0 ;
A1(N+2, N+2) = 0 ;
% Zero band Jacobian Term
A0 = zeros(N+2) ;
%
%% Create Overall Jacobian based on individual segments
J_pres = [ A4, A4, A1, A1, A4;...
A4, A4, A1, A1, A4;...
A1, A1, A1, A0, A1;...
A1, A1, A0, A1, A1;...
A4, A1, A1, A1, A4 ] ;
%
%% Modify based on boundary conditions
% Pressure Inlet
J_pres(1, :) = 0 ;
J_pres(1, 1) = 1 ;
% Pressure Outlet
J_pres(N+2, :) = J_pres(N+1, :) ;
J_pres(:, N+2) = 0 ;
% Mole Fraction Inlet
J_pres(N+3, :) = 0 ;
J_pres(:, N+3) = 0 ;
% Mole Fraction Outlet
J_pres(2*N+4, :) = J_pres(2*N+3, :) ;
J_pres(:, 2*N+4) = 0 ;
% Temperature Inlet
J_pres(4*N+9, :) = 0 ;
J_pres(:, 4*N+9) = 0 ;
% Temperature Outlet
J_pres(5*N+10, :) = J_pres(5*N+9, :) ;
J_pres(:, 5*N+10) = 0 ;
J_pres=sparse(J_pres);
%
end
function J_ads = JacAdsorption(N)
%JacAdsorption: Calculates a Jacobian pattern for the step
% This function calculates a jacobian pattern for the pressurization
% step. This is to be used with the ode solver to decrease the
% computational time
%
% Input:
% N : Number of finite volumes used in the column
% Output:
% J_ads: The sparse Jacobian pattern scheme
%
%% Create individual segments
% Four band Jacobian scheme for advection terms
B4 = ones(N+2, 4) ;
A4 = full(spdiags(B4, -2:1, N+2, N+2)) ;
% One band Jacobian scheme for adsorption/ desoprtion term
B1 = ones(N+2, 1) ;
A1 = full(spdiags(B1, 0, N+2, N+2)) ;
A1(1, 1) = 0 ;
A1(N+2, N+2) = 0 ;