Global optimization of general constrained grey-box models: new method and its application to constrained PDEs for pressure swing adsorption

Abstract

This paper introduces a novel methodology for the global optimization of general constrained grey-box problems. A grey-box problem may contain a combination of black-box constraints and constraints with a known functional form. The novel features of this work include (i) the selection of initial samples through a subset selection optimization problem from a large number of faster low-fidelity model samples (when a low-fidelity model is available), (ii) the exploration of a diverse set of interpolating and non-interpolating functional forms for representing the objective function and each of the constraints, (iii) the global optimization of the parameter estimation of surrogate functions and the global optimization of the constrained grey-box formulation, and (iv) the updating of variable bounds based on a clustering technique. The performance of the algorithm is presented for a set of case studies representing an expensive non-linear algebraic partial differential equation simulation of a pressure swing adsorption system for $\hbox {CO}_{2}$. We address three significant sources of variability and their effects on the consistency and reliability of the algorithm: (i) the initial sampling variability, (ii) the type of surrogate function, and (iii) global versus local optimization of the surrogate function parameter estimation and overall surrogate constrained grey-box problem. It is shown that globally optimizing the parameters in the parameter estimation model, and globally optimizing the constrained grey-box formulation has a significant impact on the performance. The effect of sampling variability is mitigated by a two-stage sampling approach which exploits information from reduced-order models. Finally, the proposed global optimization approach is compared to existing constrained derivative-free optimization algorithms.

Appendices

Appendix 1: Model equations

See Tables 11 and 12.

Table 11 NAPDE model equations for PSA process

Table 12 PSA process constrains, operating constraints and cost objective

Appendix 2: Nomenclature

Indices

i :

Component

n :

Packed column

Sets

The set of all components is given as follows: $i\in I=\{ CO _2,N_2\}$

Parameters

The following parameters are defined:

F :

Feed flow rate

$y_{i,f}$ :

Mole fraction of component i in the feed gas

$T_o$ :

Feed temperature

$P_f$ :

Feed flow rate

$u_o$ :

Feed velocity

$T_a$ :

Outside air temperature

$y_{i,0}$ :

Initial mole fraction of component i in the bed

$P_{atm}$ :

Atmospheric pressure

$K_z$ :

Effective heat conductivity

$K_w$ :

Thermal heat conductivity of column wall

$h_{in}$ :

Heat transfer coefficient inside the column

$h_o$ :

Heat transfer coefficient for outside the column

$C_{pw}$ :

Specific heat capacity of column wall

$C_{pg}$ :

Specific heat capacity of gas mixture

$C_{pa}$ :

Specific heat capacity of adsorbed gas in solid

$C_{ps}$ :

Specific heat capacity of adsorbent

R :

Gas constant

$q_s$ :

Saturation capacity

$b_i^o$ :

Isotherm parameter for component i

$\Delta U_i$ :

Heat of adsorption for component i

$\tau _p$ :

Tortuosity factor

$D_M$ :

Molecular diffusivity of ${ CO }_2 -N_2\,mixture$

$D_p$ :

Macropore diffusivity, $D_p =\frac{D_M }{\tau _p }$

$D_L$ :

Axial dispersion coefficient defined by the correlation: $D_L =0.7D_M +0.5u_o d_p$

UC :

Unit operating cost

$d_p$ :

Particle diameter

$r_p$ :

Particle radius, $\frac{d_p }{2}$

$\rho _w$ :

Density of column

$\rho _s$ :

Density of solid particle

$\mu$ :

Gas viscosity

$\varepsilon _p$ :

Particle porosity

$\varepsilon$ :

Bed porosity

$\phi$ :

Annualization factor

$\alpha$ :

Unit capital cost

$\beta$ :

Exponent of power cost

$k_p$ :

Bed permeability defined by the correlation: $k_p =\frac{d_p^2 }{150}\left( {\frac{\varepsilon }{1-\varepsilon }} \right) ^{2}$

Variables

We consider the variation both in time and along the column length. To denote this, we define:

l :

Bed length

t :

Time

The following system variables are defined to describe the system:

P(l, t):

Pressure

T(l, t):

Temperature

$T_W (l,t)$ :

Column wall temperature

$u_z (l,t)$ :

Velocity

$c_i (l,t)$ :

Concentration of component i in the gas phase

$y_i (l,t)$ :

Mole fraction of component i in the gas phase

$x_i (l,t)$ :

Fractional loading of component i in the solid phase

$x_i^*(l,t)$ :

Equilibrium fractional loading of component i in the solid phase

$k_i (l,t)$ :

Mass transfer coefficient of component i

$b_i (l,t)$ :

Temperature dependent parameter for component i

$B_i (l,t)$ :

An isotherm parameter for component i

$\Delta H_i (l,t)$ :

Change in internal energy for component i

$\rho _g (l,t)$ :

Density of gas obtained from the ideal gas law

The decision variables for process optimization are as follows:

N :

Number of columns

$P_{ads}$ :

Highest pressure of the system

L :

Column length

D :

Column diameter

$P_{bd}$ :

Blowdown pressure

$P_{evac}$ :

Evacuation pressure

$t_{pr}$ :

Duration of the pressurization step, 20

$\hbox {s}t_{ads}$ :

Duration of the adsorption step

$t_{bd}$ :

Duration of the blowdown step

$t_{evac}$ :

Duration of the evacuation step

Variable scaling

The model is described in terms of these dimensionless variables:

$$\begin{aligned}&\overline{{P}}=\frac{P}{P_{ads} }; \quad \overline{{P}}_{evac} =\frac{P_{evac} }{P_{ads} }; \quad \overline{{P}}_{bd} =\frac{P_{bd} }{P_{ads} }; \quad \overline{{T}}=\frac{T}{T_0 }; \quad \overline{{T}}_w =\frac{T_w }{T_0 }; \quad \overline{{T}}_a =\frac{T_a }{T_0 }; \quad \\&x_i =\frac{\overline{{q}}_i }{q_s };\quad x_i^*=\frac{q_i^*}{q_s };\quad \overline{{u}}_z =\frac{u_z }{u_0 }; \quad Z=\frac{l}{L};\quad \tau =\frac{tu_o }{L} \end{aligned}$$

We also need appropriate boundary and initial conditions to describe the four steps, since they are different for different steps. These are defined subsequently.

Initial and boundary conditions

Pressurization The initial conditions for the first pressurization step are given by the following equation:

$$\begin{aligned} y_i= & {} y_{i,0} \\ \overline{{P}}= & {} \overline{{P}}_{evac} \\ \overline{{T}}= & {} 1 \\ x_i= & {} \left. {x_i^*} \right| _{y_{i,0} } \end{aligned}$$

In a 4-step cyclic process, the initial conditions for each subsequent step are the final conditions of the previous step, so initial conditions are not specified for the remaining steps.

Let $\hbox {Z} = 0^{+}$ and $\hbox {Z} = 1^{-}$ be the two ends of an adsorption column where the boundary conditions must be applied. The boundary conditions at $\hbox {Z} = 0^{+}$ for the pressurization step are as follows:

$$\begin{aligned}&\displaystyle \frac{1}{Pe}\frac{\partial y_i }{\partial Z}=-\overline{{u}}_z (y_{i,f} -y_i ) \\&\displaystyle \frac{1}{Pe_H }\frac{\partial \overline{{T}}}{\partial Z}=-\overline{{u}}_z (1-\overline{{T}}) \\&\displaystyle \overline{{T}}_w =\overline{{T}}_a \\&\displaystyle \overline{{P}}=f\left( \tau \right) ;\quad \overline{{P}}_{evac} \rightarrow 1 \\ \end{aligned}$$

The boundary conditions at $\hbox {Z} = 1^{-}$ are:

$$\begin{aligned}&\displaystyle \frac{\partial y_i }{\partial Z}=0 \\&\displaystyle \frac{\partial \overline{{T}}}{\partial Z}=0 \\&\displaystyle \overline{{T}}_w =\overline{{T}}_a \\&\displaystyle \frac{\partial \overline{{P}}}{\partial Z}=0 \end{aligned}$$

Adsorption The boundary conditions at $\hbox {Z}= 0^{+}$ and $\hbox {Z}= 1^{-}$ for the adsorption step are the following:

$$\begin{aligned}&\displaystyle \frac{\partial y_i }{\partial Z}=-Pe(y_{i,f} -y_i ) \\&\displaystyle \frac{1}{Pe_H }\frac{\partial \overline{{T}}}{\partial Z}=-(1-\overline{{T}}) \\&\displaystyle \overline{{T}}_w =\overline{{T}}_a \\&\displaystyle \overline{{P}}=1+\frac{\Delta }{P_{ads} } \\&\displaystyle \frac{\partial y_i }{\partial Z}=0 \\&\displaystyle \frac{\partial \overline{{T}}}{\partial Z}=0 \\&\displaystyle \overline{{T}}_w =\overline{{T}}_a \\&\displaystyle \overline{{P}}=1 \end{aligned}$$

where $Pe_H =\frac{\varepsilon u_o L\rho _g C_{pg} }{K_z }$

Blowdown The boundary conditions at $\hbox {Z} = 0^{+}$ and $\hbox {Z} = 1^{-}$ for the blowdown step are given by:

$$\begin{aligned}&\displaystyle \frac{\partial y_i }{\partial Z}=0 \\&\displaystyle \frac{\partial \overline{{T}}}{\partial Z}=0 \\&\displaystyle \overline{{T}}_w =\overline{{T}}_a \\&\displaystyle \frac{\partial \overline{{P}}}{\partial Z}=0 \\&\displaystyle \frac{\partial y_i }{\partial Z}=0 \\&\displaystyle \frac{\partial \overline{{T}}}{\partial Z}=0 \\&\displaystyle \overline{{T}}_w =\overline{{T}}_a \\&\displaystyle \overline{{P}}=f(\tau ){:}~1\rightarrow \overline{{P}}_{bd} \end{aligned}$$

Evacuation The boundary conditions at $\hbox {Z} = 0^{+}$ and $\hbox {Z} = 1^{-}$ for the evacuation step are:

$$\begin{aligned}&\displaystyle \frac{\partial y_i }{\partial Z}=0 \\&\displaystyle \frac{\partial \overline{{T}}}{\partial Z}=0 \\&\displaystyle \overline{{T}}_w =\overline{{T}}_a \\&\displaystyle \frac{\partial \overline{{P}}}{\partial Z}=0 \\&\displaystyle \frac{\partial y_i }{\partial Z}=0 \\&\displaystyle \frac{\partial \overline{{T}}}{\partial Z}=0 \\&\displaystyle \overline{{T}}_w =\overline{{T}}_a \\&\displaystyle \overline{{P}}=f(\tau ):\overline{{P}}_{bd} \rightarrow \overline{{P}}_{evac} \end{aligned}$$