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Processes 2022, 10, 812 8 of 19 Table 2. Mathematical model of 1D non-isothermal adsorption bed. Category Component mass balance Total mass balance Gas and Solid phase Bedwall Momentum balance Mass transfer Langmuir isotherm Ideal gas equation Category Equation ∂Ci +(1−ε)ρ ∂wi −D ∂2Ci +∂(uCi) =0 ∂T ∂Tx 4DB ∂2T 4(DB +WT )2 ∂t ε S ∂t ax ∂z2 ∂z ∂ C + ( 1 − ε ) ρ s ∑n ∂ w i + ∂ ( u C ) = 0 ∂t ε i=1 ∂t ∂z n ∂qi ερgcg+(1−ε)ρscs ∂t −(1−ε)ρs ∑(−∆Hi)∂t −λex∂z2 +ρgcg ∂z +rb,i(T−Tw)=0 ∂(uT) 2hi ρwcp ∂t −Hw(DB+WT)2−DB2 (Tε−Tw)+Hamb(DB+WT)2−DB2 (Tw−Tam)=0 i=1 ∂P ∂z =− 1.5×10−3 (1−εi )2 2 3 −5 (1−εi ) 2 μvg+1.75×10 Mρg2r ψε3vg p i (2rpψ) εi ∂wi = MTCsiw∗ − wi ∂t i w = i IP1ieIP2i/TsPi 1+Σk(IP3keIP4k/Ts Pk) Pyi = RTci Table 3. Model equation of compressor, buffer and valve. Equation Vacuum pump and compressor Buffer tank Valve Purity (%) Recovery (%) Energy consumption (MJ/kg) γ−1 inlet. P =tcycle dW, γ=1.3, η =0.8. if outlet. P > inlet. P, then dW = inlet.p inlet. V ηp ifoutlet.P≤ inlet.P, thendW =0;W total γ−1 γ outlet. P γ − 1 0 p Mass balance : ∂ni − ∑ Fiyi + ∑ FOyO = 0Energy balance : ∂t ∂T(∑niCpi) − ∑N FiTi ∑ykCpg,i + ∑N FOTO∑ykCpg,i=0 ∂t i=1 O=1 Unidirection: if inlet. P > outlet. P, then F = Cv(inlet. P − outlet. P); if inlet. P ≤ outlet. P, then F = 0 Bidirection: F = Cv(inlet. P > outlet. P) tcycle F y 0 tcycle PoutVin γ Pout (γ−1)/γ 0 ηp γ−1 Pin −1 dt E = t Mtarget 0 cycle Fout yout , target dt dt Purity target (%) = Recovery target (%) = t cycle Ffeed,target yfeed,target dt 0 dt The pressure-swing-adsorption process includes design parameters and operating variables such as bed size, selection of adsorbents, timing settings, duration of each step, adsorption and desorption pressures, P/F ratios, etc. [58]. Finding the best design and operation variables is an effective way to improve the efficiency of PSA process. How- ever, design parameters and operating variables are always highly coupled with purity, recovery rate, productivity, energy consumption and other performance indicators, making it difficult to determine its optimal conditions through empirical methods. In addition, as PSA optimization is a typical multi-objective optimization problem, several objective functions may conflict with each other; for example, purity and recovery vary in the op- posite direction [34,59,60]. Due to the mutual influence and tradeoff between different objective functions, a multi-objective optimization problem has a series of optimal solutions product,target product,target t cycle Fproduct dt 0 tcycle F y 0 product,target product,target 3. PSA Numerical Optimization and Control 3.1. Optimization StrategiesPDF Image | Numerical Research on the Pressure Swing Adsorption Process
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