• PARTICLE CONSERVATION OF MASS (PARTICLE DIFFUSIVITY)
• GAS STREAM CONSERVATION OF MASS
• GAS-ADSORBENT CONSERVATION OF ENERGY
• VESSEL CONSERVATION OF ENERGY
• GAS CONSERVATION OF MOMENTUM
• NOMENCLATURE

PARTICLE CONSERVATION OF MASS (PARTICLE DIFFUSIVITY)

        GOVERNING EQUATION

        ASSUMPTIONS

                • Gas phase storage term is negligible
                • D_eff  is constant through particle
                • D_eff  is independent of loading

        INITIAL CONDITIONS

        BOUNDARY CONDITIONS

        CONSTITUTIVE RELATIONS

                • Desiccant Capacity
                       

                • Mass Transfer Film Coefficient

                        Sherwood Number
                               

                         Reynold Number
                               

                        Schmidt Number
                               

                • Ideal Gas Law
                       

                • Effective Diffusivity
                       

                    where
                       

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GAS STREAM CONSERVATION OF MASS

        GOVERNING EQUATION

        ASSUMPTIONS
                • Gas phase storage is neglected -

                • One dimensional – no gradients in radial direction

                • Plug flow is assumed – D_L = 0

        INITIAL CONDITIONS

        BOUNDARY CONDITIONS

        CONSTITUTIVE RELATIONS

                • Superficial Velocity
                       

                • Gas Concentration
                       

                • Average Loading
                       

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GAS-ADSORBENT CONSERVATION OF ENERGY

        GOVERNING EQUATION

        ASSUMPTIONS

                • Conduction heat transfer is neglected

                • Energy storage term in gas phase is neglected

                • Thermal equilibrium between fluid and solid is assumed, i.e. T_f (Z) = T_s (Z)

                • No radial temperature gradient in sorbent bed

        INITIAL CONDITIONS

        BOUNDARY CONDITIONS

        CONSTITUTIVE RELATIONS

                • Film heat transfer coeficient to vessel

                        Nusselt Number

                        Reynold Number

                        Prantl Number

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VESSEL CONSERVATION OF ENERGY

        GOVERNING EQUATION

        ASSUMPTIONS

                • No radial temperature gradient through vessel

                • No axial conduction through vessel

        INITIAL CONDITIONS

        BOUNDARY CONDITIONS

        CONSTITUTIVE RELATIONS – none

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GAS CONSERVATION OF MOMENTUM

        GOVERNING EQUATION

        ASSUMPTIONS – none

        INITIAL CONDITIONS – none

        BOUNDARY CONDITIONS

        CONSTITUTIVE RELATION

                • Ergun friction factor

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NOMENCLATURE

        a – sorbent external surface area [ft²/ft³]
        A_c – Bed cross sectional area [ft²]
        c – gas phase concentration [lb_H2O/ft³_g]
        c_B – gas phase concentration in bulk stream [lb_H2O/ft³_g]
        c_gas – concentration of gas (density) [lb/ft³]
        c_initial – initial gas phase concentration  [lb_H2O/ft³_g]
        c_inlet – gas phase concentration at bed inlet [lb_H2O/ft³_g]
        C_P,f – fluid specific heat [Btu/lb-R]
        C_P,s – sorbent specific heat [Btu/lb-R]
        C_v – vessel specific heat [Btu/lb-R]
        c_R – gas phase concentration at sorbent particle surface [lb_H2O/ft³_g]
        D_eff – effective diffusivity through adsorbent particle [ft²/min]
        d_i – vessel inside diameter [ft]
        D_L – axial dispersion coefficient – [ft²/min]
        D_m – molecular diffusivity [ft²/min]
        D_m,o – molecular diffusivity at standard temperature and pressure [ft²/min]
        d_o – vessel outside diameter [ft]
        D_p – sorbate pore diffusivity [ft²/min]
        Flow_std – flow referenced to standard temperature and pressure [scfm]
        K – thermal conductivity of sorbent bed [Btu/min-ft-R]
        K_f – external fluid film mass transfer coefficient [ft/min]
        h_v – film coefficient to vessel [Btu/min-ft²-R]
        P – pressure [lb_f/ft²]
        P_inlet –pressure at bed inlet [lb_f/ft²]
        P_o – standard pressure [lb_f/ft²]
        P_T – total gas pressure [lb_f/ft²]
        P_v – sorbate vapor pressure [lb_f/ft²]
        q – solid phase concentration [lb_H2O/ft³_s]
        - average solid phase concentration [lb_H2O/ft³_s]
        q_initial – initial solid phase concentration [lb_H2O/ft³_s]
        _initial – average initial solid phase concentration [lb_H2O/ft³]
        r – radial coordinate for adsorbent [ft]
        R_a – resistance to ambient heat loss from vessel [min-ft²-R/Btu]
        R_gas – gas constant of bulk gas [ft-lb_f/lb_m-R]
        R_s – adsorbent particle radius [ft]
        R_v – gas constant of sorbate [ft-lb_f/lb_m-R]
        t – time [min]
        T – temperature [R]
        T_amb – ambient temperature [R]
        T_f – fluid temperature [R]
        T_inlet – temperatue of inlet fluid [R]
        T_o – standard temperature [R]
        T_s – sorbent temperature [R]
        T_v – vessel temperature [R]
        T_v,initial – initial vessel temperature [R]
        V – superficial velocity [ft/min]
        V_inlet – superficial velocity at bed inlet [ft/min]
        Z – axial coordinate [ft]
        DH – heat of adsorption [Btu/lb]
        e - voidage of sorbent bed [ft³/ft³]
        e_p – porosity of adsorbent particle [ft³/ft³]
        r_f – fluid density [lb/ft³]
        r_s – sorbent density [lb/ft³]
        r_std – fluid density at standard pressure and temperature [lb/scf]
        r_v – vessel density [lb/ft³]
        t - tortuosity [ft/ft]
        m - viscosity [lb_m/ft-min]

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