Some of the five equations (reactions R1-R5) are endothermic while some are exothermic. The overall heat balance of reacting char particles is known from a balance of a particle’s heat generation and heat loss to the gas by conduction and radiation.

where Cpc is the specific heat of the char, AHk is the heat of reaction of the kth reaction at the char surface at temperature Ts, ep is the emissivity of the char particle, Xg is the thermal conductivity of the gas, and a is the Stefan-Boltzmann constant.

A similar heat balance for the gas in an element dz in length can be carried out as

— [hcom(Tg — Tw ) + ewo(Tg — TW)] kDr

where A|k is the extent of the gas-phase kth reaction with the heat of reaction, AHk (Tg); hconv is the gas-wall convective heat transfer coefficient; and Dr is the reactor’s internal diameter.

The first term on the right of Eq. (5.91) is the net heat absorption by the gas-phase reaction, the second is the heat transfer from the gas to the char particles, and the third is the heat loss by the gas at temperature Tg to the wall at temperature Tw.

The equations are solved for an elemental volume, AdLr, with boundary conditions from the previous upstream cell. The results are then used to solve the next downstream cell.

Symbols and Nomenclature

A = cross-sectional area of bed or reactor (m2)

A0 = pre-exponential coefficient in Eq. (5.42) (s-1)

Ab, Aw = pre-exponential coefficients in Eqs. (5.44) and (5.47), respectively (bar-" s-1) Aj = total number of atoms of element j entering the reactor (-) a, = number of atoms of jth element in Ith species (-) ajk = mass of jth gas, required for the kth reaction (kg)

Cl = molar concentration of ith gas (mol/m3)

Cpc = specific heat of char (kJ/kg. K)

Cpg = specific heat of the bulk gas Dr = internal diameter of the reactor (m)

Dgj = diffusion coefficient of the jth gas in the mixture of gases (m2/s) db = diameter of the bubble (m)

E = activation energy (kJ/mol) ep = emissivity of char particle (-)

Fgl0 = initial flow rate of the gas (mol/s)

Fgl = molar flow rate of the Ith gas (mol/s)

Gtotal = total Gibbs free energy (kJ) g = acceleration due to gravity, 9.81 (m/s2)

AHk = heat of reaction of kth reaction at char surface (kJ/mol)

AH = enthalpy change (kJ)

H0, Щ = heat of formation at reference state (kJ)

hconv = gas-wall convective heat transfer coefficient (kW/m2K)

hm = mass-transfer coefficient (kg carbon/m2.bar2.s)

k = first-order reaction rate constant (s-1)

k0 = pre-exponential factor (s-1)

kiiq = rate constant for the liquid yield of pyrolysis (s-1)

kBE = bubble-emulsion mass exchange coefficient (m/s)

kc = rate constant for the char yield of pyrolysis (s-1)

kg = rate constant for the gas yield of pyrolysis (s-1)

kj = rate constant of three primary pyrolysis reactions taken together (s-1)

K = number of element in Eq (5.77)

kw1, kw2, kw3 = rate constants in Eq (5.47) (bar-1 s-1)

Ksk = surface reaction rate constant for kth reaction, mol/m2.barn Ke, Kequlbbnum = equilibrium constant (-) l = number of gaseous reactants (-)

Lr = length of the reactor (m)

L = Lagrangian function (-)

mb = mass of the biomass in the primary pyrolysis process (kg) m0 = initial mass of the biomass (kg)

mc = mass of the biomass remaining after complete conversion (kg) m = reaction order with respect to carbon conversion in Eq. 5.42 (-)

m, n, p, q = stoichiometric coefficients in Eqs. 5.27-5.29 n = reaction order with respect to the gas partial pressure, Eq. 5.44 (-)

N = number of species present (-)

Nc = number of char particles in unit gas volume (-)

Nor = number of orifices in a bed of area (Ar)

Pi = partial pressure of the species i (bar)

P = total pressure of the species (bar)

Q = char gasification rate (kg carbon/m2.s)

Qualification, Qconv, Qrad, and Qmass = energy transfer due to gasification, convection, radia­tion, respectively (kW/m3 of bed)

R = gas constant (8.314 J/mol. K, or 8.314 x 10-5 m3.bar/mol. K)

Rc = chemical kinetic reaction rate (kg carbon/m2.bar2s)

Rm, gj = rate of production or consumption of gas species j (kg/m3s)

Ті = reaction rate of the ith reaction (s-1) rc = char particle radius (m)

T = temperature (K)

Ts = surface temperature of char particles (K)

Tg = gas temperature (K)

Tw = wall temperature (K) t = time (s)

ug = superficial gas velocity in Eq. 5.80 (m/s)

U = fluidization velocity (m/s)

UB = bubble rise velocity (m/s)

Umf = minimum fluidization velocity (m/s)

X = fractional change in the carbon mass of the biomass (kg)

y = mole fraction of a species (-)

yt = mole fraction of gas in the bulk (-)

yls = mole fraction of gas on the char surface (-)

z = height above grid or distance along a reactor from fuel entry (m)

alk = stoichiometric coefficient for lth gas in kth reaction (-)

В = partition coefficient (-)

X = Lagrangian multiplier (-)

Xg = thermal conductivity of gas (kJ/m. K)

О = Stefan-Boltzmann constant (5.67 x 10-8 W m-2 K-4)

AG, AG° = change in Gibbs free energy (kJ)

AG° = change in Gibbs free energy of formation of species i (kJ)

A^k = extent of gas-phase kth reaction (-) pj = density of jth gas (kg/m3)

Emf = voidage at minimum fluidization condition pg = density of the bulk gas AS = entropy change (kJ/K)