Journal of the American Ceramic Society-Zhu et al. Vol 81. No. 3 d-c dc Equation (7)is a differential equation with the following boundary conditions (at r=R) x-i/ Unoxidized Core atr=R-x) R Q Solving Eq(7)yields dC A d C=AIn r+B where A and B are ants. Substituting Eq.(8) (10)and solving for A yield Fig. 1. Schematic drawing of the cross section of a fiber with Co-C A g.(1)Adsorption of oxygen at the outer layer from the oxI- Substituting Eqs. (9)and(I1)into Eq(4)yields dizing environment. The total oxygen flow per unit time is C-C (2) Diffusion of oxygen from the outer layer to the inter- O2=-2m/D (12) face where oxidation occurs. The total oxygen flow per unit time is denoted as O. (3)Reaction to The total oxygen con new layer of oxide at the interface. Using Eq (2), we set 21=22 and 21=23, which yiel on per unit time is denoted as O3 We assum De d Grove, 10 that the total oxyge per unit time at each of the above-mentioned three steps is the 丌Rh(C*-C0)=-2mD xidation. le Q1=Q2=Q3 2mRIh(C-Co)=2T/(R-x)kCi (13) Assuming that the fiber length, l, is much larger than the fiber radius(ie, />> R), the oxidation from the fiber ends ca Solving Eq(13)for C yields be ignored. The total oxygen flow per unit time from the oxi- dizing environment to the outer fiber surface can be written Ck(1-1),(12104 Or Substituting Eq(14)into Eq(5)yields where h is a gas-phase transport coefficient, Co the oxyger concentration at the outer surface of the oxide. and c* the 2(R一x)C* equilibrium oxygen concentration in the oxide (15) The diffusion flux of oxygen across the oxide layer can be described by Fick's first law: 1-D(R-m(1-)+h(1-R dc(r) Assuming that N is the quantity of oxygen required to form a O2=2TrID (4) unit volume of the oxide layer, the rate of oxide layer growth where r is defined in Fig. 1, C(r) is the oxygen concentration Q within the oxide laver. and d is the diffusion coefficient of oxygen in the oxid Assuming the oxidation is a first-order chemical reaction kC* the total oxygen consumption at the reaction interface is pro- portional to the interface area and the oxygen concentra the interface, C(see Fig. I for definition);1.e 1-b(2-x)n(1-2)+(1-2 O3= 2T(R-x)/kCi (5) Substituting Eq (1)into Eq. (16)yields a differential equation where k is the chemical-reaction rate constant The total oxygen flow per unit time in the oxide layer should remain constant along the diffusion path, which yield (17) 1- Eu In (6) EIn r Substituting Eq(4)into Eq.(6)and simplifying the result with an initial condi u=R( t=0) (1(1) Adsorption of oxygen at the outer layer from the oxi￾dizing environment. The total oxygen flow per unit time is denoted as Q1. (2) Diffusion of oxygen from the outer layer to the inter￾face where oxidation occurs. The total oxygen flow per unit time is denoted as Q2. (3) Reaction to form a new layer of oxide at the interface. The total oxygen consumption per unit time is denoted as Q3. We assume, as did Deal and Grove,10 that the total oxygen flow per unit time at each of the above-mentioned three steps is the same during oxidation, i.e., Q1 4 Q2 4 Q3 (2) Assuming that the fiber length, l, is much larger than the fiber radius (i.e., l >> R), the oxidation from the fiber ends can be ignored. The total oxygen flow per unit time from the oxi￾dizing environment to the outer fiber surface can be written as10 Q1 4 2pRlh(C* − C0) (3) where h is a gas-phase transport coefficient, C0 the oxygen concentration at the outer surface of the oxide, and C* the equilibrium oxygen concentration in the oxide. The diffusion flux of oxygen across the oxide layer can be described by Fick’s first law: Q2 = 2prlD dC~r! dr (4) where r is defined in Fig. 1, C(r) is the oxygen concentration within the oxide layer, and D is the diffusion coefficient of oxygen in the oxide. Assuming the oxidation is a first-order chemical reaction, the total oxygen consumption at the reaction interface is pro￾portional to the interface area and the oxygen concentration at the interface, Ci (see Fig. 1 for definition); i.e., Q3 4 2p(R−x)lkCi (5) where k is the chemical-reaction rate constant. The total oxygen flow per unit time in the oxide layer should remain constant along the diffusion path, which yields dQ2 dr = 0 (6) Substituting Eq. (4) into Eq. (6) and simplifying the result yields r d2 C dr 2 + dC dr = 0 (7) Equation (7) is a differential equation with the following boundary conditions: C = C0 ~at r = R! C = Ci ~at r = R − x! (8) Solving Eq. (7) yields dC dr = A r (9) and C = A ln r + B (10) where A and B are constants. Substituting Eq. (8) into Eq. (10) and solving for A yields A = − C0 − Ci ln S1 − x RD (11) Substituting Eqs. (9) and (11) into Eq. (4) yields Q2 = −2plD C0 − Ci ln S1 − x RD (12) Using Eq. (2), we set Q1 4 Q2 and Q1 4 Q3, which yields 2pRlh~C* − C0! = −2plD C0 − Ci ln S1 − x RD 2pRlh~C* − C0! = 2pl~R − x!kCi (13) Solving Eq. (13) for Ci yields Ci = C* 1 − k D ~R − x! ln S1 − x RD + k h S1 − x RD (14) Substituting Eq. (14) into Eq. (5) yields Q3 = 2pl~R − x!kC* 1 − k D ~R − x! ln S1 − x RD + k h S1 − x RD (15) Assuming that N is the quantity of oxygen required to form a unit volume of the oxide layer, the rate of oxide layer growth for a cylindrical fiber can be described as dx dt = Q3 2pl~R − x!N = kC* N 1 − k D ~R − x! ln S1 − x RD + k h S1 − x RD (16) Substituting Eq. (1) into Eq. (16) yields a differential equation − du dt = kC* N 1 − Eu ln u + S F R + E ln RDu (17) with an initial condition of u = R ~at t = 0! (18) Fig. 1. Schematic drawing of the cross section of a fiber with an oxide layer. 656 Journal of the American Ceramic Society—Zhu et al. Vol. 81, No. 3