1­-D Thermal Diffusion Equation and Solutions

1­D Thermal Diffusion Equation and Solutions 3.185 Fall, 2003 The 1­D thermal diffusion equation for constant k, ρ and cp (thermal conductivity, density, specific heat) is almost identical to the solute diffusion equation: ∂T ∂2T q̇ =α 2 + ∂t ∂x ρcp (1) or in cylindrical coordinates: r ∂T ∂ =α ∂t ∂r � � ∂T q̇ r +r ∂r ρcp (2) and spherical coordinates:1 2 ∂T ∂ r =α ∂t ∂r � r 2 ∂T � ∂r + r2 q̇ ρcp (3) The most important difference is that it uses the thermal diffusivity α = ρckp in the unsteady solutions, but the thermal conductivity k to determine the heat flux using Fourier’s first law qx = −k ∂T ∂x (4) For this reason, to get solute diffusion solutions from the thermal diffusion solutions below, substitute D for both k and α, effectively setting ρcp to one. 1 Most texts simplify the cylindrical and spherical equations, they divide by r and r 2 respectively and product rule the r­derivative apart. This gives � � q̇ ∂T ∂2T 1 ∂T =α + + ∂t ∂r2 r ∂r ρcp for cylindrical and ∂T =α ∂t � ∂2T 2 ∂T + ∂r2 r ∂r � + q̇ ρcp for spherical coordinates. I prefer equations 2 and 3 because they are easier to solve. 1 1­D Heat Conduction Solutions 1. Steady­state (a) No generation i. Cartesian equation: d2 T =0 dx2 Solution: T = Ax + B Flux magnitude for conduction through a plate in series with heat transfer through a fluid boundary layer (analagous to either 1st­order chemical reaction or mass transfer through a fluid boundary layer): |Tf l − T1 | |qx | = 1 L h + k (Tf l is the fluid temperature, analagous to the concentration in equilibrium with the fluid in diffusion; T1 is the temperature on the side opposite the fluid.) Dimensionless form: 1 πq = 1 − 1 + πh where πq = qx L k(Tf l −T1 ) and πh = hL k (a.k.a. the Biot number). ii. Cylindrical equation: d dr � dT r dr � =0 Solution: T = A ln r + B Flux magnitude for heat transfer through a fluid boundary layer at R1 in series with conduc­ tion through a cylindrical shell between R1 and R2 : |r · qr | = |Tf l − T2 | 2 + k1 ln R R1 1 hR1 iii. Spherical equation: � d dr r 2 dT Solution: T = dr � =0 A +B r (b) Constant generation i. Cartesian equation: k d2 T + q̇ = 0 dx2 Solution: T =− q̇x2 + Ax + B 2k ii. Cylindrical equation: k d dr � � dT r + rq̇ = 0 dr Solution: T =− 2 q̇r2 + A ln r + B 4k iii. Spherical equation: k d dr � r2 dT dr � + r2 q̇ = 0 Solution: q̇r2 A + +B r 6k (c) (Diffusion only) first­order homogeneous reaction consuming the reactant, so G = −kC i. Cartesian equation: d2 C D 2 − kC = 0 dx Solution: √k √k C = Ae D x + Be− D x T =− or: �� C = A cosh � �� � k k x + B sinh x D D ii. Cylindrical and spherical solutions involve Bessel functions, but here are the equations: � � d dC D r − krC = 0 dr dr � � d 2 dC D r − kr2 C = 0 dr dr 2. Unsteady solutions without generation based on the Cartesian equation with constant k and ρcp : ∂T ∂2T =α 2 ∂t ∂x where α = k ρcp . (a) Uniform initial condition T = Ti (or T = T∞ ), constant boundary condition T = Ts at x = 0, semi­infinite body; or step function initial condition in an infinite body. Solution is the error function or its complement: � � T − Ts x √ = erf Ti − Ts 2 αT � � T − Ti x √ = erfc T s − Ti 2 αT Semi­infinite criterion: L √ ≥2 2 αt Note: this also applies to a “diffusion couple”, where two bodies of different temperatures (or concentrations) are joined at x = 0 and diffuse into each other; the boundary condition there is halfway between the two initial conditions. This works only if the (thermal) diffusivities are the same. (b) Fixed quantity of heat/solute diffusing into a (semi­)infinite body (same semi­infinite criterion as 2a), no flux through x = 0, initial condition T = Ti (or T = T∞ ) everywhere except a layer of thickness δ if semi­infinite or 2δ if fully infinite where T = T0 . Short­time solution consists of erfs at the interfaces, like a diffusion couple. Long­time solution is the shrinking Gaussian: � � (T0 − Ti )δ x2 T = Ti + √ exp − 4αt παt 3 (c) Uniform initial condition T = Ti , constant boundary condition T = Ts at x = 0 and x = L (or zero­flux boundary condition qx = −k∂T /∂x = 0 at x = L/2), finite body; or periodic initial condition (we’ve covered sine and square waves) in an infinite body. Solution is the Fourier series: � 2 2 � ∞ � nπx � � n π αt T = Ts + (Ti − Ts ) an exp − sin 2 L L n=0 4 For a square wave or uniform IC in a finite body, an = nπ for odd n, zero for even n, Ts is the average temperature for a periodic situation or the boundary condition for a finite layer, L is the half period of the wave or the thickness of the finite layer. The n = 1 term dominates when π2 α L2 t ≥ 1. (d) *Uniform initial condition T = T∞ , constant flux boundary condition at x = 0 qx = −k dT dx = q0 , semi­infinite body (same semi­infinite criterion as 2a). Solution: � � � � � �� q0 αt z2 z T = T∞ + 2 exp − − z 1 − erf √ k π 4αt 2 αt (e) *Uniform initial condition T = T∞ , heat transfer coefficient boundary condition at x = 0 qx = −k dT dx = h(Tf l − T ), semi­infinite body (same semi­infinite criterion as 2a). Solution: √ � � � � � � T − Tf l x hx h2 αt x h αt √ + √ − exp + 2 · erfc = erfc k T∞ − Tf l k k 2 αt 2 αt *These solutions are neither covered nor required, but are here for your edification and future reference. 3. Moving body If a body is moving relative to a frame of reference at speed ux and conducting heat only in the direction of motion, then the equation in that reference frame (for constant properties) is: ∂T ∂T ∂2T q̇ + ux =α 2 + ∂t ∂x ∂x ρcp Note that this is the diffusion equation with the substantial derivative instead of the partial derivative, and nonzero velocity only in the x­direction. Recall the definition of the substantial derivative: D ∂ = + �u · � Dt ∂t Applied to temperature with uy = uz = 0: DT ∂T ∂T = + ux Dt ∂t ∂x Therefore: When this reaches steady­state, so q̇ DT ∂2T =α 2 + Dt ∂x ρcp ∂T ∂t = 0, then the solution in the absence of generation is T = A + Beux x/α 4