Transport processes Part 3b Ron Zevenhoven bo Akademi University - PDF document

Transport processes (TRP) Transport processes – Part 3b Ron Zevenhoven Åbo Akademi University Thermal and Flow Engineering / Värme- och strömningsteknik tel. 3223 ; ron.zevenhoven@abo.fi VST rz18 2/48 Transport processes (TRP) VST rz18

3/48 Transport processes (TRP) Non-stationary diffusion in 2-D VST rz18 Non-stationary diffusion in 2-D /1 • For diffusion in two dimensions, with constant   material properties: 2 2    T T T   a   Transport processes (TRP)  2 2 t    x y    • which requires 1 initial condition for the x,y plane + 4 boundary conditions. • Using variables ζ = x/√(at) and η= y/√(at): 2 2     T T T T       ½ ½     2 2     • with for example starting and boundary conditions: t = 0: x≥0, y≥0 : T = T 1 t > 0: x=0, y>0 : T = T 0 x>0, y=0 : T = T 0 x=∞, y>0 : T = T 1 x>0, y=∞ : T = T 1 VST rz18

Non-stationary diffusion in 2-D /2 • A case like this (↑) can be reduced to 2 ordinary differential equations by using separation of Transport processes (TRP) variables. • Sometimes a solution can be found by adding or multiplying the solutions for 1-dimensional problems. • Cases with a uniform starting temperature T(t=0,x,y) = T 1 over the whole (x,y) range, for example (0  L, 0  M), typically give solutions of the type T(t,x,y) = T 1 + (T 0 -T 1 )·(1 – F(x,t)·G(y,t)) • If F(x,t) and G(y,t) fullfil the boundary conditions at x=0, x=L, and at y=0, y=M, respectively, then the product function F(x,t)·G(y,t) will do that too. VST rz18 Non-stationary diffusion in 2-D /3 • Example case , start/boundary conditions: t = 0: 0≤x≤L, 0≤y≤M T = 0 Transport processes (TRP) t > 0: x=0 & x=L, 0≤y≤M T = T 0 y=0 & y=M, 0≤x≤L T = T 0 • Using the solution given in § 2.2: • For large t, using only the first eigenvalue (n = 0) including L = ∞, M = ∞ VST rz18

Non-stationary diffusion in 2-D /4 • Example case , start/boundary conditions: t = 0: 0≤x≤L, 0≤y≤M T = 0 Transport processes (TRP) t > 0: x=0 0≤y≤M -λ·∂T/∂x = q y=0 0≤x≤L -λ·∂T/∂y = q • which gives a solution of the type T = φ(x,t) + ψ(y,t), with for t = 0: φ = 0 and ψ = 0, for t > 0, ∂φ/∂x = -q/λ for x=0, ∂ψ/∂y = -q/λ for y=0 • The result (see also next page): VST rz18 Non-stationary diffusion in 2-D /5 • Temperature field T(x,y,t) in a (half-infinite) corner at time t = 0.25·L 2 /a. At the corner (0,0) the temperature Transport processes (TRP) is T c = (4·q/λ)·√(a·t/π). Isotherms are given as 2·T/T c VST rz18

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 10/48 3b 9/48 3b

11/48 3b Transport processes (TRP)         lim 0 r s a  t Heat is taken up at x = ξ VST rz18 12/48 3b Transport processes (TRP) VST rz18

Transport processes (TRP) Transport processes (TRP) see next slide… VST rz18 VST rz18 14/48 3b 13/48 3b

Moving front problems • The value for constant k in (3.73) follows via d 2 2 d Transport processes (TRP)  ( f ( x ))  making use of : erf ( f ( x )) . e . f ( x )  dx dx 2  x   T ( T T ) 2 1 4 a t a s 1  a . . e . and similar for medium b k  x  2 a t erf ( ) a 4 a a d ξ dk t k   gives with boundary condition (3.68) & dt dt 2 t 2 2  x  x   ( T T ) 2 1 k ( T T ) 2 1 4 a t 4 a t  s 1 a     s 0 b . . . e . r . . . . e . a s a b k k   2 a t 2 t 2 a t erf ( ) a erf ( ) b 4 a 4 a a b 2     and use x and also k to obtain a transceden tal equation for k t (that is not a function of x or t ! ) VST rz18 16/48 3b Transport processes (TRP) VST rz18

17/48 3b Transport processes (TRP) see next two slides… VST rz18 Moving front & mass transfer ξ c c A0 c B0 Transport processes (TRP) c A • Chemical reaction A + B  P x • Diffusion coefficient D is for A in reaction product P • At the reaction front, x = ξ:    c dV . c A . c d d A B B    0  D c . for surface A B 0  x A . dt A . dt dt   x For example: • Some solutions for H 2 + unsaturated fat  saturated fat. (3.80), z = k/√(4D) : VST rz18

Moving front & mass transfer • Alternatively to the case above, species B may also have a noticable diffusion coefficient D B , giving a Transport processes (TRP) result similar to a thermal process. • With boundary condition the result will be - see Figure 3.7 above. • Here again k from a transcedental equation: VST rz18 20/48 3b Transport processes (TRP) δ = ξ - x d VST rz18

21/48 3b Transport processes (TRP) Heat is released at x = ξ VST rz18 22/48 3b Transport processes (TRP) see next slide… VST rz18

Moving front & integral method     2 2 x 2 • From (3.85): *)     2 x  Transport processes (TRP) • Integral which leads to (3.86)    2      T d d T            dx Tdx T ( , t ). T ( , t ). a dx   2 t t dt dt  x          • First term in (3.86) gives, using *): δ – δ + δ/3 = δ/3 • Last term gives, using *) (2/δ-0)-(2/δ-2/δ) = 2/δ • The ”boundary condition at the solidification front” to be used to give (3.88) is (3.83); this gives    d         .( T T ). r . . .( T T ) q s 0 s 1 s  x dt VST rz18 24/48 3b Transport processes (TRP)         r r Y X  s   s     t c q t t c q t p p see next slide… VST rz18

Moving front & integral method  c q • is the dimensionless thickness of the p   X   r molten layer s Transport processes (TRP) • Dimensionless group Ste is known as Stefan number • Long times τ→∞ give a linear relation between X and τ:     2 Ste  2 Ste             X or X Ste 1    Ste 3 Ste 1 Ste 3   and the front moves with constant velocity dξ/dt: Note: Figure 3.9 is for Ste = 2 • At the start X decreases with τ because some liquid solidifies on the cold solid. This gives the minimum dX/dτ =0 for Y = 2·Ste VST rz18 26/48 3b Transport processes (TRP) VST rz18

27/48 3b Transport processes (TRP) Note: Figure 3.9 is for Ste = 2 VST rz18 28/48 A classroom exercise - 3 • Scrap steel at T 0 = 50°C, melting temperature T s = 1450°C is put into molten steel at 1550°C. Heat is tranferred from the liquid to Transport processes (TRP) solid material by convection with heat transfer coefficient α = 5000 W/m 2 ·K. C • Calculate, using the material data given below, the Stefan number Ste, and the time to melt 0.1 m and 0.2 m, respectively, from the scrap steel. The dimensionless expression for relatively long times may be used (see course material below eq. 3-93) :     2 2              X Ste or X Ste Ste 1  Ste 3 1  3  For liquid steel : a = 5.7·10 -6 m 2 /s ; λ = 20 W/m·K, ρ = 7500 • kg/m 3 . For solid steel : a = 1·10 -5 m 2 /s ; λ = 30 W/m·K, ρ = 7500 kg/m 3 . Melting heat : r s = 2.8·10 5 J/kg. • answer : Ste = 2 ; t ( ξ = 0.1 m) = 1965 s ; t ( ξ = 0.2 m) = 3226 s VST rz18

Transport processes (TRP) Transport processes (TRP) see next slide… VST rz18 VST rz18 30/48 3b

Diffusion & source terms • For a cylinder, with symmetry around θ, and L→∞:     T T q   a r Transport processes (TRP)     t r r r ρ c p • (3.95) into (3.94) gives   T T ' dG q          a T ' a F    t t dt ρ c p which gives, selecting only the terms that are a function of (x,y,z,t), an equation (3.96) for T’(x,y,z,t) and an equation (3.97) with the source term, F(x,y,z) and G(t) • The result is (3.95) T(x,y,z,t) = T’(x,y,z,t) + F(x,y,z) + G(t) VST rz18 32/48 3b Transport processes (TRP) with constants C 1 , C 2 . VST rz18

Transport processes (TRP) Transport processes (TRP) = h( T(r) – T(r > R) ) = T(r) VST rz18 VST rz18 34/48 3b 33/48 3b

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 36/48 3b 35/48 3b

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 38/48 3b 37/48 3b

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 40/48 3b 39/48 3b

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 42/48 3b 41/48 3b

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 44/48 3b 43/48 3b