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

▶
transport processes part 3b
SMART_READER_LITE
LIVE PREVIEW

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

Nov 07, 2023 189 likes •437 views

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

slide-1
SLIDE 1

Transport processes (TRP)

VST rz18

Transport processes – Part 3b

Ron Zevenhoven Åbo Akademi University Thermal and Flow Engineering / Värme- och strömningsteknik

  • tel. 3223 ; ron.zevenhoven@abo.fi

Transport processes (TRP)

VST rz18 2/48

slide-2
SLIDE 2

Transport processes (TRP)

VST rz18

Non-stationary diffusion in 2-D

3/48

Transport processes (TRP)

VST rz18

Non-stationary diffusion in 2-D /1

  • For diffusion in two dimensions, with constant

material properties:

  • which requires 1 initial condition for the x,y plane + 4

boundary conditions.

  • Using variables ζ = x/√(at) and η= y/√(at):
  • with for example starting and boundary conditions:

t = 0: x≥0, y≥0 : T = T1 t > 0: x=0, y>0 : T = T0 x>0, y=0 : T = T0 x=∞, y>0 : T = T1 x>0, y=∞ : T = T1

               

2 2 2 2

y T x T a t T

2 2 2 2

                  T T T T ½ ½

slide-3
SLIDE 3

Transport processes (TRP)

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 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) = T1 over the whole (x,y) range, for example (0L, 0M), typically give solutions of the type T(t,x,y) = T1 + (T0-T1)·(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.

Transport processes (TRP)

VST rz18

Non-stationary diffusion in 2-D /3

  • Example case, start/boundary conditions:

t = 0: 0≤x≤L, 0≤y≤M T = 0 t > 0: x=0 & x=L, 0≤y≤M T = T0 y=0 & y=M, 0≤x≤L T = T0

  • Using the solution given in § 2.2:
  • For large t, using only the first eigenvalue (n = 0)

including L = ∞, M = ∞

slide-4
SLIDE 4

Transport processes (TRP)

VST rz18

Non-stationary diffusion in 2-D /4

  • Example case, start/boundary conditions:

t = 0: 0≤x≤L, 0≤y≤M T = 0 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):

Transport processes (TRP)

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·L2/a. At the corner (0,0) the temperature is Tc = (4·q/λ)·√(a·t/π). Isotherms are given as 2·T/Tc

slide-5
SLIDE 5

Transport processes (TRP)

VST rz18 9/48 3b

Transport processes (TRP)

VST rz18 10/48 3b

slide-6
SLIDE 6

Transport processes (TRP)

VST rz18 11/48 3b

t r

a s

         lim

Heat is taken up at x = ξ

Transport processes (TRP)

VST rz18 12/48 3b

slide-7
SLIDE 7

Transport processes (TRP)

VST rz18 13/48 3b

Transport processes (TRP)

VST rz18 14/48 3b

see next slide…

slide-8
SLIDE 8

Transport processes (TRP)

VST rz18

Moving front problems

  • The value for constant k in (3.73) follows via

) ! t

  • r

x

  • f

function a not is (that k for equation tal transceden a

  • btain

to t also and x use and 2 1 . . 2 . ) 4 ( ) ( . 2 . 2 1 . . 2 . ) 4 ( ) ( . 2 & (3.68) condition boundary with gives b medium for similar and 2 1 . . 2 . ) 4 ( ) ( ) ( . . 2 )) ( ( :

  • f

use making

2 4 2 4 2 1 4 2 1 2 )) ( (

k t a e a k erf T T t k r t a e a k erf T T t k dt t dk dt dξ t a e a k erf T T x T x f dx d e x f erf dx d

b t b a x b s b a s a t a a x a s a a t a a x a s a x f

            

   

        

Transport processes (TRP)

VST rz18 16/48 3b

slide-9
SLIDE 9

Transport processes (TRP)

VST rz18 17/48 3b

see next two slides…

Transport processes (TRP)

VST rz18

Moving front & mass transfer

  • Chemical reaction A + B  P
  • Diffusion coefficient D is for A in reaction product P
  • At the reaction front, x = ξ:
  • Some solutions for

(3.80), z = k/√(4D) :

A surface for dt d c dt A d c A dt A c dV x c D

B B B x A

       

 

. . . . .

c x cB0 cA0 cA ξ For example: H2 + unsaturated fat  saturated fat.

slide-10
SLIDE 10

Transport processes (TRP)

VST rz18

Moving front & mass transfer

  • Alternatively to the case above, species B may also

have a noticable diffusion coefficient DB, giving a 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:

Transport processes (TRP)

VST rz18 20/48 3b

δ = ξ - xd

slide-11
SLIDE 11

Transport processes (TRP)

VST rz18 21/48 3b

Heat is released at x = ξ

Transport processes (TRP)

VST rz18 22/48 3b

see next slide…

slide-12
SLIDE 12

Transport processes (TRP)

VST rz18

Moving front & integral method

  • From (3.85): *)
  • Integral which leads to (3.86)
  • 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

2

2 2 2          x x

  

           

                dx x T a dt d t T dt d t T Tdx t dx t T

2 2

). , ( ). , ( q T T dt d r x T T

s s s

            ) .( . . ). .(

1

Transport processes (TRP)

VST rz18 24/48 3b

see next slide…

t X q c r t t Y q c r t

p s p s

              

slide-13
SLIDE 13

Transport processes (TRP)

VST rz18

Moving front & integral method

  • is the dimensionless thickness of the

molten layer

  • Dimensionless group Ste is known as Stefan number
  • Long times τ→∞ give a linear relation between X and

τ: and the front moves with constant velocity dξ/dt:

  • 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

    

s p

r q c X

 

1 3 2 1 3 2                     Ste Ste Ste X Ste Ste Ste X

  • r

Note: Figure 3.9 is for Ste = 2

Transport processes (TRP)

VST rz18 26/48 3b

slide-14
SLIDE 14

Transport processes (TRP)

VST rz18 27/48 3b

Note: Figure 3.9 is for Ste = 2

Transport processes (TRP)

VST rz18

A classroom exercise - 3

  • Scrap steel at T0 = 50°C, melting temperature Ts = 1450°C is put

into molten steel at 1550°C. Heat is tranferred from the liquid to solid material by convection with heat transfer coefficient α = 5000 W/m2·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) :

  • For liquid steel : a = 5.7·10-6 m2/s ; λ = 20 W/m·K, ρ = 7500

kg/m3. For solid steel : a = 1·10-5 m2/s ; λ = 30 W/m·K, ρ = 7500 kg/m3. Melting heat : rs = 2.8·105 J/kg.

  • answer : Ste = 2 ; t (ξ = 0.1 m) = 1965 s ; t (ξ = 0.2 m) = 3226 s

28/48

 

1 3 2 1 3 2                 Ste Ste X Ste Ste X  

  • r
slide-15
SLIDE 15

Transport processes (TRP)

VST rz18

Transport processes (TRP)

VST rz18 30/48 3b

see next slide…

slide-16
SLIDE 16

Transport processes (TRP)

VST rz18

Diffusion & source terms

  • For a cylinder, with symmetry around θ, and L→∞:
  • (3.95) into (3.94) gives

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)

p

c ρ q r T r r r a t T          

p

c ρ q F a ' T a dt dG t ' T t T            

 

Transport processes (TRP)

VST rz18 32/48 3b

with constants C1, C2.

slide-17
SLIDE 17

Transport processes (TRP)

VST rz18 33/48 3b

Transport processes (TRP)

VST rz18 34/48 3b

= h( T(r) – T(r > R) ) = T(r)

slide-18
SLIDE 18

Transport processes (TRP)

VST rz18 35/48 3b

Transport processes (TRP)

VST rz18 36/48 3b

slide-19
SLIDE 19

Transport processes (TRP)

VST rz18 37/48 3b

Transport processes (TRP)

VST rz18 38/48 3b

slide-20
SLIDE 20

Transport processes (TRP)

VST rz18 39/48 3b

Transport processes (TRP)

VST rz18 40/48 3b

slide-21
SLIDE 21

Transport processes (TRP)

VST rz18 41/48 3b

Transport processes (TRP)

VST rz18 42/48 3b

slide-22
SLIDE 22

Transport processes (TRP)

VST rz18 43/48 3b

Transport processes (TRP)

VST rz18 44/48 3b

slide-23
SLIDE 23

Transport processes (TRP)

VST rz18 45/48 3b

r c r q c r T c r T r

2 2 3 2

6 ) ( :           

) 3 ( 3

2 3 env R r

T qR c h Rq r T       

  

Transport processes (TRP)

VST rz18 46/48 3b

slide-24
SLIDE 24

Transport processes (TRP)

VST rz18 47/48 3b

Transport processes (TRP)

VST rz18

Sources used

(besides course book Hanjalić et al.)

  • Beek, W.J., Muttzall, K.M.K., van Heuven, J.W. ”Transport phenomena”

Wiley, 2nd edition (1999)

  • R.B. Bird, W.E. Stewart, E.N. Lightfoot ”Transport phenomena” Wiley,

New York (1960)

  • * C.J. Hoogendoorn ”Fysische Transportverschijnselen II”, TU Delft /

D.U.M., the Netherlands 2nd. ed. (1985)

  • * C.J. Hoogendoorn, T.H. van der Meer ”Fysische Transport-

verschijnselen II”, TU Delft /VSSD, the Netherlands 3nd. ed. (1991)

  • D. Kaminski, M. Jensen ”Introduction to Thermal and Fluids

Engineering”, Wiley (2005)

  • S.R. Turns ”Thermal – Fluid Sciences”, Cambridge Univ. Press (2006)

* Earlier versions of Hanjalić et al. book but in Dutch

48/48 3b