Transport processes Part 2a Ron Zevenhoven bo Akademi University - PDF document

1/58 2a Transport processes (TRP) Page 58 added 22.1.2018 Transport processes – Part 2a Ron Zevenhoven Åbo Akademi University Thermal and Flow Engineering / Värme- och strömningsteknik tel. 3223 ; ron.zevenhoven@abo.fi VST rz18 2/58 2a Transport processes (TRP) VST rz18

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 4/58 2a 3/58 2a

5/58 2a Transport processes (TRP) Thermal diffusivity α = λ /(ρꞏc p ) VST rz18 6/58 2a Transport processes (TRP) more general: T=T* more general: θ = (T -T*)/(T 0 -T*) VST rz18

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 8/58 2a 7/58 2a

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 10/58 2a 9/58 2a

11/58 2a Orthogonality Note :     sin xdx cos x c    cos xdx sin x c Transport processes (TRP) sin( Ax )    cos ( Ax ) dx c A  sin( Ax )         cos ( Ax ) dx  ½ ½ cos ( Ax ) dx ½ x c  A          m m       cos ( ) π xdx sin( ) π x ( ) m         ( m ) π ( m ) π      for integer m 0,1,2,3...      and sin( π / ) 1, sin(3 π / ) 1 , etc.      m                cos ( ) π xdx ½ x sin( m ) π x        ( m ) π         sin( π ) 0, sin( π ) 0, sin(3 π ) 0 , etc. VST rz18 12/58 2a Transport processes (TRP) VST rz18

13/58 2a EXAMPLE Transport processes (TRP) VST rz18 14/58 2a EXAMPLE Transport processes (TRP) This can be since the concrete has an 8x higher heat capacity ρ∙c p , i.e. enthalpy / volume. VST rz18

Transport processes (TRP) Transport processes (TRP) more general: h(T-T*) θ = (T -T*)/(T 0 -T*) more general: VST rz18 VST rz18 16/58 2a 15/58 2a

Transport processes (TRP) Transport processes (TRP) Note: µ 0 = 0 VST rz18 VST rz18 18/58 2a 17/58 2a

19/58 2a Transport processes (TRP) VST rz18 20/58 2a Separation of variables – example /1 • Q: A steel plate at 900°C is cooled by spraying 40°C water on one side of it. This Transport processes (TRP) gives convective heat transfer with constant heat transfer coefficient h = 5000 W/(m 2 .K). The other side of the plate may be considered thermally insulated. • For a plate with thickness d = 4 mm, calculate the temperature on both plate surfaces 5 seconds after the spray cooling has started. • For the steel, assume conductivity λ=20 W/(m.K) and thermal diffusivity a = 6ꞏ10 -6 m 2 /s VST rz18

21/58 2a Separation of variables – example /2 • A: For the Biot number: – Bi = 5000ꞏ0.004/20 = 1; Transport processes (TRP) – First eigenvalue µ 1 = 0.860 (from Figure 2.2) • Using only the first eigenvalue: – @ x=d : T(x=d) = 40+860ꞏ0.73ꞏexp(-0.28ꞏt) – This gives T = 195°C @ t = 5 s – @ x=0 : T(x=0) = 40+860ꞏ1.12ꞏexp(-0.28ꞏt) – This gives T = 277°C @ t = 5 s • It is readily seen that the second eigenvalue can be neglected. VST rz18 22/58 2a Transport processes (TRP) VST rz18

Transport processes (TRP) Transport processes (TRP) = Fourier number, Fo VST rz18 VST rz18 24/58 2a 23/58 2a

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 26/58 2a 25/58 2a

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 28/58 2a 27/58 2a

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 30/58 2a 29/58 2a

31/58 2a Transport processes (TRP) VST rz18 32/58 2a The functions Yk(x) are in practice (in the field addressed by this course) not needed. Transport processes (TRP) VST rz18

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 34/58 2a 33/58 2a

Bessel functions data Transport processes (TRP) • Source: Introduction to Thermal and Fluid Engineering by Deborah Kaminski and Michael Jensen 2005 by John Wiley & Sons, Inc. VST rz18 Using Bessel functions – example /1 • Q: A cylindrical column with diameter d = 0.05 m is initially at T = T 0 when at Transport processes (TRP) time t = 0 suddenly the surface temperature is brought to T = 0 (with respect to some reference temperature). • Similar to the case for a plane surface, determine the time until the centre temperature T c is equalised to 0.05 = (T c -0)/(T 0 -0) VST rz18

Using Bessel functions – example /2 • A: For long times use the first eigenvalue, with n = 0 this gives (see p. 37) Transport processes (TRP)  2 r a t 2      T T J ( µ ) exp( µ ) 0 0 0 0  2 µ J ( µ ) R R 0 1 0 µ 0 = the first zero of J 0 (..), which is 2.405, and with J 1 (2.405) = 0.519  for r = 0: 0.05 = 1.60ꞏexp(-0.0037ꞏt) • This gives the result t = 937 s, which is ~ 2x faster than a plate with d= 0.05 m • The heat flux (W/m 2 ) can be calculated using -λꞏdT/dr and differentiated Bessel functions VST rz18 38/58 2a Separation of variables simplification Fo > 0.2 Transport processes (TRP) (”long times”) VST rz18

1-dimensional transient conduction /1 • Using separation of variables, convective cooling/heating (see above) by a medium flow at Transport processes (TRP) temperature T flow with Bi = h.L char /k, with convective heat transfer coefficient h (Wm 2 .K), characteristic length scale L char (m) and material conductivity k (W/m.K), gives for dimensionless time τ = Fo > 0.2, using only the first eigenvalue λ 1 :  T ( x , t ) T λ x     plane wall : flow C exp( λ τ ) cos(  )    T T L see tabelised data start flow  T ( r , t ) T λ r on next page for      cylinder : flow C exp( λ τ ) J ( )     T T r first eigenvalue λ 1  start flow λ r and constant C 1 sin(  )  T ( r , t ) T r      sphere : flow C exp( λ τ )   λ r  T T  start flow r  VST rz18 1 -dimensional transient conduction /2 Transport processes (TRP) Wall with thickness 2L Cylinder with radius r 0 Sphere with radius r 0 • Source: Introduction to Thermal and Fluid Engineering by Deborah Kaminski and Michael Jensen 2005 by John Wiley & Sons, Inc. VST rz18

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 42/58 2a 41/58 2a

43/58 2a 1 t     F ( ) d F ( p ) p o Transport processes (TRP) VST rz18 44/58 2a More general: T 1 Transport processes (TRP) T 1 T 1 -T 1 + + T 1 / p VST rz18

45/58 2a + T 1 / p Transport processes (TRP) T 1 / p (T 0 - T 1 ) + T 1 / p p. 51 12. T 1 + (T 0 - T 1 )ꞏerfc(..) VST rz18 46/58 2a Transport processes (TRP) (T 0 - T 1 )  pt e 1    dt ,  t p o   1 1  or : £     t  p (T 0 - T 1 ) VST rz18

47/58 2a Transport processes (TRP) T 1 T 1 T 1 T 0 VST rz18 48/58 2a -T 1 / α Transport processes (TRP) (T 0 - T 1 ) (T 0 - T 1 ) + T 1 / p VST rz18

49/58 2a (T 0 - T 1 ) + T 1 / p Transport processes (TRP) =1 – x + x 2 – x 3 + x 4 … (T 0 - T 1 ) + T 1 / p + T 1 (T 0 - T 1 ) VST rz18 50/58 2a (T 0 - T 1 ) + T 1 Transport processes (TRP) ! Fo >> 0.2 or simply (T 0 - T 1 ) (T 0 - T 1 ) Fo > 0.2 + T 1 VST rz18

51/58 2a q = √ p / a Transport processes (TRP) VST rz18 Transformation simplification  2  T 2 T     can be simplified with T r Transport processes (TRP)   2 r r r             d dT dr r dT T dr   T r    T    r T   r r  2   2   T T T     r    2  2 r r r r  2   2  1 T 2 T   2 2  r   r r r r 2 terms reduced to 1 term VST rz18