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

Transport processes (TRP)

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Transport processes – Part 3a

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

• tel. 3223 ; ron.zevenhoven@abo.fi

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more general: T=T1

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more general: θ = (T –T1)/(T0 –T1)

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Orthogonality

) cos( ) sin( ) ( cos

1 2 k k k k

d        



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Note: if the source gives transport in

• nly one direction, then take 2x in

that direction and 0x in opposite direction

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) , ( ) , ( ) ( ) ( ) ( ) ( 1 t x c Q t x T x c Q x T Dirac dx x dx Q T c Tdx c Q

p p p p

              

  

        

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b is a position ! But: note the Second Law of Thermodynamics!!!

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

 

t

dt dt dT t t x G ' ' ) ' , (

p w p w

c dt dT dt dT c dt dQ    

" "

' ' ' "     

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2

t t a r q  

~ (3.22) ~ (3.23)

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   

1 1 2 4 4 1 " ) ( ) ( 2 4 4 1 " 4 4 4 1 " 4 1 " : ) 39 . 3 (

4 4 4 4 4 4

2 2 2 2 2 2

       

           

 

          at at e c Q erf erf at at e c Q at y d e at at e c Q dy e e at c Q

at x p at x p at y at x p at y at x p

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at T at T dx dT : x @ at e T dx dT

at x

             

 1 1 4 1

2 1 2 2 1 2

2

) ( 2 ) (

2

dx x derfc e dx x derf

x

  





Penetration depth

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dt dc dx d D dx dc dt d D dt d

" m

    

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~ (2.87)

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Example contact temperature

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Contact temperature – example /1

• A (human) tongue may freeze to a bridge-rail

depending on the material.

• Consider the contact temperature for two

types of bridge-rail material: steel: λb = 40 W/(m.K), ab = 10-5 m2/s wood: λb = 0.2 W/(m.K), ab = 1.2·10-7 m2/s

• Tongue data: Ta0 = 35°C, λa = 0.55 W/(m.K),

aa = 1.7·10-7 m2/s

• Bridge temperature: Tb0 < 0°C
• Calculate Tb0 giving contact temperature 0°C

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Contact temperature – example /2

• Use (equation 3.59):
• This gives

for steel: Tb0 = - 3.7°C for wood: Tb0 = - 81°C

• This shows that a wooden bridge need not

be a problem.

6 6

10 7 1 55 10 7 1 55 35

 

             . . . .

b b b b b

a T a

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x = L x = b x = 0

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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

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