Physical transport phenomena /1 Transfer of mass and/or energy in a - - PDF document

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Transport processes – Part 1

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

• tel. 3223 ; ron.zevenhoven@abo.fi

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Introduction / re-wrap of concepts

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Physical transport phenomena /1

• Transfer of mass and/or energy in a system

that is not in thermodynamic equilibrium, towards such equilibrium.

• Systems are usually not very far away from

equilbrium, which results in (practically) linear driving forces: transport = coefficient × driving force

– heat flux (W/m2)= conductivity (W/m2.K)×temperature gradient (K/m) et cetera.

T

heat

     "

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Physical transport phenomena /2

• Continuum approach:

a small volume dV where system properties are constant

– For example dx = 0.1 µm  dV = 10-21 m3 still contains in liquid water ~106 molecules

• Not considered here: cross-correlations such

as

– Mass transfer = coefficient × temperature gradient (“thermal diffusion”) – For example Seebeck effect, Peltier effect – See also so-called ”irreversible thermodynamics”

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Fourier’s Law /1

• In a non-moving medium (i.e. a solid, or

stagnant fluid) in the presence of a temperature gradient, heat is transferred from high to low temperature as a result of molecular movement: heat conduction

(sv: värmeledning)

• For a one-dimensional temperature gradient

ΔT/Δx or dT/dx, Fourier’s Law gives the conductive heat transfer rate Q through a cross-sectional area A (m2). If λ is a constant: with thermal conductivity λ, unit: W/(mK)

(sv: termisk konduktivitet eller värmeledningsförmåga)

) (W/m (W)

2

dx dT A Q " Q dx dT A Q           

Pictures: T06

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Fourier’s Law /2

• For a general case with a 3-dimensional temperature

gradient T = (∂T/∂x,∂T/∂y,∂T/∂z), Fourier’s Law gives (for constant λ) for the heat flux Q” = - λ T

• The temperature field inside the

conducting medium can be written as T = T(t, x) with time t and 3-dimensional location vector x

• For stationary (sv: stationärt, tidsinvariant)

heat transfer ∂T/∂t = 0 at each position x

• The heat transfer vector is perpendicular

(sv: vinkelrätt) to the isothermal surfaces

• Note that material property λ is, in fact, a

function of temperature: more accurately Q” = -

λ(T)T

∆ ∆

.

Figure: KJ05

∆

Q is a vector with direction - T

∆

. .

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Non-steady heat conduction

where in principle heat Q is a 3-dimensional vector Q that creates (or is the result of !) a vector temperature gradient:

(in Cartesian coordinates)

• Non-steady or transient (sv: övergående) heat

conduction through a stagnant medium depends not

• nly on heat conductivity λ but also on heat capacity c

(or cp, cv). A general energy balance for mass m gives

t T c m Q Q

• ut

in

                        z T , y T , x T T

Picture: ÖS96

. .

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Transient heat conduction 1-D /1

• For 1-dimensional transient heat

conduction in a balance volume dV with mass dm = ρ·dV = ρ·A·dx :

2 2 2 2

x T a t T x T A x x T A t T A c x T A x Q t T c A dx x Q t T c dm Q Q

• ut

in

                                                                

• Q

Law s Fourier' with A ρ dm/dx with     

x w L dx Q . A = L·w

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Transient heat conduction 1-D /2

• The initial and boundary conditions

(sv: start- och randvillkor) determine a heat

transfer process

• The three most important cases are:

t for t)) T(0,

• (T

h x t) T(0,

• and

t for T T(x,0) : t at h convection surface

• f

change Sudden 3. t for Q x t) T(0,

• and

t for T T(x,0) : t at Q flux heat surface

• f

change Sudden 2. t for T t) T(0, and t for T T(x,0) : t at T T e temperatur surface

• f

change Sudden 1.

surr " x " x 1 1

                        

 

x w L dx Q . A = L·w x w L dx Q . Q . A = L·w

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Transient heat conduction 1-D /3

• Case 1: Assume a material with flat boundary at x=0,

infinite length in x-direction, with T=T0 at all x

• At time t≥0 the temperature at x=0 is increased to

T=T1 and heat starts to enter (diffuse into) the

• material. At x→∞, T stays at T0.

conditions initial and boundary      

2 2

x T a t T

Picture: BMH99

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Transient heat conduction 1-D /4

• With dimensionless variables

θ = (T-T0)/(T1-T0) and ξ = x / (4at)½ this gives the following solution:

) (y erf d e d e T T T T

y at x

          

 

    4 1 1

2 2

2 2 1 with

ÖS96: erf(x) ≈ 1 - exp(- 1.128x - 0.655x2 - 0.063x3)

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Transient heat conduction 1-D /5

• At x = 0 the slope of the

penetration profile lines equals ∂T/∂x = -(T1-T0)/(πat)½ where x = (πat)½ is referred to as penetration depth.

• Fourier number Fo is (for heat transfer) defined as

Fo = at/d2 = t /(d2/a)) for a medium with thickness d

• Fo gives the ratio between time t and the penetration time d2/a
• The penetration depth concept is valid for Fo < 0.1

Picture: BMH99

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Diffusion and heat conduction

• Heat conduction is in principle diffusion of heat
• Since a ”temperature balance” does not exist, an

energy balance must be used: T → ρcpT (unit: J/m3)

Fick’s Law Fourier’s Law

p p p p x , heat

c ρ λ a dx T c ρ d a dx T c ρ d c ρ λ " Φ    y diffusivit thermal with

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Internal friction in fluid flow /1

• Diffusion of momentum subscript ”xy” means in y-direction in

plane of fixed x

• Kinematic viscosity = dynamic viscosity/density, ν = η/ρ

xy y y y xy momentum

dx v d

• dx

v d

• dx

dv

• 

          

,

"

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Internal friction in fluid flow /2

• Concentration, c, temperature, T, and energy, E, are

scalars, and their gradient is a vector dc/dx or c = (∂c/ ∂x, ∂c/ ∂y, ∂c/ ∂z), etc.

• Velocity is a vector v, for example v = (vx, vy, vz) and

it’s gradient is a (second order) tensor: dvx/dy (gradient

• f vx in y-direction)

                                      z v z v z v y v y v y v x v x v x v v

z y x z y x z y x

) ( . z v y v x v v : note

z y x

         

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Internal friction in fluid flow /3

• v results in compressive stresses xx, yy and

zz and shear stresses xy, xz, yz, zx, yx and zy:

etc. dy v d dy dv dy v d dy dv

z z yz x x yx

; ;                

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The course book; Chapters 1 – 6 are used for this course

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VST rz18 22/46 1 Note here: W > 0 if work is done BY the system.

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Note: mass = density · volume m = ρ·V  dm = ρ·dV + dρ·V thus: dm = 0 ≠ dV =0

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Left: Cartesian (x,y,z) Centre: Cylindrical (r,θ,z) with r2 = x2+y2 Right: Spherical (r,φ,θ) with r2 = x2+y2+z2

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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)
• R. Zevenhoven ”Principles of process engineering” (Processteknikens

grunder), course material ÅA (compendium Aug. 2013, 214 pp.):

http://users.abo.fi/rzevenho/PTG%20Aug2013.pdf

• R. Zevenhoven ”Massöverföring & separationsteknik” (2016),

”Processteknik” (2017) course material ÅA

* Earlier versions of Hanjalić et al. book but in Dutch 46/46 1