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

1/46 1 Transport processes (TRP) Transport processes – Part 1 Ron Zevenhoven Åbo Akademi University Thermal and Flow Engineering / Värme- och strömningsteknik tel. 3223 ; ron.zevenhoven@abo.fi VST rz18 2/46 1 Transport processes (TRP) Introduction / re-wrap of concepts VST rz18

3/46 1 Physical transport phenomena /1 • Transfer of mass and/or energy in a system that is not in thermodynamic equilibrium, Transport processes (TRP) 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/m 2 )= conductivity (W/m 2 .K)×temperature gradient (K/m)  "      T heat et cetera . VST rz18 4/46 1 Physical transport phenomena /2 • Continuum approach: a small volume dV where system properties Transport processes (TRP) are constant – For example dx = 0.1 µm  dV = 10 -21 m 3 still contains in liquid water ~10 6 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” VST rz18

Fourier’s Law /1 • In a non-moving medium (i.e. a solid, or stagnant fluid) in the presence of a Transport processes (TRP) 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 (m 2 ). If λ is a constant:  dT Q dT   2         Q A (W) Q " (W/m ) dx A dx with thermal conductivity λ, unit: W/(mK) Pictures: T06 (sv: termisk konduktivitet eller värmeledningsförmåga) VST rz18 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 . Transport processes (TRP) ∆ (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 Figure: KJ05 function of temperature: . . Q is a vector ∆ more accurately Q” = - λ(T)T ∆ with direction - T VST rz18

Non-steady heat conduction • Non-steady or transient (sv: övergående) heat conduction through a stagnant medium depends not only on heat conductivity λ but also on heat capacity c Transport processes (TRP) (or c p , c v ). A general energy balance for mass m gives  T       Q Q m c in out  t . 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)      T T T     T , ,   Picture: ÖS96    x y z   VST rz18 Transient heat conduction 1-D /1 A = L·w • For 1-dimensional transient heat w conduction in a balance volume Transport processes (TRP) dV with mass dm = ρ·dV = ρ·A·dx : .  L   Q T Q         Q Q dm c dx in out   t x    T Q          with dm/dx ρ A A c   t x dx  T      with Fourier' s Law Q - A x  x   T       - A  2 2      T T T T  x                 c A A a    2 2 t x  t  x x VST rz18

Transient heat conduction 1-D /2 A = L·w A = L·w w w • The initial and boundary conditions (sv: start- och randvillkor) determine a heat Transport processes (TRP) . . . L L transfer process Q Q Q • The three most important cases are : dx dx x x   1. Sudden change of surface temperatur e T T at t 0 : 0 1     T(x,0) T for t 0 and T(0, t) T for t 0 0 1 "   2. Sudden change of surface heat flux 0 Q at t 0 :  x 0  T(0, t) "      T(x,0) T for t 0 and - Q for t 0  0 x 0  x   3. Sudden change of surface convection 0 h at t 0 :  T(0, t)       T(x,0) T for t 0 and - h (T - T(0, t)) for t 0 0 surr  x VST rz18 Transient heat conduction 1-D /3 • Case 1: Assume a material with flat boundary at x=0, infinite length in x-direction, with T=T 0 at all x Transport processes (TRP) • At time t≥0 the temperature at x=0 is increased to T=T 1 and heat starts to enter (diffuse into) the material. At x→∞, T stays at T 0 . 2   T T  a  2 t  x  boundary and initial conditions Picture: BMH99 VST rz18

Transient heat conduction 1-D /4 • With dimensionless variables θ = (T-T 0 )/(T 1 -T 0 ) Transport processes (TRP) and ξ = x / (4at) ½ this gives the following solution: x 4 at  T T 2  2      1   1 e d  T T  1 0 0 y 2  2     with e d erf ( y )  0 ÖS96: erf(x) ≈ 1 - exp(- 1.128x - 0.655x 2 - 0.063x 3 ) VST rz18 Transient heat conduction 1-D /5 • At x = 0 the slope of the penetration profile lines Transport processes (TRP) equals ∂T/∂x = -(T 1 -T 0 )/(πat) ½ where x = (πat) ½ is referred to as penetration depth. • Fourier number Fo is (for heat transfer) defined as Fo = at/d 2 = t /(d 2 /a)) for a medium with thickness d • Fo gives the ratio between time t and the penetration time d 2 /a • The penetration depth concept is valid for Fo < 0.1 Picture: BMH99 VST rz18

Diffusion and heat conduction Transport processes (TRP) Fick’s Law Fourier’s Law • Heat conduction is in principle diffusion of heat • Since a ”temperature balance” does not exist, an energy balance must be used: T → ρc p T (unit: J/m 3 ) d ρ c T d ρ c T λ λ  p  p  Φ " - - a with thermal diffusivit y a heat , x ρ c dx dx ρ c p p VST rz18 Internal friction in fluid flow /1 Transport processes (TRP) • Diffusion of momentum subscript ”xy” means in y-direction in plane of fixed x • Kinematic viscosity = dynamic viscosity/density, ν = η/ρ   dv d v d v  y y y         - - - " momentum xy xy ,  dx dx dx VST rz18

Internal friction in fluid flow /2 • Concentration, c, temperature, T, and energy, E, are scalars, and their gradient is a vector dc/dx or Transport processes (TRP)  c = (∂c/ ∂x, ∂c/ ∂y, ∂c/ ∂z), etc. • Velocity is a vector v, for example v = (v x , v y , v z ) and it’s gradient is a (second order) tensor: dv x /dy (gradient of v x in y-direction)    v   v v  y  x z note :      x x x     v   v v v   v v y x z   v y     x   z v . ( )    y y y      x y z  v     v v y x z       z z z  VST rz18 Internal friction in fluid flow /3 •  v results in compressive stresses  xx ,  yy and  zz and shear stresses  xy ,  xz ,  yz ,  zx ,  yx and Transport processes (TRP)  zy :   dv d v dv d v x x z z               etc. ; ; yx yz dy dy dy dy VST rz18

Transport processes (TRP) Transport processes (TRP) The course book; are used for this Chapters 1 – 6 course VST rz18 VST rz18 18/46 1 17/46 1

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 20/46 1 19/46 1

Transport processes (TRP) Transport processes (TRP) BY the system. W > 0 if work is done Note here: VST rz18 VST rz18 22/46 1 21/46 1

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 24/46 1 23/46 1

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 26/46 1 25/46 1

27/46 1 Transport processes (TRP) Note: mass = density · volume m = ρ·V  dm = ρ·dV + dρ·V thus: dm = 0 ≠ dV =0 VST rz18 28/46 1 Transport processes (TRP) VST rz18

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 30/46 1 29/46 1

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 32/46 1 31/46 1

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 34/46 1 33/46 1

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 36/46 1 35/46 1

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 38/46 1 37/46 1

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 40/46 1 39/46 1

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 42/46 1 41/46 1