1 / 30

Transient Heat Diffusion: Thermal Boundary Layer

This lecture discusses steady-state, quasi-1D heat conduction, transient heat diffusion, and thermal boundary layers. It covers the combination-of-variables and separation-of-variables methods for solving heat diffusion problems.

stevenperez
Download Presentation

Transient Heat Diffusion: Thermal Boundary Layer

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Advanced Transport Phenomena Module 5 Lecture 20 Energy Transport: Transient Heat Diffusion Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

  2. STEADY-STATE, QUASI-1D HEAT CONDUCTION • Energy diffusion predominantly in one direction • e.g., ducts of slowly varying area, within slender “fins” on gas-side of primary heat-transfer surfaces to increase heat-transfer area per unit volume of heat exchanger • Fin efficiency factor

  3. STEADY-STATE, QUASI-1D HEAT CONDUCTION

  4. STEADY-STATE, QUASI-1D HEAT CONDUCTION • Pin fin of slowly varying area, A(x), wetted perimeter P(x), length L • Losing heat by convection to surrounding fluid of uniform temperature T∞ over entire outer surface • T(x)  cross-sectional-area-averaged fin material temperature • Neglecting transverse temperature nonuniformities

  5. STEADY-STATE, QUASI-1D HEAT CONDUCTION • Fin/ fluid heat exchange rate for slice of fin material between x and x+Dx where  dimensional perimeter-mean htc • Steady-flow energy balance on semi-differential control volume, A(x) . Dx

  6. STEADY-STATE, QUASI-1D HEAT CONDUCTION • Dividing both sides by Dx and passing to the limit Dx  0, and introducing the Fourier law: leads to

  7. STEADY-STATE, QUASI-1D HEAT CONDUCTION • Boundary values: • At x = 0, T = T(0) (root temperature) • At fin tip (x = L), some condition is imposed, e.g., (dT/dx)x=L = 0 (negligible heat loss at tip), then: where numerator could also be written as

  8. STEADY-STATE, QUASI-1D HEAT CONDUCTION • Special case: k, , A, P are all constant wrtx; then: Hence: where the governing dimensionless parameter  effective diameter of fin

  9. TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER • Region initially at uniform temperature T0 • Suddenly altered by changing boundary temperature or heat flux • Methods of solution: • Combination of variables (self-similarity) • Fourier method of separation of variables

  10. TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER • Combination-of-Variables: • Two important special cases: • Semi-infinite wall, with sudden change in boundary temperature to a new constant value (T0 to Tw > T0) • Semi-infinite wall with periodic heat flux at boundary • In both, only one spatial dimension, one simple PDE T(x,t) • In the absence of convection, volume heat sources, variable properties: wherea  thermal diffusivity of medium

  11. TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER • Combination-of-Variables: • Case 1: Sudden change in boundary temperature: • Resulting temperature profiles are always “self-similar”, i.e., [Tw – T(x,t)]/[Tw – T0] depends on x and t only through their combination and

  12. TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER • Combination-of-Variables: • Case 1: Sudden change in boundary temperature: • Thermal effects are confined to a thermal BL of nominal thickness When t  0, dh  0, wall heat flux  ∞ (~ t-1/2), accumulated heat flow up to time t ~ t1/2

  13. TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER • Combination-of-Variables: • Case 2: Periodic heat flux at x = 0: • e.g., cylinder walls in a reciprocating (IC) engine • Thermal penetration depth frequency-dependent: wherew circular frequency 2pf of imposed heat flux • e.g., for aluminum (a0.92 cm2/s), f = 3000 rpm, dh ~ 1 mm

  14. TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER • Separation-of-Variables: Transient energy diffusion in a solid of finite thickness • -b < x < b • Initial temperature, T0 • Outer surfaces @ x= +/- b • Suddenly brought to Twat t = 0+ • bc’s become:

  15. TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER • Separation-of-Variables: Transient energy diffusion in a solid of finite thickness • Define non-dimensional variables: • satisfying

  16. TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER • Separation-of-Variables: Transient energy diffusion in a solid of finite thickness • IC: T*(x*,0) = 1 • BC’s: • T*( 1, t*) = 0 • (T*/  x*)y*=0 = 0 • Fourier’s solution of separable form:

  17. TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER • Separation-of-Variables: Transient energy diffusion in a solid of finite thickness

  18. TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER • Separation-of-Variables: Transient energy diffusion in a solid of finite thickness • Inserting ODE into earlier PDE: Equation satisfied if corresponding terms on LHS & RHS equal– i.e., for each integer n or (collecting like terms)

  19. TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER • Separation-of-Variables: Transient energy diffusion in a solid of finite thickness • LHS function of t* alone • RHS function of x* alone • Hence, both sides must equal same constant: and

  20. TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER • Separation-of-Variables: Transient energy diffusion in a solid of finite thickness • Hence: • and

  21. TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER • Separation-of-Variables: Transient energy diffusion in a solid of finite thickness • Constants are selected by applying appropriate boundary conditions: • Bn = 0 • Cn eigen values • Dn chosen to satisfy initial conditions, yielding:

  22. TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER • Separation-of-Variables: Transient energy diffusion in a solid of finite thickness • Non-explicit BC example: • Heat flux from surrounding fluid approximated via a dimensional htc, h • Yields linear interrelation:

  23. TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER T*(x*, t*) within solid then depends on the non-dimensional parameter, Biot number: (ratio of thermal resistance of semi-slab to that of external fluid film)

  24. STEADY LAMINAR FLOWS • (Re . Pr)1/2 or (RahPr)1/4 not negligibly small => energy convection & diffusion both important • Re or Rah below “transition” values => laminar flow • Stable wrt small disturbances • Steady if bc’s are time-independent • Examples: • Flat plate (external) • Isolated sphere (external) • Straight circular duct (internal flow)

  25. THERMAL BOUNDARY LAYER ADJACENT TO FLAT PLATE • Forced convection (constant properties, Newtonian fluid): • T(x,y) satisfies: (neglecting streamwise heat diffusion) and are known Blasius functions of similarity variable

  26. THERMAL BOUNDARY LAYER ADJACENT TO FLAT PLATE • Pohlhausen, 1992: (in the absence of viscous dissipation, when T∞ and Tw are constants) • Local dimensionless heat transfer coefficient

  27. THERMAL BOUNDARY LAYER ADJACENT TO FLAT PLATE Reference heat flux in forced-convection surface-transfer Comparing prevailing heat flux to this reference value yields a dimensionless htc, Stanton number, Sth: When Pr = 1, Sth= cf/2 • Strict analogy between momentum & heat transfer for forced-convection flows with negligible streamwise pressure gradients • In general, thermal BL thicker than momentum BL

  28. THERMAL BOUNDARY LAYER ADJACENT TO FLAT PLATE • Natural convection: • Velocity & T-fields are coupled • Need to be solved simultaneously • In case of constant properties, • Buoyancy force where fluid thermal expansion coefficient Local dimensionless heat transfer coefficient Area-averaged heat-transfer coefficient on a vertical plate of total height L

  29. CONVECTIVE HEAT TRANSFER FROM/ TO ISOLATED SPHERE • Forced convection (constant properties, Newtonian fluid): • For Re < 104 and Pr ≥ 0.7, a good fit for data yields: (reference length  dw) • Frequently applied to nearly-isolated liquid droplets in a spray • Analogous correlations available for isolated circular cylinder in cross-flow

  30. CONVECTIVE HEAT TRANSFER FROM/ TO ISOLATED SPHERE • Natural convection: • For Rah < 109 in the absence of forced convection: • Local htc’s highly variable (rear wake region quite different from upstream “separation”) • For buoyancy to be negligible, Grh1/4/Re1/2 << 1 • Not true in CVD reactors, in large-scale combustion systems

More Related