Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

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