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

2. 2  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 STEADY-STATE, QUASI-1D HEAT CONDUCTION

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4. 4  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 STEADY-STATE, QUASI-1D HEAT CONDUCTION

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

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

7. 7  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. 8  Special case: k,, A, P are all constant wrt x; then: Hence: where the governing dimensionless parameter  effective diameter of fin STEADY-STATE, QUASI-1D HEAT CONDUCTION

9. 9 TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER  Region initially at uniform temperature T 0  Suddenly altered by changing boundary temperature or heat flux  Methods of solution:  Combination of variables (self-similarity)  Fourier method of separation of variables

10. 10  Combination-of-Variables:  Two important special cases:  Semi-infinite wall, with sudden change in boundary temperature to a new constant value (T 0 to T w > T 0 )  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: where   thermal diffusivity of medium TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER

11. 11  Combination-of-Variables:  Case 1: Sudden change in boundary temperature:  Resulting temperature profiles are always “self- similar”, i.e., [T w – T(x,t)]/[T w – T 0 ] depends on x and t only through their combination and TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER

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

13. 13  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: where   circular frequency 2  f of imposed heat flux  e.g., for aluminum (  0.92 cm 2 /s ), f = 3000 rpm,  h ~ 1 mm TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER

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

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

16. 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. 17  Separation-of-Variables: Transient energy diffusion in a solid of finite thickness TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER

18. 18  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) TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER

19. 19  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 TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER

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

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

22. 22  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: TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER

23. 23 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) TRANSIENT HEAT DIFFUSION: THERMAL BOUNDARY LAYER

24. 24 STEADY LAMINAR FLOWS  (Re. Pr) 1/2 or (Ra h Pr) 1/4 not negligibly small => energy convection & diffusion both important  Re or Ra h 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. 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. 26  Pohlhausen, 1992: (in the absence of viscous dissipation, when T ∞ and T w are constants)  Local dimensionless heat transfer coefficient THERMAL BOUNDARY LAYER ADJACENT TO FLAT PLATE

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

28. 28  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 THERMAL BOUNDARY LAYER ADJACENT TO FLAT PLATE

29. 29 CONVECTIVE HEAT TRANSFER FROM/ TO ISOLATED SPHERE  Forced convection (constant properties, Newtonian fluid):  For Re < 10 4 and Pr ≥ 0.7, a good fit for data yields: (reference length  d w )  Frequently applied to nearly-isolated liquid droplets in a spray  Analogous correlations available for isolated circular cylinder in cross-flow

30. 30 CONVECTIVE HEAT TRANSFER FROM/ TO ISOLATED SPHERE  Natural convection:  For Ra h < 10 9 in the absence of forced convection:  Local htc’s highly variable (rear wake region quite different from upstream “separation”)  For buoyancy to be negligible, Gr h 1/4 /Re 1/2 << 1  Not true in CVD reactors, in large-scale combustion systems