Numerical Heat and Mass Transfer

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1 Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 03 Finned Surfaces Fausto Arpino

2 Outline Introduction Straight fin with constant circular cross section Long fins Fins with adiabatic tip Fins with convection at the tip Remarks Limits for application of finned surfaces Fin efficiency 2

3 Introduction Heat transfer by convection between a surface and the fluid surrounding it can be increased by attaching to the surface thin strips of metal called fins. Heat transfer from one system to the environment is proportional to the heat transfer coefficient U, the exchange area A and temperature difference DT: Q = U A ΔT Finned surfaces present an increased exchange area surface, so enhancing the heat transfer to the environment. A fin is a solid of a fixed shape stretching out from the primary surface to the surrounding environment. When heat transfer takes place by convection from both interior and exterior surfaces of a tube or a plate, generally fins are used on surface where the heat transfer coefficient is low. 3

4 Introduction A large number of finned surfaces applications is available form the everyday life: Air treatment units Fan coils 4

5 Introduction Head of a two-stroke motorbike engine Laptop CPU cooling system 5

6 Introduction CPU liquid cooling system 6

7 Introduction Fins are mainly classified as straight or circular. The fin profile is the perimeter of the cross section (perpendicular to the fin axis in the case of a straight fin and containing the symmetry axis in the case of a circular fin). 7

8 Straight fin of uniform circular cross section To determine the temperature profile distribution through a fin, it is necessary to develop the governing equation by performing an energy balance of a differential volume element in the fin. The governing equation is written on the basis of the following hypothesis ü Steady state regime. ü The material thermal conductivity k is assumed to be constant. ü Absence of heat generation in the fin. ü Heat transfer coefficient h is assumed to be constant and uniform over the fin surface. ü Cross thermal gradient (in radial direction) are assumed to be negligible. ü Uniform temperature at the fin base. ü Negligible contact thermal resistance between the fin and the primary surface. Heat gain by conduction ka dt dx = kadt dx + d dx kadt dx dx + hpdx T T Heat out by convection Heat out by conduction Fin perimeter 8

9 Straight fin of uniform circular cross section Rearranging the equation: d 2 T It can be observed that: dx 2 = m2 T T ka where m = hp d ( T T ) = dt It follows that the above equation can be rewritten as: d 2 T dx 2 = m2 T with T = T T 9

10 Straight fin of uniform circular cross section The written equation is called one-dimensional fin equation for fins of constant cross section. Such equation is a linear, homogeneous, second-order ordinary differential equation (ODE) with constant coefficient. The general solution may be taken in the form The final solution assumes the form: λ 2 m 2 = 0 λ 1,2 = ±m T x T = C 1 emx +C 2 e mx The two constants C 1 e C 2 are determined from the two Boundary Conditions (BCs) specified for the fin problem (fin base and fin tip). The first BC comes from the basis hypothesis of constant and uniform temperature at the fin base: T ( 0) = T s ( T s T ) = C 1 +C 2 10

11 Long fins For a sufficient long fin, it is reasonable to assume that the temperature at the fin tip approaches T. The two integration constant C 1 and C 2 are then T ( ) = T C 1 = 0 and C 2 = T s While the temperature field in the fine is given by Finally, the heat flow rate through the fin is: Q a = ka dt dx x=0 = T T s = e mx hp T dx = hpka T s 0 11

12 Fins with adiabatic tip In the case, since the heat transfer area at the fin tip is generally small compared to the lateral area of the fin for heat transfer. The heat loss at the fin tip is then assumed negligible, and the boundary condition at the fin tip is: T = cosh m L x T s cosh ml dt dx x=l = 0 C 2 = C 1 e 2mL C 1 = T s 1+ e 2mL C 2 = T s 1+ e 2mL Q a = ka dt dx x=0 = hpka ( T s ) tgh( ml) x tanh(x)

13 Fins with convection at the tip This is the physically more realistic boundary condition. In the following, the heat transfer coefficient at the fin tip is indicated ad h L ka dt = h dx L A( T L ) x=l T T s = cosh m( L x) + h L mk sinh m L x cosh( ml) + h L mk sinh( ml) Q a = sinh( ml) + h L km cosh( ml) + h L hpka T s km cosh( ml) sinh( ml) 13

14 Remarks consistency of the temperature profiles in the three analyzed cases It can be easily noticed that the temperature profiles determined by applying the three different boundary conditions are consistent: Fins with convection at the tip h L = 0 Fins with adiabatic tip L T T s = cosh m( L x) + h L mk sinh m L x cosh( ml) + h L mk sinh( ml) T = cosh m L x T s cosh ml Long fins T T s = e mx 14

15 Remarks qualitative temperature profiles 15

16 Limits for application of finned surfaces Dividing by cosh(ml) the fin heat transfer equation becomes: obtaining: L = 0 Q a = sinh( ml) + h L km cosh( ml) + h L hpka T s Q a = hpka T s tanh ml 1+ h L km km + h L cosh( ml) sinh( ml) km tanh( ml) L Q a = Q = ha T s Limits Q a = hpka( T s ) 16

17 Limits for application of finned surfaces From the obtained relations it can be observed that the convenience of a finned surface is related to the following condition: Q a ( L ) > Q a ( L = 0) hpka ( Ts T ) > ha ( T ) s T kp k 1 = = > 1 ha hl Bi Bi <1 It means that, in order to be sure to increase the convective heat transfer of a finned surface compared to the same surface without fins, the fin thermal resistance must be lower than the surface resistance 17

18 Fin efficiency The heat transfer analysis has been performed for a variety of fin geometries, and the results are available in the literature in terms of a parameter called fin efficiency η = Q a Q id = Q a Exchange surface of any fin shape hs T s Assuming that the fin temperature is uniform and equal T s corresponds to the case of infinite thermal conductivity material. The fin efficiency definition allows a fast calculation of the heat flow rate through the fin by using appropriate charts. Q a = η hs ( T s ) 18

19 Fin efficiency fin with adiabatic tip tanh ml ml ml tanh(ml)/ll ml 19

20 Fin efficiency Efficiency of axial fins where the fins thickness y varies with the distance x from the root of the fin where y=t 20

21 Fin efficiency y y ~ x 2 t/2 x L c = L A p = Lt/3 60 L η f (%) 40 L c = L + t/2 A p = L c t L t/2 y ~ x 20 y t/2 x L c = L A p = Lt/2 L /2 L c (h/ka p ) 1/2 T. Bergman, A.S. Lavine, F.P. Incropera, D.P. Dewitt, Introduction to Heat Transfer 21

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