NATURAL CONVECTION No mechanical force to push the fluid pump, fan etc. No predefined fluid flowrate and velocity can t prescribe Reynolds

Transcription

1 NATURA CONVECTION No mechanical force to psh the flid pmp, fan etc. No predefined flid flowrate and velocit can t prescribe Renolds nmber Flid moves as a reslt of densit difference Flid velocit established as a reslt of the temperatre field Flid can move downward and pward Examples: Cold air cools the egg Warm air heats the can

2 Frther examples: high temperatre Atmospheric circlation low temperatre z ρ(z) T(z) z ρ(z) T(z) low temperatre high temperatre T(z), ρ(z) Atmospheric inversion - no vertical exchange of mass of air Room circlation T(z), ρ(z) Vertical exchange of mass of air Cold window Space heater radiator

3 3 Heated vertical wall Which force makes the flid to raise? The force called the boanc (vztlak) Which role does the gravit pla?? How the velocit is established P.D.E. for momentm conservation ν g x p ρ v x ν g x p ρ v x simpl add gravit in x direction Flid moves pwards Heat flx gravit Individal terms can be expressed in nits of force per mass [N/kg] or in nits of acceleration [m/s ]

4 x v ρ p x g ν Flid moves pwards No movement in - direction p Same p x in the bondar laer and otside it p x ρ ρ p x g g g ρ Volme expansion coefficient β ρ ρ T p or ( ρ ρ) We need temperatre difference how to replace Δρ b ΔT? 0 Pressre difference reslts from the weight of the flid colmn β ρ ρ T ρ T For ideal gas β 4 gravit T

5 5 Flid moves pwards gravit Momentm conservation eqation T a T v x T Energ conservation eqation Continit eqation 0 v x ( ) ν T T gβ v x boanc force So called copled problem can t solve velocit field nless we know temperatre field which is a fnction of velocit field.

6 Heat transfer coefficient Similarit parameter dimensionless nmber. Can t se Renolds nmber flid velocit or flow rate not defined a priori. Grashof nmber for vertical wall Gr gβ ( T T ) ν 3 boanc viscos.force Fnctional relation for Heat Transfer coefficient Nsselt nmber N f ( Gr.Pr ) f(ra ) Ra Raleigh nmber 6

7 aminar verss trblent Under certain conditions, laminar regime can change to trblent. Vertical wall: ( ) 3 T x 9 gβ T Ra Gr.Pr w x,krit x,krit νa 0 Characteristic length is alwas dimension in the direction of the flid movement: Vertical clinder: ength Vertical wall: Height of the wall of the clinder if: d 35 4 Horizontal clinder: Diameter of the clinder Gr 7

8 Horizontal plates inslation A. Upper srface of a cold plate T w <T B. ower srface of a cold plate T w <T B. Upper srface of a hot plate T w >T A. ower srface of a hot plate T w >T Nsselt nmber Ra gβ ( T T ) w νa A. B. Characteristic dimension: 3 N 0,7Ra N 0,54Ra N 0,5Ra Can o sketch graphicall? A/P srface area/srface perimeter 4 0 < Ra < 7 0 Ra

9 Inclined plates Use vertical plate eqations for the pper srface of a cold plate and the lower srface of a hot plate eqations A. Nsselt nmber A. N 0,7Ra 4 Ra gβ ( T T ) w νa 3 Replace g b gcosθ θ angle from the vertical 9

10 Cavities Applications: plate solar collectors, doble glazed windows, sandwich walls, etc. Air trapped inside good inslator Complications: air doesn t remain stationar it moves pwards and downwards 0

11 Bt if Ra gβ Horizontal cavit When the hotter plate at the top no convection occrs pre condction transfer of heat When the hotter plate at the bottom tendenc for the lighter air to rise to the top ( T T ) 3 < aν 708 boanc force too week compared to viscos force still condction heat transfer λ ( T T ) ( T ) α T distance between hot and cold plates αλ/ and N. For Ra > 708 natral convection occrs Bénard cells For Ra > Bénard cells break down trblence occrs

12 For air: Horizontal cavit 4 N 0.95Ra for 0 4 < Ra < Ra for < Ra < 0 7 N 8

13 Vertical cavit For Ra < 000, no natral convection pre condction heat transfer across the cavit N For Ra > 000, natral convection occrs - along the hot srface air rises, along the cold srface air flows down Convection enhances heat transfer As Ra increases, circlation region gets closer to walls, a centre is created with almost no movement 0.8 Pr H N 0. Ra H < < 0 0. Pr H H N 0.4Ra Pr 0 < <

14 Heat Transfer Rate T T Q& αs(t T ) λns α λn It resembles eqation for heat condction T T Q& λ S cond eff Effective condctivit λ eff λ.n Conclsions: Heat transfer rate can be determined from heat condction sing effective thermal condctivit λ eff 4

15 Natral verss forced convection Forced convection mch higher heat transfer coefficients Tendenc to ignore natral convection Error in ignoring natral convection negligible at high velocities Error considerable at low velocities Parameter representing the importance of natral convection If Gr Re < 0, Natral convection negligible Gr Re If Gr 0 Re Forced convection negligible If Gr 0, 0 Re Both convections important 5

16 Natral verss forced convection Natral convection ma help or hrt forced convection depending on relative directions of boanc - indced and forced convection motion Assisting flow sign Opposing flow sign - n n forced N ( N ± N n natral ) Exponent n recommended 3 6