Forced Convection

Convection Convection is classified as i) Natural or free convection ii) Forced convection. Natural convection occurs when a quiescent fluid is exposed to a hot or cold surface. If the surface is hot, the fluid next to the surface will be heated, its temperature will increase and its density will decrease. Due to the decreased density of the fluid next to the surface, it will rise due to buoyancy. If the surface is cold, then the temperature of the fluid will be colder than the bulk fluid, its density will decrease and will fall due to buoyancy. In forced convection, the fluid is forced to flow over a surface or in a tube by external means such as a pump or a fan. Convection is also classified as external or internal depending on whether the fluid is forced to flow over a surface or inside a channel. Forced Convection Concept of boundary layer is central to understanding of forced convection heat transfer. Velocity boundary layer: Consider flow over a flat plate.When the fluid particles make contact with the surface,they assume zero velocity.These particles then act to retard the motion of the particles in the next layer,until at a distance y   from he surface, the effect becomes negligible.

Laminar region u  -free stream velocity, u -velocity at a distance y from plate surface  4.64 For laminar flow, boundary layer thickness is given by,  where, Re , x is x Re , x1/ 2 Reynolds number at distance x from leading edge. ux Reynolds number Re , x  where  -kinematic viscosity of fluid and x-distance from  leading edge. In many cases laminar and turbulent flow conditions occur.In laminar boundary layer the flow is highly ordered.the highly ordered behaviour continues until a transition zone is

reached.Conversion from laminar to turbulent condition occurs.Flow in fully turbulent boundary layer is highly irregular and is characterized by random three dimensional motion of large particles of fluid.The conversion from laminar to turbulent zone depends on Reynolds number.The critical Reynolds number for flow over a plate is Re  5 105 .

Laminar & turbulent region Thermal boundary layer: Just as velocity boundary layer develops when there is fluid flow over a surface, a thermal boundary layer develops if the fluid free stream and plate surface temperatures differ.At the leading edge the temperature profile is uniform. The fluid particles that come into contact with the plate achieve thermal equilibrium at the plate’s surface temperature. In turn these particles exchange energy with those in the adjoining fluid layer and the temperature gradients develop in the fluid. The region of the fluid in which the temperature gradient exists is called thermal boundary layer and its thickness is T .It is TS  T  0.99 where TS -surface temperature defined as the value of y for which the ratio TS  T  of plate. By Polhausen solution for energy equation, T  1/ 3 where Pr C  Prandtl number, Pr  p k

T -free stream temperature, T-velocity at a distance y from plate surface At any distance x from the leading edge the local surface heat flux may be obtained by applying Fourier’s law to the fluid at y=o since at the surface there is fluid motion and energy transfer occurs by conduction. g T q  k y Above surface there is fluid motion and the energy transfer is by convection. By Newton’s law of cooling, g

q  h(TS  T ) Combining the above two equations, T k y h (TS  T ) g

q and h decrease with increasing x. g

Heat transfer rate, q   qdAS   h(TS  T )dAS

Or, q  (TS  T )  hdAS where h -local heat transfer coefficient If h is the average heat transfer coefficient for the entire surface, q  hAS (TS  T ) 1 hdAS Or, h  AS  Flow over flat surface Heat transfer to surface in a flowing fluid is dependent on: 1. Geometry of the body. 2. The position or orientation of the body (parallel, perpendicular to flow). 3. Proximity of other bodies. The heat transfer coefficient varies across the surface of the object. But the average heat transfer coefficient can be determined from an equation of the form: Nu  C Re m Pr1/ 3 where C & m -constants, Nu -Nusselt number, Pr-Prandtl number

The constants are determined through experiments. 5 Laminar flow over a flat plate ( Recr  5 10 ): At a distance x from leading edge, heat transfer coefficient h is given by, 1 1 hx Nu x  x  0.332 Re x 2 Pr 3 k Average Nu  0.664 Re

1 2

Average Nu  0.037 Re

4 5

1 3

L Pr & Heat transfer rate, Q  hA(T  TS ) where A -surface area of plate, Ts -surface temperature, T -flowing medium temperature Turbulent flow over a plate: At a distance x from leading edge, heat transfer coefficient h is given by, 1 4 hx Nu x  x  0.0296 Re x 5 Pr 3 K

L

1

Pr 3 ;

Combined Laminar & Turbulent flow (5 105  Re L  107 ) : 1

4

Average Nu  (0.037 Re 5  871) Pr 3 L Flow across a cylinder Reynolds number is given by, Re 

V D 

In case of other shapes, D is replaced by D 

4 AC P

AC -cross section area, P -perimeter Critical Reynolds number, Recritical  2300 Heat transfer rate, q  hA(Ts  T ) , A =surface area, Cross-flow over cylinders (Fluid properties at average temperature= (Ts  T ) / 2 ) 1

1

4

hD 0.62 Re 2 Pr 3  Re 85 5 Nucyl   0.3  1  ( ) 1  K 28200  ---Churchill and Bernstein equation 2 4    3  1  (0.4 / Pr)    Flow over spheres 2 1   hD  1 Nusph   2   0.4 Re 2  0.06 Re 3 Pr 0.4 (  ) 4 ---Whitaker equation K s  

(Fluid properties at fluid temperature, T ;  s alone at surface temperature, Ts ) hD Nu  e ; De  D for circular crossection, De  4 Ac / P for others ; Ac =cross-sectional area K Flow though tubes q  mC p (Te  Ti ) ;

Constant heat flux( q&) q  q& A ; A =surface area & qA Te  Ti  & p mC Constant surface temp T T LMTD  e i T T & q  h( LMTD ) ; ln s e Ts  Ti  hA & p mC

 hA /( mC )

p Te  Ts  (Ts  Ti )e ; Te  Ti e Laminar flow through tube Nu  4.36 for constant heat flux, Nu  3.66 for constant surface temperature Turbulent flow through tube (for both constant heat flux and constant temperature) q& h(Ts  T f )

Nu  0.023Re0.8 Pr n ; n=0.4 for heating & 0.3 for cooling---- Dittus-Boelter equation