HEI 3.1 Fundamentals Part1 Correctedvucoe.drbriansullivan.com/wp-content/uploads/Convection-Summary-1.pdfConvection • Convection is the mode of energy transfer between a solid surface

Hypersonic Educational Initiative

Convection

• Convection is the mode of energy transfer between a solid surface and the adjacent liquid or gas that is in motion.

• Convection is commonly classified into three sub-modes:– Forced convection,– Natural (or free) convection,– Change of phase (liquid/vapor,

solid/liquid, etc.)

Convection = Conduction + Advection(fluid motion)

Sect 3.1.1


Convection

• The rate of convection heat transfer is expressed by Newton’s law of cooling as

• h is the convection heat transfer coefficient in W/m2°C.

• h depends on variables such as the surface geometry, the nature of fluid motion, the properties of the fluid, and the bulk fluid velocity.

( ) (W)conv s sQ hA T T∞= −&

Sect 3.1.1


Convection Boundary Condition

[ ]2 2( , ) ( , )T L tk h T L t Tx ∞

∂− = −

∂

Heat conductionat the surface in aselected direction

Heat convectionat the surface in

the same direction=

and[ ]1 1

(0, ) (0, )T tk h T T tx ∞

∂− = −

∂

Sect 3.1.1.1


Convection

Frederick FergusonNorth Carolina A&T State University

September 13, 2007

Sect 3.1.1.2


Physical Mechanism of Convection

• Conduction and convection are similar in that both mechanisms require the presence of a material medium.

• But they are different in that convection requires the presence of fluid motion.

• Heat transfer through a liquid or gas can be by conduction or convection, depending on the presence of any bulk fluid motion.

• The fluid motion enhances heat transfer, since it brings warmer and cooler chunks of fluid into contact, initiating higher rates of conduction at a greater number of sites in a fluid.

( )12 TThq −−=&

Sect 3.1.1.2


• Experience shows that convection heat transfer strongly depends on the fluid properties:– dynamic viscosity µ, – thermal conductivity k, – density ρ, and – specific heat cp, as well as the – fluid velocity V.

• It also depends on the geometry and the roughness of the solid surface.• The rate of convection heat transfer is observed to be proportional to the

temperature difference and is expressed by Newton’s law of cooling as

• The convection heat transfer coefficient h depends on the several of the mentioned variables, and thus is difficult to determine.

( ) 2 (W/m )conv sq h T T∞= −&

Newton’s law of cooling

Sect 3.1.1.2


• All experimental observations indicate that a fluid in motion comes to a complete stop at the surface and assumes a zero velocity relative to the surface (no-slip).

• The no-slip condition is responsible for the development of the velocity profile.

• The flow region adjacent to the wall in which the viscous effects (and thus the velocity gradients) are significant is called the boundary layer.

Boundary Layer

Sect 3.1.1.2


• An implication of the no-slip condition is that heat transfer from the solid surface to the fluid layer adjacent to the surface is by pure conduction, and can be expressed as

• The convection heat transfer coefficient, in general, varies along the flow direction.

2

0

(W/m )conv cond fluidy

Tq q ky =

∂= = −

∂& &

( ) 0 2 (W/m C)fluid y

s

k T yh

T T=

∞

− ∂ ∂= ⋅

−o

Boundary Layer

Sect 3.1.1.2


The Nusselt Number• It is common practice to nondimensionalize the heat transfer coefficient h

with the Nusselt number:

• Heat flux through the fluid layer by convection and by conduction can be expressed as, respectively:

• Taking their ratio gives

• The Nusselt number represents the enhancement of heat transfer through a fluid layer as a result of convection relative to conduction across the same fluid layer. Note, for Nu=1 pure conduction.

chLNuk

=

convq h T= ∆& condTq kL∆

=&

/conv

cond

q h T hL Nuq k T L k

∆= = =

∆&

&

Sect 3.1.1.2


Classification of Fluid Flows• Viscous versus inviscid regions of flow• Internal versus external flow• Compressible versus incompressible flow• Laminar versus turbulent flow• Natural (or unforced) versus forced flow• Steady versus unsteady flow• One-, two-, and three-dimensional flows

Sect 3.1.1.2


Definition: Surface Shear Stress

• Consider the flow of a fluid over the surface of a plate.• The fluid layer in contact with the surface tries to drag the plate along via

friction, exerting a friction force on it.• Friction force per unit area is called shear stress, and is denoted by τ .• Experimental studies indicate that the shear stress for most fluids is

proportional to the velocity gradient.• The shear stress at the wall surface for these fluids is expressed as

• The fluids that that obey the linear relationship above are called Newtonian fluids.

• The viscosity of a fluid is a measure of its resistance to deformation.

2

0

(N/m )sy

uy

τ µ=

∂=

∂

Sect 3.1.1.2


Definition: Velocity Boundary Layer

• The region of the flow above the plate bounded by δ is called the velocity boundary layer.

• δ is typically defined as the distance y from the surface at which u=0.99V.

• The hypothetical line of u=0.99V divides the flow over a plate into two regions:– the boundary layer region, and– the irrotational flow region.

Sect 3.1.1.2


Definition: Thermal Boundary Layer

• Like the velocity a thermal boundary layer develops when a fluid at a specified temperature flows over a surface that is at a different temperature.

• Consider the flow of a fluid at a uniform temperature of T∞ over an isothermal flat plate at temperature Ts.

• The fluid particles in the layer adjacent assume the surface temperature Ts. • A temperature profile develops that ranges from Ts at the surface to T∞ sufficiently far

from the surface.• The thermal boundary layer ─ the flow region over the surface in which the temperature

variation in the direction normal to the surface is significant.• The thickness of the thermal boundary layer dt at any location along the surface is

defined as the distance from the surface at which the temperature difference T(y=dt)-Ts= 0.99(T∞-Ts).

• The thickness of the thermal boundary layer increases in the flow direction.• The convection heat transfer rate anywhere along the surface is directly related to the

temperature gradient at that location.

Sect 3.1.1.2


Definition: Prandtl Number

• The relative thickness of the velocity and the thermal boundary layers is best described by the dimensionless parameter Prandtl number, defined as

• Heat diffuses very quickly in liquid metals (Pr«1) and very slowly in oils (Pr»1) relative to momentum.

• Consequently the thermal boundary layer is much thicker for liquid metals and much thinner for oils relative to the velocityboundary layer.

Molecular diffusivity of momentumPrMolecular diffusivity of heat

pckµν

α= = =

Sect 3.1.1.2


Laminar and Turbulent Flows

• Laminar flow ─ the flow is characterized by smooth streamlines and highly-ordered motion.

• Turbulent flow ─ the flow is characterized by velocity fluctuations and highly-disordered motion.

• The transition from laminarto turbulent flow does not occur suddenly.

Sect 3.1.1.2


• The velocity profile in turbulent flow is much fuller than that in laminar flow, with a sharp drop near the surface.

• The turbulent boundary layer can be considered to consist of four regions:– Viscous sublayer– Buffer layer– Overlap layer– Turbulent layer

• The intense mixing in turbulent flow enhances heat and momentum transfer, which increases the friction force on the surface and the convection heat transfer rate.

Turbulent Boundary Layer

Sect 3.1.1.2


Reynolds Number

• The transition from laminar to turbulent flow depends on the surface geometry, surface roughness, flow velocity, surface temperature, and type of fluid.

• The flow regime depends mainly on the ratio of the inertia forces to viscous forces in the fluid.

• This ratio is called the Reynolds number, which is expressed for external flow as

• At large Reynolds numbers (turbulent flow) the inertia forces are large relative to the viscous forces.

• At small or moderate Reynolds numbers (laminar flow), the viscous forces are large enough to suppress these fluctuations and to keep the fluid “inline.”

• Critical Reynolds number ─ the Reynolds number at which the flow becomes turbulent.

Inertia forcesReViscous forces

c cVL VLρν µ

= = =

Sect 3.1.1.2


Heat and Momentum Transfer in Turbulent Flow

• Turbulent flow is a complex mechanism dominated by fluctuations,and despite tremendous amounts of research the theory of turbulent flow remains largely undeveloped.

• Knowledge is based primarily on experiments and the empirical orsemi-empirical correlations developed for various situations.

• Turbulent flow is characterized by random and rapid fluctuations of swirling regions of fluid, called eddies.

• The velocity can be expressed as the sum of an average value u and a fluctuating component u’

'u u u= +

Sect 3.1.1.2


• It is convenient to think of the turbulent shear stress as consisting of two parts: – the laminar component, and– the turbulent component.

• The turbulent shear stress can be expressed as

• The rate of thermal energy transport by turbulent eddies is

• The turbulent wall shear stress and turbulent heat transfer

• ─ turbulent (or eddy) viscosity.• kt ─ turbulent (or eddy) thermal conductivity.

' 'turb pq c v Tρ=&

' 'turb u vτ ρ= −

' ' ; turb t turb p tu Tu v q c vT ky y

τ ρ µ ρ∂ ∂= − = = = −

∂ ∂&

Shear Stress and Heat Flux in Turbulent Flows

Sect 3.1.1.2

tµ


• The total shear stress and total heat flux can be expressed as

and

• In the core region of a turbulent boundary layer ─ eddy motion (and eddy diffusivities) are much larger than their molecular counterparts.

• Close to the wall ─ the eddy motion loses its intensity. • At the wall ─ the eddy motion diminishes because of the no-slip condition.

( ) ( )turb t tu uy y

τ µ µ ρ ν ν∂ ∂= + = +

∂ ∂

( ) ( )turb t p tT Tq k k cy y

ρ α α∂ ∂= − + = − +

∂ ∂&

Shear Stress and Heat Flux in Turbulent Flows

Sect 3.1.1.2


• Reynolds Analogy (Chilton─Colburn Analogy) ─ under some conditions knowledge of the friction coefficient, Cf, can be used to obtain Nu and vice versa. It follows that the average Nu and Cf depends on

• These relations are extremely valuable:– The friction coefficient can be expressed as a function of

Reynolds number alone, and – The Nusselt number can be expressed as a function of

Reynolds and Prandtl numbers alone.

• The experiment data for heat transfer is often represented by a simple power-law relation of the form:

( ) ( )3 4Re , Pr ; ReL f LNu g C f= =

Re PrL

m nNu C= ⋅

Empirical Relations for Shear Stress and Heat Flux

Sect 3.1.1.2


The Reynolds analogy can be extended to a wide range of Pr by adding a Prandtl number correction.

,Re (Pr=1)

2L

f xC Nu=Reynolds analogy

1/ 2, 0.664Ref x xC −= 1/3 1/ 20.332Pr Re Pr>0.6xNu =

1 3,

Re Pr 0.6 Pr 602f x xC Nu= < <

Analogies Between Momentum and Heat Transfer

Sect 3.1.1.2

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HEI 3.1 Fundamentals Part1 Correctedvucoe.drbriansullivan.com/wp-content/uploads/Convection-Summary-1.pdfConvection • Convection is the mode of energy transfer between a solid surface