FLUID MECHANICS

C

HAPTER

6:

I

NTRODUCTION

TO

C

ONVECTION

For conduction through solids, we derived the general heat diffusion

equation based on all the mechanisms by which heat can be transferred or generated in a solid control volume:

= +

   

 

   

 

∂ ∂ ∂

∂ +

   

 

∂ ∂ ∂

∂ +

   

 

∂ ∂ ∂

∂

q z

T k z y

T k y x

T k

x & _t

T c_p

∂ ∂

ρ

(Ein – Eout)cond + Egen = Estorage

qx q_x+dx

qz

qz+dz

qy

qy+dy

What happens if instead we use a fluid

(gas, liquid) control volume? All these terms remain BUT… heat transfer by

bulk fluid flow also becomes possible:

k T q cp T_t + cpu ⋅∇T ∂

∂ =

+

∇2 _& ρ ρ

(Ein – Eout)cond + Egen = Estorage + Efluid flow

where u ⋅∇T = u ∂_∂T_x + v ∂_∂T_y + w ∂_∂T_z

u = velocity in x-direction; v = velocity in

y-direction; w = velocity in z-direction

During our studies of conduction, we treated convection as a boundary

condition (i.e. if all the convective

medium has T = T∞ ! dT/dx = dT/dy

= dT/dz = 0 in the fluid itself)

However, if dealing with heat transfer through fluids, the inertial (flow) term becomes important to the analysis.

This is particularly true for convection analysis given that convection

involves heat transfer via both

conduction and advection, the

physical movement of heated (or

cooled) fluid from a solid interface or within a fluid.

Advection may arise from different types of fluid flows:

" External flows

• Forced, i.e. flowing at a

significant velocity (pumps, fans)

• Free, i.e. stagnant flow where

diffusion effects dominate

" Internal flows (pipes & conduits)

Thus, in order to develop a framework for convection analysis, we need to

know something about how fluids flow under these different circumstances ! FLUID MECHANICS

F

LUID

M

ECHANICS

The flow properties of a fluid are

defined by its density (

ρ

) and viscosity

(

µ

) [Note: both strong functions of T]

Density – mass per unit volume

! Property of atoms/molecules

High density (liquids) ! larger inertial force required to move a given

volume of fluid (F = ma)

Viscosity - internal resistance of a

fluid to flow (unit: Pa s = N/m2_s)

! Intermolecular interactions

! Molecular tangling (polymers)

High viscosity (oil, polymers)! large

inertia/resistance to flow

! Flow in highly viscous fluids may generate frictional heat

(viscous dissipation)

Velocity Boundary Layer

When a velocity gradient exists in a flow field, the fluid experiences shear forces. A measure of the shear force per unit area is called shear stress,τ.

∂u/∂y = fluid velocity

For a fluid traveling in the x-direction over a surface, the force diagram

acting on the fluid is shown below:

y u Area

Force

∂ ∂ =

  



= µ

τ

FLUID FLOW

Friction

Applied Force

SURFACE

Key Feature: No-Slip Surface where net force = 0 (applied flow force =

surface frictional drag) !

! !

! wall velocity u = dx/dt|x=0 = 0

Thus, the velocity profile of a flowing fluid over or within an object exhibits a parabolic shape near the interface:

u∞ = free stream fluid velocity parallel

to the surface (no frictional drag)

δ = boundary layer thickness (defined as y value at which u = 0.99u∞)

The parabolic shape of the velocity profile is called the “velocity

boundary layer” of fluid flow.

δ

_u

u∞

u∞

Plug Flow Surface Flow

x

y _{Surface shear stress}

Frictional effect No surface effect Bulk fluid flow

Consider the case where a plug flow of fluid contacts a wall (x = 0). With increasing distance from the wall

interface, the viscosity (frictional drag

within fluid) transmits the surface

frictional drag deeper into the fluid ! development of boundary layer

Boundary layer development is

governed by the fluid velocity and the frictional drag at the wall interface,

which can be expressed in terms of

the local friction coefficient Cf:

high Cf = high friction

Plug Flow

δ

δ(x)

Fully developed flow Steady State

x

2 2

∞ ≡

u

C _f s

ρ τ

Thermal Boundary Layer

Thermal boundary layers form as a result of two contributions:

! Advection contributions to

convection change as the fluid

velocity changes as a function of y

• directly proportional to u profile

! Conductive contributions to

convection vary according to the fluid’s thermal conductivity (kf)

• independent of u profile

• if heat can’t diffuse into fluid (u=0

at surface), advection is useless!

δ

t

q x

y

T_{s u = 0}

u_∞

Fl

u

id

La

y

er

s

T_∞ T_∞ No heat flow between these two layers

δ

t is the thickness of the thermal boundary layer, defined as the y

position at which

↑

δ

t ! ↓

∂

T/

∂

y ! slower heat transfer

Q: Will

δ

t and

δ

be the same value?

At the surface, u=0 ! conduction only Fourier’s Law applies:

-heat must first diffuse out of solid surface!

Newton’s Law of Cooling (convection from surface):

By energy balance, q”s1 = q”s2; thus:

h depends on kf

(conduction) and boundary

layer temperature gradient

(advection)

Thus, h varies with the fluid velocity

profile defining the boundary layer

99 . 0 = −

− ∞

T T

T T

s s

0 "

1

=

∂ ∂ −

=

y f

s

y T k

q

(

− _∞

)

= h T T q_s"_{2 s}

(

− _∞

)

=

∂ ∂ −

=

T T

y T k

h

s

y f

0

We have already applied different h

values in the same convective fluid for fin analysis (comparing h vs. hL)

h = average convection coefficient

hL = localized convection coefficient

To account for localized h values due to boundary layer variation, the total heat transfer rate over any arbitrary surface is the integral of the local heat fluxes:

Define the heat transfer rate in terms of h , the average convection

coefficient over surface: q = hA_s(T_s −T_∞)

where =

∫

⋅

s

A

s s

dA h

A

h 1

(general) or

(for a flat plate) !main problem of convection analysis

(

)

_∫

∫

= − ∞

=

A

s A

s

s T T hdA

dA q

q "

∫

=

L

dx x h L h

0

) ( 1

C

ONVECTION

P

ROBLEMS

Step 1. Solve the appropriate

equations of fluid mechanics to define the velocity profile

1) Start with the general continuity equation for fluid flow through a

differential control volume (2D flow):

(flow in = flow out)

2) Apply Newton’s 2nd _{Law of Motion}

(ΣF acting on control volume = net rate at which momentum leaves the control volume). In the x-direction:

0 = ∂ ∂ + ∂ ∂ y v x u X y u x v x P y v v x u

u _ +

     ∂ ∂ + ∂ ∂ + ∂ ∂ − =       ∂ ∂ + ∂ ∂ 2 2 2 2 µ ρ

Net rate of = momentum

flow from CV

Net + pressure force

Step 2. Solve the appropriate form of the heat diffusion

equation to define the temperature profile

For steady, 2D flow of an

incompressible fluid with constant properties:

where the viscous dissipation term (mechanical friction during fluid flow that generates heat) is given by:

… solutions can be very ugly!

Φ + +       ∂ ∂ + ∂ ∂ =       ∂ ∂ + ∂ ∂ µ ρ q y T x T k y T v x T u

c_p &

2 2 2

2

Net rate of heat = flow due to bulk fluid flow

(advection)

Net heat + flow by conduction Net heat generation by viscous dissipation Net heat +

Step 3. Estimate the convection

heat transfer coefficient h based

on the energy balance

Since the velocity at the wall surface is zero, Ts,wall = Tfluid, y=0 ! heat

transfer by conduction only at surface. Thus, by energy balance at y>0:

(

wall fluid

)

wall

T T

h y

T k

q" _ = − _∞,

  

 

∂ ∂ −

=

Solve for h to get average convection coefficient over a given surface

Step 4. Use Newton’s Law of

Cooling to estimate heat flow/flux

or

Heat Flux Heat Flow

We need

δ

T/

δ

y ∝

δ

u/

δ

y ! functional approximations of v/T profile solutions

(

Twall T fluid

)

h

q"= − _∞, q = hA_s

(

T_wall −T_{∞, fluid}

)

L

AMINAR

/T

URBULENT

F

LOW

The velocity and thermal boundary layers strongly depend on the nature of the fluid flow over the surface.

Slower Velocities:

LAMINAR FLOW –

! fluid will travel along continuous stream-lines

Adjacent layers of fluid move pass one another along parallel paths.

Faster Velocities: TURBULENT FLOW

! fluid flow lines are discontinuous

around the object Swirling flow regions (“eddies”) occur Random 3D motion of large fluid V

How does the flow regime impact surface friction and h?

Laminar: Ordered flow;

∂

u/

∂

y ↓ as x↑

since

δ

increases as a function of x

Turbulent: Disordered flow; typically

comprised of distinct layers

# Close to surface: viscous sublayer

– diffusion dominant, linear

∂

u/

∂

y

# Intermediate: buffer layer – both

diffusion/turbulent mixing occur

# Outer layer: turbulent mixing

predominant, near-zero

∂

u/

∂

y

How do these flow regimes impact heat transfer?

!

Compare the surface velocity profiles in laminar and turbulent flow:

Laminar Flow: Velocity

gradient (y direction) occurs over full

boundary layer

Turbulent Flow:

Significant y velocity gradient observed only in viscous sublayer

>

(turbulent) (laminar) >

(turbulent) (laminar) 0 = ∂ ∂ = y s y u µ

τ ₂ 0

2 2 2 ∞ = ∞       ∂ ∂ = ≡ u y u u

C _f s y

Laminar-to-turbulent flow is triggered by surface roughness, surface

vibrations, or stream fluctuations.

The transition between laminar and turbulent regimes is identified based on a dimensionless number, the

Reynolds Number (Re), the ratio

between inertial and viscous forces.

µ

ρ

µ

ρ

µ

ρ

_vL

L v

L v v

dx v d

dx dv v

F F

viscous

inertial = ≈ =

=

2 2

2

Re

v = bulk fluid velocity over object/surface

In a tube: Re = ρu_µ∞D

Laminar Transition Turbulent

Re < 2100 2100 – 10000 > 10000

Over a plate: Re = ρu_µ∞x

Laminar Transition Turbulent

Re < 5x105 _5x105 _{– 3x10}6 _{> 3 x 10}6

D

IMENSIONLESS

N

UMBERS

See Table 6.2 for other dimensionless numbers used in convection analysis.

For laminar flow, the thickness of the velocity boundary layer (

δ

) and the temperature boundary layer (

δ

t) are related by the Prandtl Number (Pr)

Notes: cp, k values are for the fluid

ν = µ/ρ is the kinematic viscosity

Pr therefore represents the relative effectiveness of momentum transport

in the velocity boundary layer to that

of energy transport (by heat diffusion)

in the thermal boundary layer.

f p

p

f k

c

c k

y diffusivit thermal

y diffusivit

momentum µ

ρ ρ

µ

= =

= Pr

The boundary layer thicknesses in the laminar flow regime are related to the Prandtl number by the expression:

n = positive exponent (often ~1/2)

For gases, low k, cp,

µ

; Pr ~ 1 !

δ

≅

δ

t For water, mid k,

µ

; high cp

Pr ~ 7 !

δ

>

δ

t

# Limiting step of heat transfer is conduction of heat into water (i.e. through no-slip layer at surface) For oils, low k, high cp,

µ

Pr > 1000 !

δ

>>

δ

t

# Advection effective, but relatively little heat diffuses into oil

For molten metal, high k, mid cp,

µ

Pr < 0.01!

δ

<<

δ

t

# Large amount of heat diffuses into metal but advection is less important for heat dissipation

n

t

Pr ≈ δ

δ

Dimensionless characteristic lengths, velocities, and temperatures can also be defined, as in Chapter 5:

Dimensionless length L = characteristic length

Dimensionless velocity

V = upstream velocity = u∞

Dimensionless Temperature

These dimensionless parameters are useful since they reduce every

variable to a unitless variable with a value between 0-1

# simplify boundary condition evaluation in ODE solving # enable combination of these

parameters in any form (e.g.

RePr would also be dimensionless – we will use this later)

L x

x* ≡

L y

y* ≡

V u

u* ≡

V v

v* ≡

s s

T T

T T

T

− − ≡

∞ *

We can also apply these definitions to simplify the expressions defining other dimensionless parameters.

For example, rewriting the friction

factor Cf in terms of the dimensionless parameters u = u∞u* and y = Ly*:

# Reduce the number of variables in a given problem.

o Three variables (ρ,

µ

, u∞) are

be reduced to one single,

dimensionless variable with a universal applicability to various fluids/flow conditions (Re).

# Differential boundary conditions

now u*=0 at y* = 0 and u* = 1 at

y* = 1 ! easier to evaluate

0 * 0

* 2

0 * 2

* *

Re 2

* * 2

2

= =

∞ ∞

=

∞ ∂

∂ =

∂ ∂ =

∂ ∂ ≡

y L

y y

f

y u y

u u

L u y

u u

C

ρ

µ

ρ

µ

Alternately, substituting dimensionless parameters into expressions may

expose new dimensionless constants of use to further simplify problems.

Substituting T = T*(T∞ - Ts) + Ts and

y = Ly* into the definition of the

convection coefficient derived earlier:

!

!

!

!

Simplifying,

Nusselt Number

(Nu)

May be expressed locally (Nu=hx/kf) or averaged over a surface (Nu=hL/kf)

f s

f

s

k hL L

T T

k

T T

h

Nu =

− −

=

∞ ∞

/ ) (

) (

Physical definition: Nu is the ratio of

convection to conduction heat transfer through the fluid normal to surface

Compare to the Biot Number:

s

k hL

Bi =

Biot Number – conduction in solid

# Use k value for solid, L is the dimension over which the

conduction occurs (x for a plate, r

for infinite cylinder/sphere, or

V/As for other shapes)

Nusselt Number – heat conduction

occurs through liquid phase

# Use k value for fluid, L is the dimension associated with

boundary layer development (L

for plates, D for cylinder/sphere, or V/As for an arbitrary shape)

We can apply these dimensionless

numbers to evaluate the velocity and temperature profiles and calculate an average h value for convection.

Re-write the velocity and temperature profile equations in terms of

dimensionless parameters to simplify boundary conditions for evaluation (see Table 6.1 for full expressions):

Momentum:

Temperature:

dP*/dx* = pressure gradient on surface (determined by geometry of object)

We can then solve these functions with reference to dimensionless

parameters to reduce the number of variables which need be considered.

(

_{*, *,Re ,} * _*

)

*

dx dP

y x f

u = _L

(

_{*, *,Re ,Pr,} * _*

)

*

dx dP

y x f

T = _L

For a prescribed geometry, dP*/dx* is fixed (pressure distributions can be

independently obtained by considering flow conditions in the free stream)

Thus, at y* = 0 (surface) and remembering that

the thermal boundary layer can be described as a function of Nu:

Or, integrating over all x for an average surface property:

Thus, a problem involving 7 variables

(h, kf, cp, ρ,

µ

, V, L) is reduced to one

containing 3 universally applicable

dimensionless constants (Nu, Re, Pr).

(

x*,Re_L,Pr

)

f

Nu =

Nu k

hL y

T

f y

= =

∂ ∂

=0 * *

*

(

)

f L

k L h f

Nu = Re ,Pr =

The basic dimensionless parameter correlations developed here will be applied to solve for h (and thus

effectively solve convection problems) using expressions of the form

Nu=ARem_Prn_{, where A, m, and n are}

all constants for a given geometry (i.e. fixed dP*/dx*), flow regime, etc.

Such expressions are derived:

(i) From rigorous mathematical

solutions for a limited number of problems (laminar flow)

(ii) From semi-empirical analysis (turbulent flow)

(iii) Correlations of experimental

data (for majority of problems)

We will learn specific correlations for different geometries and flow

conditions (laminar vs. turbulent,

external vs. internal) in Chapters 7-8.

EXAMPLE: Consider a car windshield of length L=800mm, thickness t=6mm, and

k = 1.4W/mK.

The car moves at 111km/h through

ambient air at -15°C. Hot air at 50°C is blown on the inside of the window to

defrost the windshield, a process which requires that the inner surface

temperature of the windshield remain at least at 10°C. If the average convection coefficient on the outside of the

windshield is correlated by the expression

NuL = 0.030ReL0.8 Pr1/3, what is the

smallest value of the internal convection coefficient (hi) which will maintain a

defrosted windshield?