(10) introduction to convection

8/13/2019 (10) Introduction to Convection

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ME 327(10)Introduction to Convection 1 of 51

Heat Transfer

Introduction to Convection Heat and Mass

Transfer


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Convection Fundamental Concepts-Convection Transfer

Consider the flow of a fluid past a flat surface.

The local heat flux is:

Where h is the local heat transfer coefficient (W/m2K). q

and h vary along the surface. The total heat transfer rateover the entire surface is:

Lx

q Ts, AsU, T

TThq s

TTAhq ss where L

0

hdxL

1h

Average

convection heat

transfer coefficient


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Results similar to convection heat transfer may be

obtained for convection mass transfer when a fluid with

molar concentration C A, (kmol/m3) or density A,

(kg/m3

) for species A flows over a surface with uniformconcentration C A,sor density A,sfor species A as shown

below:

The molar flux NA (kmol/m2s) and mass flux mA (kg/ m

2

s) are expressed, respectively as:

Lx

NACA,s, As

U, CA,

dx

,As,AmA CChN ,As,AmA hm

Where hm= local convection mass transfer coefficient

A= McCA= density

MA= molecular weight of species A


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Total molar transfer rate NA(kmol/s) and total mass transfer

rate mA(kg/s) are also expressed, respectively, as follows:

Where = average mass transfer coefficient, m/s

Assuming saturated state of species A at surface temperature

Ts, the density A,may be obtained directly from

thermodynamic tables ( for water: table A.6(text))

At the corresponding saturation pressure Psatthe molar

concentration may be obtained from equation of state for anideal gas

,As,AmA CChN ,As,AmA hm

L

0

hdxL

1h

h

s

sats,A

RT

PC Where R = universal gas constant

= 8.314 kJ / kmol K


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Convection boundary layer

Convection can be forced or natural

It is due to the random motion of molecules (diffusion) and

the bulk motion of the fluid particles Convection could be internal or external

ExamplesAirflow over wings, buildings, a bank of heat

exchangers (external)

1. The Velocity boundary Layer Velocity B.L. Region with velocity gradient dU / dy 0

u

x

Free Stream

U

y

(x)

B.L.

U

s Velocity gradient

due to shear

stress,


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is the boundary layer thickness defined as the value of y at

which u = 0.99 U. At y = dU/dy and are negligible.

The local friction coefficient is calculated from:

The surface frictional drag, FD, is calculated using Cf

Assuming a Newtonian fluid, sis calculated from:

Where is the dynamic viscosity of the fluid, and du/dy is the

velocity gradient perpendicular to the wall.

2

UC

2s

f Wall shear .

Kinetic energy of fluid

0ys

y

u

s

2f

ssD A2

UCAF

also


7/51ME 327(10)Introduction to Convection 7 of 51

2. Thermal Boundary Layer

Thermal B.L. Region with temperature gradient dT/dy

tis the thermal boundary layer thickness, defined as the

value of y at which:

x

T

yt

t(x)U

T

Ts

Isothermal Surface

99.0TT

TT

s

s



The local convective heat transfer coefficient may be

calculated by first finding qs from energy balance at the

surface. Fourier Law (No fluid motion at the surface,

conduction only)

For a constant surface temperature,Ts Ts- T= constant

Since tincreases with x, T/y must decrease with x

Therefore h and q also decrease with x

0ys

y

Tkq

TThq ss

TT

yTk

hs

0yf

also

0yy

T

decreases with xSince



,As,A

AAB

m CC

y

CD

h

,As,A

AAB

my

D

h

Concentration Boundary Layer

Ficks Law

Where DABor Dv= mass diffusion coefficient, m/s2

Mass diffusivity or mass diffusion coefficient has the same

units as thermal diffusivity, (m/s2)

As with heat transfer, we can write

x

yc

c(x)

U

CA,s

Concentration

Boundary Layer

CA,

CA

CA, Free Stream

Mixtureof A & B

y

CDN AABA

yDm AABA

Heat flux (qx)

Fouriers LawMolar or Mass flux

Ficks Law



SUMMARY:

Three boundary layers:

Velocity B.L. ()

Velocity distribution (V)

Wall friction ()

Thermal B.L. (t)

Temperature distribution (T)

Convection heat transfer (h, Nu)

Concentration B.L. (c)

Concentration distribution (C)

Convection mass transfer (hm, Sh)



Problem 6.1



Laminar & Turbulent Flow & Reynolds Number

Convection transfer rates and surface friction depend on

whether the flow is laminar or turbulent. Laminar orturbulent flow is determined by the value of Re

For internal flow: Reynolds Number (Re)

where ReD,cis the critical Reynolds number

For external flow: Reynolds Number (Re)

where Rex,cis the critical Reynolds number

xU

Rex5c

c,x 105xU

Re

DURe

D

2300Rec,D



Laminar Ordered fluid motion with identifiable streamlines on which

fluid particles move.

Turbulent Irregular fluid motion with velocity fluctuations which increase

energy, mass, and momentum transfer. Larger twith flatter

velocity, temperature and concentration profiles.

Transition Turbulent

Laminar

xc

U



Regions in turbulent Boundary Layer

Turbulent regionTransport mainly by

turbulent mixing

Buffer ZoneTransport by diffusion and

turbulent mixing

Viscous or Laminar Sub layerTransport

by diffusion and linear velocity profile

Variation of Velocity B.L.

Thickness () and local heat

transfer coefficient (h) for flowover an isothermal surface

h(x)

(x)

x

h ,

Laminar Transition Turbulent



Problem 6.14



Boundary Layer Equations

Consider a steady , 2-D flow of a viscous incompressible

fluid in a Cartesian coordinate as shown below. We wish to

obtain a set of differential equations that governs the

velocity and temperature distributions in the fluid to solve

V, T, C and (force), q (heat transfer), m (mass transfer).

x

T

U

Thermal B.L.

Concentration B.L.

Velocity B.L.

qs

NA,s

T

U

CA,

CA

y

Ts

t

C

CA, s

Mixture of

A & B

dx

dy



The equations are based on application of conservation

principles on a differential control volume of the fluid as

demonstrated below for conservation of mass over a control

volume: The velocity Boundary Layer

Continuity equation

Conservation of mass over a control volume, inout = 0

u

v

y

x

0yv

xu

0y

v

x

u

dyyvv

dx

x

uu

u

vGu = mass velocity (kg / m2s)

For an incompressible fluid where = constant:

See Appendix E for detailed development of conservation ofmass, momentum, energy and chemical species



The following differential equations are therefore obtained for

steady, 2-D flow for an incompressible fluid with constant

properties (, , cv, cp, k, etc.)

Continuity Equation: Conservation of Mass

Momentum Equation in x-direction: Velocity B.L.

Momentum Equation in y-direction: Velocity B.L.

0y

v

x

u

Xy

u

x

u

x

P

y

uvx

uu2

2

2

2

Yy

v

x

v

y

P

y

vv

x

vu

2

2

2

2

Where X = Bodyforce in x direction

X=gx

Where Y = Bodyforce in y direction

Y=gy


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Energy Equation: Thermal B.L.

Viscous Energy Dissipation Equation

Species Transfer or Continuity Equation: Species

Concentration B.L.

q

y

T

x

Tk

y

Tv

x

Tuc

2

2

2

2

p

222

y

v

x

u

x

v

y

u

A2

A2

2

A2

AB

AA myx

Dy

vx

u

A2A

2

2A

2

ABAA N

y

C

x

CD

y

Cv

x

Cu

Where = viscous

energy dissipation


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Approximations and Special Conditions

A most common situation is one in which the 2-D boundary

layer can be characterized as Steady (time independent)

Incompressible ( is constant)

Having constant properties (, , k, etc.) with temperature

Having negligible body forces ( X = Y = 0)

Non-reacting (no chemical reaction) without internal heat

generation

In addition, since is very small, the following B.L.

approximations apply:

x

v,y

v,

x

u

y

u

vu Velocity B.L.


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This means that normal stresses in x-direction are negligible,

and the wall shear stress reduces to:

These simplification reduce the B.L. equations to the

following equations.

Continuity Equation:

x

T

y

T

xCor

xyCor

yAAAA

yu

yxxy

0y

v

x

u

Thermal B.L.

Concentration B.L.


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Momentum equationin x-direction, and y-direction

The continuity and momentum equations can be used to solve

for the spatial variations of u and v.

Energy Equation:

Species Continuity Equation

2

2

y

uv

x

P1

y

uv

x

uu

0y

P

2

p2

2

y

u

c

v

y

T

y

Tv

x

Tu

,

This can be used to

solve for the temperature

2A

2

ABAA

yD

yv

xu

2A

2

ABAA

y

CD

y

Cv

x

Cu

This can be used to solve for concentration variation

or


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Boundary Layer Similarity and Normalized

Convection Equations

Examination of the simplified convection equations

repeated below shows a strong similarity between themomentum equation and energy equation:

Similarly, there is a strong similarity between mass transferand momentum and energy equations as indicated below:

2

2

y

uv

x

P1

y

uv

x

uu

2

p2

2

y

u

c

v

y

T

y

Tv

x

Tu

Momentum equation

in x-direction for

velocities u and v

Energy equation for

temperature T

DiffusionAdvection

2A

2

ABAA

y

CD

y

Cv

x

Cu

Mass TransferEquation

Diffusion

Advection


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Similarity Parameters for Boundary Layers

Dimensionless independent variables defined as:

Dependent dimensionless variables are:

The dimensionless independent and dependent variables may besubstituted in to the momentum and energy equations to obtain

dimensionless forms of conservation equations (see table 6.1)

Similarity parameters make it possible to apply results obtained for a

surface experiencing one set of conditions to geometrically similar

surfaces experiencing entirely different conditions.

L

x*x

L

y*y Where L = characteristic

length for the surface

V

u*u

V

v*v

s

s

TT

TT*T

s,A,A

s,AAA

CC

CC*C

2V

P*P

and

, ,

,

Where

V = Upstream velocity

CA= mass concentration

of species A


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Dimensionless groups

1. Reynolds Number (Re)

Is the viscosity (kg / s m)

is the kinematic viscosity (m2/s)

2. Prandtl Number (Pr)

3. Nusselt Number (Nu) = The dimensionless temperature

gradient at the surface.

4. Stanton Number (St)

cc VLVL

Re

k

cPr

p

0*yf

c

*y

*T

k

hL

Nu

pVc

h

PrRe

NuSt

= Inertia Forces

Viscous Forces

= Momentum diffusivity

Thermal diffusivity

(A modified

Nusselt Number)

is the thermal diffusivity (m2/s)


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5. Schmidt Number (Sc)

6. Sherwood Number (Sh) = Dimensionless concentration

gradient at the surface

In terms of dimensionless groups, the complete set of

dimensionless convection equations are therefore:

ABDSc

0*y

A

AB

mL

*y

*C

D

LhSh

= Momentum diffusivity

Mass diffusivity

DAB= mass diffusivity (m2/s)

0*y

*v

*x

*u

2

2

L *y

*u

Re

1

*x

*P

*y

*u*v

*x

*u*u

Velocity (Continuity equation)

Velocity

(Momentum

equation)


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Solution Forms:

2

2

L *y

*T

PrRe

1

*y

*T*v

*x

*T*u

2A2

L

AA

*y

*CScRe1

*y*C*v

*x*C*u

*dx*dp

,Re*,y*,xf*u L1

*dx

*dp,Re*,xf

*y

*uL2

0*y0*y

L2f *y

*u

Re

2

2V

C

*dx

*dpPr,,Re*,y*,xf*T L3

Thermal

(Energy Equation)

Mass Concentration(Mass Diffusion

Equation)


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Nusselt Number:

Convection mass transfer:

0*y

f

0*ys

sf

*y

*T

L

k

*y

*T

TT

TT

L

kh

L,V,,,c,kfh p

0*yf *y

*T

k

hLNu

Pr,Re*,xfNu L4

Pr,Refk

LhNu L5

f

Pr,Re,Nufh L

*dx

*dp,Sc,Re*,y*,xf*C L6A

0*yAAB

0*y

A

,As,A

s,A,AABm

*y

*C

L

D

*y

*C

CC

CC

L

Dh

Where hm= mass transfer coefficient, m/s (convection)

,

,

,


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Sherwood Number:

Physical Significance of the Dimensionless Parameters

Reynolds Number

Inertia forces therefore dominate for large values of Re andviscous forces dominate for small Re values. Viscous forces

dominate in laminar flow but become progressively less

important than inertia forces as Re increases.

0*y

A

AB

m

*y

*C

D

LhSh

Sc,Re*,xfSh L7

Sc,RefD

LhSh L8

AB

m L,V,,,c,DfSc,Re,Shfh pABLm

L2

2

2

1 ReVL

LV

LV

F

F

= Inertia Forces

Viscous Forces


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Prandtl Number, Pr

Large differences in Pr are associated with large variations in the

fluid viscosity, . The spectrum of Prandtl Numbers of fluids isgiven below.

Normally

is the velocity B.L. thickness and tis the thermal B.L.thickness.

k

cPr

p

= Momentum diffusivityThermal diffusivity

10-2 10-1 100 10 102 103

Liquid metals Gases WaterLight Organics

Oils

n

t

Pr

Where n is a positive integer


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For gases,

For liquid metals (very high k, low Pr)

For oils (very high , high Pr)

Mass transfer and Lewis Number, Le

Table 6.2 (text) lists several dimensionless parameters that

are generally relevant in heat and mass transfer

t

t

t

n

c

Sc

Pr

Sc

DLe

AB

n

c

t Le


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34/51



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Boundary Layer Analogies


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Boundary Layer Analogies

Heat and mass transfer analogy

Heat and mass transfer are analogous

Heat and mass transfer relations are therefore interchangeablefor a particular geometry

For example, it the form of relationship for a convection heat

transfer problem involving x*, Re, and Pr, that is f4(x*, Re, Pr) has

been obtained for a particular surface geometry, the results may

be used for convection mass transfer for the same geometrysimply by replacing Nu with Sh and Pr with Sc

The analogy can also be used to relate two convection

coefficients. By analogy:

nL7L4n Sc

Sh

Re*,xfRe*,xfPr

Nu

nABm

n Sc

DLh

Pr

khL

n1pn

ABm

LecLeD

k

h

h


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Problem 6.46


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Reynolds Analogy

When dP*/dx*= 0 and Pr = Sc = 1, the boundary layer equations

(momentum, energy and mass concentration) are exactly of the

same form. If dP*/dx*= 0 then U= V = upstream fluid velocity, and the

boundary conditions are also of the same form.

Solutions for U*,T*, CA* must also be of the same form.

It follows that:

ShNu2

ReC Lf

pVc

h

PrRe

NutS

V

h

ScRe

ShSt mm

(1)

Modified Nusselt Number

Mass Transfer Stanton Number


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From (1) we have: Reynolds Analogy,

The Reynolds analogy can be used, provided these restrictionsare satisfied

dP*/dx* 0 , Pr 1 , Sc 1

In general application, the modified Reynolds analogy is:

Where jH, and jmare the Colburn j factor for heat and masstransfer

Equations (2) and (3) are approximate for laminar flow, butmore accurate or valid for turbulent flow.

mf StSt2

C If one parameter is known, it can be

used to obtain others.

Hf jPrSt2

C32

mmf jScSt2

C32

For 0.6 < Pr < 60

For 0.6 < Sc < 3000

(2)

(3)

Evaporative Cooling: Simultaneous Heat &


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Evaporative Cooling: Simultaneous Heat &

Mass Transfer

As shown in the figure below, the Evaporative cooling

process of heat and mass transfer occurs when a gasflows over a liquid surface.

There are numerous industrial and environmental

applications of evaporative cooling process.

In the evaporative cooling process the energy required for

evaporation of the liquid comes from the internal energy of

the liquid. As a consequence, the temperature of the liquid

reduces or cooling effect occurs

Liquid Layer (species A)

Gas Layer(species B)

qevap

qadd

qconv

Latent and sensible heat exchange at a gas liquid interface


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Under steady state operation, the latent energy lost by the

liquid must be replenished by energy transfer to the liquid from

its surroundings or by energy addition by other means (e.g.

electrical heating of the fluid By energy conservation on a control surface on the liquid

surface, we have:

Where

qevap = evaporative cooling load

qconv = convection sensible heat transfer from the gas to the

liquid

qadd = heat addition to the liquid

If there is no heat addition we have

Where A,satis the saturated vapor density at the surface

temperature, Ts.

evapaddconv qqq

fgs,s,Amfgaevap hhhmq

s,ss,Afgms ThhTTh

d


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expressed as:

Since by heat and mass transfer analogy, and for exponent n=1/3

With mA


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Solution Methods in Internal and External

Convection Transfer

The primary objective in internal and external convection

transfer is to obtain convection coefficients for different flow

geometries. The convection coefficients are subsequently used

to obtain heat or mass transfer rates.

Solution methods in obtaining convection coefficients include:

Experimental or Empirical Method

Theoretical Method Non-dimensionalized approach is generally followed, and the

local and average convection coefficients are generally

correlated by the following form of equations:

Heat Transfer Mass Transfer

Pr,Re*,xfNu x4x

Pr,RefNu x5x

Sc,Re*,xfSh x7x

Sc,RefSh x8x


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Experimental or Empirical Method

The empirical method involves heat and mass transfer

measurements and data correlation using appropriate

dimensionless parameters For fixed Prandtl numbers for a given fluid, log-log plots of Re

versus Nu generally are straight lines as illustrated below (left).

The straight line plots may be represented by an equation of

the form: nmLL PrReCNu Where C, m, n are independent

constants


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Log-log plot of Re versus the ratio Nu / Prncombine into a

single straight line for all the Pr as illustrated above (right).

The corresponding correlation equation for mass transfer

is of the form:

Where the independent constants C, m, and n are the

same as obtained for heat transfer for similar geometry

nmLL ScReCSh


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Theoretical method

Special case of exact solution to convection transfer equations(example 6.4 in text)

Parallel flow or Couette flow is one of the situations where exactsolutions can be obtained for convection transfer equations. Aspecial case of parallel flow involves stationary and movingplates of infinite extent separated by a distance L with theintervening space filled by an incompressible fluid as shownbelow:

With the assumption of steady state conditions, incompressiblefluid with constant properties, no body forces (I.e. X=0, andY=0), and no internal energy generation the convection transferequations reduce to the following:

Engine oil

Moving Plate

Stationary Plate

LTL

T0 x,uy,v

U


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1. Continuity Equation

For incompressible fluid (= constant) and parallel flow(v = 0) the continuity equation reduces to:

The x velocity component is therefore independent of x, and

the velocity field is said to be fully developed.

2. Momentum Equation With u / v = 0, v = 0 and X = 0, the momentum equation

reduces to:

In parallel flow the pressure gradient P / x = 0, and the x-

momentum equation reduces to:

0xu

0y

u

yx

P0

0y

u2

2


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Finally, the velocity distribution is given by:

3. Energy equation For the 2-D, S.S. condition with y=0, energy generation q=0 and

u/x = 0 for fully developed temperature field, the energy

equation reduces to:

Integrating twice and solving, the temperature distribution is

given by:

0)0(u U)L(u UL

y)y(u

2

2

2

y

u

y

Tk0

L

yTT

L

y

L

yU

k2

TyT 0L

22

o

2

22

bL

y1U

k2TyT

B.C.s

T(0) = ToT(L) = TL

Tb= bearing Temp.


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4. Surface Heat Fluxes

The surface heat fluxes are obtained from the temperature

distribution by applying the Fouriers Law:

At the bottom and top surfaces, the heat fluxes are respectively

given be:

5. Location of Maximum Temperature

The location of the maximum temperature may be found from

the requirement:

LTT

Ly2

L1U

k2k

dydTkq 0L

22

y

0L2

0 TT

L

k

L2

Uq

0L

2

L TT

L

k

L2

Uq

0L

TT

L

y2

L

1

Uk2dy

dT 0L2

2

L2

1TT

U

ky 0L2max


51/51

The maximum temperature occurs in the fluid and there is heat

transfer to the hot and cold plates. The temperature distribution

is a string function of the velocity of the moving plate.

6. Viscous Energy Dissipation The viscous energy dissipation was defined as:

With v = 0, and du/dx = 0, the viscous energy dissipation

equation reduces to:

For U = 0 there is no viscous dissipation, and the temperature

distribution is linear.

2222

y

v

x

u

3

2

y

v

x

u2

x

v

y

u

2

dy

du

(10) introduction to convection

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