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Heat Transfer in Microchannels. Applications. Cooling of microelectronics. Inkjet printer. 11.1 Introduction:. Medical research. Micro-electro-mechanical systems (MEMS): Micro heat exchangers, mixers, pumps, turbines, sensors and actuators.

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heat transfer in microchannels

Heat Transfer in Microchannels

Applications

Cooling of microelectronics

Inkjet printer

11.1 Introduction:

Medical research

  • Micro-electro-mechanical systems (MEMS): Micro heat exchangers, mixers, pumps, turbines, sensors and actuators
slide2
11.1.1 Continuum and Thermodynamic Equilibrium Hypothesis

Properties: (pressure, temperature, density, etc) are macroscopic manifestation of molecular activity

Continuum:material having sufficiently large number of molecules in a given volume to give unique values for properties

Validity of continuum assumption: the molecular-mean-free path, , is small relative to the characteristic dimension of the system

Mean-free-path: average distance traveled by molecules before colliding

slide3
(1.2)

= characteristic length

(1.3a)

(1.3b)

Knudson number Kn:

Gases: the criterion for the validity of the continuum assumption is:

Thermodynamic equilibrium: depends on collisions frequency of molecules. The condition for thermodynamic equilibrium is:

slide4
Microchannels: Channels where the continuum assumptionand/or thermodynamic equilibrium break down
slide5
(11.1)

9.1.2. Surface Forces. Examine ratio of surface to volume for tube:

  • For D = 1 m, A/V = 4 (1/m)
  • For D = 1 μm, A/V = 4 x 10 6 (1/m)
  • Consequence:
  • Surface forces may alter the nature of surface boundary conditions

(2) For gas flow, increased pressure drop results in large density changes. Compressibility becomes important

slide6
(11.2)

9.1.3 Chapter Scope

  • Classification
  • Gases vs. liquids
  • Surface boundary conditions
  • Heat transfer in Couette flow
  • Heat transfer in Poiseuille flow

9.2 Basic Consideration

9.2.1 Mean Free Path. For gases:

slide7
Pressure drops along a channel

increases

Kn increases

is very small, expressed in terms of the micrometer,

p = pressure

R = gas constant

T = temperature

μ = viscosity

NOTE:

slide8
(6.57)

(11.3)

As

11.2.2 Why Microchannels?

Nusselt number: fully developed flow through tubes at uniformsurface temperature

Application:

Water cooled microchips

slide9
(11.4)

11.2.3 Classification

Based on the Knudsen number:

Four important factors:

(1) Continuum

(2) Thermodynamic equilibrium

(3) Velocity slip

(4) Temperature jump

slide10
(1)Kn < 0.001: Macro-scale regime (previous chapters):

Continuum: valid

Thermodynamic equilibrium: valid

No velocity slip

No temperature jump

(2) 0.001

Continuum: valid

Thermodynamic equilibrium: fails

  • Velocity slip
  • Temperature jump

Continuity, Navier-Stokes equations, and energy equations are validNo-velocity slip and No-temperature jump conditions, conditions failReformulate boundary conditions

slide11
(3) 0.1< Kn<10: Transition flow:

Continuity and thermodynamic equilibrium fail Reformulate governing equations and boundary conditions Analysis by statistical methods

(4) Kn>10: Free molecular flow: analysis by kinetic theory of gases

11.2.4 Macro and Microchannels

Macrochannels: Continuum domain, no velocity slip, no temperature jump

Microchannels: Temperature jump and velocity slip, with or without failure of continuum assumption

slide12
Distinguishing factors:

(1) Two and three dimensional effects

(2) Axial conduction

(3) Viscous dissipation

(4) Compressibility

(5) Temperature dependent properties

(6) Slip velocity and temperature

(7) Dominant role of surface forces

slide13
(1) Mean free path:

11.2.5 Gases vs. Liquids

Macro convection:

  • No distinction between gases and liquids
  • Solutions for both are the same for the same geometry, governing parameters (Re, Pr, Gr,…) and boundary conditions

Micro convection:

  • Flow and heat transfer of gases differ from liquids

Gas and liquid characteristics:

  • Continuum assumption may hold for liquids but fail for gases
slide14
Typical MEMS applications: continuum assumption is valid for liquids

(2) Knudsen number: used as criterion for thermodynamic equilibrium and continuum for gases but not for liquids

(3) Onset of failure of thermodynamic equilibrium and continuum: not well defined for liquids

(4) Surface forces: liquid forces are different from gas forces

(5) Boundary conditions: differ for liquids from gases

(6) Compressibility: liquids are almost incompressible while gases are not

(7) Flow physics: liquid flow is not well known. Gas flow is well known

slide15
(8) Analysis: more complex for liquids than gases

11.3 General Features

  • Flow and heat transfer phenomena change as channel size is reduced:

Rarefaction: Knudsen number effect

Compressibility: Effect of density change due to pressure drop along channel

Viscous dissipation: Effect of large velocity gradient

Examine: Effect of channel size on:

  • Velocity profile
  • Flow rate
slide16
Friction factor
  • Transition Reynolds number
  • Nusselt number

Consider:

Fully developed microchannel gas flow as the Knudsen number increases from the continuum through the slip flow domain

slide17
(11.5)
  • Define friction coefficient

(4.37a)

11.3.1 Flow Rate

Slip flow: increased velocity and flow rate

e = determined experimentally

t = from macrochannel theory or correlation equations

11.3.2 Friction Factor f

slide18
= wall shear stress

= mean velocity

(11.6)

= pressure drop

  • Fully developed flow through channels: define friction factor f

D = diameter

L = length

slide19
(11.7)

Microchannels: compare experimental data,

, with

theoretical value,

, (macroscopic, continuum)

Macrochannels: fully developed laminar flow:

(1) f is independent of surface roughness

(2) Product of f and Reynolds number is constant for each channel geometry:

Po = Poiseuille number

(3) Po is independent of Reynolds number

slide20
(11.8)

(1)

departs from unity:

Conclusion:

(2) Unlike macrochannels, Po for fully developed flow depends on the Re

(3) Conflicting findings due to: difficulties in measurements of channel size, surface roughness, pressure distribution, uncertainties in entrance effects, transition, and determination of properties

slide21
(6.1)

Factors affecting the determination of

11.3.3 Transition to turbulent flow

Macrochannels: smooth macrotubes

Microchannels: reported transition

  • Variation of fluid properties
  • Measurements accuracy
  • Surface roughness
slide22
(11.9)

11.3.4 Nusselt number. For fully developed conditions:

Macrochannel: Nusselt number is constant

Microchannels: In general, Nusselt number is not well established:

  • Nu varies along microchannels
  • Nu depends on:
  • Surface roughness
  • Reynolds number
  • Nature of gas
  • Widely different reported results:
slide23
= experimental

= macrochannel theory

Factors affecting the determination of

where:

  • Variation of fluid properties
  • Measurements accuracy
slide24
Slip flow regime:

M =

11.4 Governing Equations

Factors to be considered:

  • Compressibility
  • Axial conduction
  • Dissipation

11.4.1 Compressibility: Expressed in terms of Mach number

slide25
(6.30)

Macrochannels:

  • Incompressible flow, M < 1
  • Linear pressure drop

Microchannels:

  • Compressible flow
  • Non-linear pressure drop
  • Decrease in Nusselt number

11.4.2 Axial Conduction

Macrochannels: neglect axial conduction for

slide26
(11.10)

Pe = Peclet number

Microchannels: low Peclet numbers, axial conduction may be important, it increases the Nusselt number

11.4.3 Dissipation

Microchannels: large velocity gradient, dissipation may become important

11.5 Slip Velocity and Temperature Jump Boundary Conditions

Slip velocity for gases:

slide27
= fluid axial velocity at surface

surface axial velocity

= tangential momentum accommodating coefficient

(11.11)

= surface temperature

x = axial coordinate

n = normal coordinate measured from the surface

Temperature jump for gases

T(x,0) = fluid temperature at the boundary

slide28
, specific heat ratio

= energy accommodating coefficient

NOTE

(1) Eq. (11.10) and (11.11) are valid for gases

(2) Eq. (11.10) and (11.11) are valid forKn < 0.1

(3) σu and σT, are:

  • Empirical factors
  • They depend on the gas, geometry and surface
  • Values range from zero (perfectly smooth) to unity
slide29
Difficult to determine experimentally
  • Values for various gases are approximately unity
slide30
11.4. 8 Analytic Solutions: Slip Flows

Two common flow types, extensive use in MEMS:

(1) Couette flow (shear driven): fluid is set in motion by a moving surface

Examples:

slide31
(2) Poiseuille flow (pressure driven): fluid is set in motion by an axial pressure gradient

Examples:

Micro heat exchangers, mixers, microelectronic heat sinks

NOTE

  • No pressure drop in Couette flow
  • Signifiant pressure drop in Poiseuille flow

Boundary conditions: two types:

(1) Uniform surface temperature

(2) Uniform surface heat flux

slide32
11.6.1 Assumptions

(1) Steady state

(2) Laminar Flow

(3) Two-dimensional

(4) Slip flow regime (0.001 < Kn < 0.1)

(5) Ideal gas

(6) Constant viscosity, conductivity and specific heats

(7) Negligible lateral variation of density and pressure

(8) Negligible dissipation (unless otherwise stated)

(9) Negligible gravity

slide33
(10) The accommodation coefficients are equal to unity,

11.6.2 Couette Flow with Viscous Dissipation:

Parallel Plates with Surface Convection

  • Infinitely large parallel plates
  • Gas fills gap between plates
  • Upper plate: moves with velocity us
  • Lower plate: stationary, insulated
  • Convection at the upper plate
  • Consider dissipation and slip conditions
slide34
(11.12)

Determine:

(1) Velocity distribution

(2) Mass flow rate

(3) Nusselt number

Find flow field and temperature distribution

Flow Field

  • Normal velocity and all axial derivatives vanish
  • Axial component of the Navier-Stokes equations, (2.9), simplifies to
slide35
Boundary conditions: use (11.10), Set
  • Lower plate: n = y = 0 and

(11.10) gives

(g)

(h)

(11.14)

  • Upper plate: n = H – y, (9.10) gives

Solution

slide36
(11.13)

Kn is the local Knudsen number

NOTE

(1) Fluid velocity at the moving plate: set y = H in (11.14)

Effect of slip:

  • Decrease fluid velocity at the moving plate
  • Increase fluid velocity at the stationary plate
slide37
(k)

(11.15)

(2) Velocity distribution is linear

(3) Setting Kn = 0 in (11.14) gives the no-slip solution

Mass Flow Rate m

W = channel width

Neglect variation of ρ along y, (11.14) into (11.15)

slide38
(11.16)

(11.17)

(11.18)

  • Flow rate is independent of the Knudsen number
  • Compare with macrochannel flow rate mo

(k) into (11.15)

This is identical to (11.16), thus

slide39
(l)

Nusselt Number

  • Equivalent diameter for parallel plates,De = 2H
  • Nusselt number

Heat transfer coefficient h:

slide40
(11.19)

k = conductivity of fluid

T = fluid temperature

Ts = plate temperature

NOTE

(1) Fluid temperature at the moving plate, T (x,H), is not equal to surface temperature

(2) h is defined in terms of surface temperature Ts

slide41
(11.20)

(11.21)

(3) Use temperature jump, (11.11), to determine Ts

(4) For the upper plate, n =H – y, eq. (11.11) gives

  • Mean temperature Tm: defined in Section 6.6.2
  • Neglect variation of cp and ρ along y, use (11.14) . for u and (11.15) for m
slide42
(11.22)

(11.23)

Determine temperature distribution:

  • Use energy equation, (2.15)
  • Apply above assumptions, note that axial derivatives vanish, (2.15) gives
slide43
(2.17) gives the dissipation function

which simplifies to

(11.24)

(11.25)

(m)

(9.24) into (9.23)

Boundary conditions

Lower plate:

slide44
(n)

(11.26)

Upper plate:

Use (920) to eliminate Ts

Use velocity solution (9.14), solve for T

slide45
(p)

(u)

(w)

where

Velocity solution (11.14), temperature solution (11.26) giveTs , Tm and Nu

slide46
(11.27)

Note the following regarding the Nusselt number

(1) It is independent of Biot number

(2) It is independent of the Reynolds number

(3) Unlike macrochannels, it depends on the fluid

(4) First two terms in the denominator of (11.27) represent rarefaction (Knudsen number). The second term represents effect of temperature jump

slide47
(11.28)

(11.29)

(5) Nusselt number for macrochannels, Nuo: set Kn = 0 in (11.27):

Ratio of (11.27) and (11.28)

NOTE: Ratio is less than unity

slide48
Uniform surface flux,

11.6.3 Fully Developed Poiseuille Channel Flow: Uniform Surface Flux

  • Pressure driven flow between parallel plates
  • Fully developed velocity and temperature
  • Inlet and outlet pressures are pi and po

Determine:

(1) Velocity distribution

(2) Pressure distribution

(3) Mass flow rate

(4) Nusselt number

slide49
Note:

(3) Invariant axial velocity

(4) Linear axial pressure

Major difference between macro and micro fully developedslip flow:

Macrochannels: incompressible flow

(1) Parallel streamlines

(2) Zero lateral velocity component (v = 0)

slide50
density changes
  • (2) Large axial pressure drop

compressible flow

(3) Rarefaction: pressure decreases

increases

Kn increases with x

Microchannels: compressibility and rarefaction change above flow pattern:

  • (1) None of above conditions hold
  • (4) Axial velocity varies with axial distance

(5) Lateral velocity v does not vanish

  • (6) Streamlines are not parallel

(7) Pressure gradient is not constant

slide51
(3)

(9)

Assumptions

(1) Steady state

(2) Laminar flow

(4) Two-dimensional

(5) Slip flow regime (0.001 < Kn < 0.1)

(6) Ideal gas

(7) Constant viscosity, conductivity and specific heats

(8) Negligible lateral variation of density and pressure

slide52
(12) Negligible inertia forces:

= 0

(13) The dominant viscous force is

(c)

(10) Negligible dissipation

Flow Field

Additional assumptions:

(11) Isothermal flow

Navier-Stokes equations (2.9) simplify to:

slide53
(e)

(f)

(11.30)

Boundary conditions:

Symmetry at y = 0

For the upper plate, n = H – y

Solution to u

slide54
(11.33)

(2.2a)

For an ideal gas

Pressure Distribution p:

To determine p(x), must determine vertical component v: start with continuity (2.2a)

Apply above assumptions

slide55
(h)

(11.31)

(i)

Use ideal gas to eliminate ρ:

(11.31) into (h), assuming constant temperature

(11.30) into (i)

slide56
(j)

(k)

(l)

(m)

Boundary conditions:

Multiply (j) by dy, integrate and using (k)

slide57
(11.32)

(n)

(11.33)

Evaluate the integrals

Determination of p(x): Apply boundary condition (l) to (11.32)

Express Kn in terms of pressure. Equations (11.2) and (11.13) give

slide58
(o)

(p)

Evaluate (n) at y = H/2, substitute (11.33) into (n) and integrate

Integrate again (T is assumed constant)

Solve for p

slide59
(q)

,

Pressure boundary conditions

Apply (q) to (p)

Substitute into (p) and normalize by po

slide60
(r)

(11.34)

(11.35)

Introduce outlet Knudsen number Knousing (11.2) and (11.13)

Substitute (11.34) into (r)

slide61
(11.36)

NOTE:

(1) Unlike macrochannel Poiseuille flow, pressure variation along the channel is non-linear

(2) Knudsen number terms represent rarefaction effect

(3) The terms (pi/po)2 and [1- (pi/po)2](x/L) represent the effect of compressibility

(4) Application of (11.35) to the limiting case of Kno =0 gives

This result represents the effect of compressibility alone

slide62
(s)

(t)

(11.37)

Mass Flow Rate

W = channel width

(11.30) in (s)

Density ρ :

slide63
(11.33)

(11.38)

(11.39)

(11.33) gives Kn(p)

(11.33) and (11.37) in (t)

(11.35) into (11.38) and let T=To

slide64
(11.40)

(11.41)

Compare with no-slip, incompressible macrochannel case:

Taking the ratio

NOTE

(1) Microchannels flow rate is very sensitive to H

(2) (11.39) shows effect of rarefaction (slip) and compressibility on m

slide65
(11.41) shows that neglecting

(3) Since

compressibility and rarefaction underestimates m

(u)

For uniform surface flux

(v)

Nusselt Number

Substitute into (g)

slide66
(11.42)

(11.43)

Plate temperature Ts: use (11.11)

Mean temperatureTm:

Needu(x,y) and T(x,y)

Velocity distribution: (11.30) givesu(x,y)for isothermal flow

slide67
(15) No dissipation,

(16) No axial conduction,

(18) Nearly parallel flow,

(11.44)

Additional assumption:

(14) Isothermal axial velocity solution is applicable

(17) Negligible effect of compressibility on the energy equation

Energy equation: equation (2.15) simplifies to

slide68
(w)

(x)

(11.45)

Boundary conditions:

To solve (11.44), assume:

(19) Fully developed temperature

Solution:T(x,y) and Tm(x): Define

slide69
Fully developed temperature:

is independent of x

(11.46)

(11.47)

Thus

(11.45) and (11.46) give

slide70
(11.48)

and

(y)

Expanding and use (11.45)

Determine:

Heat transfer coefficient h:

slide71
(z)

(11.49)

(11.42) gives Ts(x). (11.45) gives temperature gradient in (y)

Differentiate

(z) into (y), use (11.42) forTs(x)

slide72
(11.50)

Newton’s law of cooling:

Equate with (11.49)

Differentiate

Combine this with (11.48)

slide73
(11.51)

(11.51) replaces

with

in (11.44)

Determine

:

NOTE:

Conservation of energy for element:

slide74
= constant

(aa)

(bb)

Conservation of energy for element:

Simplify

However

(bb) into (aa)

slide75
= constant

(11.52)

(11.53)

(11.54)

(11.52) into (11.51)

(11.53) into (11.44)

slide76
(cc)

(11.55)

Mean velocity:

(11.30) gives velocity u. (11.30) into (cc)

Integrate

slide77
(11.56)

(11.57)

(dd)

Combining (11.30) and (11.55)

(11.56) into (11.54)

Integrate twice

slide78
(11.58)

f(x) and g(x) are “constants” of integration

Boundary condition (w) gives

Solution (dd) becomes

NOTE:

  • Boundary condition (x) is automatically satisfied

(2) g(x) is determine by formulating Tm using two methods

slide79
(11.59)

(11.60)

Method 1: Integrate (11.52)

where

Evaluate the integrals

slide80
(11.61)

(11.62)

Method 2: Use definition of Tm. Substitute (11.30) and(11.58) into (11.43)

Evaluate the integrals

Equating (11.60) and (11.61) gives g(x)

slide81
(11.63)

(v)

(11.64)

(11.58) into (11.42) gives Ts

The Nusselt number is given in (v)

(11.61) and (11.63) into (v)

slide82
(11.65)

NOTE:

(1) Kn in (11.64) depends of local pressurep

(2) Pressure varies with x, Kn varies with x

(3) Unlike macrochannels, Nu is not constant

(4) Unlike macrochannels, Nu depends on the fluid

  • No-slip Nu for macrochannel flow, Nuo: set Kn = 0 in (11.64)
slide83
This agrees with Table 6.2

(6) Rarefaction and compressibility decrease the Nusselt number

slide84
11.6.4 Fully Developed Poiseuille Channel Flow: Uniform Surface Temperature

(11.30)

u(y)

(11.35)

(11.35)

m

Repeat Section 11.6.3 with plates at uniform surfacetemperature Ts

  • Flow field: same for both cases:
  • Energy equation: (11.44) is modified to include axial conduction
slide85
(11.66a)

Solve the Graetz channel entrance problem and set

to obtain the fully developed solution

  • Boundary conditions: different for the two cases

Nusselt number:

Need T(x,y) and Tm(x)

Solution approach:

slide86
(7.50)

Small Reynolds

Small Peclet number

Axial

conduction is important

(11.67a)

Axial conduction: can be neglected for:

Microchannels:

Include axial conduction: modify energy equation (11.44)

slide87
(11.68a)

(11.69a)

(11.70a)

(11.71a)

Boundary and inlet conditions:

slide88
(11.56)
  • Specialize to fully developed: set

Axial velocity

Solution

  • Use method of separation of variables

Result: Fig. 11.11 shows Nu vs. Kn

slide89
(4) Limiting case: no-slip (Kn = 0) and no axial conduction

:

(11.73)

Heat Transfer Rate,

:

NOTE

(1) Nu decreases as the Kn is increased

(2) No-slip solution overestimates microchannels Nu

(3) Axial conduction increases Nu

This agrees with Table 6.2

Following Section 6.5

slide90
(6.14)

(6.13)

is determine numerically using (6.12)

(6.12)

Tm(x) is given by

slide91
11.6.5 Fully Developed Poiseuille Flow in Micro Tubes: Uniform Surface Flux

Consider:

  • Poiseuille flow in micro tube
  • Uniform surface flux
  • Fully developed velocity and temperature
  • Inlet and outlet pressures are pi and po
slide92
Determine

(1) Velocity distribution

(2) Nusselt number

  • Rarefaction and compressibility affect flow and heat transfer
  • Velocity slip and temperature jump
  • Axial velocity variation
  • Lateral velocity component
  • Non-parallel stream lines
  • Non-linear pressure
slide93
(a)

= axial velocity

Assumptions

Apply the 19 assumptions of Poiseuille flow between parallel plates (Sections 11.6.3)

Flow Field

  • Follow analysis of Section 11.6.3
  • Axial component of Navier-Stokes equations in cylindrical coordinates:
slide94
(b)

(c)

(11.74)

Boundary conditions:

Assume symmetry and set σu= 1

Solution

slide95
(11.75)

(11.76)

Knudsen number

Mean velocity vzm

Use (11.74), integrate

slide96
(11.77)

(11.78)

(11.79a)

(11.74) and (11.76)

Solution to axial pressure

(11.76) and (11.78) give m

slide97
(11.79b)

(d)

For incompressible no-slip (macroscopic)

Nusselt Number

  • Follow Section 11.6.3

Heat transfer coefficient h:

slide98
(e)

(f)

(11.80)

Substituting into (d)

Ts = tube surface temperature, obtained from temperaturejump condition (11.11)

Mean temperature:

slide99
(11.81)

(g)

(h)

Energy equation:

Boundary conditions:

Define

slide100
(11.82)

(11.83)

(11.84)

Fully developed temperature:

Thus

(11.82) and (11.84) give

slide101
(11.85)

and

(i)

Expand (11.82)

Determine:

Heat transfer coefficient h,:

slide102
Differentiate and evaluating at

(j)

(k)

Rewrite (11.81)

(j) into (i)

slide103
(11.86)

Newton’s law of cooling h

Equate with (k)

Differentiate

slide104
(11.87)

Will use (11.87) to replace

in (11.81) with

Combine with (11.85)

Conservation of energy to dx

slide105
(l)

(m)

(11.88)

(11.89)

Simplify

However

(m) into (l)

(11.88) into (11.87)

slide106
(11.90)

(11.77) is used to eliminate

in the above

(11.91)

(n)

(11.89) into (11.81)

Integrate

slide107
(11.92)

Condition (g) gives

Solution (n) becomes

Condition (h) is automatically satisfied

Determine g(z): Use two methods to determine Tm

Method 1: Integrate (11.88)

slide108
(11.93)

(11.94)

where

Evaluate the integral

Method 2: Use definition ofTm in (11.80). Substitute (11.74)and (11.92) into (11.80)

slide109
(11.95)

(11.96)

Integrate

Equate (11.94) and (11.95), solve for g(z)

slide110
Use (f) and (11.92) to determine

(11.97)

(11.98)

Nusselt number: (11.95) and (11.97) into (e)

slide111
(11.99)

Results: Fig. 11.14

  • Fig. 11.14 gives Nu vs. Kn for air
  • Rarefaction and compressibility decrease the Nusselt number
  • Nusselt number depends on the fluid
  • Nu varies with distance along channel
  • No-slip Nusselt number,Nuo,is obtained by setting Kn = 0 in (11.97)
slide112
This agrees with (6.55) for macro tubes

11.9.6 Fully Developed Poiseuille Flow in Micro Tubes: Uniform Surface Temperature

  • Repeat Section 11.6.5 with the tube at surface temperatureTs
  • Apply same assumptions
  • Boundary conditions are different
  • Flow field solution is identical for the two cases
slide113
(11.100a)

Nusselt number:

  • DetermineT(r,z) and Tm(z)
  • Follow the analysis of Section 11.6.4
  • Solution is based on the limiting case of Graetz tube entrance problem
  • Axial conduction is taken into consideration
  • Energy equation (11.81) is modified to include axial conduction:
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(11.102a)

(11.103a)

(11.104a)

(11.105a)

(11.101a)

Boundary and inlet conditions

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(11.76)
  • Neglecting axial conduction: set

(11.76) gives axial velocity

  • Solution by the method of separation of variables
  • Solution is specialized for fully developed conditions at large z
  • Result for air shown in Fig. 11.16
  • Axial conduction increases the Nu
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Limiting case: no slip and no axial conduction: at Kn = 0and

(11.72)

This agrees with (6.59)

  • Limiting case: no slip with axial conduction:at Kn =0and Pe = 0:
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