Convection Heat Transfer in Micro-channels

by:Behzad Mohajer

Supervisor:Dr. Hamedi

1) Introduction:

The need for efficient cooling methods for high heat flux components

focused attention on the cooling features of microchannels.

Engineering and scientific applications:

Inkjet printer

Medical applications

Micoelecro-mechanical systems (MEMS)

Micro heat exchangers, Mixers, Pumps, Turbines, Sensors, Actuators

2) Continuum and Thermodynamic Hypothesis

The Knudsen number is used to establish a criterion for the validity of the continuum and thermodynamic assumptions.

Continuum : continuity equation, Navier-Stokes equations, and the energy equation

Thermodynamic Equilibrium : no-velocity slip and no-temperature

jump at a solid boundary

Channels that function in the continuum domain, with no velocity slip and temperature jump, and whose flow and heat transfer behavior can be predicted from continuum theory, are referred to as macrochannels. On the other hand, channels for which this approach fails to predict their flow and heat transfer characteristics are known as microchannels.

For gases:

3) Knudsen Number

λ : The molecular mean free path

De : Characteristic length, such as channel equivalent diameter

4) Classification of Flow In Microcannels According to The Knudsen Number:

Continuum, No-Slip Boundary Conditions Kn < 0.001

Continuum, Slip Boundary Conditions 0.001 < Kn < 0.1

Transition Flow 0.1 < Kn < 1

Free Molecular Flow Kn > 1

5) Presentation Scope

distinction between gases and liquids

rarefaction and compressibility

velocity slip and temperature jump phenomena

The effect of compressibility and axial conduction

Analytic solutions to Couette and Poiseuille flows and heat transfer

convection of gases in microchannels

In this presentation, we will limit ourselves to the slip flow regime.

6) Why Microchannels?

In tubes:

The smaller the diameter, the larger the heat transfer coefficient.

Pressure drop through channels increases as channel size becomes smaller.

For constant surface temperature, continuum and no-slip solution,

the Nusselt number is constant in the fully developed region.

7) Gases vs. Liquids

No distinction is made between gases and liquids In the analysis of macro flow.

In general, the physics of liquid flow in microdevices is not well known. Analysis of liquid flow and heat transfer is more complex for liquids than for gases

1. the mean free paths of liquids are much smaller than those of gases

2. The classification of flow using Knudsen number is just valid for gases

3. The onset of failure of thermodynamic equilibrium and continuum is not

well defined for liquids

4. liquids are almost incompressible while gases are compressible.

Differences between gases and liquids in microchannels:

8) General Features

Knudsen number effect is referred to as rarefaction

Density change due to pressure drop along microchannels gives rise to

compressibility effects.

Viscous dissipation affects temperature distribution

channel size affects the velocity profile, flow rate, friction factor,

transition Reynolds number, and Nusselt number

9) Flow Rate

Velocity slip at the surface results in an increase in the flow rate

Experiment

Macrochannel Theory

10) Friction Factor

For continuum flow through tubes, Po = 64

For fully developed laminar flow in macrochannels Po is independent of Reynolds number.

The departure of C* from unity represents the degree to which macroscopic theory fails to predict microscopic conditions.

Reported values for C* ranged from much smaller than unity to much larger than unity

1. The Poiseuille number Po appears to depend on the Reynolds number. This is in contrast to macrochannels

2. Both increase and a decrease in the friction factor are reported.

3. The conflicting findings are attributed to the difficulty in making accurate measurements of channel size, surface roughness, pressure distribution, as well as uncertainties in entrance effects, transition to turbulent flow, and the determination of thermophysical properties.

11) Transition to Turbulent Flow

The Reynolds number is used as the criterion for transition from laminar to turbulent flow.

In macrochannels, transition Reynolds number depends on cross-section geometry and surface roughness

For microchannels, reported transition Reynolds numbers ranged from 300 to 16,000.

Fluid property variationOutlet Reynolds number can be significantly different from inlet.

The effect of size and surface roughness on the transition Reynolds numberReasons

12) Nusselt Number

The Nusselt number for fully developed laminar flow in macrochannels is constant (depending on the channel geometry and thermal boundary conditions - independent of Reynolds number)

Difficulties in accurate measurements of temperature and channel size, as well as inconsistencies in the determination of thermophysical properties

Microchannel Nusselt number depends on surface roughness and Reynolds number

The behavior of the Nusselt number for microchannels is not well understood

Compressibility

Compressibility in microchannel flow results in nonlinear pressure drop

Its effect depends on Mach number as well as the Reynolds number

Fluid axial conduction in macrochannels was neglected for Peclet numbers greater than 100.

Axial Conduction

Microchannels are typically operated at low Peclet numbers

The effect of axial conduction in microchannels is to increase the Nusselt number in the velocity-slip domain

The increase in Nusselt number diminishes as the Knudsen number is increased

The maximum increase is of order 10%, corresponding to Kn = 0

Dissipation

The dimensionless form of the energy equation

Since Ec is proportional to V2 , so it is proportional to the square of Mach number, M2

As long as M is small compared to unity, the effect of dissipation can be neglected in microchannels

Velocity Slip and Temperature Jump Boundary Conditions

An approximate equation for the velocity slip for gases is referred to as the Maxwell slip model :

Gas temperature at a surface is approximated by

u(x,0): fluid axial velocity at surfaceus : surface axial velocity

T(x,0): fluid temperature at surfaceTs : surface temperature

Couette and Poiseuille flows in MEMS

The electrostatic comb-drive used in microactuators and microsensors

Lubrication of micromotors, rotating shafts and microturbines

Fluid cooled micro heat sink

Assumptions

(1) Steady state,

(2) laminar flow,

(3) two-dimensional,

(4) ideal gas,

(5) slip flow regime (0.001 < Kn < 0.1),

(6) constant viscosity, conductivity, and specific heats,

(7) negligible lateral variation of density and pressure,

(8) negligible dissipation (unless otherwise stated),

(9) negligible gravity,

(10) the accommodation coefficients are assumed to be equal to unity,(σT = σu = 1)

1 )Couette Flow with Viscous Dissipation

(1) The velocity distribution?

(2) The Nusselt number?

Flow Field:

Navier-Stokes equations for compressible, constant viscosity flow:

The axial component

The continuity equation

v = constant v = 0

Boundary Conditions:

Knudsen Number

Flow Field:

The following observations are made regarding this result:

)1 (Fluid velocity at the moving plate, y = H, is

the effect of slip is to decrease fluid velocity at the moving plate and increase it at the stationary plate.

(2) Setting Kn = 0 gives the limiting case of no-slip.

Nusselt Number:

k : thermal conductivity of fluidT : fluid temperature function (variable)Tm : fluid mean temperatureTs : plate temperature

Surface temperature is determined using temperature jump equation:

Energy Equation

According to assumptions:

This energy equation requires two boundary conditions:

The Nusselt number is independent of Biot number. This means that changing the heat transfer coefficient h0 does not affect the Nusselt number.

The Nusselt number is independent of the Reynolds number. This is also the case with macrochannel flows.

Unlike macrochannels, the Nusselt number depends on the fluid.

The first two terms in the denominator represent the effect of rarefaction (Kn number) while the second term represents the effect of temperature jump. Both act to reduce the Nusselt number.

Kn=0 →Nu0 (Macrochannels) → The ratio is less than unity.

If dissipation is neglected (φ=0) → T= T∞ → The temperature is uniform and no heat transfer takes place.

Nusselt Number:

2) Fully Developed Poiseuille Channel Flow )Uniform Surface Flux)

How does microchannel Poiseuille flow differ from fully developed, no-slip macrochannel flow?!

Macrochannels:

1. Parallel streamlines,

2. Zero lateral velocity component (v = 0)

3. Invariant axial velocity with axial distance (∂u / ∂x = 0)

4. Linear axial pressure

Microchannels:

Compressibility and Rarefaction change this flow pattern

Because of the large pressure drop in microchannels, density change in gaseous flows becomes appreciable and the flow can no longer be assumed incompressible

a decrease in pressure in microchannels results in an increase in the mean free path . Thus the Kn number increases along a microchannel

axial velocity varies with axial distance

lateral velocity component does not vanish

streamlines are not parallel

pressure gradient is not constant

Flow Field:

Negligible inertia forces. This approximation is justified for low Reynolds numbers. (The Reynolds number in most microchannels is indeed small because of the small channel spacing or equivalent diameter.)

Scale analysis shows that The dominant viscous force is

For steady state and negligible gravity

Boundary Conditions:

The Knudsen number, which varies with pressure along the channel, represents rarefaction effect on axial velocity.

To complete the flow field solution, the lateral velocity component v and pressure distribution p must be determined:

Continuity Equation:

Ideal Gas:

Knudsen number must be expressed in terms of pressure:

Boundary Conditions (Inlet and Outlet Pressures):

This result can be expressed in terms of the Knudsen number at the outlet:

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

Knudsen number terms represent rarefaction effect on the pressure distribution.

The terms represent the effect of compressibility.

Nusselt number

Energy Equation:

Negligible dissipation,

Negligible axial conduction,

Negligible effect of compressibility on the energy equation

Nearly parallel flow, v = 0

Dimensionless temperature:

Fully developed temperature is defined as a profile in which φ is independent of x

Differentiating the above and evaluating the derivative at y=H/2 :

The heat transfer coefficient:

Differentiating

Conservation of Energy:

Integrating twice:

Boundary conditions:

First method:

Mean Temperature

(1) and (2)

Second method:

(1)

(2)

(3)

(4)

(1), (2), (3), (4)

The Knudsen number in is a function of local pressure. Since pressure varies along the channel, it follows that the Nusselt number varies with distance x. This is contrary to the no-slip macrochannels case where the Nusselt number is constant.

Unlike macrochannels, the Nusselt number depends on the fluid, as indicated by Pr and γ

The effect of temperature jump on the Nusselt number is represented by the last term in the denominator

Rarefaction and compressibility have the effect of decreasing the Nusselt number. Depending on the Knudsen number, using the no-slip solution can significantly overestimate the Nusselt number.

Reference:

Latif M. Jiji, “Heat convection”, Springer