I am working in area of micro scale heat transfer but there is one question which has puzzled me for long. I have read in several papers that beyond this limits continuum theory is obeyed etc.
When we say at micro scale, the continuum theory is applicable, what is the criteria for the same?
Is it applicability of conventional macro scale relations, like Po number for flow or Nu number for heat transfer etc. which support this factor or one looks for something else from experimental data?
What relations or experimental data needs to be studied for the same?
The definition of continuum flow is not linked to gases or liquids, or to microchannels, since it applies also to rarefied gasdeynamics, as that encountered in the flow around satellites or high altitude rockets, or to flows in a vacuum chamber.
The need arises from the necessity of describing the matter which is intrinsically discontinuous with average quantities and not molecule by molecule. So having the need of an average macroscopic description and not with a microscpic one. E.g., the exchange of momentum between a fluid and the wall, that contains it, is not described by the exchange that occurs at a molecular level but by two average quantities which are the pressure and the shear stress at the wall. So the latter represent in a continuum vision the integral effects of the momentum exchanged by a great number of molecules. The temperature is representing nothing else than the average energy of a large number of molecules.
An important parameter to check when it is possible to apply the continuum hypothesis is the molecular mean free path which represent the average distance between the molecular collisions.
Please, remember that in each collision molecules exchange momentum and enegy which are the quantities of interest.
In a gas the molecular mean free path can be expressed as:
1/lamda = = const x dsquare x n
where d is the diameter of the molecule and n the number of molecules per unit volume. So for a given gas, the mean free path changes with the number of molecules per unit volume changing.
Given a fluid flow that has to be described in a control volume, one may use the continuum hypothesis as long as in such a volume there exist a very large number of much smaller volumes (flow field resolution) which in turn have their own volume much larger than the cube of the molecular mean free path. This latter conditions allows to have a large number of molecules in the small volume and to make averages of the microscopic quantities.
More simply, the condition is fullfilled as long as the Knudsen number defined as
Kn = lamda/L
where L is a characteristic lenght of the examined flow field
is small.
If you take the Shuttle, at its orbiting altitude is in a regime of free molecular flow (the mean free path is large) so you have to take into account momentum and energy echanges molecule by molecule (newtoniaqn approach). But, as its altitude gets much lower, you can use the continuum assumption because the molecular mean fre path gets much lower and the Knudsen nuber decreases. Naturally, between this two limiting cases other regimes do exist such as slip flow and transition regime.
For your microchannels, the characteristic lenght may be represented by the channel diameter
for gas flow there is the Knudsen number, Kn. If the Kn 5 you have to use free molecular approximation.
That is theoretical.
My question was say I have tubes of dia varying as 1000, 100,10, 1 micron.
I have to decide in which case there is transition from continuum to slip and so on. Then in case of fluid (both liquid and gases) what parameters or data I need to study in experiment to conclude the same.
Like to study whether the flow is laminar or turbulent, we plot f*Re versus Re and say if its linear then we say its laminar else go for some turbulence.
Similarly what to look at while studying whether continuum or slip/no slip is obeyed.
No, it is not theoretical. Fisenko is right. The Knudsen number (Kn) helps you to decide how to treat micro-flows in terms of continuum approach and boundary conditions (slip/no-slip). The Kn number for continuum/statistical molecular approach is like the Reynolds for laminar/turbulent flows.
It is true that microflow theory is not so systematic as macroflows (as you mentioned the Moody chart).
Perhaps the following books can help you somehow:
GP Celata - heat transfer and fluid flow in microchannels
SG Kandlikar et al. - heat transfer and fluid flow in minichannels and microchannels
- Badges
- Science topic