in context of heat transfer, how can we get actual time and distance?
In order to make a physical quantity dimensionless such (x, distance from certain point) we divide it by some relevant reference value (already known, such as plate length L). to get x back we should multiply the dimensionless x/L by L.
Non-Dimensioning leads to possible results:
1- Change the differential equation into a simpler form (e.g. separable ODE), so the analytical solution can be obtained.
2- Get a dimensionless numbers (e.g. Reynold's no., Nusselt no., and Weber no.) which are useful in determining the weight of some terms compared to the others and/or the relation between some parameters and the change in mode (e.g. Laminar and Turbulent flow).
3- Show the comparison of relative change of specific variable (e.g. T) to a distinguished parameter (To). Comparison is helpful is determining the value of the change and if it's really physical especially nowadays with the numerical solutions.
4- Finally a non-dimensioned comparison is more significant in relating the results to more general conditions, not only solving the specific problem studied.
Directly the physical significance of any dimensionless number is going to be dependent on what parameters are used to non-dimensionalize it. So not all non-dimensional distances or times are necessarily the same. One example would be non-dimensionalizing the location of point along the length a 2D pipe in the X direction. Non-dimensional X in this case could be x (actual location) / Length of Pipe. This would essentially result in a number between 0 and 1 corresponding to the location along the pipe.
This sort of approach is very handing when trying to solve sets of equations, the starting value in the pipe would become zero making and the ending value would be 1 in this case making the differential equations much easier to solve.
Another example is the Reynolds number, it is non-dimensional and is described by:
(density) * (velocity) * L (characteristic length, ie. pipe diameter)
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(viscosity)
This takes the inertial terms found in the Navier Stokes equations and divides them by the viscous terms to create a non-dimensional relationship between inertia and viscosity. With this information the type of flow laminar or turbulent can be determined since viscously dominated flows are laminar in nature and flows with very high intertial terms are turbulent in nature (Re > ~2000).
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