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# LawApproach

For the Law Approach one has to identify the relevant single effects, influences respectively, and their basic "constitutional" laws. The single effects are in most cases contributions to the force acting on the system or energies entering the overall energy balance. However, in this approach these effects are not balanced like in the Equation Approach, where the actual balancing equations and implicitly the interrelations of all effects are considered. Therefore the Law Approach might be seen as a methodology in between the Parameter Approach and the Equation Approach.

#### Step 1

Decide what are the important effects. In a gas mixing process this will be inertia, friction, buoyancy and diffusion.

#### Step 2

Give for each term identified in Step 1 the law defining the related forces or energies. For inertia this would be the Newton's law which allows to determine the inertial forces as the product of mass m and acceleration a : $$F_{inertia} = m a$$

#### Step 3

Now we replace the terms in the laws provided in step 2 by suitable representative quantities. By this replacement we formally also replace "equality" by a "correspondence". So for the example above this will give and "estimate" for the representative inertial force $$F_{r,inertia} \simeq m_r l_r / t_r^2 = M L / T^2$$ where capital letter denominate the representative quantities (here representative mass M , length L and time T . Or - as one should not have force and mass terms simultaneously - one could introduce some material properties here: $$F_{r,inertia} \simeq \rho L^4 / T^2 = \rho L^2 V^2$$ This could be also the preferred form for any problem in the field of continuum flow.

#### Step 4

Now we relate the term on the right hand side, representing the intertial forces in this case, to a fictive external force F . This yields a dimensionless \Pi -number. $$\Pi_{inertia} = {{ \rho L^2 V^2 } \over F }$$

These dimensionless \Pi -numbers resemble these numbers one could have found alternatively via the parameter approach. Like in Step 7 of the Dimensional Analysis combinations of these \Pi -numbers - in most cases ratios of them - will eliminate the artificially introduced fictive external force and will yield the more conventional \Pi -numbers.

For example, in step 2 we might have additionally introduced for the frictional forces in a fluid $$F_{friction} = \mu {{ \partial v } \over { \partial y }} A$$ According to the described procedure, this will give $$\Pi_{friction} = {{ \mu V L } \over F }$$ The ratio of the two \Pi -numbers is the well known Reynolds-number Re $${{\Pi_{inertia}} \over {\Pi_{friction}}} = {{ L V } \over { \mu / \rho } } = Re$$

The law approach provides a reasonable compromise between the difficult selection of parameters, formal mathematical dimension analytical approach and the too general Equation approach. Still one has to chose the right laws relevant for this phenomena.

The application in the second part of this report will concentrate on this methodology.