Dimensionless variables Subject

Stability relates to the behaviour of a system when it is subjected to a small perturbation away from a given stationary state (or if fluctuations occur naturally). If the perturbation decays to zero, the system has some in-built tendency to return back to the same state. In this case it is described as locally stable. (The qualification local means that very large perturbations may have different consequences.) We will introduce the relatively simple mathematical techniques required to determine this local stability of a given state in Chapter 3. It will also be useful before then to reduce the reaction rate equations (2.1)—(2.3) to their simplest possible form by introducing dimensionless variables and quantities. 

Another important dimensionless variable is the ratio of chamber-to-pintle diameters, djdp. Typical values for this quantity range from 3 to 5 [2]. Finally, the skip distance is defined as the length that the annular flow must travel before impacting the radial holes divided by the pintle diameter, LJd. A typical value for this parameter is around 1 larger skip distances are subject to substantial deceleration of the liquid due to friction against the pintle post whUe very short skip distances may lead to spray impingement on the head-end of the combustion chamber. 

The dimensionless model equations are programmed into the ISIM simulation program HOMPOLY, where the variables, M, I, X and TEMP are zero. The values of the dimensionless constant terms in the program are realistic values chosen for this type of polymerisation reaction. The program starts off at steady state, but can then be subjected to fractional changes in the reactor inlet conditions, Mq, Iq, Tq and F of between 2 and 5 per cent, using the ISIM interactive facility. The value of T in the program, of course, refers to dimensionless time. 

Note that the bar over the symbol y represents the fact that the dimensionless concentration was transformed. All variables not subjected to direct transformation remain as they were before applying the transform. 

We consider fully developed incompressible laminar flow, considering slip at the walls, inside a circular micro-tube or a parallel plates micro-channel subjected to a pressure gradient dp/dz that varies in an arbitrary functional form with the time variable. The velocity field is represented by u(r,t), which varies with the transversal coordinate, r, and time, t. The related time-dependent axial momentum equation (z-direction) is then written in dimensionless form as ... 

Dimensionless numbers are needed whenever nonlinear equations are encountered because transcendental and polynomial functions cannot have units. Taking the logarithm of 10 moles or raising e to the power of 20 minutes is a meaningless calculation. This does not mean that there cannot be units within these functions but it does mean that those units must cancel out. For example, it is perfectly acceptable to raise e to kt where t is time and the rate constant, k, has the dimension of reciprocal time so that the product kt has no units. Recasting a variable into a dimensionless number for use in a nonlinear equation can be done in various ways, but all those ways are subject to the restriction of the Buckingham n theorem. 

Those dimensionless parameters that are ratios of material variables, such as n , which is the ratio of fluid densities, and He/Hi 3, which is the ratio of fluid viscosities, are not an issue for this scaling effort. The criteria for scaling these dimensionless parameters are (ri7)M = (ri7)p and (ne/nn) = (n6/rii3)p, where the subscript M denotes model and the subscript P denotes prototype. These criteria must hold true. The other combinations of material variables are also subject to the above restriction in other words, they do not complicate... 

---