z-axis. Other dimensionless combinations Demonstrated in the table below is another value of dimensional analysis. Any number of new dimensionless parameters can be calculated through exponents and multiplication. For example, a variation of the first parameter shown below is popularly used in aircraft engine system analysis. The third parameter is a simplified dimensional variation of the first and second. This third definition is applicable with strict limitations. The fourth parameter, specific speed, is very well known and useful in that it removes diameter. The fifth parameter, specific diameter, is a less often discussed dimensionless parameter found useful by Balje. It may be found interesting that the specific speed coefficient may be used in place of speed to define the y-axis of Figure 1.2, while at the same time, the specific diameter coefficient may be in place of diameter to define the z-axis. Affinity laws The following affinity laws are derived from the five Π-parameters shown above. They provide a simple basis for scaling turbomachinery from one application to the next. Aero-thermodynamic fundamentals The following equations outline a fully three-dimensional mathematical problem that is very difficult to solve even with simplifying assumptions. Until recently, limitations in computational power, forced these equations to be simplified to an Inviscid two-dimensional problem with pseudo losses. Before the advent of computers, these equations were almost always simplified to a one-dimensional problem. Solving this one-dimensional problem is still valuable today and is often termed mean-line analysis. Even with all of this simplification it still requires large textbooks to outline and large computer programs to solve practically. Conservation of mass Also termed continuity, this fundamental equation written in general form is as follows: Conservation of momentum Also termed the Navier–Stokes equations, this fundamental is derivable from Newton's second law when applied to fluid motion. Written in compressible form for a Newtonian fluid, this equation may be written as follows: Conservation of energy The first law of thermodynamics is the statement of the conservation of energy. Under specific conditions, the operation of a Centrifugal compressor is considered a reversible process. For a reversible process, the total amount of heat added to a system can be expressed as where is temperature and is entropy. Therefore, for a reversible process: Since U, S and V are thermodynamic functions of state, the above relation holds also for non-reversible changes. The above equation is known as the fundamental thermodynamic relation. Equation of