COMPRESSIBLE FLOW METHODS The major difference in the pressure drop calculations between compressible flow and incompressible flow is the great change that can occur in the density of the gas between the inlet and outlet conditions. The use of incompressible flow relationships can lead to major errors if the pressure drop in the pipe exceeds 10 % of the inlet conditions. However, the incompressible flow methods can be used with fair accuracy provided the average density between the inlet and outlet conditions are used. However, this procedure is also prone to error if the gas velocity approaches sonic limits. It also requires considerable trial and error to recalculate to determine the outlet conditions and to determine the average properties for the gas. The method used in this program is based upon exact theoretical solutions of the flow equations, and are accurate for high Mach numbers. The maximum flow rates are base upon a limitation for adiabatic flow of a Mach number of 1.0 and for Isothermal flow at it's limit of 1 / SQRT( K ). This is the most accurate method for calculation of adiabatic and isothermal flow, but it is too complex for normal hand calculation procedures due to the recursive methods needed for the solution. The equations used are presented in Table 1. MACH NUMBER The maximum possible velocity in a pipe determined by the velocity of sound in the gas. Once the velocity of sound is reached the pressure in the pipe cannot be reduced below a minimum value. This is also called the choke point. The pressure downstream of the choke point remains at this constant value until the outlet of the pipe is reached and the remaining energy is dissipated as a shock wave. The velocity of sound in the gas is a function of the gas properties and the temperature of the gas. It is calculated by the equation. Cs = Sqrt[ k*Gc*(Z*R*T/MW) ] Where Gc = 32.17 and R is gas constant 1545. The Mach Number M is the velocity of the gas in the pipe divided by Cs. For adiabatic flow the highest gas velocity possible is achieved at a Mach number of one. A Mach number of one at the source conditions determines the maximum possible flow through the inlet nozzle. The Mach number increases while the gas flows through the pipe due to the effects of friction. Friction reduces the pressure and increases the volumetric flow rate in cubic ft/min. STAGNATION STATE Flow is possible between two extremes. At one extreme, the velocity is zero and temperature is a maximum, since all the potential kinetic energy is converted to enthalpy. The speed of sound is also a maximum. At the other extreme, the velocity is a maximum and the temperature falls to zero, all the enthalpy being converted to kinetic energy. The speed of sound is then zero ( zero temperature state). Between these extremes the flow may be subsonic, transonic, or supersonic. However, Supersonic flow can only be achieved in diverging nozzles and is not the subject of this program. Friction flow in a pipe is limited by the Mach number constraints discussed above. The stagnation state is the condition where the gas is compressed and stored in a reservoir under stagnant conditions. The fluid is at rest and has zero linear volocity. These conditions correspond to the information entered at the source conditions. These conditions are given the subscript o in the Table below. CRITICAL STATE The critical state condition refered to in this program is not the thermodynamic condition of the gas at high pressures where its properties blend with the liquid state, but the conditions at maximum flow in the pipe. The critical state corresponds to the conditions fixed by the sonic velocity of the gas. These conditions are given the subscript * in the Table. FRICTION EFFECTS The pipe definition is converted to velocity head; by the equation. Nf = f(Length/Diameter) where f is the friction factor The maximum friction drop at a specified location in the pipe is defined as a function of the Mach number, and the K ratio of the gas. The equations for this are different for isothermal and adiabatic flow and are given in the Table. The friction factor is essentially constant in the pipe since it is a function of the Reynold's number and pipe roughness, and at the high velocities encountered in gas flow the friction factor is approachs a constant limit. The value of Lmax derived from the equations in the table corresponds to the maximum length where the flow reaches critical conditions. This value is reported in the program, based upon the inlet conditions. The conditions at the outlet of the pipe are calculated by subtracting the Nf of the specified pipe from the value at the inlet of the pipe, and solving for the outlet Mach number. (fLmax/D) = (fLmax/D) - (fLmax/D) M2 M1 actual The flow conditions at the outlet of the pipe are calculated by computing the ratio of the pressure to the critical pressure at the inlet and outlet Mach numbers. ie: Pout = [(P/P*)out / (P/P*)in ] * Pin Tout = [(T/T*)out / (T/T*)in ] * Tin Pressures are in Psia Temperatures are in degrees R. absolute The conditions at the inlet of the pipe are calculated by using the ratio equations for isentropic flow on the conditions at the source. The derivation of these equations and methods are presented in Volume 1 of : Shapiro 'The Dynamics and Thermodynamics of Compressible Fluid Flow' , The Ronald Press Co, New York, 1954 TABLE 1 ISENTROPIC FLOW PROPERTY EQUATION Mach N M = 1 Temp To/T = 1 + ((k-1)/2) * M^2 Pressure Po/P = (1 + (k-1)/2*M^2 )^((k/k-1)) Friction f*LENmax /D = 0 ISOTHERMAL FLOW PROPERTY EQUATION Mach N M = 1 / SQRT(k) Temp Constant Pressure P/P* = 1/ Sqrt(k) Friction f*LENmax / D = (1-k*M^2)/k*M^2 + ln[k*M^2] ADIABATIC FLOW PROPERTY EQUATION Mach N M = 1 Temp T/T* = (k+1)/2*(1+((k-1)*M^2) / 2 ) Pressure P/P* = (1/M)*SQRT [ (k+1)/(2*(1+((k-1)*M^2)/2] Friction f*Lmax/D = (1-M^2)/(k*M^2) + ((k+1)/2k)* Ln [(k+1)*M^2 / 2*(1+(k-1/2)*M^2)]