EBSILON®Professional Online Documentation
EBSILON®Professional uses the Stodola’s law of the ellipse, a summary of which is given below.
For an uncontrolled multistage expansion till a high vacuum, it is normal that at any point in the expansion downstream the pressure-flow rate ratio relation can be approximated for each expansion point
PHI=Mi / SQRT(Pi / Vi)=const with i = Expansion point
PHI is set as
PHI=SQRT(1-( POUTj / PINj )**2) with j = Extraction group
according to Stodola Ellipse. With Pi=PINj=P1 and POUTj=P2 the result is
M1**2=(P1**2-P2**2) / (P1*V1)
For the comparison of the design and the off-design case, we get
(M1 / M1N) **2 = (P1**2 - P2**2) / (P1N**2 - P2N**2) * (P1N * V1N) / (P1*V1)
M - mass flow
P - pressure
V - specific volume
index 1: inlet
index 2: output
index N: nominal value from design calculation
Umgewandelt ergibt sich:
M1 = S * SQRT ( P1**2 - P2**2 ) / ( P1 * V1 ) (1) with
S = M1N * SQRT ( P1N * V1N) / SQRT ( P1N**2 - P2N**2 )
The coefficient S of the turbine is determined during the design calculation. Sometimes it is also referred to as the "swallowing capacity" of a turbine.
Converted, the result is:
P1 = SQRT ( P2**2 + ( (M1 / S)**2 * P1 * V1 ) )
This equation is solved iteratively and applies to components 6, 56 and 58.
It is valid for real and ideal gases, for steam and wet steam.
Component 122 uses an improved version of equation (1) recommended by Traupel.
M1 = S * SQRT ( P1 * V1 ) * SQRT ( 1 - (P2/P1) ** ((n+1)/n) ) (2) with
S = M1N * SQRT ( P1N * V1N) / SQRT ( 1 - (P2N/P1N) ** ((n+1)/n) )
with n: Polytropic exponent n = kappa / ( kappa- etap * (kappa - 1) )
with
kappa: Isentropic exponent
etap: Polytropic efficiency of the turbine
Equation (1) is a special case of equation (2) when the polytropic exponent n becomes 1.