In the problem below, a thermodynamics student wants to know how to find , in order for her to calculate the isentropic efficiency of the turbine. What is the step she must do to eventually find ?
400 3 MPa Actual process Isentropic process Steam turbine 100 50 kPa
she has to find the efficiency of the compressor in order for her to find the efficiency of the turbine through the direct x calculation she first has to use at 3MPa as well as sf and sfg at 50 kPa to find x she has to find and use it with hf and hfg at 3 MPa using x she first has to use at 50 kPa as well as hf and hfg at 3 MPa to find to find x she first has to use at 50 kPa as well as hf and hfg at 3 MPa to find directly she must first find hf at 3 MPa and hg at the 50 kPa to solve for she has to first realize this is an actual (real) process and therefore she must use the value of h at 2a together with in the same equation to find she can’t there is missing information
The Correct Answer and Explanation is:
Correct Answer:
She first has to use s₁ at 3 MPa as well as sf and sfg at 50 kPa to find x.
Explanation
To calculate the isentropic efficiency of the turbine, the student must find h₂s, the enthalpy at the turbine outlet assuming an isentropic process (idealized, no entropy generation). The isentropic efficiency is defined as: ηturbine=h1−h2ah1−h2s\eta_{turbine} = \frac{h_1 – h_{2a}}{h_1 – h_{2s}}
Where:
h1h_1 = Enthalpy at turbine inlet (given or can be found from T₁ and P₁)
h2ah_{2a} = Enthalpy at turbine outlet (actual, given as 100°C at 50 kPa)
h2sh_{2s} = Enthalpy at turbine outlet under isentropic conditions (to be found)
Step-by-Step to Find h2sh_{2s}:
Determine s₁ (entropy at state 1): Since state 1 is superheated steam at 3 MPa and 440°C, use steam tables to find entropy s1s_1.
Use s₁ = s₂s (isentropic assumption): For isentropic processes, entropy remains constant. Therefore, s2s=s1s_2s = s_1.
At 50 kPa, determine the quality (x): Look up saturated properties at 50 kPa:
sfs_f: entropy of saturated liquid
sfgs_{fg}: difference in entropy between saturated vapor and liquid
Use the formula: x=s1−sfsfgx = \frac{s_1 – s_f}{s_{fg}}
Find h2sh_{2s}: Use the quality (x) found above to determine h2sh_{2s} from: h2s=hf+xhfgh_{2s} = h_f + x h_{fg}
Where:
hfh_f: enthalpy of saturated liquid at 50 kPa
hfgh_{fg}: latent heat of vaporization at 50 kPa
This method uses the isentropic relation of constant entropy and applies steam table data to compute the thermodynamic quality and enthalpy of the steam under ideal conditions at turbine exit pressure.
Hence, the correct step is:
She first has to use s₁ at 3 MPa as well as sf and sfg at 50 kPa to find x.
