Additional hints on solving the project problem

27 Nov 2020

Mathematical background

1) You can use the first law analysis of open systems in order to write down the equations for net power generation in the combined cycle and efficiency. It will be a set of simple equations with many unknown parameters. (i.e. the enthalpies of many states, e.g. 8-11, the mass flow rates, etc.)

2) You will notice immediately that you have many more unknown variables than known or given parameters. You will need to formulate a system of equations to solve the problem. For a well constrained/defined system, you will end up with the same number of equations and unknown variables. Some of these equations you can write explicitly (like using the first law of thermodynamics in an open system to link power, mass flow rates, and enthalpy differences). Other properties are linked by fluid-specific equations, which may be tabulated (like an equation of state that links the pressure, density, and temperature of a fluid).

3)

Eventually, you will find that you need to know four additional parameters in order to close the system (i.e. have a number of unknowns equal to the number of equations). These four parameters in this project are given to be

*P2/P1 , T8 , P8 *, and

*P9.* This is a selection that was made for you, so that the problem be more tractable. After this point, you will need to guess values of these four parameters (

*P2/P1, T8 , P8 *, and

*P9*) in order to solve the cycle. Different choices for these parameters will lead to different solutions for the cycle. In principle, some choice of these parameters would lead to maximum values for power and efficiency, but finding that choice is not trivial (it involves an iterative process, that is frequently used in engineering). But remember, in all cases,

*all your parameters would have to satisfy the equations that apply for the problem*.

4)

**For the purposes of this project, you might just try a few iterations to observe trends (even if you have to solve the calculations by hand, i.e. determine enthalpies and other properties from the applicable tables) rather than exhaustively finding the optimal parameters, using CoolProp that can determine the properties in an automated fashion. In engineering, one is better served by keeping all parameters fixed and changing another parameter, to designate how the main outcome variable is affected. For example, does a higher value of ***T8* when *P2/P1 , P8 *and *P9* **are fixed, increase or decrease the efficiency of the cycle? This method would guide you towards improving efficiency as you modify the assumed value of each of the four parameters.**
Practical example of the solution procedure:

Identify the properties necessary for efficiency and power, as above. The system of equations between the given properties, guessed properties, and efficiency and power will have to be solved one time, after which time you can just plug in different values for the guesses. Here are the details for one iteration

*for R-134a.* (I pulled the numbers from CoolProp, your values for enthalpy will be different if you use the tables from the back of your textbook, but the differences in enthalpies should be very close)

1)

*Guess the four parameters identified from the start:* a) P2/P1 = 10, b) T8 = 450 K, c) P8 = 20 bar, d) P9 = 12 bar.

2) From the given/guessed properties and constant pressure processes, you can solve for the pressures everywhere P2, P3, P4, etc.

3) Calculate isentropic state 2s, i.e. T2s

4) Calculate state 2 accounting for isentropic efficiency of compressor. (in this case, T2 = 629K)

5) Calculate isentropic state 5s

6) Calculate state 5 accounting for the isentropic efficiency of the turbine (in this case, T5 = 708K)

7) Calculate state 3 using regenerator equation. (I got T3 = 692 K)

8) Calculate state 6 using the energy balance on the regenerator.

9) Identify mass flow rate of the air in the gas turbine part of the cycle by comparing net power and the information from the turbine and compressor. (I got 0.65 kg/s)

10) Specific enthalpy and entropy of the R-134a state going into the turbine of the Rankine cycle can be obtained from your guess.

*You need to make sure it is a vapor state…*
11) Calculate the isentropic state 9 using your guessed P9 and the fact that s8 = s9s

12) Use the isentropic efficiency of the turbine to calculate the real state 9 (I got T9 = 432 K)

13) Identify the density of the saturated liquid leaving the condenser. You can use this along with the pressure difference to calculate the isentropic power of the pump.

14) Use the isentropic efficiency of the pump to calculate the real power of the pump.

15) Use the energy balance on the heat exchanger in order to calculate the mass flow throughout the Rankine cycle. (I got 0.39 kg/s)

16) The above steps should result in values for all the parameters you need in order to calculate total thermal efficiency and total power. (I got 102.1 kW, and an efficiency of 31%)

17)

**Evaluate the feasibility of the result. Does the system violate the second law of thermodynamics anywhere? Are the values of temperature and pressure at different states appropriate?**
18) Change parameters (sensibly) and repeat the above steps to determine whether you could find another set of the four parameters that produce a higher efficiency and/or maximum net power.