THERMODYNAMIC COMPONENTS FOR IDEAL GAS CYCLE (BRAYTON)

This section demonstrates how to evaluate several thermodynamic components for the ideal gas cycle (Brayton). We assume components operate as steady-flow devices, meaning there is no time-rate of change in the intrinsic properties of the device. Relevant to proposed Brayton cycles for FPP, the analysis can be further simplified if the working fluid has constant specific heats. In the case of Helium, constant specific heats can be assumed.

TURBINES AND COMPRESSORS

Turbines and compressors are components that convert energy between fluid and mechanical forms. Turbines convert pressure into mechanical work, while compressors use work to increase the pressure of the working fluid. In an isentropic expansion or compression of an ideal gas with constant specific heats, the following relationships can be found.

Where subscripts \(1\) and \(2\) correspond to the initial and final values. Moreover, the parameter \((P_2/P_1)\) is known as the compression ratio, (\(r_p\)).

The isentropic efficiency of a device quantifies the deviation of a real device compared to the idealized process. Intuitively, for turbines, this quantity reflects the (Actual turbine work)/(Isentropic turbine work). While for compressors, the quantity conveys the (Minimum work required)/(Actual work required).

In the above eq, and for the remainder of this report, the subscript (\(s\)) indicates an isentropic process and the subscript indicates the actual process. The use of isentropic efficiency in the energy balance for these components is shown below as an example. A typical control volume energy balance can be performed on all components in a Brayton Cycle. Usually, it is safe to assume \(\Delta ke = \Delta pe = 0\) (i.e. negligible change in the kinetic and potential energy of the fluid). Then, the only mechanisms of energy transfer remaining are through heat (\(Q\)) and work (\(W\)). For SISO control volumes, \(\dot{m}_{in} = \dot{m}_{out}\) and it is common to remove \(\dot{m}\) from the equation, leaving results in specific form with units \(kJ/kg\). The specific heat and work transfer are shown by lower case letters \(\dot{q}, \dot{w}\). The specific and true values are given by \(\dot{Q} = \dot{m}\dot{q}\) and \(\dot{W} = \dot{m}\dot{w}\). Since there is insignificant heat transfer associated with compressor and turbine components, the energy balance reduces to: \[w_{s} = \dot{m}c_p(T_{2s}-T_{1})\]

Where (\(T_{2s}\)) is the outlet temperature for an isentropic process, found by (\ref{idealGas}). We can also say that the actual work is of a similar form:

Substituting the above equations into (\ref{isenEff}), we can solve for the actual outlet temperature as well as component work.

INTERCOOLING AND REHEAT

In power cycles with multiple stages of compression and or turbines, there is an intermediate stage between components. The working fluid is either intercooled (compressors) or reheated (turbines) before entering the next component. These processes are isotropic heat removal and addition, respectively. In either case, the temperature of the working fluid is returned to its initial temperature, before compression or expansion. To maintain consistency in numbering, intercooling or reheating describes the process from state 2 to state 3. The state variables for state 3 and associated heat transfer are easily found by:

REGENERATOR OR RECUPERATOR

In a Brayton cycle, a regenerator (also known as a recuperator) is simply a heat exchanger that transfers excess heat from gases at the turbine circuit outlet to the colder gas at the compressor circuit outlet. Effectively reducing the amount of waste heat by siphoning some back into the system. The regenerator is evaluated in the same manner as a traditional heat exchanger (HX). In this section, to avoid confusion, we will drop the numbering used in the preceding sections. Instead, we will let \(T_{ci}\) and \(T_{co}\) be the inlet and outlet temperatures on the cold side of the HX, and \(T_{hi}\) and \(T_{ho}\) be the in and outlet temperatures on the turbine side. As will be clearer in later sections, \(T_{ci}\) and \(T_{hi}\) are not initially known but can easily be calculated.

Heat exchangers are characterized by a new quantity, known as the effectiveness \(\epsilon\). Similar to isentropic efficiency:

Furthermore, we can say:

Therefore, if the effectiveness and inlet temperatures are known, we can calculate \(q_{max}\) which can be used to calculate \(q_{actual}\). In the case of constant specific heats, the equation reduces even further.