Question

Electricity and process heat requirements of a manufacturing facility are to be met by a cogeneration plant consisting of a gas turbine and a heat exchanger for steam production. The plant operates on the simple Brayton cycle between the pressure limits of 100 and 1000 kPa with air as the working fluid. Air enters the compressor at \(20^{\circ} \mathrm{C}\). Combustion gases leave the turbine and enter the heat exchanger at \(450^{\circ} \mathrm{C},\) and leave the heat exchanger of \(325^{\circ} \mathrm{C},\) while the liquid water enters the heat exchanger at \(15^{\circ} \mathrm{C}\) and leaves at \(200^{\circ} \mathrm{C}\) as a saturated vapor. The net power produced by the gas-turbine cycle is \(1500 \mathrm{kW}\). Assuming a compressor isentropic efficiency of 86 percent and a turbine isentropic efficiency of 88 percent and using variable specific heats, determine \((a)\) the mass flow rate of air, \((b)\) the back work ratio and the thermal efficiency, and \((c)\) the rate at which steam is produced in the heat exchanger. Also determine \((d)\) the utilization efficiency of the cogeneration plant, defined as the ratio of the total energy utilized to the energy supplied to the plant.

Short answer

Question: Calculate (a) the mass flow rate of air, (b) the back work ratio and thermal efficiency, (c) the rate at which steam is produced in the heat exchanger, and (d) the utilization efficiency of the cogeneration plant based on the given step-by-step solution. Answer: (a) To calculate the mass flow rate of air, use the equation: \(\dot{m}_{air} = \frac{W_{net}}{w_t - w_c}\) (b) The back work ratio can be found using the equation: \(BWR = \frac{w_c}{w_t}\) And the thermal efficiency can be calculated using: \(\eta_{th} = \frac{W_{net}}{Q_{in}}\) (c) To find the rate of steam production in the heat exchanger, use the energy balance equation: \(\dot{m}_{air}(h_4 - h_3) = \dot{m}_{steam}(h_6 - h_5)\) (d) The utilization efficiency of the cogeneration plant can be calculated with: \(\eta_{utilization} = \frac{W_{net} + \dot{m}_{steam}(h_6 - h_5)}{\dot{m}_{air}(h_3 - h_2)}\)

Step-by-step solution

Calculate actual adiabatic efficiencies

Since the compressor and turbine have specified isentropic efficiencies, we need to find the actual adiabatic efficiencies. For the compressor, we have: \(\eta_c = 0.86\) For the turbine, we have: \(\eta_t = 0.88\)

Calculate specific work in the compressor and turbine

We can use the adiabatic efficiencies to find the specific work in the compressor and turbine. The specific work in the compressor (\(w_c\)) is given by: \(w_c = \frac{h_2 - h_1}{\eta_c}\) Similarly, the specific work in the turbine (\(w_t\)) is given by: \(w_t = \eta_t (h_3 - h_4)\) We need to find \(h_2, h_1, h_3,\) and \(h_4\) using the given temperature and pressure information. Using the air properties table, we can find these values using the temperatures and pressures given. However, since the properties of air vary with temperature, we should use variable specific heats.

Calculate mass flow rate of air

Given the net power produced by the gas-turbine cycle (\(1500\,\text{kW}\)), we can find the mass flow rate of air (\(\dot{m}_{air}\)) using the following equation: \(\dot{m}_{air} = \frac{ W_{net}}{w_t - w_c}\)

Calculate back work ratio and thermal efficiency

To find the back work ratio, we use the formula: \(BWR = \frac{w_c}{w_t}\) The thermal efficiency (\(\eta_{th}\)) can be calculated as: \(\eta_{th} = \frac{W_{net}}{Q_{in}}\) where \(Q_{in}\) is the heat input in the combustion chamber, given by: \(Q_{in} = \dot{m}_{air}(h_3 - h_2)\)

Calculate the rate of steam production in the heat exchanger

We need to find the energy balance in the heat exchanger to determine the rate at which steam is produced. We set up an energy balance equation as follows: \(\dot{m}_{air}(h_4 - h_3) = \dot{m}_{steam}(h_6 - h_5)\) The rate of steam production, \(\dot{m}_{steam}\), can be found by solving the equation for the given heat exchanger temperatures.

Calculate utilization efficiency of the cogeneration plant

To find the utilization efficiency, we need the total energy utilized in both the electricity production and the process heat: \(\eta_{utilization} = \frac{W_{net} + \dot{m}_{steam}(h_6 - h_5)}{\dot{m}_{air}(h_3 - h_2)}\) Using the values calculated in the previous steps, we can find the utilization efficiency of the cogeneration plant.

Key concepts

Brayton cycle

The Brayton cycle is the thermodynamic process used by gas turbines and is characterized by constant pressure combustion and expansion. It's comprised of four steps: isentropic compression in a compressor, constant pressure heat addition in a combustion chamber, isentropic expansion through a turbine, and constant pressure heat rejection in a heat exchanger or the environment.

This cycle is pivotal in power generation applications, especially within gas turbine engines for aircraft propulsion and power plants. When we talk about a cogeneration plant utilizing a Brayton cycle, it means that not only is electricity being generated, but also useful heat is captured and utilized, increasing the overall system's efficiency.

Isentropic efficiency

Isentropic efficiency is a measure of the performance of a compressor or turbine compared to an ideal isentropic process. Isentropic means a process that is both adiabatic (no heat transfer) and reversible (no entropy generation).

The isentropic efficiencies of the compressor (\( \text{typically noted as} \text{\tiny \text{\tiny }eta}_c\)) and turbine (\( \text{typically noted as} \text{\tiny \text{\tiny }eta}_t\)) are critical for determining the actual work input or output, as real machines do not operate ideally. Higher isentropic efficiency means less energy loss and closer performance to the ideal scenario.

Thermal efficiency

Thermal efficiency (\( \text{typically noted as} \text{\tiny \text{\tiny }eta}_{th}\)) is a crucial factor in power generation, reflecting the fraction of heat converted to work. It is defined as the ratio of net work output to heat input. A higher thermal efficiency indicates a more effective engine or system, as it means more of the fuel's heat is transformed into useful work and less is wasted.

In cogeneration plants, the calculation of thermal efficiency is essential to gauge how effectively the plant is running. This efficiency is a significant factor when comparing different power systems or when looking to optimize existing systems.

Mass flow rate

The mass flow rate (\( \text{typically noted as} \text{\tiny }\dot{m}\)) represents the quantity of mass passing through a given surface per unit time. In the context of the Brayton cycle, it's the amount of air passing through the compressor and turbine.

Calculating the mass flow rate of air is crucial because it directly impacts the plant's power output and efficiency. In cogeneration systems, it also affects the rate of steam production for heating purposes. Thus, understanding mass flow rates is essential for determining the plant's energy distribution and overall performance.

Back work ratio

The back work ratio (BWR) is used in thermodynamics to assess the proportion of turbine work utilized to drive the compressor. Specifically, it is the ratio of the compressor work to the turbine work. A low BWR indicates a more efficient cycle, as less of the work produced by the turbine is needed to power the compressor, thereby leaving more net work for output or other uses.

For cogeneration plants, minimizing the BWR is desirable to maximize electricity generation efficiency while retaining enough energy to produce process heat.

Specific heats

Specific heats, denoted as (\( c_p\)) for constant pressure and (\( c_v\)) for constant volume, are properties of substances that describe how much heat is needed to increase the temperature of a unit of mass by one-degree Celsius (or one Kelvin).

In thermodynamic cycles like the Brayton cycle, the variable specific heats of the working fluid — air in the case of our example — must be considered because the air's temperature changes substantially during the cycle. As a result, the specific heats at the actual temperatures during the processes should be used for accurate calculations rather than assuming constant values.

Energy balance

Energy balance is the principle of conservation of energy applied to thermodynamic systems. For a cogeneration plant running on the Brayton cycle, maintaining a balance between the energy entering and leaving each component of the system is critical for calculations and efficient operation.

An energy balance requires accounting for the work done by the system and the heat addition or rejection, ensuring that the total energy in equals the total energy out. In cogeneration, the balance includes the electricity produced, the steam generated for heating, and the fuels consumed, and is essential for determining steam production rates and utilization efficiency of the plant.