Question

An old gas turbine has an efficiency of 21 percent and develops a power output of \(6000 \mathrm{kW}\). Determine the fuel consumption rate of this gas turbine, in L/min, if the fuel has a heating value of \(42,000 \mathrm{kJ} / \mathrm{kg}\) and a density of \(0.8 \mathrm{g} / \mathrm{cm}^{3}\)

Short answer

Answer: The fuel consumption rate of the gas turbine is approximately 51.02 L/min.

Step-by-step solution

Calculate the energy required from the fuel

Based on the efficiency of the gas turbine, we can determine the amount of energy required from the fuel. The efficiency of a gas turbine is calculated as: Efficiency = (Power Output / Energy Input) * 100 We can rearrange the formula to find the Energy Input: Energy Input = Power Output / Efficiency Plugging in the given values: Energy Input = (6000 kW) / (0.21) Energy Input = 28,571.43 kW

Convert energy input to kJ/min

We need to convert the energy input to kilojoules per minute (kJ/min) since the heating value is in kJ/kg. 1 kW = 1 kJ/s, so we can write: Energy Input = 28,571.43 kJ/s Now convert to kJ/min: Energy Input = (28,571.43 kJ/s) * (60 s/min) Energy Input = 1,714,285.71 kJ/min

Calculate fuel consumption in kg/min

Now, we can use the heating value of the fuel to find the fuel consumption rate in kg/min: Fuel Consumption (kg/min) = Energy Input (kJ/min) / Heating Value (kJ/kg) Fuel Consumption (kg/min) = (1,714,285.71 kJ/min) / (42,000 kJ/kg) Fuel Consumption (kg/min) = 40.8197 kg/min

Convert kg/min to L/min

Finally, we can use the given density of the fuel to convert the fuel consumption rate from kg/min to L/min: 1 kg = 1000 g, so: Fuel Density = 0.8 g/cm³ = 0.8 kg/L Fuel Consumption (L/min) = Fuel Consumption (kg/min) / Fuel Density (kg/L) Fuel Consumption (L/min) = 40.8197 kg/min / 0.8 kg/L Fuel Consumption (L/min) = 51.0246 L/min The fuel consumption rate of this gas turbine is approximately 51.02 L/min.

Key concepts

Gas Turbine Efficiency

Understanding the efficiency of a gas turbine is crucial when analyzing its performance. In simple terms, efficiency is a measure of how well the turbine converts the energy from fuel into useful work. It is defined as the ratio of the power output to the energy input, expressed as a percentage. This can be represented by the equation:
\[ \text{Efficiency} = \left(\frac{\text{Power Output}}{\text{Energy Input}}\right) \times 100 \% \]
For an old gas turbine with 21% efficiency, this means that only 21% of the energy provided by the fuel is converted into electricity or mechanical power, while the rest is lost, primarily as heat. The evaluation of efficiency is a starting point for calculations involving fuel consumption, as higher efficiency turbines will require less fuel to produce the same amount of power. In this context, methods to enhance the efficiency of gas turbines, such as regenerative heating or combined cycle applications, are highly valuable for energy savings and emission reductions.

Energy Conversion in Gas Turbines

The process of energy conversion in gas turbines involves several steps where fuel energy is transformed into mechanical energy and then into electricity. In the simplest form, a gas turbine works by compressing air and mixing it with fuel, which is then ignited. The high-pressure combustion gases expand through the turbine, spinning the blades and producing mechanical work.
In our exercise, the conversion of the provided fuel energy into power output is assessed to determine the fuel consumption rate. The formula highlighting the relationship between energy input, power output, and efficiency reveals the interdependence of these variables. To find the energy input when power output and efficiency are known, we use the rearranged formula:
\[ \text{Energy Input} = \frac{\text{Power Output}}{\text{Efficiency}} \]
Since gas turbines operate continuously and not in bursts, it's necessary to know the fuel consumption rate per time unit, prompting the conversion from energy per second to energy per minute. Understanding this conversion allows us to relate the fuel's energy content to the amount needed to sustain the turbine's power output, completing the energy transformation cycle.

Heating Value of Fuel

Heating value, also known as calorific value, is a vital concept when calculating fuel requirements for a gas turbine. It represents the amount of heat released when a certain mass of fuel is completely burned. It is usually expressed in energy units per mass, such as kilojoules per kilogram (kJ/kg).
When we have the heating value of the fuel (42,000 kJ/kg), we can determine how much fuel is consumed to achieve the desired energy input. This is crucial in our exercise to find the fuel consumption rate. The equation to find the fuel consumption in kg per minute is as follows:
\[ \text{Fuel Consumption} (\text{kg/min}) = \frac{\text{Energy Input} (\text{kJ/min})}{\text{Heating Value} (\text{kJ/kg})} \]
This equation allows us to translate the energy required by the gas turbine into a mass of fuel consumed per time unit. Once we have the mass, we can convert it to volume, taking into account the fuel's density. In practical terms, knowing the heating value helps in estimating fuel costs, planning fuel logistics, and analyzing the overall economy of turbine operation. For engineers and operators, optimizing the use of fuel with high heating values is an ongoing challenge to improve the efficiency and sustainability of power generation.