18.7
Conservation of Energy in Control Volume
The first law of thermodynamics states that the rate of change of total stored energy in a system equals the net rate of heat and work transferred into a system.
For a control volume that coincides with the system at a given moment, the net heat and work entering the system equal the net heat and work entering the control volume.
For this system and coincident control volume, the Reynolds Transport Theorem is applied.
The total rate of energy change within the control volume is determined by both the time rate of change in stored energy and the net energy flow through the boundary.
The energy flowing across the boundary includes contributions from heat and work, such as shaft work and pressure work.
Shaft work results from rotating components like turbines, while fluid stresses produce pressure work.
The general energy equation combines all elements: stored energy, heat, shaft work, and pressure work due to normal stresses.
This equation represents energy conservation in fluid systems, incorporating heat transfer, shaft work, and pressure forces at the control surface.
Consider a turbine operating under steady-flow conditions. The control volume is drawn around the turbine, with fluid entering at one point and exiting at another. The turbine extracts energy from the fluid, which performs mechanical work (shaft work).
For steady flow systems, the time derivative of the stored energy becomes zero since there is no energy accumulation within the control volume. This simplifies the energy equation to:
Since we focus on work done by the turbine shaft and assume negligible heat transfer, Q̇ ≈ 0. Therefore, the energy equation simplifies further to:
The flow is divided into two components: the fluid entering the turbine (inlet) and exiting the turbine (outlet). The mass flow rate at both points is constant, and the energy flux at the inlet and outlet is given by:
Where:
-ṁ is the mass flow rate,
- ũ1, V1, and z1 refer to the internal energy, velocity, and height at the inlet,
- ũ2, V2, and z2 refer to the internal energy, velocity, and height at the outlet,
In a typical turbine, the fluid's internal and kinetic energy decreases as it passes through the turbine, which results in useful shaft work being done. Suppose a turbine operates with an inlet velocity of V1=50 m/s, an outlet velocity of V2= 20 m/s, an inlet height z1=100 m, and an outlet height z2=90m. Assume the internal energy change is small compared to the kinetic and potential energy changes, and the mass flow rate is 10 kg/s.
As a result, the turbine produces approximately 11.48 kW of mechanical power by extracting energy from the flowing fluid.
The first law of thermodynamics states that the rate of change of total stored energy in a system equals the net rate of heat and work transferred into a system.
For a control volume that coincides with the system at a given moment, the net heat and work entering the system equal the net heat and work entering the control volume.
For this system and coincident control volume, the Reynolds Transport Theorem is applied.
The total rate of energy change within the control volume is determined by both the time rate of change in stored energy and the net energy flow through the boundary.
The energy flowing across the boundary includes contributions from heat and work, such as shaft work and pressure work.
Shaft work results from rotating components like turbines, while fluid stresses produce pressure work.
The general energy equation combines all elements: stored energy, heat, shaft work, and pressure work due to normal stresses.
This equation represents energy conservation in fluid systems, incorporating heat transfer, shaft work, and pressure forces at the control surface.
- Thermodynamics Control Volume - This refers to a specific, delimited space in which energy exchanges occur during physical transformations.
- Shaft Work - The work done or energy transmitted when a force is applied to rotate an object.
- Conservation of Energy - A fundamental concept in physics that states the total energy of an isolated system remains constant.
- First Law of Thermodynamics - An expression of the principle of conservation of energy, stating that energy can be transformed, but not created or destroyed.
- Shaft work Equation Thermodynamics - Equation describing the amount of shaft work done under given system conditions.
- Define Thermodynamics Control Volume – Explain what is a thermodynamics control volume (e.g., Thermodynamics Control Volume).
- Contrast Conservation of Energy vs First Law of Thermodynamics – Explain key differences (e.g., energy cannot be created nor destroyed vs energy transformation ways).
- Explore Shaft Work Example – Describe a scenario where shaft work is understandable (e.g., turbine extracting energy from a fluid).
- Explain Shaft Work Equation in Thermodynamics – This involves interpreting and using the equation to solve problems in thermodynamics.
- Apply First Law of Thermodynamics in Context – Use the law in practical scenarios considering the control volume and accompanying conditions.
- What is a thermodynamics control volume and how does it relate to the conservation of energy and the first law of thermodynamics?
- What is the key difference between the conservation of energy and the first law of thermodynamics?
- How does a turbine perform shaft work to extract energy from a fluid?
From Chapter 18:
Now Playing
18.7 : Conservation of Energy in Control Volume
Finite Control Volume Analysis
1.1K Views
18.1 : Conservation of Mass in Finite Cotrol Volume
Finite Control Volume Analysis
1.7K Views
18.2 : Conservation of Mass in Fixed, Nondeforming Control Volume
Finite Control Volume Analysis
1.5K Views
18.3 : Conservation of Mass in Moving, Nondeforming Control Volume
Finite Control Volume Analysis
1.2K Views
18.4 : Linear Momentum in Control Volume
Finite Control Volume Analysis
1.2K Views
18.5 : Application of the Linear Momentum Equation
Finite Control Volume Analysis
1.0K Views
18.6 : Moment-of-Momentum Equation
Finite Control Volume Analysis
610 Views
18.8 : Application of the Energy Equation
Finite Control Volume Analysis
2.5K Views
18.9 : Design Example: Forces in Sluice Gate
Finite Control Volume Analysis
4.4K Views