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Energy balance equation for the closed system

Energy balance equation for the closed system
Let’s formulate the first law for the closed system. For a closed system, there is no material exchange between our system and surroundings. Only energy exchange happens. Energy transfer between system and surrounding is divided into two categories, Q (the heat) and W (the work). heat and work are the only forms of energy in transit. They cannot be stored in the system. Other forms of energy such as thermal energy can be stored in the system, but the energy in transit can have only two forms. Heat or work. Then, what is heat and work in engineering terminology? Heat is the energy transfer due to temperature difference. And work is all the other forms of energy transferring.
The first law of thermodynamics for the closed system in differential form is dU=dQ+dW, if the potential and kinetic energy are ignorable. Here we need a sign convention to determine in whisch case heat and work assume positive values or negative values. By convention, heat and work is positive when they are coming into the system. So Q is positive when heat is coming from the surrounding into the system. W is positive when the surrounding is doing work on the system. Let’s think about the property of energies. Internal energy U is a state function or point function and they are path independent.
It means If two different processes have the same initial and final states, internal energy changes are the same for the two different process histories. On the other hand, heat and work are path dependent, so their values can be different for different process histories although these processes have the same initial and final conditions. Mathematically, state function is the exact function, so internal energy change is independent of the order of differentiation thus path indes P bar dV. As we intuitively know, there are various types of work. Here let’s focus on the most representative work, the mechanical work. In traditional thermodynamics, it is sometimes called P-V work since many mechanical work is done by the expansion of gases with pressure P.
Let’s calculate the work done on the gas inside this cylinder. Define our system as the gas inside. The piston moves by dx by the expansion of gas. The mechanical work done by this expansion is the is force F times displacement dx. Here the force is pressure times area, and the pressure here is the resisting pressure P bar, not the system pressure. P bar is the pressure resisting against the gas pressure P. The gas expansion is doing work against the resisting pressure P bar, thus the work is expressed with resisting pressure, not the system pressure. The work is thus P bar Adx and Adx is the volume change dV. This is the work done by the system.
Complying to the sign convention, the work done “on” the system is thus minus P bar dV. We said the work is a reversible work when the pressure resisting an expansion differs only infinitesimally from the pressure of gas in the system. In that case we set the resisting pressure P bar the same with the gas pressure or system pressure P. And the work is expressed with the system pressure, the gas pressure P. Delta W is -PdV in this case.

As a mathematical interpretation on the first law, we can formulate the conservation of energy concept into equations, called energy balance equation.

The first and the most simple formulation is for the closed system, where there is no material exchange with surroundings. In energy balance equation for the closed system, the energy change of the system is described as the energy change of internal energy, potential energy and the kinetic energy. For energies in transit (the energies transferring between system and surrounding), only two types of energies are involved: the heat and work.

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Thermodynamics in Energy Engineering

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