Conservation Of Energy

• Fluid Mechanics Fundamentals and Applications, Fourth Edition, Yunus Cengel, John Cimbala

pp-219

One of the most fundamental laws in nature is the first law of thermodynamics, also known as the conservation of energy principle, which provides a sound basis for studying the relationships among the various forms of energy and energy interactions. It states that energy can be neither created nor destroyed during a process; it can only change forms. Therefore, every bit of energy must be accounted for during a process.

conservation of energy principle,

The energy content of a fixed quantity of mass (a closed system) can be changed by two mechanisms: heat transfer Q and work transfer W. Then the conservation of energy for a fixed quantity of mass can be expressed in rate form as

\[\cfrac{d({m}E_{sys})}{dt}=\cfrac{d}{dt}\int_{CV}(\rho E)dV+\int_{CS}(\rho E)(\vec{V}-\vec{V}_{CS})dA=\dot{Q}_{\text{net in}}+\dot{W}_{\text{net in}}\]

\[\begin{split}\begin{align} \frac{d ({m} E)_{\mathrm{sys}}}{d t} & = \frac{d}{d t}\int_{\mathrm{CV}}(\rho E)dV +\int_{\mathrm{CS}} (\rho E) ((\vec{V}-\vec{V}_{\text{CS}}) \cdot \vec{n}) d A \\ & = \int_{\mathrm{CV}} (\text{div }(\boldsymbol{\sigma}\cdot\mathbf{v})-\text{div }\mathbf{ q}+\mathbf{v}\cdot\rho \mathbf{b}+\rho s) d V \end{align}\end{split}\]