## Energy Conservation

**Introduction: **The First Law of Thermodynamics: the energy conservation which is the first law of thermodynamics1. The fluid, as all phases and materials, obeys this law which creates strange and wonderful phenomena such as a shock and choked flow. Moreover, this law allows solving problems. For example, the relationship between height and flow rate was assumed previously, here it will be derived. Additionally a discussion on various energy approximations is presented.

**The energy rate equation for a system is**

**This equation can be rearranged to be:**

Equation is similar to equation in which the right hand side has to be interpreted and the left hand side interpolated using the Reynolds’s Transport Theorem (RTT)^{. }The right hand side is very complicated and only some of the effects will be discussed (It is only an introductory material).

The energy transfer is carried (mostly3) by heat transfer to the system or the control volume. There are three modes of heat transfer, conduction, convection4 and radiation. In most problems, the radiation is minimal. Hence, the discussion here will be restricted to convection and conduction. Issues related to radiation are very complicated and considered advance material and hence will be left out. The issues of convection are mostly covered by the terms on the left hand side. The main heat transfer mode on the left hand side is conduction. Conduction for most simple cases is governed by Fourier’s Law which is

Where dq˙ is heat transfer to an infinitesimal small area per time and kT is the heat conduction coefficient. The heat derivative is normalized into area direction. The total heat transfer to the control volume is

The work done on the system is more complicated to express than the heat transfer. There are two kinds of works that the system does on the surroundings. The first kind work is by the friction or the shear stress and the second by normal force. As in the previous chapter, the surface forces are divided into two categories: one perpendicular to the surface and one with the surface direction.

**The work done by system on the surroundings:**

**The change of the work for an infinitesimal time (excluding the shaft work) is:**

**The total work for the system including the shaft work is:**