Total energy, equation

A useful simphfication of the total energy equation applies to a particular set of assumptions. These are a control volume with fixed solid boundaries, except for those producing shaft work, steady state conditions, and mass flow at a rate m through a single planar entrance and a single planar exit (Fig. 6-4), to whi(m the velocity vectors are perpendicular. As with Eq. (6-11), it is assumed that the stress vector tu is normal to the entrance and exit surfaces and may be approximated by the pressure p. The equivalent pressure, p + pgz, is assumed to be uniform across the entrance and exit. The average velocity at the entrance and exit surfaces is denoted by V. Subscripts 1 and 2 denote the entrance and exit, respectively. 

By subtracting the mechanical-energy contributions from the total energy equation, a thermal energy equation can be derived. It is this equation that proves to be most useful in the solution of chemically reacting flow problems. By a vector-tensor identity for symmetric tensors, the work-rate term in the previous sections can be expanded as... 

It is apparent that all the terms in Eq. 3.190 also appear directly on the left-hand side of the total energy equation Eq. 3.187. Therefore, subtracting Eq. 3.190 from Eq. 3.187 serves to remove the mechanical-energy components from the total-energy equation. 

As illustrated in Fig. 4.18, a uniform pressure gradient causes flow between a central rod and an outer guide. Assume the purely axial flow of an incompressible fluid, v = w = 0. The axial velocity varies as a function of r alone, u r). Assume that dp/dz is a constant, and that dp/dr = 1 /r(dp/dO) = 0. For this one-dimensional parallel-flow situation, develop an expression for the work term in the total energy equation that is, for this special case, expand... 

Again, based on the differential cylindrical control volume, derive the total energy equation. 

Assume that the water contains a radioactive substance that causes internal volumetric heat generation q (W/m3). With the objective of deriving a total energy equation for the annular flow, state the first law. Be careful with the signs and the definitions of positive heat transfer and work. 

Assemble a differential-equation form of the steady-state total-energy equation. 

Adopting this simplified formulation (i.e., (1.112)), the total energy equation, (1.96), may be transformed to the equation of internal- and kinetic [or mechanical] energy... 

To proceed in our model derivation, the equation for internal energy, e, is obtained by subtracting the mechanical energy equation from the total energy equation. The first step in this mathematical exercise, is to formulate the equation for mechanical energy. 

Now we are in the position of being able to subtract the mechanical energy equation (1.124) from the total energy equation (1.113). The result is the equation of internal [or thermal] energy. 

Another equation that resembles the Bernoulli equation very much is derived from the total energy balance [168]. For one component fluids the total energy equation (1.96) can be written ... 

This is the total energy equation, for which the potential energy term is expressed in terms of the external force F. By use of the momentum equation we can derive a transport equation for the mean kinetic energy, and thereafter extract the mean kinetic energy part from the equation (i.e, the same procedure was used manipulating the continuum model counterpart in chap. 1, sect. 1.2.4). The result is ... 

In addition to the total energy equation given above, we also have the condition of conservation of angular momentum,... 

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