the basic - aeweb.tamu.eduaeweb.tamu.edu/haisler/engr214/word_lecture_notes_b… · web viewthis...

2001, W. E. Haisler Chapter 6: Conservation of Energy 1

Conservation of Energy (chapter 6)

The total energy must be conserved for a volume element x y z during a time period t. Energy may take on several forms which are generally associated with mechanical and thermal terms such as internal energy (due to deformation; pressure,

temperature or volume change, etc.), kinetic energy (due to motion), external energy or work (due to external forces, tractions

or body forces), energy flow due to temperature gradients, internal energy generation (due to inelastic deformation,

chemical reactions, point thermal sources, etc).


Consider the various forms of energy flux associated with the continuum differential volume : Only the x component listed below (flow through dy dz):Internal energy of the mass flux: J/ m2 /s

Energy/mass, Energy Joule 0.74 ft-lb, watt=Joule/secKinetic energy of the mass flux: J/ m2 /sWork done by boundary tractions: J/ m2 /sHeat energy flux by conduction: J/ m2 /sWork done by the body forces: J/ m3 /sRadiation source or heat sink in volume: J/ m3 /s

Note: Other heat energy flux terms which could be considered include convection and radiation.


Considering the x, y and z components, we have


In the limit as , we obtain Conservation of Energy

Using the product rule on the first and last terms gives


The underlined terms [multiplied by ( )] sum to zero by Conservation of Mass.


Thus we have the Conservation of Total Energy (Mechanical + Thermal Energy)


We can rewrite the energy, convection and traction terms in vector form:

Each of the above vector products is a scalar quantity (energy)!


The final result in vector form for Conservation of Total Energy (Mechanical + Thermal Energy) is


It is useful to separate the total energy into mechanical energy (that due to tractions and body forces) and thermal energy. The mechanical energy can be thought of as being associated with the linear momentum. Consider the conservation of linear momentum equation

Take the dot product of both sides with the velocity vector


Note: =kinetic energy per unit mass =1/2 mass /mass = 1/2 .Thus, terms on left side can be written in terms of as follows

Note:

Then:

The mechanical energy portion of Conservation of Energy becomes


The Thermal Energy portion of Conservation of Energy may be obtained by subtracting the mechanical energy portion from the total energy: We obtain

where P is the average compressive pressure (hydrostatic stress),

is the deviatoric or extra stress tensor ([] = [S] - P [I] where [I] = identity matrix), as defined before, and


tr (trace) is a scalar quantity and is defined as the sum of the diagonal terms of a square matrix.

Thus is the sum of the diagonal terms of the 3x3 matrix formed by the vector operation ( ), where and are each 3x3 matrices.


Conservation of Thermal (internal) Energy

We previously obtained the thermal energy equation:


Problem: Internal Energy, ,is not a directly observable quantity. If we wish to solve real problems, we must relate

to observable (measurable) quantities like

heat capacity, (at either constant volume or pressure)

pressure, P volume, temperature, T

This requires the use of the thermodynamics equations which relate and other thermodynamic properties to the measurable variables of P, V and T.


The thermodynamic energy functions of current interest are: = internal energy = entropy=enthalpy

From your previous work in thermodynamics, we can write thermodynamic functions in terms of the PVT measurable variables:


Consider the internal energy written in terms of two of the three measurable variables (say, T and V):

then the total differential of the internal energy is given by

The term implies change in with respect to

T at constant volume, .


We assume that that is related to T (during a constant

volume process) by a constant such that

and, from previous thermodynamics,

Thus, the total change in internal potential energy becomes


Note that since , the total change, ,is

The four partial derivatives above are given by


The sum of the last three equations is the gradient of , i.e.,

Substituting the last two equations (in boxes) into the thermal energy equation (in terms of deviatoric or extra stress) yields the following Conservation of Thermal Energy Equation (for a constant volume process):


For incompressible continua function of time, (from conservation of

mass), , and the thermal energy equation is

For the static case (no mass velocity, )

Heat Conduction in Solid

Later, we will show that (Fourier's Law where k

depends on the material). In 1-D, , where k =

thermal conductivity (for an isotropic material).


Second Law of Thermodynamics

The second law of thermodynamics states that entropy is related to the amount of reversible heat transfer and that entropy is not conserved.

T = absolute temperature (R or K) = entropy/unit mass

= reversible heat transfer/unit mass


The second law states that entropy is not conserved, but only generated. Considering heat and entropy flow through a control volume, we obtain

where is the entropy generation rate. If we use the energy conservation equation from chapter 7, we can show that the 2nd Law becomes


The entropy generation terms from the right side of the previous equation state:

Thus we conclude that entropy generation may occur as a result of the heat flux by conduction or the conversion of mechanical energy of tractions into heat energy.

Due to heat flux by conduction

Conversion of mechanical energy to heat energy

the basic - aeweb.tamu.eduaeweb.tamu.edu/haisler/engr214/word_lecture_notes_b… · web viewthis...

Documents