Energy Balances

As with mass balances it is useful to start with our initial definition of a balance equation:

IN - OUT + GENERATION - CONSUMPTION = ACCUMULATION

Where IN and OUT correspond to energy flowing into and out of the system, respectively. In contrast to mass balances, however, energy is a conserved quantity (First Law of Thermodynamics) and therefore GENERATION=CONSUMPTION=0! (We will see as we go that this is also true for total mass, and even for atoms, but not for specific molecular species). This gives us:

IN - OUT = ACCUMULATION

In order to finish this balance we need to look at exactly how energy can move IN and OUT of a system...

Also, as we did with mass balances, we need to start out with some definitions.

Types of Energy

DEFINITION:

Kinetic energy is energy due to the translational (or rotational) motion relative to some frame of reference.

DEFINITION:

Potential energy is energy due to something's position in a potential field (electromagnetic or gravitational, for example).

DEFINITION:

Internal energy is where we lump everyone else! (Thermal energy, chemical energy, etc.)

Each bit of mass within our balances will contain some amount of each of these forms of energy, so that we can write the total energy of some bit of mass as:

Etotal = U + Ek + Ep

where U is the internal energy, Ek is the kinetic energy, and Ep is the potential energy.

How Energy Moves

Obviously, we can imagine that STUFF has energy (a hot potato, for example).

One way for the energy to move would be to move the entire potato! Therefore, like we had in mass balances we can have an IN-flow and OUT-flow in the system through moving mass. (Some potatoes may be hotter or colder, moving faster or slower, etc.)

In contrast to mass balances, however, we can also change the energy that is in the potatoes already in the system. Imagine putting a bucket of potatoes in the oven (we use heat to change the internal energy of the potatoes already in the bucket). Or perhaps we vigorously shake the bucket of potatoes; here, we convey mechanical (kinetic) energy right through the walls!

Energy Balance Equations

As with mass balances, energy balances will have an integral and differential form.

Recalling that our general energy balance equation looks like this:

IN - OUT = ACCUMULATION

We can start with our differential form by writing:

$\frac{dE_{total}}{dt} = \dot E_{{total}_{in}} - \dot E_{{total}_{out}}$

Recalling that total energy is given by:

Etotal = U + Ek + Ep

we now simply need to account for the possibility of sticking stuff in an oven, or "shaking" it. We will denote the net flow of heat into the system as Q, and the net transfer of mechanical energy as W. The only catch is that we will consider a negative W to be going into the system, while a negative Q will be coming out of the system. This gives us:

General Differential Energy Balance

$\frac{dU}{dt} + \frac{dE_k}{dt} + \frac{dE_p}{dt} = \dot U_{in} - \dot U_{out} + \dot E_{k_{in}} - \dot E_{k_{out}} + \dot E_{p_{in}} - \dot E_{p_{out}} + Q - W$

For a steady state problem this reduces to:

$\dot U_{out} - \dot U_{in} + \dot E_{k_{out}} - \dot E_{k_{in}} + \dot E_{p_{out}} - \dot E_{p_{in}} = Q - W$

As we did with mass, we can integrate this to yield this integral form of the balance:

General Integral Energy Balance

$\dot U_{final} - \dot U_{initial} + \dot E_{{k}_{final}} - \dot E_{{k}_{initial}}+ \dot E_{{p}_{final}} - \dot E_{{p}_{initial}} =$
$\int_{t_{initial}}^{t_{final}} \dot U_{in} dt - \int_{t_{initial}}^{t_{final}} \dot U_{out} dt + \int_{t_{initial}}^{t_{final}} \dot E_{k_{in}} dt - \int_{t_{initial}}^{t_{final}} \dot E_{k_{out}} dt + \int_{t_{initial}}^{t_{final}} \dot E_{p_{in}} dt - \int_{t_{initial}}^{t_{final}} \dot E_{p_{out}} dt + \int_{t_{initial}}^{t_{final}} Q dt - \int_{t_{initial}}^{t_{final}} W dt$

which for a closed system simplifies to:

$U_{final} - U_{initial} + E_{{k}_{final}} - E_{{k}_{initial}}+ E_{{p}_{final}} - E_{{p}_{initial}} = \int_{t_{initial}}^{t_{final}} Q dt - \int_{t_{initial}}^{t_{final}} W dt$

OUTCOME:

Explain the origin and physical meaning of each of the terms in the General Energy Balance Equation