Energy Balance Equations

Now that we know the different types of energy and the different ways in which energy can move, it is time to look (formally) at how to do the balances.

Closed Systems

Recall that a closed system is one where no mass moves across the boundaries. (So we can neglect the mode of energy transport by moving the STUFF)

Becasue of this is, it is (obviously) silly to talk about a continuous balance (Since a process that is closed is by definition a batch process.)

Like the material balance, the governing equation for an integral balance (as opposed to a continuous balance) is given by:

Final - Initial = Accumulation (change)

Why did the generation and consumption terms disappear? (HINT: They will reappear in heat transfer!)

If we represent the amount of energy as the amount that is in each of the energy's forms (kinetic, potential, internal), we might right the IN and OUT terms like this:

INITIAL (FINAL) = U_{INITIAL (FINAL)} + E_{k,INITIAL (FINAL)} + E_{p,INITIAL (FINAL)}

where U is the internal energy, E_k is the kinetic energy, and E_p is the potential energy.

As I am SURE you remeber, for a closed (batch) process, if the thing ISN'T run at unsteady-state it makes for a really boring problem (NOTHING CHANGES!).

SO, what does the accumulation term look like?!

We already talked about how energy can be transfered to or from a closed system by heat and work! The only trick now is to give them a symbol and to decide whether we think of it as positive going into the system or positive coming out of the system.

NOTE:

Sign conventions differ from book to book and person to person. In practice you should always write down your convention and be consistent with that convention. In this book we will use the convention outlined below.

Heat rarely is difficult to decide a sign convention for: heat going into the system is positive, heat leaving the system is therfore negative. We use just a Q then in the accumulation term.

Work is more difficult. The reason is that often you are interested in getting work OUT of a system. For this reason, the work coming out of the system is often considered positive and work input to the system is thought of as negative. For our accumulation term, we then use a -W.

Our balance for closed systems then is given by:

U_FINAL-U_INITIAL + E_k,FINAL-E_k,INITIAL + E_p,FINAL-E_p,INITIAL = Q - W

Closed System Balance Rules

You should know something about these equations:

• U depends on chemical composition, phase, and temperature (only a little on pressure). If no reactions, phase changes or temperature changes occur, the change in U is zero! (except a little if the pressure changes a lot)
• Kinetic energy needs an acceleration (positive or negative) to change.
• Potential energy needs a change in position. (We will not really do any of these.)
• If the system and surroundings are at the same T (or if we say that the system is (perfectly) insulated) then Q = 0 (the system is adiabatic).
• Work can only be done by moving parts (a compressive/expansion piston of rotating shaft) or by electrical or radiation energy.

Open Systems

In an open system, we now have material moving across the boundaries. As we discussed previously, this means that energy can now move with the STUFF! (So we can have contin uous and/or steday state processes.

How does energy move with the stuff?

• If the incoming stuff is hotter/colder than the outgoing stuff
• If the chemical species coming in is higher on the "food chain" than that coming out (obviously if something came in as methane and oxygen and left as water and carbon dioxide there is differing energy there)

Both of these are forms of internal energy!

We can also have differences in kinetic and potential (if the stuff coming in is moving faster/slower than the stuff coming out or if the position is higher/lower), but this will be less important here.

So, just like in the closed case, we will have changes in U, E_k, and E_p. However, there is one other thing about flowing systems ... It takes energy (work) to make them flow!!!!

The work term, therefore has a "shaft" component and a "flow" component: W = W_s + W_f = W_s + (PV)

OBJECTIVES:

Explain the difference between shaft work and flow work

The simplest way to account for this is to use: H = U + PV, instead of U to cover both internal energy and flow work at the same time. This leaves only shaft work (pistons or totalling things) and electrical, etc. work. We will call all of these shaft work, W_s

So, at steady state (where we will do most of our continuous stuff), the open system energy balance is given by: IN = OUT, or

(H_OUT-H_IN) + (E_kOUT-E_kIN) + (E_pOUT-E_pIN) = Q - W_s

NOTE

For an open system, all of the above should be flows. Moreover, it is possible for the system to be at steady state and still have non-zero Q and W_s