Introduction to Chemical Engineering Processes/Steady state energy balance

General Balance Equation Revisited

Recall the general balance equation that was derived for any system property:

$In-Out+Generation-Consumption=Accumulation$

When we derived the mass balance, we did so by citing the law of conservation of mass, which states that the total generation of mass is 0, and therefore $Accumulation=In-Out$.

There is one other major conservation law which provides an additional equation we can use: the law of conservation of energy. This states that if E denotes the entire amount of energy in the system,

Law of Conservation of Energy

$E_{in}-E_{out}=E_{accumulated}$

Types of Energy

In order to write an energy balance, we need to know what kinds of energy can enter or leave a system. Here are some examples (this is not an exhaustive list by any means) of the types of energy that can be gained or lost.

1. A system could gain or lose kinetic energy, if we're analyzing a moving system.
2. Again, if the system is moving, there could be potential energy changes.
3. Heat could enter the system via conduction, convection, or radiation.
4. Work (either expansion work or shaft work) could be done on, or by, the system.

The total amount of energy entering the system is the sum of all of the different types entering the system. Here are the expressions for the different types of energy:

1. From physics, recall that $KE={\frac {1}{2}}mv^{2}$. If the system itself is not moving, this is zero.
2. The gravitational potential energy of a system is $GPE=mgh$ where g is the gravitational constant, m is mass in kg and h is the height of the center of mass of the system. If the system does not change height, there is no change in GPE.
3. The heat entering the system is denoted by Q, regardless of the mechanism by which it enters (the means of calculating this will be discussed in a course on transport phenomenon). According to this book's conventions, heat entering a system is positive and heat leaving a system is negative, because the system in effect gains energy when heat enters.
4. The work done by or on the system is denoted by W. Work done BY a system is negative because the system has to "give up" energy to do work on its surroundings. For example, if a system expands, it loses energy to account for that expansion. Conversely, work done ON a system is positve.

Energy Flows due to Mass Flows

Accumulation of anything is 0 at steady state, and energy is no exception. If, as we have the entire time, we assume that the system is at steady state, we obtain the energy balance equation:

$E_{in}=E_{out}$

This is the starting point for all of the energy balances below.

Consider a system in which a mass, such as water, enters a system, such as a cup, like so:

The mass flow into (or out of) the system carries a certain amount of energy, associated with how fast it is moving (kinetic energy), how high off the ground it is (potential energy), and its temperature (internal energy). It is possible for it to have other types of energy as well, but for now let's assume that these are the only three types of energy that are important. If this is true, then we can say that the total energy carried in the flow itself is:

${\dot {E}}_{i}=({\frac {1}{2}}{\dot {m}}v^{2}+{\dot {m}}gh+{\dot {U}})_{i}$

However, there is one additional factor that must be taken into account. When a mass stream flows into a system it expands or contracts and therefore performs work on the system. An expression for work due to this expansion is:

$W_{exp}=P*{\dot {V}}_{i}$

Since this work is done on the system, it enters the energy balance as a positive quantity. Therefore the total energy flow into the system due to mass flow is as follows:

${\dot {E}}_{i}=({\frac {1}{2}}{\dot {m}}v^{2}+{\dot {m}}gh+{\dot {U}})_{i}+P*{\dot {V}}_{i}$

Now, to simplify the math a little bit, we generally don't use internal energy and the PV term. Instead, we combine these terms and call the result the enthalpy of the stream. Enthalpy is just the combination of internal energy and expansion work due to the stream's flow, and is denoted by the letter H:

Definition of enthalpy

$H=U+PV$

Therefore, we obtain the following important equation for energy flow carried by mass:

In stream i, if only KE, GPE, internal energy, and expansion work are considered,

the energy carried by mass flow is:
${\dot {E}}_{i}=({\frac {1}{2}}{\dot {m}}v^{2}+{\dot {m}}gh+{\dot {H}})_{i}$

Other energy flows into and out of the system

The other types of energy flows that could occur in and out of a system are heat and work. Heat is defined as energy flow due to a change in temperature, and always flows from higher temperature to lower temperature. Work is defined as an energy transferred by a force (see here for details).

• If there is no heat flow into or out of a system, it is referred to as adiabatic.
• If there are no mechanical parts connected to a system, and the system is not able to expand, then the work is essentially 0.

Some systems which have mechanical parts that perform work are turbines, mixers, engines, stirred tank reactors, agitators, and many others. The type of work performed by these parts is called shaft work to distinguish it from work due to expansion of the system itself (which is called expansion work).

An "insulated system" is generally interpreted as being essentially adiabatic, though how good this assumption is depends on the quality of the insulation. A system that cannot expand is sometimes described as "rigid".

The notation for these values are as follows:

• Heat flows: ${\dot {Q}}_{j}$, at the "j"th location.
• Shaft work: ${\dot {W}}_{s}$
• Expansion work: $P*{\frac {\Delta {V}}{\Delta {t}}}$

Note that the above implies that there is no expansion work at steady state because at steady state nothing about the system, including the volume, changes with time, i.e. ${\frac {\Delta {V}}{\Delta {t}}}=0{\mbox{ at steady state}}$.

Overall steady - state energy balance

If we combine all of these components together, remembering that heat flow into a system and work done on a system are positive, we obtain the following:

Steady State Energy Balance on an Open System

$\Sigma ({\frac {1}{2}}{\dot {m}}v^{2}+{\dot {m}}gh+{\dot {H}})_{i,in}-\Sigma ({\frac {1}{2}}{\dot {m}}v^{2}+{\dot {m}}gh+{\dot {H}})_{i,out}+\Sigma {\dot {Q}}_{j}+{\dot {W}}_{s}=0$

Some important points:

1. If the system is closed AND at steady state that means the total heat flow must equal the total work done in magnitude, and be opposite in sign. However, according to another law of thermodynamics, the second law, it is impossible to change ALL of the heat flow into work, even in the most ideal case.
2. In an adiabatic system with no work done, the total amount of energy carried by mass flows is equal between those flowing in and those flowing out. However, that DOES NOT imply that the temperature remains the same, as we will see in a later section. Some substances have a greater capacity to hold heat than others, hence the term heat capacity.
3. If the conditions inside the system change over time, then we CANNOT use this form of the energy balance. The next section has information on what to do in the case that the energetics of the system change.