Energy and Mass Balance 101

The simple truth for anyone interested in extraction is this: scientific rabbit holes are unavoidable on the journey to perfecting any extraction process. A casual exploration of thermodynamics evolves into an endless pursuit of understanding; a continual clash between confusion and clarity, frustration and reward. At any rate, it becomes tempting to forfeit the pursuit of scientific understanding to the experts—engineers, chemists, and the like. But what is it exactly that these experts are doing? How can someone who is detached from the hours of countless, painstaking effort, make decisions that will positively benefit your extraction? The answer is, without a doubt: Energy and Mass Balance.

Energy and Mass Balance (EMB) is a concise, step-wise method by which a graphical representation of a process (and its equipment) can be transformed into a singular numeric outcome. For example, say you’ve got a known amount of a fluid flowing into a heat exchanger, how much energy is required for this heat exchanger to heat the fluid to the desired temperature? How much electricity will that require? How much does that cost? All of this begins with an EMB. Given that an EMB is a step-wise process, it makes sense to explain it a step-by-step manner.

Step 1) Define A Thermodynamic System

A quick Wikipedia query provides the following definition for thermodynamic system: “the material and radiative content of a macroscopic volume in space, that can be adequately described by thermodynamic state variables such as temperature, entropy, internal energy and pressure.” In the context of basic understanding, this definition is already unsettling; radiative content, macroscopic volume, entropy—scientific rabbit holes aplenty. A thermodynamic system (or system for short) can be more simply characterized as a box (macroscopic volume) where the effects of temperature, pressure, etc. (thermodynamic state variables) can be interpreted in terms of how they affect the contents of the box. In extraction (or any chemical process for that matter) the thermodynamic system/box is equivalent to an individual piece process equipment: a heat exchanger, reactor, evaporator, etc. Moreover, the thermodynamic state variables are equivalent to the flow rates, temperatures and pressures of the various liquids interacting with the equipment at a given time.

Now, with the understanding that an individual piece of equipment is essentially a man-made thermodynamic system, one has a system to balance.

Step 2) Define A System Boundary

In order to balance a system, one must define the inputs and outputs of the system as well as the ‘thermodynamic state variables’ (see Step 1) of the inflowing and outflowing materials. For this discussion, let’s focus on the example of the heat exchanger where we can determine the following:

m – Flow Rate
T – Temperature
P – Pressure

Note: The subscripts “in” and “out” represent components flowing into or out of the system, respectively.

The final step in creating a system boundary is graphically demarcating the system from its surroundings; in other words, draw a box around the piece of equipment. Now the hard part.

Step 3) Understanding the Mathematics of an EMB

In academia, the mathematics of an EMB are generally presented (to the utter anguish of unsuspecting engineering students) as follows:

Note: This equation, in and of itself, evokes strong feelings of self-doubt, despair and anxiety in the author; a chemical engineer.

Now, this is about as terrible an equation as any; a total of ten variables, differentiable terms, double and triple integrals, subscripts and the rest, but the intent the author in providing this equation is to demonstrate the depth and robustness one can achieve with an energy balance after a great deal of practice. Clearly, this is beyond the scope of this article, but the following equation (which is directly related to the above equation) is well within the scope of the discussion:

Author Suggestion: Memorize the heck out of this equation.

Where:

This equation is essentially the mathematical equivalent of the First Law of Thermodynamics for a system. The First Law of Thermodynamics can be articulated as follows: the total energy of a thermodynamic system is constant; energy can be transformed from one form to another, but cannot be created nor destroyed.

As luck would have it, the symbol ∆ designates a change, and in the context of the entire preceding conversation, that change occurs between the inflow and the outflow across the system boundary. So, how does one determine the value of each of the variables listed above?

Step 4) Calculating the Variables

In this step, it is often prudent to begin with the simplest variable and gradually move to the most difficult variable…

W – Work

Using the example of the heat exchanger, one can reasonably assume that the system is doing no work. For now, that will suffice—and the work done by the system is equal to zero!

∆E_P– Potential Energy

The change in Potential Energy, in the context of this discussion, is dictated by the change in height (h) of the inflowing and outflowing fluid:

In the example of the heat exchanger, let’s make the assumption that the flow rates in and out are equal and the heights at which the fluid flows in and out are equal. g, the acceleration of gravity can be assumed to stay constant—which means that the change in potential energy is zero!

∆E_K– Kinetic Energy

The change in Kinetic Energy, in the context of this discussion, is dictated by the change in velocity (v) of the inflowing and outflowing fluid:

Again, let’s make the assumption that the flow rates in and out are equal and that the velocities are equal—the change in kinetic energy is zero!

Now, taking a brief pause to recollect, the original equation has reduced to the following:

At this point, it becomes necessary to define the intent of creating an EMB in the first place, perhaps the most realistic possibility (in the example of the heat exchanger) is how much will it cost to heat the fluid from one temperature to another? The answer can be derived by directly determining the value of Q (in units like kilowatts) and multiplying by the cost of electricity (in units of $ per kilowatt).

We’ll save that for next time.