The General Balance Equation

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DOFPro group

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Intro to The General Balance Equation

The General Balance Equation is the foundation for all material, energy, entropy, and exergy balances. The mass and mole balance equations are simply restatements of the general balance equation applied to mass and moles, respectively. Mass is conserved; moles are not conserved when chemical reactions occur.

When applied to energy, the general balance equation becomes the First Law of Thermodynamics. When applied to entropy, it becomes the Second Law of Thermodynamics. Like mass, energy is conserved; entropy is not.

The General Balance Equation is applied to systems, and the specific form of the equation depends on whether the system is open or closed, and whether it is steady-state or transient. The videos introduce systems and the definitions needed to apply balances correctly.

How to Keep Your Balance

How to Keep Your Balance begins with bathtub analogies to explain system definitions: open or closed, and steady-state or transient. It then introduces the General Balance Equation and defines each of its terms.

Visuals

Speed Balancing

Speed Balancing continues with definitions for system terms such as PFD, recycle, bypass, purge, tie component, and overall balances. It then provides hints and suggestions for solving balance equations quickly and correctly, including a worked bathtub example.

Visuals

Examples and Definitions

“Engineering is solving story problems where you often have to make up the story.”

“Bookkeeping is just arithmetic and definitions, but it needs to be done correctly.”

“There are many ways to solve a problem correctly. You want to develop a systematic approach and not treat each problem as an isolated issue.”

First – Define Your System

  1. Transient — Process variables change with time.

  2. Steady-State — Process variables do not change with time, although they may vary with position.
    (What electrical engineers would call a DC steady state.)

  3. Batch (closed) — Add material, allow the process to occur, then remove the products.

  4. Semibatch — Neither purely batch nor purely continuous.

  5. Continuous (open) — Material continuously enters and leaves the system.

Control Volume

For an open system, whether transient or steady state, the system boundary sometimes encloses a fixed region in space. This fixed boundary is called a control volume (CV).

Material and energy may cross the control surface, but the boundary itself does not move.

The General Balance Equation

\[ \mathrm{Input + Generation = Output + Consumption + Accumulation} \]

  • Any quantity that can enter or leave a system can be described with the general balance equation.
  • For conserved quantities (mass, energy, atoms), the generation and consumption terms are zero.
  • For steady-state systems, the accumulation term is zero.

Differential and Integral Balances

Differential Balances

  • All terms are rates (e.g., mol/s, kg/hr, gal/min).
  • Usually used for continuous processes.

Integral Balances

  • All terms are amounts (e.g., mol, kg, gal).
  • Usually used for batch processes.

Intensive and Extensive Properties

An intensive property does not depend on the amount of material present.
Examples include temperature, pressure, density, and composition.

An extensive property does depend on the amount of material present.
Examples include mass, volume, and total energy.

It is often useful to convert an extensive property into an intensive one by dividing by mass or moles. These are called specific or molar properties.

For example,

\[ \hat{V} = \frac{V}{n} \]

is the molar volume.

Definitions

PFD (Process Flow Diagram)
A schematic drawing of the mixers, dividers, separators, reactors, and other unit operations used in a process.
Recycle
A portion of a product stream returned to the feed of a process.
Bypass
A portion of the feed routed around part of the process and later recombined with the product.
Purge
A stream removed from a recycle loop to prevent buildup of trace or inert components.
Tie Component
A component of the feed that is unchanged by the process. It is often useful for simplifying calculations but is never strictly necessary.
Overall Balance
Treating an entire process or flowsheet as a single “black box” with only overall inputs and outputs.

One Useful Technique

  1. Draw a flowchart and fill in all given variable values.
  2. Choose a basis of calculation (usually a flow rate or amount of one stream).
  3. Label unknown stream variables clearly.
  4. Convert volumetric quantities to mass or molar quantities.
  5. Convert mixed mass and mole units to either all mass or all mole units.
  6. Write the material balance equations.
  7. Write any additional constraints as equations.
  8. Perform a degree-of-freedom analysis.
    Count unknowns and independent equations.
  9. Solve the equations.
  10. Convert answers to the requested units.

Speed Tips

  1. Don’t solve for everything.
  2. Write equations in order of fewest unknowns first.
  3. Try overall balances first.
  4. Look for tie components.
  5. Remember that mass fractions and mole fractions sum to 1.
  6. Recycle loops usually require a complete set of equations around the loop.