Conservation and Accounting Laws

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Conceptual Framework for Accounting
Process to use Accounting equations
Forms of Accounting Equations
Accounting Equation Terms
Example Conservation Problem

Conceptual Framework for Accounting

In the realm of engineering, conservation refers to the principle that certain physical quantities remain constant as they move or change form within a system. While many laws of physics are governed by these principles, not every property is conserved. For those that can be created or lost (like specific chemical species or certain types of energy), we use an accounting equation to track their status.

An accounting equation acts as a mathematical ledger. It tracks how an extensive property enters, leaves, is produced, or is used up within a specific boundary over a set timeframe.

All the extensive properties listed in Extensive Properties can be counted in accounting equations, but only a subset of these extensive properties is always conserved. Below is a complete list of extensive properties that are conserved in all situations (except nuclear reactions):

Total mass can be counted for individual species
Mass of individual element can be counted per element
Moles of individual element can be counted per element
Total energy can be counted according to thermal and mechanical energy
Net charge can be counted across a system
Linear momentum and Angular momentum can be counted across a system

Process to Use Accounting Equations

To be able to write an accounting statement, three items are necessary:

The extensive property to be counted must be specified.
The system and its surroundings must be defined by specifying a boundary.
A time period must be specified.

System Diagram: A schematic representation of a system illustrating its key features, including system boundaries and depictions of energy and mass flows.

Heads up!

Check out Foundations of Transport & Flow.

Forms of Accounting Equations

Mathematically, we illustrate both these concepts:

Accounting:

$$ ~\textrm{Input} - ~\textrm{Output} + ~\textrm{Generation} - ~\textrm{Consumption} = ~\textrm{Accumulation} $$

and Conservation:

$$ ~\textrm{Input} - ~\textrm{Output} = ~\textrm{Accumulation} $$

$$ ~\textrm{Accumulation} = ~\textrm{Final Condition} - ~\textrm{Initial Condition} $$

Accounting equations can be represented in three forms:

Algebraic
Differential
Integral

The following equations will be shown with $ \Psi $ representing any extensive property.

Algebraic Accounting Equations

Algebraic accounting equations are generally applied to extensive properties within a defined system and time period. Algebraic equations can be applied when discrete quantities or “chunks” of extensive property are involved. They cannot be applied when rates or time dependent terms are involved.

Algebraic Accounting Equation

$$ \Psi_{in} - \Psi_{out}+ \Psi_{gen} - \Psi_{cons} = \Psi^{sys}_{acc} $$

Differential Accounting Equations

Differential accounting equations are used when we have flow rates. A flow rate describes the transport of an extensive property over a period of time. We can represent the rate with a dot over the variable, $ \dot{\Psi} $.

Differential Accounting Equation

$$ \dot{\Psi}_{in}-\dot{\Psi}_{out}+\dot{\Psi}_{gen}-\dot{\Psi}_{cons}=\dot{\Psi}^{sys}_{acc}=\frac{d\Psi}{dt} $$

Integral Accounting Equations

Integral balances are most useful when trying to evaluate conditions between two discrete time points. Integral accounting equations can be written to incorporate rates of change of an extensive property. When developing an integral balance, you can write the differential balance equation and integrate it between the initial and final times.

Integral Accounting Equation

$$ \int_{t_f}^{t_0}{\dot{\Psi}_{in}dt}-\int_{t_f}^{t_0}{\dot{\Psi}_{out}dt}+ \int_{t_f}^{t_0}{\dot{\Psi}_{gen}dt}-\int_{t_f}^{t_0}{\dot{\Psi}_{cons}dt}= \int_{t_f}^{t_0}{\dot{\Psi}^{sys}_{acc}dt} $$

Use the table below to ask yourself critical questions about the problem to identify what type of accounting equation is needed.

	Algebraic	Differential	Integral
Can it incorporate discrete transfer of extensive property?	Yes	No	Sometimes
Does it involve a time interval?	Finite	Ongoing	Finite
Can it incorporate rates?	No	Yes	Yes
What is the dimension of the equation?	Extensive Property	Extensive Property/time	Extensive Property

Accounting Equation Practice Problem

Example Problem: Identifying When to Use Types of Accounting Equations #AccountingEq

For each of the following scenarios, identify which type of accounting equation (algebraic, differential, or integral) would be applied:

Scenario 1: A cell in the kidney works constantly to keep the ion balance in the blood correct. The cell membrane contains ion pumps and channels that move Na+ ions across the membrane. Looking specifically at one type of pump, Na+ ions are transported from the inside of the cell to the outside of the cell. Assume that the cell does not generate or consume Na+ ions. You are interested in writing a model on the positive charge contributed by Na+ ions as they move across the cell membrane through the pumps (Bioengineering Fundamentals).
Scenario 2: A patient receives a drug through an IV infusion. The drug enters the bloodstream at a rate that varies throughout treatment according to the infusion pump settings. At the same time, the drug is eliminated by the kidneys at a rate that changes as the drug concentration changes. You are interested in the amount of drug in the bloodstream over the period of infusion.
Scenario 3: Sprinters and long-distance runners rely on muscles to propel them as they run. Muscles require oxygen and glucose for contraction. Metabolism of these reactants generates carbon dioxide and lactate. Consider chemical compounds in the adductor longus muscle in the leg of a sprinter during a 100-m race (Bioengineering Fundamentals).
Scenario 4: During a hemodialysis treatment, a patient's blood passes through a dialyzer that contains dialysate solution to remove urea. Consider the dialyzer as the system. To prepare for the treatment, the nurse has to mix an acid concentrate, a bicarbonate concentrate, and purified water at ratios that align with the patient's clinical needs. You want to model the dialysate solution after it has been prepared.

Scenario 1: Differential. This model represents Na+ ions as they flow through the cell membrane. This is an ongoing process that will contain flow rates of the Na+ ions.

Scenario 2: Integral. This model focuses on a specific period: the period of infusion.

Scenario 3: Integral. This model focuses on a specific period: the duration of the 100-m race.

Scenario 4: Algebraic. This model represents a mixture of three separate batches (acid concentrate, bicarbonate concentrate, and water). Additionally, the model represents a finite point in time after the dialysate is prepared, rather than a process occurring over a time period.

Accounting Equation Terms

Input and Output

The input and output terms describe the movement of an extensive property across the system boundary. Systems can be classified as either open or closed based on whether mass is permitted to cross this boundary.

An open system allows mass to enter and leave through its system boundary. Consequently, the accounting equation must include the input and output terms. Open systems may contain multiple inlets and outlets, each contributing to the overall balance. Blood flowing through the heart represents an open system. Blood enters through the superior and inferior vena cava and pulmonary veins and exits through the pulmonary artery and the aorta, transporting mass across the system boundary.

A closed system does not allow mass to cross its system boundary. Therefore, the input and output terms are zero. However, energy may still be transferred across the boundary in the form of heat or work. A sealed container holding a cell culture can be modeled as a closed system if no material is added or removed during the specified period.

Generation and Consumption

The generation and consumption terms describe the production or elimination of an extensive property within a system. Generation occurs when a property is produced within the system boundary, while consumption occurs when a property is destroyed, converted, or eliminated.

Chemical reactions provide a common example of both processes. During a chemical reaction, reactants are consumed as they are used to generate products. These changes are represented by the generation and consumption terms in the accounting equation.

Attention!

A nonreacting system does not necessarily have zero generation and consumption terms. For example, population models describing cell cultures or human populations can include generation through division/reproduction and consumption through death.

Accumulation

The Accumulation term describes the net gain or loss of an extensive property contained within a system. When an Accumulation term is present, the amount of extensive property in the system has changed during the time period of interest. There are two terms that describe the characteristics of the Accumulation term of a system: steady-state or dynamic.

A system is in steady-state when its internal conditions (temperature, pressure, etc.) remain constant over time, although minor fluctuations about constant mean values may occur. A steady state system has no accumulation.

The Misconception: Many believe steady-state means nothing is moving.
The Reality: A system can have massive amounts of input and output, but as long as the Accumulation is zero, it is in steady-state.

A system is dynamic when the variables describing the system, such as temperature, pressure, or mass, change with respect to time. This results in the accumulation term present in the accounting equation.

Illustrating Steady-State vs Dynamic

Let us illustrate using a photography analogy.

If you take multiple, imaginary “snapshots” of a steady-state system over a time period, each snapshot should look the same as the previous one. The initial and final conditions and all the intermediate snapshots of the system are identical or nearly so. The snapshots show that no quantity of the extensive property has accumulated in the system.

For a dynamic system, each snapshot would look very different

Heads up!

Steady State and Unsteady State build on these concepts in Heat & Mass Transfer.

Example Conservation Problem

Example Problem: Consider the chemical reaction for photosynthesis. #ChemicalRxn

Given the chemical formula for photosynthesis, balance the moles of glucose and oxygen. Does this system follow the rules of conservation?

$$ 6 CO_2 + 6 H_2O + light \rightarrow C_6H_{12}O_6 + 6O_2 $$

The product chemical species—1 mole of glucose and 6 moles of oxygen—are counted as generated. The total amount of glucose and oxygen gas has increased in the system and the universe. On the other hand, the reactant chemical species—6 moles of carbon dioxide and 6 moles of water—have been consumed simultaneously, because the total amount of carbon dioxide and water has decreased in the system and in the universe.

Learn More

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