Often before a design or process is brought to fruition it is necessary to simulate aspects of it to ensure that it will behave as anticipated.

Mathematical Modeling, broadly, is the term used when the behavior or performance of a static or steady state process or system is described using a series of equations (or even a single equation).

An example of this would be the determination the heat loss of a lagged pipe embedded in the ground with a constant flow of geothermal steam passing through it. The temperature conditions surrounding the pipe could be considered static, as they will take a long time to change with climate, so the calculations can be performed with steady temperatures. The heat loss from the steam to the surroundings can be calculated using an iterative procedure, and as such would be a set of equations with a predictor-corrector mechanism that converge on a solution.

In terms of developing new models this could be achieved in a number of ways:

A correlated model that draws on a databank of values for the process under consideration to provide future estimations to be made. This is useful when understanding of the process is limited, or insight is not practically achievable (e.g. Inside a nuclear reactor).

A model based on pre-determined physical considerations on/ of the system that can be used to predict behavior or movement. This is useful for a system that is readily understandable (e.g. To predict the maximum droop of a vehicle's suspension)

A combination of these two methods is most often used to solve practical engineering problems. Although a method of solution may often exist for a complex problem, it can sometimes be complicated due to the need for specialized data, or be too expensive or time consuming (i.e. Finite element analysis (F.E), computational fluid dynamics (CFD)) to warrant the required effort.

Simulation, broadly, is the term used when a dynamic process is being examined. In engineering applications this will often be time based, and could be considered the next step on from steady state modeling.

A classic example of simulation is illustrated below:

This Figure shows the cylinder gas pressure and temperature in a four stroke spark ignition engine. The dynamic term here comes from the crank angle, which is dependant on engine speed. This simulation was performed by solving a series of differential equations that describe the gas properties, and equations describing the physical attributes of the engine such as piston motion and valve opening. These all vary according to time, and so 'an answer' is not possible, or even desirable, hence simulation is required.

Other examples of where simulations are necessary could be to predict the freezing time of fish fillets in the hold of a deep-sea trawler to allow the process to be adjusted for optimum throughput, or the time to fill a vessel from an unsteady source, such as to determine the maximum evacuation time for a sinking boat.

Modeling can be a complex business, with data to analyse and so many considerations of system behavior, possible limitations, accuracy and the most often encountered problem in day to day engineering: "What will happen when we change this?" The answers can be far from clear.

Using a wide range of mathematical techniques, software and programming tools we can develop models and simulations to allow you to predict and extrapolate the behavior of your systems and processes. From a simple series of calculations in a booklet, all the way to bespoke software that can be integrated into your computer desktop, we can provide affordable solutions suited to you and your working practices.

The handbook development page may also be of interest. Click here to jump to it.

For enquiries about mathematical modeling and simulation please email information@mcilwainandassociates.com

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