We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 05 June 2012
Compartment modelling is a means of constructing a differential equation for a complicated process by considering just the inputs and outputs of the process, during a small time interval. The basic ideas are developed in the context of a model describing the mixing of a dye and water. Compartment models are then formulated for the pollution in a lake and the temperature of a domestic hot water system. The latter model uses ideas about the flow of heat from Chapter 12. The differential equations obtained are mainly of the first-order linear constant-coefficient type.
A mixing problem
One of the aims of modelling is to isolate the most important factors in a problem and ignore those which may not be important. Even very complicated processes can initially be analysed using very simple mathematical models which may later be extended to more complex and realistic models by incorporating more features. In problems involving the mixing of two or more substances, simple models may be formulated by considering the input and output to a compartment containing the quantity of interest.
The following problem will be used to illustrate these ideas. The problem is illustrated in Figure 13.1.1.
Statement of problem
In a dye factory a large vat is used to mix dye and water. The water flows in at a rate of 6 litres/minute and the dye flows in at a rate of 2 litres/minute.
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
