BASICS MATHEMATICAL MODELING

INTRODUCCIÓN

For size and calculate the characteristics of a control system is necessary to know the relationship between input and output blocks that constitute this relationship can be expressed as transfer functions or differential equations.

MODELS TYPE

GRAPHIC MODEL: the set of images and graphics support to help locate the functional relations that prevail in system to be studied.

PHYSICAL MODEL: consists of an assembly that can operate as a real testbed. They are usually laboratory-scale simplifications that enablemore detailed observations. Not always easy to build.

MATHEMATICAL MODEL: seeks to formalize in mathematical language relations and variations you want to represent and analyze. Normally expressed as differential equations and this reason may also be known as differential model.

ANALYTICAL MODEL: arises when the differential model has solution. This is not always possible, especially when differential models are formulated on differential Partial.

NUMBER MODEL : arises when using numerical techniques to solve differential models. They are very common when differential model has no analytical solution.

COMPUTATIONAL MODEL: refers to a program computer that allows analytical or numerical models can be solved more quickly. They are very useful for implement numerical models because these are based on iterative process can be long and tedious.

MATHEMATICAL MODEL COMPONENTS

Dependent variables Independent variables Parameters Functions of force Operators

TERMICAL SISTEM

Thermal systems have processes that somehow exchange heat energy with its environment as chemical processes. the input and output signals for these systems are temperature, heat energy and heat output. Basic natural law for the systems is the thermal energy balance, which tells us that the change in heat energy per unit time is equal to the power minus the extracted inferred.

when using the energy balance relationship for simple systems can be used the ratio of heat energy in a certain field with temperature.

As

T=Temperature

V= Volume

E= Heat energy C= Thermal capacity

P= Density

assuming that the volume V, the capacity C and density p are constant we obtain:

both the extra power can be inferred expresarce as a function of temperature as:

where Q is the flow of fluid to be treated

is important to note that both the capacitance and the density may be a function of temperature and in special situations there is heat exchange between gases such as between water and steam.

EXAMPLE

The figure below shows a water tank insulatedV= 4 cubic metersQ= 0,1 cubic meters for second Inlet Temperature varies

Find the transfer function between the inlet temperature Ti and the temperature of the tank T. Assume that the tank is a good mix, so there is the same temperature in the liquid.

Inferred energy in the tank is removed from the, the balance gives us:

As Water Heat Energy is: Thermal power of water is inferred: Thermal power is extracted from water:

Substituting in the energy balance gives:

Since the volume V, the capacitance density p c are constants, gives:

By using the Laplace transform is obtained:

Taking Ti as the input and T output is obtained as:

by substituting the values, gives a transfer function of first degree:

BIBLIOGRAFIA

• Material del Ingeniero Elkin Rodolfo Santafe

Elementos de modelamiento matemático.•http://www.slideshare.net/ptah_enki/modelamiento-matemtico.