Let us consider float division first.
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We are going to give several forms of the heat equation for reference purposes, but we will only be really solving one of them. Note that with this assumption the actual shape of the cross section i. Note that the 1-D assumption is actually not all that bad of an assumption as it might seem at first glance.
If we assume that the lateral surface of the bar is perfectly insulated i. This means that heat can only flow from left to right or right to left and thus creating a 1-D temperature distribution.
The assumption of the lateral surfaces being perfectly insulated is of course impossible, but it is possible to put enough insulation on the lateral surfaces that there will be very little heat flow through them and so, at least for a time, we can consider the lateral surfaces to be perfectly insulated.
As indicated we are going to assume, at least initially, that the specific heat may not be uniform throughout the bar. Note as well that in practice the specific heat depends upon the temperature.
As noted the thermal conductivity can vary with the location in the bar. Also, much like the specific heat the thermal conductivity can vary with temperature, but we will assume that the total temperature change is not so great that this will be an issue and so we will assume for the purposes here that the thermal conductivity will not vary with temperature.
First, we know that if the temperature in a region is constant, i. Next, we know that if there is a temperature difference in a region we know the heat will flow from the hot portion to the cold portion of the region.
For example, if it is hotter to the right then we know that the heat should flow to the left. Finally, the greater the temperature difference in a region i. In this case we generally say that the material in the bar is uniform.
There are four of them that are fairly common boundary conditions. The first type of boundary conditions that we can have would be the prescribed temperature boundary conditions, also called Dirichlet conditions.
These are usually used when the bar is in a moving fluid and note we can consider air to be a fluid for this purpose. Here are the equations for this kind of boundary condition.
Note that the two conditions do vary slightly depending on which boundary we are at. If the heat flow is negative then we need to have a minus sign on the right side of the equation to make sure that it has the proper sign. Note that we are not actually going to be looking at any of these kinds of boundary conditions here.Engineering Sciences 22 — Systems 2nd Order Systems Handout Page 2 Some of the common possibilities for Y(s) are given in table entries (If the input is constant, zero, or a step, Y(s) will in fact be one of these, or a combination of them, depending on initial conditions.).
A polynomial equation, also called algebraic equation, is an equation of the form + − − + ⋯ + + + = For example, + − = is a polynomial equation.
When considering equations, the indeterminates (variables) of polynomials are also called unknowns, and the solutions are the possible values of the unknowns for which the equality is true (in general more than one solution may exist).
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