In this module we look at more complex and more realistic models for the flow of heat through the walls of an adobe house -- or, more generally, in an object whose temperature varies continuously from one point to another. One of our goals is to shed some light on the question raised earler -- can the walls of an adobe house be built thick enough to produce a 12 hour (one-half period) time lag between the oscillations of the exterior temperature and the interior temperature?
If you and other students in your class have tried enough experiments with different models based on two or more bricks you have probably seen examples in which the oscillations of the interior temperature lag more than one-fourth period behind the oscillations of the exterior temperature. In fact, you may even have seen examples in which the lag is more than one-half period. Based on these models it appears that Ross Finney's statement is OK. In fact, experimental evidence confirms that lags of more than one-fourth period and even lags of more than one-half period are possible. Your CAS window has a simulation of a three brick model with values of the constants chosen so that we do get a delay of roughly 12 hours (one-half period). Look at the work in your CAS window. This is not a proof but it is fairly persuasive evidence that a time lag of 12 hours is possible.
The remainder of this module is CALCULUS-Rated and is suitable for students who have completed or are in the second semester of a calculus course. It requires familiarity with derivatives and partial derivatives.
Now we want to build another, mathematically more sophisticated model for heat flow through the walls of an adobe house. This model is based on a key picture. Because we refer to this picture often as we build the model, we need to have it remain on the screen throughout our discussion. Click here to open a new window with this key picture and arrange these two windows so that they overlap and you can move easily back-and-forth between them by clicking on the exposed portion of the inactive window to make it active.
The key figure shows the cross section of a wall. We can place a series of recording thermometers at different points within the wall and on the interior and exterior surfaces of the wall. If the thickness of the wall is L centimeters and we use n thermometers inside the wall plus two more at the two surfaces of the wall then the interior thermometers will be placed h units apart where h = L/n as shown in the figure. The thermometer on the exterior surface of the wall is 0.5 h centimeters from the first thermometer inside the wall. The thermometer on the inside surface of the wall is 0.5 h centimeters from the last thermometer inside the wall.
The following points in and on the wall are important.
The function E(t) will represent the readings of the thermometer on the outside surface of the wall; the function I(t) will represent the readings of the thermometer on the inside surface of the wall; and the functions
will represent the readings of the n thermometers in the interior of the wall. We sometimes use the notation T(t, x) for the readings of the thermometer placed at the point x at time t. Notice that if one interior thermometer is placed at the point x then the two nearest thermometers are placed at the two points x - h and x + h unless x = h/2 or x = L - h/2.
We will use the notation T(x, t) for the actual temperature of the brick at the point x at time t. Notice that
Our model will be based on three key considerations. Click on the links below for a discussion of these basic considerations.
How the temperature of an interior subbrick changes
We begin by looking at one of the interior subbricks -- that is, we look at the temperature being measured by one of the functions
In order to understand the change in these temperatures we need to consider two factors -- the heat energy crossing the two "surfaces" of this subbrick and the effect this heat energy has on the temperature of the subbrick.
We have been using Newton's model of cooling. This model describes the temperature change when two objects whose temperatures are different are in direct contact.
In this situation there is a temperature difference at the interface between the two objects and the rate at which heat energy flows from the hotter object to the colder one is proportional to this temperature difference. In Newton's model of cooling we think of the temperature of each object as uniform. Thus, the placement of the thermometers measuring the temperature of each object is not important.
Our current situation quite different. Now the temperature changes continuously as we move through the brick. But we are still interested in the amount of heat energy crossing a surface.
If we were to measure the temperature on both sides of the surface then we would get the same reading. In this situation the transfer of heat energy across a surface is proportional to the derivative of the temperature measured at the surface.
Now consider the i-th subbrick, whose temperature is measured by Bi. Notice this subbrick has two surfaces -- one at (i - 0.5)h and the other at (i + 0.5) h. The energy entering this subbrick is proportional to
Notice the signs on each of the two terms. Heat energy passes from the left of a surface to the right of the surface when the derivative of the temperature is negative. Heat energy is passing into the brick across the left hand surface when it is passing from left to right. Heat energy is passing out of the brick across the right hand surface when it is passing from left to right. We approximate the quantity above in the usual way by
The rate at which the temperature of a subbrick changes is proportional to the rate at which heat energy enters the subbrick divided by the mass of the subbrick. Since the original brick is homogeneous and has a constant cross-section, the mass of each subbrick is proportional to its thickness, h. Thus the rate at which the temperature of the i-th subbrick is changing can be approximated as shown below
This motivates the continuous model (partial differential equation) below in the usual way, by mentally dividing the original brick into a very large number of very thin subbricks and then observing that the right factor on right hand side of the equation above is the usual approximation for the second derivative. The equation below is called the heat equation or the diffusion equation. It has been very successful in modeling the flow of heat.
The Heat Equation describes what is happening inside the adobe wall. To build a complete model we would also need to describe what is happening on the outside surface and on the inside surface of the wall. Like all courses, this course has two goals -- to study some material and also to indicate some questions for further study. No course is ever complete -- in part, because of time limits and, in part, because we don't know all the answers. It is important that your students leave each course with more questions than answers. In this spirit we will not discuss how we model the situation on the interior and exterior surface of the wall. But we will use the heat equation to gain some insight into the question that was raised at the Carroll College meeting -- is it possible to build the walls of an adobe house in such a way that the oscillation of the temperature inside the house is one-half period, or 12 hours, behind the oscillation of the temperature outside the house?
We use a simple model that cannot answer this question completely but can give us some insight. We consider a wall that is infinitely thick -- so that we don't need to consider the heat transfer from the inside surface of the wall to the interior of the house. We also assume that the exterior temperature E(t) is transfered immediately to the outside surface of the wall.
We choose the simplest exterior function, E(t), that can give us some insight into this situation.
Thus, the temperature of the wall will be described by a function of the form
where
and
Now we invoke one of the most important methods for solving differential equations -- guess-and-check. We can often make a pretty good guess what a solution looks like and then we can try our guess out and see if it satisfies the differential equation.
Before reading on, you should try to guess what the solution might look like. Your guess should be based on the following ideas.
You may have several different guesses. That's fine. You can try out a series of guesses to see which one(s) work.
Our guess is
Notice
The next step is to calculate the necessary derivatives.
and
Thus, the heat equation requires that
Thus,
and
The solution we want has the negative sign because the amplitude of the oscillations should decrease as we move further into the wall. Thus,
is a solution of the heat equation that satisfies the condition
This is not a 100% answer to the question raised at the Carroll College meeting but it is very persuasive. It looks as if there will be a time lag of one-half period when
and at that depth in the wall the amplitude of the oscillations will be
or 4.32% the amplitude of the exterior oscillations. That would be very comfortable.
This is far from a proof. There are some practical questions -- what is the value of k and is it practical to build a wall of this thickness? In addition, we have essentially ignored what happens at the two surfaces of a real wall -- does this make a significant difference?
All of this shows that modeling is a tricky business. Sometimes models that are too simple give us answers that can be misleading.
