In Problem 36, if a small furnace that generates 1000 Btu/hr is placed in zone B, determine the coldest it would eventually get in zone B has a heat capacity of per thousand Btu.
Therefore, the cold of zone B is eventually got .
Elimination Procedure for 2 x 2 Systems:
To find a general solution for the system
Where and L4 are polynomials in :
Vieta’s formulas for finding roots:
For function y(t) to be bounded, we need both roots of the auxiliary equation to be non-positive if they are reals and, if they are complex, then the real part has to be non-positive. In other words,
Section 3.3: Heat transfer, it is modelled by the following equation where is the time constant for the building given in hours, is the outside temperature, is the inside temperature, is the heating in the building, is the cooling.
Given that, is placed in zone B.
Referring to problem 36:
Only zone A is heated by a furnace, which generates .
The heat capacity of zone A is per thousand Btu.
The time constant for heat transfer between zone A and the outside is 4 hours, between the unheated zone B and the outside is 5 hours, and between the two zones is 2 hours.
Let x(t) be denoted as the temperature in A at time t and y(t) be denoted as the temperature in B at time t.
Using the given information create the system of equation.
The above equations can be rewritten as,
Rewrite the system in operator form:
Multiply 5 on equation (3) and multiply on equation (4). Then, add them together to get.
Since the auxiliary equation to the corresponding homogeneous equation is:
So, the roots are and .
Then, the general solution of y is
Let us assume that,
Substitute equation (7) in equation (5).
Substitute the value of C in equation (7).
So, the general solution is
To find: .
Implement the limits on equation (8).
So, the solution is founded
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