# Applications of Systems of Differential Equations

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1 Brine Tank Cascade Cascade Model Recycled Brine Tank Cascade Recycled Cascade Model Home Heating Newton Cooling Model Applications of Systems of Differential Equations Homogeneous Solution and Particular Solution Underpowered Heater Forced air furnace and Furnace Cycling

2 Brine Tank Cascade Let brine tanks A, B, C be given of volumes 20, 40, 60, respectively, as in Figure 1. water A B C Assumptions and Notation Figure 1. Three brine tanks in cascade. It is supposed that fluid enters tank A at rate r, drains from A to B at rate r, drains from B to C at rate r, then drains from tank C at rate r. Hence the volumes of the tanks remain constant. Let r = 10, to illustrate the ideas. Uniform stirring of each tank is assumed, which implies uniform salt concentration throughout each tank. Let x 1 (t), x 2 (t), x 3 (t) denote the amount of salt at time t in each tank. We suppose added to tank A water containing no salt. Therefore, the salt in all the tanks is eventually lost from the drains.

3 Cascade Model The cascade is modeled by the chemical balance law rate of change = input rate output rate. Application of the balance law results in the triangular differential system x = x 1, x = x x 2, x = x x 3.

4 Cascade Model Solution The solution is justified by the integrating factor method for first order scalar differential equations. x 1 (t) = x 1 (0)e t/2, x 2 (t) = 2x 1 (0)e t/2 + (x 2 (0) + 2x 1 (0))e t/4, x 3 (t) = 3 2 x 1(0)e t/2 3(x 2 (0) + 2x 1 (0))e t/4 + (x 3 (0) 3 2 x 1(0) + 3(x 2 (0) + 2x 1 (0)))e t/6.

5 Recycled Brine Tank Cascade Let brine tanks A, B, C be given of volumes 60, 30, 60, respectively, as in Figure 2. A B C Assumptions and Notation Figure 2. Three brine tanks in cascade with recycling. Suppose that fluid drains from tank A to B at rate r, drains from tank B to C at rate r, then drains from tank C to A at rate r. The tank volumes remain constant due to constant recycling of fluid. For purposes of illustration, let r = 10. Uniform stirring of each tank is assumed, which implies uniform salt concentration throughout each tank. Let x 1 (t), x 2 (t), x 3 (t) denote the amount of salt at time t in each tank. No salt is lost from the system, due to recycling.

6 Recycled Cascade Model Using compartment analysis, the recycled cascade is modeled by the non-triangular system x 1 = 1 6 x x 3, x 2 = 1 6 x x 2, x 3 = 1 3 x x 3.

7 Recycled Cascade Solution x 1 (t) = c 1 + (c 2 2c 3 )e t/3 cos(t/6) + (2c 2 + c 3 )e t/3 sin(t/6), x 2 (t) = 1 2 c 1 + ( 2c 2 c 3 )e t/3 cos(t/6) + (c 2 2c 3 )e t/3 sin(t/6), x 3 (t) = c 1 + (c 2 + 3c 3 )e t/3 cos(t/6) + ( 3c 2 + c 3 )e t/3 sin(t/6). At infinity, x 1 = x 3 = c 1, x 2 = c 1 /2. The meaning is that the total amount of salt is uniformly distributed in the tanks, in the ratio 2 : 1 : 2. The solution of the system was found by the eigenanalysis method. It can also be found by the resolvent method in Laplace theory.

8 Home Heating Consider a typical home with attic, basement and insulated main floor. Attic Main Floor Basement Figure 3. Typical home with attic and basement. The below-grade basement and the attic are un-insulated. Only the main living area is insulated.

9 Heating Assumptions and Variables It is usual to surround the main living area with insulation, but the attic area has walls and ceiling without insulation. The walls and floor in the basement are insulated by earth. The basement ceiling is insulated by air space in the joists, a layer of flooring on the main floor and a layer of drywall in the basement. We will analyze the changing temperatures in the three levels using Newton s cooling law and the variables z(t) = Temperature in the attic, y(t) = Temperature in the main living area, x(t) = Temperature in the basement, t = Time in hours.

10 Newton Cooling Model Assume it is winter time and the outside temperature in constantly 35 F during the day. Also assumed is a basement earth temperature of 45 F. Initially, the heat is off for several days. The initial values at noon (t = 0) are then x(0) = 45, y(0) = z(0) = 35. Portable heater. A small electric heater is turned on at noon, with thermostat set for 100 F. When the heater is running, it provides a 20 F rise per hour, therefore it takes some time to reach 100 F (probably never!). Newton s cooling law Temperature rate = k(temperature difference) will be applied to five boundary surfaces: (0) the basement walls and floor, (1) the basement ceiling, (2) the main floor walls, (3) the main floor ceiling, and (4) the attic walls and ceiling. Newton s cooling law gives positive cooling constants k 0, k 1, k 2, k 3, k 4 and the equations x = k 0 (45 x) + k 1 (y x), y = k 1 (x y) + k 2 (35 y) + k 3 (z y) + 20, z = k 3 (y z) + k 4 (35 z).

11 Insulation Constants and the Final Model The insulation constants will be defined as k 0 = 1/2, k 1 = 1/2, k 2 = 1/4, k 3 = 1/4, k 4 = 1/2 to reflect insulation quality. The reciprocal 1/k is approximately the amount of time in hours required for 63% of the temperature difference to be exchanged. For instance, 4 hours elapse for the main floor. The model: x = 1 2 (45 x) + 1 (y x), 2 y = 1 2 (x y) (35 y) + 1 (z y) + 20, 4 z = 1 4 (y z) + 1 (35 z). 2

12 Homogeneous Solution The homogeneous solution in vector form is given in terms of constants a = (7 21)/8, b = (7 + 21)/8, c = , d = 1 21 and arbitrary constants c 1, c 2, c 3 by the formula x h(t) y h (t) = c 1 e t/2 z h (t) c 2 e at 4 c 2 + c 3 e bt 4 d 2.

13 Particular Solution A particular solution is an equilibrium solution x p(t) y p (t) = z p (t) The homogeneous solution has limit zero at infinity, hence the temperatures of the three spaces hover around x = 57, y = 69, z = 46 degrees Fahrenheit. Specific information can be gathered by solving for c 1, c 2, c 3 according to the initial data x(0) = 45, y(0) = z(0) = 35. The answers are c 1 = 85 24, c 2 = , c 3 =

14 Underpowered Heater To the main floor each hour is added 20 F, but the heat escapes at a substantial rate, so that after one hour y 51 F. After five hours, y 65 F. The heater in this example is so inadequate that even after many hours, the main living area is still under 69 F.

15 Forced air furnace Replacing the space heater by a normal furnace adds the difficulty of switches in the input, namely, the thermostat turns off the furnace when the main floor temperature reaches 70 F, and it turns it on again after a 4 F temperature drop. We will suppose that the furnace has four times the BTU rating of the space heater, which translates to an 80 F temperature rise per hour. The study of the forced air furnace requires two differential equations, one with 20 replaced by 80 (DE 1, furnace on) and the other with 20 replaced by 0 (DE 2, furnace off). The plan is to use the first differential equation on time interval 0 t t 1, then switch to the second differential equation for time interval t 1 t t 2. The time intervals are selected so that y(t 1 ) = 70 (the thermostat setting) and y(t 2 ) = 66 (thermostat setting less 4 degrees). Numerical work gives the following results.

16 Furnace Cycling Time in minutes Main floor temperature Model Furnace DE 1 on DE 2 off DE 1 on DE 2 off The reason for the non-uniform times between furnace cycles can be seen from the model. Each time the furnace cycles, heat enters the main floor, then escapes through the other two levels. Consequently, the initial conditions applied to models 1 and 2 are changing, resulting in different solutions to the models on each switch.

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