19. Consider the two interconnected tanks shown in Figure 7.1.6. Tank 1 initially contains 30 gal of water and 25 oz of salt, and Tank 2 initially contains 20 gal of water and 15 oz of salt. Water containing 1 oz/gal of salt flows into Tank 1 at a rate of 1.5 gal/min. The mixture flows from Tank 1 to Tank 2 at rate of 3 gal/min. Water containing 3 oz/gal of salt also flows into Tank 2 at a rate of 1 gal/min (from the outside). The mixture drains from Tank 2 at a rate of 4 gal/ min, of which some flows back into Tank 1 at a rate of 1.5 gal/min, while the remainder leaves the system.
19. Consider the two interconnected tanks shown in Figure 7.1.6. Tank 1 initially contains 30 gal of water and 25 oz of salt, and Tank 2 initially contains 20 gal of water and 15 oz of salt. Water containing 1 oz/gal of salt flows into Tank 1 at a rate of 1.5 gal/min. The mixture flows from Tank 1 to Tank 2 at rate of 3 gal/min. Water containing 3 oz/gal of salt also flows into Tank 2 at a rate of 1 gal/min (from the outside). The mixture drains from Tank 2 at a rate of 4 gal/ min, of which some flows back into Tank 1 at a rate of 1.5 gal/min, while the remainder leaves the system.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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If someone could help with the questions outlined in the images attached, that would be greatly appreciated!
Transcribed Image Text:d
or
al
1.5 gal/min
1 oz/gal
Q₁ (t) oz salt
30 gal water
Tank 1
3 gal/min
1.5 gal/min
Q₂(t) oz salt
20 gal water
2.5 gal/min Tank 2
1 gal/min
3 oz/gal
FIGURE 7.1.6 Two interconnected tanks (Problem 19).
a. Let Q₁(t) and Q2(1), respectively, be the amount of salt in
each tank at time t. Write down differential equations and initial
conditions that model the flow process. Observe that the system
of differential equations is nonhomogeneous.
b. Find the values of Q₁ and Q2 for which the system is in
equilibrium-that is, does not change with time. Let Q and
Q be the equilibrium values. Can you predict which tank will
approach its equilibrium state more rapidly?
c. Let x₁ = Q₁(t) - Q and x₂ = Q2(1) - Q. Determine an
initial value problem for x₁ and x2. Observe that the system of
equations for
is homogeneous.
X1
and x2
Transcribed Image Text:19. Consider the two interconnected tanks shown in Figure 7.1.6.
Tank 1 initially contains 30 gal of water and 25 oz of salt, and Tank 2
initially contains 20 gal of water and 15 oz of salt. Water containing
1 oz/gal of salt flows into Tank 1 at a rate of 1.5 gal/min. The
mixture flows from Tank 1 to Tank 2 at a rate of 3 gal/min. Water
containing 3 oz/gal of salt also flows into Tank 2 at a rate of 1 gal/min
(from the outside). The mixture drains from Tank 2 at a rate of 4 gal/
min, of which some flows back into Tank 1 at a rate of 1.5 gal/min,
while the remainder leaves the system.
sbos tabis of wa
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