You are a process control engineer in a water purification plant. In a certain subprocess, water enters a tank through an inlet at a rate of q; and leaves through an outlet at a rate of q.. Liquid flow is maintained such that q; > q, at any given time. It has been found that the difference between q; and q, is equal to the product of the surface area of the tank bottom A and the rate of change of the liquid column height in the tank h with respect to time. If the variation of h is continuous with respect to time, a. formulate a first-order differential equation to describe the situation. b. It was found that the outlet flow rate depends on the height of the liquid column and is given by q. = 4.429 x 10-3h m³/s. Furthermore, given that A = (0.72) m², q; = 0.003 m³ /s and that h = 1 m when t = 0 s, solve the differential equation formulated (a) to find the relationship between the liquid column height h and the time taken t. c. Verify your answer in (b) using another method of solving the differential equation (e.g. Laplace transforms). d. The rate of change of liquid volume r in the primary tank of the water purification plant was measured for 4 days. If r can be approximated to the function r = 5 sin(0.5t + 0.5) + In (t + 1); where r is in m/day and t is in days, find the approximate volume of water in the tank at 1sts4 the end of the 4 days correct to 3 decimal places using the trapezoidal rule with 4 intervals. e. Calculate the same distance as in (d) above by using the appropriate definite integral. f. Given that the permissible error in the calculation of liquid volume in the primary tank is +0.1 m3, state whether the trapezoidal rule is appropriate for future calculations of the
You are a process control engineer in a water purification plant. In a certain subprocess, water enters a tank through an inlet at a rate of q; and leaves through an outlet at a rate of q.. Liquid flow is maintained such that q; > q, at any given time. It has been found that the difference between q; and q, is equal to the product of the surface area of the tank bottom A and the rate of change of the liquid column height in the tank h with respect to time. If the variation of h is continuous with respect to time, a. formulate a first-order differential equation to describe the situation. b. It was found that the outlet flow rate depends on the height of the liquid column and is given by q. = 4.429 x 10-3h m³/s. Furthermore, given that A = (0.72) m², q; = 0.003 m³ /s and that h = 1 m when t = 0 s, solve the differential equation formulated (a) to find the relationship between the liquid column height h and the time taken t. c. Verify your answer in (b) using another method of solving the differential equation (e.g. Laplace transforms). d. The rate of change of liquid volume r in the primary tank of the water purification plant was measured for 4 days. If r can be approximated to the function r = 5 sin(0.5t + 0.5) + In (t + 1); where r is in m/day and t is in days, find the approximate volume of water in the tank at 1sts4 the end of the 4 days correct to 3 decimal places using the trapezoidal rule with 4 intervals. e. Calculate the same distance as in (d) above by using the appropriate definite integral. f. Given that the permissible error in the calculation of liquid volume in the primary tank is +0.1 m3, state whether the trapezoidal rule is appropriate for future calculations of the
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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only need part d and part e and part f
Transcribed Image Text:4. You are a process control engineer in a water purification plant. In a certain subprocess, water
enters a tank through an inlet at a rate of q; and leaves through an outlet at a rate of q. Liquid
flow is maintained such that q; > 9, at any given time. It has been found that the difference
between q; and q, is equal to the product of the surface area of the tank bottom A and the rate
of change of the liquid column height in the tank h with respect to time. If the variation of h is
continuous with respect to time,
a. formulate a first-order differential equation to describe the situation.
b. It was found that the outlet flow rate depends on the height of the liquid column and is
(0.72) m², q; =
0.003 m³ /s and that h = 1 m when t = 0 s, solve the differential equation formulated in
(a) to find the relationship between the liquid column height h and the time taken t.
c. Verify your answer in (b) using another method of solving the differential equation (e.g.
given by q, = 4.429 x 10-3h m³/s. Furthermore, given that A =
Laplace transforms).
d. The rate of change of liquid volume r in the primary tank of the water purification plant
was measured for 4 days. If r can be approximated to the function
r = 5 sin(0.5t + 0.5) + In (t + 1);
1sts 4
where r is in m?/day and t is in days, find the approximate volume of water in the tank at
the end of the 4 days correct to 3 decimal places using the trapezoidal rule with 4
intervals.
e. Calculate the same distance as in (d) above by using the appropriate definite integral.
f. Given that the permissible error in the calculation of liquid volume in the primary tank is
+0.1 m³, state whether the trapezoidal rule is appropriate for future calculations of the
same quantity.
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