Two cylindrical cross section water tanks are arranged so that the outflow from tank I falls into tank 2. The outflow from tank 2 flows to the drain. 4 is the volumetric flow rate out of a tank, and h is the height of the water in a tank. (a) State the fundamental physical law, and provide the general equation for a tank used to model the time history of the tank levels, hị and hz. Your equation should contain the following terms: A (cross sectional area of a water tank), p (density of water), and h (rate of change of the height of the water in a tank) (b) Both outflows have the characteristic volumetric flow rate being directly proportional to the head in the tank. ie. 41 = a,h, and 42 = azh2. Show that the differential equation governing the head in tank 2 has the form of: h, + ah, = be" Time, t, is the independent variable. Take the initial values of the heads to be h(0) = H, and h,(0) -H,. Determine a, b and c. (Note: There is no d, or hi in the equation governing tank 2. You should Figure 1: Schematic Diagram of Flow of 2 tanks solve for hi in tank 1 first, then solve for tank 2). (c) Suppose that H, has a value of Sm (H- Sm), H; has a value of Im (H; - Im), the area of tank I is 5 m', the arca of tank 2 is 2 m', a, is 0.2, and a; is 0.04. From the differential equation in part (b), solve for the head in tank 2, h, , as a function of time. Do not use Laplace transform methods from Leeture 6. (d) State the equation for the head in tank 1, hi,as a function of time, t. Make a rough sketch of hi, showing clearly on the sketch the initial value at t-0, the shape of the graph, the equilibrium value, and how long it takes to reach the equilibrium value to within 1%.
Two cylindrical cross section water tanks are arranged so that the outflow from tank I falls into tank 2. The outflow from tank 2 flows to the drain. 4 is the volumetric flow rate out of a tank, and h is the height of the water in a tank. (a) State the fundamental physical law, and provide the general equation for a tank used to model the time history of the tank levels, hị and hz. Your equation should contain the following terms: A (cross sectional area of a water tank), p (density of water), and h (rate of change of the height of the water in a tank) (b) Both outflows have the characteristic volumetric flow rate being directly proportional to the head in the tank. ie. 41 = a,h, and 42 = azh2. Show that the differential equation governing the head in tank 2 has the form of: h, + ah, = be" Time, t, is the independent variable. Take the initial values of the heads to be h(0) = H, and h,(0) -H,. Determine a, b and c. (Note: There is no d, or hi in the equation governing tank 2. You should Figure 1: Schematic Diagram of Flow of 2 tanks solve for hi in tank 1 first, then solve for tank 2). (c) Suppose that H, has a value of Sm (H- Sm), H; has a value of Im (H; - Im), the area of tank I is 5 m', the arca of tank 2 is 2 m', a, is 0.2, and a; is 0.04. From the differential equation in part (b), solve for the head in tank 2, h, , as a function of time. Do not use Laplace transform methods from Leeture 6. (d) State the equation for the head in tank 1, hi,as a function of time, t. Make a rough sketch of hi, showing clearly on the sketch the initial value at t-0, the shape of the graph, the equilibrium value, and how long it takes to reach the equilibrium value to within 1%.
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Transcribed Image Text:Two cylindrical cross section water tanks are arranged so that the outflow from tank 1 falls into
tank 2. The outflow from tank 2 flows to the drain. ģ is the volumetric flow rate out of a tank,
and h is the height of the water in a tank.
(a) State the fundamental physical law, and provide the general equation for a tank used to
model the time history of the tank levels, h, and hz. Your equation should contain the
following terms: A (eross sectional area of a water tank), p (density of water), and h (rate
of change of the height of the water in a tank)
(b) Both outflows have the characteristic
volumetric flow rate being directly
proportional to the head in the tank.
ie. ġ1 = a, h, and q2 = azh2.
Show that the differential equation
governing the head in tank 2 has the form
of :
h, + ah, = be"
Time, t, is the independent variable.
Take the initial values of the heads to be
A
h(0) = H, and h,(0)= H,. Determine a,
b and c. (Note: There is no d, or hi in the
equation governing tank 2. You should
solve for hi in tank I first, then solve for
tank 2).
Figure 1: Schematic Diagram of Flow of 2 tanks
Suppose that H, has a value of 5m (H= Sm), H; has a value of Im (H2 = Im), the area of
tank I is 5 m, the area of tank 2 is 2 m', a, is 0.2, and a; is 0.04. From the differential
equation in part (b), solve for the head in tank 2, h, , as a function of time. Do not use
Laplace transform methods from Lecture 6.
(c)
State the equation for the head in tank 1, hi,as a function of time, t. Make a rough sketch
of hi, showing clearly on the sketch the initial value at t= 0, the shape of the graph, the
equilibrium valuc, and how long it takes to reach the equilibrium value to within 1%.
(d)
Show that the argument of the exponential functions of your solution for h:(t) are
(e)
dimensionally correct
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