EMPTY A TANK Assignment: You were email a file that discusses Bernoulli\'s Equat

Question

EMPTY A TANK Assignment: You were email a file that discusses Bernoulli's Equation and an ODE for emptying a tank. You are to construct the initial value problem for the data given below for for the ODE developed in the file, then find an EXPLICIT SOLUTION for the IVP, and compute the time required for the tank to empty. (SIMPLIFY YOUR SOLUTION.) DO NOT LEAVE ANY FRACTIONS; write numerical values as decimals with at most 4 decimal places rounded. The units involved can be English or metric. For English units use g = 32ft/sec2 and for metric units use g = 9.8m/sec2. SHOW ALL WORK; NO WORK, NO CREDIT. YOUR DATA:

Explanation / Answer

Cross-section area of tank = A_tank = pi/4 *D^2

Cross-section area of exit pipe = A_pipe = pi/4 *d^2

Velocity in pipe = V

Volume flow rate Q = A_tank*dh/dt = A_pipe*V

dh/dt = (A_pipe / A_tank)*V

dh/dt = (d/D)^2 *V

dh/dt = (15/200)^2 *V

dh/dt = 0.005625*V

(dh/dt) / V = 0.005625

Bernoulli eqn: P_atm + 1/2*rho*(dh/dt)^2 + rho*g*h = P_atm + 1/2*rho*V^2 + 0

rho*g*h = 1/2*rho*V^2 [1 - ((dh/dt) / V)^2]

g*h = 1/2*V^2 [1 - 0.005625^2]

Ignoring 0.005625^2 compared to 1 we get,

g*h = 1/2*V^2

or V = sqrt (2gh)

dh/dt = (d/D)^2 *sqrt (2gh)

dh / sqrt (h) = (d/D)^2 *sqrt(2g) *dt

Integrating LHS from h = h0 to h = 0 and RHS from t = 0 to t = t we get

2*sqrt(h0) = (d/D)^2 *sqrt (2g) *t

t = (D / d)^2 *sqrt (2*h0 / g)

Putting values,

t = (200 / 15)^2 *sqrt (2*0.5 / 9.8)

t = 56.789 seconds

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