Would like to see the following problem worked out. Having trouble solving. Explanation would be helpful.
Given:Water at 20 Degrees C is filled to the top of a 15 m high, 15 m diameter tank. The tank is connected to a 15 m section of 5 cm diameter pipe which contains a fully open globe valve.
a) Determine the initial volumetric flow rate from the tank when the value is opened.
b) If the flow rate from the tank was constant in time, determine the amount of time required to empty the tank.
c) Apply and simplify the appropriate equations that could be solved to calculate the volumetric flow rate from the tank as a function of time.
d) Solve the equation in part c. to calculate the flow rate as a function of time.
To solve this problem, we need to consider several factors including the height and diameter of the tank, the diameter of the pipe, and the flow rate. Let's break down the problem step by step and explain how to arrive at the solution for each part.
a) Determining the initial volumetric flow rate from the tank when the valve is opened:
To find the initial flow rate, we can use the principle of continuity, which states that the volumetric flow rate is constant at any point in an incompressible fluid. In this case, we can apply the equation:
Q1 = Q2
Where Q1 is the initial volumetric flow rate from the tank and Q2 is the volumetric flow rate through the pipe. Since the tank is initially full and the valve is fully open, the flow rate through the pipe is the same as the flow rate from the tank.
To calculate Q2, we can use Bernoulli's principle, which relates pressure, height, and velocity. The equation is as follows:
P1 + 0.5 * ρ * v1^2 + ρ * g * h1 = P2 + 0.5 * ρ * v2^2 + ρ * g * h2
Where P1 and P2 are the pressures at two different points, v1 and v2 are the velocities at those points, ρ is the density of the fluid, g is the acceleration due to gravity, and h1 and h2 are the heights at those points.
In this case, P1 is atmospheric pressure, P2 is atmospheric pressure plus the pressure difference due to the height and velocity of the fluid. We can assume the velocity at the top of the tank is zero, so v1 is zero. We know the height of the tank is 15 m, and the diameter is 15 m, so the radius is 7.5 m.
Using these values, we can calculate the initial flow rate Q1.
b) Determining the time required to empty the tank:
To find the time required to empty the tank, we need to calculate the volume of the tank and divide it by the flow rate.
The volume of a cylinder is given by the equation:
V = π * r^2 * h
Where V is the volume, r is the radius, and h is the height.
We can calculate the volume of the tank using this equation, and then divide it by the flow rate Q1 to find the time.
c) Applying and simplifying the appropriate equations to calculate the flow rate as a function of time:
To calculate the flow rate as a function of time, we can use Torricelli's law, which states that the velocity of fluid flowing out of an orifice is proportional to the square root of the height of the fluid above the orifice.
We can use this law to derive an equation to calculate the flow rate as a function of time. By rearranging the equation and integrating, we can obtain an expression for the flow rate as a function of time.
d) Solving the equation obtained in part c to calculate the flow rate as a function of time:
To solve the equation obtained in part c, we can substitute the necessary values and integrate to find the flow rate as a function of time. This integral can be evaluated numerically or using appropriate software.
By following these steps and using the appropriate equations, you should be able to solve each part of the problem and obtain the desired solutions.