Water at temperature of 15°C flows through a pipe of diameter D = 2 cm and into a
glass. Determine
i. The minimum time taken to fill in a 0.5 L glass full with water if the flow in
the pipe is to be laminar.
ii. The maximum time taken to fill the glass if the flow is to be turbulent.
To determine the minimum and maximum time taken to fill the glass with water, we need to calculate the volumetric flow rate for both laminar and turbulent flow.
i. For laminar flow, we can use Poiseuille's law to calculate the volumetric flow rate:
Q_laminar = (π * r^4 * ΔP) / (8 * μ * L),
where Q_laminar is the volumetric flow rate, r is the radius of the pipe (D/2), ΔP is the pressure difference across the pipe, μ is the dynamic viscosity of water, and L is the length of the pipe.
We can rearrange the equation to solve for ΔP:
ΔP = (Q_laminar * 8 * μ * L) / (π * r^4).
Since we want to determine the minimum time taken, we need to maximize the flow rate. The flow rate Q_laminar is equal to the volume V divided by the time t:
Q_laminar = V / t_min,
where V is the volume of the glass (0.5 L) and t_min is the minimum time taken to fill the glass.
Substituting this into the rearranged equation for ΔP:
ΔP = (V * 8 * μ * L) / (t_min * π * r^4).
We can use the fact that the pressure difference is equal to the hydrostatic pressure of the water column in the glass:
ΔP = ρ * g * h,
where ρ is the density of water, g is the acceleration due to gravity, and h is the height of the water column in the glass.
Equating the two expressions for ΔP:
(V * 8 * μ * L) / (t_min * π * r^4) = ρ * g * h,
and solving for t_min:
t_min = (V * 8 * μ * L) / (π * r^4 * ρ * g * h).
Now, let's substitute the given values:
- V = 0.5 L = 0.5 * 1000 cm^3 = 500 cm^3,
- μ ≈ 0.001 kg/(m*s),
- L = length of the pipe (not given),
- r = D/2 = 2 cm / 2 = 1 cm = 0.01 m,
- ρ = 1000 kg/m^3,
- g ≈ 9.8 m/s^2,
- h = height of the water column in the glass (not given).
ii. For turbulent flow, we can use the equation for volumetric flow rate through a pipe:
Q_turbulent = A * V_turbulent,
where Q_turbulent is the volumetric flow rate, A is the cross-sectional area of the pipe (π * r^2), and V_turbulent is the velocity of the water.
Again, since we want to determine the maximum time taken, we need to minimize the flow rate. The flow rate Q_turbulent is equal to the volume V divided by the time t:
Q_turbulent = V / t_max,
where V is the volume of the glass (0.5 L) and t_max is the maximum time taken to fill the glass.
Substituting this into the equation for the volumetric flow rate:
V / t_max = A * V_turbulent,
and solving for t_max:
t_max = V / (A * V_turbulent).
Now, let's substitute the given values:
- V = 0.5 L = 500 cm^3,
- r = 0.01 m,
- A = π * r^2,
- V_turbulent = velocity of the water (not given).
It's important to note that to determine the maximum time taken for turbulent flow, we need to know the velocity of the water, which is not provided in the question. Without that information, we cannot calculate t_max accurately.
To determine the minimum time taken to fill a 0.5 L glass with water in laminar flow and the maximum time taken in turbulent flow, we need to apply the equations related to flow rate and flow conditions.
i. Minimum Time for Laminar Flow:
In laminar flow, the flow rate, Q, is calculated using Poiseuille's Law:
Q = (π * ΔP * r^4) / (8 * η * L)
where:
- Q is the flow rate (volume per unit time),
- ΔP is the pressure difference across the pipe,
- r is the radius of the pipe (half of the diameter),
- η is the viscosity of the fluid, and
- L is the length of the pipe.
To calculate the minimum time, we need to find the time taken to fill the 0.5 L glass using the flow rate.
1 L = 1000 cm^3, so 0.5 L = 500 cm^3.
Given:
- Temperature of water, T = 15°C (which we can convert to Kelvin for viscosity calculation),
- Diameter of the pipe, D = 2 cm (so the radius, r = 1 cm = 0.01 m),
- Length of the pipe, L = ? (not provided),
- Viscosity of water at 15°C, η = ? (not provided).
To determine the viscosity of water at 15°C, we need to refer to a table or use empirical equations. For water at 15°C, the viscosity is approximately 1.14 * 10^(-3) Pa.s or kg/(m.s).
With the given information, we also need to assume the pressure difference ΔP. If the pressure difference is not given, we assume it to be constant across the pipe.
Now, let's assume a length for the pipe, L, and calculate the flow rate using Poiseuille's Law.
Q_min = (π * ΔP * r^4) / (8 * η * L)
By rearranging the equation, we can solve for time, t_min:
t_min = (Volume of the glass) / Q_min = 500 cm^3 / [(π * ΔP * r^4) / (8 * η * L)]
You can enter the values into the equation and calculate the minimum time taken to fill the glass based on the assumptions made.
ii. Maximum Time for Turbulent Flow:
In turbulent flow, the flow rate can be approximated using the Darcy-Weisbach equation:
Q = (π/4) * (2g/ΔH) * D^2 * C
where:
- Q is the flow rate,
- g is the acceleration due to gravity,
- ΔH is the head loss (difference in height between the pipe inlet and the outlet),
- D is the diameter of the pipe,
- C is the coefficient of discharge.
To calculate the maximum time, we need to find the time taken to fill the 0.5 L glass using the flow rate.
Assuming the same information provided for the laminar flow case, you will have to calculate the head loss, ΔH, and the coefficient of discharge, C, to calculate the flow rate, Q_turbulent.
Then, you can calculate the maximum time as:
t_turbulent = (Volume of the glass) / Q_turbulent = 500 cm^3 / Q_turbulent
By using the Darcy-Weisbach equation and the given information, you can calculate the maximum time taken to fill the glass based on the assumptions made.
Remember to check if the flow conditions are laminar or turbulent based on the Reynolds number (Re) for more accurate results.