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Example Questions
Example Question #1 : Poiseuille Flow
The mass flow rate through a cylindrical pipe of cross sectional area is
. What is the pressure drop over a distance
?
Assume
We will use Poiseuille's law to solve this problem:
We can determine the volumetric flow rate from the mass flow rate:
Now to determine an expression for radius:
Plugging this into the first expression, we get:
Now plugging in all of our values:
Example Question #1 : Poiseuille Flow
Water enters a circular tube at a velocity of . If the pressure drops by
after a distance of
, what is the diameter of the tube?
Let's begin with the generic form of Poiseulle's law:
Where:
Substituting this in:
And:
Substituting this in:
Rearranging for diameter:
Plugging in values:
Example Question #3 : Poiseuille Flow
Dynamic viscosity of water:
Find the pressure change due to the Hagen-Poiseuille law in a long pipe of radius
, where the water is flowing at
.
None of these
Using
Where is the dynamic viscosity
is the length of the pipe
is the velocity of the fluid
is the diameter of the pipe
Plugging in values:
Example Question #4 : Poiseuille Flow
Dynamic viscosity of water:
Find the pressure change due to the Hagen-Poiseuille law in a long pipe of radius
, where the water is flowing at
.
None of these
Using
Where is the dynamic viscosity
is the length of the pipe
is the velocity of the fluid
is the diameter of the pipe
Plugging in values:
Example Question #5 : Poiseuille Flow
Suppose two pipes made out of an identical material have the same length and the same liquid flowing through them. If pipe A has a cross-sectional diameter that is twice as great as pipe B, how does the flow rate in pipe A differ from the flow rate in pipe B?
The flow rate in pipe A is greater than pipe B by a factor of
The flow rate in pipe B is greater than pipe A by a factor of
The flow rate in pipe B is greater than pipe A by a factor of
The flow rate is the same in both pipes
The flow rate in pipe A is greater than pipe B by a factor of
The flow rate in pipe A is greater than pipe B by a factor of
For this question, we're asked to consider two pipes. Each pipe has the same length, is made out of the same material, and has the same fluid moving through it. The only difference is the cross-sectional diameter of these pipes. We're asked to find how the flow rate will differ between the two pipes.
In order to solve this question, we'll need to use Poiseuille's equation.
This equation tells us that the volume flow rate is directly proportional to two things: the pressure gradient between the ends of the pipe and the radius of the pipe raised to the fourth power. Moreover, the volume flow rate is inversely proportional to the viscosity of the fluid and also to the length of the pipe.
Now we need to ask the question - which of the variables in this equation is different in pipes A and B? Both pipes have the same length. Since each pipe has the same fluid moving through it, the viscosity will also be the same. Furthermore, we can assume that the pressure gradient at the end of each pipe is the same. The only thing that's left is the radius.
We're told that the diameter of pipe A is twice as great as pipe B. Since the radius is just half of the diameter, this also means that the radius of pipe A is twice as great as pipe B. Because the volume flow rate is dependent on the radius of the pipe raised to the fourth power, we can see that doubling the radius will result in a fold difference in the volume flow rate. Thus, the flow rate in pipe A will be
times the flow rate in pipe B.
Example Question #1 : Turbulence
Turbulent flow is characterized by __________.
All of these
irregular and chaotic flow
low flow rates
regular streamlines
irregular and chaotic flow
Turbulent flow is caused by sufficiently high flow rates, and is characterized by chaotic and irregular flow patterns. Streamlines represent the curves tangent to the point of the direction of flow; thus, if a flow is turbulent the streamlines will be chaotic as well.
Example Question #1 : Fluid Dynamics
Which of the following is an example of turbulent flow?
Pouring olive oil into a pan
The wisps of smoke from a blown-out candle
The flow of water through a hose
The flow of water in the center of a calm stream
The wisps of smoke from a blown-out candle
The wisps of smoke from a blown out candle start out a smooth, but eventually swirl and become chaotic and turbulent. The other examples can be intuitively thought of as smooth flow, and thus are laminar.
Example Question #3 : Turbulence
Methanol is traveling through a cylindrical tube with a diameter of . What is the maximum velocity of methanol that maintains laminar flow?
We will use the expression for Reynold's number for this problem:
Where is the hydraulic diameter, and for a cylindrical tube,
Rearranging for velocity:
For laminar flow:
Plugging in our values into the expression, we get:
Methanol has a relatively low viscosity, so it only has a very small range for laminar flow and quickly becomes turbulent
Example Question #1 : Fluid Dynamics
What is the Reynold's number of water flowing through a circular tube of diameter at a rate of
?
Assume and
We will use the expression for Reynold's number for this problem:
Where is the hydraulic diameter, and for a cylindrical tube,
Plugging in our values, we get:
Example Question #1 : Fluid Dynamics
What is the Reynold's number of water flowing through a fully filled rectangle duct that is at a velocity
?
Assume and
None of the other answers
We will use the expression for Reynold's number for this problem:
For a fully filled rectangular duct, the hydraulic diameter is:
Plugging in values:
Now we can plug our values into the original expression:
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