In this lesson, we will cover the following topics:
1. Laminar Fully Developed Pipe Flow & Equations
2. Example 1 – Inclined Pipe: Is the Flow Laminar?
3. Turbulent Pipe Flow & Equations
4. Turbulent Pipe Flow – Effect of Rough Walls, Moody Chart
5. Example 2 – Finding Head Loss and Pressure Drop
6. Head Loss – Friction Factor
video coming later
Note: This lesson plan has a lot of equations. All variable definitions and their respective units can be found in the very first lesson plan for this unit, linked here.
Laminar Fully Developed Pipe Flow & Equations
We can derive analytical solutions for laminar pipe flow (Reynolds number is < 2300). When flow is fully developed, laminar pipe flow, we can use certain equations to help us solve these problems.
Example 1 – Inclined Pipe: Is the Flow Laminar?
(a) To determine whether the flow is up, we need to know that fluid flows in the direction of a high Hydraulic Grade Line (HGL) to a low one. An HGL is a line which gives the pressure head of a fluid in a pipe at every point, and both HGL and pressure head are measurements of length. Pressure head is the height difference between the fluid at one point versus the discharge point of that fluid. This is what it would look like visually:
To calculate HGL, use the following equation:
Make two equations, one for each points 1 and 2 in the image given in the question:
Substituting in the values shown in the picture above, and using the trigonometric rates to find Z2 gives the following:
Solving for HGL1 and HGL2 gives:
Because HGL1 is higher, the flow will go from this higher HGL to the lower one. This validates that the flow goes from the lower point to the higher point, and it does indeed go up.
(b) Hf is the head loss, and this is computed as the difference between the two HGL values we found in (a).
(c) To find Q, I have rearranged one of the equations that used Q, given in the list above. To find the dynamic viscosity (curly U), we multiplied p*v (density * kinematic viscosity). Converting the diameter to meters and substituting in the other variables for Q, we get the following:
(d) To find the velocity, we divide Q/A, as given in the equation list above. A is the cross-sectional area of the pipe that the fluid flows through. The cross-section of the pipe is a circle, so the area of a circle is used.
(e) Then for the Reynolds Number, we use the equation velocity*diameter/kinematic viscosity. The Reynolds Number is dimensionless.
Because the Reynolds Number is below 2300, the flow is laminar.
Turbulent Pipe Flow & Equations
Before we get started on all the equations, let’s talk about the shear distributions that occur when there is turbulent pipe flow. When the flow is near a wall, the shear distribution is as follows:
(1) Wall Layer: Laminar/viscous shear dominates near a wall.
(2) Outer Layer: Turbulent shear dominates in the outer layer.
(3) Overlap Layer: Both laminar and turbulent shear are important.
This is what the regions would look like, with the viscous shear stress (Tv) being the highest at the viscous wall layer and lowest at the outer turbulent layer. The turbulent shear stress (Tt) is the only dominating shear at the outer turbulent layer. Look at the horizontal blue arrows below – the space from the y-axis to the first blue curve (going left to right) is the viscous shear stress, and then the space from the first blue curve to the second blue curve is the turbulent shear stress:
What is the viscous shear and turbulent shear that I mention above? There are two types of shear stress that can occur in a fluid, and the total shear stress (tau) is actually made up of these two terms. I have shown this below, as well as the kinematic eddy viscosity, which is not used in the total shear stress, but just good to know:
There are many equations that can be used with turbulent pipe flow. Let’s go through them – they will be numbered so you can keep track:
(1) Time-Averages Variables
For turbulent flow, because the flow has so many fluctuations, the velocity and pressure terms are random varying functions to time and space. Because we cannot handle these quick changes in flow with math modeling at the moment, we instead find the average (the mean) values for velocity, pressure, shear stress, and other variables when the Reynolds Number is >2300 (turbulent flow).
These variables can then be rewritten in terms of the mean or the time-averaged turbulent variables. The u signifies any variable you are finding for turbulent flow, so you will replace this with your variable instead. You will take the integral of this variable, and T is the averaging period (the time over which you want to take the variable’s average). T is time, so it is usually in seconds. A time of 5 seconds for turbulent gas and water flows is usually long enough.
1(a) Time Mean
1(b) Fluctuation from the Mean
Here is the fluctuation equation, which can be done with any variable and not specifically u:
The fluctuation of the variable is how much it deviates from the average, or mean, value. It is denoted with an apostrophe (eg. u’) and is equal to the variable subtracted by its mean value.
1(c) Square of the Time-Mean Variables
You can find the square of a time mean variable from part (a) by using this equation above. We used u’ as the example variable. If you try to find the average of the fluctuation value in part (b) using this integral equation in part (c), it will be zero.
1(d) Momentum Equations for Time-Averaged Variables
Using these previous equations, we can make substitutions into the momentum equations and change them to account for time-averaged variables in turbulent flow. Here is what the momentum equation in the x-direction would look like with the necessary substitutions. The black underlined terms are known as the turbulent stresses.
(2) Set of Equations relating to the Law of the Wall
2(a) Law of the Wall
2(b) Velocity Defect Law
2(c) Logarithmic Overlap Layer
Or you may also see the same equation, but with y replaced by (R-r) and u is u(r):
2(d) Average Velocity (found using equation 1(g))
2(e) Linear Viscous Relation
(3) Other Relevant Equations
3(a) Smooth-Walled Pipe Flow
Here is an equation that is used for turbulent flow in a smooth-walled pipe as the relation between the dimensionless friction factor f and the Reynolds Number Re, which should be large (>2300) since this equation is for turbulent flow:
3(b) Maximum Velocity
3(c) Average Velocity
This equation related maximum velocity umax with average velocity V.
3(d) Pressure Drop in a Horizontal Pipe
Or you may see it written as:
Turbulent Pipe Flow – Effect of Rough Walls, Moody Chart
Roughness of the surface on which a fluid is flowing has an effect on the resistance due to friction that the fluid will face. This effect can be ignored for laminar flow, but surface roughness strongly affects turbulent flow and will need to be acknowledged. Here are some main points:
(a) Sublayer Thickness Ys
Here is an equation for the sublayer thickness Ys that can be used with rough walls. In the equations for the previous section, we only considered smooth walls.
(b) Surface Roughness and Friction
Take a look at these two example images. Both pipes have rough walls. In the top image there is laminar flow through the pipe because roughness does not have too much of an effect on laminar flow, so the maximum velocity (Umax) can go further. With the bottom image, there is turbulent flow and the maximum velocity does not reach as far as with laminar flow, even with the same volume flowing through it as the first pipe. Therefore, the effect of rough walls is larger in turbulent flow.
(c) Defining the Effect of Roughness
How do we know if we have “rough flow?” You can calculate the value of Eu*/v, and compare it to these values in the table:
(d) Log Law Equations if Eu*/v > 70 (fully rough flow)
The Logarithmic Law of the Wall equation that was discussed earlier in this lesson changes, as well as the value for the variable B which that equation uses. These are now:
(e) Average Velocity if Eu*/v > 70 (fully rough flow)
(f) Friction Factor if Eu*/v > 70 (fully rough flow)
(g) Friction Factor if Eu*/v is in transitional roughness
To cover the transitional roughness section, the smooth wall and rough flow relations are combined to make an equation that works for turbulent friction.
(h) Moody Chart
The equation in part (g) was then plotted into a chart called the Moody Chart for fluid friction in pipe flow (one of the most useful charts in fluid mechanics)! The pink shaded squares on the left of the graph are the areas where laminar flow changes to turbulent. We will learn how to read this graph when we do some examples later. The Moody Chart is only applicable when we have pipes/ducts/etc of constant cross-section (eg. constant diameter)!
(i) Types of Problems
There are usually four types of problems you will run into. These are given below, where you can be given a different combination of variables each time. We will practice examples below with all of these different scenarios:
Now let’s do an example!
Example 2 – Finding Head Loss and Pressure Drop
(a) The first step of finding the loss of head (head loss) and the pressure drop is to find the Reynolds Number and the relative roughness. These are both variables on the x- and y-axis of the Moody Chart, and will help us in finding the answers we need (since both the head loss and pressure drop can be found on the Moody Chart, we can use it)!
(b) Here are the equations for the Reynolds Number and the relative roughness from the Moody Chart. I have substituted in the values already but you can check part (c) to see how I got these number – some of them were constants taken from textbook tables.
(c) There are some variables which can be found online or in tables in the textbook.
(d) We need to now find the friction factor, because this will help to find the head loss and pressure drop (it is used in those equations). We can use either the equation in part (g) above, or the Moody Chart to do this, since they are equivalent. I will use the Moody chart to follow the pink line (which I colored in purple) from the roughness value we found of 0.0008.
Keep going until you hit the Reynolds Number vertical line of 2*10^5, which is approximately what we found for it above. This horizontal line at the intersection of the roughness value and Reynolds Number is the friction factor, which I’ve shown in a blue line to be 0.02.
(e) Now we finish this by finding what we originally were asked to find – the head loss and pressure drop. The head loss equation is given below in the section “Head Loss – Friction Factor” and the pressure drop equation is what we used in Example 1 above. I will put them here again and we will calculate the final answer:
Head Loss – Friction Factor
Head loss is the loss in pressure head due to fluid viscosity, friction and other obstructions (eg. elbows, valves). Pressure head is the height of a liquid column that corresponds to a particular pressure, as shown in the image below. Note that both head loss and pressure head are measured in units of distance, usually meters.
There is an equation we can use to measure head loss in a pipe. There is also an equation to find the friction factor that is used in the head loss equation:
