Pipe flow with friction

hanson

Hi all!
Let's say we are finding the friction loss for a pipe. And the reynolds number is low enough to use Blasius formula for the coefficient of pipe friction.

What if I introduce some kind of fouling inside the pipe? Would the actual friction loss greater or smaller than the estimation given by the blasius formula?

My thought is that, the actual diameter of the pipe would be smaller than the value used in the prediction if the pipe has fouling. Then, the estimated value should be smaller than the actual one, what do you think?

hanson

I am using the Darcy-Weisbach equation to find the head loss and the friction factor is found by Blasius formula.
http://en.wikipedia.org/wiki/Darcy-Weisbach_equation

p.s. I don't know how to type mathematical symbols here, anyone can tell me how?

Astronuc

for writing mathematical symbols and equations, see -

Introducing LaTeX Math Typesetting
http://www.physicsforums.com/showthread.php?t=8997

Astronuc

Fouling would have two effects - possibly changing the effective roughness, but also changing the hydraulic diameter, as you correctly stated.

hanson

thx Astronuc!
So, is the consequence of fouling producing a greater head loss than the estimated value?

Astronuc

Well if one's estimation or model of the effect of fouling is reasonable accurate, then one's estimate of head loss should be reasonably accurate. If one assumes a particular roughness and pipe geometry, yet the fouling produces a different (greater) roughness and reduced internal clearance, then one's calculation would underestimate the actual head loss.

f = 0.316/(Re^1/4) (2000< Re< 100000) which is a broad range

and for Re > 100,000 one uses the Karman-Prandtl relationship.

The use of the Blasius equation assumes very low roughness - probably about 0.00001 - for Re < 100000. One should refer to the Moody diagram.

Does the problem of interest involve gas, liquid or both?

hanson

Water is flowing in the pipe actually.
The problem is that the predicted values using the friction coefficient obtained by Blasius equation f = 0.316/(Re1/4) actually overestimated the real head loss obtained. And I am trying to find an explanantion for this.

Initially I thought of fouling in the pipe, but as you've mentioned, that would produce even larger head loss. So what kind of argument could be used to explain the difference between the theoretical estimation and the actual results? (Instead of fouling, we have corrosion of pipe? so that the actual pipe diameter is greater than that used in the estimation? That seems quite a childish argument, right?)

Q_Goest

Hi hanson, Could you give us more information about your set up? For the pipe in question, it sounds as if you're calculating a pressure drop that is more than what you're measuring. Can you give us the ID, length, elevation changes and any other restrictions to flow? If the system is too complex, it might be easier to draw a picture and post it. Also need the water flow rate, inlet and outlet pressure and temperature you are measuring. With that it would be easier to check your calculations and figure out where the discrepancy might be.

Astronuc

As Q_Goest indicated, it would be helpful to have more information, particularly pipe geometry (diameter and length), water chemistry (i.e. pure or solution), water temperature, flowrate or mean velocity, pipe fittings (45's, L's, etc.).

What was the magnitude by which the head loss was overestimated?

What is the Re for your problem?

FredGarvin

You got that right. Pipers in industry have their own methods for throwing in fudge factors for dealing with pipe fouling over time. That way their installations do not have to be perfectly clean to operate as designed. It is a tough thing to estimate and depends on a lot of factors.

hanson

hi all!
Sorry for this late reply!
Actually I was doing an experiment to check the head loss of various pipe components and the head loss due to friction etc.

And now I obtained the experimental results and are trying to interpret them.

The first thing is the head loss by friction along a pipe.
The experimental head loss are generally smaller than that predicted theoretically. I found this very strange since I expect different kinds of additional loss in the actual experiment. As mentioned, like the fouling, which cause extra head loss. But it turned out that the theory overestimate the actual loss, I just can't figure out the reaons (just one thing come to my mind, that's corrosion of pipe, hence reduction in diameter, but that's seems ridiculous)

Secondly, the head loss due to expansion and contraction of pipe diameter were also studied.
And it is found that the head loss due to expansion is much smaller that by contraction of cross-section of pipe. Does this make sense and how would you interpret it? One thing comes to my mind: For expansion, there will be turbulence occuring, am I right?

What do you guys think?

Clausius2

I think that's wrong. The head loss due to expansion must be much larger than that of contraction. Revise your results.

Whereas a contraction is almost an isentropic process, an expansion produces great irreversibility in the flow.

hanson

Yes..that's strange.

For contraction, the bigger pipe is of diameter 17mm and the smaller is 14.6mm.

For expansion, the smaller pipe is 17mm and the bigger is 28.6mm

According to theory, the coefficient of resistance should be
[tex]K_L = (1-(\frac{A_2}{A_1})^2)^2[/tex] for contraction
[tex]K_L = (1-(\frac{A_2}{A_1})^2)^2[/tex] for expansion

And I find them to be 0.06887 for contraction and 0.4182 for expansion.
So given the same velocity, the head loss due to expansions should be theoretically greater, right?

But the experimental results are as follow:
For contraction,
the pressure loss measured at the two cross sections are:
17mbar for a flow rate of 12 liter/min
25.8mbar for a flow rate of 15 liter/min
43.3 mbar for a flow rate of 20 liter/min

Since the velocity is not constant throughout the pipe due to the change of pipe diameter, to find the head loss due to contraction, i use
[tex]\frac{P_1}{\gamma}+\frac{V_1^2}{2g}=\frac{P_2}{\gamma}+\frac{V_2^2}{2g} +h_L[/tex]
[tex]h_L=\frac{P_1}{\gamma}+\frac{V_1^2}{2g}-\frac{P_2}{\gamma}-\frac{V_2^2}{2g}[/tex]

by finding V_1 and V_2, I get the head loss. Is my method of finding the head loss appropriate?

For expansion
the pressure loss measure at the two cross sections are:
1.1 mbar for a flow rate of 12 liter/min
0.3 mbar for a flow rate of 15 liter/min
-1 mbar for a flow rate of 20 liter/min

And using my calculation,
the measured head loss for contraction are:
For contraction
0.14m for 12liter/min
0.21m for 15liter/min
0.35m for 20liter/min

For expansion
0.046m for 12 liter/min
0.057m for 15liter/min
0.086m for 20liter/min

So the head losses by expansion are much smaller than those by contraction....

How come?

Clausius2

I am not going to recalculate your experimental data. Maybe your data is right, so be careful because head loss is inversely proportional to the diameter powered to five. Therefore, an small change in the diameter (pay attention because your expansion-contraction diameters are not equal) may be the solution to your puzzle.

As I said I am not going to check your experimental data, but the misconception you posted just above these lines is worth of a reply. Up to today, nobody has found a theoretical head loss coefficient for a contraction, so it seems strange to me you are so happy using your [tex]K_L[/tex] for a contraction. The approximate theoretical head loss coefficient of an expansion cannot be used in reverse. Revise how it is derived, and try to understand why you cannot derive it in the same way for a contraction (the key is just in the boundary layer growth at the throat).

Q_Goest

Hi Hanson. You asked:

Yes, the head loss you've calculated that way accounts for the irreversible frictional loss inside the restriction. Use Bernoulli's equation to separate out the various pressure components of the flow.

Regarding the equations you're using for head loss, the one for contraction is incorrect, and I'm not sure what restrictions are on the expansion, so I wouldn't use it there either. I'd suggest using Crane paper #410 which a terrific resource for these types of things. Unfortunately I'm on vacation and mine is at work so I won't be able to look it up till Monday.

Regardless, I can tell you it says that for a contraction, K can be as small as 0.04 for a well rounded nozzle (ie: contraction). The coefficient is dependant on the geometry of the restriction so simply using a single equation as you've given above is insufficient. It doesn't take into account the various different ways a restriction such as this can be created. For example, Crane gives the loss coefficient at 0.5 for a sharp edge entrance loss and 0.78 for an inward projecting pipe (I have these listed in a computer program). But for a well rounded entrance, the loss can be as small as 0.04.

The reason the well rounded restriction doesn't cause as much head loss as the sharp edge or inward projecting pipe for example can be explained from two phenomena. First, there is a "vena contracta" at (or immediately downstream of) the restriction. The restriction will cause the streamlines of the flow to contract depending on how the streamlines negotiate the restriction. For inward projecting pipes, the vena contracta is much smaller than the ID of the smaller pipe. In comparison, a well rounded reducer won't result in a significant vena contracta. The VC for a well rounded entrance will be roughly the same size as the small pipe ID.

The second reason is that the turbulence introduced by the restriction will result in irreversible losses. An inward projecting pipe creates significant turbulence as seen by the streamlines as the flow enters the smaller pipe. A well rounded restriction in comparison, allows the conversion of kinetic to potential energy within the streamline. In other words, as the flow in a well rounded restriction enters the smaller pipe, more of the potential energy (rho*g*h) in the flow is converted to kinetic energy (.5*rho*v2), thus less irreversible head loss. The more turbulence you have, the less conversion of energy there will be between these two, and the more irreversible pressure loss you will have.

hanson

um...can I know the fundamental reasons for the head loss in cross-section contraction and cross-section expansion?

For contraction, can I say it is because some fluid cannot turn the corner, producing a vena contracta, and the velocity is the greatest in the vena contracta. When the flow continues to flow out form the vena contracta, it slow down but not as efficienct as it was accelerated. Therefore, the kinetic energy could not be converted into pressure. So, there is head loss. Am I correct to put in this way?

For the expansion, the situation is even worse since this is entirely a deceleration process and the kinetic energy cannot be converted into pressure efficiently because of the viscous dissipation. Since the change in kinetic energy for expanasion is usually greater than the change in kinetic energy from the vena contracta to the original pipe, so the head loss due to expansion should be greater than that in contraction?

FredGarvin

You have a mistaken factor in there.
According to Clausius' most hated reference (Crane's). The first is obtained through the momentum equation with Bernoulli and the second is "the continuity equation and a close approximation of the contraction coefficients determined by Julius Weisbach." The K values are as follows:
For sudden enlargenments:
[tex]K_L = (1-\beta^2)^2[/tex]
For sudden contractions:
[tex]K_L = \frac{1}{2}(1-\beta^2)[/tex]