Durand, Condolios, Cave, Thomas, Wilson, Fuhrboter, Newitt. If you have any experience working with slurries or if you are just researching this topic, you probably have already come across these names. This is no coincidence, since for the last 70 years, these authors (among many others) have been at the forefront of research, investigation, and development of models to determine everything related to hydraulic slurry transportation.
From obtaining the minimum velocity a slurry needs to be pumped at, to hydraulic gradients and many things in between, it is very likely that one of those authors has developed a model for what you are trying to find.
It is undeniable that having as many researchers as possible investigating these topics is good for everyone. However, with so many authors publishing their own models and equations to, in many times, calculate the same variables, a big question is raised:
To answer this question, two alternatives will be discussed in this article:
This is not an easy question to answer since there are many models around, each with its own set of assumptions and limitations. To simplify the answer a bit, we are going to limit the compared models in this article to some of the most widely used:
But first: Are these models always applicable?
The answer is no. Slurry correlation models are in most cases only applicable for settling slurries. Slurries with big amounts of diluted particles under 50 to 100 μm are considered non settling, where the solids are homogeneously mixed with the carrier liquid and cause it to exhibit non-Newtonian properties.
To determine the characteristics of these slurries a rheology study must be carried out, and you can contact us for more information.
Durand and Condolios ran their initial tests in 1952 with a d50 between 0.18 and 22.5 mm in pipes with diameters ranging from 40 to 580 mm. The tests were run with concentrations of up to 22%.
One important factor to keep in mind is that for their tests, Durand and Condolios used slurries with a closely graded particle sizing, meaning that d80/d20 < 2.
Subsequent tests have proven that Durand and Condolios model provides conservative (therefore less accurate) results when the particles are not closely graded, or when the slurries contain significant proportions of particles finer than 100 μm.
Durand, Cave & Wilson 2LM Calculator
To account for the limitations (or inaccuracies) of Durand’s model when the pumped slurries do not have closely graded particle sizing, Cave took Durand’s findings and created a modified set of curves to provide more accurate results in those cases.
Therefore, in when slurries have a good spread of particle sizes (d80/d20 > 5), or when concentrations exceed the 15% Cv that Durand’s curves present, Cave’s set of modified curves might provide results more in line with reality.
Durand, Cave & Wilson 2LM Calculator
In 1979, Wilson first introduced the Two-Layer model. Compared to the previously discussed models, the Two-Layer model considers that the stratified slurry flow consists of two distinct layers. A bottom layer that rests on the bottom of the pipe, where coarser solids are packed together, and a top layer where the finer solids are mixed with the liquid and are considered to be floating in it. Each layer has its own flow velocity and volumetric concentration, but it is considered that there is no slip between both phases.
It is very important to note that the limit velocity obtained from Wilson’s model is defined differently from the LDV that Durand’s model provides, and you can read more about this in the LDV vs LDVS section below.
The development of this model was made in pipes of 200 and 440 mm, with medium to coarse sands, and concentrations up to 16%.
We provide a calculator for Wilson’s original Two-Layer model in the following link:
Durand, Cave & Wilson 2LM Calculator
In 2001, Wilson and Sellgren created a new 4-component model. The model consists of dividing the slurry into four basic components, each based on their particle size. The friction losses are calculated for each component using the most appropriate model for their corresponding particle sizes, and then a final friction loss is obtained by calculating a weighted average. This model is a bit more complex than the other ones, so you can find more information about it in our article:
Wilson's 4 Component Model Article
And you can also try out our calculator for this model:
The limit velocity provided by the Two-Layer model is actually the Limit of Stationary Deposit Velocity, or LSDV for short. The LSDV represents the velocity at which the bottom solids layer (or bed) starts to move, i.e., stops being stationary and starts sliding.
The limit velocity provided by Durand’s model on the other hand, is the Limit Deposit Velocity, or LDV for short. The LDV represents the velocity at which no solid particles exist at the bottom of the pipe, independent of if they are sliding or stationary.
For this reason, usually the LSDV obtained by Wilson’s model is lower than the LDV, and it is very important to know the difference when comparing these models.
There is no definitive answer to which model is better than the others. Many successful pipelines have been designed and built using either model, and it’s up to the slurry engineer to choose wisely based on their particular set of starting variables.
The person in charge of the pipeline design must see each model as a different tool, each with its own set of strengths and weaknesses, and choose accordingly. For example, if Durand’s experiments were made with volumetric concentrations up to 22%, it would be unwise to try to apply that model to slurries of higher concentration.
One would think that the newer the model, the more “accurate” it would be, especially with researchers such as Wilson who have been studying the field and iterating over their findings for more than 30 years now. However, while this might certainly be the case, when it comes to real work most companies still use modified Durand or Wilson (1979) models. The reason for this is that these companies with lots of experience run their own experiments with the exact slurries they are going to be pumping and then they modify some values in the existing models to make them fit their data.
In conclusion, if you need a good estimation for a preliminary design, try to choose a model that was developed with a slurry similar to the one you expect to be pumping. If you already have a working pipeline, the best idea might be to take some measurements of your own and start playing around with all models to see which one correlates the best to your data.
As we’ve discussed, there are countless models out there that have been developed over the years, so it is understandable that even the most experienced might doubt which one fits their needs the best.
With this in mind, authors Miedema & Ramsdell developed the Delft Head Loss & Limit Deposit Velocity Framework (DHLLDV Framework for short). This framework was created with the aim of providing a comprehensive and consistent model for slurry transport in pipelines. The framework covers different flow regimes and particle sizes and is based on numerous experimental data from literature.
The DHLLDV framework has been tweaked and adjusted based on the data obtained from real experiments that the authors from the original models (Durand, Wilson, Fuhrboter, SRC, Shook, etc.) made available in their publications. This was done with the objective of getting the best agreement possible with all the experiments from the 30+ existing models considered.
What this means is that the DHLLDV framework will tend to agree with the real data obtained from experiments:
For example: Durand’s model provides good results for small to medium sized pipes. If one tried to apply this model to a 1 meter diameter pipe, the results would probably have a big deviation from reality. In a case like this, a more suitable model might be Silin’s et al. since the authors ran experiments on pipes up to 0.9 meters in internal diameter.
Now, if one were to use the DHLLDV framework to calculate the LDV for a 0.2 meter pipe and then for a 1 meter pipe, the first result would be in good agreement with Durand’s model while the second would be in good agreement with Silin’s et al. model. That is the main advantage of this framework.
More information about this framework can be found in the open-access book published by the authors:
The Delft Head Loss Limit Deposit Velocity Framework - Open Access Book
You can also check out our LDV calculator based on the DHLLDV framework:
And finally, the authors have provided open-source spreadsheets and Python programs for everyone to try out the framework:
Miedema, Ramsdell (2016). Slurry Transport Fundamentals, A Historical Overview & The Delft Head Loss & Limit Deposit Velocity Framework (3rd edition).
Baha E. Abulnaga, P.E. (2002). Slurry Systems Handbook.
B.E.A. Jacobs (1991). Design of Slurry Transport Systems.
Warman (2009). Slurry Pump Handbook Fifth Edition.
Weir (2002). Slurry Pumping Handbook First Edition.
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