2 Layer Model

The 2 layer model as proposed by Wilson (1992) is based on a hydrostatic wall pressure approach and a bed shear stress based on the Darcy-Weisbach friction factor with the particle diameter as the roughness, multiplied by a factor 2 (Televantos (1979)). Wilson (1992) assumes that there are just 2 layers, the bed, either stationary or sliding, and water above the bed. Others have extended the model to a 3 layer model with sheet flow between the bed and the water. Here the 2 layer model graphs are shown.

Miedema & Ramsdell (2014) discuss the validity of the hydrostatic wall pressure approach and suggest two alternative approaches. The normall wall pressure carrying the bed weight approach and the weight approach. They suggest to use the weight approach.

Miedema & Matousek (2014) derived an explicit equation for the Darcy-Weisbach bed shear friction factor.

With 3 wall pressure approaches and 2 bed shear friction factor approaches, a total of 6 possibilities exists. For each of these possibilities a set of graphs is shown.



   01 Wilson (1992) Hydrostatic Stress Approach with the Televantos (1979) Bed Shear Stress Approach

The original Wilson (1992) model based on the hydrostatic pressure distribution, with a Televantos factor of 2 for the bed shear stress.

The graphs support the following papers:

  1. Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASME-OMAE 2014, San Francisco, USA, June 2014.
  2. Miedema, S.A., Ramsdell, R.C., “An Analysis of the Hydrostatic Approach of Wilson for the Friction of a Sliding Bed”. WEDA 34/TAMU 45, Toronto, Canada, June 2014.
  3. Miedema, S.A. & Matousek V., “An explicit formulation for the Darcy-Weisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 24-26 June 2014, Prague, Czech Republic
>> Read more

   02 Miedema & Ramsdell (2014) Normal Stress Approach with the Televantos (1979) Bed Shear Stress Approach

The Miedema (2014) model based on a normal stress distribution, with a Televantos factor of 2 for the bed shear stress. 

The graphs support the following papers:

  1. Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASME-OMAE 2014, San Francisco, USA, June 2014.
  2. Miedema, S.A., Ramsdell, R.C., “An Analysis of the Hydrostatic Approach of Wilson for the Friction of a Sliding Bed”. WEDA 34/TAMU 45, Toronto, Canada, June 2014.
  3. Miedema, S.A. & Matousek V., “An explicit formulation for the Darcy-Weisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 24-26 June 2014, Prague, Czech Republic.

>> Read more

   03 Miedema & Ramsdell (2014) Weight Approach with the Televantos (1979) Bed Shear Stress Approach

The Miedema (2014) model based on the weight approach, with a Televantos factor of 2 for the bed shear stress. 

The graphs support the following papers:

  1. Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASME-OMAE 2014, San Francisco, USA, June 2014.
  2. Miedema, S.A., Ramsdell, R.C., “An Analysis of the Hydrostatic Approach of Wilson for the Friction of a Sliding Bed”. WEDA 34/TAMU 45, Toronto, Canada, June 2014.
  3. Miedema, S.A. & Matousek V., “An explicit formulation for the Darcy-Weisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 24-26 June 2014, Prague, Czech Republic.

>> Read more

   04 Wilson (1992) Hydrostatic Stress Approach with the Miedema & Matousek (2014) Bed Shear Stress Approach

The original Wilson (1992) model based on the hydrostatic pressure distribution, with the Miedema & Matousek (2014) bed shear stress approach. 

The graphs support the following papers:

  1. Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASME-OMAE 2014, San Francisco, USA, June 2014.
  2. Miedema, S.A., Ramsdell, R.C., “An Analysis of the Hydrostatic Approach of Wilson for the Friction of a Sliding Bed”. WEDA 34/TAMU 45, Toronto, Canada, June 2014.
  3. Miedema, S.A. & Matousek V., “An explicit formulation for the Darcy-Weisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 24-26 June 2014, Prague, Czech Republic.

>> Read more

   05 Miedema & Ramsdell (2014) Normal Stress Approach with the Miedema & Matousek (2014) Bed Shear Stress Approach

The Miedema (2014) model based on a normal stress distribution, with the Miedema & Matousek (2014) bed shear stress approach. 

The graphs support the following papers:

  1. Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASME-OMAE 2014, San Francisco, USA, June 2014.
  2. Miedema, S.A., Ramsdell, R.C., “An Analysis of the Hydrostatic Approach of Wilson for the Friction of a Sliding Bed”. WEDA 34/TAMU 45, Toronto, Canada, June 2014.
  3. Miedema, S.A. & Matousek V., “An explicit formulation for the Darcy-Weisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 24-26 June 2014, Prague, Czech Republic.

>> Read more

   06 Miedema & Ramsdell (2014) Weight Approach with the Miedema & Matousek (2014) Bed Shear Stress Approach

The Miedema (2014) model based on the weight approach, with the Miedema & Matousek (2014) bed shear stress approach. 

The graphs support the following papers:

  1. Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASME-OMAE 2014, San Francisco, USA, June 2014.
  2. Miedema, S.A., Ramsdell, R.C., “An Analysis of the Hydrostatic Approach of Wilson for the Friction of a Sliding Bed”. WEDA 34/TAMU 45, Toronto, Canada, June 2014.
  3. Miedema, S.A. & Matousek V., “An explicit formulation for the Darcy-Weisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 24-26 June 2014, Prague, Czech Republic.

>> Read more

   07 Summary: The Hydraulic Gradient, Cvs

Summary: The Hydraulic Gradient, Cvs

The graphs support the following papers:

  1. Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASME-OMAE 2014, San Francisco, USA, June 2014.
  2. Miedema, S.A., Ramsdell, R.C., “An Analysis of the Hydrostatic Approach of Wilson for the Friction of a Sliding Bed”. WEDA 34/TAMU 45, Toronto, Canada, June 2014.
  3. Miedema, S.A. & Matousek V., “An explicit formulation for the Darcy-Weisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 24-26 June 2014, Prague, Czech Republic

Hydrostatic approach with Televantos bed friction factor

Hydrostatic approach with Miedema & Matousek bed friction factor

Wall pressure carrying the weight approach with Televantos bed friction factor

Wall pressure carrying the weight approach with Miedema & Matousek bed friction factor

Weight approach with Televantos bed friction factor

Weight approach with Miedema & Matousek bed friction factor
>> Read more

   08 Summary: The Relative Excess Hydraulic Gradient, Cvs

Summary: The Relative Excess Hydraulic Gradient, Cvs

The graphs support the following papers:

  1. Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASME-OMAE 2014, San Francisco, USA, June 2014.
  2. Miedema, S.A., Ramsdell, R.C., “An Analysis of the Hydrostatic Approach of Wilson for the Friction of a Sliding Bed”. WEDA 34/TAMU 45, Toronto, Canada, June 2014.
  3. Miedema, S.A. & Matousek V., “An explicit formulation for the Darcy-Weisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 24-26 June 2014, Prague, Czech Republic

Hydrostatic approach with Televantos bed friction factor

Hydrostatic approach with Miedema & Matousek bed friction factor

Wall pressure carrying the weight approach with Televantos bed friction factor

Wall pressure carrying the weight approach with Miedema & Matousek bed friction factor

Weight approach with Televantos bed friction factor

Weight approach with Miedema & Matousek bed friction factor
>> Read more

   09 Summary: The Hydraulic Gradient, Cvt

Summary: The Hydraulic Gradient, Cvt

The graphs support the following papers:

  1. Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASME-OMAE 2014, San Francisco, USA, June 2014.
  2. Miedema, S.A., Ramsdell, R.C., “An Analysis of the Hydrostatic Approach of Wilson for the Friction of a Sliding Bed”. WEDA 34/TAMU 45, Toronto, Canada, June 2014.
  3. Miedema, S.A. & Matousek V., “An explicit formulation for the Darcy-Weisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 24-26 June 2014, Prague, Czech Republic

Hydrostatic approach with Televantos bed friction factor

Hydrostatic approach with Miedema & Matousek bed friction factor

Wall pressure carrying the weight approach with Televantos bed friction factor

Wall pressure carrying the weight approach with Miedema & Matousek bed friction factor

Weight approach with Televantos bed friction factor

Weight approach with Miedema & Matousek bed friction factor
>> Read more

   10 Summary: The Relative Excess Hydraulic Gradient, Cvt

Summary: The Relative Excess Hydraulic Gradient, Cvt

The graphs support the following papers:

  1. Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASME-OMAE 2014, San Francisco, USA, June 2014.
  2. Miedema, S.A., Ramsdell, R.C., “An Analysis of the Hydrostatic Approach of Wilson for the Friction of a Sliding Bed”. WEDA 34/TAMU 45, Toronto, Canada, June 2014.
  3. Miedema, S.A. & Matousek V., “An explicit formulation for the Darcy-Weisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 24-26 June 2014, Prague, Czech Republic

Hydrostatic approach with Televantos bed friction factor

Hydrostatic approach with Miedema & Matousek bed friction factor

Wall pressure carrying the weight approach with Televantos bed friction factor

Wall pressure carrying the weight approach with Miedema & Matousek bed friction factor

Weight approach with Televantos bed friction factor

Weight approach with Miedema & Matousek bed friction factor
>> Read more

   11 Summary: The Normalised ExcessHydraulic Gradient, Cvt

Summary: The Normalised Excess Hydraulic Gradient, Cvt

The graphs support the following papers:

  1. Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASME-OMAE 2014, San Francisco, USA, June 2014.
  2. Miedema, S.A., Ramsdell, R.C., “An Analysis of the Hydrostatic Approach of Wilson for the Friction of a Sliding Bed”. WEDA 34/TAMU 45, Toronto, Canada, June 2014.
  3. Miedema, S.A. & Matousek V., “An explicit formulation for the Darcy-Weisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 24-26 June 2014, Prague, Czech Republic

Hydrostatic approach with Televantos bed friction factor

Hydrostatic approach with Miedema & Matousek bed friction factor

Wall pressure carrying the weight approach with Televantos bed friction factor

Wall pressure carrying the weight approach with Miedema & Matousek bed friction factor

Weight approach with Televantos bed friction factor

Weight approach with Miedema & Matousek bed friction factor
>> Read more

   12 The Wilson (1992) & Wilson (1997) Fit Functions

The original Wilson (1992) & Wilson (1997) fit functions, reconstructed.


>> Read more



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