
 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 DarcyWeisbach 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 DarcyWeisbach 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:

Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASMEOMAE 2014, San Francisco, USA, June 2014.

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.

Miedema, S.A. & Matousek V., “An explicit formulation for the DarcyWeisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 2426 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:

Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASMEOMAE 2014, San Francisco, USA, June 2014.

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.

Miedema, S.A. & Matousek V., “An explicit formulation for the DarcyWeisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 2426 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:

Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASMEOMAE 2014, San Francisco, USA, June 2014.

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.

Miedema, S.A. & Matousek V., “An explicit formulation for the DarcyWeisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 2426 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:

Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASMEOMAE 2014, San Francisco, USA, June 2014.

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.

Miedema, S.A. & Matousek V., “An explicit formulation for the DarcyWeisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 2426 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:

Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASMEOMAE 2014, San Francisco, USA, June 2014.

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.

Miedema, S.A. & Matousek V., “An explicit formulation for the DarcyWeisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 2426 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:

Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASMEOMAE 2014, San Francisco, USA, June 2014.

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.

Miedema, S.A. & Matousek V., “An explicit formulation for the DarcyWeisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 2426 June 2014, Prague, Czech Republic.
>> Read more
07 Summary: The Hydraulic Gradient, Cvs 
Summary: The Hydraulic Gradient, Cvs
The graphs support the following papers:

Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASMEOMAE 2014, San Francisco, USA, June 2014.

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.

Miedema, S.A. & Matousek V., “An explicit formulation for the DarcyWeisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 2426 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:

Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASMEOMAE 2014, San Francisco, USA, June 2014.

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.

Miedema, S.A. & Matousek V., “An explicit formulation for the DarcyWeisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 2426 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:

Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASMEOMAE 2014, San Francisco, USA, June 2014.

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.

Miedema, S.A. & Matousek V., “An explicit formulation for the DarcyWeisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 2426 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:

Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASMEOMAE 2014, San Francisco, USA, June 2014.

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.

Miedema, S.A. & Matousek V., “An explicit formulation for the DarcyWeisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 2426 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:

Miedema, S.A., “An Analysis of Slurry Transport at Low Line Speeds”. ASMEOMAE 2014, San Francisco, USA, June 2014.

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.

Miedema, S.A. & Matousek V., “An explicit formulation for the DarcyWeisbach friction factor of sheet flow”. 15th International Freight Pipeline Society Symposium 2014. 2426 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

