The 3 layer model is based on the 2 layer model but with sheet flow between the bed layer and the water layer. Here the 3 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. We believe that the Miedema & Ramsdell (2014) weight approach combined with the Miedema & Matousek (2014) bed shear stress approach is the closest to reality. The 2LM/3LM  Excel Workbook can be downloaded from the following link (this will be updated regularly):

 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 and including the Pugh & Wilson (1999) equations for sheet flow.

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.

 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 and including the Pugh & Wilson (1999) equations for sheet flow.

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.

 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 and including the Pugh & Wilson (1999) equations for sheet flow.

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.

 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 and including the Pugh & Wilson (1999) equations for sheet flow.

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.

 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 and including the Pugh & Wilson (1999) equations for sheet flow.

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.

 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 and including the Pugh & Wilson (1999) equations for sheet flow.

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.

 07 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

 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

 09 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

 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

 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

 12 Comparing 2LM & 3LM

Comparing 2LM & 3LM

The 2LM model includes the Miedema & Ramsdell (2014) model based on the weight approach, with the Miedema & Matousek (2014) bed shear stress approach. The 3LM model also includes the Pugh & Wilson (1999) equations for sheet flow.

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