To calculate the friction loss of a Bingham fluid in a pipe, you can use the Bingham plastic model, which is an empirical equation that describes the relationship between shear stress and shear rate for non-Newtonian fluids with a yield stress. The Bingham plastic model has the form:

τ = τ_0 + k * γ^n

Where:

τ = shear stress (Pa)

γ = shear rate (s^-1)

k = flow behavior index (Pa.s^n)

n = flow behavior index exponent (unitless)

τ_0 = yield stress (Pa)

The friction factor (f) can be calculated using the following equation:

f = (2 * g * D * (τ_0 + k*γ^n))/(V^2 * L)

Where:

f = friction factor

g = acceleration due to gravity

D = inside diameter of pipe

L = length of pipe

V = velocity of flow

To calculate the friction loss, you will need to know the density and viscosity of the fluid, the inside diameter of the pipe, and the velocity of the fluid flow. You can then use the Darcy-Weisbach equation, which states that the friction loss (head loss) in a pipe is equal to the friction factor (f) times the length of the pipe (L) times the velocity of the flow (V) squared, divided by the inside diameter of the pipe (D) and a constant (2g).

The Darcy-Weisbach equation for the friction loss in a pipe is given by:

ΔP = f * L * V^2 / (2 * g * D)

Where:

ΔP = Friction loss (pressure drop)

f = friction factor

L = Length of pipe

V = Velocity of flow

g = acceleration due to gravity

D = inside diameter of pipe

The friction factor for Bingham fluids can be calculated using the model proposed by Herschel-Bulkley.

It's important to note that the Bingham plastic model only describes the fluid behaviour in the yield and plastic region, for the turbulent region you'll need to use other correlation such as the Colebrook-White equation. Additionally, it's important to measure or estimate the properties of the Bingham fluid, such as yield stress and viscosity, at the time of the experiment or design, since these properties can change with temperature and pressure.

It's also worth noting that the friction factor is highly dependent on the properties of the fluid and the measurement conditions, such as temperature and pressure, and the measurement equipment have to be considered as well to get accurate results.

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