Shell-MIT Equation

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Shell-MIT Equation

The Shell-MIT equation, also known as the MIT equation, was initially used by the Shell pipeline company for modeling the flow of high viscosity heated crude oil pipelines. This equation for pressure drop uses a modified Reynolds number R_m, which is a multiple of the normal Reynolds number. From R_m a friction factor is calculated depending on whether the flow is laminar or turbulent. The calculation method is as follows. The Reynolds number of flow is first calculated from

Equation 1
R = 92.24 Q / ( D v)

From the preceding, a modified Reynolds number is defined as

Equation 2
R_m = R / 7742

where R= Reynolds number, dimensionless
R_m = modified Reynolds number, dimensionless
Q = flow rate, bbl/day
D = pipe inside diameter, in
ν = liquid kinematic viscosity, cSt

Next, a friction factor is calculated from one of the following equations:

Equation 3
F = 0.00207 / R_m -> for laminar flow

Equation 4
0.0018 + 0.00662 ( 1 / R_m )^0.355 -> for turbulent flow

The laminar flow limit is the same as before: Reynolds number R < 2100 approximately.

The friction factor f in Eqs. (3) and (4) is not the Darcy friction factor we have used so far with the Colebrook equation. Therefore we cannot directly use it in the Darcy equation to calculate the pressure drop.

The pressure drop due to friction with the Shell-MIT equation is then calculated as follows:

Equation 5
P_m = 0.241 ( f Sg Q² ) / D⁶

where

P_m = pressure drop due to friction, psi/mi
f = Shell-MIT equation friction factor, dimensionless
Sg = liquid specific gravity
Q = liquid flow rate, bbl/day
D = pipe inside diameter, in

With flow rate in bbl/h, the pressure drop due to friction is calculated using the following modified version of the Darcy equation:

Equation 6
P_m = 138.82 ( f Sg Q² ) / D⁵

where

P_m = pressure drop due to friction, psi/mi
f = Shell-MIT equation friction factor, dimensionless
Sg = liquid specific gravity
Q = liquid flow rate, bbl/h
D = pipe inside diameter, in

In SI units the MIT equation is expressed as follows:

P_m = ( 6.2191 x 10¹⁰ ) ( f Sg Q² ) / D⁵

where

P_m = frictional pressure drop, kPa/km
f = Shell-MIT equation friction factor, dimensionless
Sg = liquid specific gravity
Q = liquid flow rate, m³/h
D = pipe inside diameter, mm

Reference:

Piping Calculations Manual,
E. Shashi Menon
SYSTEK Technologies, Inc