fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...

, plug flow is a simple model of the velocity profile of a

fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...

flowing in a

pipe Pipe(s), PIPE(S) or piping may refer to: Objects * Pipe (fluid conveyance), a hollow cylinder following certain dimension rules ** Piping, the use of pipes in industry * Smoking pipe ** Tobacco pipe * Half-pipe and quarter pipe, semi-circular ...

. In plug flow, the velocity of the fluid is assumed to be constant across any cross-section of the pipe perpendicular to the axis of the pipe. The plug flow model assumes there is no

boundary layer In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condi ...

adjacent to the inner wall of the pipe. The plug flow model has many practical applications. One example is in the design of

chemical reactor A chemical reactor is an enclosed volume in which a chemical reaction takes place. In chemical engineering, it is generally understood to be a process vessel used to carry out a chemical reaction, which is one of the classic unit operations in chem ...

s. Essentially no back mixing is assumed with "plugs" of fluid passing through the reactor. This results in

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

s that need to be integrated to find the reactor conversion and outlet temperatures. Other simplifications used are perfect radial mixing and a homogeneous bed structure. An advantage of the plug flow model is that no part of the solution of the problem can be perpetuated "upstream". This allows one to calculate the exact solution to the differential equation knowing only the initial conditions. No further iteration is required. Each "plug" can be solved independently provided the previous plug's state is known. : The flow model in which the velocity profile consists of the fully developed

is known as

pipe flow In fluid mechanics, pipe flow is a type of liquid flow within a closed conduit, such as a pipe or tube. The other type of flow within a conduit is open channel flow. These two types of flow are similar in many ways, but differ in one important ...

. In laminar pipe flow, the velocity profile is parabolic.

Determination

For flows in pipes, if flow is turbulent then the

laminar sublayer The laminar sublayer, also called the viscous sublayer, is the region of a mainly-turbulent flow that is near a no-slip boundary and in which viscous shear stresses are important. As such, it is a type of boundary layer. The existence of the viscou ...

caused by the pipe wall is so thin that it is negligible. Plug flow will be achieved if the sublayer thickness is much less than the pipe diameter (

\delta_s

<<''D''). :

\delta_s = \frac

u^* = \left (\frac   \right )^

\tau_w = \frac

\frac   = \frac

= -2.0 \log_ \left(\frac +  \right)  , \text

where

f

is the

Darcy friction factor Darcy, Darci or Darcey may refer to: Science * Darcy's law, which describes the flow of a fluid through porous material * Darcy (unit), a unit of permeability of fluids in porous material * Darcy friction factor in the field of fluid mechanics * ...

(from the above equation or the

Moody Chart In engineering, the Moody chart or Moody diagram (also Stanton diagram) is a graph in non-dimensional form that relates the Darcy–Weisbach friction factor ''f'D'', Reynolds number Re, and surface roughness for fully developed flow in a cir ...

\delta_s

is the sublayer thickness,

D

is the pipe diameter,

\rho

is the

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...

u^*

is the

friction velocity Shear velocity, also called friction velocity, is a form by which a shear stress may be re-written in units of velocity. It is useful as a method in fluid mechanics to compare true velocities, such as the velocity of a flow in a stream, to a veloci ...

(not an actual velocity of the fluid),

V

is the average velocity of the plug (in the pipe),

\tau_w

is the shear on the wall, and

\Delta P

is the pressure loss down the length

L

of the pipe.

\epsilon

is the

relative roughness Relative may refer to: General use * Kinship and family, the principle binding the most basic social units society. If two people are connected by circumstances of birth, they are said to be ''relatives'' Philosophy *Relativism, the concept that ...

of the pipe. In this regime the pressure drop is a result of inertia-dominated turbulent shear stress rather than viscosity-dominated laminar shear stress.

Notes

{{reflist Fluid dynamics

Determination

See also

Notes