Isothermal flow in a pipe

For isothermal flow in a pipe, the mass flux (G = till A) can be determined by integrating the differential form of the Bernoulli to give... 

This equation now expresses the macroscopic energy balance for isothermal flow in a pipe. It may further be simplified in the special case of an incompressible fluid. In this case Fis constant and the energy balance equation becomes... 

Compressibility of a gas flowing in a pipe can have significant effect on the relation between flowrate and the pressures at the two ends. Changes in fluid density can arise as a result of changes in either temperature or pressure, or in both, and the flow will be affected by the rate of heat transfer between the pipe and the surroundings. Two limiting cases of particular interest are for isothermal and adiabatic conditions. 

In considering the flow in a pipe, the differential form of the general energy balance equation 2.54 are used, and the friction term 8F will be written in terms of the energy dissipated per unit mass of fluid for flow through a length d/ of pipe. In the first instance, isothermal flow of an ideal gas is considered and the flowrate is expressed as a function of upstream and downstream pressures. Non-isothermal and adiabatic flow are discussed later. 

The velocity profile for isothermal, laminar, non-Newtonian flow in a pipe can sometimes be approximated as... 

Non-Newtonian Flow For isothermal laminar flow of time-independent non-Newtonian hquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate-pressure drop relations. For the Bingham plastic flmd described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length AP/L, the flow rate is given by... 

Isothermal flow of gas in a pipe with friction is shown in Figure 4-15. For this case the gas velocity is assumed to be well below the sonic velocity of the gas. A pressure gradient across... 

Levenspiel13 showed that the maximum velocity possible during the isothermal flow of gas in a pipe is not the sonic velocity, as in the adiabatic case. In terms of the Mach number the maximum velocity is... 

In order to maintain isothermal flow it is necessary for heat to be transferred across the pipe wall. From equation 6.7, for flow in a section with no shaft work and negligible change in elevation, the energy equation takes the form... 

An ideal gas in which the pressure P is related to the volume V by the equation PV = 75 m2/s2 flows in steady isothermal flow along a horizontal pipe of inside diameter d, = 0.02 m. The pressure drops from 20000 Pa to 10000 Pa in a 5 m length. Calculate the mass flux assuming that the Fanning function factor/= 9.0 x 10 3... 

Steam at the rate of 7000 kg/hr with an inlet pressure of 23.2 barabs and temperature of 220°C flows in a line that is 77.7 mm dia and 305 m long. Viscosity is 28.5(10 6)N sec/m2 and specific heat ratio is k = 1.31. For the pipe, e/D = 0.0006. The pressure drop will be found in (a) isothermal flow (b) adiabatic flow. Also, (c) the line diameter for sonic flow will be found. 

Moving on to compressible flow, it is first of all necessary to explain the physics of flow through an ideal, frictionless nozzle. Chapter S shows how the behaviour of such a nozzle may be derived from the differential form of the equation for energy conservation under a variety of constraint conditions constant specific volume, isothermal, isentropic and polytropic. The conditions for sonic flow are introduced, and the various flow formulae are compared. Chapter 6 uses the results of the previous chapter in deriving the equations for frictionally resisted, steady-state, compressible flow through a pipe under adiabatic conditions, physically the most likely case on... 

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