Adiabatic flow in a pipe

In Equation 2.61, it is clear that with increase in velocity (low-pressure downstream side) the specific enthalpy will reduce, and with decrease in velocity, the specific enthalpy will increase. The specific enthalpy will be a maximum when the velocity is zero, and this maximiun value is termed as stagnation enthalpy, h . The Equation 2.61 can further be modified as 

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Integrating Equation 2.63 and replacing the velocity with the Mach number. 

The limiting conditions xmder which an initially subsonic flow is choked are given by putting M2 = 1. Then, 

The general equation for a compressible fluid in a circular pipe is 

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The conditions existing during the adiabatic flow in a pipe may be calculated using the approximate expression Pi/ = a constant to give the relation between the pressure and the specific volume of the fluid. In general, however, the value of the index k may not be known for an irreversible adiabatic process. An alternative approach to the problem is therefore desirable.(2,3)... 

FIG, 6"22 Adiabatic compressible flow in a pipe with a well-rounded entrance. 

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. 

In an adiabatic process, Sq = 0, and the equation may then be written for the flow in a pipe of constant cross-sectional area A to give ... 

Integrating, a relation between P and v for adiabatic flow in a horizontal pipe is obtained ... 

The micro-channels utilized in engineering systems are frequently connected with inlet and outlet manifolds. In this case the thermal boundary condition at the inlet and outlet of the tube is not adiabatic. Heat transfer in a micro-tube under these conditions was studied by Hetsroni et al. (2004). They measured heat transfer to water flowing in a pipe of inner diameter 1.07 mm, outer diameter 1.5 mm, and 0.600 m in length, as shown in Fig. 4.2b. The pipe was divided into two sections. The development section of Lj = 0.245 m was used to obtain fully developed flow and thermal fields. The test section proper, of heating length Lh = 0.335 m, was used for collecting the experimental data. 

For adiabatic flow in a horizontal pipe with no shaft work, equation 6.7 reduces to... 

The preceding is a special case, but for an adiabatic flow with friction the relation that pvk is a constant does not apply, and instead the relation between p and v is more complicated. As an illustration, consider flow in a pipe of area A at the rate of W lb/s, and assume that at some point the values p 1, tq, and T1 are known, while at some other point p2, v2, and T2 are replaced by any values p, v, and T. As V= Wv/A, it follows from Eq. (10.16) that ... 

A topic within the purview of thermodynamics is the maximum velocity attainable in pipe flow. Consider a gas in steady-state adiabatic flow in a horizontal pipe of constant cross-sectional area. Equation (7.10) is the applicable energy balance, and it here becomes ... 

EXAMPLE 2.6 Calculation of Final Temperature for a Transient Process Steam at 10 MPa, 450°C is flowing in a pipe, as shown in Figure E2.6A. Connected to this pipe through a valve is an evacuated tank. The valve is opened and the tank fills with steam until the pressure is 10 MPa, and then the valve is closed. The process takes place adiabatically. (a) Determine the final temperature of the steam in the tank. (b) Explain why the final temperature in the tank is not the same as that of the steam flowing in the pipe. 

You wish to measure the temperature and pressure of steam flowing in a pipe. To do this task, you connect a well-insulated tank of volume 0.4 m to this pipe through a valve. This tank initially is at vacuum. The valve is opened, and the tank flUs with steam until the pressure is 9 MPa. At this point the pressure of the pipe and tank are equal, and no more steam flows through the valve. The valve is then closed. The temperature right after the valve is closed is measured to be 800°C. The process takes place adiabatically. Determine the temperature (in [K]) of the steam flowing in the pipe. You may assume the steam in the pipe stays at the same temperature and pressure throughout this process. 

It will now be shown from purely thermodynamic considerations that for, adiabatic conditions, supersonic flow cannot develop in a pipe of constant cross-sectional area because the fluid is in a condition of maximum entropy when flowing at the sonic velocity. The condition of the gas at any point in the pipe where the pressure is P is given by the equations ... 

For adiabatic, steady-state, and developed gas-liquid two-phase flow in a smooth pipe, assuming immiscible and incompressible phases, the essential variables are pu, pG, Pl, Pg, cr, dh, g, 9, Uls, and Uas, where subscripts L and G represent liquid and gas (or vapor), respectively, p is the density, p is the viscosity, cr is the surface tension, dh is the channel hydraulic diameter, 9 is the channel angle of inclination with respect to the gravity force, or the contact angle, g is the acceleration due to gravity, and Uls and Ugs are the liquid and gas superficial velocities, respectively. The independent dimensionless parameters can be chosen as Ap/pu (where Ap = Pl-Pg), and... 

The boundary conditions (10.12-10.14) correspond to the flow in a micro-channel with a cooled inlet and adiabatic receiver (an adiabatic pipe or tank, which is established at the exit of the micro-channel). Note, that the boundary conditions of the problem can be formulated by another way, if the cooling system has another construction, for example, as follows x = 0, Tl = IL.o, x = L, Tg = Tg.oo, when the inlet and outlet are cooled x = 0, dT /dx = 0, x = L, Tg = Tg.oo in case of the adiabatic inlet and the cooled outlet, etc. 

Figure 9-2 Expansion factor for adiabatic flow in piping systems, (a) k = 1.3 (b) k- 1.4. (From Crane Co., 1991.)...

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... 

An ideal gas flows in steady state adiabatic flow along a horizontal pipe of inside diameter d, = 0.02 m. The pressure and density at a point are P = 20000 Pa and p = 200 kg/m3 respectively. The density drops from 200 kg/m3 to 100 kg/m3 in a 5 m length. Calculate the mass flux assuming that the Fanning friction factor /= 9.0 x 10 3 and the ratio of heat capacities at constant pressure and constant volume y = 1.40. 

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. 

These data are plotted in Fig. 4a. It is shown in Section 4.5.4, Volume 1, that the maximum velocity which can occur in a pipe under adiabatic flow conditions is the sonic velocity which is equal to V/ 2 2-... 

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