Adiabatic Frictional Flow

THE FRICTION PARAMETER. The basic quantity that measures the elfect of friction is the friction parameter /L/r . This arises from the integration of Eq. 

Friction factors in supersonic flow are not well established. Apparently, they are approximately one-half those in subsonic flow for the same Reynolds number,  

In all the integrated equations in the next section it is assumed that the entrance to the conduit is rounded to form an isentropic convergent nozzle. If supersonic flow in the conduit is required, the entrance nozzle must include a divergent section to generate a Mach number greater than 1, 

EQUATIONS FOR ADIABATIC FRICTIONAL FLOW. Equation (6.8) is multiplied by p/p, giving 

It is desired to obtain an integrated form of this equation. The most useful integrated form is one containing the Mach number as the dependent variable and the friction parameter as an independent variable. To accomplish this, the density factor is eliminated from Eq. (6.34) using Eq. (6.18), and relationships between JV a, rfp/p, and duju are found from Eqs. (6.2) and (6.11). Quantity dT/T, when it appears, is eliminated by using Eqs. (6.4), (6.13), and (6.18), The results are 

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Adiabatic frictional flow through a conduit of constant cross section. This process is irreversible, and the entropy of the gas increases, but as shown by Eq. (6.22), since Q 0, the stagnation temperature is constant throughout the conduit. This process is shown in Fig. 6.1b. 

In adiabatic frictional flow, the temperature of the gas changes. The viscosity also varies, and the Reynolds number and friction factor are not actually constant. In gas flow, however, the effect of temperature on viscosity is small, and the effect of Reynolds number on the friction factor / is still less. Also, unless the Mach number is nearly unity, the temperature change is small. It is satisfactory to use an average value for /as a constant in calculations. If necessary,/ can be evaluated at the two ends of the conduit and an arithmetic average used as a constant. 

In adiabatic frictional flow the ratio of the inlet and outlet pressures is found by direct integration of Eq. (6.35) between the limits p , p, and iVMa,o> to give... 

Because the gas viscosity is not highly sensitive to pressure, for isothermal flow the Reynolds number and hence the friction factor will be very nearly constant along the pipe. For adiabatic flow, the viscosity may change as the temperature changes, but these changes are usually small. Equation (9-15) is valid for any prescribed conditions, and we will apply it to an ideal gas in both isothermal and adiabatic (isentropic) flow. 

Figure 4-11 Adiabatic nonchoked flow of gas through a pipe. The gas temperature might increase or decrease, depending on the magnitude of the frictional losses.

Hot spot formation mechanisms have been studied in some detail by Field and his co-workers at Cambridge University (15-19), following on from the pioneering work of Bowden and Yoffe (13). Among the mechanisms proposed have been (i) fracture of explosive crystals (20,21), (ii) plastic flow with dislocation pile-up (22) and adiabatic shear flow (23-26), (iii) friction, (27,28), (iv) adiabatic compression of gas pockets (27) and (v) shock-void interactions (29). Chaudhri has very recently examined these possibilities (30) and concludes that intercrystalline friction and the interaction of strong shocks with voids producing fast jets are dominant mechanisms. Plastic flow can also be effective when the strain is high. 

Adiabatic Flow with Friction in a Duct of Constant... 

Adiabatic Frictionless Nozzle Flow In process plant pipelines, compressible flows are usually more nearly adiabatic than isothermal. Solutions for adiabatic flows through frictionless nozzles and in channels with constant cross section and constant friction factor are readily available. 

These equations are consistent with the isentropic relations for a perfect gas p/po = (p/po), T/To = p/poY. Equation (6-116) is valid for adiabatic flows with or without friction it does not require isentropic flow However, Eqs. (6-115) and (6-117) do require isentropic flow The exit Mach number Mi may not exceed unity. At Mi = 1, the flow is said to be choked, sonic, or critical. When the flow is choked, the pressure at the exit is greater than the pressure of the surroundings into which the gas flow discharges. The pressure drops from the exit pressure to the pressure of the surroundings in a series of shocks which are highly nonisentropic. Sonic flow conditions are denoted by sonic exit conditions are found by substituting Mi = Mf = 1 into Eqs. (6-115) to (6-118). 

The equations for nozzle flow, Eqs. (6-114) through (6-118), remain valid for the nozzle section even in the presence of the discharge pipe. Equations (6-116) and (6-120), for the temperature variation, may also be used for the pipe, with Mo, po replacing Mi, pi since they are valid for adiabatic flow, with or without friction. 

To calculate the heat duty it must be remembered that the pressure drop through the choke is instantaneous. That is, no heat is absorbed or lost, but there is a temperature change. This is an adiabatic expansion of the gas w ith no change in enthalpy. Flow through the coils is a constant pressure process, except for the small amount of pressure drop due to friction. Thus, the change in enthalpy of the gas is equal to the heat absorbed. 

Compressible fluid flow occurs between the two extremes of isothermal and adiabatic conditions. For adiabatic flow the temperature decreases (normally) for decreases in pressure, and the condition is represented by p V (k) = constant. Adiabatic flow is often assumed in short and well-insulated pipe, supporting the assumption that no heat is transferred to or from the pipe contents, except for the small heat generated by friction during flow. Isothermal p Va = constant temperature, and is the mechanism usually (not always) assumed for most process piping design. This is in reality close to actual conditions for many process and utility service applications. 

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. 

Fig. 3.7a,b Friction coefficients as function of solvent Reynolds number Rewat in (n) adiabatic flow, and (b) diabatic flow. Reprinted from Hetsroni et al. (2004) with permission... 

Annular flow. In annular flow, as mentioned in Section 3.4.6.1, modeling of the interfacial shear remains empirical. For adiabatic two-phase flow, Asali et al. (1985) suggested that the friction factor, fjfs, is dependent on a dimensionless group for the film thickness, 8+, as defined in Eq. (3-136), and the gas Reynolds number, Rec ... 

The adiabatic condition occurs, for example, when the residence time of the fluid is short as for flow through a short pipe, valve, orifice, etc. and/or for well-insulated boundaries. When friction loss is small, the system can also be described as locally isentropic. It can readily be shown that an ideal gas under isentropic conditions obeys the relationship... 

In the case of adiabatic flow we use Eqs. (9-1) and (9-3) to eliminate density and temperature from Eq. (9-15). This can be called the locally isentropic approach, because the friction loss is still included in the energy balance. Actual flow conditions are often somewhere between isothermal and adiabatic, in which case the flow behavior can be described by the isentropic equations, with the isentropic constant k replaced by a polytropic constant (or isentropic exponent ) y, where 1 < y < k, as is done for compressors. (The isothermal condition corresponds to y= 1, whereas truly isentropic flow corresponds to y = k.) This same approach can be used for some non-ideal gases by using a variable isentropic exponent for k (e.g., for steam, see Fig. C-l). 

Putting k = y gives an approximate equation for adiabatic flow. The result is only approximate because it implies an isentropic change, ie a reversible adiabatic change, but this is not the case owing to friction. A rigorous solution for adiabatic flow is given in Section 6.5. 

Equation 6.19 is the basic equation relating the pressure drop to the flow rate. The difficulty that arises in the case of adiabatic flow is that the equation of state is unknown. The relationship, PVy = constant, is valid for a reversible adiabatic change but flow with friction is irreversible. Thus a difficulty arises in determining the integral in equation 6.19 an alternative method of finding an expression for dPIV is sought. 

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. 

The differential energy balances of Eqs. (6.10) and (6.15) with the friction term of Eq. (6.18) can be integrated for compressible fluid flow under certain restrictions. Three cases of particular importance are of isentropic or isothermal or adiabatic flows. Equations will be developed for them for ideal gases, and the procedure for nonidcal gases also will be indicated. 

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