2.11. COMPRESSIBLE FLOW OF GASES

2.11A. Introduction and Basic Equation for Flow in Pipes

When pressure changes in gases occur which are greater than about 10%, the friction-loss equations (2.10-9) and (2.10-10) may be in error since compressible flow is occurring. Then the solution of the energy balance is more complicated because of the variation of the density or specific volume with changes in pressure. The field of compressible flow is very large and covers a very wide range of variation in geometry, pressure, velocity, and temperature. In this section we restrict our discussion to isothermal and adiabatic flow in uniform, straight pipes and do not cover flow in nozzles, which is discussed in some detail in other references (M2, P1).

The general mechanical-energy-balance equation (2.7-27) can be used as a starting point. Assuming turbulent flow, so that α = 1.0, and no shaft work, so that W_S = 0, and writing the equation for a differential length dL, Eq. (2.7-27) becomes

Equation 2.11-1

For a horizontal duct, dz = 0. Using only the wall shear frictional term for dF and writing Eq. (2.10-6) in differential form,

Equation 2.11-2

where V = 1/ρ. Assuming steady-state flow and a uniform pipe diameter, G is constant and

Equation 2.11-3

Equation 2.11-4

Substituting Eqs. (2.11-3) and (2.11-4) into (2.11-2) and rearranging,

Equation 2.11-5

This is the basic differential equation that is to be integrated. To do this the relation between V and p must be known so that the integral of dp/V can be evaluated. This integral depends upon the nature of the flow, and two important conditions used are isothermal and adiabatic flow in pipes.

2.11B. Isothermal Compressible Flow

To integrate Eq. (2.11-5) for isothermal flow, an ideal gas will be assumed, where

Equation 2.11-6

Solving for V in Eq. (2.11-6), and substituting it into Eq. (2.11-5) and integrating, assuming f is constant,

Equation 2.11-7

Equation 2.11-8

Substituting p₁/p₂ for V₂/V₁ and rearranging,

Equation 2.11-9

where M = molecular weight in kg mass/kg mol, R = 8314.34 N · m/kg mol · K, and T = temperature K. The quantity RT/M = p_av/p_av, where p_av = (p₁ + p₂)/2 and ρ_av is the average density at T and p_av. In English units, R = 1545.3 ft · lb_f/lb · mol · °R and the right-hand terms are divided by g_c. Equation (2.11-9) then becomes

Equation 2.11-10

The first term on the right in Eqs. (2.11-9) and (2.11-10) represents the frictional loss as given by Eqs. (2.10-9) and (2.10-10). The last term in both equations is generally negligible in ducts of appreciable length unless the pressure drop is very large.

EXAMPLE 2.11-1. Compressible Flow of a Gas in a Pipe Line Natural gas, which is essentially methane, is being pumped through a 1.016-m-ID pipeline for a distance of 1.609 × 105 m (D1) at a rate of 2.077 kg mol/s. It can be assumed that the line is isothermal at 288.8 K. The pressure p2 at the discharge end of the line is 170.3 × 103 Pa absolute. Calculate the pressure p1 at the inlet of the line. The viscosity of methane at 288.8 K is 1.04 × 10−5 Pa · s. Solution: D = 1.016 m, A = πD2/4 = π(1.016)2/4 = 0.8107 m2. Then, From Fig. 2.10-3, ε = 4.6 × 105 m. The friction factor f = 0.0027. In order to solve for p1 in Eq. (2.11-9), trial and error must be used. Estimating p1 at 620.5 × 103 Pa, R = 8314.34 N · m/kg mol · K, and ΔL = 1.609 × 105 m. Substituting into Eq. (2.11-9), Now, P2 = 170.3 × 103 Pa. Substituting this into the above and solving for p1, p1 = 683.5 × 103 Pa. Substituting this new value of p1 into Eq. (2.11-9) again and solving for p1, the final result is p1 = 683.5 × 103 Pa. Note that the last term in Eq. (2.11-9) in this case is almost negligible.

When the upstream pressure p₁ remains constant, the mass flow rate G changes as the downstream pressure p₂ is varied. From Eq. (2.11-9), when p₁ = p₂, G = 0, and when p₂ = 0, G = 0. This indicates that at some intermediate value of p₂, the flow G must be a maximum. This means that the flow is a maximum when dG/dp₂ = 0. Performing this differentiation on Eq. (2.11-9) for constant p₁ and f and solving for G,

Equation 2.11-11

Using Eqs. (2.11-3) and (2.11-6),

Equation 2.11-12

This is the equation for the velocity of sound in the fluid at the conditions for isothermal flow. Thus, for isothermal compressible flow there is a maximum flow for a given upstream p₁, and further reduction of p₂ will not give any further increase in flow. Further details as to the length of pipe and the pressure at maximum flow conditions are discussed elsewhere (D1, M2, P1).

EXAMPLE 2.11-2. Maximum Flow for Compressible Flow of a Gas For the conditions of Example 2.11-1, calculate the maximum velocity that can be obtained and the velocity of sound at these conditions. Compare with Example 2.11-1. Solution: Using Eq. (2.11-12) and the conditions in Example 2.11-1, This is the maximum velocity obtainable if p2 is decreased. This is also the velocity of sound in the fluid at the conditions for isothermal flow. To compare with Example 2.11-1, the actual velocity at the exit pressure p2 is obtained by combining Eqs. (2.11-3) and (2.11-6) to give Equation 2.11-13

2.11C. Adiabatic Compressible Flow

When heat transfer through the wall of the pipe is negligible, the flow of gas in compressible flow in a straight pipe of constant cross section is adiabatic. Equation (2.11-5) has been integrated for adiabatic flow and details are given elsewhere (D1, M1, P1). Convenient charts for solving this case are also available (P1). The results for adiabatic flow often deviate very little from isothermal flow, especially in long lines. For very short pipes and relatively large pressure drops, the adiabatic flow rate is greater than the isothermal, but the maximum possible difference is about 20% (D1). For pipes of length about 1000 diameters or longer, the difference is generally less than 5%. Equation (2.11-8) can also be used when the temperature change over the conduit is small by using an arithmetic-average temperature.

Using the same procedures for finding maximum flow that were used in the isothermal case, maximum flow occurs when the velocity at the downstream end of the pipe is the sonic velocity for adiabatic flow. This is

Equation 2.11-14

where, γ = c_p/c_ν, the ratio of heat capacities. For air, γ = 1.4. Hence, the maximum velocity for adiabatic flow is about 20% greater than for isothermal flow. The rate of flow may not be limited by the flow conditions in the pipe, in practice, but by the development of sonic velocity in a fitting or valve in the pipe. Hence, care should be used in the selection of fittings in such pipes for compressible flow. Further details as to the length of pipe and pressure at maximum flow conditions are given elsewhere (D1, M2, P1).

A convenient parameter often used in compressible-flow equations is the Mach number, N_Ma, which is defined as the ratio of ν, the speed of the fluid in the conduit, to ν_max, the speed of sound in the fluid at the actual flow conditions:

Equation 2.11-15

At a Mach number of 1.0, the flow is sonic. At a value less than 1.0, the flow is subsonic, and supersonic at a number above 1.0.