Continuity Equation for Steady One Dimensional Flow

CONTINUITY EQUATION:

When a fluid is in motion, it must move in such a way that mass is conserved. To see how mass conservation places restrictions on the velocity field, consider the steady flow of fluid through a duct (that is, the inlet and outlet flows do not vary with time). The inflow and outflow are one-dimensional, so that the velocity V and density r are constant over the area A (figure 14).

Figure- One-dimensional duct showing control volume

Now we apply the principle of mass conservation. Since there is no flow through the side walls of the duct, what mass comes in over area A₁ goes out of area A₂, (the flow is steady so that there is no mass accumulation). Over a short time interval Δt,

This is a statement of the principle of mass conservation for a steady, one-dimensional flow, with one inlet and one outlet. This equation is called the continuity equation for steady one-dimensional flow. For a steady flow through a control volume with many inlets and outlets, the net mass flow must be zero, where inflows are negative and outflows are positive.

MOMENTUM EQUATION:

Conservation of momentum requires that the time rate of change of momentum in a given direction is equal to the sum of the forces acting in that direction. This is known as Newton's second law of motion and in the model used here the forces concerned are gravitational (body) forces and the surface forces.

Consider a fluid in steady flow, and take any small stream tube as in Fig. 2.4. s is the distance measured along the axis of the stream tube from some arbitrary origin. A is the cross-sectional area of the stream tube at distance s from the arbitrary origin. p, ρ, and v represent pressure, density and flow speed respectively.

A, p, r, and v vary with s, i.e. with position along the stream tube, but not with time since the motion is steady.

Now consider the small element of fluid shown in Fig. 2.5, which is immersed in fluid of varying pressure. The element is the right frustrum of a cone of length ds, area A at the upstream section, area A+dA on the downstream section. The pressure acting on one face of the element is p, and on the other face is p + (dp/ds)ds around the curved surface the pressure may be taken to be the mean value

Fig. 2.4 The stream tube and element for the momentum equation

Fig. 2.5 The forces on the element

P+1/2(dp/ds)ds. In addition the weight W of the fluid in the element acts vertically as shown in figure. Shear forces on the surface due to viscosity would add another force, which is ignored here. As a result of these pressures and the weight, there is a resultant force F acting along the axis of the cylinder where F is given by

Where a is the angle between the axis of the stream tube and the vertical.

From Eqn (2.5) it is seen that on neglecting quantities of small order such as (dp/ds)dsdA and cancelling,

since the gravitational force on the fluid in the element is pgA Ss, i.e. volume x density x g.

Now, Newton's second law of motion (force = mass x acceleration) applied to the element of Fig. 2.5, gives

where

t represents time. Dividing by A ds this becomes

But

and therefore

Integrating along the stream tube; this becomes

but since

And

Then

This result is known as Bernoulli's equation and is discussed below.

ENERGY EQUATION:

Conservation of energy implies that changes in energy, heat transferred and work done by a system in steady operation are in balance. In seeking an equation to represent the conservation of energy in the steady flow of a fluid it is useful to consider a length of stream tube, e.g. between sections 1 and 2 (Fig. 2.6), as

Fig. 2.6 Control volume for the energy equation

constituting the control surface of a 'thermodynamic system' or control volume. At sections 1 and 2, let the fluid properties be as shown. Then unit mass of fluid entering the system through section will possess internal energy c_vT1, kinetic energy v₁ ²/2 and potential energy gz₁, i.e.

Likewise on exit from the system across section 2 unit mass will possess energy

Now to enter the system, unit mass possesses a volume 1/r₁ which must push against the pressure p₁ and utilize energy to the value of p₁ x 1/r₁ pressure x (specific) volume. At exit p₂ /r₂ is utilized in a similar manner.

In the meantime, the system accepts, or rejects, heat q per unit mass. As all the quantities are flowing steadily, the energy entering plus the heat transfer must equal the energy leaving. Thus, with a positive heat transfer it follows from conservation of energy

However, enthalpy per unit mass of fluid is c_vT +p/r = c_p T. Thus

or in differential form

For an adiabatic (no heat transfer) horizontal flow system, Eqn (2.10) becomes zero and thus

EQUATION OF STATE :

The equation of state for a perfect gas is

Substituting for p/r in Eqn (1.11) yields Eqn (1.13) and (1.14), namely

The first law of thermodynamics requires that the gain in internal energy of a mass of gas plus the work done by the mass is equal to the heat supplied, i.e. for unit mass of gas with no heat transfer

Differentiating Eqn (1 .lo) for enthalpy gives

and combining Eqns (2.12) and (2.13) yields

But

Therefore, from Eqns (2.14) and (2.15)

which on integrating gives

where k is a constant. This is the isentropic relationship between pressure and density, and has been replicated for convenience from Eqn (1.24).

SPEED OF SOUND

The bulk elasticity is a measure of how much a fluid (or solid) will be compressed by the application of external pressure. If a certain small volume, V, of fluid is subjected to a rise in pressure, dp, this reduces the volume by an amount -dV, i.e. it produces a volumetric strain of -dV/V. Accordingly, the bulk elasticity is defined as

The propagation of sound waves involves alternating compression and expansion of the medium. Accordingly, the bulk elasticity is closely related to the speed of sound, a, as follows:

Let the mass of the small volume of fluid be M , then by definition the density, r = M/V. By differentiating this definition keeping M constant, we obtain

Therefore, combining this with Eqns (l.6ayb), it can be seen that

The propagation of sound in a perfect gas is regarded as an isentropic process.Accordingly, (see the passage below on Entropy) the pressure and density are related by Eqn (1.24), so that for a perfect gas

where g is the ratio of the specific heats. Equation (1.6d) is the formula usually used to determine the speed of sound in gases for applications in aerodynamics.

ISENTROPIC ONE-DIMENSIONAL FLOW :

For many applications in aeronautics the viscous effects can be neglected to a good approximation and, moreover, no significant heat transfer occurs. Under these circumstances the thermodynamic processes are termed adiabatic.

Provided no other irreversible processes occur we can also assume that the entropy will remain unchanged, such processes are termed isentropic . We can, therefore, refer to isentropic flow. At this point it is convenient to recall the special relationships between the main thermodynamic and flow variables that hold when the flow processes are isentropic.

In Section 1.2.8 it was shown that for isentropic processes p = k r g (Eqn (1.24)), where k is a constant. When this relationship is combined with the equation of state for a perfect gas (see Eqn (1.12)), namely p/( r T) = R, where R is the gas constant, we can write the following relationships linking the variables at two different states (or stations) of an isentropic flow:

From these it follows that

A useful, special, simplifed model flow is one-dimensional, or more precisely quasi-one- dimensional flow. This is an internal flow through ducts or passages having slowly varying cross-sections so that to a good approximation the flow is uniform at each cross-section and the flow variables only vary with x in the streamwise direction. Despite the seemingly restrictive nature of these assumptions this is a very useful model flow with several important applications. It also provides a good way to learn about the fundamental features of compressible flow.

The equations of conservation and state for quasi-one dimensional, adiabatic flow in differential form become

where u is the streamwise, and only non-negligible, velocity component.

Expanding Eqn (6.3) and rearranging

From Eqn (6.4), using eqn (6.3)

which, on dividing through by d A and using the identity M² = u² / a² = ru² /(gp),using Eqn (1.6d) for the speed of sound in isentropic flow becomes

Likewise the energy Eqn (6.5), with cpT = d/(-y– 1) found by combining Eqns (1.15) and (1.6d), becomes

Then combining Eqns (6.7) and (6.8) to eliminate dp/p and substituting for dp/p and d T/ T gives

Equation (6.12) indicates the way in which the cross-sectional area of the stream tube must change to produce a change in velocity for a given mass flow. It will be noted that a change of sign occurs at M = 1.

For subsonic flow dA must be negative for an increase, i.e a positive change, in velocity. At M = 1, dA is zero and a throat appears in the tube. For acceleration to supersonic flow a positive change in area is required, that is, the tube diverges from the point of minimum cross-sectional area.

Eqn (6.12) indicates that a stream tube along which gas speeds up from subsonic to supersonic velocity must have a converging-diverging shape. For the reverse process, the one of slowing down, a similar change in tube area is theoretically required but such a deceleration from supersonic flow is not possible in practice.

Other factors also control the flow in the tube and a simple convergence is not the only condition required. To investigate the change of other parameters along the tube it is convenient to consider the model flow shown in Fig. 6.1. In this model the air expands from a high-pressure reservoir (where the conditions may be identified by suffix 0), to a low-pressure reservoir, through a constriction, or throat, in a convergent-divergent tube. Denoting conditions at two separate points along the tube by suffices 1 and 2, respectively, the equations of state, continuity, motion and energy become

Fig: One-dimensional isentropic expansive flow

The last of these equations, on taking account of the various ways in which the acoustic speed can be expressed in isentropic flow (see Eqn (1.6cYd)),i .e.