SYSTEM MODELS
4.3 HYDRAULIC AND ACOUSTIC CIRCUITS
4.3.1 Fluid Resistance
Figure 3.1 showed a 1-port fluid resistor that relates an effort, a pressure drop, P, to a flow, a volume flow rate, or volume velocity, Q. Figure 4.23 shows how resistors typically appear in a bond graph, together with example devices represented by a resistor. Partaof Figure 4.23 is a combination of a resistor and a 1-junction. This combination implies that the pressure drop, P3, is related to the pressuresP1 andP2by the equation
P3=P1−P2, (4.19)
and that all volume flow rates are equal,
Q1=Q2=Q3. (4.20)
Following Table 3.1, the resistor implies that the pressure drop is related to the volume flow rate by a nonlinear function,
P3=R(Q3), (4.21)
or, if a linear relation is assumed, by a resistance coefficient R3
P3=R3Q3. (4.22)
It is an unfortunate fact that in many hydraulic and acoustic systems it is not easy to predict resistance functions or coefficients for parts of the system before it is constructed. However, experimentally it not difficult to measure pressure drops
(c)
P1 P2
Q1 Q2
(d) (e)
Q1
Q1 Q2 Q2
P1 P2 P1 P2
x (a)
P1 P2
Q1 Q2
(b) 1
R P3 Q3 P1 Q1
P2 Q2
FIGURE 4.23. Fluid resistors: (a) bond graph; (b) porous plug; (c) long pipe; (d) orifice;
(e) valve with variable area,A(x).
and flow rates under steady conditions and to characterize resistance effects. This means that there are at least guidelines in the engineering literature for suitable resistance law assumptions.
Partbof Figure 4.23 is intended to represent a porous plug in a pipe for which viscous forces on the fluid from the plug might be assumed to dominate. In this case it would be logical to use Eq. (4.22). In the absence of experimental data relating to the plug, one would have to experiment with values of the coefficient R3 in studying a system model.
Another case in which a linear resistance can be assumed is shown in partc of Figure 4.23, which is supposed to represent a long, thin tube in which laminar flow of an incompressible fluid develops. In this case a theoretical value for the resistance can be given, (see Reference [4], Section 7.4):
R3=128μl/π d4, (4.23)
where μ[Pa·s] is the fluid viscosity coefficient, l [m] is the length and d [m]
is the inside diameter. (Note that the Appendix lists typical property values, such as viscosity, for a number of materials. These are particularly useful for estimating 1-port element parameters and functions for the types of systems treated in this section.)
For incompressible flow in long pipes, it is useful to compute a Reynolds number, Re, defined as
Re=4ρQ/π dμ, (4.24)
where ρ [kg/m3] is the fluid mass density. When the Reynolds number is low, say about 200 or less, viscous forces predominate and Eqs. (4.22) and (4.23) can be used. At a higher Reynolds number, the flow becomes turbulent and the pressure–flow relationship under steady-flow conditions is nonlinear. The transition to turbulent flow depends on the pipe dimensions l and d, the pipe surface roughness, and fluid properties.
For Reynolds numbers greater than about 5000, the flow is likely to be turbu- lent, and a general function for the nonlinear relation from Eq. (4.21) is
P3=atQ3|Q3|3/4. (4.25) In this formula, the absolute value sign is necessary to make sure that P3is negative if Q3 is negative. The constant at is often determined experimentally (see Reference [5]). Even when Eq. (4.25) is known to be valid for steady flow, its use in studying transient conditions is not necessarily valid. For oscillatory flows or other dynamic conditions, turbulence may not develop fully, so in these conditions, Eq. (4.25) should be regarded only as an approximation.
Partsdandeof Figure 4.23 show two important cases in which a pressure drop occurs over short lengths. The orifice is assumed to have a fixed area A0, while the valve has a variable area A(x), where x is a position coordinate. Although the valve is shown as if it were a gate valve, the basic equation for the pressure drop applies to a variety of other valve configurations. It is a standard exercise in fluid mechanics texts to derive the laws of an orifice using considerations of energy, momentum, and continuity. The main result is that the pressure drop is proportional to the square of the volume flow rate. One form of the law is (see Reference [6], Section 3.8)
P3=(ρQ3|Q3|)/2Cd2A20, (4.26) where again the use of the absolute value sign corrects the sign of the pressure drop P3 if Q3 happens to be negative andCd is a discharge coefficient. For a round, sharp-edged orifice, Cd can be predicted to have a value of 0.62, but for other shapes of holes, the values of Cd vary somewhat.
For valves, the area depends on a position coordinate and the discharge coef- ficient may also vary, so the relation equivalent to Eq. (4.26) can be written P3=(ρQ3|Q3|)/2Cd2(x)A2(x). (4.27) In case it turns out that a causal analysis of the system model ultimately requires that the flow must be expressed in terms of pressure drop, the
relationships of Eqs. (4.26) and (4.27) must be inverted. For example, the inverse version of Eq. (4.26) is
Q3=CdA0(2|P3|/ρ)1/2sgnP3. (4.28) This form is slightly more complicated than the form usually given to allow the pressure drop and volume flow rate to take on both positive and negative values.
(Thesignum function “sgn” is just+1 ifP3is positive and−1 if it is negative.) In dynamic systems in which oscillating flows may exist, one must make sure that both plus and minus values of the variables are computed correctly.
Equation (4.28) is the inverse form of a nonlinear relation as indicated by Eq. (4.21). When a linear relation can be assumed as shown in Eq. (4.22), the inverse relation is much simpler:
Q3=P3/R3. (4.29)
Chapter 5 will present methods for deciding which causal form of resistance laws are required for formulating the state equations of a system to be analyzed or simulated.