# Static Systems

## Abstract

This chapter deals with the important special case of static systems, which are systems described by static conservation equations, and which are extensively used, e.g., for the purpose of sizing. The differential term in the static conservation equations vanishes, which means that the state variables that describe the physical state of the system do not appear explicitly in the equations. In spite of this difficulty, it is shown how the physical state can still be efficiently computed.

## Nomenclature

- \([G]\)
Unit of quantity \(G\)

- \(A\)
Flow cross-sectional area (m

^{2})- \(A_{b:a}\)
Flow cross-sectional area of boundary

*b*:*a*(m^{2})*b*:*a*Boundary between volumes \(b\) and \(a\)

- \(g\)
Gravity constant (m s

^{−2})- \(g_{a}\)
Average value of specific extensive quantity \(g\) inside volume \(a\) (\([G]\) kg

^{−1})- \(P_{e}\)
Peclet number

- \(G_{x}\)
Total value of specific extensive quantity \(g\) in the tube from the origin up to coordinate \(x\) (\([G]\))

- \(h\)
Specific enthalpy (J kg

^{−1})- \(h_{a}\)
Average specific enthalpy inside volume \(a\) (J kg

^{−1})- \(h_{b:a}\)
Average specific enthalpy over \(A_{b:a}\) (J kg

^{−1})- \(J(b \to a)\)
Total thermal diffusion through boundary \(b:a\), positively from \(b\) to \(a\) (W)

- \(\dot{m}(b \to a)\)
Mass flow rate through boundary

*b*:*a*, positively from \(b\) to \(a\) (kg s^{−1})- \(P\)
Fluid pressure (Pa)

- \(P_{a}\)
Average fluid pressure in volume \(a\) (Pa)

- \(s(x)\)
Step function

- \(\hat{s}(x)\)
Hyperbolic function

- \(\text{sgn} (x)\)
Sign function

- \(\dot{W}_{a}\)
Total heating power received by volume \(a\) (J s

^{−1})- \(V_{a}\)
Volume of volume \(a\) (m

^{3})- \(z_{a}\)
Altitude of volume \(a\) (m)

- \(z_{x}\)
Pipe altitude at coordinate \(x\) (m)

- \(\alpha\)
Diffusion constant (s kg

^{−1}m^{−1})- \(\gamma_{b:a}\)
Diffusion mass flow rate through \(A_{b:a}\) (kg s

^{−1})- \(\Delta P\)
Variation of pressure (Pa)

- \(\Delta P_{\text{f}} (a \to b)\)
Pressure loss due to friction between volumes \(a\) and \(b\) oriented positively from \(a\) to \(b\) (Pa)

- \(\rho_{a}\)
Average fluid density in volume \(a\) (kg m

^{−3})- \(\rho_{b:a}\)
Average fluid density over \(A_{b:a}\) (kg m

^{−3})

## Reference

- Wylie EB, Streeter VL (1993) Fluid transients in systems. Prentice HallGoogle Scholar