# Averaged Physical Quantities

## Abstract

The present chapter introduces the mathematical definitions and notations for averaged physical quantities in control volumes, which are central to the 0D/1D modeling methodology. The intention is to show that the quantities used in 0D/1D modeling are exact quantities which represent average values in space and time for the sake of computational efficiency. This is done at the expense of resolution in space (but not in time). In Chap. 6 and the following sections, it will be shown that this limitation still gives acceptable simulation results.

## Nomenclature

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

- \(A_{b:a}\)
Cross-sectional area of boundary \(b{\text{:}}a\) (m

^{2})- \(A_{k,x}\)
Cross-sectional area of phase \(k\) at coordinate \(x\) along the tube (m

^{2})- \(A_{x}\)
Flow cross-sectional area at coordinate \(x\) (m

^{2})- \(b{\text{:}}a\)
Boundary between control volumes \(b\) and \(a\)

- \(f_{x}\)
Average of quantity \(f\) over \(A_{x}\) (\(\left[ f \right]\))

- \(f_{k,x}\)
Phase average of phase \(k\)-related quantity \(f_{k}\) over \(A_{k,x}\) (\(\left[ {f_{k} } \right]\))

- \(g\)
Specific extensive quantity (\(\left[ G \right]\).kg

^{−1})- \(g_{a}\)
Average value of specific extensive quantity \(g\) inside volume \(a\) (\(\left[ G \right]\).kg

^{−1})- \(g_{b:a}\)
Average value of specific extensive quantity \(g\) over \(A_{b:a}\) (\(\left[ G \right]\).kg

^{−1})- \(g_{x}\)
Average value of specific extensive quantity \(g\) over \(A_{x}\) for the left-hand side of the energy balance equation (\(\left[ G \right]\).kg

^{−1})- \(g_{x:}\)
Average value of specific extensive quantity \(g\) over \(A_{x}\) for the right-hand side of the energy balance equation (\(\left[ G \right]\).kg

^{−1})- \(G_{a}\)
Total value of specific extensive quantity \(g\) inside \(a\) (\(\left[ G \right]\))

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

- \(L\)
Tube length (m)

- \(m_{a}\)
Fluid mass inside volume \(a\) (kg)

- \(m_{x}\)
Total value of fluid mass in the tube from the origin up to coordinate \(x\) along the tube (kg)

- \(\dot{m}(b \to a)\)
Mass flow rate through boundary \(b{\text{:}}a\), counted positively when fluid flows from \(b\) to \(a\) (kg s

^{−1})- \(\dot{m}(x)\)
Mass flow rate through \(A_{x}\), counted positively when fluid flows along increasing \(x\) (kg s

^{−1})- \(\vec{n}(b \to a)\)
Orientation of boundary \(b{\text{:}}a\), positively from \(b\) to \(a\)

- \(v_{x}\)
Average superficial velocity (or volumetric flux) over \(A_{x}\) (m s

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

^{3})- \(x\)
Coordinate \(x\) along the tube or the fluid vein (m)

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

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

^{−3})- \(\varphi_{g} (b \to a)\)
Flux of quantity \(g\) through boundary \(b{\text{:}}a\), oriented positively from \(b\) to \(a\) (\(\left[ g \right]\).m

^{−2}s^{−1})- \(\Phi _{g} (b \to a)\)
Flow of specific extensive quantity \(g\) through boundary \(b{\text{:}}a\), oriented positively from \(b\) to \(a\) (\(\left[ g \right]\).s

^{−1})- \(\Phi _{g} (x)\)
Flow of specific extensive quantity \(g\) at coordinate \(x\) along the tube, oriented positively with increasing \(x\) (\(\left[ g \right]\).s

^{−1})- \(\left\langle f \right\rangle_{x}\)
Average value of quantity \(f\) over \(A_{x}\) (\(\left[ f \right]\))

- \(\left\langle {f_{k} } \right\rangle_{x}\)
Phase average of phase \(k\)-related quantity \(f_{k}\) over \(A_{k,x}\) (\(\left[ {f_{k} } \right]\))