What is Mass flow rate and volume flow rate? — Physics

Definition 2.7 University Physics — Year 2 · Chapter 2 — Fluid Kinematics

The mass flow rate through an oriented surface $S$ is the mass crossing it per unit time, $D_m = \sum\rho\,\vect v\cdot\vect n\,\dd S$ — the flux of the mass current density $\vect j = \rho\vect v$ (unit $\mathrm{kg}\,\mathrm{m}^{-2}\,\mathrm{s}^{-1}$ ); the volume flow rate is $D_V = \sum\vect v \cdot\vect n\,\dd S$ ( $\mathrm{m}^{3}/\mathrm{s}$ ). For a uniform velocity normal to a plane section of area $S$ , $D_m = \rho vS$ and $D_V = vS$ .

Left: the mass balance of a fixed box — the net outflow through its six faces, per unit volume, is the divergence of v. Right: in a steady flow the mass flow rate is the same through every section of a stream tube. — Left: the mass balance of a fixed box — the net outflow through its six faces, per unit volume, is the divergence of $\rho\vect v$ . Right: in a steady flow the mass flow rate is the same through every section of a stream tube.

Examples

Example 2.11 (Blood)

The aorta (section $2.5\,\mathrm{cm}^{2}$ ) carries $5\,\mathrm{L}/\mathrm{min}$ $= 83\,\mathrm{cm}^{3}/\mathrm{s}$ : $v = 33\,\mathrm{cm}/\mathrm{s}$ . The same flow passes through some $10^{10}$ capillaries of total section about $2500\,\mathrm{cm}^{2}$ : $v = 0.3\,\mathrm{mm}/\mathrm{s}$ — slow enough for exchanges through the walls, the point of the branching.

Read in context →