Study Question Set 01
Mass Balances
Let us write the general conservation equation for humidity in the air:
\[ 0 = \frac{\partial \rho_v}{\partial t} + u \frac{\partial \rho_v}{\partial x} + v \frac{\partial \rho_v}{\partial y} + w \frac{\partial \rho_v}{\partial z} \]
where \(\rho_v\) is vapor density (same as absolute humidity). For the purpose of this set of questions, we assume there is no condensation or vaporization happening.
- What does the term \(\frac{\partial \rho_v}{\partial t}\) describe, and what is the unit of the term?
The term \(\frac{\partial \rho_v}{\partial t}\) describes the change in absolute humidity \(\partial \rho_v\) in time \(\partial t\) and as such is the change in storage within the `volume’. Its unit is partial density of water vapor (\(\rm{g}\,\rm{m}^{-3}\)) divided by time (in \(\rm{s}\)), i.e. \(\rm{g} \rm{m}^{-3} \rm{s}^{-1}\)
- What does the term \(u \frac{\partial \rho_v}{\partial x}\) describe, and what is the unit of the term?
The term \(u\frac{\partial \rho_v}{\partial x}\) describes transport of a humidity gradient along the \(x\)-axis by the wind. Its unit is wind speed (\(\rm{m}\,\rm{s}^{-1}\)) times partial density of water vapor (\(\rm{g}\,\rm{m}^{-3}\)) divided by distance (in \(\rm{m}\)), i.e. again \(\rm{g}\,\rm{m}^{-3}\,\rm{s}^{-1}\)
- Assume horizontally homogeneous conditions, and \(\frac{\partial \rho_v}{\partial z} = -1\,\rm{g}\,\rm{m}^{-3}\,\rm{s}^{-1}\). \(u = 2\,\rm{m}\,\rm{s}^{-1}\), \(v = 0\,\rm{m}\,\rm{s}^{-1}\) and \(w = 0.1 \,\rm{m}\,\rm{s}^{-1}\) Is the air drying out, becoming more humid, or is the humidity staying constant?
Horizontally homogeneous conditions mean \(\frac{\partial \rho_v}{\partial x} = 0\) and \(\frac{\partial \rho_v}{\partial y} = 0\). So the conservation equation simplifies to: \[ 0 = \frac{\partial \rho_v}{\partial t} + w \frac{\partial \rho_v}{\partial z} \] or \[ \frac{\partial \rho_v}{\partial t} = - w \frac{\partial \rho_v}{\partial z} \]
Inserting \(\frac{\partial \rho_v}{\partial z} = -1\,\rm{g}\,\rm{m}^{-3}\,\rm{m}^{-1}\) and \(w = 0.1 \,\rm{m}\,\rm{s}^{-1}\) results in: \[ \frac{\partial \rho_v}{\partial t} = 0.1\,\rm{g}\,\rm{m}^{-3}\,\rm{s}^{-1}. \]
So the `volume’ becomes more humid over time.