Are You Dense? Crank Up the Volume and Go With the Flow!
Just the Facts
DOFPro Team
Mass Density and Specific Volume
Mass Density (Density), \(\rho\ \equiv\) mass per unit volume
SI units: \(\mathrm{kg/m^3}\)
\(\rho = \frac{m}{V} = \frac{1}{\hat{V}}\)
Specific Volume, \(\hat{V}\ \equiv\) volume per unit mass
SI units: \(\mathrm{m^3/kg}\)
\(\hat{V} = \frac{V}{m} = \frac{1}{\rho}\)
Molar Density and Molar Volume
Molar Density, \(\rho\ \equiv\) moles per unit volume
SI units: \(\text{g-mol}/\mathrm{m^3}\) (same units as concentration)
\(\rho = \frac{n}{V} = \frac{1}{\hat{V}}\)
Molar Volume, \(\hat{V}\ \equiv\) volume per mole
SI units: \(\mathrm{m^3/}\text{g-mol}\)
\(\hat{V} = \frac{V}{n} = \frac{1}{\rho}\)
We reuse the mass-density symbol for molar density and the specific-volume symbol for molar volume.
Check the units to determine which is being used!
Specific Gravity
Specific Gravity \(\equiv\) ratio of density to a reference density
\(= \frac{\rho}{\rho_\mathrm{{ref}}}\)
Most common reference density is liquid water at \(4\ ^{\circ}\text{C}\)
\(\rho\mathrm{_{ H_2O(\textit{l})}}(4\ ^{\circ}\mathrm{C}) = 1000\ \mathrm{kg/m^3}\)
\(= 1.000\ \mathrm{g/cm^3}\)
\(= 62.43\ \mathrm{lb_m/ft^3}\)
Concentration
Molar Concentration (Concentration), \(C\ \equiv\) moles per unit volume
\(C_\mathrm{A} = \frac{n_\mathrm{A}}{V}\)
For an ideal gas, can be calculated as:
\(C_\mathrm{A} = \frac{n_\mathrm{A}}{V} = \frac{p_\mathrm{A}}{RT}\)
Mass Concentration, \(\rho\ \equiv\) mass per unit volume (same units as density)
Most often used for quantities such as salinity
\(\rho_\mathrm{A} = \frac{m_\mathrm{A}}{V}\)
Mass, Molar, and Volumetric Flow Rates
Mass Flow Rate, \(\dot{m} \equiv \frac{\text{mass flowing past a given location}}{\text{period of time}}\)
\(\sum \dot{m}_{i_\mathrm{in}} = \sum \dot{m}_{i_\mathrm{out}}\) (open, steady state)
Molar Flow Rate, \(\dot{n} \equiv \frac{\text{moles flowing past a given location}}{\text{period of time}}\)
\(\sum \dot{n}_{i_\mathrm{in}} = \sum \dot{n}_{i_\mathrm{out}}\) (open, steady state, no reactions)
Volumetric Flow Rate, \(\dot{V} \equiv\) \(\frac{\text{volume flowing past a given location}}{\text{period of time}}\)
\(\sum \dot{V}_{i_\mathrm{in}} \ne \sum \dot{V}_{i_\mathrm{out}}\) (in general)
Mass, Molar, and Volumetric Flow Rates (cont.)
Given a specific location in space and instant in time, the following relationship exists between the the volumetric flow rate (\(\dot{V}\)), the average fluid velocity (\(u\)), and the cross-sectional area of the flow (\(A\)):
\(\dot{V}\) = \(uA\)
For an open, steady state system, the previous expression is true at any location in the system.
Mass, Molar, and Volumetric Flow Rates (cont.)
Relationship between mass flow rate (\(\dot{m}\)) and volumetric flow rate (\(\dot{V}\)):
\(\dot{m} = \rho \dot{V}\)
\(\rho_\mathrm{in} \dot{V}_\mathrm{in} = \rho_\mathrm{out} \dot{V}_\mathrm{out}\) (open, steady-state system, 1 inlet & 1 outlet)
Don’t Forget! Volume is not a conserved quantity. Just because the mass or molar flowrate is constant, it does not follow that the volumetric flow rate is constant.
Fluid Velocity
Recall \(\dot{V}\) = \(uA\)
\(\implies u=\frac{\dot{V}}{A}\)
Measuring cup of volume, \(V\), fills in time, \(t\).
Average fluid flow rate, \(\dot{V} = \frac{V}{t}\).
Cross-sectional area, \(A\ (=\pi\frac{D^2}{4})\) for a tube or circular pipe
\(\implies u=\frac{V}{At}\)
For reference, for laminar flow in a circular tube, \(u_\mathrm{wall} = 0\), \(u_{CL} = 2 u\).
The Takeaways
- Mass density or density and specific volume are reciprocals.
- Molar density and molar volume are reciprocals.
- The mass flow rate is equal to the density times the volumetric flow rate.
- Fluid velocity is defined as the volumetric flow rate divided by the cross-sectional area.
Thanks for watching!
The Full Story companion video is in the link in the upper left. The next video in the series is in the upper right. To learn more about Chemical and Thermal Processes, visit the website linked in the description to find previous and following videos in this series.
The DOFPro Team
