Introduction & Context

The calculation determines the convective mass‑transfer coefficient for a spherical particle immersed in a flowing fluid. In process engineering this coefficient, \(h_m\), is required to size gas‑liquid contactors, design packed‑bed reactors, and predict the rate of species transport from the bulk fluid to the surface of catalyst pellets or droplets. Accurate estimation of \(h_m\) enables reliable scale‑up from laboratory to industrial scale and ensures that mass‑transfer limitations are properly accounted for in reaction and separation models.

Methodology & Formulas

The procedure follows the standard dimensionless‑group approach for forced convection around a sphere. All input quantities are first expressed in SI units using the conversion factors defined in the code.

Unit conversions
- Length: \(d_{\text{m}} = d_{\text{mm}} \times \text{MM\_TO\_M}\)
- Viscosity: \(\mu_{\text{kg·m}^{-1}\text{s}^{-1}} = \mu_{\text{cP}} \times \text{CP\_TO\_KG\_M\_S}\)
- Pressure (if needed): \(P_{\text{Pa}} = P_{\text{bar}} \times \text{BAR\_TO\_PA}\) or \(P_{\text{Pa}} = P_{\text{atm}} \times \text{ATM\_TO\_PA}\)
Reynolds number (Re) – characterises the ratio of inertial to viscous forces: \[ \text{Re} = \frac{\rho\,u\,d}{\mu} \] where \(\rho\) is the fluid density, \(u\) the free‑stream velocity, \(d\) the sphere diameter, and \(\mu\) the dynamic viscosity.
Schmidt number (Sc) – ratio of momentum diffusivity to mass diffusivity: \[ \text{Sc} = \frac{\mu}{\rho\,D_{AB}} \] with \(D_{AB}\) the binary diffusion coefficient of species \(A\) in \(B\).
Sherwood number (Sh) – dimensionless mass‑transfer coefficient for a sphere in external flow. The empirical correlation of Ranz and Marshall is used: \[ \text{Sh} = 2.0 + 0.6\,\text{Re}^{1/2}\,\text{Sc}^{1/3} \] The first term (2.0) represents pure diffusion for a stationary sphere, while the second term accounts for convective enhancement.
Mass‑transfer coefficient (hm) – obtained by rearranging the definition of the Sherwood number: \[ h_m = \frac{\text{Sh}\,D_{AB}}{d} \]

Flow regime classification

Regime	Reynolds number range	Applicable correlation
Laminar (creeping flow)	\(\text{Re} < 1\)	\(\text{Sh} = 2.0\) (pure diffusion)
Transitional	\(1 \le \text{Re} \le 10^3\)	Ranz–Marshall correlation shown above
Turbulent	\(\text{Re} > 10^3\)	More advanced correlations (e.g., Frössling) may be required

The presented steps provide a quick, first‑order estimate of the external mass‑transfer coefficient for a sphere under forced convection. For high‑temperature gases, compressibility corrections, and variable property effects should be incorporated as needed.

Match the Reynolds-number range covered by the correlation to your operating Re.
Check that the Schmidt-number range fits your system (gases ≈ 1, liquids 100–1000).
Prefer correlations that explicitly account for surface blowing/suction when mass injection or evaporation is present.
When in doubt, use the widely validated Ranz–Marshall equation: Sh = 2 + 0.6 Re^0.5 Sc^0.33 for 0 ≤ Re ≤ 200 and 0.6 ≤ Sc ≤ 400.

Internal circulation reduces the liquid-side resistance but leaves the external gas-side coefficient essentially unchanged. Continue to calculate the external Sh with the same sphere correlations; couple the two resistances in series to obtain the overall mass-transfer coefficient.

Replace the free-stream velocity with the superficial velocity divided by the bed void fraction ε.
Multiply the single-sphere Sh by a packing factor such as Sh_bed = Sh_iso (1 + 1.5(1 – ε)) valid for 0.35 < ε < 0.7.
Verify that the particle Reynolds number is still within the correlation range after the velocity adjustment.

Yes, but use the film temperature and film composition to evaluate all physical properties. Because water vapor is transferred into the boundary layer, the blowing factor (1 + B_m)^0.7 should be applied to the correlated Sh to account for mass-transfer-driven thickening of the boundary layer.

Worked Example – Forced Convection Mass Transfer from a Spherical Particle

Scenario: A 5 mm diameter polymer bead is suspended in a gas stream flowing at 3 m s⁻¹. The gas temperature is 60 °C and the pressure is 1 bar. The binary diffusion coefficient of the species of interest in the gas is \(D_{AB}=2.6\times10^{-5}\;{\rm m^{2}\,s^{-1}}\). The gas has a density of 1.05 kg m⁻³ and a dynamic viscosity of \(2.0\times10^{-8}\;{\rm kg\,m^{-1}\,s^{-1}}\). Determine the convective mass‑transfer coefficient \(h_m\) for the sphere.

Knowns

Particle diameter, \(d = 5.0\;{\rm mm}=0.005\;{\rm m}\)
Gas velocity, \(u = 3.0\;{\rm m\,s^{-1}}\)
Gas temperature, \(T = 60.0\;^{\circ}{\rm C}\) (used only for property selection)
Gas pressure, \(P = 1.0\;{\rm bar}=1.0\times10^{5}\;{\rm Pa}\)
Binary diffusion coefficient, \(D_{AB}=2.6\times10^{-5}\;{\rm m^{2}\,s^{-1}}\)
Gas density, \(\rho = 1.05\;{\rm kg\,m^{-3}}\)
Dynamic viscosity, \(\mu = 2.0\times10^{-8}\;{\rm kg\,m^{-1}\,s^{-1}}\)

Step‑by‑Step Calculation

Calculate the Reynolds number for flow over a sphere \[ {\rm Re}= \frac{\rho\,u\,d}{\mu} =\frac{1.05\;(3.0)\;(0.005)}{2.0\times10^{-8}} =7.88\times10^{5}\;( \approx 787500 ) \]
Calculate the Schmidt number \[ {\rm Sc}= \frac{\mu}{\rho\,D_{AB}} =\frac{2.0\times10^{-8}}{1.05\;(2.6\times10^{-5})} =7.33\times10^{-4}\;( \approx 0.001 ) \]
Apply the Ranz–Marshall correlation for a sphere \[ {\rm Sh}=2+0.6\,{\rm Re}^{1/2}{\rm Sc}^{1/3} \] \[ {\rm Re}^{1/2}= \sqrt{7.88\times10^{5}}=8.87\times10^{2} \] \[ {\rm Sc}^{1/3}= (7.33\times10^{-4})^{1/3}=9.25\times10^{-2} \] \[ {\rm Sh}=2+0.6\,(8.87\times10^{2})(9.25\times10^{-2}) =2+49.115\approx 50.115 \]
Determine the convective mass‑transfer coefficient \[ h_m = \frac{{\rm Sh}\;D_{AB}}{d} =\frac{50.115\;(2.6\times10^{-5})}{0.005} =2.61\times10^{-1}\;{\rm m\,s^{-1}} \approx 0.261\;{\rm m\,s^{-1}} \]

Final Answer

The convective mass‑transfer coefficient for the 5 mm sphere under the stated conditions is \(h_m = 0.261\;{\rm m\,s^{-1}}\).

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle