Question

Heat convection is expressed by Newton's law of cooling as $\dot{Q}=h A_{s}\left(T_{s}-T_{\infty}\right)$. Express mass convection in an analogous manner on a mass basis, and identify all the quantities in the expression and state their units.

Short answer

Question: Express mass convection in a similar manner to heat convection and identify the quantities and units involved. Answer: Mass convection can be expressed analogously to heat convection as \(\dot{m} = k_c A_p \left(C_s - C_\infty\right)\), with the quantities and units being: - Mass transfer rate (\(\dot{m}\)): kg/s - Mass transfer coefficient (\(k_c\)): kg/m²s - Area perpendicular to mass flow direction (\(A_p\)): m² - Concentration at the surface (\(C_s\)): kg/m³ - Concentration in the surrounding (\(C_\infty\)): kg/m³

Step-by-step solution

Define the quantities in the heat convection expression

In Newton's law of cooling, the heat convection is given by \(\dot{Q}=hA_{s}\left(T_{s}-T_{\infty}\right)\). The quantities in this expression are: - \(\dot{Q}\): heat transfer rate (W, watts) - \(h\): heat transfer coefficient (W/m²K, watts per square meter-kelvin) - \(A_s\): surface area (m², square meters) - \(T_s\): surface temperature (K, kelvin) - \(T_{\infty}\): surrounding temperature (K, kelvin)

Develop an analogous expression for mass convection

We know that mass transfer is driven by a concentration gradient, similarly to how heat transfer is driven by a temperature gradient. Therefore, an analogous expression for mass convection can be written as: \(\dot{m} = k_c A_p \left(C_s - C_\infty\right)\) Where the quantities are: - \(\dot{m}\): mass transfer rate (kg/s, kilograms per second) - \(k_c\): mass transfer coefficient (kg/m²s, kilograms per square meter-second) - \(A_p\): area perpendicular to mass flow direction (m², square meters) - \(C_s\): concentration at the surface (kg/m³, kilograms per cubic meter) - \(C_{\infty}\): concentration in the surrounding (kg/m³, kilograms per cubic meter)

Identify the quantities and units in the mass convection expression

The quantities and units in the mass convection expression are: - Mass transfer rate (\(\dot{m}\)): the rate at which mass is transferred across the surface, measured in kilograms per second (kg/s) - Mass transfer coefficient (\(k_c\)): a proportionality constant that relates the mass transfer rate to the concentration gradient, measured in kilograms per square meter-second (kg/m²s) - Area perpendicular to mass flow direction (\(A_p\)): the area through which the mass is transferred, measured in square meters (m²) - Concentration at the surface (\(C_s\)): the mass concentration at the surface, measured in kilograms per cubic meter (kg/m³) - Concentration in the surrounding (\(C_\infty\)): the mass concentration far away from the surface, measured in kilograms per cubic meter (kg/m³)