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

Answer: The analogous expression for mass convection is \(\dot{m}=K_{m} A_{s}\left(C_{s}-C_{\infty}\right)\), where: - \(\dot{m}\) is the mass transfer rate (kg/s, kilograms per second) - \(K_{m}\) is the mass transfer coefficient (kg/m²s, kilograms per square meter second) - \(A_{s}\) is the surface area through which mass is being transferred (m², square meters) - \(C_{s}\) is the concentration of the component at the surface (kg/m³, kilograms per cubic meter) - \(C_{\infty}\) is the concentration of the component in the bulk fluid (kg/m³, kilograms per cubic meter)

Step-by-step solution

Analyzing given heat convection expression

First, let's understand the given heat convection equation: Newton's law of cooling: $\dot{Q}=h A_{s}\left(T_{s}-T_{\infty}\right)$ Where: - \(\dot{Q}\) is the heat transfer rate (W, watts) - \(h\) is the convective heat transfer coefficient (W/m²K, watts per square meter Kelvin) - \(A_{s}\) is the surface area through which heat is being transferred (m², square meters) - \(T_{s}\) is the temperature of the surface (K, Kelvin) - \(T_{\infty}\) is the temperature of the surrounding fluid (K, Kelvin)

Determine the mass transfer analogy for heat transfer

We can now express mass convection in an analogous manner to heat convection. Mass transfer is expressed as: \(\dot{m}=K_{m} A_{s}\left(C_{s}-C_{\infty}\right)\) Where: - \(\dot{m}\) is the mass transfer rate (kg/s, kilograms per second) - \(K_{m}\) is the mass transfer coefficient (kg/m²s, kilograms per square meter second) - \(A_{s}\) is the surface area through which mass is being transferred (m², square meters) (similar to heat convection) - \(C_{s}\) is the concentration of the component at the surface (kg/m³, kilograms per cubic meter) - \(C_{\infty}\) is the concentration of the component in the bulk fluid (kg/m³, kilograms per cubic meter) We have now expressed mass convection in an analogous manner to heat convection and identified all the quantities in the expression along with their units.

Key concepts

Newton's Law of Cooling

Newton's Law of Cooling describes how the temperature of an object changes over time as it comes into thermal equilibrium with its surroundings. This law is crucial in understanding how quickly an object will cool down or heat up, a phenomenon commonly observed in various engineering applications, such as cooling of electronic devices or heating of fluids in a heat exchanger.

According to this law, the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment. Mathematically, it's expressed as \(\dot{Q}=hA_{s}(T_{s}-T_{\infty})\), where \(\dot{Q}\) is the rate of heat transfer from the object, \(h\) represents the convective heat transfer coefficient, \(A_{s}\) stands for the surface area of the object, \(T_{s}\) is the object's surface temperature, and \(T_{\infty}\) is the surrounding temperature.

The formula provides a simple yet powerful tool for predicting the cooling rate in practical situations. It enables designers to calculate the necessary conditions for achieving desired temperatures within a given timeframe.

Convective Heat Transfer Coefficient

The convective heat transfer coefficient, denoted by \(h\), is a measure of the ability of a fluid medium to convey heat away from a surface. The units of \(h\) are W/m²K, which accounts for the heat transfer per unit area and per unit temperature difference.

This coefficient is affected by several factors including fluid velocity, fluid viscosity, surface roughness, and the nature of the fluid flow (laminar or turbulent). A higher \(h\) value indicates a more efficient heat transfer and is often desirable in systems designed to dissipate heat swiftly, such as radiators or air conditioning units.

Role of Surface Area and Temperature Gradient

It's important to note that in Newton's law of cooling, \(h\) is used in conjunction with \(A_{s}\) and the temperature gradient \(T_{s}-T_{\infty}\) to calculate the total heat transfer rate. The larger the surface area, or the greater the temperature difference, the higher the rate of heat transfer. Under standing how these variables interact is key in optimizing systems for efficient cooling or heating.

Mass Transfer Rate

Analogous to heat transfer, mass transfer involves the movement of a particular component, such as a gas or solute, from one location to another. The mass transfer rate, represented by \(\dot{m}\), indicates how much mass is moving through a specified cross-sectional area in a unit of time, with the typical units being kg/s.

The expression for mass transfer is given by \(\dot{m}=K_{m} A_{s}(C_{s}-C_{\infty})\), where \(K_{m}\) represents the mass transfer coefficient in kg/m²s, \(A_{s}\) is the area over which the mass transfer occurs, and \(C_{s}\) and \(C_{\infty}\) are the concentrations of the component at the surface and in the bulk fluid, respectively.

Significance in Engineering and Design

In the context of engineering and environmental sciences, understanding mass transfer is vital for designing equipment like scrubbers and biofilters, and for dealing with pollution control. For instance, in a scrubber used for air purification, knowing the mass transfer rate helps in assessing the efficiency of the system in removing contaminants. By analyzing the components akin to heat transfer in Newton's law of cooling, engineers can derive meaningful insights for mass transfer applications.