Question

Air at \(40^{\circ} \mathrm{C}\) and 1 atm flows over a \(5-\mathrm{m}\)-long wet plate with an average velocity of \(2.5 \mathrm{~m} / \mathrm{s}\) in order to dry the surface. Using the analogy between heat and mass transfer, determine the mass transfer coefficient on the plate.

Short answer

The mass transfer coefficient for the given situation is approximately \(9.86 \times 10^{-10} \frac{\mathrm{kg}}{\mathrm{m^2} \cdot \mathrm{s}}\).

Step-by-step solution

1. Find the Reynolds number

The Reynolds number (Re) is a dimensionless number that helps to predict flow patterns in fluid flow situations. It is given by the formula: $$Re=\frac{\rho v L}{\mu}$$ where \(\rho\) is the fluid density, \(v\) is the velocity, \(L\) is the length, and \(\mu\) is the fluid viscosity. For air at \(40^{\circ} \mathrm{C}\), we can look up the values of \(\rho\) and \(\mu\) to be approximately \(1.127 \frac{\mathrm{kg}}{\mathrm{m^3}}\) and \(1.95\times10^{-5} \frac{\mathrm{kg}}{\mathrm{m} \cdot \mathrm{s}}\), respectively. The velocity, \(v\), is given in the problem statement as \(2.5 \frac{\mathrm{m}}{\mathrm{s}}\), and the length, \(L\), is given as \(5 \mathrm{m}\). Therefore, we can calculate the Reynolds number to be: $$Re = \frac{(1.127)(2.5)(5)}{1.95\times10^{-5}} = 1.445\times10^5$$

2. Determine the heat transfer analogy for given Reynolds number

Since we are using an analogy between heat and mass transfer, we need to use the appropriate heat transfer analogy based on the Reynolds number of the flow. Since the Reynolds number is in the range of forced convection on a flat plate (\(Re < 5\times10^5\)), we will use the average Nusselt number for a smooth flat plate, which can be determined using the following equation: $$Nu = \frac{hL}{k} = \frac{0.664 \times Re^{1/2} \times Pr^{1/3}}{1+(0.492/Pr)^{9/16}^{2/3}}$$ where \(Nu\) is the Nusselt number, \(h\) is the heat transfer coefficient, \(k\) is the thermal conductivity, and \(Pr\) is the Prandtl number.

3. Calculate the Nusselt number

For air at \(40^{\circ} \mathrm{C}\), we can look up the Prandtl number to be approximately 0.70, and the thermal conductivity, \(k\) to be approximately \(0.0282 \frac{\mathrm{W}}{\mathrm{m} \cdot \mathrm{K}}\). We can then substitute the Reynolds number and the Prandtl number into the Nusselt number equation: $$Nu = \frac{0.664 \times (1.445\times10^{5})^{1/2} \times 0.7^{1/3}}{1+(0.492/0.7)^{9/16}^{2/3}} = 174.69$$

4. Determine the heat transfer coefficient

We can now determine the heat transfer coefficient by rearranging the Nusselt number equation and substituting in the Nusselt number, length, and thermal conductivity: $$h = \frac{Nu \times k}{L} = \frac{(174.69)(0.0282 \frac{\mathrm{W}}{\mathrm{m} \cdot \mathrm{K}})}{5 \mathrm{m}} = 0.982 \frac{\mathrm{W}}{\mathrm{m}^2 \cdot \mathrm{K}}$$

5. Calculate the mass transfer coefficient

Now we can use the analogy between heat and mass transfer to find the mass transfer coefficient, which is given by the following equation: $$k_{m} = h \times \frac{\rho_{L}}{C_{pL}\rho_{G}}$$ For air, the specific heat at constant pressure \(C_{pL}\) is approximately \(1005 \frac{ \mathrm{J} } { \mathrm{kg} \cdot \mathrm {K}}\), so we can determine the mass transfer coefficient to be: $$k_{m} = (0.982 \frac{\mathrm{W}}{\mathrm{m^2} \cdot \mathrm{K}}) \times \frac{(1.127 \frac{\mathrm{kg}}{\mathrm{m^3}})}{(1005 \frac{\mathrm{J}}{\mathrm{kg} \cdot \mathrm{K}} \times 1.127 \frac{\mathrm{kg}}{\mathrm{m^3}})} = 9.86 \times 10^{-10} \frac{\mathrm{kg}}{\mathrm{m^2} \cdot \mathrm{s}}$$ So the mass transfer coefficient on the plate is approximately \(9.86 \times 10^{-10} \frac{\mathrm{kg}}{\mathrm{m^2} \cdot \mathrm{s}}\).

Key concepts

Reynolds Number

The Reynolds number (Re) is a crucial dimensionless quantity in fluid mechanics that helps to characterize the type of flow—whether it is laminar or turbulent—around an object. It is calculated by the formula \(Re = \frac{\rho v L}{\mu}\), where \(\rho\) is the density of the fluid, \(v\) represents the flow velocity, \(L\) signifies a characteristic linear dimension (like the plate length in the given exercise), and \(\mu\) is the dynamic viscosity of the fluid. A high Reynolds number typically suggests turbulent flow, while a low Reynolds number indicates laminar flow.

In the given problem, the air is flowing over a wet plate, and calculating the Reynolds number helped determine which heat transfer analogy to use, leading to the accurate determination of the mass transfer coefficient. Understanding Reynolds number facilitates the prediction of fluid flow patterns and associated heat and mass transfer rates.

Nusselt Number

The Nusselt number (Nu) is another vital dimensionless parameter in the study of heat transfer, especially in forced convection scenarios. It represents the ratio of convective to conductive heat transfer across a boundary. Calculated as \(Nu = \frac{hL}{k}\), with \(h\) being the convective heat transfer coefficient, \(L\) is a characteristic length, and \(k\) is the thermal conductivity of the fluid.

In practical applications, such as drying a wet surface with air flow as in our exercise, the Nusselt number assisted in determining the heat transfer coefficient. Once established, this coefficient was pivotal in applying the analogy between heat and mass transfer to find the mass transfer coefficient, showcasing the intertwined relationship between these parameters.

Forced Convection

Forced convection occurs when a fluid's flow is induced by external means—such as a fan, pump, or in our case, the air moving over a plate. It differs from natural convection as it does not rely on buoyancy forces caused by temperature differences within the fluid.

The study of forced convection is essential when designing systems involving heat exchange, as understanding how the fluid moves is key to predicting how heat is transferred. This concept helps in calculating both Reynolds and Nusselt numbers, which then leads to the estimation of the heat and mass transfer coefficients. Hence, forced convection provides a basis for the analogy that bridges the gap between heat and mass transfer in the exercise.

Prandtl Number

The Prandtl number (Pr) is a dimensionless number that represents the ratio of momentum diffusivity to thermal diffusivity, or, in simpler terms, the ratio of viscosity to thermal conductivity scaled by the specific heat. It is expressed as \(Pr = \frac{\mu C_p}{k}\), where \(\mu\) is the fluid's viscosity, \(C_p\) is the specific heat at constant pressure, and \(k\) is the thermal conductivity.

What makes the Prandtl number so valuable in the context of our drying plate exercise is its role in the calculation of the Nusselt number—an essential step towards mapping from heat to mass transfer coefficients. A carefully determined Prandtl number not only affects the heat transfer calculations but also ensures accuracy when using analogies to deduce the mass transfer coefficient.