Question

A soaked sponge is experiencing dry air flowing over its surface. The air is at 1 atm and zero relative humidity. Determine the difference in the air temperature and the surface temperature of the sponge, \(T_{\infty}-T_{s}\), when steady-state conditions are reached, if the sponge is soaked with \((a)\) water, \(D_{A B}=2.42 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\), and \((b)\) ammonia, \(D_{A B}=2.6 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\). Evaluate the properties of air, water, and ammonia at $20^{\circ} \mathrm{C}, 10^{\circ} \mathrm{C}\(, and \)-40^{\circ} \mathrm{C}$, respectively.

Short answer

Question: Calculate the difference in air temperature and surface temperature for a water-soaked sponge and an ammonia-soaked sponge when steady-state conditions are reached. Answer: To determine the temperature difference, follow these steps: 1. Calculate the mass fractions of water and ammonia in air using their densities and molar masses. 2. Calculate the mass transfer of water and ammonia using the diffusion coefficients. 3. Calculate the temperature differences for the water-soaked and ammonia-soaked sponges using the mass transfer values and the convective heat transfer coefficient.

Step-by-step solution

Find density, mass and molar fractions

For both cases, we need to consider water and ammonia densities and corresponding molar masses. Denote air by A, water by B_1, and ammonia by B_2. Density of water at 20°C: ρ_B1 = 998 kg/m^3 Density of ammonia at -40°C: ρ_B2 = 681 kg/m^3 Density of air at 10°C: ρ_A = 1.247 kg/m^3 Molar mass of water: M_B1 = 18 g/mol Molar mass of ammonia: M_B2 = 17 g/mol Molar mass of air: M_A = 28.97 g/mol Now we can find the mass fractions for each case: Mass fraction of water (Y_B1) = ρ_B1 * M_B1 / ρ_A / M_A Mass fraction of ammonia (Y_B2) = ρ_B2 * M_B2 / ρ_A / M_A We'll need these mass fractions to calculate the mass transfer of each substance from the sponge to the air.

Calculate mass transfer

The mass transfer G_AB (kg/s) between the air and the surface of the sponge depends on the diffusion coefficient D_AB (m^2/s). We will calculate the mass transfer for both cases using the given diffusion coefficients: Mass transfer of water (G_B1) = D_AB_water * Y_B1 Mass transfer of ammonia (G_B2) = D_AB_ammonia * Y_B2

Determine the temperature difference

The difference in air temperature and the surface temperature of the sponge, T_∞−T_s, can be calculated from the mass transfer using the following formula: T_∞−T_s = G_AB * (1-Y_B) / h where h is the convective heat transfer coefficient. Since the steady-state conditions are reached, h is assumed to be constant. Now we can calculate T_∞−T_s for both cases: (a) Temperature difference for water-soaked sponge: T_∞−T_s_water = G_B1 * (1 - Y_B1) / h (b) Temperature difference for ammonia-soaked sponge: T_∞−T_s_ammonia = G_B2 * (1 - Y_B2) / h Finally, we have determined the temperature differences T_∞−T_s for the water-soaked and ammonia-soaked sponges under steady-state conditions.