Question

Saturated water vapor is condensing on a \(0.5 \mathrm{~m}^{2}\) vertical flat plate in a continuous film with an average heat transfer coefficient of $5 \mathrm{~kW} / \mathrm{m}^{2} \cdot \mathrm{K}$. The temperature of the water is $80^{\circ} \mathrm{C}\left(h_{f g}=2309 \mathrm{~kJ} / \mathrm{kg}\right)\(, and the temperature of the plate is \)60^{\circ} \mathrm{C}$. The rate at which condensate is being formed is (a) \(0.022 \mathrm{~kg} / \mathrm{s}\) (b) \(0.048 \mathrm{~kg} / \mathrm{s}\) (c) \(0.077 \mathrm{~kg} / \mathrm{s}\) (d) \(0.16 \mathrm{~kg} / \mathrm{s}\) (e) \(0.32 \mathrm{~kg} / \mathrm{s}\)

Short answer

(a) 0.022 kg/s (b) 0.045 kg/s (c) 0.067 kg/s (d) 0.090 kg/s Answer: (a) 0.022 kg/s

Step-by-step solution

Calculate the temperature difference

First, we have to find the temperature difference between the water and the plate, which is given as follows: $$ \Delta T = T_{water} - T_{plate} $$

Calculate the heat transfer rate

Now, we will calculate the heat transfer rate using the given average heat transfer coefficient (h), the temperature difference (ΔT), and the area (A): $$ Q = h \cdot A \cdot \Delta T $$

Calculate the mass flow rate of condensate

Next, we will find the mass flow rate (ṁ) of the condensate by dividing the heat transfer rate (Q) by the latent heat of vaporization (h_fg): $$ \dot{m} = \frac{Q}{h_{fg}} $$ Now, let's plug in the given values and solve:

Calculate the temperature difference

We are given: $$ T_{water} = 80^{\circ}C \\ T_{plate} = 60^{\circ}C $$ Using the temperature difference formula: $$ \Delta T = 80 - 60 = 20^{\circ}C $$

Calculate the heat transfer rate

We are given: $$ h = 5 \frac{kW}{m^2 K} \\ A = 0.5 m^2 \\ \Delta T = 20^{\circ}C $$ Using the heat transfer rate formula: $$ Q = (5 \frac{kW}{m^2 K}) \cdot (0.5 m^2) \cdot (20^{\circ}C) = 50 kW $$

Calculate the mass flow rate of condensate

We are given: $$ Q = 50 kW \\ h_{fg} = 2309 \frac{kJ}{kg} $$ Note that we need to convert kW to kJ/s: $$ 50 kW = 50 kJ / s $$ Using the mass flow rate formula: $$ \dot{m} = \frac{50 kJ/s}{2309 kJ/kg} \approx 0.0216 \frac{kg}{s} $$ The calculated mass flow rate is approximately equal to 0.022 kg/s, which corresponds to the answer (a).