Question

Saturated steam at 1 atm condenses on a vertical plate that is maintained at \(90^{\circ} \mathrm{C}\) by circulating cooling water through the other side. If the rate of heat transfer by condensation to the plate is \(180 \mathrm{kJ} / \mathrm{s}\), determine the rate at which the condensate drips off the plate at the bottom.

Short answer

In this problem, we are asked to find the rate at which the condensate drips off a vertical plate experiencing saturated steam condensation under given conditions. By relating the heat transfer rate to the mass flow rate of the condensate and using the latent heat of vaporization, we calculated the mass flow rate to be approximately \(0.0797\,\mathrm{kg/s}\). Therefore, the condensate drips off the plate at the bottom at a rate of \(0.0797\,\mathrm{kg/s}\).

Step-by-step solution

Find the latent heat of vaporization at 1 atm

To proceed, we need to know the latent heat of vaporization of water at \(1\,\text{atm}\). You can find this value on steam tables or textbooks on thermodynamics. Another option is using online resources such as the Engineering Toolbox or NIST webbook. At \(1\,\text{atm}\), the latent heat of vaporization of water is approximately \(2257\,\mathrm{kJ/kg}\).

Write the equation relating heat transfer rate and mass flow rate of condensate

The mass flow rate of the condensate is related to the rate of heat transfer by the following equation: \(\dot{q}=\dot{m}\cdot L_v\) where \(\dot{q}\) is the heat transfer rate \((180\, \mathrm{kJ/s})\) \(\dot{m}\) is the mass flow rate of the condensate (kg/s) - which we want to determine \(L_v\) is the latent heat of vaporization of water at \(1\, \mathrm{atm}\) \((2257\,\mathrm{kJ/kg})\)

Determine the mass flow rate of the condensate with the given heat transfer rate

We can solve the equation for the unknown mass flow rate \(\dot{m}\): \(\dot{m} = \frac{\dot{q}}{L_v}\) Now plug in the given values: \(\dot{m} = \frac{180\,\mathrm{kJ/s}}{2257\,\mathrm{kJ/kg}} = 0.0797\,\mathrm{kg/s}\)

Conclusion

The rate at which the condensate drips off the plate at the bottom is \(0.0797\,\mathrm{kg/s}\).