Heat and Mass Transfer in Outdoor Spa Bath?

[Yr11Kid]

Hi everyone I've got a problem about the convective heat loss from a spa
any help or pointers would be greatly appreciated

A person installed a circular outdoor spa bath in the open area. In a typical operating condition, when the spa water is maintained at 37°C, and when the ambient air has a temperature of 17°C with a relative humidity of 30%, he has to top up the water at a rate of 0.002 kg/s so as to maintain a fixed water level in the bath. The tub is well insulated on all sides and its bottom, and the makeup water enters at 17°C.

a) Determine the value of hmA that commensurate with the makeup water flow rate. The symbols have their usual meaning.

b) Determine the extent of convective heat loss from the spa bath.

c) Determine the electrical power needed to maintain the spa bath temperature at 37°C.


At 300K, rhoair = 1.161 kg/m3, cp,air = 1.007 kJ/kgK, alpha = 22.5X 10-6 m2/s.
At 310K, rhowater vapour,sat = 0.0436 kg/m3, hfg = 2414 kJ/kg.
At 290K, rhowater vapour,sat = 0.0143 kg/m3.
DAB = 26X 10-6 m2/s, cp,water = 4.2 kJ/kgK



For part a i found hmA by using the evaporative cooling rate, the humidity and the density of saturated water vapor at 310k.

part b i don't quite understand, do i simply use q=hmA(Ts-Tambient) or do i need to find the temperature gradient between both mediums utilising the heat and mass transfer coefficients. This would make sense as i would have to use more of the given information like the diffusivity etc

for part c am i just doing a simple energy balance on the system?

Thanks

 

[KataKoniK]

for reaching out for help with your spa heat loss problem. Let me break down the steps for you to solve each part of the problem.

a) To determine the value of hmA, you will need to use the evaporative cooling rate, humidity, and density of saturated water vapor at 310K. The evaporation rate is given as 0.002 kg/s, so you can use the following equation to solve for hmA:

hmA = (m_dot * h_fg) / (rho_air * (w_sat - w_air))

Where:
- m_dot is the evaporation rate (0.002 kg/s)
- h_fg is the latent heat of vaporization (2414 kJ/kg)
- rho_air is the density of air at 310K (1.161 kg/m3)
- w_sat is the humidity of saturated water vapor at 310K (0.0436 kg/m3)
- w_air is the humidity of air at 310K (0.3 * w_sat = 0.01308 kg/m3)

b) To determine the extent of convective heat loss from the spa bath, you can use the heat transfer equation, q = hmA(Ts - Tambient). However, you will need to calculate the surface temperature of the spa bath, Ts, using the heat and mass transfer coefficients and the temperature gradient between the spa water and ambient air. The temperature gradient can be calculated using the following equation:

dT/dx = (Ts - Tambient) / (hmA * (1 / DAB + 1 / alpha))

Where:
- dT/dx is the temperature gradient (in K/m)
- Ts is the surface temperature of the spa bath (in K)
- Tambient is the ambient air temperature (310K)
- hmA is the mass transfer coefficient calculated in part a (in kg/m2s)
- DAB is the diffusion coefficient of water vapor in air (26 * 10^-6 m2/s)
- alpha is the thermal diffusivity of air (22.5 * 10^-6 m2/s)

Once you have calculated the temperature gradient, you can use it in the heat transfer equation to solve for the convective heat loss, q.

c) To determine the electrical power needed to maintain the spa bath temperature at 37°C, you can use the energy balance equation:

Power = m_dot * cp_water * (Tspawater -