Mass and heat transfer - evaparation.xls

KNOWN: Operating temperature, ambient air conditions and make-up water requirements for a hot tub.

FIND: Heater power required to maintain prescribed conditions.

ASSUMPTIONS: 1) Side wall and bottom are adiabatic.

2) Heat and mass transfer analogy is applicable.

Calculation Reference
Fundamentals of Heat and Mass Transfer - Frank P. Incropera

To determine the heater power required to maintain prescribed conditions in a hot tub, considering heat and mass transfer during evaporation, you can follow these steps:

1. Determine the prescribed conditions: Identify the desired temperature (T_desired) of the hot tub and the ambient air conditions, including the temperature (T_ambient) and relative humidity (RH_ambient). Also, determine the make-up water requirements, such as the water temperature (T_water) and the flow rate (Q_water).

2. Calculate the water evaporation rate: Use the heat and mass transfer analogy to calculate the water evaporation rate (m_dot) from the hot tub. The equation for evaporation rate is:

   m_dot = h_m * A_w * (P_w - P_v)

   Where h_m is the mass transfer coefficient, A_w is the water surface area, P_w is the partial pressure of water vapor at the water surface temperature, and P_v is the partial pressure of water vapor in the ambient air. The mass transfer coefficient (h_m) can be obtained from correlations or experimental data.

3. Determine the heat transfer required: The heat transfer required (Q) to maintain the prescribed conditions is given by:

   Q = m_dot * h_vap + Q_water

   Where h_vap is the latent heat of vaporization of water and Q_water is the heat transferred by the make-up water. The latent heat of vaporization can be obtained from thermodynamic data.

4. Calculate the heater power: The heater power required is given by:

   P_heater = Q / efficiency

   Where efficiency is the efficiency of the heating system.

By following these steps and incorporating the heat and mass transfer analogy, you can determine the heater power required to maintain the prescribed conditions in the hot tub. It takes into account the evaporation process and the heat transfer required to compensate for the energy loss due to evaporation and the make-up water requirements.