Triple-Effect Evaporation

This simulation shows the efficacy of a triple-effect evaporator. A 5.0 wt% sugar solution is fed into a tank, which contains a steam-fed heat exchanger. The evaporated steam from the first evaporator is used to heat the solution in the second evaporator, and the evaporated steam from the second evaporator is used to heat the solution in the third evaporator. The user adjusts the flow rate and temperature of the feed solution. The conditions of inlet and outlet sugar solution streams (and the vapor formed) are shown on the image. Hover your mouse over the heat exchangers, valves, or arrows to see more details about them.

The rate of a heat exchange $ (Q) $ in Watts in an evaporator is:

$$ [1] \quad \dot{ Q } = U A \; ( T_{ steam } - T_{ conc } ) $$

where $ U $ is the overall heat transfer coefficient [W/(m²K)], $ A $ is the area of heat transfer (m²), $ T_{conc} $ is the temperature of the concentrated sugar solution (K), and $ T_{ steam } $ is the temperature of the steam (K), which is saturated. The value of $ U $ depends on the fluids' density, viscosity, velocity, thermal conductivity, and heat capacity. The values of $ T_{ steam } $ and $ T_{ conc } $ are obtained from Antoine's equations:

$$ [2] \quad \mathrm{ log }_{ 10 } \left( P \right) = A - \frac{ B }{ T + C } $$

where $ P $ is the pressure (Pa), $ A $, $ B $, and $ C $ are constants (from reference 1), and $ T $ is the temperature (K). The required mass flow rate of steam $ S $ (kg/s) in the first evaporator was calculated from:

$$ [3] \quad S = \frac{ \dot{ Q } }{ \Delta H_{ evap, s } } $$

where $ \Delta H_{ evap, s } $, which is the heat of vaporization (J/kg) of steam, was determined using an approximation from reference 2. The mass flow rate ($ V_{ 1 } $, kg/s) of water vapor leaving the first evaporator ($ V_{1} $) is calculated from

$$ [4] \quad V_{1} = \frac{ \dot{ Q } - F_{ in } \, C_{ p } \, \left( T_{ conc } - T_{ in } \right) }{ \Delta H_{ vap, 1 } } $$

where $ F_{ in } $ is the mass flow rate of the feed solution (kg/s), $ C_{p} $ is the heat capacity of the solution [J/(kg K)] (from reference 3), $ T_{in} $ is the temperature of the feed solution (K), and $ \Delta H_{vap,1} $ is the heat of vaporization of the solution in the first evaporator (from reference 1). The mass flow rate ($ V_{2} $, kg/s) of water vapor leaving the second evaporator is the sum of the flash-vaporized water from the liquid stream of the first evaporator ($ V_{flash,2} $) and water evaporated from the second heat exchanger ($ V_{hx,2} $):

$$ [5] \quad V_{2} = V_{flash,1} + V_{hx,2} $$

where

$$ [6] \quad V_{flash,1} = (1 - q) * L_{1} $$

and

$$ [7] \quad V_{hx,2} = \frac{ U A \; ( T_{1} - T_{2} ) }{ \Delta H_{vap, 2} } $$

where $ q $ is the quality of the vapor-liquid mixture (mostly liquid) entering the second evaporator, $ L_{1} $ is the liquid flow rate leaving the first evaporator, $ \Delta H_{vap,2} $ is the heat of vaporization of the liquid in the second evaporator, and $ T_{1} $ and $ T_{2} $ are the temperatures in the first and second evaporators, respectively. Quality is determined using steam tables and an energy balance,

$$ [8] \quad H_{L1} = q * H_{L2} + (1 - q) * H_{V2} $$

where $ H_{L1} $ and $ H_{L2} $ are the specific enthalpies of the liquid in the first and second evaporators, respectively, and $ H_{V2} $ is the specific enthalpy of the vapor in the second evaporator. Once $ V_{2} $ is found, the liquid flow rate from the second evaporator ($ L_{2} $) is found using a mass balance:

$$ [9] \quad L_{2} = L_{1} - V_{2} $$

The same procedure is used to find the vapor and liquid flow rates from the third evaporator. The concentration of sugar in the concentrate stream of the third evaporator ($ x_{ 3 } $) is found with a mass balance:

$$ [10] \quad x_{ 3 } = \frac{ x_{in} * F_{in} }{ L_{3} } $$

The steam economy $ SE $ is the ratio of the total water evaporated to the amount of steam used:

$$ [11] \quad SE = \frac{ L_{3} - F_{ in } }{ S } $$

More accurate boiling points in each evaporator can be obtained using Raoult's law, based on the mole fraction of water in each evaporator, but this small correction was ignored since the mole fractions of sugar in the concentrate were low.

References:

(1) National Institute of Standards and Technology (2021). Water phase change data. Retrieved from NIST's website.
(2) My Chemical Engineering Musings (2019). Handy equations to calculate Heat of Evaporation and Condensation of Water/Steam.
Click here to download the article.
(3) Mohos, F.Á. (2017). Data on Engineering Properties of Materials Used and Made by the Confectionery Industry. In Confectionery and Chocolate Engineering, F.Á. Mohos (Ed.). https://doi.org/10.1002/9781118939741.app1.