Instant Notes: 7. Evaporation
Temperature Drop in Each Effect
Heat transfer in each effect of multiple effect evaporation is given by:
\[Q_i = U_i A_i \Delta T_i\]
It is desired to have uniform equal heat transfer rates and areas, due to the following reasons:

1 kg of condensing steam can generate nearly 1 kg of water vapor. Hence from the point of consideration of steam it is better to have equal heat transfer rates in every effect of the multiple effect evaporation.

With uniform heat transfer area, every effect will be of same size, leading to reduction in initial investment due to economy of scaling.
Since \(Q_i\), and \(A_i\) of every effect is the same, i.e.,
\[\frac{Q_i}{A_i} = \frac{Q_1}{A_1} = \frac{Q_2}{A_2} = \cdots = \text{constant}\] we get
\[U_i\Delta T_i = U_1\Delta T_1 = U_2\Delta T_2 = \cdots = U_n\Delta T_n = \text{constant}\]
The total temperature drop across the effects is given as
\[\Delta T = T_s  T_n\]
where \(T_s\) and \(T_n\) are the temperatures of the steam to the first effect and the vapor formed in the last effect, \(n\). This temperature drop is also equal to the sum of temperature drops in every effect, given as: \[\begin{aligned} \Delta T &= \Delta T_1 + \Delta T_2 + \Delta T_3+ \cdots + \Delta T_n \\ &= \frac{\Delta T_1U_1}{U_1} + \frac{\Delta T_1U_1}{U_2} + \frac{\Delta T_1U_1}{U_3} + \cdots + \frac{\Delta T_1U_1}{U_n} \\ \Longrightarrow \quad \Delta T_1 &= \frac{\Delta T}{U_1\,{\displaystyle\sum_i}\dfrac{1}{U_i}}\end{aligned}\]
The above result can be generalized, for any effect \(i\), as below:
\[\Delta T_i = \frac{\Delta T}{U_i\,{\displaystyle\sum_i}\dfrac{1}{U_i}}\]
This formula would be useful to find the temperature difference in each effect for triple and higher effects evaporator.