Looking for $A_1=A_2=A_3=\dots$
- The case of $BPE=0$ -
The interpretation of zero boiling point elevation $BPE$ could be that the boilint temperature of the solvent and that of the solution are the same. This could possibly occure if concentration by evaporation would occur in a short range in dilute systems. One simple example could be found in the distillation of water.
On the other hand, the most feasible scenario is to get,
$A_1\approx A_2\approx A_3\approx \dots$ Eq. (01)
since the heating surfaces depend in a very intricate way on the temperature.
For the present post, the situtation about a triple effect evaporator in backward feed mode Example 4.3 presented in the textbook of Donald Q. Kern Process heat transfer is used.
General approach to estimate $A_1\approx A_2\approx A_3\approx \ldots$
In general, the requirement of heating surface for certain operating conditions are calculated at the end of all estimations so that if in a multiple effect evaporator, the heating surfaces were the same, big troubles come out. From this last scenario, only one thing can be done: to concile very close heating surfaces of all effects with the operating conditions, from mass and heat balances, of the equiment.
Let us dig a little on the above statement. Heating surfaces are estimated from (for the present case),
$A_1=\dfrac{q_1}{U_1\left( T_{W1}-T_{S1} \right)}$ Eq. (02)
$A_2=\dfrac{q_2}{U_2\left( T_{L1}-T_{S2} \right)}$ Eq. (03)
$A_3=\dfrac{q_3}{U_3\left( T_{L2}-T_{S3} \right)}$ Eq. (04)
and so on for more effects. As you can see from Eq. (03), changing $A_1$ requires modify $T_{S1}$, which will in turn modify $\lambda_{W1}$, a similar situation is found in Eqs. (03-04). Furthermore, changing the temperatures will also change the flows of heat $q$'s into each effect. Next, all temperature changes must satisfy the mass and heat balances. What all this means, is that the dependency of the heating surfaces on the temperatures is not straight but rather complex.
The only way of approaching Eq. (01) is by trial and error. Here, the procedure to achieve Eq. (01) in a seudo iterative manner is outlined. Consider first that the set of heating surfaces $A_1$, $A_2$ and $A_3$ are modified a correction factor into $A_1^{(new)}$, $A_2^{(new)}$ and $A_3^{(new)}$ as follows,
$A_1^{(new)}=C_1A_1$ Eq. (05)
$A_2^{(new)}=C_2A_2$ Eq. (06)
$A_3^{(new)}=C_3A_3$ Eq. (07)
where the correction factors $C_1$, $C_2$ and $C_3$ are such that the heating surfaces approach Eq. (01). Next, since the heating surfaces areas and the dirving temperature gradients are inversely proportional, these should be modified as,
$T_{W1}-T_{S1}=\dfrac{T_{W1}-T_{S1}^{(new)}}{C_1}$ Eq. (08)
$T_{L1}-T_{S2}=\dfrac{T_{L1}^{(new)}-T_{S2}^{(new)}}{C_2}$ Eq. (09)
$T_{L2}-T_{S3}=\dfrac{T_{L2}^{(new)}-T_{L3}^{(new)}}{C_3}$ Eq. (10)
Yo should recall that for the case as presented by Kern: $T_{S1}=T_{L1}$, $T_{S2}=T_{L2}$ and $T_{S3}=T_{L3}$, which makes the calculation of the new temperatures in Eqs. (08-10) much easier. All new temperatures can be obtained in a recursive manner starting from Eq. (08), so that a new set of temperatures $T_{S1}^{(new)}$, $T_{S1}^{(new)}$ and $T_{S1}^{(new)}$, are obtained.
Once, the new temperatures hace been set, the fluid properties need to be recalculated along with the mass and heat balances. These calculations would produce a thrid set of new heating surfaces: $A_1$, $A_2$ and $A_3$, which are supposed to be closed to Eq. (01). If not, the process needs to be repeated for a new set of correction factors until a reasonable approximation to Eq. (01) is achieved. You should notice that these new correction factors could not be related to the previous ones since the dependence on temperature is complex.
This is the end of the post. I hope you find it useful.
Other stuff of interest
- LE01 - AC and DC voltage measurement and continuity test
- LE 02 - Start and stop push button installation 24V DC
- LE 03 - Turn on/off an 24V DC pilot light with a push button
- LE 04 - Latch contact with encapsulated relay for turning on/off an AC bulb light
- LE 05 - Emergency stop button installation
- About PID controllers
- Ways to control a process
- About pilot lights
- Solving the Colebrook equation
- Example #01: single stage chemical evaporator
- Example #02: single stage process plant evaporator
- Example #03: single stage chemical evaporator
- Example #04: triple effect chemical evaporator
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Ildebrando.
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