LMTD Model

If the condenser uses the flow rate of a cooling medium (typically cooling water) or if the reboiler uses the flow rate of a heating medium (hot oil), a model, using a log-mean temperature differential driving force (temperature differentials at outlet and inlet ends), can be used. The inlet medium temperature and the minimum approach temperature difference between the process and the medium are specified. Then Aspen calculates the required UA product (overall heat-transfer coefficient U and heat-transfer area A) and the required flow rate of the medium from the known heat-transfer rate. 

The flow rate of the medium is manipulated in the Aspen Dynamic simulations, not Qc or (2r directly. However, this model contains no d5mamics. The holdup of medium in the heat exchanger is not considered. Medium flow rate changes produce instantaneous changes in temperature driving forces and subsequent heat-transfer rates. 

---

In the first case (constant QC), the condenser duty was held constant. In the second case (constant TC), the reflux-drum temperature was controlled by manipulating condenser heat duty. In the third case (constant CW), the flow rate of cooling water was held constant (using the LMTD model). Disturbances in the flow rate and composition of the feed to the column were made. 

The resulting model is a MINLP with linear constraints and nonlinear objective function. The objective function terms can beconvexified using the exponential transformation with the exception of the condenser-cold utility expressions. If we replace the LMTDs with 2-3 times the ATmln, then, the whole objective can be convexified Floudas and Paules (1988). This implies that its global solution can be attained with the OA/ER or the v2-GBD algorithms. 

Solving dynamic models that employ the LMTD can, however, be problematic because the LMTD is not well defined when the temperature gradient along the heat exchanger is constant i.e., Tr = Tc and Tr = Tm (note that heat transfer in a physical exchanger would still take place under these circumstances). Moreover, the LMTD is not well defined in the case of a temperature cross-over (e.g., Tr > Tc and Tr < Tn), a situation that can arise temporarily during transient operation. These issues were recognized relatively early (Paterson 1984), and several approximate formulations with improved numerical properties have been... 

Figure 8.21 Select LMTD dynamic model in partial condenser.

The most realistic option for partial condenser modeling is the LMTD option. The cooling medium is a liquid that enters a counter-current heat exchanger at a specified inlet temperature. The minimum approach differential temperature is specified. The process inlet and outlet temperatures are known, so the log-mean temperature differential driving force is known. With the known condenser duty, the required product of the overall heat-transfer coefficient and the condenser heat-transfer area (UA) is calculated. The required flow rate of the cooling medium can also be calculated. 

The LMTD option produces a dynamic model in which the flow rate of the cooling medium is the manipulated variable. This is the model that will reahstically provide prediction of how the partial condenser system responses to disturbances. 

The variable U, A, and 0 represent the overall heat transfer coefficient, the area of the heat exchange, and the temperature difference between the polymer and water streams at different points. The logarithmic-mean-temperature difference (LMTD) is defined in the second part of equation (3). The primary modeling effort was to use this equation to calculate the temperature difference at point 1 since all other variables were known. Unfortunately, this equation has a discontinuity when the temperature difference at points 1 and 2 are equal. This discontinuity makes simulation difficult since it separates the feasible space for the temperature difference into two regions. Equation (3) can be written to avoid this discontinuity. 

---