The Problem Table Algorithm

It is important to note that shifting the curves vertically does not alter the horizontal overlap between the curves. It therefore does not alter the amount by which the 

This shifting technique can be used to develop a strategy to calculate the energy targets without having to construct composite curves1,9  

In each shifted temperature interval, calculate a simple energy balance from  

This basic approach can be developed into a formal 

First determine the shifted temperature intervals (T ) from actual supply and target temperatures. Hot streams are shifted down in temperature by ATmul/2 and cold streams up by ATmin/2 as detailed in Table 16.3. 

Energy Targets for Heat Exchanger Network and Utilities 175 

It is important to note that shifting the curves vertically does not alter the horizontal overlap between the curves. It therefore does not alter the amount by which the cold composite curve extends beyond the start of the hot composite curve at the hot end of the problem and the amount by which the hot composite curve extends beyond the start of the cold composite curve at the cold end. The shift simply removes the problem of ensuring temperature feasibility within temperature intervals. 

Set up shifted temperature intervals from the stream supply and 

---

This basic approach can be developed into a formal algorithm known as the problem table algorithm. To jllustrate the algorithm, it can be developed using the data from Fig. 6.2 given in Table 6.2 for AT ,i = 10°C. 

The composite curves are useful in providing conceptual understanding of the process, but the problem table algorithm is a more convenient calculation tool. 

The question now is, given that there are often constraints to deal with, how do we evaluate the effect of these constraints on the system performance The problem table algorithm cannot be used directly if constraints are imposed. However, often the effect of constraints on... 

The shaded areas in Fig. 6.24, known as pockets, represent areas of additional process-to-process heat transfer. Remember that the profile of the grand composite curve represents residual heating and cooling demands after recovering heat within the shifted temperature intervals in the problem table algorithm. In these pockets in Fig. 6.24, a local surplus of heat in the process is used at temperature differences in excess of AT ,in to satisfy a local deficit. ... 

Solution The first problem is that a different value of AT ,i is required for difi erent matches. The problem table algorithm is easily adapted to accommodate this. This is achieved by assigning AT ,i contributions to streams. If the process streams are assigned a contribution of 5 C and flue gas a contribution of 25°C, then a process-process match has a of 5 -H 5 = 10 C and a... 

Having decided that no exchanger should have a temperature difference smaller than ATmi, two rules were deduced. If the energy target set by the composite curves (or the problem table algorithm) is to be achieved, there must be no heat transfer across the pinch by... 

The final design shown in Fig. 16.7 amalgamates the hot-end design from Fig. 16.5c and the cold-end design from Fig. 16.6c. The duty on hot utility of 7.5 MW agrees with Qnmm and the duty on the cold utility of 10 MW agrees with Qcmm predicted by the composite curves and the problem table algorithm. 

When carrying out the problem table algorithm, the temperatures were shifted according to ATmin/2 being added to the cold streams and subtracted from the hot streams. This value of ATmin/2 can be considered to be a contribution to the overall A Tmin between the hot and the cold streams. Rather than making the A Tmin contribution equal for all streams, it could be made stream-specific ... 

The pinch design method, as discussed so far, has assumed the same A Tmin applied between all stream matches. In Chapter 16, it was discussed how the basic targeting methods for the composite curves and the problem table algorithm can be modified to allow stream-specific values of A Tmin. The example was quoted in which liquid streams were required to have a A Tmin contribution of 5°C and gas streams a ATmin contribution of 10°C. For liquid-liquid matches, this would lead to a ATmin = 10°C. For gas-gas matches, this would lead to a ATmin = 20°C. For liquid-gas matches, it will lead to a ATmin = 15°C 2. Modifying the problem table and the composite curves to account for these stream-specific values of ATmin is straightforward. But how is the pinch design method modified to take account of such A Tmin contributions Figure 18.9 illustrates the approach. Suppose the interval pinch temperature from the problem table is 120°C. This would correspond with hot stream pinch temperatures of 125°C and 130°C for hot streams with ATmin contributions of 5°C and 10°C respectively. For... 

Suppose that the actual behavior of temperature versus enthalpy is known and is highly nonlinear, as shown in Figure 19.4. How can the nonlinear data be linearized so that the construction of composite curves and the problem table algorithm can be performed Figure 19.4 shows the nonlinear streams being represented by a series of linear segments. The linearization of the hot streams should... 

To ensure that both concentration and time constraints are met, this analysis should be applied in each of the concentration intervals. It would be expected that a similar approach to the Problem Table Algorithm (Linnhoff and Flower, 1978) should be applied in order to ensure that a specific minimum concentration difference holds in each of the intervals. This would entail shifting of the inlet and outlet concentrations for the process streams as it was done in setting the energy targets. However, the fact that the limiting concentration constraints have been built into the problem makes the shifting of concentrations irrelevant. The concentration intervals are demarcated by the inlet and outlet concentrations as shown in Fig. 12.3. 

The problem table to find the minimum utility requirements and the pinch temperature can be built in a spreadsheet. The calculations in each cell are repetitive, and the formula can be copied from cell to cell using the cell copy commands. A spreadsheet template for the problem table algorithm is available in MS Excel format in the online material at http //books.elsevier.com/companions. The use of the spreadsheet is illustrated in Eigure 3.31 and described here. 

Step 1. The first step in the problem table algorithm is to adjust the temperatures to include a finite temperature difference for heat transfer in the subsequent analysis. Hot streams are adjusted by reducing their temperature by AT., /2. Cold streams are adjusted by increasing their temperature by AT., /2. These adjusted temperatures are known as interval temperatures. They are not real temperatures, and the adjustment is made purely for the sake of the algorithm. If a hot and a cold stream have the same interval temperature, then they are in reality separated by just ATmin- Table 2 shows the supply and target temperatures of the streams from Table 1 adjusted for ATmin- In this case, the temperatures need to be adjusted by 5°C. 

The other feature to note regarding the cascade in Fig. IIB is the point where the heat flow becomes zero. This is at an interval temperature of 75°C. This corresponds to the location of the pinch. Given that the interval temperature for the pinch is 75°C, the hot stream pinch temperature is therefore 80° C and the cold stream pinch temperature 70° C. This agrees with the location of the pinch from the composite curves. Thus, the problem table algorithm provides the information required for minimum hot and cold utility targets and the location of the pinch that is critical in the next step, which is to design to achieve the targets. 

The fundamental computational tool is the Problem Table algorithm. This tool allows the identifications of the Pinch, as well as of targets for hot and cold utilities. 

---