Heat Transfer Constraints

The amount of boilup vapor produced from the reboiler is constrained by the heat transfer surface in the reboiler and the fouling in combination with the temperature of the heating medium compared to the boiling temperature. The amount of condensation is constrained by the amount of heat transfer surface in the condenser and fouling in combination with the temperature of the cooling medium compared to the bubble point of the overhead vapor stream at the column pressure. 

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One of the challenging aspects of distillation column control is the many limitations imposed on the operation of the column. There are hydraulic constraints, separation constraints, heat-transfer constraints, pressure constraints, and temperature constraints. We recommend the excellent books by Kister (1992 and 1990) on distillation design and operation. 

Heat transfer constraints Heat must be transferred into the liquid in the reboiler to boil off the vapor needed to provide the vapor-liquid contacting in the column. If the base temperature becomes too high and approaches the temperature of the heating source, the heat transfer rate will decrease and vapor boilup will drop. The same result occurs if the reboiler fouls and the heat transfer coefficient drops. In the condenser, heat must be transferred from the hot vapor into the coolant stream to remove the heat of condensation. If the column is operating at its maximum pressure, capacity maybe limited by condenser heat removal. 

Because of the rate limitations of the tower and tube-tank processes that were primarily heat transfer constraints, further developments in the continuous solution process for crystal polystyrene (GP) were aimed at improving heat transfer. One obvious solution was to incorporate agitation of some type in the reactor. Although at Dow the incorporation of agitation in the reactors came about with the development of rubber-modified polystyrene [11], and this aspect will be discussed in a later section, agitation also significantly raises the heat transfer... 

Find a way to overcome the constraint while still maintaining the areas. This is often possible by using indirect heat transfer between the two areas. The simplest option is via the existing utility system. For example, rather than have a direct match between two streams, one can perhaps generate steam to be fed into the steam mains and the other use steam from the same mains. The utility system then acts as a buffer between the two areas. Another possibility might be to use a heat transfer medium such as a hot oil which circulates between the two streams being matched. To maintain operational independence, a standby heater and cooler supplied by utilities is needed in the hot oil circuit such that if either area is not operational, utilities could substitute heat recovery for short periods. 

Thus loops, utility paths, and stream splits offer the degrees of freedom for manipulating the network cost. The problem is one of multivariable nonlinear optimization. The constraints are only those of feasible heat transfer positive temperature difference and nonnegative heat duty for each exchanger. Furthermore, if stream splits exist, then positive bremch flow rates are additional constraints. 

Third, design constraints are imposed by the requirement for controlled cooling rates for NO reduction. The 1.5—2 s residence time required increases furnace volume and surface area. The physical processes involved in NO control, including the kinetics of NO chemistry, radiative heat transfer and gas cooling rates, fluid dynamics and boundary layer effects in the boiler, and final combustion of fuel-rich MHD generator exhaust gases, must be considered. 

The best quahty to be found may be a temperature, a temperature program, a concentration, a conversion, a yield of preferred product, a cycle period for a batch reaction, a daily production level, a land of reactor, a size for a reactor, an arrangement of reactor elements, provisions for heat transfer, profit or cost, and so on—a maximum or minimum of some of these factors. Among the constraints that may be imposed on the process are temperature range, pressure range, corrosiveness, waste disposal, and others. 

The radiant section of an industrial boiler may typically contain only 10 per cent of the total heating surface, yet, because of the large temperature difference, it can absorb 30-50 per cent of the total heat exchange. The mean temperature difference available for heat transfer in the convective section is much smaller. To achieve a thermally efficient yet commercially viable design it is necessary to make full use of forced convection within the constraint of acceptable pressure drop. 

So far, consideration has been limited to chemistry physical constraints such as heat transfer may also dictate the way in which reactions are performed. Oxidation reactions are highly exothermic and effectively there are only two types of reactor in which selective oxidation can be achieved on a practical scale multitubular fixed bed reactors with fused salt cooling on the outside of the tubes and fluid bed reactors. Each has its own characteristics and constraints. Multitubular reactors have an effective upper size limit and if a plant is required which is too large to allow the use of a single reactor, two reactors must be used in parallel. 

The differential equation describing the temperature distribution as a function of time and space is subject to several constraints that control the final temperature function. Heat loss from the exterior of the barrel was by natural convection, so a heat transfer coefficient correlation (2) was used for convection from horizontal cylinders. The ends of the cylinder were assumed to be insulated. The equations describing these conditions are ... 

Once the optimum profile(s) has been established, its practicality for implementation must be assessed. For a continuous process, the equipment must be able to be designed such that the profile can be followed through space by adjusting rates of reaction, mass transfer, heat transfer, and so on. In a dynamic problem, a control system must be designed that will allow the profile to be followed through time. If the profile is not practical, then the optimization must be repeated with additional constraints added to avoid the impractical features. 

Once the initial network structure has been defined, then loops, utility paths and stream splits offer the degrees of freedom for manipulating network cost in multivariable continuous optimization. When the design is optimized, any constraint that temperature differences should be larger than A Tmin or that there should not be heat transfer across the pinch no longer applies. The objective is simply to design for minimum total cost. 

Reaction kinetics, catalyst handling, mass and heat transfer, corrosion and many other practical industrial chemistry and engineering considerations impact the success of scaleup from lab to commercial for batch processing. Since the starting point for scaleup studies is the ultimate intended commercial unit, the professional should scaledown from the design parameters and constraints of the proposed commercial unit. 

Constraints (11.3) stipulates the quantity of heat transferred to storage from a hot unit at the beginning of the time horizon. Constraints (11.4) and (11.5) quantify the amount of heat transferred and received from storage unit, respectively. They ensure that if there is no heat integration between a processing unit and storage, then the amount of heat related to storage is not disturbed. 

This approach in essence assumes a temperature discontinuity at the wall. Alternative methods of formulating this constraint have also been proposed (94, 95, 115), together with empirical methods of evaluating the heat transfer coefficients introduced by each method. 

If a more complex mathematical model is employed to represent the evaporation process, you must shift from analytic to numerical methods. The material and enthalpy balances become complicated functions of temperature (and pressure). Usually all of the system parameters are specified except for the heat transfer areas in each effect (n unknown variables) and the vapor temperatures in each effect excluding the last one (n — 1 unknown variables). The model introduces n independent equations that serve as constraints, many of which are nonlinear, plus nonlinear relations among the temperatures, concentrations, and physical properties such as the enthalpy and the heat transfer coefficient. 

Peterson, J. and Y. Bayazitoglu. Optimization of Cost Subject to Uncertainty Constraints in Experimental Fluid Flow and Heat Transfer. J Heat Transfer 113 314-320 (1991). 

Change the temperature from T0 to Tf in minimum time subject to heat transfer rate constraints. 

The process has four separate subsystems for the degree-of-freedom analysis. Redundant variables and redundant constraints are removed to obtain the net degrees of freedom for the overall process. The 2 added to Nsp refers to the conditions of temperature and pressure in a stream +1 represents the heat transfer Q. 

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