Heat exchangers variable coefficient

It is important to have the correct set of variables specified as independent and dependent to meet the modeling objectives. For monitoring objectives observed conditions, including the aforementioned independent variables (FICs, TICs, etc.) and many of the "normally" (for simulation and optimization cases) dependent variables (FIs, TIs, etc.) are specified as independent, while numerous equipment performance parameters are specified as dependent. These equipment performance parameters include heat exchanger heat transfer coefficients, heterogeneous catalyst "activities" (representing the relative number of active sites), distillation column efficiencies, and similar parameters for compressors, gas and steam turbines, resistance-to-flow parameters (indicated by pressure drops), as well as many others. These equipment performance parameters are independent in simulation and optimization model executions. 

The time constant of any process is the result of its capacitance and resistance. Usually, the heat exchanger outlet temperature is the controlled variable, and the flow rate of the heat transfer fluid is the manipulated variable. The time constant of an exchanger is a function of the mass and the specific heat of the tube material, the mass flow, and the specific heat of the process and utility streams and their heat transfer coefficients. 

The above derivation for LMTD involves two important assumptions (1) the fluid specific heats do not vary with temperature, and (2) the convection heat-transfer coefficients are constant throughout the heat exchanger. The second assumption is usually the more serious one because of entrance effects, fluid viscosity, and thermal-conductivity changes, etc. Numerical methods must normally be employed to correct for these effects. Section 10-8 describes one way of performing a variable-properties analysis. 

Because of the fouling effects, there may be a limit on the velocity of one of the fluids in a heat exchanger. For example, the velocity of cooling water in tubes of a shell-and-tube exchanger is often specified as 3 ft/s. If the velocity of one fluid is specified, the coefficient for that fluid is set, and the independent variables become At, and the film coefficient of the other fluid. 

The interconnections between the unit modules may represent information flow as well as material and energy flow. In the mathematical representation of the plant, the interconnection equations are the material and energy balance flows between model subsystems. Equations for models such as mixing, reaction, heat exchange, and so on, must also be listed so that they can be entered into the computer code used to solve the equation. Table 5.1 lists the common type of equations that might be used for a single subsystem. In general, similar process units repeatedly occur in a plant and can be represented by the same set of equations, which differ only in the names of variables, the number of terms in the summations, and the values of any coefficients in the equations. 

Regression analysis is often employed to fit experimental data to a mathematical model. The purpose may be to determine physical properties or constants (e.g., rate constants, transport coefficients), to discriminate between proposed models, to interpolate or extrapolate data, etc. The model should provide estimates of the uncertainty in calculations from the resulting model and, if possible, make use of available error in the data. An initial model (or models) may be empirical, but with advanced knowledge of reactors, distillation columns, other separation devices, heat exchangers, etc., more sophisticated and fundamental models can be employed. As a starting point, a linear equation with a single independent variable may be initially chosen. Of importance, is the mathematical model linear In general, a function,/, of a set of adjustable parameters, 3y, is linear if a derivative of that function with respect to any adjustable parameter is not itself a function of any other adjustable parameter, that is. 

Often, the quantity of interest in an experiment is not measured directly, but is computed via a mathematical equation or model. Eor example, the overall heat-transfer coefficient (say, C) of a specific heat exchanger might be determined indirectly by measuring the inlet and outlet temperatures. Repeated experiments provide an estimate of the variance of U, but this variance does not account for possible experimental errors (e.g., the thermocouple errors in the temperature measurements). The error in each measurement accumulates in the overall error of a calculated quantity in a manner known as propagation of error. Measurement error can arise from random variability or instrument sensitivity. There is, however, a mathematical approach to deal with these errors. 

This equation is encountered in heat transfer problems with spherical symmetry (e.g., heat exchange between a ball and the ambient medium, with r being the dimensionless radial coordinate). Note that the change of variables u(r, r) = rT(r,r) leads to the equation dTu = drru with constant coefficients, considered in Subsection S.l-1. 

If the heat transfer coefficients, h, were constant, the curves in figure 5.18 would be logarithmic. As was shown in chapter 2, however, there is considerable variation in the value of the coefficient, depending on the temperature of gas and air, density and velocity of gas and air, after-burning, radiation, leakage, and the character of the heat exchanging surface. In view of these many variables, the necessity for approximation is no drawback. 

In these equations, Q and A, are the heat duty, heat transfer coefficient, and heat transfer area for heat exchanger /. Values for the latter two are assumed known, so they are not process variables. Similar equations are written for the other two heat exchangers, making a total of nine equations. Consequently, the number of manipulated variables is computed as - A/ExKmauy Deaned =... 

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. 

Removing the heat of reaction necessitates an internal heat exchanger. This exchanger will also help to limit the bubble diameter. Take vertical tubes of 0.06 m outer diameter on a 0.14-m triangular pitch. This limits the effective diameter of the bubbles to 0.1 m. Note that this is a very crude way of determining the (average) bubble diameter, which is the main variable in the Kunii and Levenspiel model. Select the superficial velocity of the feed to be 1800 m/h. Calculate the mass transfer coefficients from (13.5.3-4) and (13.5.3-5) and use (13.5.3-13) to calculate the bed height. 

If offset is intolerable, some means must be provided for recalibration while the system is operating. In general, this can be done most directly by readjusting the set point, which is already scaled in terms of the controlled variable. Other adjustments could be made, such as the coefficient K in the heat-exchanger control system, but with less predictable results. 

This is not always an easy question to resolve. The feedback controller may be asked to perform a number of different services. In the heat-exchanger application it can be useful in correcting for heat loss, in which employment it should add an increment of heat to the process at all loads this would amount to a zero adjustment. Or its principal function might be to correct for variable steam enthalpy, in which case it should apply a span adjustment by setting the coefficient K. In another process, linearity could be the largest unknown factor. But a single feedback controller can hardly be called upon to do all these things. 

The complex flow pattern on the shell-side, and the great number of variables involved, make it difficult to predict the shell-side coefficient and pressure drop with complete assurance. In methods used for the design of exchangers prior to about 1960 no attempt was made to account for the leakage and bypass streams. Correlations were based on the total stream flow, and empirical methods were used to account for the performance of real exchangers compared with that for cross flow over ideal tube banks. Typical of these bulk-flow methods are those of Kern (1950) and Donohue (1955). Reliable predictions can only be achieved by comprehensive analysis of the contribution to heat transfer and pressure drop made by the individual streams shown in Figure 12.26. Tinker (1951, 1958) published the first detailed stream-analysis method for predicting shell-side heat-transfer coefficients and pressure drop, and the methods subsequently developed... 

VARIABLE HEAT-TRANSFER COEFFICIENT. If the heat-transfer coefficient varies with temperature, one can assume that the complete exchanger consists of a number of smaller exchangers in series and that the coefficient varies linearly with temperature in each of these sections. When the last five conditions listed in the preceding section hold and the overall coefficient varies linearly with temperature, integration of Eq. (17) gives... 

---