Heat gain curve

The above equation then represents the balanced conditions for steady-state reactor operation. The rate of heat loss, Hl, and the rate of heat gain, Hq, terms may be calculated as functions of the reactor temperature. The rate of heat loss, Hl, plots as a linear function of temperature and the rate of heat gain, Hq, owing to the exponential dependence of the rate coefficient on temperature, plots as a sigmoidal curve, as shown in Fig. 3.14. The points of intersection of the rate of heat lost and the rate of heat gain curves thus represent potential steady-state operating conditions that satisfy the above steady-state heat balance criterion. 

Fig. 3.14 shows the heat gain curve, Hq, for one particular set of system parameters, and a set of three possible heat loss, Hl, curves. Possible curve intersection points. A] and C2, represent singular stable steady-state operating curves for the reactor, with cooling conditions as given by cooling curves, 1 and III, respectively. 

Table 8.4 compares heat tfansfer rates for 6" (152 mm) square billets in a Curve 2 versus Curve 4 situation, both with spacing ratios of 3 1 and 2000 F (1090 C) furnace gas. Gains from wider spacing have diminishing returns (especially for four-side heating). All curves droop at low spacing-to-thickness ratios because all radiation is less with narrower spacing. 

Differential scanning calorimetry, DSC, is a technique which combines the ease of measurement of heating and cooling curves as displayed in Fig. 4.9 with the quantitative features of calorimetry (see Sect. 4.2). Temperature is measured continuously, and a differential technique is used to assess the heat flow into the sample and to equalize incidental heat gains and losses between reference and sample. Calorimetry is never a direct determination of the heat content. Measuring heat is different from volume or mass determinations, for example. In the latter cases the total amount can be established with a single measurement. The heat content, in contrast, must be measured by beginning at zero kelvin where the heat content is zero, and add all heat increments up to the temperature of interest. 

If the reactor is unstable, bbth the gain and the time constant will be negative. The denominator in both expresaons is the difference between the slopes of the heat-removal and heat evolution curves, as in Fig. 10.3. If both denominators are positive, the reactor behaves as a simple first-order lag. If both are negative, positive feedback dominates the dynamic gain is the same as a simple lag, but the phase angle goes from -90 at zero period to -180 at an Infinite period ... 

In principle, differential thermal analysis, DTA is a technique which combines the ease of measurement of the cooling and heating curves discussed in Chapter 3 with the quantitative features of calorimetry which are treated in Chapter 5. Temperature is measured continuously and a differential technique is used in an effort to compensate for heat gains and losses. In the case of DTA as also in calorimetry, the actual heat measurement does not rely on a direct measurement of the heat content. A heat meter, as such, does not exist. In volume or mass determinations (see Chapters 6 and 7, respectively), the total quantity of interest can be established with one simple measurement. In the determination of heat content, in contrast, one must start at zero kelvin and measure all heat increments and add them up to the temperature of interest. In DTA one derives the flow of heat, AQ/dt, from a measurement of the temperature difference between a reference material and the sample. ... 

The right-hand side (RHS) of Equations (9.116) and (9.119) represent the net heat loss and the left-hand side (LHS) represents the energy gain. The gain and the loss terms can be plotted as a function of the flame temperature for both the diffusion and premixed flames as Semenov combustion diagrams. Intersection of the gain and loss curves indicates a steady solution, while a tangency indicates extinction. 

Fig. 1. Schematic thermal analysis results on fusion and devitrification (top and bottom curves, respectively). The area under the peak in the top curve represents the heat of fusion Hf and can be used to calculate the entropy of fusion ASf = AHf/Tm in case of equilibrium melting. The increase in heat capacity ACP at Tg is related to the number of motifs that gain mobility...

Figure 3.26 compares the Nyquist plots for four different cases. The jacket-cooled Nyquist plot is much closer to the critical (—1,0) point. As more area is used in the external heat exchanger, the curves move deeper into the third quadrant, indicating the potential for improved closedloop control. However, the point where they cross the negative real axis moves further to the left. The ultimate gains for the three areas with the external heat exchanger are = 39.8/24.9/15.1 (dimensionless) for areas of 45.2/58.1/ 100 m2. These results are counter-intuitive since we would expect the controllability to improve with increasing area. 

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