Capacitance, thermal

Heat flows from a heat source at temperature 6 t) through a wall having ideal thermal resistance Ri to a heat sink at temperature 62(1) having ideal thermal capacitance Ct as shown in Figure 2.14. Find the differential equation relating 6 t) and 02(0-... 

Wall thermal capacitance effects. The wall thermal capacitance effect on CHF in a boiling water flow can be observed only at low pressures, where the bubble size is large and the wall temperature fluctuation period is long. These conditions were satisfied in a test in water at 29-87 psia (200-600 kPa) (Fiori and Bergles, 1968). Two test sections of 0.094-in. (2.39-mm) I.D. with wall thicknesses... 

As the thermal capacitance and resistance of the hotplate provide a thermal low-pass transfer function (with the dominant pole corresponding to a characteristic time of 10-20 ms, depending on the fabrication process), the ZA modulator driving the hotplate constitutes a linear noise-shaping DAC with an output in the thermal domain. 

J.B. Planeaux and K.F. Jensen. Bifurcation phenomena in CSTR dynamics A system with extraneous thermal capacitance. Chem. Eng. Sci., 41 (6) 1497-1523, 1986. 

A couple points are emphasized 1) the authors have found no published effort to determine the forces on the barrel as a function of material type and solids conveying discharge pressure, and 2) unlike conventional barrels that are thick to induce a maximum thermal capacitance and strength, the barrel in this device was machined to a minimum thickness in order to enhance heat transfer and thus allow a good estimate of the inner wall temperature from the outside temperature measurements and the local heat flux. 

Due to the relatively small radius of the well, the thermal capacitance Vpcp of the bed with the thermal well (16 cal/K) is only slightly higher than that without (14 cal/K) and the loss in reaction volume is only about 2%. 

Another potential model simplification involves assuming negligible energy accumulation in the gas phase as compared to that in the solid, which is equivalent to the earlier approximation [Eq. (66)] based on the relative magnitude of the energy accumulation in the gas and solid. For our system, the accumulation of energy in the solid is approximately 250 to 300 times that in the gas phase due to the relative thermal capacitance of the gas [Eq. (65)] and the similarity of the temporal behavior of the gas and catalyst temperatures (e.g., Fig. 19). Thus the accumulation term in the energy balance... 

Where TG = gas temperature and 7, = liquid temperature. The number of transfer units NG can also be calculated as the capability for change divided by the thermal capacitance of the flowing streams. 

The tubular reactor can be an empty vessel if no catalyst is used. If a solid catalyst is required, the vessel is packed with catalyst, either in a bed or inside tubes. The dynamic behavior of the reactor is significantly affected by the presence of catalyst in the reactor because the thermal capacitance of the catalyst is usually greater than that of the process fluid, particularly if the system is gas-phase. The temperatures of both the process fluid and the catalyst change with time. Of course, under steady-state conditions, the two temperatures are equal at any axial position. 

However, the exit temperature of the reactor with catalyst (rout)cat takes about 2 hours to attain the same steady state. And in fact, it initially actually decreases The inverse response or wrongway effect is caused by the thermal capacitance of the catalyst. 

Later in the rigorous simulations, the dynamics of the heat exchanger are included because of the significant thermal capacitance of the metal tubes in this very large unit. 

The thermal capacitance of the gas in the reactor is assumed negligible compared to that of the solid catalyst. Therefore a single dynamic energy balance is used for each lump, and the gas temperature is assumed to be equal to the catalyst temperature... 

The heat exchanger is quite large because of the low heat transfer coefficient found in these gas-phase systems. Therefore the mass of metal in the tubes is quite significant in terms of thermal capacitance. Table 7.3 gives design details of the heat exchanger. The tube diameter is 0.0254 m, length is 5 m, wall thickness is 0.000524 m, and heat capacity is 0.05 kJ kg-1 KT1. 

The catalyst bed level in the reactor is controlled by adjusting a slide valve in the catalyst return line from the reactor to the regenerator. Since the total amount of catalyst in both vessels is essentially constant, the level in the regenerator is not controlled. The thermal capacitance in the system is mostly in the solid catalyst. The catalyst holdup in the regenerator is much larger than in the reactor, so the dynamic response of the reactor is faster than that of the regenerator. 

Most processes include some form of capacitance or storage capability. These capacitance elements can provide storage for materials (gas, liquid, or solids) or storage for energy (thermal, chemical, etc.). Thermal capacitance is directly analogous to electric capacitance and can be calculated by multiplying the mass of the object (W) with the specific heat of the material it is made of (Cp). The gas capacitance of a tank is constant and is analogous to electric capacitance. The liquid capacitance equals the cross-sectional area of the tank at the liquid surface, and if the cross-sectional area is constant, the capacitance of the process is also constant at any head. 

The thermal network for the single-capacity system is shown in Fig. 4-26. In this network we notice that the thermal capacity of the system is charged" initially at the potential To by closing the switch 5. Then, when the switch is opened, the energy stored in the thermal capacitance is dissipated through the resistance VhA. The analogy between this thermal system and an electric system is apparent, and we could easily construct an electric system which would behave exactly like the thermal system as long as we made the ratio... 

Therefore we have developed a new type of thermoelectric device which can drive above electric parts immediately by heat of a gas flame. Reducing a thermal capacitance of a device by decreasing thickness is effective for good response. It is well known that functionally graded material, a composite of ceramic and metal, can reinforce the brittleness of the ceramic decreasing the amount of weight. 

The capillary cassette was placed directly on the heating surface of the thin-film resistive heater to minimize thermal capacitance. A Plexiglas frame was used to hold the assembly and allow access of the cooling air flow from the fans to the capillary cassette. 

The control signal calculated by the PID algorithm (see Note 3) is translated into a heating profile by way of pulse width modulation. The heater receives a pulse train (frequency of 0.5 Hz) with variable on time. During the on segment of each 0.5 s pulse, the heater is on at lull power. The integrated power over the lull pulse is controlled to achieve the desired temperature. The thermal capacitance of the capillaries and the heater itself is sufficient to damp out any temperature fluctuations such that the temperature measured inside the capillaries is steady (does not show any 0.5 Hz ripple). 

FIGURE 4.38 Temperature response of a vertical plate to a step heat input including the effects of thermal capacitance (from Gebhart and Adams [108]). 

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