Temperature unsteady-state

In an unsteady-state energy balance the same principle applies. The accumulation of energy within a process where all the energy forms are considered including kinetic, potential, heat flow rates, enthalpies, and stirrer works may result in an increase in the thermal energy and a rise in temperature. Unsteady-state heat transfer involves the transfer of heat under conditions where the temperature changes with time. For the simple case of one-dimensional conduction in a solid slab, the accumulation of heat is a product of the mass and specific heat of the material and the increase in temperature where ... 

Unsteady-State Direct Oxidation Process. Periodic iatermption of the feeds can be used to reduce the sharp temperature gradients associated with the conventional oxidation of ethylene over a silver catalyst (209). Steady and periodic operation of a packed-bed reactor has been iavestigated for the production of ethylene oxide (210). By periodically varyiag the inlet feed concentration of ethylene or oxygen, or both, considerable improvements ia the selectivity to ethylene oxide were claimed. 

When temperatures of materials are a function of both time and space variables, more complicated equations result. Equation (5-2) is the three-dimensional unsteady-state conduction equation. It involves the rate of change of temperature with respect to time 3t/30. Solutions to most practical problems must be obtained through the use of digital computers. Numerous articles have been published on a wide variety of transient conduction problems involving various geometrical shapes and boundaiy conditions. 

Various numerical and graphical methods are used for unsteady-state conduction problems, in particular the Schmidt graphical method (Foppls Festschrift, Springer-Verlag, Berhn, 1924). These methods are very useful because any form of initial temperature distribution may be used. 

When this is substituted into the previous equation, both sides become functions of T and may be plotted against each other. As Fig. 23-17 of a typical case shows, as many as three steady states are possible. When generation is greater than removal (as at points A— and B-t-), the temperature will rise to the next higher steady state when generation is less than removal (as at points A-t- and B—), it will fall to the next steady state. Point B is an unsteady state, while A and C are steady. 

Heating or cooling of process fluids in a batch-operated vessel is common in the chemical process industries. The process is unsteady state in nature because the heat flow and/or the temperature vary with time at a fixed point. The time required for the heat transfer can be modified, by increasing the agitation of the batch fluid, the rate of circulation of the heat transfer medium in a jacket and/or coil, or the heat transfer area. Bondy and Lippa [45] and Dream [46] have compiled a collection of correlations of heat transfer coefficients in agitated vessels. Batch processes are sometimes disadvantageous because ... 

When an agitated bateh eontaining M of fluid with speeifie heat e and initial temperature t is heated using an isothermal eondensing heating medium Tj, the bateh temperature tj at any time 6 ean be derived by the differential heat balanee. For an unsteady state operation as shown in Figure 7-27, the total number of heat transferred is q, and per unit time 6 is ... 

The non-isothermal heating medium has a eonstant flowrate Wj, speeifie heat Cj, and inlet temperature Tj, but a variable outlet temperature. Eor an unsteady state operation ... 

Thermal runaway is a partieular problem in unsteady state bateh reaetions, where the rate of reaetion and, therefore, the rate of heat produetion varies with time. The eonsequenees of thermal runaway are sometimes severe as in the ineidents at Seveso [3]. In this ease, a bursting disk ruptured on a reaetor. The reaetor was used to manu-faeture triehlorophenol at a temperature of 170-185°C and was heated... 

The results quantitatively show the signifieanee of eonsidering both boundaries of the unsteady state operation. The load required at the end of the eyele is less than one-half the load required at the start of the eyele. Steve [13] inferred that reporting the value at only one boundary may be misleading. For example, the total heat load (provided by ehillers and heaters) from multiple reaetors with overlapping temperature adjustment eyeles may be skewed if the ehanges within those eyeles are not eonsidered. 

The general case is that of steady-state flow, and the thermal conductivity factor is a function of the temperature. In the unsteady state the temperature of the system changes with time, and energy is stored in the system or released from the system reduced. The storage capacity is... 

Takemasa, Y., S.Togati, and Y. Aral. 1996. Application of an unsteady-state model for predicting vertical temperature distribution to an existing atrium. ASHRAE Transactions, vol. 102, no. 1. 

As outlined earlier, in multizone models, perfect mixing is assumed in the individual zone. The spatial distribution of velocities, contaminant concentrations, and air temperatures in a zone can be determined only by using CFD. On the other hand, wind effects are easily accounted for in multizone models, and unsteady-state simulation is normally performed. On the combined use of the two methods, see Schaelin et al.--... 

As seen in Chapter 7, the operation of bateh erystallizers is inherently unsteady-state. Transient values oeeur of the major operating variables sueh as slurry density, supersaturation, temperature and mean partiele size. Methods of operational eontrol sueh as by use of seeding and temperature programming were also eonsidered in detail. 

A steady-state process is one in wliich there is no change in conditions (temperature, pressure, etc.) or rates of flow with time at any given point in die system. The accumulation term in Eq. (4.5.1) is dien zero. If diere is no cheniieid reaetion, the generation tenn is also zero. All other processes are unsteady state. 

All processes may be classified as batch, continuous, or semibatch depending on how materials are transferred into and out of the system. Also, the process operation may be characterized as unsteady state (i.e., transient) or steady state, depending on whether the process variables (e.g., pressure, temperature, compositions, flowrate, etc.) are changing with time or not, respectively. In a batch process, the entire feed material (i.e., charge) is added instantaneously to the system marking the beginning of the process, and all the contents of the system including the products are removed at a later time, at the end of the process. In a continuous process, the materials enter and leave the system as continuous streams, but not necessarily at the same rate. In a semibalch process, the feed may be added at once but the products removed continuously, or vice versa. It is evident that batch and semibatch processes are inherently unsteady state, whereas continuous processes may be operated in a steady or unsteady-state mode. Start-up and shut-down procedures of a steady continuous production process are examples of transient operation. 

In the theoretical treatment, the heat- and mass-transfer processes shown in Fig. 6 were considered. Simultaneous solution of the equations describing the behavior of the unsteady-state reaction system permits the temperature history of the propellant surface to be calculated from the instant of oxidizer propellant contact to the runaway reaction stage. 

In this approach, heat transfer to a spherical particle by conduction through the surrounding fluid has been the prime consideration. In many practical situations the flow of heat from the surface to the internal parts of the particle is of importance. For example, if the particle is a poor conductor then the rate at which the particulate material reaches some desired average temperature may be limited by conduction inside the particle rather than by conduction to the outside surface of the particle. This problem involves unsteady state transfer of heat which is considered in Section 9.3.5. 

Figures 9.17-9.19 clearly show that, as the Biot number approaches zero, the temperature becomes uniform within the solid, and the lumped capacity method may be used for calculating the unsteady-state heating of the particles, as discussed in section (2). The charts are applicable for Fourier numbers greater than about 0.2.

The heat transfer problem which must be solved in order to calculate the temperature profiles has been posed by Lee and Macosko(lO) as a coupled unsteady state heat conduction problem in the adjoining domains of the reaction mixture and of the nonadiabatic, nonisothermal mold wall. Figure 5 shows the geometry of interest. The following assumptions were made 1) no flow in the reaction mixture (typical molds fill in <2 sec.) ... 

A 5% CoOj/Ti02 catalyst is quite active for the wet TCE oxidation at very low reaction temperatures, such as 310 K, and our proposed model of different forms of CoO species existing with the fresh catalyst can reasonably explain the unsteady-state catalytic behavior at the initial period during the wet catalysis. 

Fig. 17. Comparison of the variation of the time-average S02 conversion and the maximum bed temperature predicted for stationary cycling condition by an unsteady-state and a steady-state kinetic model for a packed-bed S02 converter operating with periodic flow reversal...

Prepare a full instrumentation of flow-sheet of the CO conversion section of the plant, paying particular attention to the methods of controlling liquid levels in the circulating water system and temperatures in the catalyst beds. Derive the unsteady-state equations which would have to be employed in the application of computer control to the CO conversion section of the plant. 

Common situations in reactors are when the conditions can vary with time and position as independent variables. Partial derivatives of dependent variables such as concentration and temperature with respect to each of the independent variables then are involved. Such an equation, for unsteady state dispersion in a cylindrical reactor, was derived in problem 5.08.01, namely... 

Unsteady-state oxidation experiments were carried out by employing the step change in CO concentration over the preoxidized catalyst [62], Figure 7.14 represents the CO and C02 responses after a step change from He to 1 vol% CO/He over the fully oxidized Cu0 j Ce0 902 > nanostructured catalyst. At low temperatures, CO breakthrough is delayed for a few seconds as can be seen from Figure 7.14a. At a temperature of 250°C, however, 20 sec is needed for the first traces of CO exit... 

With respect to an individual food piece, the unit operation of freezing involves unsteady-state heat transfer in other words, the temperature of the food changes with time. In these circumstances heat transfer by conduction is described by Fourier s first law... 

We first recall the physical situation to facilitate this, we draw a sketch (see Fig. 1). At high oven temperatures, the heat is transferred from the heating elements to the meat surface by both radiation and heat convection. From there, it is transferred solely by the unsteady-state heat conduction that surely represents the rate-limiting step of the whole heating process (Fig. 1). 

The only published work on the diffusion of gas in coals of different rank appears to be that of Bolt and Innes (2) who studied the diffusion of carbon dioxide from eleven samples of coal at 38°C. They found the diffusion coefficient to range from 3.5 to 9.2 x 10 8 sq. cm./sec., with no apparent correlation with coal rank. Diffusion data on coals of different rank at temperatures higher than 38°C. have only been reported by the present authors (6). It has been shown (7) that the diffusion of inert or noble gases from coal above room temperatures can be rigorously analyzed by using simple diffusion theory, and that true diffusion parameters of the micropore systems can be obtained. In this paper our measurements on the unsteady state release of argon from coals of various rank, over a temperature range, are reported. 

Unsteady State Diffusion. The apparatus, experimental procedures, and the computational procedures used to calculate the diffusion parameter D /r (where D is the diffusion coefficient and r is the diffusion path length) have been described in detail previously (6, 8). A differential experimental system was used to avoid errors caused by small temperature fluctuations. In principle, the procedure consisted of charging the sample under consideration with argon to an absolute pressure of 1204 12 torr (an equilibrium time of about 24 hours was allowed) and then measuring the unsteady state release of the gas after suddenly reducing the pressure outside the particles back to atmospheric. 

The life-time, r, of the radicals can be determined from the ratio of overall rates of polymerization measured at the steady- and unsteady state as a result of intermittent illumination by the rotating sector. In Fig. 3.3-10 the rate constant, kp, of chain propagation (left) and kh that of termination (right), are plotted versus the pressure. Both rate constants increase with increasing temperature. The energy of activation of chain propagation is Ep = 37 kJ/mol, and that of chain termination is E, = 9.9 kJ/mol. The influence of pressure is... 

Mathematical models derived from mass-conservation equations under unsteady-state conditions allow the calculation of the extracted mass at different bed locations, as a function of time. Semi-batch operation for the high-pressure gas is usually employed, so a fixed bed of solids is bathed with a flow of fluid. Mass-transfer models allow one to predict the effects of the following variables fluid velocity, pressure, temperature, gravity, particle size, degree of crushing, and bed-length. Therefore, they are extremely useful in simulation and design. 

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