Heat transfer initial conditions

A second-order reaction proceeds in a batch reac tor provided with heat transfer. Initial conditions are To = 350 and Cq = 1. Other data are ... 

JVeeset Heat Transfer Initial Condition Equipment Heat Tiansfet Cendant duty w ... 

The temperature field is solved using a backward implicit difference approximation. The nodal temperature and variable rate of temperature of element are given by the interpolation of shape function. In the solution of temperature field, the equation of heat transfer, initial conditions and boundary conditions have to be satisfied. Based on the variational principle, the problem can be converted into solving the extremum of a functional. The implicit difference equation is written as ... 

Baxter (B3) uses an enthalpy-flow temperature method, due originally to Dusinberre (D5, D6) and Eyres et al. (E4), whereby the movingboundary effect is reduced to a property variation. To begin with, the melting of a slab of finite thickness initially at the fusion temperature is considered. At the surface of the melt, which is of the same density as the solid, a heat transfer boundary condition is applied. The technique takes into account latent heat effects by allowing the specific heat to become infinite at the fusion temperature in such a way that... 

Copolymerization is effected by suspension or emulsion techniques under such conditions that tetrafluoroethylene, but not ethylene, may homopolymerize. Bulk polymerization is not commercially feasible, because of heat-transfer limitations and explosion hazard of the comonomer mixture. Polymerizations typically take place below 100°C and 5 MPa (50 atm). Initiators include peroxides, redox systems (10), free-radical sources (11), and ionizing radiation (12). 

The solution to this equation, with initial condition /if= 0 at Ti = 0 and boundaiy condition cf= 1 at = 0, originally obtained for an analogous heat transfer case [Anzelius, Z. Angew Math. Mech., 6, 291 (1926) Schumann, y. Franklin Jn.st., 208,405 (1929)], is... 

Endurance Burn Under certain cou(itious, a successfully arrested flame may stabilize on the unprotected side of an arrester element. Should this condition not be corrected, the flame will eventually penetrate the arrester as the channels become hot. An endurance burn time can be determined by testing, which specifies that the arrester has withstood a stabilized flame without penetration for a given period. The test should address either the actual or worst-case geometry, since heat transfer to the element will depend on whether the flame stabilizes on the top, bottom, or horizontal face. In general, the endurance burn time identified by test should not be regarded as an accurate measure of the time available to take remedial action, since test conditions will not necessarily approximate the worst possible practical case. Temperature sensors may be incorporated at the arrester to indicate a stabilized flame condition and either alarm or initiate appropriate action, such as valve closure. 

In the above example, 1 lb of initial steam should evaporate approximately 1 lb of water in each of the effects A, B and C. In practice however, the evaporation per pound of initial steam, even for a fixed number of effects operated in series, varies widely with conditions, and is best predicted by means of a heat balance.This brings us to the term heat economy. The heat economy of such a system must not be confused with the evaporative capacity of one of the effects. If operated with steam at 220 "F in the heating space and 26 in. vacuum in its vapor space, effect A will evaporate as much water (nearly) as all three effects costing nearly three times its much but it will require approximately three times as much steam and cooling water. The capacity of one or more effects in series is directly proportional to the difference between the condensing temperature of the steam supplied, and the temperature of the boiling solution in the last effect, but also to the overall coefficient of heat transfer from steam to solution. If these factors remain constant, the capacity of one effect is the same as a combination of three effects. 

The mechanics of deposition in a boiler are often a cycle of cause and effect, wherein an initial low level of scale deposited on boiler surfaces causes a rapid localized rise in wall temperature. The temperature increase leads to localized steam blanketing, which in turn prevents the deposit from resolubilizing. Consequentially, conditions then exist for the further buildup of deposit on the heat transfer surface. 

NOTE Steam blanketing is the static steam-water condition that occurs when boiler water circulation is impeded byfoulant or when initial deposition prevents adequate heat transfer and the dispersion of lower density steam or water. 

In most cases, however, heat transfer and mass transfer occur simultaneously, and the coupled equation (230) thus takes into account the most general case of the coupling effects between the various fluxes involved. To solve Eq (230) with the appropriate initial and boundary conditions one can decouple the equation by making the transformation (G3)... 

As we have seen before, exact differentials correspond to the total differential of a state function, while inexact differentials are associated with quantities that are not state functions, but are path-dependent. Caratheodory proved a purely mathematical theorem, with no reference to physical systems, that establishes the condition for the existence of an integrating denominator for differential expressions of the form of equation (2.44). Called the Caratheodory theorem, it asserts that an integrating denominator exists for Pfaffian differentials, Sq, when there exist final states specified by ( V, ... x )j that are inaccessible from some initial state (.vj,.... v )in by a path for which Sq = 0. Such paths are called solution curves of the differential expression The connection from the purely mathematical realm to thermodynamic systems is established by recognizing that we can express the differential expressions for heat transfer during a reversible thermodynamic process, 6qrey as Pfaffian differentials of the form given by equation (2.44). Then, solution curves (for which Sqrev = 0) correspond to reversible adiabatic processes in which no heat is absorbed or released. 

Bi very small, (say, <0.1). Here the main resistance to heat transfer lies within the fluid this occurs when the thermal conductivity of the particle in very high and/or when the particle is very small. Under these conditions, the temperature within the particle is uniform and a lumped capacity analysis may be performed. Thus, if a solid body of volume V and initial temperature Oo is suddenly immersed in a volume of fluid large enough for its temperature 0 to remain effectively constant, the rate of heat transfer from the body may be expressed as ... 

A stirred reactor contains a batch of 700 kg reactants of specific heat 3.8 kJ/kg K initially at 290 K, which is heated by dry saturated steam at 170 kN/m2 fed to a helical coil. During the heating period the steam supply rate is constant at 0.1 kg/s and condensate leaves at the temperature of the steam. If heat losses arc neglected, calculate the true temperature of the reactants when a thermometer immersed in the material reads 360 K. The bulb of the thermometer is approximately cylindrical and is 100 mm long by 10 mm diameter with a water equivalent of 15 g, and the overall heat transfer coefficient to the thermometer is 300 W/m2 K. What would a thermometer with a similar bulb of half the length and half the heat capacity indicate under these conditions ... 

The LDPE reactor is sometimes termed heat transfer limited in conversion. While this is true, the molecular weight (or melt index)—conversion relationship is not since this work shows that a selected initiator can allow conversion improvements to be made under adiabatic conditions for a specified molecular weight. The actual limitation to conversion is the decomposition temperature of the ethylene and given that temperature as a maximum limitation, an initiator (not necessarily commercial or even known with present initiator technology) can be found which will allow any product to be made at the rate dictated by this temperature. Conceptually, this is a constant (maximum) conversion reactor, runnirg at constant operating conditions where the product produced dictates the initiator to be used. 

The heat transfer model, energy and material balance equations plus boundary condition and initial conditions are shown in Figure 4. The energy balance partial differential equation (PDE) (Equation 10) assumes two dimensional axial conduction. Figure 5 illustrates the rectangular cross-section of the composite part. Convective boundary conditions are implemented at the interface between the walls and the polymer matrix. 

Figure 4. Heat transfer model, energy and material balance equations, boundary and initial conditions plus physical properties.

In order to illustrate how the mode of operation can positively modify selectivity for a large reactor of poor heat-transfer characteristics, simulations of the reactions specified in Example 5.3.1.4 carried out in a semibatch reactor were performed. The reaction data and process conditions are essentially the same as those for the batch reactor, except that the initial concentration of A was decreased to cao = 0.46 mol litre, and the remaining amount of A is dosed (1) either for the whole reaction time of 1.5 h with a rate of 0.1 mol m s", or (2) starting after 0.5 h with a rate of 0.15 mol m " s". It is assumed that the volume of the reaction mixture and its physical properties do not change during dosing. The results of these simulations are shown in Fig. 5.3-15. The results of calculation for reactors of both types are summarized in Table 5.3-3. 

---