Unsteady-state heat transfer

Steady-state heat transfer Unsteady-state heat transfer Convective heat transfer (heat transfer coefficient) Convective heat transfer (heat transfer coefficient) Radiative heat transfer (not analogous with other transfer processes) Steady-state molecular diffusion Unsteady-state molecular diffusion Convective mass transfer (mass transfer coefficients) Equilibrium staged operations (convective mass transfer using departure from equilibrium as a driving force) Mechanical separations (not analogous with other transfer processes) ... 

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 ... 

In predicting the time required to cool or heat a process fluid in a full-scale batch reactor for unsteady state heat transfer, consider a batch reactor (Figure 13-2) with an external half-pipe coil jacket and non-isothermal cooling medium (see Chapter 7). From the derivation, the time 6 to heat the batch system is ... 

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. 

Equation 10.66 is referred to as Fick s Second Law. This also applies when up is small, corresponding to conditions where C, is always low. This equation can be solved for a number of important boundary conditions, and it should be compared with the corresponding equation for unsteady state heat transfer (equation 9.29). 

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.) ... 

The calculations for the experimental reaction rates are based on an unsteady state heat transfer analysis. We computed the overall heat transfer coefficient of the system and estimated the experimental rates as follows dT... 

There is also another key parameter linked to the choice of the material for the reactor. First, the choice is obviously determined by the reactive medium in terms of corrosion resistance. However, it also has an influence on the heat transfer abilities. In fact, the heat transport depends on the effusivity relative to the material, deflned by b = (XpCp) the effusivity b appears in the unsteady-state conduction equation. 

While the particle is experiencing the accelerating motion as described above, heat is being transferred between it and the surrounding gas stream also in an unsteady state ... 

The input and output terms of equation 1.5-1 may each have more than one contribution. The input of a species may be by convective (bulk) flow, by diffusion of some kind across the entry point(s), and by formation by chemical reaction(s) within the control volume. The output of a species may include consumption by reaction(s) within the control volume. There are also corresponding terms in the energy balance (e.g., generation or consumption of enthalpy by reaction), and in addition there is heat transfer (2), which does not involve material flow. The accumulation term on the right side of equation 1.5-1 is the net result of the inputs and outputs for steady-state operation, it is zero, and for unsteady-state operation, it is nonzero. 

The general characteristics of a batch reactor (BR) are introduced in Chapter 2, in connection with its use in measuring rate of reaction. The essential picture (Figure 2.1) in a BR is that of a well-stirred, closed system that may undergo heat transfer, and be of constant or variable density. The operation is inherently unsteady-state, but at any given instant, the system is uniform in all its properties. 

In the common case of cylindrical vessels with radial symmetry, the coordinates are the radius of the vessel and the axial position. Major pertinent physical properties are thermal conductivity and mass diffusivity or dispersivity. Certain approximations for simplifying the PDEs may be justifiable. When the steady state is of primary interest, time is ruled out. In the axial direction, transfer by conduction and diffusion may be negligible in comparison with that by bulk flow. In tubes of only a few centimeters in diameter, radial variations may be small. Such a reactor may consist of an assembly of tubes surrounded by a heat transfer fluid in a shell. Conditions then will change only axially (and with time if unsteady). The dispersion model of Section P5.8 is of this type. 

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 higher the thermal conductivity 2 of the body, the faster the heat spreads out. The higher its volume-related heat capacity pCp, the slower the heat transfer. Therefore, unsteady-state heat conduction is characterized by only one material property, the thermal diffusivity, a = A/pCp of the body. 

Sect. 5.4), the heat transfer process can be modeled using classical unsteady state heat conduction theory [142-144]. From the mathematical solutions to heat conduction problems, a thermal diffusivity can be extracted from measurements of temperatures vs. time at a position inside a gel sample of well-defined geometry. 

Thermocouple junction with protective sheath. Suppose the resistance to heat transfer of the sheath surrounding the thermocouple described in Section 7.5.2 is not negligible. The unsteady-state heat transfer mechanism must then be considered in two stages. 

The model based on the concept of pure limiting film resistance involves the steady-state concept of the heat transfer process and omits the essential unsteady nature of the heat transfer phenomena observed in many gas-solid suspension systems. To take into account the unsteady heat transfer behavior and particle convection in fluidized beds, a surface renewal model can be used. The model accounts for the film resistance adjacent to the heat transfer... 

In the emulsion phase/packet model, it is perceived that the resistance to heat transfer lies in a relatively thick emulsion layer adjacent to the heating surface. This approach employs an analogy between a fluidized bed and a liquid medium, which considers the emulsion phase/packets to be the continuous phase. Differences in the various emulsion phase models primarily depend on the way the packet is defined. The presence of the maxima in the h-U curve is attributed to the simultaneous effect of an increase in the frequency of packet replacement and an increase in the fraction of time for which the heat transfer surface is covered by bubbles/voids. This unsteady-state model reaches its limit when the particle thermal time constant is smaller than the particle contact time determined by the replacement rate for small particles. In this case, the heat transfer process can be approximated by a steady-state process. Mickley and Fairbanks (1955) treated the packet as a continuum phase and first recognized the significant role of particle heat transfer since the volumetric heat capacity of the particle is 1,000-fold that of the gas at atmospheric conditions. The transient heat conduction equations are solved for a packet of emulsion swept up to the wall by bubble-induced circulation. The model of Mickley and Fairbanks (1955) is introduced in the following discussion. 

Consider a packet of emulsion phase being swept into contact with the heating surface for a certain period. During the contact, the heat is transferred by unsteady-state conduction at the surface until the packet is replaced by a fresh packet as a result of bed circulation, as shown in Fig. 12.6. The heat transfer rate depends on the rate of heating of the packets (or emulsion phase) and on the frequency of their replacement at the surface. To simplify the model, the packet of particles and interstitial gas can be regarded as having the uniform thermal properties of the quiescent bed. The simplest case is represented by the problem of one-dimensional unsteady thermal conduction in a semiinfinite medium. Thus, the governing equation with the boundary conditions and initial condition can be imposed as... 

The particle-to-gas heat transfer can be obtained by unsteady-state experiments which measure the time required for cold particles of temperature TpO) mass M, and surface area 5P to reach the bed temperature when they are introduced into the bed. Assuming that the bed is well mixed and the bulk solids temperature in the bed is the same as the gas temperature, the heat balance gives... 

ASTM F 2700 Standard Test Method for Unsteady-State Heat Transfer Evaluation of Flame Resistant Materials for Clothing with Continuous Heating... 

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