Unsteady thermal conduction

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... 

The exact mathematical solution of problems involving unsteady thermal conduction may be very difficult, and sometimes impossible, especially where bodies of irregular shapes are concerned, and other methods are therefore required. 

The equation is most conveniently solved by the method of Laplace transforms, used for the solution of the unsteady state thermal conduction problem in Chapter 9. 

Unlike at adiabatic conditions, the height of the liquid level in a heated capillary tube depends not only on cr, r, pl and 6, but also on the viscosities and thermal conductivities of the two phases, the wall heat flux and the heat loss at the inlet. The latter affects the rate of liquid evaporation and hydraulic resistance of the capillary tube. The process becomes much more complicated when the flow undergoes small perturbations triggering unsteady flow of both phases. The rising velocity, pressure and temperature fluctuations are the cause for oscillations of the position of the meniscus, its shape and, accordingly, the fluctuations of the capillary pressure. Under constant wall temperature, the velocity and temperature fluctuations promote oscillations of the wall heat flux. 

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. 

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. 

Steady-state periodic heating and unsteady-state methods can be applied to measure the thermal conductivity and diffusivity of coal. Methods such as the compound bar method and calorimetry have been replaced by transient hot-wire/line heat source, and transient hot plate methods that allow very rapid and independent measurements of a and X. In fact, such methods offer the additional advantage of measuring these properties not only for monolithic samples but also for coal aggregates and powders under conditions similar to those encountered in coal utilization systems. 

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... 

Example 3 One-Dimensional, Unsteady Conduction Calculation As an example of the use of Eq. (5-21), Taole 5-1, and Table 5-2, consider the cooking time required to raise the center of a spherical, 8-cm-diameter dumpling from 20 to 80°C. The initial temperature is uniform. The dumpling is heated with saturated steam at 95°C. The heat capacity, density, and thermal conductivity are estimated to be c = 3500 J/(kg K), p = 1000 kg/m3, and k = 0.5 W/(m K), respectively. 

We start this chapter with a description of steady, unsteady, and multidimensional heat conduction. Then we derive the differential equation that governs heat conduction in a large plane wall, a long cylinder, and a sphere, and generalize the results to three-dimensional cases in rectangular, cylindrical, and spherical coordinates. Following a discussion of the boundary conditions, we present tlie formulation of heat conduction problems and their solutions. Finally, we consider lieat conduction problems with variable thermal conductivity. 

Due to the high thermal conductivity of sphere, the conductive resistance within the sphere can be neglected in comparison to the convective resistance at its surface. Accordingly, this unsteady state heat transfer situation could be analyzed as a lumped system. 

Carslaw, H. S and Jaeger, J. C. (1959) Cottduciion of Heat in Solids, Oxford University Press. Oxford, 2nd ed. This work includes a compilation of solutions to the equation of unsteady heat conduction in the absence of flow for many different geometries, initial and boundaiy conditions. The basic equation is of the same form as the diffusion equation with the thermal diffusivity. K/pC, in place of the diffusion coefficient. (Here k, p, and C are the thermal conductivity, density, and specific heatof the continuous fluid.) Like D, the thcnnal diffusivity has cgs- dimensions of cm"/sec,... 

Various simplified models can be used with varying degrees of accuracy for the simulation of the transient behaviour of non-porous catalyst pellets. The most suitable unsteady state model for this problem is that with infinite thermal conductivity. This simplified model is quite accurate for metal and metal oxide catalysts. In this model, equation (5.45) disappears and the model becomes strictly lumped parameter described only by ordinary initial value differential equations. 

Convection of heat via blood depends primarily on the local blood flow in the tissue and the vascular morphology of the tissue. Thermal diffusion is determined by thermal conductivity in the steady state, and thermal diffusivity in the unsteady state. In addition to these transport parameters, we need to know the volumes and geometry of normal tissues and tumor. In general, tumor volume changes as a function of time more rapidly than normal tissue volume. In special applications, such as hyperthermia induced by electromagnetic waves or radiofrequency currents, we need electromagnetic properties of tissues—the electrical conductivity and the relative dielectric constant. In the case of ultrasonic heating, we need to specify the acoustic properties of the tissue—velocity of sound and attenuation (or absorption) coefficient. 

Although the heat flow and fluid flow in packed beds are quite complex, the heat transfer characteristics can be described by a simple concept of effective thermal conductivity Ke that is based on the assumption that on a macroscopic scale the bed can be described by a continuum. Effective thermal conductivity is a continuum property that depends on temperature, bed material, and structure. It is usually determined by evaluating the steady-state heat flux between two parallel plates separated by a packed bed. The effective thermal conductivity applies very accurately to steady-state heat transfer and to unsteady-state heat transfer if (t d2p) > 1.94 x 107 s/m2 [27] in other cases, for unsteady state heat transfer the thermal... 

Hence, the local mass transfer coefficient scales as the two-thirds power of a, mix for boundary layer theory adjacent to a solid-liquid interface, and the one-half power of A, mix for boundary layer theory adjacent to a gas-liquid interface, as well as unsteady state penetration theory without convective transport. By analogy, the local heat transfer coefficient follows the same scaling laws if one replaces a, mix in the previous equation by the thermal conductivity. 

With an exothermic reaction in the bed, clusters or packets of hot solid come in contact with the cooler surface of the wall or the tubes, and they give up some of their heat in the short time before they are swept away. A model based on the penetration theory of unsteady-state heat transfer and some supporting data were presented by Mickley and Fairbanks [26]. The average coefficient is predicted to vary with the square root of the thermal conductivity, density, and heat capacity of the clusters and inversely with the square root of the average contact time. The fraction of the surface in contact with clusters is taken to be (1 — a), where a is the volume fraction bubbles in the bed. Heat transfer to the bare surface... 

Flg. 8a, b. Types of dependences of P(Q) (dimensionless quantities) (a) the presence of a strong dissipative heat output, crosses denote the transition to a high-temperature process 1 — a more strong dependence Ti(P) thanT)(T) 2 — opposite case (b) high intensity of axial thermal conductivity dashed lines denote absolute unsteady states. Arrows show possible hysteresis transitions upon variation of P(a) and Q(b)... 

Transient-state or unsteady-state methods make nse of either a line source of heat or plane sources of heat. In both cases, the usual procedure is to apply a steady heat flux to the specimen, which mnst be initially in thermal eqnUibrinm, and to measnre the tanperatnre rise at some point in the specimen, resnlting from this applied flux [83]. The Fitch method is one of the most common transient methods for measuring the thermal conductivity of poor conductors. This method was developed in 1935 and was described in the National Bureau of Standards Research Report No. 561. Experimental apparatus is commercially available. 

MEASUREMENT OF THERMAL CONDUCTIVITIES OF ORGANIC ALIPHATIC LIQUIDS BY AN ABSOLUTE UNSTEADY-STATE METHOD. 

THERMAL CONDUCTIVITY OF LIQUID REFRIGERANTS MEASURED BY AN UNSTEADY-STATE HOT-WIRE METHOD. 

The major sources of error in using the unsteady-state charts are the inadequate data on the density, heat capacity, and thermal conductivity of the foods and the prediction of the convective coefficient. Food materials are irregular anisotropic substances, and the physical properties are often difficult to evaluate. Also, if evaporation of water occurs on chilling, latent heat losses can affect the accuracy of the results. 

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