Heat conduction-heating products

Consider a small control volume V = SxSySz (Fig. 4.27), where the inner heat generation is Q "(T) (heat production/volume) and the heat conductivity is A(T). The material is assumed to be homogeneous and isotropic, and the internal heat generation and thermal conductivity are functions of temperature. 

For single reactions, conversion at any point in time is proportional to the extent of the heat generation. Checking by analytical methods can be conducted to verify the heat production/conversion relationship. Conversion can also be checked by an on-line balance which is possible with well instrumented systems. 

Figure 2. Heat conduction (Seebeck effect) batch mixing calorimeter for three samples and one reference channel. After loading and establishing baselines, the assembly is inverted to mix reactants and start heat production. (Reproduced with permission from Ref. 2. 1983, Alan R. Liss, Inc.)...

T(r,t) is the spatial and temporal temperature distribution, I)th the thermal diffusivity, p the density, cp the specific heat at constant pressure, and Q(r,t) the local heat production per volume. A general solution of Eq. (12) with the appropriate boundary conditions, including thermal conductivity of the cell windows and heat transition to the ambient air, can be a challenging task. The whole problem is simplified, since the experiment is set up in such a way that it only... 

Fig. 3. Heat production is an important consideration for devices using electric fields in the liquid near cells. This figure shows the theoretical distribution of heat production in and around a spherical cell at the centre of a quadrupole electrode chamber in a solution of low electrical conductivity (top) and high conductivity (bottom). The heat production is given by gE2 where g is the conductivity of the solution or cell component and E is the (local) electric field strength. The contour interval is 7% of the maximum in each case. The cell is modelled as an electrically conductive sphere enveloped by an insulating but capacitive membrane.

Heat must also flow through the walls of a device. DEP utilises electric field inhomogeneities which means that the fields (and heat production) within the liquid tend to be quite localised. The external heat flows can occur over much wider areas so it is sometimes possible to use even poorly conducting materials such as ordinary glasses and quartz. In critical cases, silicon, which has a high thermal conductivity (150 W/m s °C), is used. 

Fig. 15. The possible mechanisms by which a strong electric field can affect cells in suspension or adherently growing. Most of the heat is produced near the electrodes and, therefore, tends not to be a direct problem as it can be easily dissipated into the substrate. This heating can, however, induce convection currents which, in turn, may impose mechanical stress on an adherent cell. There is also some heating between the electrodes. At low frequencies, this occurs only in the medium although it may be concentrated in regions surrounding the cell. At high frequencies, this heating becomes more uniform but, because high frequency currents can flow inside the cell, there is some internal heat production. The total amount of heat evolved depends on the conductivity of the medium and on the square of the applied voltage.

Equation (17) is usually called the Tian equation. In cases where significant temperature gradients are present within the reaction vessel, two or more time constants must be used. When the change in rate of a process is small, the value for X(dU/dt) will often be insignificant compared to the value for U (equation (17)). With heat conduction calorimeters used in work on cellular systems, this is typically the case and the heat production rate is then, with a good approximation, given by the simple expression... 

The magnitude of the energy conversion can be calculated as follows. The amount of electrical energy converted to heat per unit time is given by the product of voltage V and current i. This heat production is spread over the volume of the conducting medium LA, where L is length and A the cross-sectional area of the medium. Thus the rate of heat input H per unit volume is... 

This criterion resembles much those for the temperature gradients on a particle level. The first term again represents the dimensionless activation energy yw, based on the reactor wall temperature Tw. The second term represents the ratio of the heat production rate and the heat conduction rate in radial direction. The last term accounts for the relative contributions of the radial conductivity and the heat transfer at the reactor wall. The latter contains the particle to bed radius ratio and the Biot number for heat transport at the wall, defined as ... 

For non-stationary heat conduction in a semi-infinite stationary medium the onedimensional transient heat conduction without heat production, we have next parabolic differential equation... 

For a dust piled on a surface or thick on the walls of a duct, heat is lost by conduction and thus is linear with temperature whereas the reaction rate (and thus heat production) increases exponentially with temperature. This implies the existence of a temperature level, known as the critical temperature Tc, above which more heat will accumulate in the reacting material than will be lost from it. Such factors as the size or thickness of the dust deposit, airflow across the dust which may cany away some heat, thermal conductivity and packing of the dust, and temperature distribution within the dust can all influence the ultimate value for Tc. 

At low cirabient temperatures a greater portion of the metabolic heat production (depending upon exercise intensity and clothing) is dissipated by convection and radiation and a minor portion by evaporation of sweat and respiratory water. As ambient temperature rises, the portion of heat dissipated by convection and radiation decreases progressively in concert with a proportional increase in the rate of sweating and evaporative heat loss. The coordination of the rate of heat loss between conduction, radiation, and evaporation is so precise that, for ambient dry-bulb temperatures between 5 C and 29 C, the equilibrium level of core (rectal) temperature is related directly to the intensity of the exercise load and is independent of environmental temperature (25). 

The temperature distribution in a reacting mixture is stabilized when the rate of loss of heat by conduction or convection from any volume element is equal to that produced by the reaction itself in that volume element. In the case that the rate of heat loss cannot compensate for the rate of heat production, a stationary or quasi-stationary temperature distribution is impossible and the temperature of the reaction mixture increases exponentially, causing the reaction rate to do likewise, and a thermal explosion results. This is illustrated in Fig. XIV. 1, which follows... 

The heat produced in this manner is transferred to the surrounding explosive material. The heat transfer rate is dependent upon temperature as well as thermal conductivity, heat capacity, and density. One of the classical three-dimensional heat transfer equations that relates the rate of heat production to the rate of temperature rise of the reacting material and to its surroundings is the Frank-Kamenetskii (FAT) equation (Ref 1). 

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