Thermal Biot number

Correlations for hy, and Xeb can be found in the literature. Values of 10 W m" for hy, and of 1 W m K for Xgb are typical. Thermal Biot numbers are generally around 1. It follows from Eqn. (8.115) that much dilution with inert particles and small tube diameters favour radial isothermicity. Also, an inert with a good thermal conductivity should be used whenever possible. 

The geometry of the tubes allows the heat transfer being considered one dimensional, and each tube to be a lumped system in front of the ambient air. This two conditions are fulfilled when Bi < 0.1 (Biot number Bi = a /(/(2a ), where R is the radius of the sample, X its thermal conductivity and a the heat transfer coefficient between the tube and the environment). Once the temperature-time curves of the PCM and the reference substance are obtained (Figure 160), the data can be used to determine the thermophysical properties of the PCM. 

The more recent Thomas model [209] comprises elements of both the Semenov and Frank-Kamenetskii models in that there is a nonuniform temperature distribution in the liquid and a steep temperature gradient at the wall. Case C in Figure 3.20 shows a temperature distribution curve from self-heating for the Thomas model. The appropriate model (Semenov, Frank-Kamenetskii, or Thomas) is determined by the ratio of the heat removal from the vessel and the thermal conductivity in the vessel. This ratio is determined by the Biot number (Nm) which has been described previously as hx/X, in which h is the film heat transfer coefficient to the surroundings (air, cooling mantle, etc.), x is the distance such as the radius of the vessel, and X is the effective thermal conductivity. 

The discussion so far has concentrated on mass transfer. The transfer of the heat liberated on adsorption or consumed on desorption may also limit the rate process or the adsorbent capacity. Again the possible effects of the boundary-film and the intra-pellet thermal properties have to be considered. A Biot number for heat transfer is hri/ke. In general, this is less than that for mass transfer because the boundary layer offers a greater resistance to heat transfer than it does to mass transfer, whilst the converse is true in the interior of the pellet. 

For suspension-to-gas (or bed-to-gas) heat transfer in a well-mixed bed of particles, the heat balance over the bed under low Biot number (i.e., negligible internal thermal resistance) and, if the gas flow is assumed to be a plug flow, steady temperature conditions can be expressed as... 

To determine whether the thin body approximation may be used, one should compare the surface heat transfer coefficient, and the thermal conductance of the solid, ksom/8. Their ratio is the Biot number,... 

For Rep < 100 and 0.05 < rp/r, < 0.2, wall Biot numbers range between 0.8 and 10 [28], so this means that wall effects cannot be neglected a priori [38]. Also this criterion contains procurable parameters. For the wall heat transfer coefficient hw and the effective heat conductivity in the bed Abc(r, the correlations in Table 2, eqs. 44-47 can be used [8, 39]. These variables are assumed to be composed of a static and a dynamic (i.e. dependent on the flow conditions) contribution. Thermal heat conductivities of gases at 1 bar range from 0.01 to 0.5 Js m l K l, depending on the nature of the gas and temperature. 

The Biot number can be thought of as die ratio of (n) The conduction thermal resistance to the convective thermal resistance. 

In the thermal conduction theory, such a distribution in general is thought to be caused on condition that the rate of thermal conduction in the self-heating solid chemical placed in the atmosphere under isothermal conditions is far less than the rate of heat transfer from the solid chemical through the whole surface to the atmosphere. In other words, this condition is expressed as [/> > A, which is equivalent to that the Biot number takes a large value. 

The temperature held is dependent on this number when heat transfer takes place into a fluid. The Biot number has the same form as the Nusselt number defined by (1.36). There is however one very significant difference, A in the Biot number is the thermal conductivity of the solid whilst in the Nusselt number A is the thermal conductivity of the fluid. The Nusselt number serves as a dimensionless representation of the heat transfer coefficient a useful for its evaluation, whereas the Biot number describes the boundary condition for thermal conduction in a solid body. It is the ratio of L0 to the subtangent to the temperature curve within the solid body, cf. Fig. 2.4, whilst the Nusselt number is the ratio of a (possibly different choice of) characteristic length L0 to the subtangent to the temperature profile in the boundary layer of the fluid. 

Interpreting Lo/X as the specific thermal conduction resistance of the solid and l/a as the specific heat transfer resistance at its surface also allows the Biot number to be interpreted as the ratio of the two resistances... 

A small Biot number means that the resistance to thermal conduction in the body, for example due to its high thermal conductivity, is significantly smaller than the heat transfer resistance at its boundary. With small Biot numbers the temperature difference in the body is small in comparison to the difference ( w — f) between the wall and fluid temperatures. The reverse is valid for large Biot numbers. Examples of these two scenarios are shown for a cooling process in Fig. 2.5. Very large Biot numbers lead to very small values of — J), and for Bi —> oo, according to (2.34) we get (it/y — i p) —> 0. The heat transfer condition (2.34) can be replaced by the simpler boundary condition = -dj. 

A simple calculation for the heating or cooling of a body of any shape is possible for the limiting case of small Biot numbers (Bi — 0). This condition is satisfied when the resistance to heat conduction in the body is much smaller then the heat transfer resistance at its surface, cf. section 2.1.5. At a fixed time, only small temperature differences appear inside the thermally conductive body, whilst... 

One surface (x = 0) of a cooled plate, which has thickness 6, is insulated, whilst at the other surface heat is transferred to a fluid at a temperature of p. Sketch the temperature profile = (, ) in the plate for a fixed time t. Which conditions must the curve satisfy close to the two surfaces x = 0 and x = 8, if the Biot number is Bi = a8/ = 1.5 In addition sketch the temperature profile of the fluid temperature in the boundary layer on the surface of the plate, taking account of the condition Nu = atf/Ap = 10 Ap is the thermal conductivity of the fluid. 

The Biot number is decisive for the ratio of the thermal resistance of the cylinder to that of the air... 

Thus, we take advantage of the accuracy, robustness and efficiency of the direct problem solution, to tackle the associated inverse heat transfer problem analysis [26, 27] towards the simultaneous estimation of momentum and thermal accommodation coefficients in micro-channel flows with velocity slip and temperature jump. A Bayesian inference approach is adopted in the solution of the identification problem, based on the Monte Carlo Markov Chain method (MCMC) and the Metropolis-Hastings algorithm [28-30]. Only simulated temperature measurements at the external faces of the channel walls, obtained for instance via infrared thermography [30], are used in the inverse analysis in order to demonstrate the capabilities of the proposed approach. A sensitivity analysis allows for the inspection of the identification problem behavior when the external wall Biot number is also included among the parameters to be estimated. 

In Table 1.10 those dimensionless groups that appear frequently in the heat and mass transfer literature have been listed. The list includes groups already mentioned above as well as those found in special fields of heat transfer. Note that, although similar in form, the Nus-selt and Biot numbers differ in both definition and interpretation. The Nusselt number is defined in terms of thermal conductivity of the fluid the Biot number is based on the solid thermal conductivity. 

Internal one-dimensional transient conduction within infinite plates, infinite circular cylinders, and spheres is the subject of this section. The dimensionless temperature < ) = 0/0/ is a function of three dimensionless parameters (1) dimensionless position C, = xlZF, (2) dimensionless time Fo = otr/i 2, and (3) the Biot number Bi = hiElk, which depends on the convective boundary condition. The characteristic length IF, is the half-thickness L of the plate and the radius a of the cylinder or the sphere. The thermophysical properties k, a, the thermal conductivity and the thermal diffusivity, are constant. 

Heat Transfer on Convection Duct Walls. For this boundary condition, denoted as , the wall temperature is considered to be constant in the axial direction, and the duct has convection with the environment. An external heat transfer coefficient is incorporated to represent this case. The dimensionless Biot number, defined as Bi = heDhlkw, reflects the effect of the wall thermal resistance, induced by external convection. 

For large solid and liquid conductivities, within the cell the solid, the interdendritic liquid, and the cell liquid can all be in a near-thermal equilibrium. For a temperature change occurring on the boundaries of the cell, the assumption of a uniform temperature within the cell requires that the cell Biot number, that is,... 

---