Heat Biot number

An appreciation of the relative magnitudes of heat and mass transfer effects in internal and external diffusion is useful. A measure of the relative magnitudes is the ratio of the mass to heat Biot numbers ... 

Component B ratio of mass to heat Biot numbers. 

Equation 10.51 has been obtained in Chapter 5 by equating the diffusion-limited transport rate through the completely deactivated outer shell to the diffusion-limited reaction within the fresh core of the pellet. The temperature drop in the deactivated outer shell has been accounted for in terms of the heat Biot number (B/)a and y. Equation 10.50 is the Euler version of Eq. 10.49 while Eq. 10.52 is that of Eq. 10.35 for numerical calculation. For nonadiabatic reactors, Eq. 10.18 needs to be added to the above set of equations with Eq. 10.22 in place of Eq. 10.23. These equations are summarized in Table 10.6 for various modes of operation while detailed analysis procedures for adiabatic reactors are given in Figure 10.11 in the form of a flow chart. The structure of the procedures shown in Figure... 

A large Biot Number means that conduction controls the energy transfer to/from the plastic and large temperature gradients will exist in the plastic. A small Biot Number means that convection is the dominant factor. The above analysis was for conduction heat transfer (B, - oo). When the plastic moulding is taken out of the mould we need to check the value of B,. In this case... 

Analytical solutions of equation 9.44 in the form of infinite series are available for some simple regular shapes of particles, such as rectangular slabs, long cylinders and spheres, for conditions where there is heat transfer by conduction or convection to or from the surrounding fluid. These solutions tend to be quite complex, even for simple shapes. The heat transfer process may be characterised by the value of the Biot number Bi where ... 

The Biot number is essentially the ratio of the resistance to heat transfer within the particle to that within the external fluid. At first sight, it appears to be similar in form to the Nusselt Number Nu where ... 

Bi very large. The resistance to heat transfer in the fluid is then low compared with that in the solid with the temperature of the surface of the particle being approximately equal to the bulk temperature of the fluid, and the heat transfer rate is independent of the Biot number. Equation 9.44 then simplifies to ... 

Temperatures at off-centre locations within the solid body can then be obtained from a further series of charts given by Heisler (Figures 9.17-9.19) which link the desired temperature to the centre-temperature as a function of Biot number, with location within the particle as parameter (that is the distance x from the centre plane in the slab or radius in the cylinder or sphere). Additional charts are given by Heisler for the quantity of heat transferred from the particle in a given time in terms of the initial heat content of the particle. 

Figures 9.17-9.19 clearly show that, as the Biot number approaches zero, the temperature becomes uniform within the solid, and the lumped capacity method may be used for calculating the unsteady-state heating of the particles, as discussed in section (2). The charts are applicable for Fourier numbers greater than about 0.2.

Isothermal and adiabatic heat transfer conditions can be obtained with different values of the Biot number. The wall temperature, 0w, is assumed to be a piecewise linear function of the axial position and is treated as a known quantity based on experimental evidence. 

Most of the actual reactions involve a three-phase process gas, liquid, and solid catalysts are present. Internal and external mass transfer limitations in porous catalyst layers play a central role in three-phase processes. The governing phenomena are well known since the days of Thiele [43] and Frank-Kamenetskii [44], but transport phenomena coupled to chemical reactions are not frequently used for complex organic systems, but simple - often too simple - tests based on the use of first-order Thiele modulus and Biot number are used. Instead, complete numerical simulations are preferable to reveal the role of mass and heat transfer at the phase boundaries and inside the porous catalyst particles. 

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. 

Figure 8. Relative resistance to heat and mass transfer as a function of Biot number or hDdp/Dr...

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. 

During the flight of droplets in the spray, the forced convective and radiative heat exchanges with the atomization gas lead to a rapid heat extraction from the droplets. A droplet undergoing cooling and phase change may experience three states (a) fully liquid, (b) semisolid, and (c) fully solid. If the Biot number of a droplet in all three states is smaller than 0.1, the lumped parameter model 1561 can be used for the calculation of droplet temperature. Otherwise, the distributed parameter model 1541 should be used. 

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. 

Modeling of the packed bed catalytic reactor under adiabatic operation simply involves a slight modification of the boundary conditions for the catalyst and gas energy balances. A zero flux condition is needed at the outer reactor wall and can be obtained by setting the outer wall heat transfer coefficients /iws and /iwg (or corresponding Biot numbers) equal to zero. Simulations under adiabatic operation do not significantly alter any of the conclusions presented throughout this work and are often used for verification... 

There are two new parameters—the Biot numbers for heat and mass transfer at the surface, (Bi)T and (Bi)M. We can also distinguish their quotient <7 = (Bi)x/(Bi)M. The two cases a > 1 and a < 1 have different stationary-state possibilities. 

A dimensionless criterion, the Biot number, is often used in transient heat transfer problems by comparing the heat transfer resistance within the body with... 

A high Biot number means that the conductive transfer is small compared to convection and the situation is close to that considered by a Frank-Kamenetskii situation (Section 13.4.1). Inversely, a small Biot number, that is Bi < 0.2, means that the convective heat transfer dominates and the situation is close to a Semenov situation. 

Figure 5.26 Center-line temperature histories of finite thickness plates during convective heating for various Biot numbers.

Figure 5.27 Center-line temperature history of an 8 mm thick PMMA plate during convective heating inside an oven set at 155°C. The initial temperature was 20°C. The predictions correspond to a Biot number, Bi=1.3 or a corresponding heat transfer coefficient, 7i=33 W/m2/K.[7]...

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,... 

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