Biot-number

These expressions can also be used for the case of external mass transfer and solid diffusion control by substituting D, for 8pDpi/( p + ppK)) and/c rp/(ppK)D,i) for the Biot number. 

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 ... 

However, the Nusselt number refers to a single fluid phase, whereas the Biot number is related to the properties of both the fluid and the solid phases. 

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 ... 

It will be noted that the relevant characteristic dimension in the Biot number is defined as the ratio of the volume to the external surface area of the particle (V/Ae), and the higher the value of V/Ae, then the slower will be the response time. With the characteristic dimension defined in this way, this analysis is valid for particles of any shape at values of the Biot number less than 0.1... 

Figures 9.14-9.16 enable the temperature 9C at the centre of the solid (centre-plane, centre-line or centre-point) to be obtained as a function of the Fourier number, and hence of time, with the reciprocal of the Biot number (Bil) as parameter.

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...

External mass transfer resistance was neglected, as reported by [63] in biofilm reactors with granular particles (fluidized bed, airlift) the Biot number was generally larger than 100. 

For completely adiabatic systems (Biot Number = O, see Chapter 3), the induction time can be expressed as ... 

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 Semenov model applies when the Biot number is close to zero, and the Frank-Kamenetskii model applies when the Biot number is large. The... 

Thomas [209] describes the effect of Biot number on the critical 8 in calculations of runaway temperatures. Biot numbers for right cylinders with various 1/d ratios are available. 

---