• Biotechnology and Bioprocess Engineering:BBE
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• v.7 no.6
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• pp.362-366
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• 2002

A mathematical model has been developed for describing the oxygen concentration during the exponential growth of microorganisms, in a static solid substrate bed supported on a tray fermentor. The model equations comprise of one partial differential equation for mass transfer and an ordinary differential equation of growth. After nondimensionlisation, analytical solution tn the model has been obtained by the method of Laplace transforms. An expression for critical thickness of bed is deduced from the model equation. The significance of the model in the design of tray fermentors is discussed. The validity of the discussion is verified by taking an illustration from the literature.

• Journal of Magnetics
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• v.10 no.4
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• pp.157-162
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• 2005

We report the magnetic-field dependence of the irreversible magnetization of binary superconductor $MgB_2$. For the temperature region of $T\;<\;0.9T_c$, the contribution of the bulk pinning to the magnetization overwhelms that of the surface pinning. This was evident from the fact that the magnetization curves, M(H), were well described by the critical-state model without considering the reversible magnetization and the surface pinning effect. It was also found that the M(H) curves at various temperatures scaled when the field and the magnetization were normalized by the characteristic caling factors H$\ast$(T) and M$\ast$(T), respectively. This feature suggests that the pinning mechanism determining the hysteresis in M(H) is unique below $T\;=\;T_c$.