Kinetics, Dynamics and Yield of H₂ Production by HPB

Abstract

Knowledge of the kinetics of H₂ production is a necessary step towards a thorough understanding of the mechanisms involved in the complex system of microorganisms’ metabolic pathways in the substrate, including H₂-producing bacteria, electron shuttles and hydrogenase enzymes.

Appendix

3.1.1 Macro-approach and Relaxation Time

A so called macro-approach is a description of a complex system using the parameters which can be measured in order to take into account the observable behavior of the system. As the complexity of a system increases, the numbers of differential equations required to describe it grow. The systems encountered in biotechnology as well as in many of the engineering disciplines are of a complex nature and a rigorous description of their behavior leads to large sets of mathematical expressions (the state equations) describing the time evolution of a large number of relevant variables (the state variables). These equations generally contain a massive number of parameters, which in most cases cannot be readily obtained experimentally. Hence a consistent approach must be developed to simplify the picture of reality to a less complex one which can still describe the aspects of behaviour that are relevant to the desired application with sufficient accuracy. A useful approach to such simplification can be based on an analysis of the internal mechanisms of a system and their so-called relaxation times.

When considering a system in contact with an environment and exchanging matter and energy with it, a specific number of intensive quantities at its boundary need to be considered, e.g. concentrations of a number of chemical substances, temperature and pressure. After a sufficiently long time, all the processes of the system will have reached rates which no longer change with time, i.e. the state of the system has become time-independent. This state can be of two different types:

• The system finally reaches a state of thermodynamic equilibrium; if this occurs the final rates of the internal processes become zero, i.e. a lack of phenomena.

• The system reaches a so-called steady-state in which internal processes with a non-zero rate may still take place. In fact, if this system reaches a different state from thermodynamic equilibrium, at least one internal process must have a non-zero rate, whether or not a stationary state can exist. Depending on the magnitude of the dynamics of the system, this can be considered a pseudo-steady state or an “evolutionary” system which can move towards stable or unstable conditions.

Following Röels (1983) [30], a time-scale separation approach or relaxation time approach is followed, to reduce the description of the dynamics of the system without losing its validity in accordance with the desired applications. In simple words this approach considers, as affecting the dynamics of a phenomenon, only those mechanisms which present a relaxation time of the same order of magnitude independent of whether the phenomenon is internal to the system or belongs to the environment. The relaxation time must not be confused with the intrinsic constant of a different dynamic phenomenon. It represents the dynamics of interaction between the system and its surrounding environment. A brief mathematics is shown to clarify the concept. Considering a first-order system, there is a constant of proportionality between a flux and a driving force:

$$ J = k\Delta F $$

(A.1)

the relaxation time is

$$ \tau_{r} = 1/k $$

(A.2)

but unfortunately first-order systems rarely describe the dynamics of the real world. The kinetics of a biosystem or a more complex system may be described by:

$$ r = f(M) $$

(A.3)

in which f is not a first-order relationship, and hence the relaxation time of the observed mechanism cannot be rigorously defined. A way to overcome this difficulty is to define a relaxation time that depends on the actual values of the space parameters. Linearizing Eq. (A.3), it follows that:

$$ r = \frac{f(M)}{{\prod\nolimits_{i = 1,n} {M_{i}^{1/n} } }}\prod\nolimits_{i = 1,n} {M_{i}^{1/n} } $$

(A.4)

and the relaxation time for the j-th situation is:

$$ \tau_{rj} = \frac{{\prod\nolimits_{i = 1,n} {M_{i}^{1/n} } }}{{r_{j} }} $$

(A.5)

where M _i is the i-th state variable and r _j is the probe parameter, i.e. the experimentally valued parameter to follow the dynamics of the phenomena selected in accordance with the purpose of the study and the available experimental instrumentation. Equation (A.5) means that for a non-linear system it is possible to introduce the concept of actual relaxation time in the sense that it gives the time to reach a fraction (1 – 1/e) of the difference between the actual state (thermodynamic equilibrium or steady-state) and a new one as a consequence of change of matter and/or energy between the environment and the system which determines the dynamics. The rate and magnitude of exchange of matter and energy between system and environment determine the dynamics of the system towards a new stable or unstable state.

3.1.2 Application to First-Order Kinetics

The following simple example is intended to clarify the approach. Consider a well-stirred reactor in which a first-order chemical reaction takes place. Into the system a substrate solution of concentration C _s0 is introduced at a rate F. The volume of the system V is considered constant and no volume generation is assumed to take place (note that, in the case of a microorganism population, this is not a valid assumption!). The concentration of the substrate in the flow leaving the system is assumed equal to the substrate concentration in the system (ideal mixing hypothesis). The following kinetic equation is assumed to specify the rate of the first-order process inside the system:

$$ r = kC_{si} $$

(A.6)

in which r is the rate of the system evolution per unit of volume, k is the kinetic constant and C _si is the substrate concentration inside the system. The following balance equation describes the dynamics of the substrate concentration in the system as a CSTR (continuous stirred tank reactor):

$$ \frac{{{\text{d}}C_{si} }}{{{\text{d}}t}} = {\raise0.7ex\hbox{$F$} \!\mathord{\left/ {\vphantom {F V}}\right.\kern-0pt} \!\lower0.7ex\hbox{$V$}}(C_{s0} - C_{si} ) - KC_{si} $$

(A.7)

where C _s0 is the inlet concentration. In a stationary state or steady-state the left-hand side vanishes, hence:

$$ C_{ssi} = FC_{s0}/\left(F + kV \right) $$

(A.8)

Thus the following expression describes the rate of reaction in a stationary state:

$$ r = kFC_{s0} /(F + kV) $$

(A.9)

The rate of reaction is expressed in quantities which can be externally measured (macro-parameters), e.g. the system volume V, the substrate concentration in the feed solution C _s0, the flow rate F and through knowledge of such intensive parameters as the kinetic constant k of the system. The actual substrate concentration inside the system C _si does not appear. Considering a situation in which at a given moment in time the substrate concentration in the inlet is shifted from C _s0 to C _se , the following differential equation describes the system’s behaviour during the transient:

$$ V\frac{{{\text{d}}C^{'}_{si} }}{{{\text{d}}t}} = F(C_{s0} - C^{'}_{si} ) - kC_{si}^{'} V $$

(A.10)

with the initial condition:

$$ C_{ssi} = FC_{s0} /(F + kV) $$

(A.11)

Hence the dynamics of the system is the solution of the equation (A.10), in the time domain:

$$ C_{si} = \frac{(F/V)}{(F/V) + k}\left[ {C_{se} + (C_{s0} - C_{se} )\exp \left\{ { - (F/V + k)t} \right\}} \right] $$

(A.12)

The concentration C _si increases asymptotically to its new steady-state value. The relaxation time is defined as the time which elapses before the difference between C _si and its initial steady-state value reaches a fraction (1 – 1/e) of the difference between the old and the new steady-state value, which in this case is given by:

$$ \tau_{R} = 1/(F/V + k) $$

(A.13)

The relaxation time contains two contributions: the relaxation time V/F is the characteristic residence time in the reactor and 1/k is the relaxation time of the chemical reaction taking place. The last equation can be written as:

$$ 1/\tau_{R} = 1/\tau_{1} + 1/\tau_{2} $$

(A.14)

From it one can argue that the relative numerical value of the relaxation time determines the behaviour of the system: if the permanence time in the reactor is larger than the relaxation time of the system, the system dynamics is similar of that of a batch reactor (thermodynamic equilibrium will be reached); on the contrary, when the permanence time is less than the relaxation time of the system, the system evolves to reach a concentration equal to that of the feed, i.e. the reaction disappears. Only if the values are of the same order of magnitude there will be such an interaction between the system and the environment (Deborah number equal ~ 1). In the second case the evolution of the system is governed only by the permanence time:

$$ C_{si} = C_{se} + (C_{s0} - C_{se} )\exp \left\{ { - (F/V)t} \right\} $$

(A.15)

As one can see, this dynamic equation is a simpler one describing the behaviour of the system than equation (A.12).

Major details can be found in:

• J.A. Röels, “Energetic and kinetics in biotechnology”, Elsevier Biomedical Press, Amsterdam (1983)

• C. Ouwerker, “Theory of Macroscopic System”, Springer Verlag, Berlin (1991).