Let us consider arbitrary reactor volume element. From the conservation of energy for a reactor system we obtain the following:

Equation (1) can be represented mathematically as:

where means energy per unit mass, is the mass inflow, is the mass outflow and is the rate of heat.

The total rate of work done on a reactor system is expressed as follows:

where = total rate of work done,

= rate of work done by flowstream

= rate of work done by shaft

= rate of work done by boundary The Rate of work done Flowstream can be represented by such that

where = area of reactor (inflow), = area of reactor (outflow), = inflow volume of reactor, = outflow volume of reactor, = inflow pressure, = outflow Pressure, = flowrate (in), = flowrate (out), m = mass = inflow density, = outflow density and = general density, also

Substituting (5) in (4) and using the result in (3), we obtain

The energy terms of total energy composed in Internal U, Kinetic K and Potential energy is expressed as:

Substituting (7) in (2), we obtain

but in chemical reactors, only the internal energy is considered with the enthalphy per unit mass, hence 8 becomes

The batch reactors have no flowstream (i.e. ). Therefore, Equation (9) in terms of rate of heat becomes

Neglecting the work done by stirrer because the mixture is not highly viscous, so the stirring operation do not draw significant power, (10) yield

and we know that, hence (11) becomes

For Batch reactor in terms of enthalpy, we have

Taking the differential of (13) for and substituting in (12), we obtain

We now consider enthalpy as a function of temperature T, pressure P and number of moles n_j, and express its differentials as

The first partial derivative is the definitions of the heat capacity, , that is

The second partial derivative can be expressed as

where is the coefficient of expansion of mixture.

The final partial derivatives are the partial molar enthalpies,

Substituting (16), (17) and (18) in (15) and using the result in (14), we obtain

But the material balance for batch reactor is

where is the stoichiometric coefficient for species j and reaction i, is the production rate for jth species and is the reaction rate for ith reaction.

And the heat of reaction is

Substituting (20) and (21) in (19) we obtain

The constant-pressure batch reactor is the incompressible-fluid and for then Equation (22), becomes

If the heat removal is manipulated to maintain constant reactor temperature, the time derivative in Equation (23) vanishes leaving

When C_A = concentration of species A, k = reaction rate constant, and is the enthalpy change on reaction then Equation (24) becomes

For the constant-volume batch reactor, we considered the pressure as function of temperature, volume and number of moles, and also expressed its differentials as:

For reactor operation at constant volume, and forming time derivatives, just as we did in (15) to (17) and substituting into Equation (19) gives

Note that the first term in brackets is the total constant-volume heat capacitythat is    (28)

Substitution (28) and the material balance in (20), yields the rate of heat for the energy balance of the Constant-Volume batch Reactor. That is

If we consider a constant volume-ideal gas, where and. Substituting these into (29) gives

where

In order to describe the dynamic operation of a CSTR, the energy balance equation must be developed. The CSTR has flowstream, hence using the Equations (8)

As in (9) only the internal energy is considered. The out flow stream is flowing out of a well-mixed reactor, thus, the CSTR rate of heat equation using (32) is

where = volumetric flow rate, = flow density, = flow enthalpy, = flow concentration with component j and Q = flow rate.

As before, if sharf work is neglected for the CSTR, Equation (33) becomes

and if the enthalpy is considered, we obtain

We consider the change in enthalpy of the continuous stirred tank reactor (CSTR) as a function of temperature, pressure and number of moles, and express its differentials as

and substituting into Equation (35) gives

The material balance for the CSTR is

After substituting (38) in (37) and re-arrangement yields

The equation of rate of heat for constant-pressure in CSTR that is Incompressible-fluid and its mean in Equation (39) is and hence we have

From Equation (40), we obtained the equation of rate of heat for constant-volume in CSTR as follows:

Also, from Equation (41), the equation of rate of heat for ideal gas is:

For steady state constant, we have

, and   (43)

If we re-arrange Equation (39) in the form

By setting the Right hand side of (44) equals zero and substituting (43) in the result gives

The heat removal rate of CSTR required bringing CSTR reactor out-flow stream from final temperature T_f to temperature T and is given (from 45) by

The development of the semi-batch reactor energy balance follows directly from the CSTR energy balance derivation of the rate of heat by setting Q = 0. The main results in this paper are therefore summarized below:

Neglecting the Kinetic Energy in Equation (33) of the CSTR, when Q = 0, we obtain

Also, by neglecting the Sharf work and consider the Enthalpy when Q = 0 in (34) and (35) yields

and if the enthalpy is used, we obtain

By setting Q = 0 in Equations (37) and (39) respectively, we have the enthalpy change of semi-batch reactor as

and

The constant pressure semi-batch reactor is the incompressible-fluid batch reactor and in Equation (51)

when, we obtain

For steady state semi-batch reactor when is constant, we have

The equation is derived from the energy balance equation for Plug-flow reactor (PFR) single phase for rate of heat, and is given by:

Neglecting pressure drop or ideal gas for PFR and from (54), for an Ideal gas we have

The rate of heat equation of PFR for Incompressible fluid is obtain by setting in Equation (54)

The remaining six existing equations related to the rate of heat of a reactor for temperature of heat transfer medium are as stated below:

The equations derived above from the energy balance equation of chemical reactors [7] are thirty; namely: (10), (12), (14), (19), (22)-(25), (29), (30), (33)-(35), (37), (39)-(42) and (45)-(56). These equations with the six existing equations, namely (57)-(62), were structured into mathematical model in form of quadratic functional. The model with some given existing nuclear tokens were solved by the Conjugate Gradient Method algorithm, with MATLAB as a support soft-ware.

Some of the several properties of CGM are:

1) It has a quadratic convergence property that is for a quadratic functional on an n-dimensional Hilbert space, it converges in at most n steps.

2) It requires a relatively small increase in computer time per iteration and memory space.

3) It has a well worked out theory.

The first element of the descent sequence is simply guessed. The remaining members of the sequence are then found as follows:

(3.2.2)

(3.2.4)

where g_i is the gradient at the ith element of the descent sequence X_i.

It has been proved that the algorithm converges at most, in n iteration in a well posed problem and the convergence rate is given as:

where m and M are smallest and spectrums of matrix A respectively. That is, for an n dimensional problem, the algorithm will converge in at most n iterations [9].

The following tables are Table 1 for Problem 1, Table 2 for problem 2, Table 3 for problem 3 and Table 4 for problem 4 respectively.

The initial nuclear tokens used in problems 1 and 2 to represent the vectors and control operators of the quadratic model were arbitrary. Our results clearly shown that arbitrary composition of nuclear tokens will not guarantee safety. This is evidence from Tables 1 and 2 (non convergence) which did not satisfied the properties of the CGM algorithm. See [10].

After restructuring, nuclear tokens were used as the vectors and control operators of the quadratic model to generate problems 3 and 4 and were solved using the CGM algorithm. We were able to get two results that converge (Tables 3 and 4). The convergency satisfied the properties of the CGM algorithm, which shows good results. See [11] and [12].