METHOD FOR CONTROLLING LINGERATURE OF CHEMICAL REACTION

Abstract

A method for controlling lingerature of a chemical reaction. Changes in mass of a chemical reaction are monitored and are used to calculate a lingerature of the system. The reaction can be maintained at a desired lingerature () by selective addition or removal of heat or by adjusting the surface area the reactants are exposed to during the reaction. The disclosed method is useful for reactions that occur at non-equilibrium conditions where any measured lingerature would presume steady-state conditions.

Claims

1. A method for controlling lingerature of a chemical reaction, the method comprising steps of: a) determining an initial system mass (M) of a chemical system which performs a chemical reaction between reactants in a solvent to produce products, wherein at least one of the products is an exiting product that is a gaseous product or a precipitation product, the chemical reaction having a desired temperature (T), a desired lingerature (), and the chemical system having a specific heat capacity (SHC); b) adding the reactants and the solvent to a vessel, thereby initiating the chemical reaction; c) allowing the exiting product to exit the vessel; d) measuring a current system mass (M.sub.i); e) determining an exited mass (E.sub.i) of the exiting product that exited during step c) based on the current system mass (M.sub.i); f) calculating a change in reactant mass (M.sub.i) that occurred based on the exited mass (E.sub.i); g) calculating a dynamic lingerature (.sub.i) according to: ${}_i=\frac{M^2-{\left(M_i\right)}^2}{M^2}$ if the chemical reaction is an exothermic reaction or according to: ${}_i=2-\frac{M^2-{\left(M_i\right)}^2}{M^2}$ if the chemical reaction is an exothermic reaction, wherein is a number between 0.8 and 1.2, h) calculating a change in lingerature (.sub.i) based on the desired lingerature () and the dynamic lingerature (.sub.i); i) adding additional mass to the vessel in an amount equal to the exited mass (E.sub.i); j) adjusting a temperature of the chemical system by adding or removing an amount of heat (Q.sub.i) according to: $Q_i=SHCM\left(1-\frac{\left(1+{}_i/\right)}{\left(1+M_i/M\right)}\right)T$ if the chemical reaction is am exothermic reaction or according to: $Q_i=SHCM\left(\frac{\left(1-{}_i/\right)}{\left(1+M_i/M\right)}-1\right)T$ if the chemical reaction is an endothermic reaction, wherein is a number between 0.8 and 1.2; and k) iteratively repeating steps c) to j).

2. The method as recited in claim 1, wherein the chemical reaction is exothermic and the calculating the dynamic lingerature (.sub.i) performs the calculating according to: ${}_i=\frac{M^2-{\left(M_i\right)}^2}{M^2};$ and the adjusting the temperature of the chemical system adjusts the temperature by removing the amount of heat (Q.sub.i) according to: $Q_i=SHCM\left(1-\frac{\left(1+{}_i/\right)}{\left(1+M_i/M\right)}\right)T.$

3. The method as recited in claim 2, wherein is a number between 0.95 and 1.05.

4. The method as recited in claim 2, wherein is a number between 0.95 and 1.05.

5. The method as recited in claim 1, wherein the chemical reaction is endothermic and the calculating the dynamic lingerature (.sub.i) performs the calculating according to: ${}_i=2-\frac{M^2-{\left(M_i\right)}^2}{M^2};$ and the adjusting the temperature of the chemical system adjusts the temperature by adding the amount of heat (Q.sub.i) according to: $Q_i=SHCM\left(\frac{\left(1-{}_i/\right)}{\left(1+M_i/M\right)}-1\right)T.$

6. The method as recited in claim 5, wherein is a number between 0.95 and 1.05.

7. The method as recited in claim 5, wherein is a number between 0.95 and 1.05.

8. The method as recited in claim 1, wherein the exiting product is a gaseous product.

9. The method as recited in claim 1, wherein and are both 1.

10. The method as recited in claim 1, wherein specific heat capacity (SHC) is a dynamic specific heat capacity (SHC.sub.i) found according to: $SH{C}_i=\frac{k_B}{2m_{S{G}_y,i}}$ wherein k.sub.B is a Boltzmann constant k.sub.B and $m_{S{G}_y,i}$ is a i.sup.th iteration subgyrador mass.

11. A method for controlling lingerature of a chemical reaction, the method comprising steps of: a) determining an initial system mass (M) of a chemical system which performs a chemical reaction between reactants in a solvent to produce products, wherein at least one of the products is an exiting product that is a gaseous product or a precipitation product, the chemical reaction having a desired temperature (T), a desired lingerature (); b) adding the reactants and the solvent to a vessel, thereby initiating the chemical reaction; c) allowing the exiting product to exit the vessel; d) measuring a current system mass (M.sub.i); e) determining an exited mass (E.sub.i) of the exiting product that exited during step c) based on the current system mass (M.sub.i); f) calculating a change in reactant mass (M.sub.i) based on the exited mass (E.sub.i); g) adjusting a surface area experienced by the reactants by adding or removing an insoluble mass to the vessel in an amount to provide a change in surface area (A.sub.i) according to: $A_i=\left(\frac{M^2-{\left(M_i\right)}^2}{M^2}-1\right)A$ if the chemical reaction is an exothermic reaction or according to: $A_i=\left(1-\frac{M^2-{\left(M_i\right)}^2}{M^2}\right)A$ if the chemical reaction is an endothermic reaction, wherein is a number between 0.8 and 1.2, h) adding a soluble mass in an amount sufficient to restore the initial system mass (M), the amount of the soluble mass determined based on the exited mass (E.sub.i) and the insoluble mass; and i) iteratively repeating steps c) to h).

12. The method as recited in claim 11, wherein the chemical reaction is exothermic and the step of adding or removing the insoluble mass, adds the insoluble mass according to: $A_i=\left(\frac{M^2-{\left(M_i\right)}^2}{M^2}-1\right)A.$

13. The method as recited in claim 12, wherein is a number between 0.95 and 1.05.

14. The method as recited in claim 11, wherein the chemical reaction is endothermic and the chemical system comprises the insoluble mass, the step of adding or removing the insoluble mass removes a portion of the insoluble mass according to: $A_i=\left(1-\frac{M^2-{\left(M_i\right)}^2}{M^2}\right)A.$

15. The method as recited in claim 12, wherein is a number between 0.95 and 1.05.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0010] So that the manner in which the features of the invention can be understood a detailed description of the invention may be had by reference to certain embodiments, some of which are illustrated in the accompanying drawings. It is to be noted, however, that the drawings illustrate only certain embodiments of this invention and are therefore not to be considered limiting of its scope, for the scope of the invention encompasses other equally effective embodiments. The drawings are not necessarily to scale, emphasis generally being placed upon illustrating the features of certain embodiments of the invention. In the drawings, like numerals are used to indicate like parts throughout the various views. Thus, for further understanding of the invention, reference can be made to the following detailed description, read in connection with the drawings in which:

[0011] FIG. 1 is a schematic diagram of one system for implementing the disclosed method.

[0012] FIG. 2 is a flow diagram depicting one embodiment of the disclosed method.

[0013] FIG. 3 is a flow diagram depicting one embodiment of a method for adjusting lingerature.

[0014] FIG. 4 is a graph of dynamic viscosity of water versus viscosity-temperature.

[0015] FIG. 5 is a flow diagram depicting another embodiment of a method for adjusting lingerature.

DETAILED DESCRIPTION OF THE INVENTION

[0016] The disclosed system pertains to lingerature control systems and specifically pertains to systems that control non-equilibrium systems whose mass changes over the course of a chemical reaction. The disclosed method is used with chemical reactions wherein a product, such as a gaseous or solid product, exits the chemical reaction vessel over the course of the reaction. As used in this specification, the term lingerature refers to a length of time during which the chemical reaction can be considered to have a constant degree of stress.

[0017] In a first embodiment, a calculated lingerature (.sub.i) of a system at non-equilibrium conditions during an i.sup.th iteration of the method is calculated according to one of the following two equations:

[00007]$\begin{array}{cc}{}_i=\frac{M^2-{\left(M_i\right)}^2}{M^2},\mathrm{exothermic}& \left(1\right)\end{array}$$\begin{array}{cc}{}_i=2-\frac{M^2-{\left(M_i\right)}^2}{M^2},\mathrm{endothermic}& \left(2\right)\end{array}$

wherein is a desired lingerature (i.e. a target lingerature), M is an initial system mass of the entire chemical system within a vessel, M.sub.1 is a change in reactant mass during an i.sup.th iteration, and is a positive number that is near 1 (e.g. 0.8 to 1.2). Detailed discussions of establishing values for and the desired lingerature () are provided elsewhere in this disclosure. By calculating the i.sup.th iteration change in reactant mass (M.sub.i), one can then find a calculated lingerature (.sub.i) at non-equilibrium.

[0018] A change in heat (Q.sub.i) is added or removed to maintain the desired lingerature () according to one of the following two equations:

[00008]$\begin{array}{cc}{Q}_i=\mathrm{SHC}M\left(1-\frac{\left(1+{}_i/\right)}{\left(1+M_i/M\right)}\right)T,\mathrm{exothermic}& \left(3\right)\end{array}$$\begin{array}{cc}{Q}_i=\mathrm{SHC}M\left(\frac{\left(1+{}_i/\right)}{\left(1+M_i/M\right)}-1\right)T,\mathrm{endothermic}& \left(4\right)\end{array}$

wherein .sub.i is a change in lingerature that occurred during the i.sup.th iteration (i.e. .sub.i=.sub.i), SHC is a specific heat capacity of the system, a is a positive number that is near 1 (e.g. 0.8 to 1.2), M.sub.i is the change in reactant mass during the i.sup.th iteration and T is a target temperature of the chemical reaction. A detailed discussion of establishing a value for is provided elsewhere in this disclosure.

[0019] In a second embodiment, the surface area experienced by the reactants (which changes linger viscosity) is changed to maintain the desired lingerature ().

[00009]$\begin{array}{cc}{A}_i=\left(\frac{M^2-{\left(M_i\right)}^2}{M^2}-1\right)A,\mathrm{exothermic},& \left(5\right)\end{array}$$\begin{array}{cc}{A}_i=\left(1-\frac{M^2-{\left(M_i\right)}^2}{M^2}\right)A,\mathrm{endothermic},& \left(6\right)\end{array}$

wherein A.sub.i is a change in frictional surface area, A is an initial surface area, M is an initial system mass, M.sub.i is a current system mass during an i.sup.th iteration.

[0020] FIG. 1 depicts a system 100 comprising a chemical reaction vessel 102, a mass sensor 104 that provides the current system mass (M.sub.i) to a computer 106. The computer 106 controls a heat adjustor 108 which is configured to selectively heat or cool the chemical reaction vessel 102. In one embodiment, the chemical reaction vessel 102 is thermally insulated using conventional insulating methods to minimize heat loss to the ambient environment. The mass sensor 104 may be, for example, a mass balance. The heat adjustor 108 may include conventional heating or cooling elements and the computer 106 selectively actuates the heat adjustor 108 to control the joules of heat that is added or removed.

[0021] Referring to FIG. 2, a method 200 is disclosed for controlling a lingerature of a chemical reaction by adjusting heat. The method 200 comprises step 202, wherein the initial and desired conditions are determined. For example, the initial system mass (M) may be determined. The specific heat capacity of the chemical system is also determined. In the hypothetical example that follows, the chemical system has a specific heat capacity of 4.184 J g.sup.1K.sup.1. The values of and are specific to a particular chemical reaction and are also established in step 202. In the running example, =1 and =1. The desired conditions are also determined in step 202. For example, the desired temperature (T) may be determined by optimizing the yield of a particular chemical product by repeatedly conducting the chemical reaction for a variety of different reaction temperatures. In the hypothetical example used in this disclosure, the desired temperature (T) is 353 K. Likewise, a desired lingerature () is determined in step 202. A detailed discussion of the selection of the desired lingerature () value is found elsewhere in this disclosure. In the following hypothetical example, is 275.5 sec.

[0022] The initial system mass (M) includes the reactants, solvent and inert components but does not include products. By way of illustration, and not limitation, a given chemical reaction may involve permitting predetermined quantities of reactants A and B to react in a solvent to form a desired product C and byproducts D and E. In this example byproduct D is a gaseous byproduct which exits the reaction vessel as it forms.

##STR00001##

[0023] For example, one may calculate that 1300 g of reactant A (molar mass 100.0 g mol.sup.1) will react with 910 g of reactant B (molar mass 35.0 g mol.sup.1) in the presence of 13,000 g of a solvent. The initial system mass (M) is therefor 15,210 g.

[0024] In step 204, reactants are added to a vessel, such as vessel 102, which initiates the chemical reaction. In step 206, the lingerature is adjusted based on the amount of the exiting product (e.g. product D) that exits the vessel. Lingerature may be adjusted by heat adjustment or by surface area adjustment.

Exothermic ExampleHeat Adjustment

[0025] FIG. 3 depicts a method 300 for executing step 206. Method 300 will initially be described in terms of an exothermic example. In step 302, a quantity (i.e. some or all) of one product is allowed to exit the chemical reaction vessel during the course of the chemical reaction. The exiting of this i.sup.th iteration product permits one to calculate the i.sup.th iteration change in reactant mass (M.sub.i) that gave rise to this exited mass (E.sub.i).

[0026] In step 304, the current system mass (M.sub.i) of the i.sup.th iteration is measured with the mass sensor 104. The current system mass (M.sub.i) includes the entire contents of the vessel such as reactants, solvent, inert components and any products that have not exited the vessel. Because the current system mass (M.sub.i) is measured before the mass of the vessel is adjusted to match the initial system mass (M) (step 314) it may be referred to as M.sub.i,before.

[0027] In step 306, the exited mass (E.sub.i) that exited the vessel during the i.sup.th iteration is determined by comparing the current system mass (M.sub.i) at the i.sup.th iteration to the initial system mass (M). In the current example, this corresponds to the mass of product D that has exited the vessel. For example, if the initial system mass (M) was 15,210 g, and the first iteration (i=1) system mass (M.sub.1,before) is 15,070 (as measured before the addition of any mass), then the exited mass (E.sub.1) is found to be 140 g.

[00010]$\begin{array}{cc}{E}_i=M-M_{i,\mathrm{before}}& \left(8\right)\end{array}$$\begin{array}{cc}{E}_1=15,210g-15,070g=140g& \left(9\right)\end{array}$

[0028] In step 308, the change in reactant mass (M.sub.i) is calculated for the i.sup.th iteration based on the exited mass (E.sub.i). For example, given E.sub.1=140 g, the stoichiometry of the reaction (see equation 7) permits one to calculate that 400 g of reactant A and 280 g of reactant B was consumed:

[00011]$\begin{array}{cc}{M}_A=\frac{140gD}{1}\frac{\mathrm{mole}D}{35gD}\frac{1\mathrm{mole}A}{1\mathrm{mole}D}\frac{100.gA}{\mathrm{mole}A}=400g\mathrm{of}\mathrm{reactant}A\mathrm{consumed}& \left(10\right)\end{array}$$\begin{array}{cc}{M}_B=\frac{140gD}{1}\frac{\mathrm{mole}D}{35gD}\frac{2\mathrm{mole}B}{1\mathrm{mole}D}\frac{35.gB}{\mathrm{mole}B}=280g\mathrm{of}\mathrm{reactant}B\mathrm{consumed}& \left(11\right)\end{array}$

[0029] The change in reactant mass (M.sub.i) at the i.sup.th iteration, where i=1, is therefore:

[00012]$\begin{array}{cc}{M}_i=M_{\mathrm{reactants}}=M_A+M_B& \left(12\right)\end{array}$$\begin{array}{cc}{M}_1=400g+280g=680g\mathrm{reactants}\mathrm{consumed}& \left(13\right)\end{array}$

[0030] In step 310, a dynamic lingerature (.sub.i) is calculated for the i.sup.th iteration of this exothermic example according to:

[00013]$\begin{array}{cc}{}_i=\frac{M^2-{\left(M_i\right)}^2}{M^2},\mathrm{exothermic}& \left(14a\right)\end{array}$

[0031] In the running example (=1, M=15,210 g, M.sub.i=680 g, =275.5 sec) the reaction is exothermic. Accordingly, .sub.i is found by:

[00014]$\begin{array}{cc}{}_i=\frac{M^2-{\left(M_i\right)}^2}{M^2},\mathrm{exothermic}& \left(14b\right)\end{array}$$\begin{array}{cc}{}_i=1\frac{{\left(15,210g\right)}^2-{\left(680g\right)}^2}{{\left(15,210g\right)}^2}275.5\sec =274.9\sec & \left(14c\right)\end{array}$

[0032] In step 312, a change in lingerature (.sub.i) is calculated according to:

[00015]$\begin{array}{cc}{}_i={}_i-& \left(14d\right)\end{array}$$\begin{array}{cc}{}_i=274.9\sec -275.5\sec =-0.6\sec & \left(14e\right)\end{array}$

[0033] In step 314, a soluble mass is added in an amount equal to the exited mass (E.sub.i). This maintains the initial system mass (M) as a constant. In one embodiment, the soluble mass is in the form of additional solvent. In another embodiment, the soluble mass is in the form of additional reactants. For example, 140 g of reactants may be added to restore the system to the initial system mass (M). Given the stoichiometry of equation 7, this corresponds to 82.36 g of reactant A and 57.64 g of reactant B found by examining the molar mass ratio of reactants A:B.

[00016]$\begin{array}{cc}\frac{\mathrm{molar}\mathrm{mass}\mathrm{of}A\mathrm{molar}\mathrm{ratio}}{\mathrm{molar}\mathrm{mass}\mathrm{of}B\mathrm{molar}\mathrm{ratio}}=\frac{100.1}{35.2}=\frac{1.429}{1.}& \left(15\right)\end{array}$

[0034] The temperature of the soluble mass is preadjusted so as to not negatively impact the temperature of the chemical system. In one embodiment, the temperature of the soluble mass is preadjusted so that its addition simultaneously performs step 316 and heats or cools the chemical system as appropriate.

[0035] In step 316, a temperature of the chemical system is adjusted for this exothermic example by removing an amount of heat (Q.sub.i) according to:

[00017]$\begin{array}{cc}{Q}_i=\mathrm{SHC}M\left(1-\frac{\left(1+{}_i/\right)}{\left(1+M_i/M\right)}\right)T,\mathrm{exothermic}& \left(16\right)\end{array}$

[0036] In the running example (=1, =1, M=15,210 g, .sub.i=0.6 sec, =275.5 sec, T=353 K, SHC=4.184 J g.sup.1K.sup.1, M.sub.i=0.68 kg) the reaction is exothermic. Accordingly, Q.sub.i is found by:

[00018]$\begin{array}{cc}{Q}_i=\mathrm{SHC}M\left(1-\frac{\left(1+{}_i/\right)}{\left(1+M_i/M\right)}\right)T,\mathrm{exothermic}& \left(17\right)\end{array}$$\begin{array}{cc}{Q}_i=\left(4.184J{g}^{-1}{K}^{-1}\right)\left(15,210g\right)\left(1-\frac{1(1+\left(-0,6\sec \right)/275.5\sec }{1\left(1+\left(0.68\mathrm{kg}\right)/15.21\mathrm{kg}\right)}\right)353K,& \left(18\right)\end{array}$$Q_i=1.00910^{6}J$

[0037] This shows the chemical system has increased in energy by 1.00910.sup.6 J and, accordingly, this amount of heat is removed in step 316. The heat adjustor 108 may be actuated to provide a corresponding amount of cooling. The method 300 may be continued by returning to step 302.

Endothermic ExampleHeat Adjustment

[0038] Conversely, if the reaction is endothermic, steps 310 and 312 are different, but the method depicted in FIG. 3 is otherwise substantially the same.

[0039] For example, in step 310, a dynamic lingerature (.sub.i) is calculated (=1, M=15,210 g, M.sub.i=680 g, =275.5 sec) for an endothermic reaction according to:

[00019]$\begin{array}{cc}{}_i=2-\frac{M^2-{\left(M_i\right)}^2}{M^2},\mathrm{endothermic}& \left(19\right)\end{array}$$\begin{array}{cc}{}_i=2\left(275.5\right)-1\frac{{\left(15,210g\right)}^2-{\left(680g\right)}^2}{{\left(15,210g\right)}^2}=275.5\sec ,\mathrm{endothermic}& \left(20\right)\end{array}$$\begin{array}{cc}{}_i=276.1\sec & \left(21\right)\end{array}$

[0040] In step 312, a positive change in lingerature (.sub.i) is calculated according to:

[00020]${}_i={}_i-$${}_i=276.1\sec -275.5\sec =0.6\sec$

[0041] In step 316, a temperature of the chemical system (=1, =1, M=15,210 g, .sub.i=0.6 sec, =275.5 sec, T=353 K, SHC=4.184 J g.sup.1K.sup.1, M.sub.i=0.68 kg) is adjusted by adding or removing an amount of heat (Q.sub.i) according to:

[00021]$\begin{array}{cc}{Q}_i=\mathrm{SHC}M\left(\frac{\left(1+{}_i/\right)}{\left(1+M_i/M\right)}-1\right)T,\mathrm{endothermic}& \left(22\right)\end{array}$$\begin{array}{cc}{Q}_i=\left(4.184J{g}^{-1}{K}^{-1}\right)\left(15,210g\right)\left(\frac{1\left(1+\left(-0.6\sec \right)/275.5\sec \right)}{1\left(1+\left(0.68\mathrm{kg}\right)/15.21\mathrm{kg}\right)}-1\right)353K,\mathrm{endo}& \left(23\right)\end{array}$$Q_i=-1.00910^{6}J$

[0042] This shows the chemical system has decreased in energy by 1.00910J and, accordingly, this amount heat is added in step 316. The heat adjustor 108 may be actuated to provide a corresponding amount of heating.

Exothermic ExampleSurface Area Adjustment

[0043] In another embodiment, lingerature is adjusted by adjusting the surface area the reactants experience during the reaction. Referring to FIG. 5, a method 500 is depicted that is another method for performing step 206 of FIG. 2. Method 500 is substantially similar to select steps of method 300 (see FIG. 3) except in that step 502 replaces steps 310 and 312 and step 316 is omitted. Step 504 (addition of an insoluble mass) is new. Step 506 (addition of a soluble mass) is similar to step 314 but differs in the amount of soluble mass being adjusted to compensate for the insoluble mass.

[0044] In the following example, an exothermic reaction is used. In step 502, a change in frictional surface area (A.sub.i) is found according to:

[00022]$\begin{array}{cc}{A}_i=\left(\frac{M^2-{\left(M_i\right)}^2}{M^2}-1\right)A,\mathrm{exothermic},& \left(24\right)\end{array}$

wherein A is an initial surface area, M is an initial system mass, M.sub.i is a current system mass during an i.sup.th iteration.

[0045] Referring again to the aforementioned exothermic example (=1, M=15,210 g, M.sub.i=15,070 g, E.sub.1=140 g) the initial surface area (A) of the vessel is found to be 3,414 cm.sup.2 (0.3414 m.sup.2). The change in surface area to restore the lingerature is therefore found according to:

[00023]$\begin{array}{cc}{A}_i=\left(\frac{M^2-{\left(M_i\right)}^2}{M^2}-1\right)A,\mathrm{exothermic},& \left(25\right)\end{array}$$\begin{array}{cc}{A}_i=\left(1\frac{{\left(15,210g\right)}^2-{\left(15,210g-15,070g\right)}^2}{{\left(15,210g\right)}^2}-1\right)0.3414m^2,\mathrm{exothermic},& \left(26\right)\end{array}$$A_i=-0.00002892m^2$

[0046] In step 504, an insoluble mass is added to the vessel that provides a surface area to compensate for the negative change in frictional surface area (A.sub.i) found in step 502. The insoluble mass increases the surface area experienced by the reactants. One example of a highly porous, insoluble media is activated carbon which can be prepared to provide different amounts of surface area. In other embodiments, insoluble beads (e.g. silica, glass, polymeric, etc.) with lower porosity are added that change the exposed surface area. In still other embodiments, a rod is inserted into the chemical system to a predetermined depth such that the depth of insertion controls the surface area of the rod that is exposed to the system. In the following example spherical glass beads (radius 1 mm) provide a surface area of 1.2610.sup.5m.sup.2 per bead.

[0047] Given the surface area provided by the exemplary glass beads, the amount of added beads is:

[00024]$\begin{array}{cc}\left(\frac{1\mathrm{bead}}{1.2610^{-5}{m}^2}\right)\left(\frac{0.00002892m^2}{1}\right)=4\mathrm{beads}& \left(27\right)\end{array}$

[0048] A constant system mass (M) is maintained in step 506, wherein a sufficient amount of soluble mass (e.g. additional reactants, solvent, etc.) is added. In the running example, 140 g total mass (i.e. the exited mass (E.sub.i)) is added over steps 504 and 506 and consists of the mass of the 4beads (0.017 g when the beads have a density of 1.03 g per mL) plus a sufficient amount of the soluble mass to total 140 g. By replacing a soluble mass (which does not provide a surface area) with an insoluble mass (which does provide a surface area) the surface area is altered while maintaining a constant system mass. The method 500 may be repeated by returning to step 302.

Endothermic ExampleSurface Area Adjustment

[0049] In the following example, an endothermic reaction is used. In step 502, a change in frictional surface area (A.sub.i) is found according to:

[00025]$\begin{array}{cc}{A}_i=\left(1-\frac{M^2-{\left(M_i\right)}^2}{M^2}\right)A,\mathrm{endothermic},& \left(28\right)\end{array}$

wherein A is an initial surface area, M is an initial system mass, M.sub.i is a current system mass during an i.sup.th iteration.

[0050] Referring again to the aforementioned exothermic example (=1, M=15,210 g, M.sub.i=15,070 g, E.sub.1=140 g) the initial surface area (A) of the vessel is found to be 3,414 cm.sup.2 (0.3414 m.sup.2). The change in surface area to restore the lingerature is therefore found according to:

[00026]$\begin{array}{cc}{A}_i=\left(1-\frac{M^2-{\left(M_i\right)}^2}{M^2}\right)A,\mathrm{endothermic},& \left(29\right)\end{array}$$\begin{array}{cc}{A}_i=\left(1-1\frac{{\left(15,210g\right)}^2-{\left(15,210g-15,070g\right)}^2}{{\left(15,210g\right)}^2}\right)0.3414m^2,\mathrm{endothermic},& \left(30\right)\end{array}$$A_i=0.00002892m^2$

[0051] In step 504, an insoluble mass is removed from the vessel to compensate for the positive change in frictional surface area (A.sub.i) found in step 502. To permit this removal, the vessel was pre-loaded with an insoluble mass (e.g. glass beads) before the chemical reaction began. A corresponding amount of the insoluble mass is removed from the vessel in step 504 (e.g. 4 glass beads are removed).

[0052] A constant system mass is still maintained in step 506 by compensating for the mass that exited the system. Accordingly, soluble mass (e.g. additional reactants or solvent) are also added in step 506, in addition to the insoluble mass. By replacing an insoluble mass (which provided a surface area) and with a soluble mass (e.g. solvent or reactants) the surface area is altered while maintaining a constant system mass.

Establishing

[0053] The desired lingerature () value is related to the stress experienced by the chemical system. The desired lingerature () may be found from an experimentally-obtained plot of dynamic viscosity () versus viscosity-temperature (T()). In one embodiment, the plot for the solvent is used to approximate the chemical system. For example, FIG. 4 depicts a plot of dynamic viscosity () versus viscosity-temperature (T()) for water. In the running example, the desired temperature (T) is 353 K. A range of temperatures surrounding the desired temperature (T) is selected. For example, a range of 10% of the desired temperature (T) may be selected (e.g. 318 K (T.sub.min) to 388 K (T.sub.max)). This range may be approximated by a regression equation (e.g. linear regression). In FIG. 4 the equation of the corresponding line is (t)=0.0028 mPa K.sup.1(T())+1.35 mPa in the form y=mx+b, wherein y is the dynamic viscosity (), x is the viscosity-temperature (T()), b is the y-intercept and m is the slope . . .

[0054] In a first embodiment, the desired lingerature () is found based on the desired temperature (T) according to:

[00027]$\begin{array}{cc}=\frac{b+mT}{\frac{{\mathrm{GM}}^2}{{\mathrm{rVN}}_{\mathrm{DoF}}}}& \left(31\right)\end{array}$

wherein m is the slope, b is the y-intercept, T is the temperature of the chemical reaction, M is the initial mass of the system, G is the gravitational constant (6.674310.sup.11m.sup.3kg.sup.1s.sup.2), N.sub.DoF is the number of degrees of freedom of the solvent molecules (e.g. 5 for water), V is the volume of the vessel and r is the radius of spherical volume that corresponds to the volume (V) and y is a dissimilar cells factor. When the desired lingerature () is found in this manner it may also be referred to as the normal lingerature (.sub.N).

[0055] For example, a hypothetical vessel is a cylinder with a radius of 13 cm and a height of 28.8 cm thus characterized by a volume (V) of 15,290.75 cm.sup.3 or 0.01529 m.sup.3 in SI units. The spherical radius (r) for the vessel is derived from the expression (3V/4).sup.1/3 and is 0.15386 m. When the temperature (T) is 353 K and is 1, the desired lingerature () is therefore 275.5 sec. If the temperature (T) is 353 K and y is 0.07653, t would be one hour. Thus, the larger the desired value of , the smaller the value of .

[0056] In a second embodiment, a lingerature is experimentally selected within a range of potential lingerature values established based on the equation of the line. Multiple experiments are run at the desired temperature (T) while the lingerature is varied within this range. The yield for each experiment is measured and a desired lingerature () is found based on this yield optimization. A range of acceptable desired lingerature () values is within a range given by:

[00028]$\begin{array}{cc}{}_{\max }=\frac{b+m{T}_{\min }}{\frac{{\mathrm{GM}}^2}{{\mathrm{rVN}}_{\mathrm{DoF}}}}& \left(32a\right)\end{array}$$\begin{array}{cc}{}_{\min }=\frac{b+m{T}_{\max }}{\frac{{\mathrm{GM}}^2}{{\mathrm{rVN}}_{\mathrm{DoF}}}}& \left(32b\right)\end{array}$

[0057] In the running example, this results in a range of lingeratures between 201 sec and 350 sec. The chemical reaction is then conducted (i.e. method 300 is performed) at multiple lingeratures, but constant temperature, within this range while the resulting yield is measured. By optimizing yield in this manner, the desired lingerature () is found. In other embodiments, criteria other than yield optimization are used to find the desired lingerature () provided that the desired lingerature () falls within the range given by .sub.min and .sub.max. In one such embodiment, the desired lingerature () is within 5% of the normal lingerature (.sub.N).

Establishing

[0058] The value for a given chemical reaction is related to its thermal behavior. A statistical value for can be readily found. For instance, it can be found for a hypothetical chemical reaction, such as equation 7 where a gas D (g) is being continuously released, using the following method:

[00029]$\begin{array}{cc}=\frac{{}_{i=1}^N{}_i}{N}1& \left(33\right)\end{array}$$\begin{array}{cc}{}_i=\frac{\left(\frac{M}{T}\right)}{\left(\frac{M-M_i}{T-.Math.T_i.Math.}\right)}& \left(34\right)\end{array}$

where the equation denotes the averaging of N repeated reactions of the chemical reaction. These reactions yield N realizations for specified as {.sub.i: i=1, . . . , N} where .sub.i is found at each measuring time (t.sub.i) where: a) M is the initial system mass and Tis the target temperature; b) M.sub.i is the change in reactant mass during the time interval t.sub.i that starts at the beginning of the chemical reaction; and c) |T.sub.i| is the magnitude of the temperature change experienced during the time interval t.sub.i.

[0059] A set of suitable reactant time intervals {t.sub.i: i=1, . . . , N} is selected to conduct the N independent experiments. The i.sup.th experiment yields a value for M.sub.i that can be experimentally determined using a mass sensor that measures the mass of the gas product D that exists the vessel during the time interval t.sub.i. The magnitude of the temperature change of the vessel during the time interval t.sub.i is measured using, for instance, a thermocouple. The reactions are performed under conditions where the vessel is highly thermally insulated to obtain the best results.

[0060] Tables 1-3 present values and physical characteristics pertaining to a hypothetical chemical reaction example. This example examines the results derived for the generic chemical reaction given by equation 7 where the statistical value for is found to be close to one. The hypothetical example considers the case where: a) the solvent mass has the constant value of 13,000 g; b) the initial mass for the reactant A is 1,300 g; c) the initial mass of the reactant B is 910 g; d) the initial system mass M in the vessel is 15,210 g; e) the target temperature T in the vessel is 353 K. At four different times (t.sub.1, t.sub.2, t.sub.3, t.sub.4) a mass balance is used to measure the mass of the exited gas product D, seen in Table 2 to be given by (17 g, 70 g, 123 g, 140 g). The stoichiometry of the reaction then permits one to find the total mass of the reactants A plus B (85g, 340 g, 595 g, 680 g) that are responsible for the mass of the exited gas product D (17 g, 70 g, 123 g, 140 g). During each of these times a measurement is made of the temperature to determine its change (1.9 K, 8.0 K, 13.8 K, 16.0 K) from the target temperature T. Equation 34 is then used to evaluate the value of a at each different measuring time (1.00021, 0.99968, 1.00003, 0.99935), whose average value is then found to be approximately equal to one, i.e., 1.

TABLE-US-00001 TABLE 1 Solvent Reactant A Reactant B Initial Target Desired Constant Initial Initial Mass Temp Lingerature Mass Mass Mass (M) (T) () (g) (g) (g) (g) (K) (sec) 13,000 1,300 910 15,210 353 275.5

TABLE-US-00002 TABLE 2 Change in Exiting Spent A Spent B Reactant Temper- Thermal Time Gas Reactant Reactant Mass ature Change Interval D.sub.i A.sub.i B.sub.i M.sub.i T.sub.i |T.sub.i| t.sub.i (g) (g) (g) (g) (K) (K) t.sub.1 17 50 35 85 354.9 1.9 t.sub.2 70 200 140 340 361.0 8.0 t.sub.3 123 350 245 595 366.8 13.8 t.sub.4 140 400 280 680 369.0 16.0

TABLE-US-00003 TABLE 3 Time Interval t.sub.i Change in Reactant Mass M.sub.i (g) Thermal Change |T.sub.i| (K) [00030]${}_i=\frac{\left(\frac{M}{T}\right)}{\left(\frac{M-M_i}{T-.Math.T_i.Math.}\right)}$ t.sub.1 85 1.9 1.00021 t.sub.2 340 8.0 0.99968 t.sub.3 595 13.8 1.00003 t.sub.4 680 16.0 0.99935 Average 0.999818

[0061] In one embodiment, is a number between 0.8 and 1.2. In another embodiment, is a number between 0.9 and 1.1. In another embodiment, is a number between 0.95 and 1.05.

Establishing

[0062] The value for a given chemical reaction is related to its linger behavior. A statistical value for can be readily found. For instance, it can be found for the same hypothetical chemical reaction shown in equation 7.

[00031]$\begin{array}{cc}=\frac{{}_{i=1}^N{}_i}{N}1& \left(35\right)\end{array}$$\begin{array}{cc}{}_i=\frac{\left(\frac{-.Math.{}_i.Math.}{}\right)}{\frac{M^2-{\left(M_i\right)}^2}{M^2}}& \left(36\right)\end{array}$

[0063] Experimentally-obtained plots of dynamic viscosity () versus viscosity-temperature (T()) are known for a variety of substances. See, for example, FIG. 4, which shows such a plot for water. Dynamic viscosity () provides a means to determine the calculated lingerature (.sub.i). In the hypothetical example, the target temperature T is 353 K. Referring to the plot of FIG. 4, a temperature range of 318 K to 388 K was selected that both encompassed the target temperature and may be approximated by a regression equation (e.g. linear regression). In FIG. 4 the equation of the corresponding line is ()=0.0028 mPa K.sup.1(T())+1.35 mPa in the form y=mx+b, wherein y is the dynamic viscosity () and x is the viscosity-temperature (T(). The calculated lingerature (.sub.i) is found based on the measured temperature (T.sub.i) and the reactant change (M.sub.i) according to the lingerature transducer equation:

[00032]$\begin{array}{cc}{}_i=\frac{b+m{T}_i}{\frac{{G\left(M-M_i\right)}^2}{rV{N}_{DoF}}}& \left(37\right)\end{array}$

wherein m is the slope, b is the y-intercept, T.sub.i is a measured temperature during the i.sup.th interval, M is the initial mass of the system, M.sub.i is a change in reactant mass during the i.sup.th interval found based on the exited mass (E.sub.i), G is the gravitational constant (6.674310.sup.11m.sup.3kg.sup.1s.sup.2), N.sub.DoF is the number of degrees of freedom of the solvent molecules (e.g. 5 for water), V is the volume of the vessel and r is the radius of spherical volume that corresponds to the volume (V).

[0064] For example, a hypothetical vessel is a cylinder with a radius of 13 cm and a height of 28.8 cm thus characterized by a volume (V) of 15,290.75 cm.sup.3 or 0.01529 m.sup.3 in SI units. The spherical radius (r) for the vessel is derived from the expression (3V/4).sup.1/3 and is 0.15386 m. In the following example, the desired lingerature () is 275.5 sec, corresponding to equal to one and the system has the same initial conditions as shown in Table 4. The chemical reaction is permitted to run while its change in reactant mass (M.sub.i ) and measured temperature (T.sub.i) are measured. This permits one to calculate values for .sub.i as the reaction progresses.

TABLE-US-00004 TABLE 4 Time Interval t.sub.i Change in Reactant Mass M.sub.i (g) Measured Temperature T.sub.i (K) Calculated Lingerature [00033]${}_i=\frac{b+m{T}_i}{\frac{{G\left(M-M_i\right)}^2}{{\mathrm{rVN}}_{\mathrm{DoF}}}}$ (sec) Change in Lingerature .sub.i (sec) [00034]${}_i=\frac{\left(\frac{-.Math.{}_i.Math.}{}\right)}{\frac{M^2-{\left(M_i\right)}^2}{M^2}}$ t.sub.1 85 354.9 274.47 1.03 0.99636122 t.sub.2 340 361.0 270.35 5.15 0.98179294 t.sub.3 595 366.8 266.48 9.02 0.96874193 t.sub.4 680 369.0 264.47 11.03 0.96188782 Average 0.97719597

[0065] In one embodiment, is a number between 0.8 and 1.2. In another embodiment, is a number between 0.9 and 1.1. In another embodiment, is a number between 0.95 and 1.05.

Establishing Specific Heat Capacity (SHC)

[0066] In one embodiment, the SHC of the system is found by conventional thermochemical means (e.g. calorimetry). In another embodiment, a dynamic specific heat capacity (SHC.sub.i) of the system is calculated for each i.sup.th iteration according to:

[00035]$\begin{array}{cc}SH{C}_i=\frac{k_B}{2m_{S{G}_y,i}}& \left(38\right)\end{array}$

wherein k.sub.B is the Boltzmann constant and m.sub.SGy,i is the i.sup.th iteration subgyrador mass. The i.sup.th iteration subgyrador mass is given by

[00036]$\begin{array}{cc}{m}_{S{G}_y,i}=\frac{M_i}{N_{\mathrm{Th},i}}& \left(39\right)\end{array}$

wherein M.sub.i is the change in reactant mass during the i.sup.th iteration. The N.sub.Th,i is the i.sup.th iteration number of reactant-thermotes given by:

[00037]$\begin{array}{cc}{N}_{\mathrm{Th},i}=\frac{M_i/k_{B}T}{2\ln \left(\frac{M}{M_i}\right)+1-\frac{\ln \left(q\right)+\frac{N_{DoF}}{2}-2\ln \left(\frac{M}{M_i}\right)}{\ln \left(N_{Th}\right)-1}}& \left(40\right)\end{array}$$\mathrm{wherein}$$\begin{array}{cc}{N}_{Th}=\frac{M{c}^2}{N_{DoF}{k}_{B}T/2}& \left(41\right)\end{array}$$\mathrm{and}$$\begin{array}{cc}q={\left(k_{B}T\right)}^{\frac{N_{DoF}}{2}}{\left(\frac{m}{2}\right)}^{\frac{3}{2}}\frac{2I}{{}^5}V& \left(42\right)\end{array}$

is the reduced Planck constant (1.0545710.sup.34J s); c is the speed of light (2.999710.sup.8 m s.sup.1); kg is the Boltzmann constant (1.38110.sup.23 J K.sup.1); K is a constant that relates reactant kgs to heat in joules (20.9 MJ per kg in this example); M is the initial system mass, T is the desired temperature; N.sub.DoF is the number of degrees of freedom of the solvent molecules (e.g. 5 for water); I is the average moment of inertia of the solvent (e.g. 210.sup.47 m.sup.2kg for water); is the symmetry value of the solvent (2 for water); m is the mass of a solvent molecule (e.g. 310.sup.26 kg for water); V is the volume of the vessel (e.g. 0.01529 m.sup.3). The constant is generally about 20.9 MJ per kg but may be, in one embodiment, within 5% and can be found by optimizing for the yield of the desired product (see Theoretical Background section). In another embodiment, the constant is within 2% of 20.9 MJ per kg.

[0067] For example, in a given i.sup.th iteration, the SHC.sub.i may be found to be 3.779 J g.sup.1K.sup.1 as shown below. In the following example, the change in reactant mass (M.sub.i) is 340 g, the initial system mass (M) is 15.21 kg, then a partition function (q) is given by:

[00038]$\begin{array}{cc}q={\left(k_{B}T\right)}^{\frac{N_{DoF}}{2}}{\left(\frac{m}{2}\right)}^{\frac{3}{2}}\frac{2I}{{}^5}V& \left(43\right)\end{array}$$q={\left(1.38110^{-23}J.K^{-1}353K\right)}^{\frac{5}{2}}{\left(\frac{310^{-26}\mathrm{kg}}{2}\right)}^{\frac{3}{2}}\frac{2210^{-47}{m}^2\mathrm{kg}}{2{\left(1.0545710^{-34}J.s\right)}^5}0.01529m^3$$q=1.283712910^{31}$

Likewise, N.sub.Th is found according to:

[00039]$\begin{array}{cc}\begin{array}{c}{N}_{Th}=\frac{M{c}^2}{N_{DoF}{k}_{B}T/2}\\ N_{Th}=\frac{15.21\mathrm{kg}{\left(2.999710^{8}m/s\right)}^2}{51.381{10}^{-23}J.K^{-1}353K/2}\\ N_{Th}=1.1232810^{38}\end{array}& \left(44\right)\end{array}$

In the present example, the i.sup.th iteration number of reactant-thermotes (N.sub.Th,i) is therefore given by:

[00040]$\begin{array}{cc}{N}_{\mathrm{Th},i}=\frac{M_i/k_{B}T}{2\ln \left(\frac{M}{M_i}\right)+1-\frac{\ln \left(q\right)+\frac{N_{DoF}}{2}-2\ln \left(\frac{M}{M_i}\right)}{\ln \left(N_{Th}\right)-1}}& \left(45\right)\end{array}$$N_{\mathrm{Th},i}=\frac{20.9{10}^{6}J.{\mathrm{kg}}^{-1}0.34\mathrm{kg}/\left(1.381{10}^{-23}J.K^{-1}353K\right)}{2\ln \left(\frac{15.21\mathrm{kg}}{0.34\mathrm{kg}}\right)+1-\frac{\ln \left(1.2837129{10}^{31}\right)+\frac{5}{2}-2\ln \left(\frac{15.21\mathrm{kg}}{0.34\mathrm{kg}}\right)}{\ln \left(1.1232810^{38}\right)-1}}$$N_{Th,i}=1.8609510^{26}$

[0068] The i.sup.th iteration subgyrador mass

[00041]$\left(m_{S{G}_y,i}\right)$

is found according to:

[00042]$\begin{array}{cc}{m}_{S{G}_y,i}=\frac{M_i}{N_{\mathrm{Th},i}}=\frac{0.34\mathrm{kg}}{1.86095{10}^{26}}=1.8270210^{-27}\mathrm{kg}& \left(46\right)\end{array}$

[0069] Finally, the SHC.sub.i for the i.sup.th iteration is given by

[00043]$\begin{array}{cc}SH{C}_i=\frac{k_B}{2m_{S{G}_y,i}}=\frac{1.38110^{-23}J.Math.K^{-1}}{2\left(1.8270210^{-27}\mathrm{kg}\right)}=3779J{\mathrm{kg}}^{-1}{K}^{-1}=3.779J{g}^{-1}{K}^{-1}& \left(47\right)\end{array}$

[0070] During subsequent iterations, the SHC.sub.i may be dynamically recalculated based on an updated change in reactant mass (M.sub.i).

Frequency of Lingerature Adjustment

[0071] In one embodiment, the stress a chemical reaction experiences is maintained by adjusting the lingerature (i.e. performing method 300 or method 500) frequently enough to maintain a stress value within a predetermined threshold. Examples of suitable stress values include (1) a lingerature value (.sub.i), (2) a viscosity value (.sub.i), and (3) a pressure value (P.sub.i). In one embodiment, the lingerature value and the pressure value are maintained within 10% of their normal lingerature value (.sub.N) and normal pressure value (P.sub.N), respectively. In another one embodiment, the lingerature value and the pressure value are maintained within 5% of their normal lingerature value (.sub.N) and normal pressure value (P.sub.N), respectively. The viscosity deviation is within 1% of the sum of the lingerature deviation and the pressure deviation. For example, if the lingerature deviation is 5% and the pressure deviation is 5%, then the viscosity deviation is between 9-11%. As a further example, if the lingerature deviation is 10% and the pressure deviation is 10%, then the viscosity deviation is between 19-21%. As a further example, if the lingerature deviation is 3% and the pressure deviation is 3%, then the viscosity deviation is between 5-7%. In another embodiment the lingerature and pressure value are maintained within 10% of their normal values which then results in a viscosity value within 1% of 20% of its normal value where the 20% of the viscosity deviation results from the sum of the 10% for the lingerature deviation and the 10% for the pressure deviation.

[0072] These normal values are calculated based on the desired temperature (T) which was obtained by maximizing the yield of a desired product. In the running example, the desired temperature (T) is 353 K, the initial system mass (M) is 15.21 kg, the reaction occurs in a vessel that is a cylinder with a radius of 13 cm and a height of 28.8 cm thus characterized by a volume (V) of 15,290.75 cm.sup.3 or 0.01529 m.sup.3 in SI units. The spherical radius (r) for the vessel is derived from the expression (3V/4).sup.1/3 and is 0.15386 m. The solvent is water (N.sub.DoF=5) and the dynamic viscosity () versus viscosity-temperature (T() plot of FIG. 4 applies (m=0.0028 mPa.Math.s.Math.K.sup.1, b=1.35 mPa.Math.s)).

[0073] The normal lingerature value (.sub.N) is found according to:

[00044]$\begin{array}{cc}{}_N=\frac{b+mT}{\frac{{G\left(M\right)}^2}{rV{N}_{DoF}}}& \left(48\right)\end{array}$

[0074] In the running example, the normal lingerature value (.sub.N) is 275.5 seconds.

[0075] The normal viscosity value (.sub.N) is found according to:

[00045]$\begin{array}{cc}{}_N=b+\left(mT\right)& \left(49\right)\end{array}$

[0076] In the running example, the normal viscosity value (P.sub.N) is 0.3616 mPa.Math.s.

[0077] The normal pressure value (P.sub.N) is found according to:

[00046]$\begin{array}{cc}{P}_N=\frac{b+mT}{{}_N}& \left(50\right)\end{array}$

[0078] In the running example, the normal pressure value (P.sub.N) is 0.001313 mPa.

[0079] During the course of the reaction the exited mass (E.sub.i) leaves the vessel which produces the change in reactant mass (M.sub.i) and a temperature change (T.sub.i). These values, in turn, produce the change in lingerature (.sub.i), a change in viscosity (.sub.i) and a change in pressure (P.sub.i).

[0080] The lingerature value (.sub.i) is found according to:

[00047]$\begin{array}{cc}{}_i=\frac{M^2-{\left(M_i\right)}^2}{M^2}{}_N,\mathrm{exothermic}& \left(51\right)\end{array}$$\begin{array}{cc}{}_i=2-\frac{M^2-{\left(M_i\right)}^2}{M^2},\mathrm{endothermic}& \left(52\right)\end{array}$

[0081] For example, when the change in reactant mass (M.sub.i) is 0.3628 kg for an exothermic reaction and the normal lingerature value (.sub.N) is 275.5 seconds then .sub.i is 262.5 seconds. This shows a 4.7% change in lingerature which is within a 5% threshold.

[0082] Conversely, if the user waited until the change in reactant mass (M.sub.i) is 0.4000 kg and the normal lingerature value (.sub.N) is 275.5 seconds then .sub.i is 261.2 seconds. This shows a 5.2% change in lingerature which is outside of the 5% threshold. In those embodiments that use 5% as the predetermined threshold, the user should adjust the lingerature more frequently.

[0083] Similarly, the viscosity value (.sub.i) is found according to:

[00048]$\begin{array}{cc}{}_i=b+\left({\mathrm{mT}}_S\right)& \left(53\right)\end{array}$

wherein T.sub.S is a sampling temperature found according to:

[00049]$\begin{array}{cc}{T}_s=\frac{-b+\frac{{G\left(M-M_i\right)}^2}{{\mathrm{rVN}}_{\mathrm{DoF}}}{}_i}{m}& \left(54\right)\end{array}$

[0084] For example, when the change in reactant mass (M.sub.i) is 0.3628 kg and the normal lingerature value (.sub.N) is 275.5 seconds then .sub.i is 262.5 seconds. From this, the sampling temperature (T.sub.S) is found to be 364.9 K and the viscosity value (.sub.i) is therefore 0.3283 mPa.Math.s. This shows a 9.2% change in viscosity which is within a 10% threshold.

[0085] Conversely, if the user waited until the change in reactant mass (M.sub.i) is 0.4000 kg (.sub.N=275.5 sec, .sub.i=261.2 sec, T.sub.S=364.9 K) the viscosity value (.sub.i) is 0.3250 mPa.Math.s. This shows a 10.1% change in viscosity which is outside of a 10% threshold. In those embodiments that use 10% as the predetermined threshold, the user should adjust the lingerature more frequently.

[0086] The pressure value (P.sub.i) is found according to:

[00050]$\begin{array}{cc}{P}_i=\frac{b+{\mathrm{mT}}_s}{{}_i}& \left(55\right)\end{array}$

[0087] For example, when the change in reactant mass (M.sub.i) is 0.3628 kg (.sub.N=275.5 sec, .sub.i=261.2 sec, T.sub.S=364.9 K) the pressure value (P.sub.i) is 0.001251 mPa. This shows a 4.7% change in pressure which is within a 5% threshold.

[0088] Conversely, if the user waited until the change in reactant mass (M.sub.i) is 0.4000 kg (.sub.N=275.5 sec, .sub.i=261.2 sec, T.sub.S=364.9 K) the pressure value (Pi) is 0.001244 mPa. This shows a 5.2% change in pressure which is outside of a 5% threshold. In those embodiments that use 5% as the predetermined threshold, the user should adjust the lingerature more frequently.

THEORETICAL BACKGROUND

Introduction

[0089] The lingerature of any given chemical reaction can be defined as the predicted future time-duration of constancy of the viscosity and pressure that act on the particles of the medium. In this disclosure the control of the lingerature of either exothermic or endothermic reactions under non-equilibrium conditions is addressed by looking at chemical reactions through the lens of an ascending Quantum Gravity Linger Thermo Theory (QG-LTT) in physics. In this innovative framework, lingerature control methods are derived using the medium's quantum gravitational spacetime where gravity and the gyrations of mass quantums fueled by thermal quantums, rules the mediums viscosity and pressure. In this quantum spacetime the motion of mass quantums, named gyradors, which control the gravitational collapse of the medium, are constrained by the medium's linger-viscosity, while the retention of energy quantums named thermotes which fuel the gyradors' kinetic energy, are constrained by the medium's thermal-energy. In QG-LTT, the particles where the constant pressure act are dissimilar cells (DCS) made of arbitrary-size groups of atoms and/or molecules that are created, exist, die, or exit the medium. These dissimilar cells are special since, through their collective action, they unveil a QG Spacetime-Enclosing for optimum energy efficient quantum gravity interactions. The first DCS findings were star particles and organism cells first reported at an AAS Meeting in 2019. The DCS driven QG-LTT of this application has led to maximally efficient and affordable iteration methods for the dynamic control of the lingerature of chemical reactions. The roots of the QG-LTT formulation reside in a past-uncertainty/future-certainty time-complementary duality principle of physics, named POP, that first popped up in controls. In the late 1970s the POP was discovered to have naturally surfaced in the 1960 stochastic control work of Rudolph E. Kalman who optimally controlled past-uncertainty unquantized states with future-certainty unquantized controls. This revelation was then used in a CUNY Graduate Center PhD in 1981 to advance a maximally efficient and affordable future-certainty Matched Processors (MPs) alternative to Bellman's Dynamic Programming for quantized control (see, Int' Journal of Control, Vol. 42, pp. 695-713, 1985) with applications in neural networks. The POP guided unification of the future-certainty MPs with the past-uncertainty Matched Filters (MFs) for bit detection of digital communications would then lead to the efficient control of past-uncertainty quantized states with future-certainty quantized controls. Another POP guided theory was Latency Information Theory, formulated in the mid-2000s for the efficient solution of complex detection problems such as those encountered in radar (see U.S. Pat. No. 10,101,445). In general terms the POP directs the use and/or formulation of new future-certainty methods to make predictions about future physical states, for instance, when the deterministic Laws of Motion in Physics are used to make predictions about the future position and velocity of a particle. For the present and past states, the POP instructs the use and/or formulation of new past-uncertainty methods to handle present and past uncertainty states, for instance, when the laws of thermodynamics are used to determine the amount of thermal-energy that is necessary to fuel the motion of an object whose initial position and speed is known to always be uncertain as the Heisenberg Uncertainty Principle instructs. The first principles of QG-LTT will be reviewed here and then used in the derivation of the lingerature control methods of this disclosure. In an opening section the exothermic control methods will be proven first and then in another section the endothermic control ones. In a final section the dynamic specific heat capacity of controlled chemical reactions will be derived. The derivation of the dynamic specific heat capacity will make use of a novel thermote entropy for use with general mediums, whose discovery was also guided by the POP when the entropy of flexible phase mediums was being sought for use in lifespan studies (see, The Flexible Phase Entropy and its Rise from the Universal Cybernetics Duality, IEEE Int' Conference on SMC, pp. 3221-3228, San Diego, CA, Oct. 5-8, 2014). Finally, it is highlighted that the lingerature control principles presented are novel and anchored in antigravitational quantum effects propelled by the medium's dissimilar cells (see, Principle of Universal Antigravitation is Revealed through the Guidance of Principle of Physics, Gravitational Physics, Experiment/Theoretical, Poster Abstract, American Physical Society: Global Physics Summit, Anaheim, CA, 16-21 Mar. 2025).

The Exothermic Lingerature Control Methods

The Exothermic Dynamic Lingerature Control Equation (DLCE) and its Derivation

Introductory Words

[0090] The exothermic DLCE is a dynamic i.sup.th iteration equation that allows the calculation of the change in lingerature (.sub.i) that an exothermic chemical reaction has decreased by the end of the i.sup.th iteration of processing. This .sub.i knowledge is then used in its correction.

[0091] The DLCE is derived under several physical constraints. First, the medium is noted to satisfy the following gyradors' viscosity equation of state (VEOS):

[00051]$\begin{array}{cc}=P_{\mathrm{Gy}}.& \left(56\right)\end{array}$

which equates the initial amount of the medium's dynamic viscosity () to the product of its desired lingerature () and a constant gyrador pressure (P.sub.Gy) that acts on the medium's DCS. In turn, the constant gyrador pressure (P.sub.Gy) can be derived from the following gyradors' energy equation of state (EEOS):

[00052]$\begin{array}{cc}{P}_{\mathrm{Gy}}V=k_{B}T\left(N_{\mathrm{Gy}}-N_{\mathrm{Gy}}^{}\right)=k_{B}T\left(M/m_{\mathrm{Gy}}-M^{}/{m^{}}_{\mathrm{Gy}}\right)=G{M}^2/{\mathrm{rN}}_{\mathrm{DoF}}-{G\left(M^{}\right)}^2/{\mathrm{rN}}_{\mathrm{DoF}}& \left(57\right)\end{array}$$\begin{array}{cc}{N}_{\mathrm{Gy}}=M/m_{\mathrm{Gy}}=N_{\mathrm{Cell}}={\mathrm{GM}}^2/{\mathrm{rN}}_{\mathrm{DoF}}{k}_{B}T& \left(58\right)\end{array}$$\begin{array}{cc}{N}_{\mathrm{Gy}}^{}=M^{}/{m^{}}_{\mathrm{Gy}}=N_{\mathrm{Cell}}^{}={G\left(M^{}\right)}^2/{\mathrm{rN}}_{\mathrm{DoF}}{k}_{B}T& \left(58a\right)\end{array}$$\begin{array}{cc}{m}_{\mathrm{Gy}}=\frac{{.Math.}_{i=1}^L{n}_i{m}_i}{{.Math.}_{i=1}^L{n}_i}=\frac{M}{N_{\mathrm{Cell}}}=\frac{e_{Th}}{v_{\mathrm{Gy}}^2/2}=\frac{N_{\mathrm{DoF}}{k}_{B}Tr}{\mathrm{GM}}=\frac{M}{N_{\mathrm{Gy}}}& \left(59\right)\end{array}$$\begin{array}{cc}{m}_{\mathrm{Gy}}^{}=\frac{{.Math.}_{i=1}^L{n^{}}_i{m}_i}{{.Math.}_{i=1}^L{n^{}}_i}=\frac{M^{}}{N_{\mathrm{Cell}}^{}}=\frac{e_{Th}}{{v^{}}_{\mathrm{Gy}}^2/2}=\frac{N_{\mathrm{DoF}}{k}_{B}Tr}{{\mathrm{GM}}^{}}=\frac{M^{}}{N_{\mathrm{Gy}}^{}}& \left(59a\right)\end{array}$

where: 1) V is the medium's volume; 2) k.sub.B is the Boltzmann constant; 3) T is the medium's temperature; 4) M is the medium's mass; 5) r is the radius of the assumed spherical medium's volume; 6) L is the number of the different mass amounts found in the DCS (e.g. it is two if only hydrogen and helium atoms are found in the DCS); 7) n.sub.i is the number of DCS with an m.sub.i mass; 8)

[00053]$N_{Cell}={.Math.}_{i=1}^L{n}_i$

is the total number of DCS; 9)

[00054]$M={.Math.}_{i=i}^L{n}_i{m}_i$

is the medium's total mass expressed in terms of the medium's MCS; 10)

[00055]$m_{Gy}={.Math.}_{i=1}^L{n}_i{m}_i/{.Math.}_{i=1}^L$

n.sub.i is the mass of each gyrador which is the same as the average-mass of the medium's DCS; 11) N.sub.DoF is the medium's number of degrees of freedom (DoF), for example, it is 2 for a black-hole, it is 3 for a photon-gas, and it is 5 for a flexible-phase medium made of water molecules; 12) N.sub.Gy=M/m.sub.Gy is the number of gyradors in the medium which is the same as the number of DCS (N.sub.Cell); 13)

[00056]$e_{\mathrm{Gy}}=m_{\mathrm{Gy}}{v}_{\mathrm{Gy}}^2/2$

is the kinetic-energy of the gyrador where v.sub.Gy=(GM/r).sup.1/2 is its constant orbiting-speed; 14) e.sub.Th=N.sub.DoFk.sub.BT/2 is the thermal-energy of the thermote; 15) m.sub.Gy=N.sub.DoFk.sub.BTr/GM is an equation for finding the mass of each gyrador, this result surfaces when the gyrador's kinetic-energy

[00057]$\left(e_{\mathrm{Gy}}=m_{\mathrm{Gy}}{v}_{\mathrm{Gy}}^2/2\right),$

with v.sub.Gy=(GM/r).sup.1/2 replacing v.sub.Gy, is set equal to the thermote's thermal-energy (e.sub.Th=N.sub.DoFk.sub.BT/2); 16) M is the medium's antigravitational decompressing mass composed of N.sub.Cell dissimilar cells whose average mass is given by the gyrador mass's m.sub.Gy with its value depending on the numbers {n.sub.1, . . . , n.sub.L} of the L different dissimilar cells of the medium; and 17) N.sub.GyN.sub.Gy=N.sub.CellN.sub.Cell is the number of dissimilar cells that do not exert an antigravitation effect on the medium since the centripetal motion of these cells, whose motions contribute to the medium decompression, is impaired by the medium's viscosity which is a function of the medium's temperature.

[0092] Equations 56 and 57 would then give rise to the i.sup.th iteration viscosity change (.sub.i) given by:

[00058]$\begin{array}{cc}{}_i=P_{\mathrm{Gy}}{}_i=\left(1-N_{\mathrm{Gy}}^{}/N_{\mathrm{Gy}}\right)\left({\mathrm{GM}}^2/{\mathrm{rVN}}_{\mathrm{DoF}}\right){}_l=\left({\mathrm{GM}}^2/{\mathrm{rVN}}_{\mathrm{DoF}}\right){}_i& \left(60\right)\end{array}$$\begin{array}{cc}{P}_{\mathrm{Gy}}={\mathrm{GM}}^2/{\mathrm{rVN}}_{\mathrm{DoF}}-{G\left(M^{}\right)}^2/{\mathrm{rVN}}_{\mathrm{DoF}}={\mathrm{GM}}^2/{\mathrm{rVN}}_{\mathrm{DoF}}=/& \left(61\right)\end{array}$$\begin{array}{cc}=\left(1-N_{\mathrm{Gy}}^{}/N_{\mathrm{Gy}}\right)=\left(/\right)/\left({\mathrm{GM}}^2/{\mathrm{rVN}}_{\mathrm{DoF}}\right)=\left(\left(\mathrm{mT}+b\right)/\right)/\left(\frac{{\mathrm{GM}}^2}{{\mathrm{rVN}}_{\mathrm{DoF}}}\right)& \left(61a\right)\end{array}$

where: a) .sub.i is the change in lingerature that has occurred by the end of the i.sup.th iteration; b) P.sub.Gy is the constant gyrador pressure that acts on the DCS and whose value is carefully regulated by having the change in reactant mass (M.sub.i) restored back to the chemical reaction soon after it is measured; and c) is a DCS factor whose value is found making use of the initial state of the chemical reaction (equation 61a). For an exothermic chemical reaction, a positive temperature change (T.sub.i) would have occurred at the end of the i.sup.th iteration. However, since for most chemical-reactions the viscosity of the medium decreases as i.sup.th temperature increases, it would then follow from equation 60 that the change in lingerature (.sub.i) is expected to be negative, thus indicating that the lingerature of the exothermic chemical reaction has decreased.

[0093] The stated relationship that exists between viscosity and temperature is illustrated in FIG. 4 for the case of liquid water where its dynamic-viscosity, given in SI[mPa]s units, is displayed versus temperature in SI K units where the dynamic-viscosity is noted to decrease as the temperature increases. A similar experimentally found figure may be found for the actual chemical reaction under consideration. It would then become possible to find a suitable analytical expression that relates viscosity to temperature, which then simplifies the control of the chemical reaction's lingerature. For this solvent type the following analytic expression is derived for the evaluation of the medium's temperature (T) for the range of temperatures between 350 K and 400 K:

[00059]$\begin{array}{cc}T=\frac{135\mathrm{mPa}.Math.s-}{0.0028\mathrm{mPa}.Math.s.Math.K^{-1}}=\frac{1.35\mathrm{mPa}.Math.s-P_{\mathrm{Gr}}}{0.0028\mathrm{mPa}.Math.s.Math.K^{-1}}=\frac{1.35\mathrm{mPa}.Math.s-\left({\mathrm{GM}}^2/\mathrm{rV}{N}_{\mathrm{DoF}}\right)}{0.0028\mathrm{mPa}.Math.s.Math.K^{-1}},\mathrm{liquid}\mathrm{water}& \left(62\right)\end{array}$

Statement of the Exothermic DLCE

[0094] The Statement is:

[00060]$\begin{array}{cc}{}_i=\frac{M^2-{\left(M_i\right)}^2}{M^2}=+{}_i=-\left|{}_i\right|,\mathrm{exothermic}& \left(63\right)\end{array}$$\begin{array}{cc}=\frac{\left(\frac{{}_i}{}\right)}{\frac{M^2-{\left(M_i\right)}^2}{M^2}}=\frac{\left(\frac{-.Math.{}_i.Math.}{}\right)}{\frac{M^2-{\left(M_i\right)}^2}{M^2}}1& \left(64\right)\end{array}$$\begin{array}{cc}{M}_i=M-M_i<M& \left(65\right)\end{array}$$\begin{array}{cc}{}_i=+{}_i=-.Math.{}_i.Math.<& \left(66\right)\end{array}$

where: a) M.sub.i is the change in reactant mass during the i.sup.th iteration; b) M.sub.i is the current system mass remaining at the end of the i.sup.th iteration; c) .sub.i is the change in lingerature by the end of the i.sup.th iteration; d) .sub.i is the dynamic lingerature remaining at the end of the i.sup.th iteration; and e) the value of the parameter is close to one.

Derivation of the Exothermic DLCE

[0095] The derivation is achieved in five steps.

[0096] Step 1. In this first step two fundamental quantums of QG-LTT are noted, one of mass and the other of energy. The quantum of mass is the gyrador with symbol m.sub.Gy, which is the DCS's average-mass and where M/m.sub.Gy gyradors make up the initial system mass (M). The gyrador mass (m.sub.Gy), visualized as a point-mass of zero-volume, would orbit the point-mass of the initial system mass (|M|) at the radial distance (r). This radial distance, in turn, would be linked to the mediums volume (V), say, for instance, that for a cylindrical reaction vessel, which is fitted into a minimum surface-area (A=4r.sup.2) spherical medium which in principle provides a minimum energy loss to its surroundings. The orbiting speed of the gyrador (v.sub.Gy=(GM/r).sup.1/2) would be a constant value which surfaces when the kinetic-energy of the medium's M/m.sub.Gy gyradors, given by

[00061]$\left(M/m_{\mathrm{Gy}}\right)m_{\mathrm{Gy}}{v}_{\mathrm{Gy}}^2/2,$

is first set equal to the medium's gravitational potential energy (GM.sup.2/2r), and then this equality equation is solved for v.sub.Gy. Moreover, with this result implying that the gyradors that orbit the point-mass of the initial system mass (|M|) at the radial distance (r) would extract from the surface-area (A=4r.sup.2) of the medium's sphere the thermal-energy required to fuel the kinetic-energy needed to overcome the medium's dynamic viscosity (). This thermal-energy would be provided to the gyradors through quantums of thermal-energy (e.sub.Th), named thermotes, where each gyrador would have its kinetic-energy

[00062]$\left(e_{\mathrm{Gy}}=m_{\mathrm{Gy}}{v}_{\mathrm{Gy}}^2/2\right)$

fueled by a single thermote of energy given by (e.sub.Th=N.sub.DoFk.sub.BT/2) with N.sub.DoF denoting the degrees of freedom of the medium, k.sub.B is the Boltzmann constant, and T is the temperature driving the gyradors motion. Like the gyradors have a constant gyrador orbiting-speed (v.sub.Gy=(GM/r).sup.1/2) that provides a future-certainty motion measure for their motions, the thermotes have a constant thermote surface-pace (.sub.Th=/A) that provides a past-uncertainty retention measure for their retentions. The in .sub.Th=/A is thus the amount of time (or lingerature) that a thermote in A would be available in the future to fuel the kinetic-energy of a gyrador. These .sub.Th-retention and v.sub.Gy-motion measures for thermotes and gyradors, respectively, convey a POP measure duality of QG-LTT.

[0097] Step 2. In this second step the thermotes surface-pace (.sub.Th) is expressed as follows:

[00063]$\begin{array}{cc}{.Math.}_{\mathrm{th}}=\frac{}{A}=\frac{}{4r^2}=\frac{}{4\left(\mathrm{GM}/v_{\mathrm{Gy}}^2\right)}\mathrm{GM}/v_{\mathrm{Gy}}^2=\left(v_{\mathrm{Gy}}^4/4G^2\right)/M^2=\left(e_{\mathrm{Th}}^2/G^2{m}_{\mathrm{Gy}}^2\right)/M^2& \left(67\right)\end{array}$$\begin{array}{cc}{v}_{\mathrm{Gy}}={\left(\mathrm{GM}/r\right)}^{1/2}& \left(68\right)\end{array}$$\begin{array}{cc}{e}_{\mathrm{Gy}}=m_{\mathrm{Gy}}{v}_{\mathrm{Gy}}^2/2=e_{\mathrm{Th}}& \left(69\right)\end{array}$$\begin{array}{cc}{e}_{\mathrm{Th}}=N_{\mathrm{DoF}}{k}_{B}T/2& \left(70\right)\end{array}$$\begin{array}{cc}{m}_{\mathrm{Gy}}=N_{\mathrm{DoF}}{k}_B\mathrm{Tr}/\mathrm{GM}& \left(71\right)\end{array}$$\begin{array}{cc}{e}_{G,M}={\mathrm{GM}}^2/2r& \left(72\right)\end{array}$$\begin{array}{cc}{N}_{\mathrm{Gy}}{e}_{\mathrm{Gy}}=M/m_{\mathrm{Gy}}{m}_{\mathrm{Gy}}{v}_{\mathrm{Gy}}^2/2=e_{G,M}={\mathrm{GM}}^2/2r& \left(73\right)\end{array}$

where:

[0098] .sub.Th=/A is the thermotes surface-pace of the medium given by the ratio of the desired lingerature () over the minimum surface-area (A) of the medium where gyradors are fueled by thermotes and their synergistic behavior averts the medium's gravitational collapse;

[0099] .sub.Th=/4r.sup.2 is the thermotes surface-pace expressed in term of the initial lingerature () and the radial distance (r), derived making use of the surface-area expression A=4r.sup.2;

[00064]${.Math.}_{\mathrm{Th}}=\left(v_{\mathrm{Gy}}^4/4G^2\right)/M^2$

is the thermotes surface-pace expressed in terms of the medium's initial system mass (M) and the gyradors speed (v.sub.Gy), derived making use of the radial expression

[00065]$r=\mathrm{GM}/v_{\mathrm{Gy}}^2;$

[00066]${.Math.}_{\mathrm{Th}}=\left(e_{\mathrm{Th}}^2/G^2{m}_{\mathrm{Gy}}^2\right)/M^2$

is the thermotes surface-pace expressed in terms of the medium's initial system mass (M), the thermal-energy of the thermote (e.sub.Th), and the gyradors mass (m.sub.Gy), which was derived making use of the gyrador's kinetic-energy

[00067]$e_{\mathrm{Gy}}=m_{\mathrm{Gy}}{v}_{\mathrm{Gy}}^2/2=e_{\mathrm{Th}},$

which is also set equal to the thermotes thermal-energy (e.sub.Th) to express v.sub.Gy as a function of e.sub.th and m.sub.Gy;

[0100] e.sub.Th=N.sub.DoFk.sub.BT/2 is the thermote energy equation (TEE) that was first offered in 2014 for use in the finding of the entropy of a flexible-phase medium with applications in lifespan investigations. Two years later in 2016 it was also shown that known entropies for black-holes that produce the longest possible retention of matter, and photon-gases that facilitate the fastest possible motion of matter, could also be expressed in terms of thermotes. This result significantly simplified the derivation of their entropy. Using this enabling theoretical as well as practical discovery, it was further revealed that the thermote values found for both black-hole and photon-gas mediums at the cosmic microwave background (CMB) temperature of 2.725 K had 235.14 and 352.71 eV masses, respectively, in the 50 to 1,500 eV-mass range of the axion. The axion is a theoretical particle, first surfacing in the formulation of the Standard Model of Particle Physics, that is a top dark matter candidate where a promising gravitation-based windchimes method for its detection has been offered;

[0101] m.sub.Gy=N.sub.DoFk.sub.BTr/GM is the gyrador mass equation (GME), discussed earlier when equation 59 was explained, allowing the straightforward finding of the gyrador mass (m.sub.Gy) from knowledge of the medium's desired temperature (T), the initial system mass (M), radius (r), and number of degrees of freedom (N.sub.DoF);

[0102] v.sub.Gy=(GM/r).sup.1/2 is the orbiting-speed of the gyrador that surfaces when the kinetic-energy of the medium's M/m.sub.Gy gyradors given by

[00068]$\left(M/m_{\mathrm{Gy}}\right)m_{\mathrm{Gy}}{v}_{\mathrm{Gy}}^2/2$

is first set equal to the medium's gravitational potential energy (e.sub.G,M=GM.sup.2/2r) to prevent the gravitational collapse of the medium, and then the resultant equation solved for v.sub.Gy.

[0103] Thermote quantums and gyrador quantums form together a POP quantum duality of QG-LTT that conveys a past-uncertainty thermal-energy quantum, the thermote, and a future-certainty linger-viscosity quantum, the gyrador.

[0104] The thermote's surface-pace measure and the gyrador's orbiting-speed measure form together a POP measure duality of QG-LTT that conveys a past-uncertainty thermote measure, the thermotes surface-pace, and a future-certainty gyrador measure, the gyradors orbiting-speed.

[0105] Step 3. In this third step the i.sub.th iteration thermotes surface-pace (.sub.Th,i) of the chemical reaction is noted through equation 67 to be given by the following expressions:

[00069]$\begin{array}{cc}{.Math.}_{\mathrm{Th},i}={}_i/A_i={}_i/4\left(r+r_i\right)\left(r-r_i\right)=/4\left(G\left(M+M_i\right)/v_{\mathrm{Gy},i}^2\right)\left(G\left(M-M_i\right)/v_{\mathrm{Gy},i}^2\right)=\left(v_{\mathrm{Gy},i}^4/4G^2\right){}_i/\left(M^2-M_i^2\right)& \left(74\right)\end{array}$$\begin{array}{cc}{v}_{\mathrm{Gy},i}={\left({\mathrm{GM}}_i/r_i\right)}^{1/2}{v}_{\mathrm{Gy}}={\left(\mathrm{GM}/r\right)}^{1/2}& \left(75\right)\end{array}$$\begin{array}{cc}{M}_i/r_{i}M/r& \left(76\right)\end{array}$

where the medium's i.sup.th iteration current system mass (M.sub.i), radius (r.sub.i), and surface-area (A.sub.i) values are smaller than their initial ones. Moreover, with the further assumption that the approximate mass-radius ratio equation 76 will generally apply to exothermic chemical reactions.

[0106] Step 4. In this fourth step the ratio of the i.sup.th iteration thermotes surface-pace (.sub.Th,i) of equation 74 over the initial thermotes surface-pace .sub.Th of equation 67 is taken to arrive at the following relationship:

[00070]$\begin{array}{cc}{}_i/\left(M^2-M_i^2\right)=\frac{v_{\mathrm{Gy}}^4}{v_{\mathrm{Gy},i}^4}/M^2={\left(\frac{M/r}{M_i/r_i}\right)}^2/M^2=/M^2& \left(77\right)\end{array}$$\begin{array}{cc}={\left(\frac{M/r}{M_i/r_i}\right)}^2={\left(\frac{V/r}{{}_i{V}_i/r_i}\right)}^2={\left(\frac{A}{{}_i{A}_i}\right)}^2& \left(78\right)\end{array}$

where the parameter of equation 77 is close to one in value as is seen when use is made of equation 76 in equation 78. In equation 78 =M/V is the initial mass density of the medium and .sub.i=M.sub.i/V.sub.i is the i.sup.th iteration mass density, these values, which may be assumed to be equal in value, along with the surface area values for A and A.sub.i can be used to experimentally find a value for .

[0107] Step 5. In this last step equation 77 is solved for the i.sup.th iteration lingerature (.sub.i) to arrive at the sought-after equation 63.

The Exothermic Linger-Viscosity Based DLCE and its Derivation

[0108] Introductory words: The exothermic linger-viscosity based DLCE is an i.sup.th iteration equation that states the amount of surface-area (|A.sub.i|) that must be added to the medium to compensate for the medium's surface-area decrease due to the increase of the mechanical work-energy (W.sub.Mec,i)'s pressure on the thermotes of an exothermic reaction. In turn, the addition of surface-area by |A.sub.i| would lead to an increase of the medium's lingerature by |.sub.i|, an increase of the medium's dynamic viscosity by |.sub.i|, and a decrease of the medium's temperature by |T.sub.i|. Thus, restoring the values of the surface-area, lingerature, viscosity, and temperature of the medium.

Statement of the Exothermic Linger-Viscosity Based DLCE

[0109] The Statement is:

[00071]$\begin{array}{cc}{A}_i=\left(\frac{M^2-{\left(M_i\right)}^2}{M^2}-1\right)A,\mathrm{exothermic}& \left(79\right)\end{array}$

where is the DLCE parameter of equation 64 whose value is approximately one and M.sub.i is the i.sup.th iteration current system mass that is smaller than the medium's initial system mass (M) as per equation 65.

Derivation of the Exothermic Linger-Viscosity Based DLCE

[0110] The proof is in three steps:

[0111] Step 1. In this first step the initial thermotes surface-pace (.sub.Th) given by equation 67and the i.sup.th iteration thermotes surface-pace (.sub.Th,i) given by equation 74 are noted to yield the same values under the assumption that the initial to i.sup.th iteration mass over radius approximation of equation 76 holds. When equations 67 and 74 are equated under condition 76 the following ratio of lingerature to surface-area equation will result:

[00072]$\begin{array}{cc}/A={}_i/A_i=\left(+{}_i\right)/\left(A+A_i\right)& \left(80\right)\end{array}$

where both .sub.i and A.sub.i are negative quantities that give rise to lower i.sup.th iteration lingerature and i.sup.th iteration surface-area values according to .sub.i=+.sub.i and A.sub.i=A+A.sub.i, respectively.

[0112] Step 2. Solving equation 80 for A.sub.i one then arrives at the following expression:

[00073]$\begin{array}{cc}{A}_i=\left({}_i/\right)A=-\left(.Math.{}_i.Math./\right)A& \left(81\right)\end{array}$

[0113] Step 3. In this final step, equation 63 is first solved for |.sub.i| and then used in equation 81 to yield the exothermic linger-viscosity based DLCE of equation 79.

The Exothermic Thermal-Energy Based DLCE and its Derivation

Introductory Words

[0114] The exothermic thermal-energy based DLCE is an i.sup.th iteration equation that states the amount of heat that must be removed from the medium (|Q.sub.i|), to compensate for the medium's heat increase due to the increase of the mechanical work-energy (W.sub.Mec,i)'s pressure on the thermotes of an exothermic reaction. In turn, the removal of heat by |Q.sub.i| would lead to an increase of the medium's lingerature by |.sub.i|, an increase of the medium's dynamic viscosity by |.sub.i|, and a decrease of the medium's temperature by |T.sub.i|. Thus, bringing the heat, lingerature, viscosity, and temperature of the medium back to their original values.

Statement of the Exothermic Thermal-Energy Based DLCE

[0115] The statement is:

[00074]$\begin{array}{cc}{Q}_i={\mathrm{SHC}}_{i}M\left(1-\frac{\left(1+{}_i/\right)}{\left(1+M_i/M\right)}\right)T,\mathrm{exothermic}& \left(82\right)\end{array}$$\begin{array}{cc}{T}_i=T+T_i=T+.Math.T_i.Math.=2T-\frac{M_i}{M}T,\mathrm{exothermic}& \left(83\right)\end{array}$$\begin{array}{cc}{T}_{i,\mathrm{endo}}=T-.Math.T_i.Math.=\frac{M_i}{M}T,\mathrm{endothermic}& \left(84\right)\end{array}$$\begin{array}{cc}=\frac{\left(\frac{T_{i,\mathrm{endo}}}{T}\right)}{\left(\frac{M_i}{M}\right)}=\frac{\left(\frac{T-.Math.T_i.Math.}{T}\right)}{\left(\frac{M-M_i}{M}\right)}1,\mathrm{endothermic}& \left(85\right)\end{array}$

where:

[0116] SHC.sub.i is the medium's i.sup.th iteration specific heat capacity after the change in reactant mass (M.sub.i).

[0117] M is the medium's initial system mass.

[0118] Tis the medium's desired temperature.

[0119] T.sub.i is the exothermic medium's i.sup.th iteration temperature which is greater than T and results in the temperature increase T.sub.iT=T.sub.i.

[0120] T.sub.i,endo is the endothermic medium's i.sup.th iteration temperature which is smaller than T and results in the temperature decrease

[00075]$T_{i,\mathrm{endo}}-T=\frac{M_i}{M}T-T=-.Math.T_i.Math..$

is the endothermic parameter given by equation 85 whose value is close to one.

[0121] is the exothermic parameter given by equation 64 whose value is close to one.

[0122] .sub.i is the negative lingerature change of the exothermic chemical reaction that is caused by the consumed reactant-mass (M.sub.i).

Derivation of the Exothermic Thermal-Energy Based DLCE

[0123] The derivation is in four steps:

[0124] Step 1. Firstly, it is noted that equations 83, 84 and 85 denote an exothermic i.sup.th iteration dynamic temperature control equation (DTCE).

[0125] Step 2. The exothermic medium is noted to exhibit an increase in its heat (Q.sub.i) that is due to the mechanical work-energy (W.sub.Mec,i)'s pressure on the thermotes of an exothermic reaction:

[00076]$\begin{array}{cc}{Q}_i={\mathrm{SHC}}_{i}M{T}_i& \left(86\right)\end{array}$

[0126] Step 3. Equation 83 is solved for T.sub.i and its result substituted in equation 86 to yield:

[00077]$\begin{array}{cc}{Q}_i={\mathrm{SHC}}_{i}M\left(1-\frac{M_i}{M}\right)T& \left(87\right)\end{array}$

[0127] Step 4. Finally solving equation 63 for M.sub.i/M and substituting the result in equation 87 one arrives at the exothermic thermal-energy based DLCE of equation 82.

The Endothermic Lingerature Control Methods

The Endothermic Dynamic Lingerature Control Equation (DLCE) and its Derivation

Introductory Words

[0128] The endothermic DLCE is a dynamic i.sup.th iteration equation that allows the calculation of the amount of lingerature (.sub.i) that an endothermic chemical reaction has increased by the end of the i.sup.th iteration of processing, with this .sub.i knowledge then used in its correction. In an endothermic chemical reaction, the medium's temperature decreases, which then results in the increase of both the mediums viscosity and lingerature.

Statement of the Endothermic DLCE

[0129] The statement is:

[00078]$\begin{array}{cc}\frac{M^2-{\left(M_i\right)}^2}{M^2}=+{}_i=+.Math.{}_i.Math.,\mathrm{endothermic}& \left(88\right)\end{array}$$\begin{array}{cc}{M}_i=M-M_i<M& \left(89\right)\end{array}$$\begin{array}{cc}{}_i=+{}_i=+.Math.{}_i.Math.>,\mathrm{endothermic}& \left(90\right)\end{array}$

where: a) M.sub.i is the change in reactant mass during the i.sup.th iteration; b) M.sub.i is the mediums current system mass at the end of the i.sup.th iteration which is less than the initial system mass (M); c) .sub.i is the change in lingerature at the end of the i.sup.th iteration; d) .sub.i is the dynamic lingerature of the medium at the end of the i.sup.th iteration; and e) the parameter's value is close to one.

Derivation of the Endothermic DLCE

[0130] The derivation is in three steps:

[0131] Step 1. The endothermic DLCE of equation 88 is derived under the assumption that the change in reactant mass (M.sub.i) would yield similar amounts of change in lingerature |.sub.i| for both exothermic and endothermic chemical reactions.

[0132] Step 2. Making use of equation 63 the lingerature change |.sub.i| for the exothermic chemical reaction is noted to be given by:

[00079]$\begin{array}{cc}.Math.{}_i.Math.=.Math.\frac{M^2-{\left(M_i\right)}^2}{M^2}-.Math.=-\frac{M^2-{\left(M_i\right)}^2}{M^2}& \left(91\right)\end{array}$

[0133] Step 3. Finally, the lingerature change given by equation 91 is added to the initial lingerature () of the endothermic chemical reaction to yield the endothermic DLCE of equation 88.

The Endothermic Linger-Viscosity Based DLCE and its Derivation

Introductory Words

[0134] The endothermic linger-viscosity based DLCE is an i.sup.th iteration equation that states the amount of surface-area that must be removed from the medium (|A.sub.i|), to compensate for the medium's surface-area increase due to the decrease of the mechanical work-energy (W.sub.Mec,i)'s pressure on the thermotes of an endothermic reaction. In turn, the removal of surface-area by |A.sub.i| would lead to a decrease of the medium's change in lingerature by |.sub.i|, a decrease of the medium's dynamic viscosity by |.sub.i|, and an increase of the medium's temperature by |T.sub.i|. Thus, restoring the values of the surface-area, lingerature, viscosity, and temperature of the medium.

Statement of the Endothermic Linger-Viscosity Based DLCE

[0135] The statement is:

[00080]$\begin{array}{cc}{A}_i=\left(1-\frac{M^2-{\left(M_i\right)}^2}{M^2}\right)A,\mathrm{endothermic}& \left(92\right)\end{array}$