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In Combustion, G equation is a scalar ${\displaystyle G(\mathbf{x},t)}$ field equation which describes the instantaneous flame position, introduced by Forman A. Williams in 1985^[1][2] in the study of premixed turbulent combustion. The equation is derived based on the Level-set method. The equation was studied by George H. Markstein earlier, in a restrictive form.^[3][4]

Mathematical description^[5][6][edit]

The G equation reads as

${\displaystyle \frac{\mathrm{\partial }G}{\mathrm{\partial }t}+\mathbf{v}\cdot \mathrm{\nabla }G=U_L\mathrm{\nabla }G}$

where

• ${\displaystyle \mathbf{v}}$ is the flow velocity field
• ${\displaystyle U_L}$ is the local burning velocity

The flame location is given by ${\displaystyle G(\mathbf{x},t)=G_o}$ which can be defined arbitrarily such that ${\displaystyle G(\mathbf{x},t)>G_o}$ is the region of burnt gas and is the region of unburnt gas. The normal vector to the flame is ${\displaystyle \mathbf{n}=-\mathrm{\nabla }G/\mathrm{\nabla }G}$.

Local burning velocity[edit]

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The burning velocity of the stretched flame can be derived by subtracting suitable terms from the unstretched flame speed, for small curvature and small strain, as given by

${\displaystyle U_L=S_L-S_L\mathcal{L}\kappa -\mathcal{L}S}$

where

• ${\displaystyle S_L}$ is the burning velocity of unstretched flame
• ${\displaystyle S=-\mathbf{n}\cdot \mathrm{\nabla }\mathbf{v}\cdot \mathbf{n}}$ is the term corresponding to the imposed strain rate on the flame due to the flow field
• ${\displaystyle \mathcal{L}}$ is the Markstein length, proportional to the laminar flame thickness ${\displaystyle {\delta }_L}$, the constant of proportionality is Markstein number${\displaystyle \mathcal{M}}$
• ${\displaystyle \kappa =\mathrm{\nabla }\cdot \mathbf{n}=-\frac{{\mathrm{\nabla }}^{2}G-\mathbf{n}\cdot \mathrm{\nabla }(\mathbf{n}\cdot \mathrm{\nabla }G)}{\mathrm{\nabla }G}}$ is the flame curvature, which is positive if the flame front is convex with respect to the unburnt mixture and vice versa.

A simple example - Slot burner[edit]

The G equation has an exact expression for a simple slot burner. Consider a two-dimensional planar slot burner of slot width ${\displaystyle b}$ with a premixed reactant mixture is fed through the slot with constant velocity ${\displaystyle \mathbf{v}=(0,U)}$, where the coordinate ${\displaystyle (x,y)}$ is chosen such that ${\displaystyle x=0}$ lies at the center of the slot and ${\displaystyle y=0}$ lies at the location of the mouth of the slot. When the mixture is ignited, a flame develops from the mouth of the slot to certain height ${\displaystyle y=L}$ with a planar conical shape with cone angle ${\displaystyle \alpha }$. In the steady case, the G equation reduces to

${\displaystyle U\frac{\mathrm{\partial }G}{\mathrm{\partial }y}=U_L\sqrt{{\left(\frac{\mathrm{\partial }G}{\mathrm{\partial }x}\right)}^2+{\left(\frac{\mathrm{\partial }G}{\mathrm{\partial }y}\right)}^2}}$

If a separation of the form ${\displaystyle G(x,y)=y+f(x)}$ is introduced, the equation becomes

${\displaystyle U=U_L\sqrt{1+{\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }x}\right)}^2},\text{or}\frac{\mathrm{\partial }f}{\mathrm{\partial }x}=\frac{\sqrt{U^2-U_L^2}}{U_L}}$

which upon integration gives

${\displaystyle f(x)=\frac{(U^2-U_L^2{)}^{1/2}}{U_L}x+C,\Rightarrow G(x,y)=\frac{(U^2-U_L^2{)}^{1/2}}{U_L}x+y+C}$

Without loss of generality choose the flame location to be at ${\displaystyle G(x,y)=G_o=0}$. Since the flame is attached to the mouth of the slot ${\displaystyle x=b/2,y=0}$, the boundary condition is ${\displaystyle G(b/2,0)=0}$, which can be used to evaluate the constant ${\displaystyle C}$. Thus the scalar field is

${\displaystyle G(x,y)=\frac{(U^2-U_L^2{)}^{1/2}}{U_L}\left(x-\frac{b}{2}\right)+y}$

At the flame tip, we have ${\displaystyle x=0,y=L,G=0}$, the flame height is easily determined as

${\displaystyle L=\frac{b(U^2-U_L^2{)}^{1/2}}{2U_L}}$

and the flame angle ${\displaystyle \alpha }$ is given by

${\displaystyle \tan \alpha =\frac{b/2}{L}=\frac{U_L}{(U^2-U_L^2{)}^{1/2}}}$

Using the trigonometric identity${\displaystyle {\tan }^2\alpha ={\sin }^2\alpha /(1-{\sin }^2\alpha )}$, we have

${\displaystyle \sin \alpha =\frac{U_L}{U}}$

References[edit]

1. ^Williams, F. A. (1985). Turbulent combustion. In The mathematics of combustion (pp. 97-131). Society for Industrial and Applied Mathematics.
2. ^Kerstein, Alan R., William T. Ashurst, and Forman A. Williams. 'Field equation for interface propagation in an unsteady homogeneous flow field.' Physical Review A 37.7 (1988): 2728.
3. ^GH Markstein. (1951). Interaction of flow pulsations and flame propagation. Journal of the Aeronautical Sciences, 18(6), 428-429.
4. ^Markstein, G. H. (Ed.). (2014). Nonsteady flame propagation: AGARDograph (Vol. 75). Elsevier.
5. ^Peters, Norbert. Turbulent combustion. Cambridge university press, 2000.
6. ^Williams, Forman A. 'Combustion theory.' (1985).

We claim:

1. A method of combustion comprising the steps of:

introducing a fuel into a burner manifold having at least one elongated port at a rate of about 5,000 BTUH to about 30,000 BTUH per square inch of port area, the at least one elongated port having a length to width aspect ratio greater than about 10 to 1;

introducing from about 0 to about 30 percent by volume of combustion air into the burner manifold, producing a fuel/air mixture; and

combusting said fuel/air mixture passing through the at least one elongated port, said fuel/air mixture leaving said at least one elongated port having a Froude Number in the range of about 0.05 to 4.0.

2. A method according to claim 1 wherein the enclosure has a plurality of elongated ports.

3. A method according to claim 2 wherein the plurality of elongated ports form a plurality of rows of elongated ports, and the elongated ports in a respective row are positioned generally end-to-end.

4. A method according to claim 3 wherein the plurality of rows are generally parallel with respect to each other.

5. A method according to claim 4 wherein one of the plurality of rows is positioned at a distance of less than about 1 inch from another of the plurality of rows.

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6. A method according to claim 4 wherein the elongated slots of one of the rows is in a staggered relationship with the elongated slots of an adjacent row.

7. A method according to claim 1 wherein from about 5 percent to about 20 percent by volume of combustion air is introduced into the burner manifold.

8. A method according to claim 1 wherein the at least one elongated port has a width of about 0.005 inches to about 0.15 inches.

9. A gaseous fuel fireplace burner comprising:

a burner manifold having at least one elongated port having a length to width aspect ratio exceeding about 10:1 and being port loadable in a range of about 5,000 BTUH per square inch to 30,000 BTUH per square inch of port area and passing a volume percentage of combustion air in a range of 0 percent to about 30 percent of a stoichiometric requirement for complete combustion of a gaseous fuel, whereby a fuel/air mixture passing through said at least one elongated port has a Froude Number in a range of about 0.05 to 4.0.

10. A gaseous fuel fireplace burner according to claim 9 wherein the burner manifold comprises a plurality of elongated ports.

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11. A gaseous fuel fireplace burner according to claim 10 wherein the plurality of elongated ports form a plurality of rows of elongated ports, and the elongated ports in a respective row are positioned generally end-to-end.

12. A gaseous fuel fireplace burner according to claim 11 wherein one of the plurality of rows is positioned at a distance of less than about 1 inch from another of the plurality of rows.

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13. A gaseous fuel fireplace burner according to claim 11 wherein the elongated ports of one of the rows is in a staggered relationship with the elongated ports of an adjacent row.

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