# Transport phenomena in combustion

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Combustion is an exothermic chemical reaction process between fuel and oxidant. If combustion involves a liquid fuel, the liquid fuel does not actually burn as a liquid; it is vaporized first and diffuses away from the liquid-vapor surface. Meanwhile, the gaseous oxidant diffuses toward the liquid-vapor interface. Under the right conditions, the mass fluxes of vapor fuel and gaseous oxidant meet and the chemical reaction occurs at a certain location known as the flame (Lock, 1994; Avedisian, 1997, 2000). The flame is usually a very thin region with a color dictated by the temperature of the combustion.

Figure 1: Combustion near a planar surface.

The temperature and mass concentration distributions during a combustion process can be represented as shown in Fig. 1. The initial temperature of a liquid fuel is T0, and the temperature of the liquid–vapor interface, TI, is at the dew point temperature of the fuel. The temperature reaches a maximum at the location of the flame, and decreases with increasing x until it reaches T. The mass fraction of fuel, ωf, is maximal at the liquid-vapor interface and decreases as location of the flame is approached. The mass fraction of the oxidant, ωo, on the other hand, is maximal at infinity and decreases as location of the flame is approached. The mass fractions of both vapor fuel and oxidant reach their minimum values at the location of the flame. The mass fractions of the products of combustion, ωp, are at their maximum at the location of the flame and decrease as x either increases or decreases.

The transport phenomena involved in combustion include heat and mass transfer in both the liquid fuel and the gaseous mixture. The governing equations for the gaseous phase will be discussed below. Firstly, the gaseous mixture must satisfy the continuity equation, i.e.,

$\frac{{D\rho }}{{Dt}} + \rho \nabla \cdot {\mathbf{V}} = 0 \qquad \qquad(1)$

and the gas flow is governed by the momentum equation,

$\rho \frac{{D{\mathbf{V}}}}{{Dt}} = \nabla \cdot {\mathbf{\tau }} + \rho {\mathbf{g}} \qquad \qquad(2)$

where the shear stress can be obtained from eq. ${{\mathbf{\tau }}_{rel}} = 2\mu {{\mathbf{D}}_{rel}} - \frac{2}{3}\mu (\nabla \cdot {{\mathbf{V}}_{rel}}){\mathbf{I}}$ from Momentum equation and gravity is the only body force considered.

The energy equation for combustion is

$\rho \frac{{Dh}}{{Dt}} = \nabla \cdot k\nabla T \qquad \qquad(3)$

which is a simplified version of eq. (20), with internal heat generation and viscous dissipation neglected. The effect of the substantial derivative of pressure on the energy is also neglected. The specific enthalpy for the mixture is related to the specific enthalpy of each component in the mixture by

$h = \sum\limits_{i = 1}^N {{\omega _i}} {h_i} \qquad \qquad(4)$

It is a common practice in thermodynamic and heat transfer analyses to consider the change in enthalpy during a chemical reaction process rather than the absolute values of enthalpy. For a process that does not involve chemical reaction, we can choose any reference state for an individual substance and define the enthalpy at that reference state as zero. For example, the reference state for water is often chosen at the triple point, while the reference state for an ideal gas is often chosen as zero K. However, when chemical reaction is involved in a process, as is the case with combustion, the composition of the system changes during the chemical reaction; therefore, the reference states for all reactants and products must be the same.

One convenient option in such a situation is to segregate the enthalpy of any component into two parts: (1) the enthalpy due to its chemical composition at the standard reference state (at 25 ˚C and 1 atm), and (2) the sensible enthalpy due to any temperature deviation from the standard reference state. Therefore, the enthalpy for the ith component in the mixture can be expressed as

${h_i} = h_i^ \circ + h_i^T \qquad \qquad(5)$

where $h_i^ \circ$ is the enthalpy of formation for the ith component, i.e., the enthalpy due to its chemical composition at the standard reference state. The enthalpy of formation for selected substances is shown in Table 1.

The sensible enthalpy, $h_i^T$, is related to temperature by

$h_i^T = \int_{{T^ \circ }}^T {{c_{pi}}dT} \qquad \qquad(6)$

Substituting eqs. (5) and (6) into eq. (4), the enthalpy of the mixture becomes

$h = \sum\limits_{i = 1}^N {{\omega _i}} h_i^ \circ + \int_{{T^ \circ }}^T {{c_p}} dT \qquad \qquad(7)$

where cp is the average specific heat of the mixture, defined as

${c_p} = \sum\limits_{i = 1}^N {{\omega _i}} {c_{pi}} \qquad \qquad(8)$

Substituting eq. (7) into eq. (3), the energy equation for combustion becomes

$\rho \frac{{D({c_p}T)}}{{Dt}} = \nabla \cdot k\nabla T - \rho \sum\limits_{i = 1}^N {h_i^ \circ \frac{{D{\omega _i}}}{{Dt}}} \qquad \qquad(9)$

Table 1 Enthalpy of formation ho for selected substances at 25 ˚C and 1 atm

 Substance Formula (phase) Enthalpy of formation (kJ/kmol) Acetylene C2H2 (g) 226,730 Benzene C6H6 (g) 82,930 Carbon C (s) 0 Carbon monoxide CO (g) -110,530 Carbon dioxide CO2 (g) -393,520 Ethyl alcohol C2H5OH (g) -235,310 Ethyl alcohol C2H5OH ($\ell$) -277,690 Ethylene C2H4 (g) 52,280 Ethane C2H6 (g) -84,680 Hydrogen H2 (g) 0 Methane CH4 (g) -74,850 Nitrogen N2 (g) 0 Nitrogen N (g) 472,650 n-Dodecane C12H26 ($\ell$) -291,010 n-Octane C8H18 (g) -208,450 n-Octane C8H18 ($\ell$) -249,950 Oxygen O2 (g) 0 Propane C3H8 (g) -103,850 Water H2O () -241,820 Water vapor H2O (g) -285,830

If the fuel is consumed at a rate of ${\dot m'''_f}$ per unit volume, hc – the heat of combustion – is defined by

${h_c} = - \frac{\rho }{{{{\dot m'''}_f}}}\sum\limits_{i = 1}^N {{h_i}\frac{{D{\omega _i}}}{{Dt}}} \qquad \qquad(10)$

Substituting eq. (5) into eq. (10) gives

${h_c} = - \frac{\rho }{{{{\dot m'''}_f}}}\sum\limits_{i = 1}^N {h_i^ \circ \frac{{D{\omega _i}}}{{Dt}}} - \frac{\rho }{{{{\dot m'''}_f}}}\sum\limits_{i = 1}^N {h_i^T\frac{{D{\omega _i}}}{{Dt}}} \qquad \qquad(11)$

The contribution of the second term on the right-hand side of eq. (11) is negligible, since ${h_c} \gg h_i^T$. Dropping the second term from the right-hand side of eq. (11) and substituting the result into eq. (4), the energy equation becomes

$\rho {c_p}\frac{{DT}}{{Dt}} = \nabla \cdot k\nabla T + {\dot m'''_f}{h_c} \qquad \qquad(12)$

The mass fraction of each component (fuel, oxidant, and product) is dominated by

$\rho \frac{{D{\omega _i}}}{{Dt}} = - \nabla \cdot {{\mathbf{J}}_i} + {\dot m'''_i} \qquad \qquad(13)$

where the subscript i can be f (fuel), o (oxidant), or p (product). The ratio of the rates of oxygen and fuel consumption is defined as the oxygen/fuel ratio:

$\gamma = \frac{{{{\dot m'''}_o}}}{{{{\dot m'''}_f}}} \qquad \qquad(14)$
Figure 2: Combustion near a liquid fuel droplet.

The above analysis applies to combustion occurring on a planar surface. For many applications, combustion of the liquid fuel is usually preceded by breaking up a fuel jet into liquid droplets so that combustion occurs around a spherical liquid droplet. Combustion of a falling liquid droplet, as shown in Fig. 2.14, is analyzed here. The liquid fuel droplet vaporizes at the dew point, Td, which is the saturation temperature corresponding to the partial pressure of the fuel vapor in the mixture near the liquid-vapor interface. To simplify the analysis, it is assumed that the temperature in the liquid fuel droplet is uniformly equal to the saturation temperature of the fuel at the total system pressure, i.e., the mass fraction of the fuel at the liquid-vapor interface equals one. It is further assumed that the shapes of both the liquid fuel droplet and the flame are spherical, which allows for application of a one-dimensional symmetric model (Lock, 1994), which is presented here.

If the combustion process is assumed to be in a quasisteady state (neglecting the transient term in the governing equation), the energy equation for combustion becomes

$\frac{1}{{{r^2}}}\frac{d}{{dr}}({r^2}\rho {c_p}uT) = \frac{1}{{{r^2}}}\frac{d}{{dr}}\left( {{r^2}k\frac{{dT}}{{dr}}} \right) + {\dot m'''_f}{h_c} \qquad \qquad(15)$

where u is the velocity of the gaseous mixture in the radial direction.

The mass fraction of the oxidant can be obtained by simplifying eq. (13), i.e.,

$\frac{1}{{{r^2}}}\frac{d}{{dr}}({r^2}\rho u{\omega _o}) = \frac{1}{{{r^2}}}\frac{d}{{dr}}\left[ {{r^2}\rho {D_o}\frac{{d{\omega _o}}}{{dr}}} \right] - {\dot m'''_o} \qquad \qquad(16)$

where Do is the mass diffusivity of the oxidant in the gaseous mixture. Multiplying eq. (16) by hc / γ and combining the resulting equation with eq. (15) yields

$\frac{1}{{{r^2}}}\frac{d}{{dr}}\left[ {{r^2}\rho {c_p}u\left( {T + {\omega _o}\frac{{{h_c}}}{{\gamma {c_p}}}} \right)} \right] = \frac{1}{{{r^2}}}\frac{d}{{dr}}\left[ {{r^2}k\frac{{dT}}{{dr}} + {r^2}\rho {D_o}\frac{d}{{dr}}\left( {{\omega _o}\frac{{{h_c}}}{\gamma }} \right)} \right] \qquad \qquad(17)$

where eq. (14) was used to eliminate the rates of fuel and oxygen consumption.

Assuming that the Lewis number [Le = k / (ρcpDo)] equals one and the specific heat is constant, eq. (17) can be simplified as follows:

$\frac{1}{{{r^2}}}\frac{d}{{dr}}\left( {{r^2}\rho {c_p}u{T^*}} \right) = \frac{1}{{{r^2}}}\frac{d}{{dr}}\left( {{r^2}k\frac{{d{T^*}}}{{dr}}} \right) \qquad \qquad(18)$

where

${T^*} = T + {\omega _o}\frac{{{h_c}}}{{\gamma {c_p}}} \qquad \qquad(19)$

is a modified temperature in the combustion process.

Equation (18) can be rearranged to the following form:

$\frac{d}{{dr}}\left[ {{r^2}\left( {\rho {c_p}u{T^*} - k\frac{{d{T^*}}}{{dr}}} \right)} \right] = 0 \qquad \qquad(20)$

The continuity equation requires that

$4\pi {r^2}\rho u = 4\pi r_I^2{(\rho u)_I} = \dot m \qquad \qquad(21)$

Since it has been assumed that the mass fraction of the fuel in the mixture at the liquid-vapor interface equals one, the mass flow rate $\dot m$ reflects the mass flow rate of the fuel. The mass flux is often used in combustion analysis, and it is defined as

$\dot m'' = \frac{{\dot m}}{{4\pi {r^2}}} \qquad \qquad(22)$

Equations (20) and (21) can be rewritten in terms of mass flux, i.e.,

$\frac{d}{{dr}}\left[ {{r^2}\left( {\dot m''{c_p}{T^*} - k\frac{{d{T^*}}}{{dr}}} \right)} \right] = 0 \qquad \qquad(23)$
$4\pi {r^2}\dot m'' = 4\pi r_I^2{\dot m''_I} = \dot m \qquad \qquad(24)$

Integrating eq. (23) from the liquid fuel droplet surface (r = rI) to an arbitrary radius (r > rI) yields

${r^2}\left( {\dot m''{c_p}{T^*} - k\frac{{d{T^*}}}{{dr}}} \right) = r_I^2{\left( {\dot m''{c_p}{T^*} - k\frac{{d{T^*}}}{{dr}}} \right)_I} \qquad \qquad(25)$

Substituting eq. (24) into eq. (25), one obtains

${r^2}k\frac{{d{T^*}}}{{dr}} = r_I^2{\dot m''_I}{c_p}({T^*} - T_I^*) + r_I^2{\left( {k\frac{{d{T^*}}}{{dr}}} \right)_{r = {r_I}}} \qquad \qquad(26)$

The mass flow rate at the surface of the liquid fuel droplet, ${\dot m''_I}$, is the same as the fuel burning rate, ${\dot m''_f}$, because the mass fraction at the surface of the droplet is one. Introducing the excess modified temperature,

$\theta = {T^*} - T_I^* \qquad \qquad(27)$

eq. (26) becomes

${r^2}k\frac{{d\theta }}{{dr}} = r_I^2{\dot m''_f}{c_p}\left[ {\theta + \frac{1}{{{{\dot m''}_f}{c_p}}}{{\left( {k\frac{{d\theta }}{{dr}}} \right)}_{r = {r_I}}}} \right] \qquad \qquad(28)$

Introducing a new dependent variable,

$\varphi = \theta + \frac{1}{{{{\dot m''}_f}{c_p}}}{\left( {k\frac{{d\theta }}{{dr}}} \right)_{r = {r_I}}} \qquad \qquad(29)$

eq. (28) becomes

$\frac{1}{\varphi }d\varphi = \frac{{r_I^2{{\dot m''}_f}{c_p}}}{k}\frac{1}{{{r^2}}}dr \qquad \qquad(30)$

Integrating eq. (30) over the interval of $(r,\infty )$, one obtains

$\ln \frac{{{\varphi _\infty }}}{\varphi } = \frac{{r_I^2{{\dot m''}_f}{c_p}}}{k}\frac{1}{r} \qquad \qquad(31)$

The fuel burning rate, Gf, in eq. (31) is related to the heat transfer at the liquid droplet surface as follows

${\dot m''_f}{h_{\ell v}} = {q_I}^{\prime \prime } = k{\left( {\frac{{d\theta }}{{dr}}} \right)_{r = {r_I}}} \qquad \qquad(32)$

Substituting eq. (32) into eq. (29), the new dependent variable $\varphi$ becomes

$\varphi = \theta + \frac{{{h_{\ell v}}}}{{{c_p}}} \qquad \qquad(33)$

Substituting eq. (33) into eq. (31) and letting r = rI, an equation for the fuel burning rate is obtained:

${\dot m''_f} = \frac{k}{{{r_I}{c_p}}}\ln \left( {1 + \frac{{{c_p}{\theta _\infty }}}{{{h_{\ell v}}}}} \right) \qquad \qquad(34)$

Equation (34) can also be used to determine the transient liquid fuel droplet size, because the fuel burning rate is related to the size of the droplet by

${\dot m''_f} = - \frac{{d({\rho _f}4\pi r_I^3/3)/dt}}{{4\pi r_I^2}} = - {\rho _f}\frac{{d{r_I}}}{{dt}} \qquad \qquad(35)$

where ρf is the density of the liquid fuel.

Substituting eq. (35) into eq. (34), one obtains

${r_I}\frac{{d{r_I}}}{{dt}} = - \frac{k}{{{\rho _f}{c_p}}}\ln \left( {1 + \frac{{{c_p}{\theta _\infty }}}{{{h_{\ell v}}}}} \right) \qquad \qquad(36)$

Integrating eq. (35) and considering the initial condition rI = ri at t = 0, the liquid droplet radius becomes

$r_I^2 = r_i^2 - \frac{{2kt}}{{{\rho _f}{c_p}}}\ln \left( {1 + \frac{{{c_p}{\theta _\infty }}}{{{h_{\ell v}}}}} \right) \qquad \qquad(37)$

Equation (37) can be used to estimate the time needed to completely burn the liquid fuel droplet:

${t_f} = \frac{{{\rho _f}{c_p}r_i^2}}{{2k\ln \left( {1 + \frac{{{c_p}{\theta _\infty }}}{{{h_{\ell v}}}}} \right)}} \qquad \qquad(38)$

## References

Avedisian, C.T., 1997, “Soot Formation in Spherically Symmetric Droplet Combustion,” Physical and Chemical Aspects of Combustion, edited by Irvin Glassman, I., Dryer, F.L., and Sawyer, R. F., pp. 135-160, Gordon and Breach Publishers.

Avedisian, C.T., 2000, “Recent Advances in Soot Formation from Spherical Droplet Flames at Atmospheric Pressure,” Journal of Propulsation and Power, Vol. 16, pp. 628-656.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Lock, G.S.H., 1994, Latent Heat Transfer, Oxford Science Publications, Oxford University, Oxford, UK.