# Encyclopedia of Wildfires and Wildland-Urban Interface (WUI) Fires

Living Edition
| Editors: Samuel L. Manzello

# Convection

Living reference work entry
DOI: https://doi.org/10.1007/978-3-319-51727-8_1-1

## Definition

Convection is the form of heat transfer that occurs when a fluid (liquid or gas) flows over a surface at a different temperature.

## Basic Concepts

Convection is the form of heat transfer that occurs when a fluid (liquid or gas) flows over a surface at a different temperature. This fluid flow can be forced (e.g., due to wind), natural (due to buoyancy, sometimes referred to as “free convection”), laminar, or turbulent. The amount of heat that is transferred is proportional to that temperature difference:
$${\dot{q}}^{{\prime\prime} }=h\left({T}_s-{T}_{\infty}\right)$$
(1)
where $${\dot{q}}^{{\prime\prime} }$$ is the heat transfer rate for a given surface area (heat flux – kW/m2), Ts is the surface temperature, T is the fluid temperature, and h is the proportionality constant that is known as the convective heat transfer coefficient (units: kW/m2K). Determining the heat transfer coefficient is the fundamental problem in convective heat transfer and involves taking a closer look at the actual heat transfer process. Because all real surfaces have friction, the velocity of the fluid immediately next to the surface is always zero. This is called the no-slip condition. Far away from the solid, the fluid continues to move at its bulk free-stream velocity (u), so the velocity of the fluid varies from zero at the surface to the free-stream velocity (u) over some distance called the boundary layer thickness (δ) (see Fig. 1). This difference in velocity over a given distance is referred to as a velocity gradient. Note that if the surface and the fluid are at different temperatures, a temperature gradient also exists in the fluid. When viewed this way, convection is actually the result of both heat conduction at the surface and bulk fluid motion (advection). Right at the surface, the fluid isn’t moving, so heat is transferred from the solid to the fluid purely by conduction. Because the bulk of the fluid is moving, the fluid next to the surface is constantly being replaced by fresh fluid, helping to maintain the temperature difference that drives the transfer of heat. This replacement of fluid generally results in larger heat transfer rates than pure conduction. Because conduction is truly at the heart of convection, we can calculate the convective heat transfer rate starting with Fourier’s law of heat conduction:
$${\dot{q}}^{{\prime\prime} }=-k\nabla T$$
(2)
where k is the thermal conductivity (kW/mK) and ∇T is the temperature gradient. For simplicity, let’s consider the one-dimensional case with the surface at a higher temperature than the fluid. The temperature gradient is then the difference in temperature between the surface and the free-stream fluid divided by the distance over which that temperature difference occurs (thermal boundary layer thickness, δt).
$${\dot{q}}^{{\prime\prime} }=-k\frac{dT}{dy}=k\frac{\left({T}_s-{T}_{\infty}\right)}{\delta_t}$$
(3) Fig. 1Development of velocity and thermal boundary layer (δ and δt, respectively)

By comparing Eqs. 1 and 3, it is apparent that the convective heat transfer coefficient is equivalent to the thermal conductivity of the fluid divided by the thermal boundary layer thickness. The thermal boundary layer thickness (δt), however, is a result of the geometry and size of the solid, flow conditions (such as velocity, flow direction, forced vs. natural, laminar vs. turbulent, etc.), and fluid properties. This means that the convective heat transfer coefficient is not a simple material property that can be tabulated but is a variable that must be evaluated.

## Non-dimensional Numbers and Evaluation of Convective Heat Transfer Coefficient

There are two ways to evaluate the convective heat transfer coefficient. In some rare and simple cases, it can actually be calculated from first principles (see the isothermal flat plate in parallel flow discussion below). This involves solving the conservation equations for mass, momentum, and energy in the fluid flow. For forced convection, the mass and momentum conservation equations are solved first for the velocity profile of the fluid flow. This velocity profile is then used to solve the energy equation for the temperature profile. This temperature profile is then used to evaluate the temperature gradient at the surface and thus the heat transfer rate. For natural convection, however, the flow is a result of a density gradient in the fluid usually caused by a temperature gradient. This means that the mass, momentum, and energy equations are coupled and must be solved simultaneously. As one can imagine, this process is challenging for both forced and natural convection and can only be done in a handful of situations and only in laminar flow. Because of this, the convective heat transfer coefficients are most often measured experimentally. These experimental measurements are often made for a given flow configuration under a wide range of conditions and correlated with four common dimensionless numbers. These dimensionless numbers can be found by non-dimensionalizing the conservation equations.

The first dimensionless number is the Nusselt number (Nu). It is given by
$$\mathrm{Nu}=\frac{hL}{k}$$
(4)
where L is a characteristic length (e.g., the diameter of a cylinder or the length of a plate) and k is the thermal conductivity of the fluid. The Nusselt number can be viewed as a dimensionless temperature gradient at the surface or as the ratio of convection to fluid conduction. Because the Nusselt number contains the desired convective heat transfer coefficient, this is the dimensionless number that is solved for. Conveniently, because the Nusselt number contains a length scale and the fluid thermal conductivity, it also serves to scale the problem. In other words, as long as the solid shape and flow configuration are the same (e.g., parallel flow over a flat plate or a cylinder in cross flow), correlations with the Nusselt number should hold for different solid sizes and different fluids (air, water, oil, etc.).
The Prandtl number (Pr) appears in correlations for both forced and natural convection. The Prandtl number compares the momentum and thermal diffusivities of the fluid (ν and α, respectively):
$$\Pr =\frac{\nu }{\alpha }$$
(5)
The momentum diffusivity (ν) is more commonly called the kinematic viscosity and is the ratio of the dynamic viscosity (μ) and the density (ρ). It has units of m2/s and is tabulated for most common fluids. The thermal diffusivity (α) is given by
$$\alpha =\frac{k}{\rho {c}_p}$$
(6)
where cp is the specific heat of the gas. The thermal diffusivity also has units of m2/s and is tabulated for most common fluids. For most gases, including air and those found in flames, the Prandtl number is roughly equal to one. Physically, this means that in these cases the velocity boundary layer thickness is approximately the same as the thermal boundary layer thickness.
The Reynolds number (Re) appears in correlations for forced convection. It is the ratio of the inertia and viscous forces and is given by
$${\operatorname{Re}}_L=\frac{\rho {u}_{\infty }L}{\mu }=\frac{u_{\infty }L}{\nu }$$
(7)
where L is the characteristic length. Note the “Re” will often have a subscript indicating which length is used for its evaluation. For example, in the above equation, the subscript is “L,” but if the Reynolds number is evaluated in reference to a diameter, “D” would be used both as a subscript and as the length in the right-hand side of the equation. Because the flow profile develops over the length of the solid, a local Reynolds number, Rex, can also be identified where the length (x) is the distance from the leading edge of the solid to the point of interest. Physically, the Reynolds number provides a picture of what’s going on in the flow. As a fluid flows, small perturbations form due to things like minuscule surface roughness. If the Reynolds number is small, the viscous forces are important relative to the inertial forces. These viscous forces work to minimize the perturbations in the flow so that the flow remains laminar. If the inertial forces are sufficiently large, i.e., the Reynolds number is greater than some critical value, the viscous forces can’t keep up, the flow perturbations continually grow and form vortices, and the flow becomes turbulent.
The Grashof number (Gr) appears in correlations for natural convection. It is the ratio of the buoyant and viscous forces:
$$\mathrm{Gr}=\frac{g\beta \left({T}_s-{T}_{\infty}\right){L}^3}{\nu^2}$$
(8)
where g is the acceleration due to gravity (9.8 m/s2) and β is the volumetric thermal expansion coefficient, which for an ideal gas is equal to 1/T where T is in Kelvin. The Grashof number plays the same role in natural convection that the Reynolds number plays in forced convection. If the Grashof number is larger than a critical value, the flow is turbulent. In some correlations, the Grashof number is combined with the Prandtl number to form the Rayleigh number (Ra, where Ra = GrPr).

When selecting an empirical correlation for the convective heat transfer, there are several things to take note of. The first is to make sure that the flow configuration of your problem matches the configuration the correlation was developed for. Remember that thermal properties will vary with temperature, and different correlations account for this in different ways. Many will have you calculate the thermal properties at an average or “film” temperature, but others will have you calculate the thermal properties at the bulk or free-stream fluid temperature and include a ratio inside the correlation to adjust. You will also need to pay attention to whether the correlation is for average heat transfer rates or for local heat transfer rates. Average values are typically denoted by an overbar. Because both the velocity and thermal boundary layer develop as the fluid flows over the surface, so will the heat flux. Typically, the heat flux will be the largest at the place the fluid first contacts the surface (the leading edge) because the velocity and temperature gradients are the largest. As the fluid continues the flow over the surface, the boundary layer gets thicker, shrinking the gradients and reducing the heat flux. Correlations for the average heat transfer coefficient are good for the entire length of the solid whereas correlations for the local heat transfer correlation are good for the surface starting from the leading edge to some point of interest downstream. Another thing to note is if the correlation provides limitations. Sometimes they are developed for only certain fluids or over a limited range of Reynolds numbers.

## Effect of Turbulence

Convective heat transfer from turbulent flow tends to be larger than from laminar flow. Turbulent eddies in the flow enhance mixing so that the turbulent velocity and thermal boundary layer are more uniform at the core. This results in a sharper or larger temperature gradient at the surface, increasing the heat transferred. Note that because larger amounts of momentum and energy are transported by these turbulent eddies, the momentum diffusivity (kinematic viscosity) and thermal diffusivity are somewhat meaningless. In turbulent flows, the thermal boundary layer thickness is always about the same as the velocity boundary layer thickness, regardless of the Prandtl number.

## Mixed Convection

There are times when the flow due to natural convection is comparable to the forced flow. This can be checked by calculating the ratio of the Grashof number to the square of the Reynolds number (Gr/Re2). If this ratio is on the order of one, both forced and natural convection must be considered. If the ratio is much larger than one, only forced convection needs to be considered. Alternatively, if the ratio is much less than one, only natural convection needs to be considered. If both natural and forced convection effects must be considered, their direction relative to each other dictates the resulting effect. The three possible cases are if both the natural and forced flow are in the same direction (assisting), opposing, or perpendicular (transverse). As a first approximation, the total Nusselt number can be evaluated as
$${\mathrm{N}\mathrm{u}}^{\mathrm{n}}={\mathrm{N}\mathrm{u}}_{\mathrm{F}}^{\mathrm{n}}\pm {\mathrm{N}\mathrm{u}}_{\mathrm{N}}^{\mathrm{n}}$$
(9)
where NuF and NuN are the Nusselt numbers for forced and natural convection, respectively. If the flows are assisting or transverse, the two Nusselt numbers are added. If the flows oppose, they are subtracted. The exponent, n, is often set to 3, but 7/2 or 4 may work better for transverse flows over horizontal plates and cylinders/spheres, respectively.

## Effect of Blowing

Frequently the surface of interest in wildland and WUI fires is fuel that will eventually ignite and burn. As discussed in the “ignition” entry, as a solid fuel heats, it begins to thermally degrade (or pyrolyze) and produce gaseous fuels. As the surface nears ignition, a fair amount of gaseous fuel may be escaping from the surface, generating a non-negligible flow perpendicular to the surface. If the convective heat transfer needs to be accurately evaluated, this effect of “blowing” needs to be included. In general, blowing decreases the heat transfer coefficient because it makes the boundary layer thicker. Classic film theory gives a correction to the convective heat transfer coefficient as
$$\frac{h^{\ast }}{h}=\frac{\varphi }{e^{\varphi }-1}$$
(10)
where h* is the convective heat transfer coefficient corrected for blowing and
$$\upvarphi =\frac{{\dot{\mathrm{m}}}^{{\prime\prime} }{\mathrm{c}}_{\mathrm{p}}}{\mathrm{h}}=\frac{{\uprho \upnu \mathrm{c}}_{\mathrm{p}}}{\mathrm{h}}$$
(11)
where $${\dot{\mathrm{m}}}^{{\prime\prime} }$$ is the mass flux out of the surface and cp , ρ, and v are the specific heat, density, and velocity of the gas coming out of the surface.

## Isothermal Flat Plate in Laminar Parallel Flow

One classic problem in which the convective heat transfer can be calculated exactly from first principles is the case of the isothermal flat plate in laminar parallel flow. To do this, the flow is treated as two separate problems. The first part is the region near the wall where the effect of the surface is noticed, in other words the boundary layer region. The other part is the region where the effect of the surface is not noticed and the flow is treated as a simple inviscid flow. The boundary layer assumption is that the thickness of the boundary layer is much smaller than the length scale (δ << L or x). From this assumption, several other approximations to the conservations equations can be made so that a similarity solution can be obtained for the “boundary layer equation” form of the conservation of momentum and energy equations. There are several important results of this analysis that provide insight into the convective heat transfer process. The first relates the boundary layer thickness to the Reynolds number and the length along the surface (x):
$$\delta =\frac{5x}{{\operatorname{Re}}_x^{1/2}}=\frac{5x}{{\left({u}_{\infty }x/\nu \right)}^{1/2}}=5{\left(\frac{\nu x}{u_{\infty }}\right)}^{1/2}$$
(12)
This result means that the boundary layer thickness increases as x1/2 along the length of the plate. It also shows that the boundary layer thickness increases with viscosity and decreases with free-stream velocity. Note that if the Prandtl number is approximately 1, these trends hold for the thermal boundary layer as well. As mentioned above, one of the key assumptions used to solve these equations was that the boundary layer is thin. By rearranging Eq. 12, we see that the square root of the Reynolds number is proportional to the ratio of the length scale and the boundary layer thickness. Thus, the Reynolds number here is really a “slenderness ratio,” and in order for the boundary layer assumptions to be valid, $${\operatorname{Re}}_x^{1/2}\gg 1$$. Similar relations for turbulent flow or natural convection on a vertical flat plate result in δ ∝ x4/5 and δ ∝ x1/4, respectively. If the similarity solution is performed for the conservation of energy, the local Nusselt number for the laminar case with Pr larger than 0.6 is
$${\mathrm{Nu}}_x=\frac{h_xx}{k}=0.332{\operatorname{Re}}_x^{1/2}{\Pr}^{1/3}$$
(13)

This relationship indicates that the local heat transfer coefficient is infinite at the leading edge of the plate and decreases as x−1/2 along the plate. The average Nusselt number can be found from Eq. 13 by integrating over the length of the plate.

## Summary

Convection is the form of heat transfer that occurs when a fluid (liquid or gas) flows over a surface at a different temperature. The amount of heat transferred is proportional to the temperature difference between the solid and fluid. Solving for the proportionality constant, known as the convective heat transfer coefficient, is the fundamental problem of convective heat transfer. Though it can be solved for from fundamental theory in a few rare cases, it is most often measured experimentally and correlated with dimensionless numbers. Several factors will influence this heat transfer coefficient. These include the fluid velocity; the thermal properties of the fluid; whether the flow is laminar or turbulent, forced, or naturally (buoyantly) induced; and, perhaps most important in wildland and WUI fires, the surface size and shape. With all else held constant, solids with small length scales, such as needles and grasses, will transfer much more heat convectively than solids with large length scales, such as tree trunks and siding on structures. This not only makes measuring relevant convective heat transfer with traditional sensors challenging but will also likely shift the dominant heat transfer mechanism for ignition and fire spread between convection and radiation depending on the fuel size and shape.

The discussion provided here is brief and incomplete. If more information is needed, see Bejan (2013) or Bergman et al. (2011).

## References

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5. Churchill SW, Chu HHS (1975b) Correlating equations for laminar and turbulent free convection from a vertical plate. Int J Heat Mass Transf 18:1323–1329
6. Finney MA, Cohen JD, Forthofer JM, McAllister SS, Gollner MJ, Gorham DJ, Saito K, Akafuah NK, Adam BA, English JD (2015) Role of buoyant flame dynamics in wildfire spread. PNAS 112(32):9833–9838
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