Self-heating

Self-heating is a rise in a material's temperature caused by heat generated within the material itself. In the fire-engineering treatment described by Gray, chemical reactions release heat that is retained because transfer to the surroundings is limited. The resulting behaviour can range from mild warming to conditions that lead to spontaneous ignition.

The process depends on the balance between heat production and heat loss. Material chemistry matters, but so do the size and shape of the stored body, its surroundings and its initial temperature. A small test sample and a large pile of the same material can therefore behave differently.

Self-heating provides an internal ignition source, rather than requiring all the initiating heat to come from an external flame. It does not necessarily produce a fire. Subcritical heating can settle at a moderately elevated temperature, while a supercritical condition can develop into accelerating temperature rise and ignition.

Heat generation and heat retention

Chemical reactions in a stored material can release heat before visible flames appear. If that heat escapes sufficiently quickly, the temperature rise remains limited. If heat accumulates, the higher temperature can increase the reaction rate. Gray describes this temperature-dependent feedback as the basis of spontaneous combustion and ignition.

Self-heating depends on internally generated heat and its transfer to the surroundings. Based on Gray (2002), chapter 2-10, pp. 2-211–2-214.

Heat production and heat loss respond differently to temperature. In the simplified single-reaction model, the reaction rate follows an Arrhenius dependence on absolute temperature. This gives a strongly curved heat-generation relation over the temperature range being considered. The corresponding heat-loss relation is represented approximately as a straight line.

Where the two relations intersect, heat production and heat loss balance. A lower-temperature intersection can be stable: a small temperature increase raises losses sufficiently to bring the body back towards that state. A higher-temperature intersection can be unstable. Above that second intersection, heat production exceeds losses and temperature continues to rise.

Two thermal balances can exist under subcritical surrounding conditions in the simple model. Based on Gray (2002), chapter 2-10, pp. 2-213–2-214.

The stable state is warmer than the surroundings, so a temperature rise does not by itself establish imminent ignition. Gray associates subcritical heating with effects such as degradation or discolouration of stored products. These effects can make a material unsuitable for its intended use even without flaming. The distinction concerns thermal behaviour as well as visible fire.

An accelerating temperature rise is associated with thermal runaway. In the simplified representation, the critical condition occurs when the stable and unstable balance points meet. Beyond this condition, no low-temperature balance remains. The model then predicts continued heating towards ignition under its stated chemical and thermal assumptions.

Size, shape and initial temperature

The dimensions of an accumulation influence how readily it loses heat. Gray relates the heat-loss behaviour to the ratio of surface area to volume. For geometrically similar bodies, that ratio decreases as size increases. Larger bodies can therefore retain reaction heat differently from smaller samples of the same material.

The critical condition can be expressed in different ways. For a body of fixed size and shape, the critical ambient temperature identifies the surrounding temperature at the transition. For a fixed surrounding temperature, a critical size can instead be defined. These describe different views of the same dependence on material, geometry and environment.

Stacking individual packages can change the relevant thermal body. Gray discusses bales placed in contact and products accumulated in larger quantities. A result obtained for one isolated package does not automatically describe a group whose heat-loss conditions have changed. Scaling from a laboratory specimen therefore requires a model of the larger arrangement.

The material's initial temperature introduces another distinction. Recently processed goods may already be hot when they are stacked. Gray calls the relevant initial-material threshold the critical stacking temperature. It differs from critical ambient temperature because it describes the starting temperature of the body itself, rather than the surrounding air.

Hot stacking also changes the time history. A calculation beginning with the material at ambient temperature cannot simply be transferred to a body that starts much hotter. Gray discusses this limitation when interpreting ignition delays. The initial thermal state belongs with the model inputs, alongside dimensions and surrounding conditions.

The International Fire Engineering Guidelines similarly reject a single ignition temperature for all self-heating arrangements. Their discussion of prolonged heating of wood identifies the size of the accumulation as a relevant variable. This distinguishes the problem from treating autoignition temperature as one universal material number. The cited discussion belongs to the guidelines' 2005 edition.

A simple energy-balance model

The Semenov model represents the reacting body with one uniform temperature. Gray explains that this is appropriate to well-mixed systems or cases with relatively little internal resistance to heat flow. It is less representative of poorly conducting materials with strong internal temperature gradients. Its value is that it makes the competition between generation and loss explicit.

Gray's energy balance can be written in the compact form below. The body's heat capacity multiplied by its heating rate equals internally generated power minus heat transferred to the surroundings. The generation term depends on temperature and the amount of reacting material remaining. The notation groups the detailed chemical expression into one function.

C_th dT/dt = G(T,c) − H(T − T_a)

Symbol	Description	Unit
Cth	Total heat capacity of the represented body	J/K
T, Ta	Uniform body temperature and ambient temperature	K
t	Time	s
G(T,c)	Heat-generation rate, dependent on temperature and reactant concentration	W
c	Reactant concentration	mol/m³
H	Surface heat-transfer coefficient multiplied by exposed area	W/K

If generation exceeds loss, the right-hand side is positive and the body heats. If loss exceeds generation, it cools. Equality identifies a stationary thermal condition within the model. Whether that condition is stable depends on how the two rates change when temperature is disturbed.

The accompanying reaction equation describes consumption of the reacting material. Ignoring that consumption is an additional simplification, rather than a requirement of every self-heating model. Gray discusses both the basic model and corrections when reactant depletion matters. A heat-generation curve cannot always be treated as unchanged throughout a long experiment.

The Frank-Kamenetskii treatment instead represents conduction and temperature variation inside the body. Its basic form uses a surface temperature equal to ambient and neglects reactant depletion. Geometry enters the critical condition through a characteristic dimension and a shape-dependent parameter. These assumptions explain both its usefulness for scaling experiments and its limitations.

Measuring critical behaviour

Gray describes oven tests using specimens held in gauze baskets of known dimensions. The gauze allows oxygen to enter and reaction products to leave. Temperature is measured at the specimen centre, with additional measurements possible where internal profiles are needed. The oven provides a controlled surrounding temperature for the selected specimen arrangement.

A trial below the critical condition produces limited self-heating. A trial above it produces a substantially different temperature rise. Fresh material is used when the experiment is repeated at another oven temperature. Reusing an already reacted specimen would change the starting material represented by the comparison.

The critical temperature is bracketed between the highest subcritical and lowest supercritical trials. Further tests narrow that interval. Repeating the process with different basket sizes provides information about the relationship between dimensions and critical temperature. The resulting data can support a scaling relation when its chemical and thermal assumptions are satisfied.

The basic scaling method uses a logarithmic relationship involving absolute temperature and the characteristic specimen dimension. Gray describes fitting this relationship to results from several sizes. A straight-line relation supports the assumed reaction-rate dependence over the tested range. It does not establish that the same relation remains valid across every lower temperature or different moisture condition.

The surface boundary condition is also part of the test interpretation. Heat must transfer from the basket surface to the oven air rapidly enough for the assumed surface temperature to be reasonable. Gray relates this condition to the relative internal and external thermal resistances. A well-controlled oven temperature alone does not establish the internal temperature field.

Worked example for milk-powder deposits

Gray reproduces a milk-powder example associated with a spray dryer. Laboratory tests used cubic baskets with half-side lengths of 0.025, 0.0375 and 0.050 m. Their reported critical ambient temperatures were 171, 156 and 141.5 °C, respectively. The larger specimens became critical at lower surrounding temperatures.

The example gives the following fitted scaling relation. In this numerical expression, temperature is entered in kelvin and the characteristic dimension in metres. The dimensionless critical parameter depends on the represented shape. Applying the fit to another geometry therefore requires the appropriate parameter as well as the material relation.

ln(δ_critT_a² / r²) = 41.85 − 9497 / T_a

Symbol	Description	Unit
Ta	Critical ambient temperature entered in the fitted relation	K
r	Half-thickness for the slab representation	m
δcrit	Critical geometry parameter; the example uses 0.88 for an infinite slab	dimensionless

Rearranging gives the half-thickness directly for a selected ambient temperature. The total layer thickness is twice this value. The source's slab example assumes the same ambient temperature on both faces. It therefore represents a particular idealised thermal boundary condition, rather than every deposit attached to a dryer wall.

r = T_a √δ_crit exp[(9497 / T_a − 41.85) / 2]

At 200 °C, or 473.15 K, substitution with δ_crit = 0.88 gives a half-thickness of approximately 0.00828 m. The calculated full thickness is approximately 0.0166 m. Gray reports the example at the coarser precision of 0.017 m. The result belongs to the fitted milk-powder data and the stated slab model.

At 80 °C, or 353.15 K, the same calculation gives a full thickness of approximately 0.374 m. The source reports about 0.4 m. The large difference illustrates the temperature dependence of this particular critical-size calculation. Neither result establishes a general allowable deposit thickness for other powders, geometries or operating conditions.

The fitted milk-powder example predicts a smaller critical layer thickness at the higher ambient temperature. Based on Gray (2002), chapter 2-10, p. 2-226.

The example also connects a heated deposit with a dust explosion hazard. Gray describes spontaneously igniting powder as a possible ignition source for dispersed dust in the dryer. The deposit's self-heating and the subsequent dust-cloud event are different processes. Predicting one does not by itself calculate the severity of the other.

Moisture, smouldering and limits of extrapolation

Water can alter both heat transfer and reaction chemistry. Evaporation absorbs heat, while movement and condensation redistribute moisture and energy within a body. Gray explains that small specimens tested at high temperatures can dry before substantial self-heating develops. Scaling those results to a larger moist accumulation can therefore describe the wrong material condition.

More than one heat-producing reaction may also operate. Different reactions can dominate different temperature ranges, giving a change in the fitted scaling relation. Gray illustrates this through calcium-hypochlorite behaviour and discusses additional reactions in moist cellulosic material. Extending one high-temperature fit far outside its range can consequently give a misleading critical condition.

Smouldering is a related but distinct combustion process. Ohlemiller defines it as slow, flameless combustion sustained by oxidation at the surface of condensed fuel. Self-heating concerns the accumulation of internally generated heat, including conditions before sustained combustion develops. Not every mildly warming body has established a propagating smouldering front.

The smouldering-to-flaming transition requires further consideration of the combustion conditions. Ohlemiller distinguishes initiation, sustained smouldering propagation and the potential development of flames. Conditions sufficient for the first event do not necessarily establish the later events. A self-heating threshold therefore cannot substitute for a complete description of subsequent fire development.

Interpreting storage history and fire evidence

The relevant accumulation is the material in thermal contact, rather than every combustible item within the same building. Gray distinguishes a continuous body from groups separated by aisles. Fire can spread between separated groups without making them one thermal body before ignition. This distinction affects the dimensions used when assessing the original self-heating process.

Storage duration can explain why an established process suddenly produces a fire. Gray describes materials normally moved before their ignition delay has elapsed. A longer interruption can allow the same accumulation to remain in place beyond that delay. A change in storage history can therefore matter even without a change in the nominal material or ambient temperature.

Internal charring can provide evidence of the location of heat accumulation. Gray describes hot gases leaving a reacting interior through paths of relatively low resistance. Non-uniform density and porosity can produce several such paths within one body. Multiple damaged regions consequently do not, by themselves, establish that separate external ignition sources were applied.

Unconsumed material can also remain within an accumulation that has experienced spontaneous ignition. Gray gives examples of partly protected material surviving beneath char and within regions with limited oxygen access. The surviving fuel does not independently exclude self-heating as a cause. Its position and thermal surroundings belong with the interpretation of the damaged material.

The same chapter links these observations to laboratory characterisation and the reconstructed storage conditions. Size, initial temperature, surrounding heat transfer and time undisturbed are considered together. A visual pattern cannot supply all the reaction and transport parameters needed for a quantitative prediction. Conversely, a calculation for a different material arrangement cannot establish the history of the examined accumulation.

References

• Australian Building Codes Board, National Research Council of Canada, International Code Council, & Department of Building and Housing. (2005). International fire engineering guidelines (2005 ed.). Australian Building Codes Board.
• Gray, B. (2002). Spontaneous combustion and self-heating. In P. J. DiNenno (Ed.), SFPE handbook of fire protection engineering (3rd ed., chapter 2-10, pp. 2-211–2-228). National Fire Protection Association.
• Ohlemiller, T. J. (2002). Smoldering combustion. In P. J. DiNenno (Ed.), SFPE handbook of fire protection engineering (3rd ed., chapter 2-9, pp. 2-200–2-210). National Fire Protection Association.