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Constructing third-order derivative-free iterative methods

In this work, we develop nine derivative-free families of iterative methods from the three well-known classical methods: Chebyshev, Halley and Euler iterative methods. Methods of the developed families consist of two steps and they are totally free of derivatives. Convergence analysis shows that the methods of these families are cubically convergent, which is also verified through the computational work. Apart from being totally free of derivatives, numerical comparison demonstrates that the developed methods perform better than the three classical methods.

Derivative free algorithm for solving nonlinear equations

In this work, we develop a simple yet practical algorithm for constructing derivative free iterative methods of higher convergence orders. The algorithm can be easily implemented in software packages for achieving desired convergence orders. Convergence analysis shows that the algorithm can develop methods of various convergence orders which is also supported through the numerical work. Computational results ascertain that the developed algorithm is efficient and demonstrate equal or better performance as compared with other well known methods.

Temperatures of restricted turbulent fire plumes

A study has been made of the restricted plume flow above horizontal, fine mesh screens which intercept a small-scale, turbulent fire plume. Measurements were made of centerline temperatures in both restricted and free plumes, together with screen friction forces associated with restricted plumes. Counter to intuition, the restricted plume displayed higher time-averaged mean temperatures. A model based on the return of the flow of the restricted plume to the pre-screen (universal) Froude number has been developed to predict the increased time-averaged mean temperatures above the screen, i.e. lifting of the virtual point source, as a function of the plume/screen friction force. Assuming engineering velocity distributions, the theoretical model was confirmed experimentally for screen friction forces up to at least half the total plume momentum at screen level.

Modeling Skin Injury from Hot Rice Porridge Spills

The present work analyzes skin burns from spills of hot rice and milk products. The traditional Norwegian rice porridge serves as an example. By testing spills on objects emulating an arm, it was concluded that spills were seldom thinner than 3 mm, and stayed in place due to the viscosity of the porridge for more than one minute. The Pennes bioheat equation was solved numerically for such spills, including heat conduction to the skin and convective heat losses from the porridge surface. Temperatures were analyzed in the porridge and skin layers, and the resulting skin injury was calculated based on the basal layer temperature. Parameters influencing burn severity, such as porridge layer thickness, porridge temperature, removal of the porridge and thermal effects of post scald tempered (15 °C) water cooling were analyzed. The spilled porridge resulted in a prolonged heat supply to the skin, and the skin injury developed significantly with time. The porridge temperature turned out to be the most important injury parameter. A 70 °C porridge temperature could develop superficial partial-thickness burns. Porridge temperatures at processing temperatures nearly instantly developed severe burns. It was demonstrated that prompt removal of the hot porridge significantly reduced the injury development. The general advice is to avoid serving porridge and similar products at temperatures above 65 °C and, if spilled on the skin, to remove it quickly. After such scald incidents, it is advised to cool the injured area by tempered water for a prolonged period to stimulate healing.

Modeling Skin Injury from Hot Spills on Clothing

The present work analyzes scald burns from hot beverages, such as coffee and tea, spilled on the lap, i.e., an incident that may occur in daily life. The Pennes bioheat equation is solved numerically for small spills wetting the clothing, i.e., the fabric prevents the spilled liquid from draining away. Temperatures are analyzed in the wetted fabric and the skin layers and the resulting skin injury is calculated based on the basal layer temperature. Parameters influencing burn severity, such as clothing thickness, liquid temperature, removal of fabric and thermal effects of post scald water cooling are analyzed. The fabric cools the water some but represents a threat since the entrapped water results in a prolonged heat supply. The liquid temperature turned out to be the most important injury parameter, where liquid temperature of about 80–85 °C seems to be a limit for developing superficial partial-thickness burns in the present minimum case, i.e., where the liquid just wets the fabric. Spilling water in excess of just wetting the fabric, more severe burns will develop at lower liquid temperatures due to the prolonged heat supply. Higher liquid temperatures will nearly instantly develop more severe burns. It is demonstrated that removal of the clothing within the first seconds after the spill may significantly reduce the scalding severity. The general advice is therefore to avoid excessive heating of beverages and, if the beverage is spilled, to quickly remove the wetted clothing. Prolonged tempered water cooling is advised to improve the healing processes.

Health Impacts of Climate Change-Induced Subzero Temperature Fires

General fire risk and the special risk related to cold climate cellulosic drying processes are outlined. Four recent subzero temperatures fires are studied with respect to health impacts: a wooden village fire, a single wood structure fire, a wildland urban interface (WUI) fire and a huge wildland fire. The health impacts range from stress related to loss of jobs, psychological effects of lost possessions, exposure to smoke and heat as well as immediate, or delayed, loss of lives. These four fires resulted in 32 fatalities, 385 persons hospitalized for shorter or longer periods, 104 structures lost and 1015 km2 of wildland burned north of, and just south of, the Arctic Circle. It is shown that the combination of subzero temperature dry weather, strong winds, changing agricultural activities and declining snowpack may lead to previously anticipated threats to people and the environment. There are reasons to believe that these fires are a result of the ongoing climate changes. Risk impacts are discussed. Rural districts and/or vulnerable populations seem to be most affected. Training methods to identify and better monitor critical fire risk parameters are suggested to mitigate the health impacts of a possibly increasing number of such fires.

Fire Safety Experimental Investigations of Time to Flashover as a Function of Humidity in Wood

The fire in Laerdalsoyri, Norway, on 18–19 January 2014, developed faster than the fire fighters could handle, and strong winds quickly spread the fire to neighbor houses and 150 m downwind to distant houses. 36 modern buildings and 4 historic buildings of cultural heritage were lost. The cold low relative humidity air in the deep valley dried the structures and resulted in rapid growth of fire and fire spread. This has triggered studies to understand as to how early the flashover is reached when fuel moisture content (FMC) in wood goes to low levels especially in winter when heating inhabited structures is a necessity. A study of flashover as a function of the FMC in the wood has been carried out by conducting experiments on approximate equivalent of ¼ ISO rooms in the laboratory. The relative humidity and temperature in the ambient air are also noted. It is observed that low FMC is the main factor that leads to early flashover. The temperatures rise very rapidly once flashover is reached with higher heat release and increased radiated heat. Hence, it can be said that lower humidity levels that tend to be the fact in winter can lead to fast development of fire. This has to be borne in mind, and necessary precautions are to be taken to control the development and spread of fire and reduce the risk of major fire accidents.

Ethanol and Methanol Burn Risks in the Home Environment

Biofuel heaters and fireplaces have in recent years been introduced for indoor and outdoor use. Due to their simplicity, they are usually equipped with few or no safety features. Worldwide, incidents resulting in major skin burn injury and long hospitalization periods have occurred when using such biofuel units. The present study analyses the characteristics of the liquids ethanol and methanol to get a scientific background for understanding related accidents. The comparably heavy vapors, especially from ethanol, may generate a pillow of combustible gas in the vicinity of the unit, particularly in quiescent indoor air conditions. It is also revealed that these fuels represent a potential severe risk, since the equilibrium vapor pressures are close to the stoichiometric fuel–air composition at normal room temperatures. Selected incidents were reviewed to understand the mechanisms involved when severe burns were received by the users. It turns out that the most severe incidents were related to refilling operations and included ignition of the fuel container vapor phase. When ignited, the container gas phase expansion propelled burning fuel from the bottle or container onto the user or other persons in the vicinity. Similar incidents involving refilling methanol for chemistry demonstrations and ethanol for endodontic (dentistry) treatment were also studied and it was shown that these accidents followed similar accident mechanisms. It may be concluded that the main contributors to burn risk are the near-stoichiometric vapor pressure of these liquids at room temperature and the close proximity of the fuel container to burning fuel. Research needs and possible technical barriers are suggested to reduce this risk for the future.

Modeling Burns for Pre-Cooled Skin Flame Exposure

On a television show, a pre-cooled bare-skinned person (TV host) passed through engulfing kerosene flames. The assumption was that a water film should protect him during 0.74 s flame exposure in an environment of 86 kW/m2 heat flux. The TV host got light burn inflammation on the back, arms and legs. The present work studies skin temperatures and burn damage integral of such dangerous flame exposure. The skin temperature distribution during water spray pre-cooling, transport to the flames, flame exposure, transport to the water pool, and final water pool cooling is modelled numerically. Details of the temperature development of the skin layers are presented, as well as the associated damage integral. It is shown that 5 °C water spray applied for a 30 s period pre-cooled the skin sufficiently to prevent severe skin injury. Soot marks indicate that the water layer evaporated completely in some areas resulting in skin flame contact. This exposed dry skin directly to the flames contributing significantly to the damage integral. It is further analyzed how higher water temperature, shorter pre-cooling period or longer flame exposure influence the damage integral. It is evident that minor changes in conditions could lead to severe burns and that high heat flux levels at the end of the exposure period are especially dangerous. This flame stunt should never be repeated.

Erratum to: Derivative free algorithm for solving nonlinear equations

In the original publication of the article, Eqs. 8 and 11 have been erroneously published. The correct equations are given below: x n+1 = x n − α 1 f (x n) 2 f (x n) − f (x n − α 1 f (x n)) f (x n) f (x n) − f x n − α 1 f (x n) 2 f (x n)− f (x n −α 1 f (x n)) , (8) x n+1 = x n − α 2 f (x n) 3 f (x n) − f (x n − α 2 f (x n)) f (x n) f (x n) − f x n − α 2 f (x n) 2 f (x n)− f (x n −α 2 f (x n)) 1 f (x n) − f ⎛ ⎝ x n − α 2 f (x n) 3 f (x n)− f (x n −α 2 f (x n)) 1 f (x n)− f x n − α 2 f (xn) 2 f (xn)− f (xn −α 2 f (xn))