Estimates of heat loss

Flammability limits can be truly independent of the experimental apparatus only if there is a critical point (for example, a critical value of 0) at which the exothermic chemistry is shut off completely, without heat loss. Although it is highly unlikely that this occurs, flammability limits may eflec- 

Section E.5.2, it may be seen from equation (E-50) that Ip is a local property of the gas mixture, not dependent on the experimental environment. Therefore, until the gas begins to become optically thick, which might not occur until sizes on the order of meters are reached, dimensions of the experimental apparatus do not influence any of the terms in equation (9), and predicted limits from radiant loss are apparatus-independent. 

In equation (26), I/Ip is proportional to the number density of emitting molecules, which, in turn, is proportional to pressure at constant temperature and relative composition (see Section E.5.2) when reabsorption is not negligible, the pressure dependence is weaker. These results were employed in the preceding subsection. In addition, 1/lp varies strongly with position x in the flame because of the changes in the concentrations and degrees of excitation of the emitters. For this reason, the generality of equation (21), which 

Joulin has completed the necessary analysis in recent work not yet published. 

Near rich limits of hydrocarbon flames, soot is sometimes produced in the flame. The carbonaceous particles—or any other solid particles— easily can be the most powerful radiators of energy from the flame. The function k(t) is difficult to compute for soot radiation for use in equation (21) because it depends on the histories of number densities and of size distributions of the particles produced for example, an approximate formula for Ip for spherical particles of radius with number density surface emissivity 6, and surface temperature is Ip = Tl nrle ns) [50]. These parameters depend on the chemical kinetics of soot production—a complicated subject. Currently it is uncertain whether any of the tabulated flammability limits are due mainly to radiant loss (since convective and diffusive phenomena will be seen below to represent more attractive alternatives), but if any of them are, then the rich limits of sooting hydrocarbon flames almost certainly can be attributed to radiant loss from soot. 

Estimates often indicate that although chemiluminescence may provide flames with their distinctive colors, the associated energy losses may be negligible in comparison with radiation from major stable products (such as from H2O and CO2 in hydrocarbon flames). If the concentration profiles of the products in the flame are similar to those of the normalized temperature and if the rate of radiant emission from the products is proportional to T, then we find from equations (12) and (26) that k is proportional to t(t -F — 1), whence I in equation (21) becomes proportional to Ty(Ty -F — l) + [(Ty + — 1) — (a — l) ]/5, which suggests 

Unlike the radiant loss from an optically thin flame, conductive or convective losses never can be consistent exactly with the plane-flame assumption that has been employed in our development. Loss analyses must consider non-one-dimensional heat transfer and should also take flame shapes into account if high accuracy is to be achieved. This is difficult to accomplish by methods other than numerical integration of partial differential equations. Therefore, extinction formulas that in principle can be used with an accuracy as great as that of equation (21) for radiant loss are unavailable for convective or conductive loss. The most convenient approach in accounting for convective or conductive losses appears to be to employ equation (24) with L(7 ) estimated from an approximate analysis. The accuracy of the extinction prediction then depends mainly on the accuracy of the heat-loss estimate. Rough heat-loss estimates are readily obtained from overall balances. 

The conductive heat loss per unit volume from a plane flame in a circular tube of diameter D can be estimated by the following simple reasoning. Consider an Eulerian element of gas of length dx in the tube whose walls are maintained at the temperature Tq. The energy per second conducted to the walls from this element is the product of the thermal conductivity A, a mean temperature gradient (T — To)/ D/2), and the wall area nD dx. The rate of heat loss per unit volume is then obtained through division by the volume of the element dxnD /4  

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Fig. 7. Section of solar-hcatcd building. Solar collector has an area of 3500 sqnaie feet (325 square meters) facing south ar an angle of 45 degrees. There are. about 8000 square feet (743 square meters) of working space. Estimates of heat loss indicate heat demand is in range of 40,000-70,000 Btu (10,080- ] 7,640 kcal) pei day. Locaied in die. northeastern United Stales, die building was designed to furnish between 65 and 75% of total seasonal heating load...

The procedure for design is one of trial and error. For example, as a first trial, heat losses might be neglected and trays calculated with fixed difference points, and after the size of the resulting column has been determined, the first estimate of heat losses for the two column sections can be made by the usual methods of heat-transfer calculations. The heat losses can then be apportioned among the trays and the number of trays redetermined with the appropriate difference points. This leads to a second approximation of the heat loss, and so forth. 

In all of these systems, the rate of generation at the gas-solid interface is so rapid that only a small fraction is canied away from the particle surface by convective heat uansfer. The major source of heat loss from the particles is radiation loss to tire suiTounding atmosphere, and the loss per particle may be estimated using unity for both the view factor and the emissivity as an upper limit from tlris source. The practical observation is that the solids in all of these methods of roasting reach temperatures of about 1200-1800 K. 

Estimate the heat loss per square metre of surface through a brick wall 0.5 m thick when the inner surface is at 400 K and the outside surface is at 300 K. The thermal conductivity of the brick may be taken as 0.7 W/mK. 

A furnace is constructed with 0.20 m of firebrick, 0.10 m of insulating brick, and 0.20 m of building brick. The inside temperature is 1200 K and the outside temperature is 330 K. If the thermal conductivities are as shown in Figure 9.7. estimate the heat loss per unit area and the temperature at the junction of the firebrick... 

Measurements of combustion temperatures, radiation intensity distributions in the range from 400 to 800 nm, and particle size distributions of combustion products have been made for the reaction of aluminum powder with both 02/Ar and H2O oxidizers in atmospheric dump combustors. The fraction of unburned aluminum in the combustion products was also determined for the H2O oxidizer case. An analytical study was performed to determine if the measurements are consistent with each other and with theory, and also to estimate the rate of heat loss from the combustion products. A Monte Carlo technique was used to determine the expected spectral energy distribution that would be emitted from a viewport located in the side of a combustion chamber containing products of aluminum combustion. 

Heat Balance in Estimating Critical Dimensions for Thermal Explosion. For thermal explosion to occur the rate of heat generation must exceed the rate of heat loss. The general equations for this situation can be solved only by numerical means. Specialized cases are, however, amenable to analytical solutions. An interesting case is that of a semi-infinite slab of... 

We might estimate that about 20% of this surface area is in the legs and tail, which have already been accounted for in terms of heat loss, so that the remainder, about 5.12 m, is the surface area of the body. We can further model the cow s body as roughly cylindrical in shape and can estimate that its radius is 40 cm. This allows us to calculate the length of the cylinder (including the surfaces at both ends) as... 

Of a special astronomical interest is the absorption due to pairs of H2 molecules which is an important opacity source in the atmospheres of various types of cool stars, such as late stars, low-mass stars, brown dwarfs, certain white dwarfs, population III stars, etc., and in the atmospheres of the outer planets. In short absorption of infrared or visible radiation by molecular complexes is important in dense, essentially neutral atmospheres composed of non-polar gases such as hydrogen. For a treatment of such atmospheres, the absorption of pairs like H-He, H2-He, H2-H2, etc., must be known. Furthermore, it has been pointed out that for technical applications, for example in gas-core nuclear rockets, a knowledge of induced spectra is required for estimates of heat transfer [307, 308]. The transport properties of gases at high temperatures depend on collisional induction. Collision-induced absorption may be an important loss mechanism in gas lasers. Non-linear interactions of a supermolecular nature become important at high laser powers, especially at high gas densities. 

A 6.0-cm-diameter pipe whose surface temperature is maintained at 2I0°C passes through the center of a concrete slab 43 cm thick. The outer surface temperatures of the slab are maintained at 15°C. Using the flux plot, estimate the heat loss from the pipe per unit length. 

An experiment is to be designed to demonstrate measurement of heat loss for water flowing over a flat plate. The plate is 30 cm square and it will be maintained nearly constant in temperature at 50°C while the water temperature will be about 10°C. (a) Calculate the flow velocities necessary to study a range of Reynolds numbers from 104 to 107. (b) Estimate the heat-transfer coefficients and heat-transfer rates for several points in the specified range. 

A pipeline in the Arctic carries hot oil at 50°C. A strong arctic wind blows across the 50-cm-diameter pipe at a velocity of 13 m/s and a temperature of -35°C. Estimate the heat loss per meter of pipe length. 

A 5-mm-diameter copper heater rod is submerged in water at I atm. The temperature excess is I PC. Estimate the heat loss per unit length of the rod. 

An amphipod with a body weight of 9 pg consumes 3.5 X 10 9 mol oxygen every hour at steady state and eliminates 3.5 X 10 9 mol carbon dioxide, 0.4 X 10-9 mol N (as ammonia), and 0.1 X 10-9 mol lactic acid. The external work power is 47 X 10 9 W. Estimate the heat loss of the animal when the following four net reactions contribute to the energy expenditure. 

Example 5 Comparison of the Relative Importance of Natural Convection and Radiation at Room Temperature Estimate the heat losses by natural convection and radiation for an undraped person standing in still air. The temperatures of the air, surrounding surfaces, and skin are 19,15, and 35°G, respectively. The height and surface area of the person are 1.8 m and 1.8 m2. The emissivity of the skin is 0.95. 

There are two types of convection, free and forced (Holman, 2009 Incropera et al., 2007 Kreith and Bohn, 2007). Free (natural) convection occurs when the heat transferred from a leaf causes the air outside the unstirred layer to warm, expand, and thus to decrease in density this more buoyant warmer air then moves upward and thereby moves heat away from the leaf. Forced convection, caused by wind, can also remove the heated air outside the boundary layer. As the wind speed increases, more and more heat is dissipated by forced convection relative to free convection. However, even at a very low wind speed of 0.10 m s-1, forced convection dominates free convection as a means of heat loss from most leaves (0.10 m s-1 = 0.36 km hour-1 = 0.22 mile hour-1). We can therefore generally assume that heat is conducted across the boundary layer adjacent to a leaf and then is removed by forced convection in the surrounding turbulent air. In this section, we examine some general characteristics of wind, paying particular attention to the air boundary layers adjacent to plant parts, and introduce certain dimensionless numbers that can help indicate whether forced or free convection should dominate. We conclude with an estimate of the heat conduction/convection for a leaf. 

IS6 Using infotmaiion from (he utility bills for the coldest month Iasi year, estimate the average rale of heat loss from your house for tliat month. In your analysis, consider the contribution of the internal heat sources such as people, lights, and appliances. Identify the primary sources of heat loss from your house and propose ways of improving the energy efficiency of your house,... 

The roof of a house consists of a 25-cin-lhick concrete slab (k 1.9 W/m C) that is 8-m wide and 10 m long. The emissivity of tlie outer surface of the roof is 0.8, and the convection heal transfer coefficient on (hat surface is estimated to be 18 W/m °C. On a cleat winter nighi, the ambient air is reported to be at I0 C, while the night sky temperature for radiation heat transfer is 170 K. If the inner surface temperature of tberoofisTi = 16"C, determine (he outer surface temperature of the roof and the rate of heat loss through the roof when steady operating conditions are reached. 

The above-estimated thermal requirement must be corrected for the various sources of heat loss hsted above. 

The net energy gained or lost by a body can be estimated by these laws. The simplest case is that of a gray body in black surroundings. These conditions, in which none of the energy emitted by the body is reflected back, are approximately those of a body radiating to atmosphere. If the absolute temperature of the body is Ti, the rate of heat loss is aeAT [Eq. (40)], where A is the area of the body and e its emissivity. Surroundings at a temperature emit radiation proportional to ctTI, and a fraction, determined by area and absorptivity a, is absorbed by the body this heat is aaAfi, and as absorptivity and emissivity are equal, Eq. (40) is valid. 

A water system is fed from a very large tank, large enough so that the water level in the tank is essentially constant. A pump delivers 3000 gal/min in a 12-in.-ID pipe to users 40 ft below the tank level. The rate of work delivered to the water is 1.52 hp. If the exit velocity of the water is 1.5 ft/s and the water temperature in the reservoir is the same as in the exit water, estimate the heat loss per second from the pipeline by the water in transit. 

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