Averaged area arithmetic-mean

Statistically, the particle size distribution can be characterized by three properties mode, median, and mean. The mode is the value that occurs most frequently. It is a value seldom used for describing particle size distribution. The average or arithmetic mean diameter, d, is affected by all values actually observed and thus is influenced greatly by extreme values. The median particle size, is the size that divides the frequency distribution into two equal areas. In practical application, the size distribution of a typical dust is typically skewed to the right, i.e., skewed to the larger particle size. The central tendency of a skewed frequency distribution is more adequately represented by the median rather than by the mean (see Fig. 9). Mathematically, the relationships among the mean, median, and mode diameter can be expressed as... 

The diametei of average mass and surface area are quantities that involve the size raised to a power, sometimes referred to as the moment, which is descriptive of the fact that the surface area is proportional to the square of the diameter, and the mass or volume of a particle is proportional to the cube of its diameter. These averages represent means as calculated from the different powers of the diameter and mathematically converted back to units of diameter by taking the root of the moment. It is not unusual for a polydispersed particle population to exhibit a diameter of average mass as being one or two orders of magnitude larger than the arithmetic mean of the diameters. In any size distribution, the relation ia equation 4 always holds. 

Thep and q denote the integral exponents of D in the respective summations, and thereby expHcitiy define the diameter that is being used. and are the number and representative diameter of sampled drops in each size class i For example, the arithmetic mean diameter, is a simple average based on the diameters of all the individual droplets in the spray sample. The volume mean diameter, D q, is the diameter of a droplet whose volume, if multiphed by the total number of droplets, equals the total volume of the sample. The Sauter mean diameter, is the diameter of a droplet whose ratio of volume-to-surface area is equal to that of the entire sample. This diameter is frequendy used because it permits quick estimation of the total Hquid surface area available for a particular industrial process or combustion system. Typical values of pressure swid atomizers range from 50 to 100 p.m. 

MeV. WL-R = 100% x WL/radon concentrations (pCi/1). The dose conversion factor of 0.7 rad/working level month (WLM) (Harley and Pasternack, 1982) was used to calculate the mean absorbed dose to the epithelial cells and a quality factor (OF) of 20 was applied to convert the absorbed dose to dose equivalent rate. For example, from the average value of (WL) obtained from the arithmetic mean radon concentrations measured in the living area during winter and summer in South Carolina (Table I), the calculated dose equivalent rate is 4.1 rem/yr, e.g.,... 

The average area Am is given by the average of both surface areas A t = A(rx) and j = A(r2). One gets the arithmetic mean for the flat plate, the logarithmic mean for the cylinder and the geometric mean for the sphere. It is known that... 

Table 11-2. Arithmetic Means of Total C02 Concentrations (nmol/kg) in Seawater in Three Depth Ranges of Seven Ocean Areas, and Averages for the World Ocean (Takahashi et al., 1981a) ... 

Assume that the inspection department of an organization involved in the manufacture of systems for use in the area of transportation inspected ten identical systems and discovered 5, 4, 6, 7, 9, 10, 3, 2, 8, and 1 defects in each system. Calculate the average number of defects per system (i.e., arithmetic mean). 

Pipe Insulation. In the case of pipe insulation the area through which heat is transferred is not constant. If the thickness of the material is small, compared with the diameter, the arithmetic average of the larger area and the smaller area may be used. The arithmetic mean area may... 

For the following particle size distribution data, calculate the arithmetic mean, geometric mean, count median, and diameter of the particle with average surface area. 

In a normal distribution curve 68.27% area lies between x Is, 95.45% area lies between x 2s, and 99.70% area falls between x 3s. In other words, 99.70% of replicate measurement should give values that should theoretically fall within three standard deviations about the arithmetic average of all measurements. Therefore, 3s about the mean is taken as the upper and lower control limits in control charts. Any value outside x 3s should be considered unusual, thus indicating that there is some problem in the analysis which must be addressed immediately. 

Eq (3-7) represents the summation of the surface-areas divided by the summation of the diameters. It gives a mean based on the surface observed, and the volume or total surface of the particle does not enter into the calculation. As an average it is comparable to the arithmetic and geometric means. 

Vapor at a saturation pressure pc and superheat temperature T is condensed by a counter water flow in a heat exchanger of area A. The inlet temperature of the water is Twt. The flow rates of the vapor and water are ms and m,. Assume the total heat transfer coefficients to be Uy and U2 for parts 1 and 2 of the exchanger as shown in Fig. 7P-Z Log-mean averages may be replaced by arithmetic averages. Determine the exit temperature of the water and the exit quality of the vapor. 

If the ratio of inlet and exit driving forees is less than 2, an arithmetic average can be used with little error. For larger driving-force ratios the logarithmic mean could be used, though examples show that this slightly overestimates the required area. 

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