Temperature calculating

The Newton-Raphson approach, being essentially a point-slope method, converges most rapidly for near linear objective functions. Thus it is helpful to note that tends to vary as 1/P and as exp(l/T). For bubble-point-temperature calculation, we can define an objective function... 

The bubble and dew-point temperature calculations have been implemented by the FORTRAN IV subroutine BUDET and the pressure calculations by subroutine BUDEP, which are described and listed in Appendix F. These subroutines calculate the unknown temperature or pressure, given feed composition and the fixed pressure or temperature. They provide for input of initial estimates of the temperature or pressure sought, but converge quickly from any estimates within the range of validity of the thermodynamic framework. Standard initial estimates are provided by the subroutines. 

Convergence is usually accomplished in 2 to 4 iterations. For example, an average of 2.6 iterations was required for 9 bubble-point-temperature calculations over the complete composition range for the azeotropic system ehtanol-ethyl acetate. Standard initial estimates were used. Figure 1 shows results for the incipient vapor-phase compositions together with the experimental data of Murti and van Winkle (1958). For this case, calculated bubble-point temperatures were never more than 0.4 K from observed values. 

Fraiiehuk, A. U. 1949. Method of Temperature Calculation at Different Room Heights—Studies on Building Physics (in Russian). Gosstroizdat, Moscow/Leningrad. 

Gebhart B. Surface temperature calculations in radiant surroundings of arbitrary complexity—for gray, Diffuse Radiation. Int.. Heat Mitss Transfer, vol. 3, no. 4, 19iil. 

The grand-thermodynamical potential is, like the temperature, calculated in units of b. Macroscopically, b is related to the critical temperature of the oil-water separation by kT = 3(1 — p )b. The coupling constants of O2 re... 

Martensitic phase transformations are discussed for the last hundred years without loss of actuality. A concise definition of these structural phase transformations has been given by G.B. Olson stating that martensite is a diffusionless, lattice distortive, shear dominant transformation by nucleation and growth . In this work we present ab initio zero temperature calculations for two model systems, FeaNi and CuZn close in concentration to the martensitic region. Iron-nickel is a typical representative of the ferrous alloys with fee bet transition whereas the copper-zink alloy undergoes a transformation from the open to close packed structure. ... 

To illustrate and use this equation, consider again reaction (7-10). The following are the values of AG at 77 °C, the middle temperature, calculated from the rate constant at this temperature 1... 

A large block of material of thermal diffusivity Du — 0.0042 cm2/s is initially at a uniform temperature of 290 K and one face is raised suddenly to 875 K and maintained at that temperature. Calculate the time taken for the material at a depth of 0.45 m to reach a temperature of 475 K on the assumption of unidirectional heat transfer and that the material can be considered to be infinite in extent in the direction of transfer. 

Fig. 5.13. The Q-branch spectrum in N2-Ar mixture at room temperature calculated with quantum theory [215].

Assuming that the heat capacity of an ideal gas is independent of temperature, calculate the entropy change... 

Assuming that the heat capacity of an ideal gas is independent of temperature, calculate the entropy change associated with lowering the temperature of 2.92 mol of ideal gas atoms from 107.35°C to —52.39°C at (a) constant pressure and (b) constant volume. 

Assuming that these quantities are independent of temperature, calculate the temperature at which the equilibrium constant for the hydrolysis of ATP becomes greater than 1. 

The pK for the autoprotolysis (more precisely, the autodeuterolysis, because a deuteron is being transferred) of heavy water (D20) is 15.136 at 20.°C and 13.8330 at 30.°C. Assuming AH° for this reaction to be independent of temperature, calculate A.Sr°for the autoprotolysis reaction. Suggest an interpretation of the sign. Suggest a reason why the autoprotolysis constant of heavy water differs from that of ordinary water. 

The hydrolysis of sucrose is a part of the digestive process. To investigate how strongly the rate depends on our body temperature, calculate the rate constant for the hydrolysis of sucrose at 35.0°C, given that k = 1.0 mL-mol -s 1 at 37.0°C (normal body temperature) and that the activation energy of the reaction is 108 kj-mol. ... 

Solution Aside from the temperature calculations, this example illustrates the systematic use of mass rather than molar concentrations for reactor... 

Fig. 55—Temperature calculation for different liquid films (glycerin and hexadecane) at different locations in the contact region Area A is the central area in the inset photo and area B is the edge area. The filled histogram represents the positive EEF intensity of 518.6 kV/cm, and the empty one of 667.7 V/cm. The solid (glycerin) and dotted (hexadecane) lines are variation curves of the boiling point along the radial direction in the contact region.

Although molecular bromine is a liquid at 298 K, 1 bar, it has a significant vapor pressure at this temperature. Calculate this vapor pressure in torr. 

Table 3.3. Equilibrium constants for the dissociation of H2, N2 and O2 and the partial pressures of the atoms at different temperatures calculated from fundamental data given in Tab. 3.4.

Fig. 4.3. (A) Composite multispecies benthic foraminiferal Mg/Ca records from three deep-sea sites DSDP Site 573, ODP Site 926, and ODP Site 689. (B) Species-adjusted Mg/Ca data. Error bars represent standard deviations of the means where more than one species was present in a sample. The smoothed curve through the data represents a 15% weighted average. (C) Mg temperature record obtained by applying a Mg calibration to the record in (B). Broken line indicates temperatures calculated from the record assuming an ice-free world. Blue areas indicate periods of substantial ice-sheet growth determined from the S 0 record in conjunction with the Mg temperature. (D) Cenozoic composite benthic foraminiferal S 0 record based on Atlantic cores and normalized to Cibicidoides spp. Vertical dashed line indicates probable existence of ice sheets as estimated by (2). 3w, seawater S 0. (E) Estimated variation in 8 0 composition of seawater, a measure of global ice volume, calculated by substituting Mg temperatures and benthic 8 0 data into the 8 0 paleotemperature equation (Lear et al., 2000).

Equation (5) is equivalent to stating that sublimation and subsequent transport of 1 g of water vapor into the chamber demands a heat input of 650 cal (2720 J) from the shelves. The vial heat transfer coefficient, Kv, depends upon the chamber pressure, Pc and the vapor pressure of ice, P0, depends in exponential fashion upon the product temperature, Tp. With a knowledge of the mass transfer coefficients, Rp and Rs, and the vial heat transfer coefficient, Kv, specification of the process control parameters, Pc and 7 , allows Eq. (5) to be solved for the product temperature, Tp. The product temperature, and therefore P0, are obviously determined by a number of factors, including the nature of the product and the extent of prior drying (i.e., the cake thickness) through Rp, the nature of the container through Kv, and the process control variables Pc and Ts. With the product temperature calculated, the sublimation rate is determined by Eq. (4). 

To elucidate the cause of the microwave-induced enhancement of the rate of this reaction in more detail the transformation of 2-t-butylphenol was performed at low temperatures (up to -176 °C). At temperatures below zero the reaction did not proceed under conventional conditions. When the reaction was performed under micro-wave conditions in this low temperature region, however, product formation was always detected (conversion ranged from 0.5 to 31.4%). It was assumed that the catalyst was superheated or selectively heated by microwaves to a temperature calculated to be more than 105-115 °C above the low bulk temperature. Limited heat transfer in the solidified reaction mixture caused superheating of the catalyst particles and this was responsible for initiation of the reaction even at very low temperatures. If superheating of the catalyst was eliminated by the use of a nonpolar solvent, no reaction products were detected at temperatures below zero (see also Sect. 10.3.3). 

Figure 3. Heat of pyrolysis H per mass of volatiles at various pyrolysis temperatures, calculated with Equation 2.

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