Thermal symmetry

The 27T-electrons of the carbon-nitrogen double bond of 1-azirines can participate in thermal symmetry-allowed [4 + 2] cycloadditions with a variety of substrates such as cyclo-pentadienones, isobenzofurans, triazines and tetrazines 71AHC(13)45). Cycloadditions also occur with heterocumulenes such as ketenes, ketenimines, isocyanates and carbon disulfide. It is also possible for the 27r-electrons of 1-azirines to participate in ene reactions 73HCA1351). 

Dithiadiazolyl radicals are typically prepared by reduction of the corresponding cations with SbPh3. They are unstable with respect to isomerization to the 1,2,3,5-isomers both in solution " and in the solid state. The isomerization is a photochemically symmetry-allowed process, which is thermally symmetry forbidden. A bimolecular head-to-tail rearrangement has been proposed to account for this isomerization (Scheme 11.1). This rearrangement process is conveniently monitored... 

Dipolar diazo compounds may undergo a thermal symmetry allowed [1,7]-, 87r-electrocycli-zation, followed by [1,5]- H shifts70 (cf. Section 4.1.3.1.1.2.1., p346ff). 

The 27i-electrons of the carbon-nitrogen double bond of 2H-azirines can participate in thermal symmetry-allowed [4-f2]-cycloadditions with a... 

A selective sampling of the photochemical cycloaddition and cyclization chemistry of 2H-azirines has been outlined in this chapter. Some photochemical sequences increase molecular complexity more than others, but each seems to provide complex heterocyclic structures in a very efficient manner. Indeed, many of these photoreactions rapidly construct hetero-polycyclic systems that are difficult to produce in other ways. In contrast to their photochemical behavior, the major thermal reaction of 2H-azirines generally involves C(2)-N bond cleavage to form vinyl nitrenes which further react by either insertion into an adjacent C-H bond or else undergo addition across a neighboring rc-bond. The 27i-electrons of the carbon-nitrogen double bond of 2H-azirines can also participate in thermal symmetry-allowed [4- -2]-cycloadditions with a variety of substrates. It is clear from the above discussion that the chemistry of 2H-azirines is both mechanistically complex and... 

The available experimental information does not warrant any definitive distinction between these four possibilities. However, some indirect evidence allows for some choices. Regardless of the 66 kcal/mol of strain embodied in the bicyclobutane ring, it has been shown to be a relatively stable system in the gas phase or in hydrocarbon solvents. Nevertheless, when bicyclobutane is heated at 200 C two of the peripheral C-C bonds are broken, while the central bond remains intact during the purely thermal, symmetry controlled reactions. This relative stability has been associated with the predominantly rr character of this... 

Assumptions 1 The egg is spherical in shape with a radius of r, = 2.5 cm. 2 Heat conduction in the egg is one dimensional because or thermal symmetry about the midpoint. 3 The thermal properties of the egg and the heat transfer coefficient are constant, 4 The Fourier number is t > 0,2 so that the one-term approximate solutions are applicable. 

Assumptions 1 Steady operating conditions exist since the temperature readings do not change with time. 2 Heat losses through the lateral surfaces of the apparatus are negligible since those surfaces are well insulated, and thus the entire heat generated by the heater is conducted through the samples. 3 The apparatus possesses thermal symmetry. 

Discussion Perhaps you are wondering if we really need to use two samples in the apparatus, since the measurements on the second sample do not give any additional information. It seems like we can replace the second sample by insulation. Indeed, we do not need the second sample however, it enables us to verify the temperature, measurements on the first sample and provides thermal symmetry, which reduces experimental error. 

Thermal symmetry boundary condition at the center plane of a plane wall. 

Some heat transfer problems possess thermal symmetry as a result of the symmetry in imposed thermal conditions. For example, the two surfaces of a large hot plate of thickness L suspended vertically in air is subjected to the same thermal conditions, and tlius the temperature disiribulion in one half of the plate is the same as that in the other half. That is, the heat transfer problem in this plate possesses thermal symmetry about the center plane at x = U2. Also, the direction of heat flow at any point in the plate is toward the surface closer to tlie point, and there is no heal flow across the center plane. Therefore, the center plane can be viewed as an insulated surface, and the thermal condition at this plane of symmetry can be expressed as (Fig. 2-31)... 

In the case of cylindrical (or spherical) bodies having thermal symmetry about the center line (or midpoint), the thermal symmetry boundary condition requires that the first derivative of Icmperature with respect to r (the radial variable) be zero at the centerline (or the midpoint). 

Assumptions 1 Heat transfer is steady since there is no change v/ith time. 2 Heat transfer is one dimensiopal since there Is thermal symmetry about the centerline and no variation in the axial direction, and thus T= T r). 3 Thermal conductivity is constant. 4 There is no heat generation. 

C What is a thermal symmetry boundary condition How is it expressed mathematically ... 

SOLUTION A spherical container filled with iced water is subjected to convection and radiation heat transfer at its outer surface. The rate of heat transfer ans the amount of ice that melts per day arc to be determined. Assumptions 1 Heal transfer is steady since the specified thermal conditions at the lioundaries do not change with time. 2 Heat transfer is one-dimensional since there is thermal symmetry a out the midpoint. 3 Thermal conductivity is constant. 

Consider a plane wall of thickness 2L, a long cylinder of radius r , and a sphere of radius r, initially at a nnifonn temperature T,-, as shown in Fig. 4—11. At time t = 0, each geometry is placed in a large medium that is at a constant temperature T and kepi in that medium for t > 0. Heat transfer lakes place between these bodies and their environments by convection with a uniform and constant heal transfer coefficient A. Note that all three ca.ses possess geometric and thermal symmetry the plane wall is symmetric about its center plane (,v = 0), the cylinder is symmetric about its centerline (r = 0), and the sphere is symmetric about its center point (r = 0). We neglect radiation heat transfer between these bodies and their surrounding surfaces, or incorporate the radiation effect into the convection heat transfer coefficient A. 

CoWsider a plane wall of thickness 2L initially at a uniform temperature of T , as shown in Fig. 4—1 In. At lime t = 0, the wall is immersed in a fluid at temperature 7 and is subjected to convection heal transfer from both sides with a convection coefficient of h. The height and the widlh of the wall are large relative to its thickness, and thus heat conduction in the wall can be approximated to be one-dimensional. Also, there is thermal symmetry about the inidplane passing through.x = 0, and thus the temperature distribution must be symmetrical about tlie midplane. Therefore, the value of temperature at any -.T value in - A "S. t 0 at any time t must be equal to the value at f-.r in 0 X Z, at the same time. This means we can formulate and solve the heat conduction problem in the positive half domain O x L, and then apply the solution to the other half. 

Under the conditions of constant thermophysical properties, no heat generation, thermal symmetry about the midplane, uniform initial temperature, and constant convection coefficient, the one-dimensional transient heat conduc-... 

SOLUTION A long cylindrical shaft is allowed to cool slowly. The center temperature and the heat transfer per unit length are to be determined. Assumptions 1 Heat conduction in the shaft is one-dimensional since it is long and it has thermal symmetry about the centerline. 2 The thermal properties of the shaft and the heat transfer coefficient are constant. 3 The Fourier number is t > 0.2 so that the one-term approximate solutions are applicable. Properties The properties of stainless steel 304 at room temperature are k - 14,9 W/m °C, p = 7900 kg/m r. = 477 J/kg X, and a = 3.95 X 10 mVs (Table A-3). More accurate results can be oblained by using properties at average temperature. 

When two large bodies /4 and R, initially at unifonn temperatures and Tgj are brought into contact, they instantly achieve temperature equality at the contact surface (temperature equality is achieved over the entire surface if the contact resistance is negligible). If the two bodies are of the same material with constant properties, thermal symmetry requires the contact surface temperature to be the arithmetic average, + Tbj)12 and to remain... 

Woodward-Hoffmann-rules auprafacial antarafacial /41. 46/ symmetry forbidden photochemical symmetry allowed thermal symmetry allowed thermal symmetry forbidden photochemical... 

In the last step of our formulation, we need the origin of the coordinate axis. Because of the geometric as well as the thermal symmetry of the problem, this origin is assumed to be on the midplane. Then, the first boundary condition is (Step 5)... 

Because of the thermal symmetry, we need to consider only one-eighth of the chimney (Fig. 4.15) which leads to... 

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