Heat transfers Theoretical

Pioro, L, Mokry, S., 2011. Thermophysical properties at critical and supercritical conditions. In Behniloudi, A. (Ed.), Heat Transfer. Theoretical Analysis, Experimental Investigations and Industrial Systems. INTECH, Rijeka, Croatia, pp. 573—592. Free download from http // www.intechopen.com/books/heat-transfer-theoretical-analysis-experimental-investigations-and-industrial-systems/thermophysical-properties-at-critical-and-supercritical-pressuies. 

Example 6.4 The process in Fig. 6.2 is to have its hot utility supplied by a furnace. The theoretical flame temperature for combustion is 1800°C, and the acid dew point for the flue gas is 160°C. Ambient temperature is 10°C. Assume = 10°C for process-to-process heat transfer but = 30°C for flue-gas-to-process heat transfer. A high value for for flue-gas-to-process heat... 

There is ordinarily no measurable convection in cells of diameter less than about 4 mm (143). Theoretical arguments have been in general agreement with this work (151,191). Since most available cellular polymers have cell diameters smaller than 4 mm, convection heat transfer can be ignored with good justification. Studies of radiant heat transfer through cellular polymers have been made (143,151,191,196,197). 

Theoretically, controUed deposition of calcium carbonate scale can provide a film thick enough to protect, yet thin enough to allow adequate heat transfer. However, low temperature areas do not permit the development of sufficient scale for corrosion protection, and excessive scale forms in high temperature areas and interferes with heat transfer. Therefore, this approach is not used for industrial cooling systems. ControUed calcium carbonate deposition has been used successhiUy in some waterworks distribution systems where substantial temperature increases are not encountered. 

Between 1 s and 1 min specific contact time, conduction heat-transfer performance decreases theoretically as the 0.29 power of contact time. This is consistent with empirical data from several forms of indirect-heat dryers which show performance variation as the 0.4 power of rotational speed (21). In agitator-stirred and rotating indirect-heat dryers, specific contact time can be related to rotational speed provided that speed does not affect the physical properties of the material. To describe the mixing efficiency of various devices, the concept of a mixing parameter is employed. An ideal mixer has a parameter of 1. 

Natural convection occurs when a solid surface is in contact with a fluid of different temperature from the surface. Density differences provide the body force required to move the flmd. Theoretical analyses of natural convection require the simultaneous solution of the coupled equations of motion and energy. Details of theoretical studies are available in several general references (Brown and Marco, Introduction to Heat Transfer, 3d ed., McGraw-HiU, New York, 1958 and Jakob, Heat Transfer, Wiley, New York, vol. 1, 1949 vol. 2, 1957) but have generally been applied successfully to the simple case of a vertical plate. Solution of the motion and energy equations gives temperature and velocity fields from which heat-transfer coefficients may be derived. The general type of equation obtained is the so-called Nusselt equation hL I L p gp At cjl... 

Laminar Flow Normally, laminar flow occurs in closed ducts when Nrc < 2100 (based on equivalent diameter = 4 X free area -i-perimeter). Laminar-flow heat transfer has been subjected to extensive theoretical study. The energy equation has been solved for a variety of boundaiy conditions and geometrical configurations. However, true laminar-flow heat transfer veiy rarely occurs. Natural-convecdion effects are almost always present, so that the assumption that molecular conduction alone occurs is not vahd. Therefore, empirically derived equations are most rehable. 

Limiting Nusselt numbers for laminar flow in annuli have been calculated by Dwyer [Nucl. Set. Eng., 17, 336 (1963)]. In addition, theoretical analyses of laminar-flow heat transfer in concentric and eccentric annuh have been published by Reynolds, Lundberg, and McCuen [Jnt. J. Heat Ma.s.s Tran.sfer, 6, 483, 495 (1963)]. Lee fnt. J. Heat Ma.s.s Tran.sfer, 11,509 (1968)] presented an analysis of turbulent heat transfer in entrance regions of concentric annuh. Fully developed local Nusselt numbers were generally attained within a region of 30 equivalent diameters for 0.1 < Np < 30, lO < < 2 X 10, 1.01 <... 

The physical properties of the liquid, rather than those of the vapor, are used For determining the film coefficient for condensation. Nus-selt [2. Ver. Dt.sch. Ing., 60, 541, 569 (1916)] derived theoretical relationships for predicting the film coefficient of heat transfer for condensation of a pure saturated vapor. A number of simplifying assumptions were used in the derivation. 

Flame Temperature The heat released by the chemical reaction of fuel and oxidant heats the POC. Heat is transferred from the POC, primarily by radiation and convection, to the surroundings, and the resulting temperature in the reaction zone is the flame temperature. If there is no heat transfer to the surroundings, the flame temperature equals the theoretical, or adiabatic, flame temperature. 

QRA is fundamentally different from many other chemical engineering activities (e.g., chemistry, heat transfer, reaction kinetics) whose basic property data are theoretically deterministic. For example, the physical properties of a substance for a specific application can often be established experimentally. But some of the basic property data used to calculate risk estimates are probabilistic variables with no fixed values. Some of the key elements of risk, such as the statistically expected frequency of an accident and the statistically expected consequences of exposure to a toxic gas, must be determined using these probabilistic variables. QRA is an approach for estimating the risk of chemical operations using the probabilistic information. And it is a fundamentally different approach from those used in many other engineering activities because interpreting the results of a QRA requires an increased sensitivity to uncertainties that arise primarily from the probabilistic character of the data. 

N,n = Minimum theoretical stages at total reflux Q = Heat transferred, Btu/hr U - Overall heat transfer coefficient, Btu/hrfP"F u = Vapor velocity, ft/sec U d = Velocity under downcomer, ft/sec VD(js = Downcomer design velocity, GPM/fL Vioad = Column vapor load factor W = Condensate rate, Ibs/hr Xhk = Mol fraction of heavy key component Xlk = Mol fraction of the light key component a, = Relative volatility of component i versus the heavy key component... 

The mathematical principles of convective heat transfer are complex and outside the scope of this section. The problems are often so complicated that theoretical handling is difficult, and full use is made of empirical correlation formulas. These formulas often use different variables depending on the research methods. Inaccuracy in defining material characteristics, experimental errors, and geometric deviations produce noticeable deviations between correlation formulas and practice. Near the validity boundaries of the equations, or in certain unfavorable cases, the errors can be excessive. 

The maximum temperature difference that takes place in a heat exchanger is T /, - 7. A higher temperature difference cannot occur due to the second law [Pg.691]

The fin heat transfer is determined by using fin efficiency. The fin efficiency is calculated using a theoretical approach where the whole fin is considered to be at the same temperature as the fin base. The required parameters necessary to determine the fin efficiency are shown in Fig. 9.10. 

The fin efficiency is defined by the division of the actual by the theoretical heat transfer, i.e.. 

Vertical thermosiphon Capable of very high heat transfer rates. Compact simple piping required. Low residence time in heated zone. Not easily fouled. Good controllability. Maintenance and cleaning can be awkward. Additional column skirt required. Equivalent to theoretical plate only at high recycle. 

Flooded-bundle (kettle) Easy maintenance and cleaning. Convenient when heating medium is dirty. Equivalent to theoretical plate. Contains vapor disengaging space. Lower heat transfer rates. Extra piping and space required. High residence time in heated zone. Easily fouled. 

U = overall heat transfer coefficient based on theoretical single tube, h... 

Heat transfer in the furnace is mainly by radiation, from the incandescent particles in the flame and from hot radiating gases such as carbon dioxide and water vapor. The detailed theoretical prediction of overall radiation exchange is complicated by a number of factors such as carbon particle and dust distributions, and temperature variations in three-dimensional mixing. This is overcome by the use of simplified mathematical models or empirical relationships in various fields of application. 

D, W. Blair, CombustFlame 20 (1), 105—9 (1973) CA 78, 113515 (1973) A simple heat-transfer model is coupled with an Arrhenius-type pyrolysis law to study the effect of solid-state heat-transfer losses on burning rates of solid rocket-proplnt strands. Such heat-transfer losses materially affect the burning rates and also cause extinction phenomena similar to some that had been observed exptly. Strand diam and compn, adiabatic burning rate, and the heat-transfer film coeff at the strand surface are important variables. Results of theoretical analysis are applied to AP-based composite solid proplnts... 

Another contributing mechanism is the direct cooling of hot propellant surface by contact with the injected fluid. The fluid should cause the decomposing surface to reduce its pyrolysis rate to a point where combustion cannot be sustained. In addition, the presence of water on the surface would obstruct heat transfer from the gas-phase reaction zones to the solid surface, thus augmenting the cooling of the surface. Proponents of these two approaches have correlated the injection data on the basis of mass of fluid required per unit area of surface, but theoretical justifications for the use of this particular correlating parameter have not been presented. 

The experimental and theoretical work reported in the literature will be reviewed for each of the five major types of ga s-liquid-particle operation under the headings Mass transfer across gas-liquid interface mass transfer across liquid-solid interface holdup and axial dispersion of gas phase holdup and axial dispersion of liquid phase heat transfer reaction kinetics. 

Certain hydrodynamical problems, as well as mass-transfer problems in the presence of surface-active agents, have been investigated theoretically under steady-state conditions (L3, L4, L10, R9). However, if we take into account the fact that in gas-liquid dispersions, the nonstationary term must appear in the equation of mass- or heat-transfer, it becomes apparent that an exact analysis is possible if a mixing-contacting mechanism is adopted instead of a theoretical streamline flow around a single bubble sphere. 

These expressions give point values of h and Nux at x = x. It is seen that the coefficient decreases from a theoretical value of infinity at the top as the condensate film thickens. The mean value of the heat transfer coefficient over the whole surface, between r = 0 and jc = x is given by ... 

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