Heat transfer equation

Basic Heat-Transfer Equations. Consider a simple, single-pass, parallel-flow heat exchanger in which both hot (heating) and cold (heated) fluids are flowing in the same direction. The temperature profiles of the fluid streams in such a heat exchanger are shown in Figure 2a. 

Assuming that U, and are invariant with respect to temperature and space, one can integrate equation 14 subject to equation 19, and obtain, after rearrangement, a basic heat-transfer equation for a parallel-flow heat exchanger (4). 

A similar derivation can be made for a siagle-pass counterflow heat exchanger ia which the hot and cold fluids ate flowing ia the opposite direction (see Fig. 2b). The resultiag heat-transfer equation is still... 

For heat exchangers other than the parallel and counterflow types, the basic heat-transfer equations, and particularly the effective fluid-to-fluid temperature differences, become very complex (5). For simplicity, however, the basic heat-transfer equation for general flow arrangement may be written as... 

The expressions for S can be obtained usiag basic governing heat-transfer equations, such as equations 14 and 19 with proper substitutions of equation 36. For simple, siagle-pass, parallel- and counterflow heat exchangers, the foUowiag expressions result ... 

The walls of the oven have a thermal resistanee Rt and the oven has a thermal eapaeitanee Ct and operating temperature 6o(t). The heat transfer equation for the oven may be written... 

Start up of a jaeketed hateh reaetor requires eontrol of the heat-up and eool-down rates. This involves determining and setting the jaeket heat transfer fluid temperatures. An alternative is to make a trial heat-up and ineorporate the results into a time-dependent heat transfer equation ... 

L the shell-side fluid makes one pass from inlet to outlet. With a longitudinal baffle, and with the nozzles placed 180° around the shell, the shell-side fluid would be forced to enter at the left, flow to the right to get around the baffle, and flow to the left to reach the exit nozzle. This would be required to approximate true counter-current flow, which was assumed in the heat transfer equations of Chapter 2. 

In the basic heat transfer equation it is necessary to use the log mean temperature difference. In Equation 2-4 it was assumed that the two fluids are flowing counter-current to each other. Depending upon the configuration of the exchanger, this may not be true. That is, the way in which the fluid flows through the exchanger affects LMTD. The correction factor is a function of the number of tube passes and the number of shell passes. 

Cl = constant in heat transfer equation = 0.0025 (English units)... 

Figure 10-98. Levy correlation for boiling heat transfer equation. (Used by permission Levy, S. ASME paper no. 58-HT-8, 1958. American Society of Mechanical Engineers. All rights reserved.)...

The air-cooled heat transfer exchanger is like other exchangers in that the basic heat transfer equation must be statisfied ... 

Equation 10.66 is referred to as Fick s Second Law. This also applies when up is small, corresponding to conditions where C, is always low. This equation can be solved for a number of important boundary conditions, and it should be compared with the corresponding equation for unsteady state heat transfer (equation 9.29). 

Mass transfer from a single spherical drop to still air is controlled by molecular diffusion and. at low concentrations when bulk flow is negligible, the problem is analogous to that of heat transfer by conduction from a sphere, which is considered in Chapter 9, Section 9.3.4. Thus, for steady-state radial diffusion into a large expanse of stationary fluid in which the partial pressure falls off to zero over an infinite distance, the equation for mass transfer will take the same form as that for heat transfer (equation 9.26) ... 

For mass transfer, the equations corresponding to equations 12.137 and 12.138 are obtained in the same way as the analogous heat transfer equations. 

Sedov LI (1993) Similarity and dimensional methods in mechanics, 10th edn. CRC, Boca Raton Shah MM (1982) Chart correlation for saturated boding heat transfer equation and further study. ASHRAE Trans 88 185-196... 

At temperatures above ca. 1000 K, heat transfer via radiation becomes significant, that is, the heat transfer can occur by optical energy waves (photons) as well as conduction (phonons), with the heat transfer equation expressed by... 

It is the full wetted perimeter that determines the flow regime and the velocity gradients in a channel. So, in this book, de determined using the full wetted perimeter will be used for both pressure drop and heat transfer calculations. The actual area through which the heat is transferred should, of course, be used to determine the rate of heat transfer equation 12.1. 

Bypassing will affect the pressure drop only in the cross-flow zones. The correction factor is calculated from the equation used to calculate the bypass correction factor for heat transfer, equation 12.30, but with the following values for the constant a. 

At a later stage of bubble growth, heat diffusion effects are controlling (as point c in Fig. 2.9), and the solution to the coupled momentum and heat transfer equations leads to the asymptotic solutions and is closely approximated by the leading term of the Plesset-Zwick (1954) solution,... 

Shah, M. M., 1982, Chart Correlation for Saturated Boiling Heat Transfer Equations and Further Study, ASHRAE Trans. SS(Part I). (4)... 

As the pressure increases from low values, the pressure-dependent term in the denominator of Eq. (101) becomes significant, and the heat transfer is reduced from what is predicted from the free molecular flow heat transfer equation. Physically, this reduction in heat flow is a result of gas-gas collisions interfering with direct energy transfer between the gas molecules and the surfaces. If we use the heat conductivity parameters for water vapor and assume that the energy accommodation coefficient is unity, (aA0/X)dP — 150 I d cm- Thus, at a typical pressure for freeze drying of 0.1 torr, this term is unity at d 0.7 mm. Thus, gas-gas collisions reduce free molecular flow heat transfer by at least a factor of 2 for surfaces separated by less than 1 mm. Most heat transfer processes in freeze drying involve separation distances of at least a few tenths of a millimeter, so transition flow heat transfer is the most important mode of heat transfer through the gas. 

The numerical method used for solving the heat-transfer equation is similar to that for solving the momentum equation, which is a finite-volume, ALE method (Kashiwa et al., 1994). 

Single-pass counterflow heat exchanger, heat- transfer equation for, 13 251 Single-pass heat exchangers, heat-exchanger effectiveness for, 13 254 Single-photon emission computed... 

Values of B 0.021-0.027 have been reported and a mean value of 0.023 may be taken, which means that equation 12.24 very similar to the general heat transfer equation for forced convection in tubes (Volume 1, Chapter 9). The data shown in Figure 12.4 are... 

Heat from the air stream passes to the solid material during its fall through the air stream, and also from the hot walls of the shell, although the first mechanism is much the more important. The heat transfer equation may be written as ... 

This same equation holds whether or not there is mass evolution from the surface and whether or not convective effects prevail in the gaseous stream. Even for convective atmospheres in which one is interested in the heat transfer to a surface (without mass addition of any kind, i.e., the heat transfer situation generally encountered), one writes the heat transfer equation as... 

The average temperature is used in the heat transfer equation and to represent the enthalpy of jaeket material. Equation (3.27) beeomes... 

Conduction is the rate of heat transfer through a medium without mass transfer. The basic rate of conduction heat-transfer equation is Fourier s law ... 

Radiation is the rate of heat transfer by electromagnetic waves emitted by matter. Unlike conduction and convection, radiation does not require an intervening medium to propagate. The basic rate of radiation heat-transfer equation between a high temperature (Th) black body and a low temperature Tf) black body is Stefan-Boltzmann s law ... 

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