Energy equation forced convection

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... 

The Eckert number only enters the problem when the viscous dissipation term in the energy equation is significant. For moderate velocities, the viscous dissipation term may be neglected. Under such conditions, the forced convection is characterized by... 

The turbulence kinetic energy equation for forced convection was derived in this chapter. Rederive this turbulence kinetic energy equation bv starting with the momentum... 

The continuity equation (8.9) and the energy equation (8.12) are identical to those for forced convective flow. The x- and y-momentum equations, i.e., Eqs. (8.10) and (8.11), differ, however, from those for forced convective flow due to the presence of the buoyancy terms. The way in which these terms are derived was discussed in Chapter 1 when considering the application of dimensional analysis to convective heat transfer. In these buoyancy terms, is the angle that the x-axis makes to the vertical as shown in Fig. 8.3. 

Because an explicit finite-difference procedure is being used to solve the momentum and energy equations, the solution can become unstable, i.e., as the solution proceeds it can diverge increasingly from the actual solution. The analysis of the conditions under which such an instability will develop that was given in Chapter 4 for the case of forced convection in a duct essentially applies here and shows that in order to avoid instability, AZ should be selected so that ... 

Similarly, the energy equation gives as in forced convection ... 

This is the equation of motion for the free-convection boundary layer. Notice that the solution for the velocity profile demands a knowledge of the temperature distribution. The energy equation for the free-convection system is the same as that for a forced-convection system at low velocity ... 

Convection is the transfer of energy by conduction and radiation in moving, fluid media. The motion of the fluid is an essential part of convective heat transfer. A key step in calculating the rate of heat transfer by convection is the calculation of the heat-transfer coefficient. This section focuses on the estimation of heat-transfer coefficients for natural and forced convection. The conservation equations for mass, momentum, and energy, as presented in Sec. 6, can be used to calculate the rate of convective heat transfer. Our approach in this section is to rely on correlations. 

In this section we derive the equation of motion that governs the natural convection flow in laminar boundary layer. The conservation of mass and energy equations derived in Chapter 6 for forced convection are also applicable for natural convection, but tlie momentum equation needs to be modified to incorporate buoyancy. 

Kakag S., Y. Yener, 1973, Exact solution of the transient forced convection energy equation for time wise variation of inlet temperature, Int. J. Heat Mass Transfer 16, 2205-2214. 

The third chapter covers convective heat and mass transfer. The derivation of the mass, momentum and energy balance equations for pure fluids and multi-component mixtures are treated first, before the material laws are introduced and the partial differential equations for the velocity, temperature and concentration fields are derived. As typical applications we consider heat and mass transfer in flow over bodies and through channels, in packed and fluidised beds as well as free convection and the superposition of free and forced convection. Finally an introduction to heat transfer in compressible fluids is presented. 

The physical meaning of the terms in this equation can be inferred from the above modeling analysis. The term on the LHS denotes the rate of accumulation of internal and kinetic energy within the control volume per unit volume the first term on the RHS denotes the net rate of of internal and kinetic energy increase by convection per unit volume the second term on the RHS denotes the net rate of heat addition due to heat conduction, interdiffusion effects, Dufour effects and radiation per unit volume the third term on the RHS denotes the rate of work done on the fluid within the control volume by external body forces per unit volume the fourth term on the RHS denotes the rate of work done on the fluid within the control volume by the pressure forces per unit volume and the fifth term on the RHS denotes the rate of work done on the fluid within the control volume by the viscous forces per unit volume. 

Even with these simplifications, however, it is rarely possible to obtain analytic solutions for fluid mechanics or heat transfer problems. The Navier Stokes equation for an isothermal fluid is still nonlinear, as can be seen by examination of either (2 89) or (2 91). The Bousi-nesq equations involve a coupling between u and 6, introducing additional nonlinearities. It will be noted, however, that, provided the density can be taken as constant in the body-force term (thus neglecting any natural convection), the fluid mechanics problem is decoupled from the thermal problem in the sense that the equations of motion, (2 89) or (2-91), and continuity, (2-20), do not involve the temperature 0. The thermal energy equation, (2-93), is actually a linear equation in the unknown 6, once the Boussinesq approximation has been introduced. In that case, the only nonlinear term is dissipation, but this involves the product E E and can be treated simply as a source term that will be known once Eqs. (2-89) or (2 91) and (2 20) have been solved to determine the velocity. In spite of being linear, however, the velocity u appears as a coefficient (in the convective derivative term). Even when the form of u is known (either exactly or approximately), it is normally quite a complicated function, and this makes it extremely difficult to obtain analytic solutions for 0 even though the governing equation is linear. 

With appropriate simplifying assumptions, the energy equation for steady-state forced-convection heat-transfer may be written as... 

Because of the variation of air density with temperature, motion can arise solely as a result of buoyancy effects induced by temperature nonuniformities. Since the variation of density with temperature leads to the last term on the right-hand side of (16.18), the equations of motion and energy are not uncoupled, as in the case of forced convection. This situation is called, by contrast,/ree convection, since flow can arise without the imposition of external pressure gradients. 

The energy equation arises from a balance of energy inflow and outflow by convective transport, energy inflow and outflow by heat conduction, rate of work done by pressure forces, rate of work done by viscous forces, and rate of work done by external forces [1, p. 335]. The terms on the right-hand side of Eq. (19.2) describe the flows mentioned just before in this order ... 

Abstract Evaporation of multi-component liquid droplets is reviewed, and modeling approaches of various degrees of sophistication are discussed. First, the evaporation of a single droplet is considered from a general point of view by means of the conservation equations for mass, species and energy of the liquid and gas phases. Subsequently, additional assumptions and simplifications are discussed which lead to simpler evaporation models suitable for use in CFD spray calculations. In particular, the heat and mass transfer for forced and non-forced convection is expressed in terms of the Nusselt and Sherwood numbers. Finally, an evaporation model for sprays that is widely used in today s CFD codes is presented. 

Single-phase forced convection through microchaimels can be studied by determining the velocity distribution and the temperature field in the fluid region with the aid of the mass conservation principle continuity equation), the fluid momentum balance equations, and the energy balance equation [1] ... 

If the thermal energy equation is used in order to study laminar flow forced convection in microchannels, the expression of the thermal energy equation can be simplified. 

It is the first one that will be emphasized, and can be broken into conservation of mass and energy, which are coupled with Einstein s mass-energy equivalence (E=mc ). As such, the accumulation terms of the conservation of mass are not affected. Also, we could neglect forced convection effects in the system. The resulting mass diffusion equation would be similar to that in Eq. (1.5.2), except that a so-called elastic strain energy could be added to the potential function to take into account crystal lattice differences between solid phases (De Fontaine, 1967). 

Natural convection heat transfer occurs when a solid surface is in contact with a gas or liquid which is at a different temperature from the surface. Density differences in the ffuid arising from the heating process provide the buoyancy force required to move the ffuid. Free or natural convection is observed as a result of the motion of the fluid. An example of heat transfer by natural convection is a hot radiator used for heating a room. Cold air encountering the radiator is heated and rises in natural convection because of buoyancy forces. The theoretical derivation of equations for natural convection heat-transfer coefficients requires the solution of motion and energy equations. 

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