Convection energy equation

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 temperature field is obtained by solving the convective energy equation ... 

In the finite element solution of the energy equation it is sometimes necessary to impose heat transfer across a section of the domain wall as a boundary condition in the process model. This type of convection (Robins) boundary condition is given as... 

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 convective diffusion equations for mass and energy are given detailed treatments in most texts on transport phenomena. The classic reference is... 

Solution The problem requires solution of the convective diffusion equation for mass but not for energy. Rewriting Equation (8.71) in dimensionless form gives... 

In this equation, the superscript ( ) indicates that a term is computed based upon the most recent information, which complies with the ( + l)th time level when all iterative loops have converged. Further, the convective transport and viscous generation of fluctuating kinetic energy have been collected in the explicit term D. The iterative solution procedure for the granular energy equations continues until the convergence criteria... 

There are three ways in which energy can be transported outwards in a star radiative transfer, convection and conduction. The mean free path for radiation, k = /tcp — 1 cm, greatly exceeds that for conduction (k <radiative transfer dominates in those regions where the corresponding temperature gradient is stable to convection. The equation of... 

In the first approach it is assumed, as well, that the reaction proceeds by zero-order. Since the rate term d> is not a function of concentration, the continuity equation is not required so we can deal with the more convenient energy equation. Semenov, like Mallard and Le Chatelier, examined the thermal wave as if it were made up of two parts. The unbumed gas part is a zone of no chemical reaction, and the reaction part is the zone in which the reaction and diffusion terms dominate and the convective term can be ignored. Thus, in the first zone (I), the energy equation reduces to... 

Thermal/structural response models are related to field models in that they numerically solve the conservation of energy equation, though only in solid elements. Finite difference and finite element schemes are most often employed. A solid region is divided into elements in much the same way that the field models divide a compartment into regions. Several types of surface boundary conditions are available adiabatic, convection/radiation, constant flux, or constant temperature. Many ofthese models allow for temperature and spatially dependent material properties. 

Since all properties have been assumed constant in Eqs. (1-1), (1-38), and (1-47), and the solute concentration has been assumed small, the Navier-Stokes equation may be solved independently of the species continuity and energy equations. We treat only one exception where the velocity field is considered to be affected by heat or mass transfer. This exception, natural convection, is covered in Chapter 10. 

If one assumes that the resin has a constant density and thermal conductivity, an energy equation for the IP process can be obtained by simplifying Equation 5.41. First by using the mass balance equation (i.e., V (Ur) — 0) one of the convective heat transfer terms in the resin phase can be neglected. Another term can be neglected by setting V (Uf ) = 0. This assumption is justified since in the IP process der/dt = 0. 

The algebraic equations for the orthogonal collocation model consist of the axial boundary conditions along with the continuity equation solved at the interior collocation points and at the end of the bed. This latter equation is algebraic since the time derivative for the gas temperature can be replaced with the algebraic expression obtained from the energy balance for the gas. Of these, the boundary conditions for the mass balances and for the energy equation for the thermal well can be solved explicitly for the concentrations and thermal well temperatures at the axial boundary points as linear expressions of the conditions at the interior collocation points. The set of four boundary conditions for the gas and catalyst temperatures are coupled and are nonlinear due to the convective term in the inlet boundary condition for the gas phase. After a Taylor series expansion of this term around the steady-state inlet gas temperature, gas velocity, and inlet concentrations, the system of four equations is solved for the gas and catalyst temperatures at the boundary points. 

One important purpose of the energy equation is to describe and predict the fluid temperature fields. The energy equation will be closely coupled to the Navier-Stokes equations, which describe the velocity fields. The coupling comes through the convective terms in the substantial derivative, which, of course, involve the velocities. The Navier-Stokes equations are also coupled to the energy equation, since the density and other properties usually depend on temperature. Chemical reaction and molecular transport of chemical species can also have a major influence on the thermal energy of a flow. 

The time derivative of the displacement vector r is the velocity V, which, of course, assumes that the fluid system is moving with the fluid velocity. The left-hand side of the energy equation now represents the convective transport, and it remains to develop the heat-transfer and work terms on the right-hand side... 

With the convective derivatives eliminated and the properties constant, the thermal-energy equation is completely decoupled from the system. Moreover the energy equation is a simple, linear, parabolic, partial differential equation. 

Assume that the fluid temperatrue far from the sphere is different than the sphere s temperature. Under what circumstances is it reasonable to assume that the convective terms in the energy equation may be neglected Assuming the convective terms may be neglected, determine the temperatrue distribution in the fluid around the sphere. 

This relationship is captured in differential-equation form as Eq. 7.60. Since the momentum and energy equations (Eqs. 7.59 and 7.62) explicitly involve r2, the radial coordinate has become a dependent variable, not an independent variable. A consequence of the Von Mises transformation is that the radial velocity v is removed as a dependent variable and the radial convective terms are eliminated, which is a bit of a simplification. However, the fact that the group of dependent variables pur2 appear within the diffusion terms is a bit of a complication. The factor pur2 plays the role of an apparent variable diffusion coefficient. ... 

When solving for the energy equation an implicit FDM was used with a backward (up-winded) difference representation of the convective term. The viscous dissipation term was evaluated with velocity components from the previous time step. The equation of motion was integrated using a trapezoidal quadrature. Stevenson tested his model by comparing it to actual mold filling experiments of a disc with an ABS polymer. Table 8.8 presents data used for the calculations. 

There are many ways of solving the energy equation with convection effects. One that will be presented here is the widely accepted streamline upwind Petrov-Galerkin method... 

Write a program to solve by means of RFM the equation of motion and using the velocity field, calculate viscous dissipation and solve for the energy equation. Neglect inertial and convective effects. Consider T0=200°C, Ti=150°C, /x=24000 Pa-s, k=0.267 W/mK, i o=0.1 m, i i=0.13 m, k=0.769, cc=0.496 rad/s. Compare the numerical results with the analytical solution. Hint The couette flow is constant along the angular direction, hence, it is no necessary to use the whole domain. 

Omar Estrada, Ivan Lopez, Carlos Roldan, Maria del Pilar Noriega, and Whady Florez. Solution of steady and transient 2D-energy equation including convection and viscous dissipation effects using radial basis function interpolation. Journal of Applied Numerical Mathematics, 2005. 

This simplification allows an analytical solution of the one-dimensional heat conduction energy equation. By substituting Equation 3.11 into Equation 3.10, and assuming that the total heat-transfer coefficient (hT) is equal to the sum of the convective heat-transfer coefficient (hc) and the radiative heat-transfer coefficient (hT), the following expression (Equation 3.12) defines the net heat flux (q") at the surface of the solid fuel sample. 

In this paper it is shown that the rate of deposition of Brownian particles on the collector can be calculated by solving the convective diffusion equation subject to a virtual first order chemical reaction as a boundary condition at the surface. The boundary condition concentrates the surface-particle interaction forces. When the interaction potential between the particle and the collector experiences a sufficiently high maximum (see f ig. 2) the apparent rate constant of the boundary condition has the Arrhenius form. Equations for the apparent activation energy and the apparent frequency factor are established for this case as functions of Hamaker s constant, dielectric constant, ionic strength, surface potentials and particle radius. The rate... 

The equations for convection are the continuity or conservation of mass equation, the momentum equations and the energy equation. From the dimensionless equation of energy, useful dimensionless numbers are obtained. 

The momentum and energy equations are very difficult to solve except for simple cases. For many cases of practical interest, the convective heat transfer is studied experimentally and the results are presented in the form of empirical equations that relate dimensionless numbers. 

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... 

In the last section, convection in a two-dimensional porous medium is presented as a physical problem. Porous media is important in environmental heat transfer studies, transpiration cooling, and fuel cells, as some examples. Using the slug flow assumption, the energy equation is solved using an alternating implicit method to show its effectiveness. 

The equation is linear if u is a known function. This may be considered to describe a time-dependent one-dimensional heat convection equation for a problem with a known flow field. For a fluid with constant properties in the temperature range considered, the momentum equation is decoupled from the energy equation. In the following example, the... 

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