Equation energy

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  

U is the specific internal energy, thus /2pv + pU is the total energy per unit volume as the sum of internal and kinetic energies. Further, q is the heat flux relative to the motion, x is the momentum flux tensor. The enthalpy can be introduced by the relation U = H — pV = H — pjp. 

There are several important special cases of the energy equation. If there is no kinetic energy, v = 0, the velocity becomes the null vector. Equation (19.2) then reduces at constant pressure and density to 

Relating the unsteady energy term (or advected enthalpy rate) with the firepower gives the second II group commonly referred to as Q (or the Zukoski number for Professor Edward Zukoski, who popularized its use in fire)  

This length scale can be associated with flame height and the size of large-scale turbulent eddies in fire plumes. It is a natural length scale for fires. 

This group is difficult to preserve for smoke and fire in scaling, and can be troublesome. An alternate empirical radiation loss to the ambient used for unconfined fires produces 

The heat transfer to the walls or other solid surfaces takes a parallel path from the gas phase as radiation and convection to conduction through the wall thickness, 5W. This wall heat flow rate can be expressed as 

Pi heat transfer groups can now be produced from these relationships. Let us consider these in terms of Q s, i.e. Zukoski numbers for heat transfer. Normalizing with the advection of enthalpy, the conduction group is 

Because flowing viscous liquids can generate heat, we also need to consider the energy balance. In a manner similar to that used with the mass and momentum conservation relations, we can write a balance for the rate of change of internal energy over a control volume. This integral balance can be converted to a difierential balance (Bird et al., 1987, p. 9) giving 

Here U is the internal energy per unit mass and q is the conductive energy flux. 

It is more convenient to express internal energy in terms of temperature because it can readily be measured for an incompressible material with constant conductivity and no chemical reaction 

Many non-Newtonian materials also have very high viscosity, with the result that the viscous dissipation term T D can become significant. This is illustrated in Example 2.6.1. Solution of such problems is complicated by the fact that viscosity also depends on temperature, and thus shear heating can change the velocity profile. Then the energy and momentum equations are coupled through the temperature-dependent viscosity. 

The temperature dependence of viscosity can often be as important as its shear rate dependence for nonisothermal processing problems (e.g.. Tanner, 1985). For all liquids, viscosity decreases with increasing temperature and decreasing pressure. A useful empirical model for both effects on the limiting low shear rate viscosity is 

By scalar multiplication by v of both sides of equation [3.57], we obtain the kinetic energy balance equation  

By expressing the scalar product F v as a function of the fields and the polarizations, the electromagnetic energy is introduced as  

The balance equation for the sum of the kinetic energy and the electromagnetic energy is thus written as  

The quantities on the right-hand side bearing the exponent ( ) are local quantities obtained by following the fluid s motion. The expressions of these quantities are, in the non-relativistic approximation  

We know that the total energy is conserved, so we have  

For an arbitrary volume t of the multicomponent continuum, the first law of thermodynamics states that 

Rate of increase of (internal plus kinetic) energy — rate at which work is done on T (by body forces plus surface stresses) + rate of inward transport of heat by radiation, thermal conduction, and other transport process through the surface a enclosing i + rate of generation of energy through production of species within i + rate at which work is done on material produced within i. 

Let denote the absolute internal energy of species K per unit mass of species K and let u denote the absolute internal energy per unit mass of mixture. Then 

For the mass of species K, which is generated by chemical reaction in unit volume per unit time, the sum of (1) the internal and kinetic energy carried by this mass, and (2) the work done on this mass in unit time, is w (rj -h where rj is the average specific enthalpy of generated 

The first law of thermodynamics puts forward the principle of conservation of ener. Written for a general open system (where flow of material in and out of the system can occur) it is 

Flow of internal, kinetic, and potential energy into system by convection or diflhsion 

Example 2.6. The CSTR system of Example 2.3 will be considered again, this time with a cooling coil inside the tank that can remove the exothermic heat of reaction 2 (Btu/lb. mol of A reacted or cal/g mol of A reacted). We use the normal convention that 2 is negative for an exothermic reaction and positive for an endothermic reaction. The rate of heat generation (energy per time) due to reaction is the rate of consumption of A times 2. 

The rate of heat removal from the reaction mass to the cooling coil is -Q (energy per time). The temperature of the feed stream is Tq and the temperature in the reactor is T (°R or K). Writing Eq. (2.18) for this system, 

W = shaft work done by system (energy per time) 

The analytical expression of the first law of thermodynamics, subject to the fundamental postulates of independence and conservation, is therefore 

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Guffey and Wehe (1972) used excess Gibbs energy equations proposed by Renon (1968a, 1968b) and Blac)c (1959) to calculate multicomponent LLE. They concluded that prediction of ternary data from binary data is not reliable, but that quarternary LLE can be predicted from accurate ternary representations. Here, we carry these results a step further we outline a systematic procedure for determining binary parameters which are suitable for multicomponent LLE. 

The themiodynamic properties calculated by different routes are different, since the MS solution is an approximation. The osmotic coefficient from the virial pressure, compressibility and energy equations are not the same. Of these, the energy equation is the most accurate by comparison with computer simulations of Card and Valleau [ ]. The osmotic coefficients from the virial and compressibility equations are... 

The osmotic coefficients from the HNC approximation were calculated from the virial and compressibility equations the discrepancy between ([ly and ((ij is a measure of the accuracy of the approximation. The osmotic coefficients calculated via the energy equation in the MS approximation are comparable in accuracy to the HNC approximation for low valence electrolytes. Figure A2.3.15 shows deviations from the Debye-Htickel limiting law for the energy and osmotic coefficient of a 2-2 RPM electrolyte according to several theories. 

By multiplying this result by a factor of-2, and adding the result to the conservation of energy equation, one easily finds g = gj = v j - vj. This result, taken together widi conservation of angular momentum, x.gb =... 

Th c total en ergy of th e system. called the Hamilton iari, is ih c sum of th e kin elic an d poten tial energies (equation 24). 

Classical mcchan ics involves studying motion (trajectories) on the poten lial surface vh ere the classical kin etic energy equates to th c temperature. The relationship is that the average k in etic cn ergy o f... 

VVe therefore return to the point-charge model for calculating electrostatic interactions. If sufficient point charges are used then all of the electric moments can be reproduced and the multipole interaction energy. Equation (4.30), is exactly equal to that calculated from the Coulomb summation. Equation (4.19). 

Assuming constant physical coefficients for simplicity, the steady-state energy equation is expressed as... 

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... 

Working equations of the streamline upwind (SU) scheme for the steady-state energy equation in Cartesian, polar and axisymmetric coordinate systems... 

Following the procedure described in in Chapter 3, Section 3 the streamlined-upwind weighted residual statement of the energy equation is formulated as... 

Similarly in an axisymmetric coordinate system the terms of stiffness and load matrices corresponding to the governing energy equation written as... 

The inconsistent streamline upwind scheme described in the last section is fonuulated in an ad hoc manner and does not correspond to a weighted residual statement in a strict sense. In tins seetion we consider the development of weighted residual schemes for the finite element solution of the energy equation. Using vector notation for simplicity the energy equation is written as... 

Temperature variations are found by the solution of the energy equation. I he finite element scheme used in this example is based on the implicit 0 time-stepping/continuous penalty scheme described in detail in Chapter 4, Section 5. 

Step 5 - the obtained velocity field is used to solve the energy equation (see Chapter 3, Section 3). 

Petrov-Galerkin scheme - to discretize the energy Equation (5.25) for the calculation of T. 

The comparison of flow conductivity coefficients obtained from Equation (5.76) with their counterparts, found assuming flat boundary surfaces in a thin-layer flow, provides a quantitative estimate for the error involved in ignoring the cui"vature of the layer. For highly viscous flows, the derived pressure potential equation should be solved in conjunction with an energy equation, obtained using an asymptotic expansion similar to the outlined procedure. This derivation is routine and to avoid repetition is not given here. 

ENERGY. Calculates members of the elemental stiffness matrix corresponding to the energy equations. 

PUTBCT Inserts the pi-escribed temperature boundary values at the allocated place in the vector of unknowns for the energy equation. 

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