Energy equation heat-transfer rate

An equation representing an energy balance on a combustion chamber of two surface zones, a heat sink Ai at temperature T, and a refractory surface A assumed radiatively adiabatic at Tr, inmost simply solved if the total enthalpy input H is expressed as rhCJYTv rh is the mass rate of fuel plus air and Tp is a pseudoadiabatic flame temperature based on a mean specific heat from base temperature up to the gas exit temperature Te rather than up to Tp/The heat transfer rate out of the gas is then H— — T ) or rhCp(T f — Te). The... 

For isothermal and adiabatic modes of operation the energy balance equations developed above will simplify so that the design calculations are not nearly as tedious as they are for the other modes of operation. In the case of adiabatic operation the heat transfer rate is zero, so equation 10.2.10 becomes... 

In the case of isothermal operation the material and energy balance equations are not coupled, and design equations like 10.2.1 can be solved readily, since the reaction rate can be expressed directly as a function of the fraction conversion. For operation in this mode, an energy balance can be used to determine how the heat transfer rate should be programmed to keep the system isothermal. For this case equation 10.2.12 simplifies to the following expression for the heat transfer rate... 

Eleat transfer occurs not only within the solid surface, droplet and vapor phases, but also at the liquid-solid and solid-vapor interface. Thus, the energy-balance equations for all phases and interfaces are solved to determine the heat-transfer rate and evaporation rate. 

In addition to the expression for the mass and energy fluxes, conservation equations for mass and energy are required to enable the calculation of concentration and temperature profiles. From these profiles the mass and heat transfer rates through the va-pour/gas-liquid interface can subsequently be obtained. The species conservation equations for the liquid and the vapour/gas phase are respectively given by... 

Just as the macroscopic mechanical energy equation is used to determine the relations between the various forms of mechanical energy and the frictional energy losses, so the thermal energy equation, expressed in macroscopic form, is used to determine the relation between the temperature and heat transfer rates for a flow system. 

The present book is concerned with methods of predicting heat transfer rates. These methods basically utilize the continuity and momentum equations to obtain the velocity field which is then used with the energy equation to obtain the temperature field from which the heat transfer rate can then be deduced. If the variation of fluid properties with temperature is significant, the continuity and momentum equations... 

The prediction of convective heat transfer rates will, however, always involve the solution of the energy equation. Therefore, because of its fundamental importance in the present work, a discussion of the way in which the energy equation is derived will be given here [2],[3],[5],[7]. For this purpose, attention will be restricted to two-dimensional, incompressible flow. 

A flow is completely defined if the values of the velocity vector, the pressure, and the temperature are known at every point in the flow. The distributions of these variables can be described by applying the principles of conservation of mass, momentum, and energy, these conservation principles leading to the continuity, the Navier-Stokes, and the energy equations, respectively. If the fluid properties can be assumed constant, which is very frequently an adequate assumption, the first two of these equations can be simultaneously solved to give the velocity vector and pressure distributions. The energy equation can then be solved to give the temperature distribution. Fourier s law can then be applied at the surface to get the heat transfer rates. 

It should be noted that when the enclosure contains a gas, the convective heat transfer rates can be low and radiant heat transfer can be significant. Some gases, such as carbon dioxide and water vapor, absorb and emit radiation and in such cases the energy equation has to be modified to account for this. However, even when the gas in the enclosure is transparent to radiation, there can be an interaction between the radiant and convective heat transfer. For example, for the case where the end walls can be assumed to be adiabatic, if grab and qKm are the rates at which radiant energy is being absorbed and emitted per unit wall area at any point on these end walls then the actual thermal boundary conditions on these walls are ... 

Consider a PFR operating at nonisothermal conditions (refer to Figure 9.4.1). To describe the reactor performance, the material balance. Equation (9.1.1), must be solved simultaneously with the energy balance. Equation (9.2.7). Assuming that the PFR is a tubular reactor of constant cross-sectional area and that T and C, do not vary over the radial direction of the tube, the heat transfer rate Q can be written for a differential section of reactor volume as (see Figure 9.4.1) ... 

Given any nonreactive process for which the required heal transfer Q or heat transfer rate Q is to be calculated, (a) draw and label the flowchart, including Q oi Q m the labeling (b) carry out a degree-of-freedom analysis (c) write the material and energy balances and other equations you would use to solve for all requested quantities (d) perform the calculations and (e) list the assumptions and approximations built into your calculations. 

Substitute for AH, AH, or AH in the appropriate energy balance equation to determine the required heat transfer, Q, or heat transfer rate, Q. (See Example 8.3-2.)... 

In the last example the temperatures of all input and output streams were specified, and the only unknown in the energy balance equation was the heat transfer rate required to achieve the specified conditions. You will also encounter problems in which the heat input is known but the temperature of an output stream is not. For these problems, you must evaluate the outlet stream component enthalpies in terms of the unknown T, substitute the resulting expressions in the energy balance equation, and solve for T. Example 8.3-6 illustrates this procedure. 

Once all of the 0, (or all / ,) values are determined in this manner and all of the i, (or all hi) values are determined from material balances, densities or equations of state, and phase equilibrium relations, calculate AC/, A/C, or A// and substitute the result in the energy balance to determine whichever variable is unknown (usually the heat. Q. or heat transfer rate, Q). 

Liquid metals such as mercury have high thermal conductivities, and are commonly used in applications that require high heat transfer rates. However, they have very small Prandtl numbers, and thus the thermal boundary layer develops much faster than the velocity boundary layer. Then we can assume the velocity in the thermal boundary layer to be constant at the free stream value and solve the energy equation. It gives... 

Nonuniform temperatures, or a temperature level different from that of the surroundings, are common in operating reactors. The temperature may be varied deliberately to achieve optimum rates of reaction, or high heats of reaction and limited heat-transfer rates may cause unintended nonisothermal conditions. Reactor design is usually sensitive to small temperature changes because of the exponential effect of temperature on the rate (the Arrhenius equation). The temperature profile, or history, in a reactor is established by an energy balance such as those presented in Chap. 3 for ideal batch and flow reactors. 

Equation 5.2.55 is a sinoplified dimensionless differential energy balance equation of steady-flow reactors, where each term is divided hy [(Ftot)oCpo ol, the reference thermal energy rate. The first term in the bracket of Eq. 5.2.55 represents the dimensionless heat-transfer rate and its relationship to the dimensionless driving force (6ir — 6), where... 

Low-Speed. Flow. Heat transfer is best found from the Reynolds analogy, the relationship between heat transfer and skin friction evaluated through analyses utilizing the empirical velocity distributions cited earlier. Knowledge of the flow field, which is independent of the temperature field when the fluid properties are constant, can be used directly to define the temperature field for a variety of thermal conditions and to evaluate the resulting convective heat transfer rates. For low-speed, constant-property flow, the energy equation is... 

The principles of conservation of momentum, energy, mass, and charge are used to define the state of a real-fluid system quantitatively. The conservation laws are applied, with the assumption that the fluid is a continuum. The conservation equations expressing these laws are, by themselves, insufficient to uniquely define the system, and statements on the material behavior are also required. Such statements are termed constitutive relations, examples of which are Newton s law that the stress in a fluid is proportional to the rate of strain, Fourier s law that the heat transfer rate is proportional to the temperature gradient. Pick s law that mass transfer is proportional to the concentration gradient, and Ohm s law that the current in a conducting medium is proportional to the applied electric field. 

The heat transfer rate in the reactor can be determined from the energy balance equation (10.3.6) ... 

Consider the heat transfer rate equation and possible improvements that can be done for improving energy recovery in HEN. Heat transfer rate (Q) is the product of overall heat... 

Eor now we are not making any considerations regarding the fluid-solid heat transfer rate, and therefore we leave the reaction rate as a function of T utf,o (calculated from Eqs. 3.27 and 3.34 with Co). In its current form. Equation 3.67 must be solved numerically with the energy balances in the two phases and the appropriate boimdary conditions from Table 3.1. For our purposes of estimating the deviation from the pseudo-homogeneous model, the magnitude of the 0(e) contribution is of interest The relative error associated with this model is estimated by... 

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