Energy balance expressions

Alternatively, the E-Z Solve software may be used to integrate simultaneously the material- and energy-balance expressions and solve the equation of state. 

When the heat absorbed or released by reaction can markedly change the temperature of the reacting fluid, this factor must be accounted for in design. Thus we need to use both the material and energy balance expressions, Eqs. 4.1 and 4.2, rather than the material balance alone, which was the starting point of all the analyses of isothermal operations of Chapters 5 through 8. 

The last term on the left-hand side of eq. (3.301) corresponds to the heat transfer to the external fixed-bed wall. The overall heat transfer resistance is the sum of the internal, external, and wall resistances. In an adiabatic operation, the overall heat transfer coefficient is zero so the corresponding term in the energy balance expression drops out, while in an isothermal operation this coefficient is infinite, so that 7 f 7 s 7W. 

The thermal energy balance, expressed per unit polymer mass, may be written as ... 

Write the material- and energy-balance expressions for the reactor. This problem must be solved by simultaneous solution of the material- and energy-balance relationships that describe the reacting system. Since the reactor is well insulated and an exothermic reaction is taking place, the fluid in the reactor will heat up, causing the reaction to take place at some temperature other than where the reaction rate constant and heat of reaction are known. 

So far we have considered surfaces subjected to single inode heat transfer, such as the specified heat flux, convection, or radiation for simplicity. In general, however, a surface may involve convection, radiation, and specified heat flux simultaneously. The boundary condition in such cases is again obtained from a surface energy balance, expressed as... 

The quantities h, f,. e, and do not change v/ith time, and thus we do not need to use the superscript / for them. Also, the energy balance expressions are simplified using the definitions of thermal diffusivity a = A/pCp and the dimensionless mesh Fourier number t = aUtiP, where Ax = Ay - I. 

Using the energy balance, express T at the exit of the first reactor in terms of the feed conditions and Na-... 

The Energy Balance Since the system is assumed to operate adiabatically, an adiabatic energy balance may be written over the reactor system, and an expression for temperature in terms of species concentrations x and y may then be obtained. This expression may be substituted into the temperature dependent rate constants, found in Equation 7.12. Due to the nonlinear nature of both the kinetics and energy balance terms, the final kinetic expression is typically a nonlinear function. For the purposes of demonstration, we shall assume that the resulting energy balance expression, after... 

The expression for energy balance is taken from Bird et al. (see Table 18,3-1 of Bird et al., 1960). These authors provide the details of the derivation of the energy-balance expression... 

For an isothermal system the simultaneous solution of equations 30 and 31, subject to the boundary conditions imposed on the column, provides the expressions for the concentration profiles in both phases. If the system is nonisotherm a1, an energy balance is also required and since, in... 

When the basic physical laws are expressed in this form, the formulation is greatly facilitated. These expressions are quite often given the names, material balance, energy balance, and so forth. To be a little more specific, one could write the law of conservation of energy in the steady state as... 

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... 

The continuity equation gives V2 = V AJa, and Vj = Q/A. The pressure drop measured by the manometer is pi —p2= (p — p)gA . Substituting these relations into the energy balance and rearranging, the desired expression for the flow rate is found. 

Develop via mathematical expressions a valid process or equipment model that relates the input-output variables of the process and associated coefficients. Include both equality and inequality constraints. Use well-known physical principles (mass balances, energy balances), empirical relations, implicit concepts, and external restrictions. Identify the independent and dependent variables (number of degrees of freedom). 

A similar statement, or equation, can be used to express the energy balance ... 

Energy balances differ from mass balances in that the total mass is known but the total energy of a component is difficult to express. Consequently, the heat energy of a material is usually expressed relative to its standard state at a given temperature. For example, the heat content, or enthalpy, of steam is expressed relative to liquid water at 273 K (0°C) at a pressure equal to its own vapor pressure. 

Equation 6-10 is the macroscopic energy balance equation, in which potential and kinetic energy terms are neglected. From tliermodynamics, the enthalpy per unit mass is expressed as... 

The energy balance for the bioreactor is shown by the following equation. As the fermenter is used batch wise, the heat balance is mathematically expressed as stated in (4.4.1) ... 

Since in the energy balance equation, the kinetic energy per unit mass is expressed as a2/2a, hence a = 0.5 for the streamline flow of a fluid in a round pipe. 

In considering the flow in a pipe, the differential form of the general energy balance equation 2.54 are used, and the friction term 8F will be written in terms of the energy dissipated per unit mass of fluid for flow through a length d/ of pipe. In the first instance, isothermal flow of an ideal gas is considered and the flowrate is expressed as a function of upstream and downstream pressures. Non-isothermal and adiabatic flow are discussed later. 

The value of the integral in the energy balance (equation 11.55) is again given by equation 11.60 [substituting (6S - 8o) for 0 ]. The heat flux q0 at the surface is now constant, and the right-hand side of equation 11.55 may be expressed as (—qa/Cf,p). Thus, for constant surface heat flux, equation 11.55 becomes ... 

For reactions involving heat effects, the total and component mass balance equations must be coupled with a reactor energy balance equation. Neglecting work done by the system on the surroundings, the energy balance is expressed by... 

When the process occurs nonisothermally in the reactor, energy balances must be used in conjunction with material balances. The general expression for energy balance is ... 

The energy balance of the PCM from the phase change beginning to its end can be expressed as... 

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