Energy balance equations equation

In general, when designing a batch reactor, it will be necessary to solve simultaneously one form of the material balance equation and one form of the energy balance equation (equations 10.2.1 and 10.2.5 or equations derived therefrom). Since the reaction rate depends both on temperature and extent of reaction, closed form solutions can be obtained only when the system is isothermal. One must normally employ numerical methods of solution when dealing with nonisothermal systems. 

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

Starting with the open system balance equation, derive the steady-state mechanical energy balance equation (Equation 7.7-2) for an incompressible fluid and simplify the equation further to derive the Bernoulli equation. List all the assumptions made in the derivation of the latter equation. 

We first note that the conditions of validity of the simplified closed system energy balance equation, Equation 11.3-13, are all satisfied verify)-, moreover, since the system has constant volume and the energy input due to the stirrer is presumed negligible, W = 0. The equation therefore becomes... 

The initial conditions for the energy balance equations. Equations 6.70 and 6.71, are... 

Conditions shown in Figure 18.6 represent a base case. The mechanical energy balance equation. Equation Q8.8T written between Points 1 and 2 in Figure 18.6. gives... 

Derivations for the stress tensor and energy flux vector that appear in the momentum and energy balance equations (Equations 4.3 and 4.4) can be found in Chapman and Cowling (1970, Section 16.41). The stress tensor is... 

Equations-Oriented Simulators. In contrast to the sequential-modular simulators that handle the calculations of each unit operation as an iaput—output module, the equations-oriented simulators treat all the material and energy balance equations that arise ia all the unit operations of the process dow sheet as one set of simultaneous equations. In some cases, the physical properties estimation equations also are iacluded as additional equations ia this set of simultaneous equations. 

Isothermal Gas Flow in Pipes and Channels Isothermal compressible flow is often encountered in long transport lines, where there is sufficient heat transfer to maintain constant temperature. Velocities and Mach numbers are usually small, yet compressibihty effects are important when the total pressure drop is a large fraction of the absolute pressure. For an ideal gas with p = pM. JKT, integration of the differential form of the momentum or mechanical energy balance equations, assuming a constant fric tion factor/over a length L of a channel of constant cross section and hydraulic diameter D, yields,... 

For each stage J, the following 2C -1- 3 component material-balance (M), phase-equilibrium (E), mole-fraction-summation (S), and energy-balance (H) equations apply, where C is the number of chemical species ... 

Compute a new set of values of the Vj tear variables one at a time, starting with V3, from an energy-balance equation that is obtained by combining Eqs. (13-72) and (13-74), ehminating Lj-i and Lj to give... 

Compute a new set of values of the T) tear variables by solving simultaneously the set of N energy-balance equations (13-72), which are nonlinear in the temperatures that determine the enthalpy values. When linearized by a Newton iterative procedure, a tridiagonal-matrix equation that is solved by the Thomas gorithm is obtained. If we set gj equal to Eq. (13-72), i.e., its residual, the hnearized equations to be solved simultaneously are... 

The initial inside-loop error from the solution of the normalized energy-balance equations, is found to be only 0,04624, This is reduced to 0,000401 after two iterations through the inner loop. 

Equation (13-160) is replaced by the following overall energy-balance equation if is to be specified rather than D ... 

Adsorption The design of gas-adsorption equipment is in many ways analogous to the design of gas-absorption equipment, with a solid adsorbent replacing the liqiiid solvent (see Secs. 16 and 19). Similarity is evident in the material- and energy-balance equations as well as in the methods employed to determine the column height. The final choice, as one would expect, rests with the overall process economics. 

This derivation indicates a strong coupling between the momentum equation and the energy equation, which implies that the momentum and energy balance equations should be solved as a coupled system. In particular, the dis-... 

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 from Equation 6-39 yields a heating system... 

In a similar manner, the energy balance equation can be determined ... 

Griffith derived a similar equation using an energy balance approach, equating stored energy with the energy required for crack propagation ... 

Equation (22) was obtained, essentially, with examination of the energy balance equation with respect to flows of gas-containing polymer melts. The key moment of this analysis is, in our view, comprehension of the fact that the energy of gas dissolved in the polymer is transformed into the energy of movement of the two-phase medium. 

The kinetic energy attributable to this velocity will be dissipated when the liquid enters the reservoir. The pressure drop may now be calculated from the energy balance equation and equation 3.19. For turbulent flow of an incompressible fluid ... 

If at time t the liquid level is D m above the bottom of the tank, then designating point 1 as the liquid level and point 2 as the pipe outlet, and applying the energy balance equation (2.67) for turbulent flow, then ... 

Then applying the energy balance equation between D and the liquid level in each of the tanks gives ... 

In order to obtain the kinetic energy term for use in the energy balance equation, it is necessary to obtain the average kinetic energy per unit mass in terms of the mean velocity. 

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. 

Methods have been given for the calculation of the pressure drop for the flow of an incompressible fluid and for a compressible fluid which behaves as an ideal gas. If the fluid is compressible and deviations from the ideal gas law are appreciable, one of the approximate equations of state, such as van der Waals equation, may be used in place of the law PV = nRT to give the relation between temperature, pressure, and volume. Alternatively, if the enthalpy of the gas is known over a range of temperature and pressure, the energy balance, equation 2.56, which involves a term representing the change in the enthalpy, may be employed ... 

Woou.ATT, E. Trans. Inst. Chem. Eng. 24 (1946) 17. Some aspects of chemical engineering thermodynamics with particular reference to the development and use of the steady flow energy balance equations. 

The energy balance equation can be applied between any two sections in a continuous fluid, If the fluid is not moving, the kinetic energy and the frictional loss are both zero, and therefore ... 

Considering a small filament of liquid which is brought to rest at section 2, and applying the energy balance equation between the two sections, since g Az, Wv, and F are all zero ... 

The work done by the pump is found by setting up an energy balance equation. If W, is the shaft work done by unit mass of fluid on the surroundings, then —Ws is the shaft work done on the fluid by the pump. 

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 ... 

Enthalpy-pressure diagram for water-steam 813 Enthalpy term, energy balance equation 74 Entropy 3,28... 

To describe the flow in a horizontal heated capillary we use the mass, momentum and energy balance equations. At moderate velocity, the effects due to compressibility of liquid and vapor, as well as energy dissipation in gaseous and liquid phases are negligible. Assuming that thermal conductivity and viscosity of the vapor and the liquid are independent of temperature and pressure, we arrive at the following system of equations ... 

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