General Energy Balances

Calculate P from, x and n s from material balances T from energy balance (generally nonlinear in T)... 

The approach to constructing adiabatic ARs with temperature is slightly more complicated than isothermal constructions. The energy balance generally does not allow for temperature to obey a linear mixing law, and as a result temperature cannot generally be treated as a pseudo component in the state vector C (which is possible with residence time t). If temperature is to be incorporated, T must usually be introduced into the analysis via an energy balance and treated as an extra parameter in a rate expression, in the form r(C, T). 

There is a great variety in the modes of operation of semibatch reactors and transient systems. An energy balance general enough to satisfy all these possibilities becomes unwieldy and often difficult to solve. Several authors have recommended that the energy balance be derived on a case-by-case, system-by-system basis. 

The conservation of mass gives comparatively Httle useful information until it is combined with the results of the momentum and energy balances. Conservation of Momentum. The general equation for the conservation of momentum is... 

Generally, q is small because the outside area is not large in comparison to the amount of heat being transferred, and the energy balance can be simplified. In these conditions it is also convenient to write balances over a differential section of the column. These balances yield the following ... 

A further enhancement to the HRS process whereby the exhaust from a gas fired turbine is used to superheat steam from the HRS process is also possible (129). The superheated steam is then fed through a turbogenerator to produce additional electricity. This increases the efficiency of heat recovery of the turbine exhaust gas. With this arrangement, electric power generation of over 13.6 kW for 1 t/d (15 kW/STPD) is possible. Good general discussions on the sources of heat and the energy balance within a sulfuric acid plant are available (130,131). 

In the equation-oriented approach, the executive organizes the equations and controls a general-purpose equation solver. The equations for material and energy balances may be grouped separately from those for the calculation of physical properties or phase equiHbria, or as ia the design of some simulators, the distinction between these groups of equations may disappear completely. 

Material balances, often an energy balance, and occasionally a momentum balance are needed to describe an adsorption process. These are written in various forms depending on the specific application and desire for simplicity or rigor. Reasonably general material balances for various processes are given below. An energy balance is developed for a fixea bed for gas-phase application and simphfied for liquid-phase application. Momentum balances for pressure drop in packed beds are given in Sec. 6. 

The energy balance for a general fixed-bed process, ignoring dispersion, is... 

Nonisothermal hquid-phase processes may be driven by changes in feed temperature or heat addition or withdrawal through a column wall. For these, heats of adsorption and pressure effects are generally of less concern. For this case a suitable energy balance is... 

Using the values corresponding to the states 1 and 2, is the resistance height between 1 and 2, and py and p2 3re determined by the pressure loss and initial pressure. Other quantities are determined correspondingly. It is worth noting that there are no outflow losses in this case. Generally, it is wise to use energy balances in calculations. 

General equations of momentum and energy balance for dispersed two-phase flow were derived by Van Deemter and Van Der Laan (V2) by integration over a volume containing a large number of elements of the dispersed phase. A complete system of solutions of linearized Navier-Stokes equations... 

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. 

A reasonably general energy balance for a flow reactor can be written in English as... 

A specially built conductivity probe was used to investigate the gas holdup in a novel internal-loop airlift reactor. The gas holdup generally increases with increasing solid holdup due to increased flow resistance. A model based on energy balance was developed that can be used to predict the average gas holdup in this novel interal-loop airlift reactor. 

Referring to Fig. 4.7, the general energy balance for segment n with S components and R reactions is... 

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 ... 

Himmelbau (1995) or any of the general texts on material and energy balances listed at the end of Chapter 2. The Ponchon-Savarit graphical method used in the design of distillation columns, described in Volume 2, Chapter 11, is a further example of the application of the lever rule, and the use of enthalpy-concentration diagrams. 

---