UNSTEADY-STATE MATERIAL AND ENERGY BALANCES

Your objectives in studying this section are to be able to  

Write down the macroscopic unsteady-state material and energy balances in words and in symbols. 

Solve single linear ordinary differential material or energy balance equations given the initial conditions. 

Take a word problem and translate it into a differential equation(s). 

Unsteady-State Material and Energy Balances Chap. 6 

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Using the usual assumptions, we obtain the following dimensionless unsteady-state material and energy balance equations for the dense (emulsion) phase of the bubbling fluidized bed ... 

Sec. 6.1 Unsteady-State Material and Energy Balances accumulation —... 

By its nature process control is concerned with the dynamic behaviour of systems. It is no longer sufficient to make the steady-state assumption. Material and energy balances for unsteady systems must include the accumulation terms so far omitted. Because of the extra mathematical complexity involved in a quantitatve treatment of control this section will, instead, concentrate on general concepts rather than detailed analysis of... 

Several other dehnitions need to be discussed here. Steady-state behavior pertains to the case where there is no variation in the process variables with respect to time. If the system is in equilibrium (at steady state), it can be described by algebraic equations, such as material and energy balances. Unsteady-state (dynamic) behavior occurs when the process variables change as a function of time. The required mathematical model for this case includes ordinary differential equations as well as algebraic relationships. 

The input and output terms of equation 1.5-1 may each have more than one contribution. The input of a species may be by convective (bulk) flow, by diffusion of some kind across the entry point(s), and by formation by chemical reaction(s) within the control volume. The output of a species may include consumption by reaction(s) within the control volume. There are also corresponding terms in the energy balance (e.g., generation or consumption of enthalpy by reaction), and in addition there is heat transfer (2), which does not involve material flow. The accumulation term on the right side of equation 1.5-1 is the net result of the inputs and outputs for steady-state operation, it is zero, and for unsteady-state operation, it is nonzero. 

In an unsteady-state energy balance the same principle applies. The accumulation of energy within a process where all the energy forms are considered including kinetic, potential, heat flow rates, enthalpies, and stirrer works may result in an increase in the thermal energy and a rise in temperature. Unsteady-state heat transfer involves the transfer of heat under conditions where the temperature changes with time. For the simple case of one-dimensional conduction in a solid slab, the accumulation of heat is a product of the mass and specific heat of the material and the increase in temperature where ... 

The question of whether the reactor operates at SI or S2 depends on how it is started up. In order to determine the effect of startup conditions, the unsteady-state energy and material balances must be solved simultaneously. This task is beyond the scope of this chapter. 

Discussion In order to answer this question with complete confidence, it would be necessary to simultaneously solve the unsteady-state energy and material balances for the reactor. However, a qualitative analysis can shed some light on the situation. 

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