The Unsteady-State Energy Balance

We begin by recalling the unsteady-state form of the energy balance developed in Chapter 11. 

We shall first concentrate on evaluating the change in the total energy of the system wrt time. dEny ldt. The total energy of the system is the sum of the products of specific energies, , (e.g., J/mol /) of the various species in the system volume and the number of moles. Nj (niol /) of that species  

In evaluating, we shall neglect changes in the potential and kinetic energies and substitute for the internal energy / in terms of the enthalpy //,  

We note the last term on the right-hand side of Equation (13-2) is just the total pressure times the total volume. i.e PV. and this tram is virtually always smaller than the other terms in Equation (13-2) and thus will be neglected. For brevity we shall write all summations as 

When no spatial variations are present in the system volume, and time variations in the product of the total pressure and volume [PV) are neglected, the energy balance, after substitution of Equation (13-2) into (11-9), gives 

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The unsteady-state energy balance for an open system that has n species, each entering and leaving the system at its respective molar flow rates Fi (moles of i per time) and with its respective energy (joules per mole of i), is... 

Up to now we have focused on the steady-state operation of nonisothermal reactors. In this section the unsteady-state energy balance wtU be developed and then applied to CSTRs, plug-flow reactors, and well-mixed batch and semibateh reactors. 

The mathematical model consists of the unsteady state energy balance in the dried and frozen regions ... 

Our next step is to set up the unsteady-state energy balance for the interval At. Let Ts = the steam temperature and T = the oil temperature ... 

Closure. After completing this chapter, the reader should be able to appi the unsteady-state energy balance to CSTRs, semibatch and batch reactor The reader should be able to discuss reactor safety using two examples on a case study of an explosion and the other the use of the ARSST to hel prevent explosions. Included in the reader s discussion should be how t start up a reactor so as not to exceed the practical stability liniit. After reac ing these examples, the reader should be able to describe how to operat reactors in a safe maimer for both single and multiple reactions. 

The Unsteady-State Energy Balance 591 Energy Balance on Batch Reactors 594... 

Let s analyze the response to a small, positive change in temperature, ST, as we did with point U in Figure 8-7. Both G(T) and R(T) increase. However, the increase in G(T), SG, now is smaller than the increase in R(J), SR. Therefore, R(J) is greater than G(7), and the unsteady-state energy balance requires that the temperature decrease toward the temperature of the original operating point, SI. 

The early two-node models were not widely used by people due to the lack of sufficient consideration of the complicated physiological phenomena of the human body. Gagge et al. [19] introduced a more complete two-node model for the entire body, which includes the unsteady-state energy balance equation for the entire human body and two energy balance equations for the skin node and core node, as listed in the following ... 

Another useful form of the energy balance for open systems is for unsteady-state conditions. Unsteady-state is important, for example, in start-up as the equipment warms up. In the case when the inlet and outlet streams stay constant with time, the unsteady-state energy balance becomes ... 

The temperature of the thermocouple 0, will also vary with time. It is required to determine the relationship between 0O and 0,. Writing an unsteady-state energy balance for the junction ... 

Each segment is further divided into four subsection skin, fat, muscle and core. Unlike the distributed parameter model, each subsection is assumed to have spatially uniform temperature, which is given by an unsteady-state energy-balance equation ... 

Once the number of compartments has been specified, the analysis consists of applying unsteady-state energy balance equations to each compartment. Upon substitution of suitable numerical values for the various parameters and heat flux terms, the set of coupled, nonlinear ordinary differential equations is solved numerically. 

The coefficient of dNi/dt in the summation of (6-4) is defined as the partial molar internal energy of species i, because differentiation with respect to mole numbers of component i is performed at constant T, p and mole numbers of all other species in the mixture. The unsteady-state mass balance for species i in a batch reactor,... 

With these in mind, we multiply by delta t in order to obtain the following, unsteady state energy balance. 

Process modeling is described in detail in the books by Denn (1986) and by Hangos and Cameron (2001). In particular. Derm (1986) shows that volume balance, used in some industries, is satisfactory for many liquid systems. He highlights a common mistake in unsteady-state energy balance when heat capacity is not constant, and lists the published articles with the incorrect energy balance. 

This equation gives the general unsteady-state energy balance equation in an open system. It is often referred to as the form of the first law for open systems. Below we consider a number of special cases. 

Balakrishnan and Edgar (2000) evaluated gain-scheduled control of a commercial RTF reactor. They determined that a FID controller based on a semi-empirical model of the heating process provided effective temperature control of the reactor. Derivation of a fundamental heat transfer model based on an unsteady-state energy balance yielded an approximate second-order transfer function with wafer heating time constant T and heating lamp time constant << t, ). 

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. 

Solution to Example 6.3. This program calculates and plots % the temperature profiles of a furnace wall by solving the % two-dimensional unsteady-state energy balance equation using % the function PARABOLIC2D.M. 

Unsteady-State Analysis For the choice of system shown in Figure E2.6A, we must use an unsteady-state energy balance since the mass and energy inside the tank (system) increase with time. Let us define the initial state of the tank (empty), state 1, and the final state (filled to 10 MPa), state 2. The inlet flow stream will be designated by in. The conservation of mass is given by Equation (2.16). Since we have one stream in and no streams out, we get ... 

Similarly, energy balances in the reactor and the jacket can be drawn. However, the energy balance in the reactor can be rather complicated depending on the rate law and the stoichiometry of the reaction [3], [9], [14]. The general equation of the unsteady state of the energy balance is the following ... 

We consider here the same continuous reactor of Section 2.2.3.S under unsteady-state conditions. Assuming that all temperature dependent parameters remain constant at some average values, the mass and energy balances in the CSTR are governed by the following equations ... 

Apply the unsteady-state mole and energy balances to this system. Consider the upper steady state. Use the values you obtained in pan (b) as your initial values to plot C and T versus time up to 6 hours and then to plot versus T. What did you find Do you want to change any of your answers to pan (b) ... 

Dynamic flowsheeting is based on the unsteady state balance of mass and energy, and may be formulated in general terms by the following equation ... 

When the steady-state thermal energy balance (6-16) is combined with the unsteady-state species mass balance (6-21), the time dependence of reactant conversion (i.e., dx/dt) can be calculated from the digital controller response, which monitors the rate of thermal energy removal across the outer wall of the reactor for exothermic chemical reactions ... 

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