Terms of the Heat Balance

Understanding the heat balance is essential when considering thermal process safety. This also applies to the industrial scale for reactors or storage units, as well as at laboratory scale for understanding the results of calorimetric experiments. In fact, the same heat balance terms will serve in both situations. For this reason, we first present the different terms of the heat balance of a reactor with a reacting system. This is followed by an often-used and simplified heat balance and finally we will study how reaction rate is affected by adiabatic conditions. 

In chemical thermodynamics, the convention is that exothermal effects are negative and endothermal positive. Here, since we consider the heat balance for practi- 

---

Let us consider a simplified heat balance involving an exothermal reaction with zero-order kinetics. The heat release rate of the reaction q = f(T) varies as an exponential function of temperature. The second term of the heat balance, the heat removal by a cooling system qKX =f(T), with Newtonian cooling (Equation 2.18), varies linearly with temperature. The slope of this straight line is U-A and the intersection with the abscissa is the temperature of the cooling system Tc. This... 

In order to determine the heat released by a reaction, the calorimeter can work using a simplified heat balance, as presented in Section 2.4.2. Many calorimeters are designed in such a way as to eliminate one of the three terms of the heat balance, in order to determine the heat release rate by measuring the other term. 

Figure 7.2 The different terms of the heat balance of an isothermal semi-batch reactor (in kW) as a function of time. The maximal cooling capacity of the reactor (qama,) obtained with cold water at 5°C is also represented. The difference between both curves q and qa represents the cooling effect by the feed. Its disappearance at the end of the feed at 4 hours is visible.

The fractional conversion XA in terms of the heat balance equation is ... 

Accidents happening in polymerization reactors are practically always due to a lack of control of the course of reaction caused by a disturbance of the heat balance, which results in a temperature increase leading to loss of control of the reactor and a runaway reaction. In this section a systematic procedure based on a failure scenario with six key questions, allowing assessment of the criticality of a process, is presented. Since the heat balance is at the center of our concerns in matters of thermal control of reactors, the different terms of the heat balance will be examined. Finally, aspects of the dynamic stability of reactors and of the thermal stability of reaction masses are analyzed. 

The heat balance is also at the center of the evaluation of calorimetric experiments as used for safety studies. Thus understanding the heat balance of a reactor is essential for the design of safe processes. Hereafter the different contributions to the heat balance, such as the heat release rate of the reaction, the heat exchange at the wall of the reactor, the heat dissipated by the stirrer, the heat accumulation in the reactor, the effects of the sensible heat of the feed, and the heat losses, will be discussed in detail. The different terms of the heat balance are expressed as heat release rates or thermal power. 

The first order reaction is represented by (-r ) = kC, and applying the mass balance Equation 6-120 and the heat balance Equation 6-121, respectively, gives the fractional conversion in terms of the mass balance equation ... 

It can be seen that k in Eq. (10) replaces the system-describing parameters L and Ah in Eq. (1). A direct test of the hypothesis is therefore to plot (j> against k for fixed values of P, G, and d, with L and Ah varying. For the hypothesis to be correct, the data points must all lie on a smooth curve. Experience shows, however, that plotting (f> against k often produces an undue amount of scatter which may obscure and distort any true relationship existing. This enhanced scatter is caused by the cumulative effect of experimental errors in the various terms in the heat-balance equation from which the quality k is derived. 

This is the simplest system for temperature control of a reactor only the jacket temperature is controlled and maintained constant, leaving the reaction medium following its temperature course as a result of the heat balance between the heat flow across the wall and the heat release rate due to the reaction (Figure 9.9). This simplicity has a price in terms of reaction control, as analysed in Sections 6.7 and 7.6. Isoperibolic temperature control can be achieved with a single heat carrier circuit, as well as with the more sophisticated secondary circulation loop. 

In the Frank-Kamenetskii model, the surroundings temperature is set equal to the temperature of the reacting solid. Thus, there is only a small temperature gradient between this element and the wall, so only a limited heat transfer to the surroundings. This simplification establishes the above described criteria, but it is not really representative of a certain number of industrial situations. In fact, there are numerous situations where the surrounding temperature different from the initial product temperature, for example, discharging of a hot product from a dryer to a container placed at ambient temperature, and so on. Therefore, Thomas [7] developed a model that accounts for heat transfer at the wall. He added a convective term to the heat balance ... 

The first law of thermodynamics provides a description of the energy balance for a given process the second law provides a criterion for deciding whether or not the process will occur spontaneously. The second law of thermodynamics defines the entropy change (A5, in units of J K l) associated with a change in a closed system in terms of the heat absorbed by the system at constant temperature T ... 

The major terms in the heat balance of a pyrolysis reactor are ... 

An incremental improvement in the path to practical applications of catalytic combustion was disclosed by Yasuyoshi et al., who recognized the important applications of manipulating the heat balance in Eq. (5). These authors coated the walls of alternate channels to have only half of the reactant flow passing through catalytic channels. For similar heat transfer coefficients in coated and uncoated channels, and assuming that there is no temperature gradient in the channel wall, the heat removal term in the heat balance is given by Eq. (7). 

Although the total heat flux at the surface in a binary gas is composed of the sum of a conduction term and a diffusion term, the results of analyses are expressed solely in terms of the heat conduction term. The reason is that this term is equal to the heat gained by the coolant in passing from its reservoir to the surface in contact with the boundary layer. This simple heat balance is... 

The occurrence of multiple steady states can be illustrated at best by a CSTR in which a high exothermic reaction takes place. A simple method is to examine separately the behaviour of the two terms of the energy balance heat generated by reaction, and heat transferred from the reactor. The heat generated is proportional with the reaction rate and the thermal effect ... 

In the 1904 edition there is, for example, a sample calculation of the heat balance on a Glover tower treated as an evaporator, which shows how inefficient it was then ( what a heat waster it is (23)), There is also a discussion on the efficiency of various packings, explaining in terms of surface areas why coke is 1.5 to 2 times more efiBcient than bricks (23, 26), But in general, Davis approach was still empirical the operations are described as procedures of practical utility, and are not based on fundamental physics. Neither the work of Osborne Reynolds nor dimensionless group theory had been assimilated yet into the profession. 

The second term in the heat balance accounts for heat removal by a circulating liquid at temperature from the surface of the volume element dV, assuming an overall heat transfer coefficient U across the tube wall. Note that represents the rate per unit volume of reactor. Hence... 

The term heat balance space refers to the space considered for heat exchange phenomena. The heat flow emanating from the molded article enters the balance space. The following pass-out of the heat balance space the heat flow dissipated by the heat-balancing medium, the convective- as well as the radiant heat-flow (both to the environment), and the thermal conductivity flow, which enters the platens. 

Partitioning and adsorption may be best distinguished from each other in terms of the heat effects associated with these two processes. Adsorption of a trace component, as on activated carbon (79, 77), requires a relatively high exothermic Mi in order to balance the decrease in entropy associated with solute condensation from dilute solution. On the other hand, partitioning of a solute may not be exothermic and would have a nearly constant over a wide range of concentration relative to its solubility (as evidenced by a linear isotherm). Conceptually, the enthalpy change for a partition process should follow ... 

The various terms take into account the various sinks q < 0) or sources q > 0) of the heat balance. These include, specifically, the reaction itself, transport, agitation, radiation, and heat loss through evaporation and gas stream. In most cases the last four terms can be neglected (Bronn, 1971 Cooney, Wang, and Mateles, 1968 Mou and Cooney, 1976). The term for the reaction, the heat produced, is represented in Equ. 2.4c. The kinetics are, as always, introduced in the balance equation. All heats of reaction can be calculated from a series of T/t measurements using Equ. 3.70 when the specific heat capacity of the reactor system is known... 

Sometimes the system [Eq. (2.1) (2.4)] is simplified with the rejection of the heat balance equation [Eq. (2.2)] and with the adoption of the flows Lj and V) constancy within each colunm section (the term section refers to the part of a column between the flow inlet and outlet points). 

Bolodyan and co-workers have modelled the test in terms of a heat balance equation [29]. They have used this to account for the effect of inert fillers in terms of their heat capacity and found reasonable agreement with experiment. More recently, the heat balance approach has been extended by Case and Jackson [27] and by Khalturinskii and Berlin [30] to include the thermal properties of flame retardant fillers. The Case and Jackson work has taken the analysis further than Khalturinskii and Berlin, and is reproduced in detail here. 

Rewriting Eq. (4.10.70) with the internal surface per volume of reactor (dAt.int/ dVR = 4/dint), the superficial fluid velocity us = 4V/ndj ), and substitution of the rate Ta by Cj indK ldr leads to the heat balance in terms of the heat flow per unit volume ... 

The stoichiometric specific air demand p o is defined by the relation of the stoichiometric air mass flow and the corresponding fuel mass flow. The relation of all terms of the energy balance on the mass flow fiipj of the incoming fuel allows a generalised consideration. The generated heat is... 

The analysis of the heat exchanger network first identifies sources of heat (termed hot streams) and sinks (termed cold streams) from the material and energy balance. Consider first a very simple problem with just one hot stream (heat source) and one cold stream (heat sink). The initial temperature (termed supply temperature), final temperature (termed target temperature), and enthalpy change of both streams are given in Table 6.1. 

The heat balance of a reacdor is made up of three terms Heat of reaction -I- Heat transfer = Gain of sensible and latent heats by the mixture. This estabhshes the temperature as a function of the composition... 

Mathematically, multiplicities become evident when heat and material balances are combined. Both are functions of temperature, the latter through the rate equation which depends on temperature by way of the Arrhenius law. The curves representing these b ances may intersect in several points. For first order in a CSTR, the material balance in terms of the fraction converted can be written... 

---