Lumped-Heat-Capacity System

In Sec. 4-4, this solution will be presented in graphical form for calculation purposes. For now, our purpose has been to show how the unsteady-heat-conduction equation can be solved, for at least one case, with the separation-of-variables method. Further information on analytical methods in unsteady-state problems is given in the references 

We continue our discussion of transient heat conduction by analyzing systems which may be considered uniform in temperature. This type of analysis is called the lumped-heat-capacity method. Such systems are obviously idealized because a temperature gradient must exist in a material if heat is to be conducted into or out of the material. In general, the smaller the physical size of the body, the more realistic the assumption of a uniform temperature throughout in the limit a differential volume could be employed as in the derivation of the general heat-conduction equation. 

If a hot steel ball were immersed in a cool pan of water, the lumped-heat-capacity method of analysis might be used if we could justify an assumption of uniform ball temperature during the cooling process. Clearly, the temperature distribution in the ball would depend on the thermal conductivity of the ball material and the heat-transfer conditions from the surface of the ball to the surrounding fluid, i.e., the surface-convection heat-transfer coefficient. We should obtain a reasonably uniform temperature distribution in the ball if the resistance to heat transfer by conduction were small compared with the convection resistance at the surface, so that the major temperature gradient would occur through the fluid layer at the surface. The lumped-heat-capacity analysis, then, is one which assumes that the internal resistance of the body is negligible in comparison with the external resistance. 

The convection heat loss from the body is evidenced as a decrease in the internal energy of the body, as shown in Fig. 4-2. Thus 

The thermal network for the single-capacity system is shown in Fig. 4-26. In this network we notice that the thermal capacity of the system is charged initially at the potential To by closing the switch 5. Then, when the switch is opened, the energy stored in the thermal capacitance is dissipated through the resistance VhA. The analogy between this thermal system and an electric system is apparent, and we could easily construct an electric system which would behave exactly like the thermal system as long as we made the ratio 

---

SJi. The initial startup of an adiabatic, gas-phase packed tubular reactor makes a good example of how a distributed system can be lumped into a series of CSTRs in order to study the dynamic response. The reactor is a cylindrical vessel (3 feet ID by 20 feet long) packed with a metal packing. The packing occupies 5 percent of the total volume, provides 50 ft of area per of total volume, weighs 400 ib yft and has a heat capacity of 0.1 Btu/lb °F. The heat transfer coefficient between the packing and the gas is 10 Btu/h It "F. 

In progressing through this chapter the reader will have noted analysis techniques of varying complexity, ranging from simple lumped-capacity systems to numerical computer solutions. At this point some suggestions are offered for a general approach to follow in the solution of transient heat-transfer problems. 

The resistances concentrate at the wall surface, at that position the temperature gradients are the largest. The resistance of conduction has been ignored. The heat capacity of the wall, C aih determines the heat accumulation in the wall. This system representation is called a lumped system representation. 

The second consideration is the model for the interstage heaters, product separators and compressors. In order to model these units meaningfully, we must have reasonable estimates for the key thermophysical properties of the lumps. In the case of the reformer, we must make reasonable prediction of reactant concentration (at system pressure), fC-values (for the product separator) and heat capacity (to correctly model the reactor temperature drop and product temperatures). The reforming process generally operates at temperatures and pressures where the ideal gas law applies for hydrocarbon species in the reactor section. Ancheyta-Juarez et al. [1, 2] use the ideal gas assumption to calculate the concentration of reactant species. In addition, they use the polynomial heat capacity correlations for pure components to approximate the heat capacity of the mixture. Work by Bommannan et al. [30] and Padmavathi et al. [31] uses a fixed value for the heat capacity and fC-value correlation to predict compositions in the primary product separator. 

In some cases, where the wall of the reactor has an appreciable thermal capacity, the dynamics of the wall can be of importance (Luyben, 1973). The simplest approach is to assume the whole wall material has a uniform temperature and therefore can be treated as a single lumped parameter system or, in effect, as a single well-stirred tank. The heat flow through the jacket wall is represented in... 

This equation describes the time-temperature history of the solid object. The term c pV is often called the lumped thermal capacitance of the system. This type of analysis is often called the lumped capacity method or Newtonian heating or cooling method. 

---