Lumped capacity analysis

Bi very small, (say, <0.1). Here the main resistance to heat transfer lies within the fluid this occurs when the thermal conductivity of the particle in very high and/or when the particle is very small. Under these conditions, the temperature within the particle is uniform and a lumped capacity analysis may be performed. Thus, if a solid body of volume V and initial temperature Oo is suddenly immersed in a volume of fluid large enough for its temperature 0 to remain effectively constant, the rate of heat transfer from the body may be expressed as ... 

Do not dismiss lumped capacity analysis because of its simplicity. In many cases one will not know the convection coefficient better than 25 percent, so it is not necessary to use more elaborate analysis techniques. 

First, determine if a lumped capacity analysis can apply. If so, you may be led to a much easier calculation. 

An aluminum sphere, S.O cm in diameter, is initially at a uniform temperature of 50°C. It is suddenly exposed to an outers pace radiation environment at 0 K (no convection). Assuming the surface of aluminum is blackened and lumped-capacity analysis applies, calculate the time required for the temperature of the sphere to drop to -110°C. 

An aluminum can having a volume of about 350 cm3 contains beer at 1°C. Using a lumped-capacity analysis, estimate the time required for the contents to warm to 15°C when the can is placed in a room at 22°C with a convection coefficient of 15 W/m2 - °C. Assume beer has the same properties as water. 

Assuming that the bubble in Prob. 9-27 moves through the liquid at a velocity of 4.5 m/s, estimate the time required to cool the bubble 0.3°C by calculating the heat-transfer coefficient for flow over a sphere and using this in a lumped-capacity analysis as described in Chap. 4. 

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. 

Fig. 4-2 Nomenclature for single-lump heat-capacity analysis.

We have already noted that the lumped-capacity type of analysis assumes a uniform temperature distribution throughout the solid body and that the assumption is equivalent to saying that the surface-convection resistance is large compared with the internal-conduction resistance. Such an analysis may be expected to yield reasonable estimates when the following condition is met ... 

A very low value of the Biot modulus means that internal-conduction resistance is negligible in comparison with surface-convection resistance. This in turn implies that the temperature will be nearly uniform throughout the solid, and its behavior may be approximated by the lumped-capacity method of analysis. It is interesting to note that the exponent of Gq. (4-5) may be expressed in terms of the Biot and Fourier numbers if one takes the ratio VIA as the characteristic dimension 5. Then,... 

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. 

What is meant by a lumped capacity What are the physical assumptions necessary for a lumped-capacity unsteady-state analysis to apply ... 

Derive an expression for the temperature of the sphere as a function of time and the heat-transfer coefficient from the fluid to the sphere. Assume that the temperatures of the sphere and fluid are uniform at any instant so that the lumped-capacity method of analysis may be used. 

A piece of aluminum weighing 5.5 kg and initially at a temperature of 290°C is suddenly immersed in a fluid at 15°C. The convection heat-transfer coefficient is 58 W/m2 °C. Taking the aluminum as a sphere having the same weight as that given, estimate the time required to cool the aluminum to 90°C, using the lumped-capacity method of analysis. 

Two identical 7.5-cm cubes of copper at 425 and 90°C are brought into contact. Assuming that the blocks exchange heat only with each other and that there is no resistance to heat flow as a result of the contact of the blocks, plot the temperature of each block as a function of time, using the lumped-capacity method of analysis. That is, assume the resistance to heat transfer is the conduction resistance of the two blocks. Assume that all surfaces are insulated except those in contact. 

A 5-cm-diameter copper sphere is initially at a uniform temperature of 250°C. It is suddenly exposed to an environment at 30°C having a heat-transfer coefficient h = 28 W/m2 - °C. Using the lumped-capacity method of analysis, calculate the time necessary for the sphere temperature to reach 90°C. 

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. 

Aside from the original assumption of a lumped analysis, thus far there have been no other assumptions or approximations to the model. The model relies completely on basic thermodynamic principles, a known cell performance R(I), and rigorous mathematical operations. To solve the model, we need to know the bulk mass and heat capacity of the cell, M and C, respectively the reactant supply flow rate (m = fuel flow + air flow) the inlet temperature and pressure and the change in stream composition due to the electrochemical reaction, AX, so that the change in enthalpy can be calculated the electrical load current, / and the inlet and exit temperatures, Tm and rout. 

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. 

Using a model with lumped masses at two characteristic levels (top and level 1), a nonlinear dynamic analysis was performed with a complex hysteretic (IZIIS) model (Fig. 8.5, Shendova 1998). To calibrate the computations in defining the capacity degradation in hysteretic models, the results from seismic shaking table testing of the model were used. With this, an attempt was made to model the dynamic response in a simple way, suitable for everyday analyses, resulting however in satisfactory final results on the behavior at individual levels. 

A nonlinear dynamic analysis has been performed for the three monuments (Sect. 8.2.3), with the masses lumped at characteristic levels and applying a corresponding storey hysteretic model obtained by summing up the elastoplastic characteristics of each of the bearing walls, with the load-bearing capacity of each of them limited to the bending and shear capacity, whichever is less. 

Dynamic analysis With the masses lumped at two characteristic levels, a nonlinear dynamic analysis has been performed with storey hysteretic model obtained by summing up the elastoplastic characteristics of each of the bearing walls, whereas the load-bearing capacity of each of them has been limited to the lower value of bending and shear capacity (according to Sect. 8.3.3). To obtain the dynamic response, three different types of earthquake (Petrovac 1979, Ulcinj 1979 and El Centro 1940) with maximum input acceleration of 0.24g and return period of 1,000 years have been applied. Obtained as the results from the dynamic analysis are the storey displacements and ductility ratios required by the earthquake that have to comply with the design criteria defined in Sect. 8.3.4. 

---