Lumped systems analysis

Crilcha for Lumped System Analysis 219 Some Remarks on Heat Transfer in Lumped Systems 221... 

In heat transfer analysts, some bodies are observed to behave like a lump whose interior temperature remains essentially uniform at all times during a heat transfer process. The temperature of such bodies can be taken to be a function of time only, 7(r). Heat transfer analysis that utilizes this idealization is known as lumped system analysis, which provides great simplification in certain classes of heat transfer problems without much sacrifice from accuracy. 

Now let tis go to the other extreme and consider a large roast in an oven. If you have done any roasting, you must liave noticed that the temperature distribution within the roast is not even close to being unifonn. You can easily verify this by taking the roast outbefote it is completely done and cutting it in half. You will see that the outer parts of the roast are well done while the center part is barely warm. Thus, lumped system analysis is not applicable in this case. Before presenting a criterion about applicability of lumped system analysis, we develop the formulation associated with it. 

Consider a body of arbitrary shape of mass m, volume J, surface area density p, and specific heat initially at a uniform temperature 7) (Fig. 4-2). At time t = 0, the body is placed into a medium at temperature T., and heat transfer takes place between the body and its environment, with a heat transfer coefficient h. For the sake of discussion, we assume that 7) > 7), but the analysis is equally valid for the opposite case. We assume lumped system analysis to be applicable, so lhat the temperature remains uniform wilhin the body at all times and changes with time only, T T t). 

The lumped system analysis certainly provides great convenience in heat transfer analysis, and naturally we would like to know when it is appropriate... 

Lumped system analysis assumes a uniform temperature dislribulion throughout the body, which is the case only when the thermal resistance of the body to heat conduction (the conduction resixtance) is zero. Thus, lumped system analysis is exact when Bi = 0 and approximate when Bi > 0. Of course, the smaller the Bi number, the more accurate the lumped system analysis. Then the question we must answer is, How much accuracy are we willing to sacrifice for the convenience of the lumped system analysis ... 

The first step in ilie application of lumped system analysis is the calculation of the Biot number, and the assessment of the applicability of this approach. One may still wish to use Inmped system analysis even when the criterion Bi < 0.1 is not satisfied, if high accuracy is not a major concern. 

Note that the Biot number is tlic ratio of the convection at the surface to conduction within the body, and this number should be as small as possible for lumped system analysis to be applicable. Therefore, small bodies with high tlieniial conductivity are good candidates for lumped system analysis, especially when they are in a medium that is a poor conductor of heat (such as air or another gas) and motionless. Thus, Ihe hot small copper ball placed in quiescent air, discussed eailier, is most likely to satisfy the criterion for lumped system analysis (Fig. 4-6). 

Small bodies with high thermal conductivities and low convection coefficients arc most likely to satisfy the criterion for lumped system analysis. 

Therefore, lumped system analysis is applicable, and the error involved in this approximation is negligible. 

Therefore, lumped system analysis is not applicable. However, we can still use it to get a "rough" estimate of the time of death. The exponent b in this case is... 

Discussion This example dem.onstrates how to obtain "ball park values using a simple analysis. A similar analysis is used in practice by incorporating constants to account for deviation from lumped system analysis. 

The dimensionless quantities defined above for a plane wall can also be used for a cylinder or sphere by replacing the space variable x by r and the half-thickness L by the outer radius r. Note that the characieristic length in the definition of the Biot number is taken to be the half-thickness L for the plane wall, and the radius for the long cylinder and sphere instead of 1//A used in lumped system analysis. 

Wc discussed the physical significance of the Biot number earlier and indicated that it is a measure of the relative magnitudes of the two heal transfer mechanisms convection at the surface and conduction through the solid. A small value of Bi indicates tliat the inner resistance of the body to heat conduction is smalt relative to the resistance to convection between the surface and the fluid. As a result, the temperature distribution within the solid becomes fairly uniform, and lumped system analysis becomes applicable. Recall that when Bi <0.1, the error in assuming the temperature within the body to be uniform is negligible. 

Discussion Note that the Biot number in lumped system analysis was defined differenfjy.as Bi = hLJk= h rJ3)lk. However, either definition can be used in determijiifig the applicability of the lumped system analysis unless Bi = 0.1. 

Discussion We notice that the Biot number In this case is Bi = 1/45.8 - 0.022, which is much less than 0.1. Therefore, we expect the lumped system analysis to be applicable. This is also evident from (T - TJ/tFo - TJ = 0.99, which indicates that the temperatures at the center and the surface of the plate relative to the surrounding temperature are within 1 percent of each other. Noting that the error involved in reading the Heisler charts is typically a few percent, the lumped system analysis in this case may yield just as accurate results with less effort. 

The heat transfer surface area of the plate is 2A, where A is the face area of the plate (the plate transfers heat through both of its surfaces), and the volume of the plate is V = 2L)A, where L is the half-thickness of the plate. The exponent b used in the lumped system analysis is... 

I which is practically identical to the result obtained above using the Hetsler charts. Therefore, we can use lumped system analysis with confidence when the Biot number is sufficiently small. 

The error involved in lumped system analysis is negligible when... 

When the lumped system analysis is not applicable, the variation of temperature with position as well as time can be determined using the transient temperaiure charts given in Figs, 4-15,4-16, 4 17, and 4-29 for a large plane wall, a long cylinder, a sphere, and a semi-infinite medium, respectively. These charts are applicable for one-dimensional heal transfer in those geometries. Therefore, their use is limited to situations in which the body is initially at a uniform temperature, all surfaces are subjected to the same thermal conditions, and the body docs not involve any heat geiieiation. Tliese charts can also be used to determine the total heat transfer from the body up to a specified lime I. 

C Consider heat transfer between two identical hot solid bodies and the air surrounding them. The first solid is being cooled by a fan while the second one is allowed to cool naturally. For which solid is the lumped system analysis more likely to be applicable Why ... 

C In what medium is the lumped system analysis more likely to be applicable in water or in ait Why ... 

Lumped system analysis of transient heat conduction situations is valid when the Biot number is... 

XTfe A 1.5-cm-diameler silver sphere initially at 30°C is suspended in a room filled with saturated steam at lOO C, Using the lumped system analysis, determine how long it will take for the temperature of the ball to rise to 50°C. zkJso, determine the amount of steam that condenses during this process and verify that the lumped system analysis is applicable. 

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