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Unformatted text preview: on 4–4 is plotted
in Fig. 4–3 for different values of b. There are two observations that can be
made from this figure and the relation above:
1. Equation 4–4 enables us to determine the temperature T(t) of a body at
time t, or alternatively, the time t required for the temperature to reach
a specified value T(t).
2. The temperature of a body approaches the ambient temperature T
exponentially. The temperature of the body changes rapidly at the
beginning, but rather slowly later on. A large value of b indicates that
the body will approach the environment temperature in a short time.
The larger the value of the exponent b, the higher the rate of decay in
temperature. Note that b is proportional to the surface area, but inversely
proportional to the mass and the specific heat of the body. This is not
surprising since it takes longer to heat or cool a larger mass, especially
when it has a large specific heat. b1
b3 > b2 > b1 Ti t FIGURE 4–3
The temperature of a lumped
system approaches the environment
temperature as time gets larger. Once the temperature T(t) at time t is available from Eq. 4–4, the rate of convection heat transfer between the body and its environment at that time can be
determined from Newton’s law of cooling as
·
Q (t) hAs[T(t) T] (W) (46) The total amount of heat transfer between the body and the surrounding
medium over the time interval t 0 to t is simply the change in the energy
content of the body:
Q mCp[T(t) Ti] (kJ) (47) The amount of heat transfer reaches its upper limit when the body reaches the
surrounding temperature T . Therefore, the maximum heat transfer between
the body and its surroundings is (Fig. 4–4)
Qmax mCp(T Ti) (kJ) (48) Ti
Ti Criteria for Lumped System Analysis
The lumped system analysis certainly provides great convenience in heat
transfer analysis, and naturally we would like to know when it is appropriate t→ Ti
Ti Ti We could also obtain this equation by substituting the T(t) relation from Eq.
·
4–4 into the Q (t) relation in Eq. 4–6 and integrating it from t 0 to t → . h
T t=0 T T
Ti
Ti T T T
T T Q = Qmax = mCp (Ti – T ) FIGURE 4–4
Heat transfer to or from a body
reaches its maximum value
when the body reaches
the environment temperature. cen58933_ch04.qxd 9/10/2002 9:12 AM Page 212 212
HEAT TRANSFER to use it. The first step in establishing a criterion for the applicability of the
lumped system analysis is to define a characteristic length as Convection
h
T SOLID
BODY Lc V
As Bi Conduction hLc
k and a Biot number Bi as
(49) It can also be expressed as (Fig. 4–5)
heat convection
Bi = ———————–
heat conduction FIGURE 4–5
The Biot number can be viewed as the
ratio of the convection at the surface
to conduction within the body. Bi h
T
k /Lc T Convection at the surface of the body
Conduction within the body or
Bi Lc /k
1/h Conduction resistance within the body
Convection resistance at the surface of the body When a solid body is being heated by the hotter fluid surrounding it (such as
a potato being baked in an oven), heat is first convected to the body and
subsequently conducted within the body. The Biot number is the ratio of the
internal resistance of a body to heat conduction to its external resistance to
heat convection. Therefore, a small Biot number represents small resistance
to heat conduction, and thus small temperature gradients within the body.
Lumped system analysis assumes a uniform temperature distribution
throughout the body, which will be the case only when the thermal resistance
of the body to heat conduction (the conduction resistance) 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?
Before answering this question, we should mention that a 20 percent
uncertainty in the convection heat transfer coefficient h in most cases is considered “normal” and “expected.” Assuming h to be constant and uniform is
also an approximation of questionable validity, especially for irregular geometries. Therefore, in the absence of sufficient experimental data for the specific
geometry under consideration, we cannot claim our results to be better than
20 percent, even when Bi
0. This being the case, introducing another
source of uncertainty in the problem will hardly have any effect on the overall uncertainty, provided that it is minor. It is generally accepted that lumped
system analysis is applicable if
Bi 0.1 When this criterion is satisfied, the temperatures within the body relative to
the surroundings (i.e., T T ) remain within 5 percent of each other even for
wellrounded geometries such as a spherical ball. Thus, when Bi 0.1, the
variation of temperature with location within the body will be slight and can
reasonably be approximated as being uniform. cen58933_ch04.qxd 9/10/2002 9:12 AM Page 213 213
CHAPTER 4 The first step in the application of lumped system analysis is the calculation
of the Biot number, and the assessment of the applicability...
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This note was uploaded on 01/28/2010 for the course HEAT ENG taught by Professor Ghaz during the Spring '10 term at University of Guelph.
 Spring '10
 Ghaz

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