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Thermocouple
wire Gas
T ,h Junction
D = 1 mm
T(t) FIGURE 4–9
Schematic for Example 4–1. Temperature Measurement by Thermocouples The temperature of a gas stream is to be measured by a thermocouple whose
junction can be approximated as a 1mmdiameter sphere, as shown in Fig.
4–9. The properties of the junction are k 35 W/m · °C,
8500 kg/m3, and
Cp
320 J/kg · °C, and the convection heat transfer coefficient between the
junction and the gas is h 210 W/m2 · °C. Determine how long it will take for
the thermocouple to read 99 percent of the initial temperature difference. SOLUTION The temperature of a gas stream is to be measured by a thermocouple. The time it takes to register 99 percent of the initial T is to be
determined.
Assumptions 1 The junction is spherical in shape with a diameter of D
0.001 m. 2 The thermal properties of the junction and the heat transfer coefficient are constant. 3 Radiation effects are negligible.
Properties The properties of the junction are given in the problem statement.
Analysis The characteristic length of the junction is
Lc V
As 1
6 D3 1
D
6 D2 1
(0.001 m)
6 1.67 10 4 m Then the Biot number becomes Bi hLc
k (210 W/m2 · °C)(1.67 10
35 W/m · °C 4 m) 0.001 0.1 Therefore, lumped system analysis is applicable, and the error involved in this
approximation is negligible.
In order to read 99 percent of the initial temperature difference Ti
T
between the junction and the gas, we must have
T (t ) T
Ti T 0.01 For example, when Ti 0°C and T
100°C, a thermocouple is considered to
have read 99 percent of this applied temperature difference when its reading
indicates T (t ) 99°C. cen58933_ch04.qxd 9/10/2002 9:12 AM Page 215 215
CHAPTER 4 The value of the exponent b is hAs
CpV b h
Cp Lc 210 W/m2 · °C
(8500 kg/m )(320 J/kg · °C)(1.67
3 10 4 m) 0.462 s 1 We now substitute these values into Eq. 4–4 and obtain T (t ) T
Ti T e → bt 0.01 e (0.462 s 1)t which yields t 10 s Therefore, we must wait at least 10 s for the temperature of the thermocouple
junction to approach within 1 percent of the initial junctiongas temperature
difference.
Discussion Note that conduction through the wires and radiation exchange
with the surrounding surfaces will affect the result, and should be considered in
a more refined analysis. EXAMPLE 4–2 Predicting the Time of Death A person is found dead at 5 PM in a room whose temperature is 20°C. The temperature of the body is measured to be 25°C when found, and the heat transfer coefficient is estimated to be h
8 W/m2 · °C. Modeling the body as a
30cmdiameter, 1.70mlong cylinder, estimate the time of death of that person (Fig. 4–10). SOLUTION A body is found while still warm. The time of death is to be
estimated.
Assumptions 1 The body can be modeled as a 30cmdiameter, 1.70mlong
cylinder. 2 The thermal properties of the body and the heat transfer coefficient
are constant. 3 The radiation effects are negligible. 4 The person was healthy(!)
when he or she died with a body temperature of 37°C.
Properties The average human body is 72 percent water by mass, and thus we
can assume the body to have the properties of water at the average temperature
of (37 25)/2 31°C; k 0.617 W/m · °C,
996 kg/m3, and Cp 4178
J/kg · °C (Table A9).
Analysis The characteristic length of the body is Lc V
As ro2 L
2 ro L 2 ro2 (0.15 m)2(1.7 m)
2 (0.15 m)(1.7 m) 2 (0.15 m)2 Then the Biot number becomes Bi hLc
k (8 W/m2 · °C)(0.0689 m)
0.617 W/m · °C 0.89 0.1 0.0689 m FIGURE 4–10
Schematic for Example 4–2. cen58933_ch04.qxd 9/10/2002 9:12 AM Page 216 216
HEAT TRANSFER 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 b hAs
CpV
2.79 8 W/m2 · °C
(996 kg/m )(4178 J/kg · °C)(0.0689 m) h
Cp Lc
10 5 3 1 s We now substitute these values into Eq. 4–4, T (t ) T
Ti T 25
37 bt → t e 43,860 s 20
20 e (2.79 10 5 s 1)t which yields 12.2 h Therefore, as a rough estimate, the person died about 12 h before the body was
found, and thus the time of death is 5 AM. This example demonstrates how to
obtain “ball park” values using a simple analysis. 4–2 I TRANSIENT HEAT CONDUCTION IN LARGE
PLANE WALLS, LONG CYLINDERS, AND
SPHERES WITH SPATIAL EFFECTS In Section, 4–1, we considered bodies in which the variation of temperature
within the body was negligible; that is, bodies that remain nearly isothermal
during a process. Relatively small bodies of highly conductive materials approximate this behavior. In general, however, the temperature within a body
will change from point to point as well as with time. In this section, we consider the variation of temperature with time and position in onedimensional
problems such as those associated with a large plane wall, a long cylinder, and
a sphere.
Consider a plane wall of thickness 2L, a long cylinder of radius ro, and
a sphere of radius ro initially at a uniform temperature Ti, as shown in Fig.
4–11. At time t 0, e...
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 Spring '10
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 Heat Transfer, TI

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