The thermal resistance concept or the electrical analogy can also be used to solve steady heat transfer problems that involve parallel layers or combined series-parallel arrangements. Although such problems are often two- or even three-dimensional, approximate solutions can be obtained by assuming one-dimensional heat transfer and using the thermal resistance network.
Consider the composite wall shown in Fig. 3–19, which consists of two par-allel layers. The thermal resistance network, which consists of two parpar-allel re-sistances, can be represented as shown in the figure. Noting that the total heat transfer is the sum of the heat transfers through each layer, we have
Q· Q·1 Q·
2 (T1 T2) (3-29)
Utilizing electrical analogy, we get
Q· (3-30)
where
→ Rtotal (3-31)
since the resistances are in parallel.
Now consider the combined series-parallel arrangement shown in Fig.
3–20. The total rate of heat transfer through this composite system can again be expressed as
Once the individual thermal resistances are evaluated, the total resistance and the total rate of heat transfer can easily be determined from the relations above.
The result obtained will be somewhat approximate, since the surfaces of the third layer will probably not be isothermal, and heat transfer between the first two layers is likely to occur.
Two assumptions commonly used in solving complex multidimensional heat transfer problems by treating them as one-dimensional (say, in the
1 network for two parallel layers.
T1 Thermal resistance network for combined series-parallel arrangement.
x-direction) using the thermal resistance network are (1) any plane wall nor-mal to the x-axis is isothernor-mal (i.e., to assume the temperature to vary in the x-direction only) and (2) any plane parallel to the x-axis is adiabatic (i.e., to assume heat transfer to occur in the x-direction only). These two assumptions result in different resistance networks, and thus different (but usually close) values for the total thermal resistance and thus heat transfer. The actual result lies between these two values. In geometries in which heat transfer occurs pre-dominantly in one direction, either approach gives satisfactory results.
EXAMPLE 3–6 Heat Loss through a Composite Wall
A 3-m-high and 5-m-wide wall consists of long 16-cm 22-cm cross section horizontal bricks (k 0.72 W/m · °C) separated by 3-cm-thick plaster layers (k 0.22 W/m · °C). There are also 2-cm-thick plaster layers on each side of the brick and a 3-cm-thick rigid foam (k 0.026 W/m · °C) on the inner side of the wall, as shown in Fig. 3–21. The indoor and the outdoor temperatures are 20°C and 10°C, and the convection heat transfer coefficients on the inner and the outer sides are h1 10 W/m2· °C and h2 25 W/m2· °C, respectively.
Assuming one-dimensional heat transfer and disregarding radiation, determine the rate of heat transfer through the wall.
SOLUTION The composition of a composite wall is given. The rate of heat transfer through the wall is to be determined.
Assumptions 1 Heat transfer is steady since there is no indication of change with time. 2 Heat transfer can be approximated as being one-dimensional since it is predominantly in the x-direction. 3 Thermal conductivities are constant.
4 Heat transfer by radiation is negligible.
Properties The thermal conductivities are given to be k 0.72 W/m · °C for bricks, k 0.22 W/m · °C for plaster layers, and k 0.026 W/m · °C for the rigid foam.
Analysis There is a pattern in the construction of this wall that repeats itself every 25-cm distance in the vertical direction. There is no variation in the hori-zontal direction. Therefore, we consider a 1-m-deep and 0.25-m-high portion of the wall, since it is representative of the entire wall.
Assuming any cross section of the wall normal to the x-direction to be isothermal, the thermal resistance network for the representative section of the wall becomes as shown in Fig. 3–21. The individual resistances are eval-uated as:
Ri Rconv, 1 0.4°C/ W
R1 Rfoam 4.6°C/ W
R2 R6 Rplaster, side
0.36°C/ W R3 R5 Rplaster, center
48.48°C/ W
L
kA 0.16 m
(0.22 W/m · °C)(0.015 1 m2) L
kA 0.02 m
(0.22 W/m · °C)(0.25 1 m2) L
kA 0.03 m
(0.026 W/m · °C)(0.25 1 m2) 1
h1 A 1
(10 W/m2 · °C)(0.25 1 m2) Brick
h1 T1
h2 T2
1.5 cm
22 cm
1.5 cm
3 x
2 16 cm 2
Plaster Foam
R5 R4 R3
R6
Ri R1 R2 Ro
T2 T1
FIGURE 3–21
Schematic for Example 3–6.
R4 Rbrick 1.01°C/ W
Ro Rconv, 2 0.16°C/ W
The three resistances R3, R4, and R5in the middle are parallel, and their equiv-alent resistance is determined from
1.03 W/°C
which gives
Rmid 0.97°C/ W
Now all the resistances are in series, and the total resistance is
Rtotal Ri R1 R2 Rmid R6 Ro
0.4 4.6 0.36 0.97 0.36 0.16
6.85°C/ W
Then the steady rate of heat transfer through the wall becomes
Q· 4.38 W (per 0.25 m2surface area)
or 4.38/0.25 17.5 W per m2area. The total area of the wall is A 3 m 5 m 15 m2. Then the rate of heat transfer through the entire wall becomes
Q·
total (17.5 W/m2)(15 m2) 263 W
Of course, this result is approximate, since we assumed the temperature within the wall to vary in one direction only and ignored any temperature change (and thus heat transfer) in the other two directions.
Discussion In the above solution, we assumed the temperature at any cross section of the wall normal to the x-direction to be isothermal. We could also solve this problem by going to the other extreme and assuming the surfaces par-allel to the x-direction to be adiabatic. The thermal resistance network in this case will be as shown in Fig. 3–22. By following the approach outlined above, the total thermal resistance in this case is determined to be Rtotal 6.97°C/ W, which is very close to the value 6.85°C/ W obtained before. Thus either ap-proach would give roughly the same result in this case. This example demon-strates that either approach can be used in practice to obtain satisfactory results.
T1 T2
Rtotal [20 (10)]°C 6.85°C/ W 1
Rmid 1 R3 1
R4 1 R5 1
48.48 1 1.01 1
48.48 1
h2 A 1
(25 W/m2 · °C)(0.25 1 m2) L
kA 0.16 m
(0.72 W/m · °C)(0.22 1 m2)
x
Ro T2 Ri
T1
Adiabatic lines
FIGURE 3–22 Alternative thermal resistance network for Example 3–6 for the case of surfaces parallel to the primary direction of heat transfer being adiabatic.