CHAPTER 5
HEAT TRANSFER THEORY (cont'd)

OVERALL HEAT-TRANSFER COEFFICIENTS

It is most convenient to use overall heat transfer coefficients in heat transfer calculations as these combine all of the constituent factors into one, and are based on the overall temperature drop. An overall coefficient, U, combining conduction and surface coefficients, has already been introduced in eqn. (5.5). Radiation coefficients, subject to the limitations discussed in the section on radiation, can be incorporated also in the overall coefficient. The radiation coefficients should be combined with the convection coefficient to give a total surface coefficient, as they are in series, and so:

h_s = (h_r + h_c)                                                           (5.32)

The overall coefficient U for a composite system, consisting of surface film, composite wall, surface film, in series, can then be calculated as in eqn. (5.5) from:

1/U = 1/(h_r + h_c)₁ + x₁/k₁ + x₂/k₂ + …+ 1/(h_r + h_c)₂.   (5.33)

EXAMPLE 5.11. Effect of air movement on heat transfer in a cold store
In Example 5.2, the overall conductance of the materials in a cold-store wall was calculated. Now on the outside of such a wall a wind of 6.7 m s^-1 is blowing, and on the inside a cooling unit moves air over the wall surface at about 0.61 m s^-1. The radiation coefficients can be taken as 6.25 and 1.7 J m^-2 s^-1 °C^-1 on the outside and inside of the wall respectively. Calculate the overall heat transfer coefficient for the wall.

Outside surface: v = 6.7 m s^-1.
And so from eqn. (5.28)
                       h_c = 7.4v^0.8 = 7.4(6.7)^0.8 = 34 J m^-2 s^-1 °C^-1
                 and h_r = 6.25 J m^-2 s^-1 °C^-1
       Therefore h_s1 = (34+6) = 40 J m^-2 s^-1 °C^-1

Inside surface:  v   = 0.61 m s^-1.
From eqn. (5.27)
                       h_c = 5.7 + 3.9v = 5.7 + (3.9 x 0.61)
                            = 8.1 J m^-2 s^-1 °C^-1
                  and h_r = 1.7 J m^-2 s^-1 °C^-1
        Therefore h_s2 = (8.1 + 1.7) = 9.8 J m^-2 s^-1 °C^-1

Now from Example 5.2 the overall conductance of the wall,

                     U_old = 0.38 J m^-2 s^-1 °C^-1
and so

                1/U_new = 1/h_s1 + 1/U_old + 1/h_s2
                            = 1/40 + 1/0.38 + 1/9.8
                            = 2.76.
Therefore      U_new = 0.36 J m^-2 s^-1 °C^-1

In eqn. (5.33) often one or two terms are much more important than other terms because of their numerical values. In such a case, the important terms, those signifying the low thermal conductances, are said to be the controlling terms. Thus, in Example 5.11 the introduction of values for the surface coefficients made only a small difference to the overall U value for the insulated wall. The reverse situation might be the case for other walls that were better heat conductors.

EXAMPLE 5.12. Comparison of heat transfer in brick and aluminium walls
Calculate the respective U values for a wall made from either (a) 10 cm of brick of thermal conductivity 0.7 J m^-1 s^-1 °C^-1, or (b) 1.3mm of aluminium sheet, conductivity 208 J m^-1 s^-1 °C^-1.
Surface heat-transfer coefficients are on the one side 9.8 and on the other 40 J m^-2 s^-1 °C^-1.

(a) For brick

                 k = 0.7 J m^-1 s^-1°C^-1
               x/k = 0.1/0.7 = 0.14
Therefore 1/U = 1/40 + 0.14 + 1/9.8
                    = 0.27
                 U = 3.7 J m^-2 s^-1°C^-1

(b) For aluminium

                 k = 208 J m^-1 s^-1°C^-1
               x/k = 0.0013/208
                    = 6.2 x 10^-6
              1/U = 1/40 + 6.2 x 10^-6 + 1/9.8
                    = 0.13
                 U = 7.7 J m^-2 s^-1°C^-1

Comparing the calculations in Example 5.11 with those in Example 5.12, it can be seen that the relative importance of the terms varies. In the first case, with the insulated wall, the thermal conductivity of the insulation is so low that the neglect of the surface terms makes little difference to the calculated U value. In the second case, with a wall whose conductance is of the same order as the surface coefficients, all terms have to be considered to arrive at a reasonably accurate U value. In the third case, with a wall of high conductivity, the wall conductance is insignificant compared with the surface terms and it could be neglected without any appreciable effect on U. The practical significance of this observation is that if the controlling terms are known, then in any overall heat-transfer situation other factors may often be neglected without introducing significant error. On the other hand, if all terms are of the same magnitude, there are no controlling terms and all factors have to be taken into account.


Heat-Transfer Theory > HEAT TRANSFER FROM CONDENSING VAPOURS

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