## 79 Practical correlations for heat and mass transfer

In addition to the theoretical treatments just described, many workers have measured heat and mass transfer by forced convection from bodies such as plates, spheres and cylinders. The bulk of the literature in the field has been reviewed recently [Shenoy and Mashelkar, 1982; Irvine, Jr. and Karni, 1987; Nakayama, 1988; Chhabra, 1993a, b, 1999; Ghosh et al., 1994], but only a selection of reliable correlations for spheres and cylinders is presented in this section which is based primarily on the reviews of Irvine and Karni [1987] and of Ghosh et al. [1994].

### 7.9.1 Spheres

For particle-liquid heat and mass transfer in non-Newtonian polymer solutions flowing over spheres fixed in tubes (0.25 < d/Dt < 0.5), Ghosh et al. [1992, 1994] invoked the usual heat and mass transfer analogy, that is, Sh = Nu and

Sc = Pr to correlate most of the literature data as follows: Y = 1.428Rep/3 Rep < 4

where Y = (Nu - 2)(ms/mb)1/(3n+1)Pr—1/3 for heat transfer and Y = (Sh — 2)Sc-1/3 for mass transfer.

The predicted results are shown in Figure 7.5 for both heat and mass transfer over the range of conditions : 0.32 < n < 0.93; Rep < 200; 29 000 < Scp < 4.9 x 105 and 9500 < Prp < 1.9 x 106. The effective viscosity used in the Reynolds, Prandtl and Schmidt numbers (denoted by suffix to) is m(V/d)n—1 where V is the mean velocity of flow in the tube. The average deviation between predictions and data is 17%.

- Heat transfer data o, • Mass transfer data

- Heat transfer data o, • Mass transfer data

(Nu1-2)(ms/mfc)1/3n+1 or (Sh1-2) = 0.997 Rep (Prp or Scp)1/3 for Rep > 4

(Nu1-2)(ms/mb)V3n+1 or (Sh1-2) = 1.428 Re1//3 (Prp or Scp)1/3

(Nu1-2)(ms/mfc)1/3n+1 or (Sh1-2) = 0.997 Rep (Prp or Scp)1/3 for Rep > 4

(Nu1-2)(ms/mb)V3n+1 or (Sh1-2) = 1.428 Re1//3 (Prp or Scp)1/3

Reynolds number (Rep)

102 103

Figure 7.5 Overall correlation for heat and mass transfer from a single sphere immersed in power law fluids

### 7.9.2 Cylinders in cross-flow

The limited work on heat and mass transfer between power-law fluids and cylinders with their axis normal to the flow has been summarised recently by Ghosh et al. [1994] who proposed the following correlation for heat and mass transfer:

For heat transfer:

For mass transfer:

where the effective viscosity is evaluated, as for spheres, with the diameter of the cylinder as the characteristic length. The overall correlations for heat and mass transfer are given in Figure 7.6.

Figure 7.6 Overall correlation for heat and mass transfer from cylinders in cross flow

Reynolds number (Rep)

Figure 7.6 Overall correlation for heat and mass transfer from cylinders in cross flow

### Example 7.3

A polymer solution at 25°C flows at 1.8 m/s over a heated hollow copper sphere of diameter of 30 mm, maintained at a constant temperature of 55°C (by steam condensing inside the sphere). Estimate the rate of heat loss from the sphere. The thermophysical properties of the polymer solution may be approximated by those of water, the power-law constants in the temperature interval 25 < T < 55°C are given below: n = 0.26 and m = 26 — 0.0566 T where T is in K. What will be the rate of heat loss from a cylinder 30 mm in diameter and 60 mm long, oriented normal to flow?

Solution

At the mean film temperature of (25 + 55)/2 = 40°C, the values of p, Cp and k for water:

p = 991 kg/m3; Cp = 4180 J/kg°C;k = 0.634 W/mK The consistency index, m: m = 26 - 0.0566(273 + 40) = 8.28 Pa-s" The effective viscosity,

p Meff 0.4

Therefore, equation (7.71b) applies in this case.

Substituting values, hd = 2 + 134'/22637'/^8^ k 7.44

D 172

For a cylinder, the Reynolds number of the flow is still the same and therefore equation (7.72b) applies.

= 2570 x n x 30 x 10—3 x 60 x 10—3 x (55 — 25) = 436 W

Note that the drop in the value of the heat transfer coefficient in this case has been compensated by the increase in surface area resulting in higher rate of heat loss. Also, the heat loss from the flat ends has been neglected.

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