4.6. HEAT TRANSFER OUTSIDE VARIOUS GEOMETRIES IN FORCED CONVECTION

4.6A. Introduction

In many cases a fluid is flowing over completely immersed bodies such as spheres, tubes, plates, and so on, and heat transfer is occurring between the fluid and the solid only. Many of these shapes are of practical interest in process engineering. The sphere, cylinder, and flat plate are perhaps of greatest importance, with heat transfer between these surfaces and a moving fluid frequently encountered.

When heat transfer occurs during immersed flow, the flux is dependent on the geometry of the body, the position on the body (front, side, back, etc.), the proximity of other bodies, the flow rate, and the fluid properties. The heat-transfer coefficient varies over the body. The average heat-transfer coefficient is given in the empirical relationships to be discussed in the following sections.

In general, the average heat-transfer coefficient on immersed bodies is given by

Equation 4.6-1

where C and m are constants that depend on the various configurations. The fluid properties are evaluated at the film temperature T_f = (T_w + T_b)/2, where T_w is the surface or wall temperature and T_b the average bulk fluid temperature. The velocity in the N_Re is the undisturbed free stream velocity v of the fluid approaching the object.

4.6B. Flow Parallel to Flat Plate

When the fluid is flowing parallel to a flat plate and heat transfer is occurring between the whole plate of length L m and the fluid, the N_Nu is as follows for a N_Re,L below 3 × 10⁵ in the laminar region and a N_Pr > 0.7:

Equation 4.6-2

where N_Re,L = Lνρ/μ and N_Nu = hL/k.

For the completely turbulent region at a N_Re,L above 3 × 10⁵ (K1, K3) and N_Pr > 0.7,

Equation 4.6-3

However, turbulence can start at a N_Re,L below 3 × 10⁵ if the plate is rough (K3), and then Eq. (4.6-3) will hold and give a N_Nu greater than that given by Eq. (4.6-2). For a N_Re,L below about 2 × 10⁴, Eq. (4.6-2) gives the larger value of N_Nu.

EXAMPLE 4.6-1. Cooling a Copper Fin A smooth, flat, thin fin of copper extending out from a tube is 51 mm by 51 mm square. Its temperature is approximately uniform at 82.2°C. Cooling air at 15.6°C and 1 atm abs flows parallel to the fin at a velocity of 12.2 m/s. For laminar flow, calculate the heat-transfer coefficient, h. If the leading edge of the fin is rough so that all of the boundary layer or film next to the fin is completely turbulent, calculate h. Solution: The fluid properties will be evaluated at the film temperature Tf = (Tw + Tb)/2: The physical properties of air at 48.9°C from Appendix A.3 are k = 0.0280 W/m · K, ρ = 1.097 kg/m3, μ = 1.95 × 10-5 Pa · s, NPr = 0.704. The Reynolds number is, for L = 0.051 m, Substituting into Eq. (4.6-2), Solving, h = 60.7 W/m2 · K (10.7 btu/h · ft2 · °F). For part (b), substituting into Eq. (4.6-3) and solving, h = 77.2 W/m2 · K (13.6 btu/h · ft2 · °F).

4.6C. Cylinder with Axis Perpendicular to Flow

Often a cylinder containing a fluid inside is being heated or cooled by a fluid flowing perpendicular to its axis. The equation for predicting the average heat-transfer coefficient of the outside of the cylinder for gases and liquids is (K3, P3) Eq. (4.6-1), with C and m as given in Table 4.6-1. The N_Re = Dνρ/μ, where D is the outside tube diameter and all physical properties are evaluated at the film temperature T_f. The velocity is the undisturbed free stream velocity approaching the cylinder.

Table 4.6-1. Constants for Use in Eq. (4.6-1) for Heat Transfer to Cylinders with Axis Perpendicular to Flow (N_Pr > 0.6)

NRe	m	C
1–4	0.330	0.989
4–40	0.385	0.911
40−4 × 103	0.466	0.683
4 × 103−4 × 104	0.618	0.193
4 × 104−2.5 × 105	0.805	0.0266

4.6D. Flow Past Single Sphere

When a single sphere is being heated or cooled by a fluid flowing past it, the following equation can be used to predict the average heat-transfer coefficient for a N_Re = Dνρ/μ of 1 to 70 000 and a N_Pr of 0.6 to 400:

Equation 4.6-4

The fluid properties are evaluated at the film temperature T_f. A somewhat more accurate correlation, which takes into account the effects of natural convection at these lower Reynolds numbers, is available for a N_Re range 1–17 000 from other sources (S2).

EXAMPLE 4.6-2. Cooling of a Sphere Using the same conditions as Example 4.6-1, where air at 1 atm abs pressure and 15.6°C is flowing at a velocity of 12.2 m/s, predict the average heat-transfer coefficient for air flowing past a sphere having a diameter of 51 mm and an average surface temperature of 82.2°C. Compare this with the value of h = 77.2 W/m2 · K for the flat plate in turbulent flow. Solution: The physical properties at the average film temperature of 48.9°C are the same as for Example 4.6-1. The NRe is Substituting into Eq. (4.6-4) for a sphere, Solving, h = 56.1 W/m2 · K (9.88 btu/h · ft2 · °F). This value is somewhat smaller than the value of h = 77.2 W/m2 · K (13.6 btu/h · ft2 · °F) for a flat plate.

4.6E. Flow Past Banks of Tubes or Cylinders

Many types of commercial heat exchangers are constructed with multiple rows of tubes, where the fluid flows at right angles to the bank of tubes. An example is a gas heater in which a hot fluid inside the tubes heats a gas passing over the outside of the tubes. Another example is a cold liquid stream inside the tubes being heated by a hot fluid on the outside.

Figure 4.6-1 shows the arrangement for banks of tubes in-line and banks of tubes staggered, where D is tube OD in m (ft), S_n is distance m (ft) between the centers of the tubes normal to the flow, and S_p that parallel to the flow. The open area to flow for in-line tubes is (S_n − D) and (S_p − D); for staggered tubes it is (S_n − D) and ( − D). Values of C and m to be used in Eq. (4.6-1) for a Reynolds-number range of 2000 to 40 000 for heat transfer to banks of tubes containing more than 10 transverse rows in the direction of flow are given in Table 4.6-2. For less than 10 rows, Table 4.6-3 gives correction factors.

Table 4.6-2. Values of C and m To Be Used in Eq. (4.6-1) for Heat Transfer to Banks of Tubes Containing More Than 10 Transverse Rows

Arrangement	C	m	C	m	C	m
In-line	0.386	0.592	0.278	0.620	0.254	0.632
Staggered	0.575	0.556	0.511	0.562	0.535	0.556
Source: E. D. Grimison, Trans. ASME, 59, 583 (1937).

Table 4.6-3. Ratio of h for N Transverse Rows Deep to h for 10 Transverse Rows Deep (for Use with Table 4.6-2)

N	1	2	3	4	5	6	7	8	9	10
Ratio for staggered tubes	0.68	0.75	0.83	0.89	0.92	0.95	0.97	0.98	0.99	1.00
Ratio for in-line tubes	0.64	0.80	0.87	0.90	0.92	0.94	0.96	0.98	0.99	1.00
Source: W. M. Kays and R. K. Lo, Stanford Univ. Tech. Rept. 15, Navy Contract N6-ONR-251 T.O.6, 1952.

For cases where S_n/D and S_p/D are not equal to each other, the reader should consult Grimison (G1) for more data. In baffled exchangers where there is normal leakage where all the fluid does not flow normal to the tubes, the average values of h obtained should be multiplied by about 0.6 (P3). The Reynolds number is calculated using the minimum area open to flow for the velocity. All physical properties are evaluated at T_f.

EXAMPLE 4.6-3. Heating Air by a Bank of Tubes Air at 15.6°C and 1 atm abs flows across a bank of tubes containing four transverse rows in the direction of flow and 10 rows normal to the flow at a velocity of 7.62 m/s as the air approaches the bank of tubes. The tube surfaces are maintained at 57.2°C. The outside diameter of the tubes is 25.4 mm and the tubes are in-line to the flow. The spacing Sn of the tubes normal to the flow is 38.1 mm and Sp is also 38.1 mm parallel to the flow. For a 0.305-m length of the tube bank, calculate the heat-transfer rate. Solution: Referring to Fig. 4.6-1a, Since the air is heated in passing through the four transverse rows, an outlet bulk temperature of 21.1°C will be assumed. The average bulk temperature is then The average film temperature is From Appendix A.3, for air at 37.7°C, The ratio of the minimum-flow area to the total frontal area is (Sn − D)/Sn. The maximum velocity in the tube banks is then For Sn/D = Sp/D = 1.5/1, the values of C and m from Table 4.6-2 are 0.278 and 0.620, respectively. Substituting into Eq. (4.6-1) and solving for h, This h is for 10 rows. For only four rows in the transverse direction, the h must be multiplied by 0.90, as given in Table 4.6-3. Since there are 10 × 4 or 40 tubes, the total heat-transfer area per 0.305 m length is The total heat-transfer rate q using an arithmetic average temperature difference between the wall and the bulk fluid is Next, a heat balance on the air is made to calculate its temperature rise ΔT using the calculated q. First, the mass flow rate of air m must be calculated. The total frontal area of the tube-bank assembly of 10 rows of tubes each 0.305 m long is The density of the entering air at 15.6°C is ρ = 1.224 kg/m3. The mass flow rate m is For the heat balance, the mean cp of air at 18.3°C is 1.0048 kJ/kg · K, and then Solving, ΔT = 5.37°C. Hence, the calculated outlet bulk gas temperature is 15.6 + 5.37 = 20.97°C, which is close to the assumed value of 21.1°C. If a second trial were to be made, the new average Tb to use would be (15.6 + 20.97)/2 or 18.28°C.

4.6F. Heat Transfer for Flow in Packed Beds

Correlations for heat-transfer coefficients for packed beds are useful in designing fixed-bed systems such as catalytic reactors, dryers for solids, and pebble-bed heat exchangers. In Section 3.1C the pressure drop in packed beds was considered and discussions of the geometry factors in these beds were given. For determining the rate of heat transfer in packed beds for a differential length dz in m,

Equation 4.6-5

where a is the solid-particle surface area per unit volume of bed in m^-1, S the empty cross-sectional area of bed in m², T₁ the bulk gas temperature in K. and T₂ the solid surface temperature.

For the heat transfer of gases in beds of spheres (G2, G3) and a Reynolds number range of 10–10 000,

Equation 4.6-6

where ν' is the superficial velocity based on the cross section of the empty container in m/s [see Eq. (3.1-11)], ε is the void fraction, N_Re = D_pG'/μ_f, and G' = v'ρ is the superficial mass velocity in kg/m² · s. The subscript f indicates properties evaluated at the film temperature, with others at the bulk temperature. This correlation can also be used for a fluidized bed. An alternate equation to use in place of Eq. (4.6-6) for fixed and fluidized beds is Eq. (7.3-36) for a Reynolds-number range of 10–4000. The term J_H is called the Colburn J factor and is defined as in Eq. (4.6-6) in terms of h.

Equations for heat transfer to noncircular cylinders such as hexagons and so forth are given elsewhere (H1, J1, P3).