4.13. SPECIAL HEAT-TRANSFER COEFFICIENTS

4.13A. Heat Transfer in Agitated Vessels

1. Introduction

Many chemical and biological processes are often carried out in agitated vessels. As discussed in Section 3.4, the liquids are generally agitated in cylindrical vessels with an impeller mounted on a shaft and driven by an electric motor. Typical agitators and vessel assemblies have been shown in Figs. 3.4-1 and 3.4-3. Often it is necessary to cool or heat the contents of the vessel during agitation. This is usually done by heat-transfer surfaces, which may be in the form of cooling or heating jackets in the wall of the vessel or coils of pipe immersed in the liquid.

2. Vessel with heating jacke

In Fig. 4.13-1a, a vessel with a cooling or heating jacket is shown. When heating, the fluid entering is often steam, which condenses inside the jacket and leaves at the bottom. The vessel is equipped with an agitator and in most cases also with baffles (not shown).

Correlations for the heat-transfer coefficient from the agitated Newtonian liquid inside the vessel to the jacket walls of the vessel have the following form:

Equation 4.13-1

where h is the heat-transfer coefficient for the agitated liquid to the inner wall in W/m² · K, D_t is the inside diameter of the tank in m, k is thermal conductivity in W/m · K, D_a is diameter of agitator in m, N is rotational speed in revolutions per sec, ρ is fluid density in kg/m³, and μ, is liquid viscosity in Pa · s. All the liquid physical properties are evaluated at the bulk liquid temperature except μ_w, which is evaluated at the wall temperature T_w. Below are listed some available correlations and the Reynolds-number range ().

Some typical overall U values for jacketed vessels for various process applications are tabulated in Table 4.13-1.

Table 4.13-1. Typical Overall Heat-Transfer Coefficients in Jacketed Vessels

Fluid in Jacket	Fluid in Vessel	Wall Material	Agitation	U		Ref.
				btu	W
				h · ft2 · °F	m2 · K
Steam	Water	Copper	None	150	852	(P1)
			Simple stirring	250	1420
Steam	Paste	Cast iron	Double scrapers	125	710	(P1)
Steam	Boiling water	Copper	None	250	1420	(P1)
Steam	Milk	Enameled cast iron	None	200	1135	(P1)
			Stirring	300	1700
Hot water	Cold water	Enameled cast iron	None	70	398	(P1)
Steam	Tomato purée	Metal	Agitation	30	170	(C1)

EXAMPLE 4.13-1. Heat-Transfer Coefficient in Agitated Vessel with Jacket A jacketed 1.83-m-diameter agitated vessel with baffles is being used to heat a liquid which is at 300 K. The agitator is 0.61 m in diameter and is a flat-blade turbine rotating at 100 rpm. Hot water is in the heating jacket. The wall surface temperature is constant at 355.4 K. The liquid has the following bulk physical properties: ρ = 961 kg/m3, cp = 2500 J/kg · K, k = 0.173 W/m · K, and μ = 1.00 Pa · s at 300 K and 0.084 Pa · s at 355.4 K. Calculate the heat-transfer coefficient to the wall of the jacket. Solution: The following are given: First, calculating the Reynolds number at 300 K, The Prandtl number is Using Eq. (4.13-1) with a = 0.74, b = , and m = 0.14, Equation 4.13-1 Substituting and solving for h,

A correlation to predict the heat-transfer coefficient of a power-law non-Newtonian fluid in a jacketed vessel with a turbine agitator is also available elsewhere (C6).

3. Vessel with heating coils

In Fig. 4.13-1b, an agitated vessel with a helical heating or cooling coil is shown. Correlations for the heat-transfer coefficient to the outside surface of the coils in agitated vessels are listed below for various types of agitators.

For a paddle agitator with no baffles (C5),

Equation 4.13-2

This holds for a Reynolds-number range of 300 to 4 × 10⁵.

For a flat-blade turbine agitator with baffles, see (O1).

When the heating or cooling coil is in the form of vertical tube baffles with a flat-blade turbine, the following correlation can be used (D1):

Equation 4.13-3

where D₀ is the outside diameter of the coil tube in m, n_b is the number of vertical baffle tubes, and μ_f is the viscosity at the mean film temperature.

Perry and Green (P3) give typical values of overall heat-transfer coefficients U for coils immersed in various liquids in agitated and nonagitated vessels.

4.13B. Scraped-Surface Heat Exchangers

Liquid-solid suspensions, viscous aqueous and organic solutions, and numerous food products, such as margarine and orange juice concentrate, are often cooled or heated in a scraped-surface exchanger. This consists of a double-pipe heat exchanger with a jacketed cylinder containing steam or cooling liquid and an internal shaft rotating and fitted with wiper blades, as shown in Fig. 4.13-2.

The viscous liquid product flows at low velocity through the central tube between the rotating shaft and the inner pipe. The rotating scrapers or wiper blades continually scrape the surface of liquid, preventing localized overheating and giving rapid heat transfer. In some cases this device is also called a votator heat exchanger.

Skelland et al. (S4) give the following equation to predict the inside heat-transfer coefficient for the votator:

Equation 4.13-4

where D = diameter of vessel in m, D_S = diameter of rotating shaft in m, v = axial flow velocity of liquid in m/s, N = agitator speed in rev/s, and n_B = number of blades on agitator. Data cover a region of axial flow velocities of 0.076 to 0.38 m/min and rotational speeds of 100 to 750 rpm.

Typical overall heat-transfer coefficients in food applications are U = 1700 W/m² · K (300 btu/h · ft² · F) for cooling margarine with NH₃, 2270 (400) for heating applesauce with steam, 1420 (250) for chilling shortening with NH₃, and 2270 (400) for cooling cream with water (B6).

4.13C. Extended Surface or Finned Exchangers

1. Introduction

The use of fins or extended surfaces on the outside of a heat-exchanger pipe wall to give relatively high heat-transfer coefficients in the exchanger is quite common. An automobile radiator is such a device, where hot water passes inside through a bank of tubes and loses heat to the air. On the outside of the tubes, extended surfaces receive heat from the tube walls and transmit it to the air by forced convection.

Two common types of fins attached to the outside of a tube wall are shown in Fig. 4.13-3. In Fig. 4.13-3a there are a number of longitudinal fins spaced around the tube wall and the direction of gas flow is parallel to the axis of the tube. In Fig. 4.13-3b the gas flows normal to the tubes containing many circular or transverse fins.

The qualitative effect of using extended surfaces can be shown approximately in Eq. (4.13-5) for a fluid inside a tube having a heat-transfer coefficient of h_i and an outside coefficient of h₀:

Equation 4.13-5

The resistance R_metal of the wall can often be neglected. The presence of the fins on the outside increases A₀ and hence reduces the resistance 1/h₀A₀ of the fluid on the outside of the tube. For example, if we have h_i for condensing steam, which is very large, and h₀ for air outside the tube, which is quite small, increasing A₀ greatly reduces 1/h₀A₀. This in turn greatly reduces the total resistance, which increases the heat-transfer rate. If the positions of the two fluids are reversed, with air inside and steam outside, little increase in heat transfer could be obtained by using fins.

Equation (4.13-5) is only an approximation, since the temperature on the outside surface of the bare tube is not the same as that at the end of the fin because of the added resistance to heat flow by conduction from the fin tip to the base of the fin. Hence, a unit area of fin surface is not as efficient as a unit area of bare tube surface at the base of the fin. A fin efficiency η_f has been mathematically derived for various geometries of fins.

2. Derivation of equation for fin efficiency

We will consider a one-dimensional fin exposed to a surrounding fluid at temperature T_∞ as shown in Fig. 4.13-4. At the base of the fin the temperature is T₀ and at point x it is T. At steady state, the rate of heat conducted into the element at x is q_x|x and is equal to the rate of heat conducted out plus the rate of heat lost by convection:

Equation 4.13-6

Substituting Fourier's equation for conduction and the convection equation,

Equation 4.13-7

where A is the cross-sectional area of the fin in m², P the perimeter of the fin in m, and (P Δx) the area for convection. Rearranging Eq. (4.13-7), dividing by Δx, and letting Δx approach zero,

Equation 4.13-8

Letting θ = T - T_∞, Eq. (4.13-8) becomes

Equation 4.13-9

The first boundary condition is that θ = θ₀ = T₀ − T_∞ at x = 0. For the second boundary condition needed to integrate Eq. (4.13-9), several cases can be considered, depending upon the physical conditions at x = L. In the first case, the end of the fin is insulated and dθ/dx = 0 at x = L. In the second case, the fin loses heat by convection from the tip surface, so that -k(dT/dx)_L = h(T_L − T_∞). The solution using the second case is quite involved and will not be considered here. Using the first case, where the tip is insulated, integration of Eq. (4.13-9) gives

Equation 4.13-10

where m = (hP/kA)^1/2.

The heat lost by the fin is expressed as

Equation 4.13-11

Differentiating Eq. (4.13-10) with respect to x and combining it with Eq. (4.13-11),

Equation 4.13-12

In the actual fin the temperature T in the fin decreases as the tip of the fin is approached. Hence, the rate of heat transfer per unit area decreases as the distance from the tube base is increased. To indicate this effectiveness of the fin in transferring heat, the fin efficiency η_f is defined as the ratio of the actual heat transferred from the fin to the heat transferred if the entire fin were at the base temperature T₀:

Equation 4.13-13

where PL is the entire surface area of fin. The expression for mL is

Equation 4.13-14

For fins which are thin, 2t is small compared to 2w, and

Equation 4.13-15

Equation (4.13-15) holds for a fin with an insulated tip. This equation can be modified to hold for the case where the fin loses heat from its tip. This can be done by extending the length of the fin by t/2, where the corrected length L_c to use in Eqs. (4.13-13)–(4.13-15) is

Equation 4.13-16

The fin efficiency calculated from Eq. (4.13-13) for a longitudinal fin is shown in Fig. 4.13-5a. In Fig. 4.13-5b, the fin efficiency for a circular fin is presented. Note that the abscissa on the curves is L_c(h/kt)^1/2 and not L_c(2h/kt)^1/2 as in Eq. (4.13-15).

EXAMPLE 4.13.-2. Fin Efficiency and Heat Loss from Fin A circular aluminum fin as shown in Fig. 4.13-3b (k = 222 W/m · K) is attached to a copper tube having an outside radius of 0.04 m. The length of the fin is 0.04 m and the thickness is 2 mm. The outside wall or tube base is at 523.2 K and the external surrounding air at 343.2 K has a convective coefficient of 30 W/m2 · K. Calculate the fin efficiency and rate of heat loss from the fin. Solution: The given data are T0 = 523.2 K, T∞ = 343.2 K, L = 0.04 m, rl = 0.04 m, t = 0.002 m, k = 222 W/m · K, h = 30 W/m2 · K. By Eq. (4.13-16), Lc = L + t/2 = 0.040 + 0.002/2 = 0.041 m. Then, Also, (Lc + r1)/r1 = (0.041 + 0.040)/0.040 = 2.025. Using Fig. 4.13-5b, ηf = 0.89. The heat transfer from the fin itself is Equation 4.13-17 where Af is the outside surface area (annulus) of the fin and is given by the following for both sides of the fin: Equation 4.13-18 Hence, Substituting into Eq. (4.13-17),

3. Overall heat-transfer coefficient for finned tubes

We consider here the general case similar to Fig. 4.3-3b, where heat transfer occurs from a fluid inside a cylinder or tube, through the cylinder metal wall A of thickness Δx_A, and then to the fluid outside the tube, where the tube has fins on the outside. The heat is transferred through a series of resistances. The total heat q leaving the outside of the tube is the sum of heat loss by convection from the base of the bare tube q_t and the loss by convection from the fins, q_f:

Equation 4.13-19

This can be written as a resistance since the paths are in parallel:

Equation 4.13-20

where A_t is the area of the bare tube between the fins, A_f the area of the fins, and h₀ the outside convective coefficient. The resistance in Eq. (4.3-20) can be substituted for the resistance (1/h₀A₀) in Eq. (4.3-15) for a bare tube to give the overall equation for a finned tube exchanger:

Equation 4.13-21

where T₄ is the temperature of the fluid inside the tube and T₁ the outside fluid temperature. Writing Eq. (4.13-21) in the form of an overall heat-transfer coefficient U_i based on the inside area A_i, q = U_iA_i(T₄ - T₁) and

Equation 4.13-22

The presence of fins on the outside of the tube changes the characteristics of the fluid flowing past the tube (either flowing parallel to the longitudinal finned tube or transverse to the circular finned tube). Hence, the correlations for fluid flow parallel to or transverse to bare tubes cannot be used to predict the outside convective coefficient h₀. Correlations are available in the literature (K4, M1, P1, P3) for heat transfer to various types of fins.