4.9. HEAT EXCHANGERS

4.9A. Types of Exchangers

1. Introduction

In the process industries the transfer of heat between two fluids is generally done in heat exchangers. The most common type is one in which the hot and cold fluids do not come into direct contact with each other but are separated by a tube wall or a flat or curved surface. The transfer of heat from the hot fluid to the wall or tube surface is accomplished by convection, through the tube wall or plate by conduction, and then by convection to the cold fluid. In the preceding sections of this chapter we have discussed the calculation procedures for these various steps. Now we will discuss some of the types of equipment used and overall thermal analyses of exchangers. Complete, detailed design methods have been highly developed and will not be considered here.

2. Double-pipe heat exchanger

The simplest exchanger is the double-pipe or concentric-pipe exchanger. This is shown in Fig. 4.9-1, where one fluid flows inside one pipe and the other fluid flows in the annular space between the two pipes. The fluids can be in cocurrent or countercurrent flow. The exchanger can be made from a pair of single lengths of pipe with fittings at the ends or from a number of pairs interconnected in series. This type of exchanger is useful mainly for small flow rates.

3. Shell-and-tube exchanger

If larger flows are involved, a shell-and-tube exchanger is used, which is the most important type of exchanger in use in the process industries. In these exchangers the flows are continuous. Many tubes in parallel are used, where one fluid flows inside these tubes. The tubes, arranged in a bundle, are enclosed in a single shell and the other fluid flows outside the tubes in the shell side. The simplest shell-and-tube exchanger is shown in Fig. 4.9-2a for one shell pass and one tube pass, or a 1-1 counterflow exchanger. The cold fluid enters and flows inside through all the tubes in parallel in one pass. The hot fluid enters at the other end and flows counterflow across the outside of the tubes. Cross-baffles are used so that the fluid is forced to flow perpendicular across the tube bank rather than parallel with it. The added turbulence generated by this cross-flow increases the shell-side heat-transfer coefficient.

In Fig. 4.9-2b a 1-2 parallel-counterflow exchanger is shown. The liquid on the tube side flows in two passes as shown and the shell-side liquid flows in one pass. In the first pass of the tube side, the cold fluid is flowing counterflow to the hot shell-side fluid; in the second pass of the tube side, the cold fluid flows in parallel (cocurrent) with the hot fluid. Another type of exchanger has two shell-side passes and four tube passes. Other combinations of number of passes are also used sometimes, with the 1-2 and 2-4 types being the most common.

4. Cross-flow exchanger

When a gas such as air is being heated or cooled, a common device used is the cross-flow heat exchanger shown in Fig. 4.9-3a. One of the fluids, which is a liquid, flows inside through the tubes, and the exterior gas flows across the tube bundle by forced or sometimes natural convection. The fluid inside the tubes is considered to be unmixed, since it is confined and cannot mix with any other stream. The gas flow outside the tubes is mixed, since it can move about freely between the tubes, and there will be a tendency for the gas temperature to equalize in the direction normal to the flow. For the unmixed fluid inside the tubes, there will be a temperature gradient both parallel and normal to the direction of flow.

A second type of cross-flow heat exchanger shown in Fig. 4.9-3b is typically used in air-conditioning and space-heating applications. In this type the gas flows across a finned-tube bundle and is unmixed, since it is confined in separate flow channels between the fins as it passes over the tubes. The fluid in the tubes is unmixed.

Discussions of other types of specialized heat-transfer equipment will be deferred to Section 4.13. The remainder of this section deals primarily with shell-and-tube and cross-flow heat exchangers.

4.9B. Log-Mean-Temperature-Difference Correction Factors

In Section 4.5H it was shown that when the hot and cold fluids in a heat exchanger are in true countercurrent flow or in cocurrent (parallel) flow, the log mean temperature difference should be used:

Equation 4.9-1

where ΔT₂ is the temperature difference at one end of the exchanger and ΔT₁ at the other end. This ΔT_1m holds for a double-pipe heat exchanger and a 1-1 exchanger with one shell pass and one tube pass in parallel or counterflow.

In cases where a multiple-pass heat exchanger is involved, it is necessary to obtain a different expression for the mean temperature difference, depending on the arrangement of the shell and tube passes. Considering first the one-shell-pass, two-tube-pass exchanger in Fig. 4.9-2b, the cold fluid in the first tube pass is in counterflow with the hot fluid. In the second tube pass, the cold fluid is in parallel flow with the hot fluid. Hence, the log mean temperature difference, which applies either to parallel or to counterflow but not to a mixture of both types, as in a 1-2 exchanger, cannot be used to calculate the true mean temperature drop without a correction.

The mathematical derivation of the equation for the proper mean temperature to use is quite complex. The usual procedure is to use a correction factor F_T which is so defined that when it is multiplied by the ΔT_1m, the product is the correct mean temperature drop ΔT_m to use. In using the correction factors F_T, it is immaterial whether the warmer fluid flows through the tubes or the shell (K1). The factor F_T has been calculated (B4) for a 1-2 exchanger and is shown in Fig. 4.9-4a. Two dimensionless ratios are used as follows:

Equation 4.9-2

Equation 4.9-3

where T_hi = inlet temperature of hot fluid in K (°F), T_ho = outlet of hot fluid, T_ci inlet of cold fluid, and T_co = outlet of cold fluid.

In Fig. 4.9-4b, the factor F_T (B4) for a 2-4 exchanger is shown. In general, it is not recommended to use a heat exchanger for conditions under which F_T < 0.75. Another shell-and-tube arrangement should be used. Correction factors for two types of cross-flow exchanger are given in Fig. 4.9-5. Other types are available elsewhere (B4, P1).

Using the nomenclature of Eqs. (4.9-2) and (4.9-3), the ΔT_1m of Eq. (4.9-1) can be written as

Equation 4.9-4

Then the equation for an exchanger is

Equation 4.9-5

where

Equation 4.9-6

EXAMPLE 4.9-1. Temperature Correction Factor for a Heat Exchanger A 1-2 heat exchanger containing one shell pass and two tube passes heats 2.52 kg/s of water from 21.1 to 54.4°C by using hot water under pressure entering at 115.6 and leaving at 48.9°C. The outside surface area of the tubes in the exchanger is Ao = 9.30 m2. Calculate the mean temperature difference ΔTm in the exchanger and the overall heat-transfer coefficient Uo. For the same temperatures but using a 2-4 exchanger, what would be the ΔTm? Solution: The temperatures are as follows: First making a heat balance on the cold water, assuming a cpm of water of 4187 J/kg · K and Tco − Tci = (54.4 − 21.1)°C = 33.3°C = 33.3 K, The log mean temperature difference using Eq. (4.9-4) is Next, substituting into Eqs. (4.9-2) and (4.9-3), Equation 4.9-2 Equation 4.9-3 From Fig. 4.9-4a, FT = 0.74. Then, by Eq. (4.9-6), Equation 4.9-6 Rearranging Eq. (4.9-5) to solve for Uo and substituting the known values, we have For part (b), using a 2-4 exchanger and Fig. 4.9-4b, FT = 0.94. Then, Hence, in this case the 2-4 exchanger utilizes more of the available temperature driving force.

4.9C. Heat-Exchanger Effectiveness

1. Introduction

In the preceding section the log mean temperature difference was used in the equation q = UA ΔT_1m in the design of heat exchangers. This form is convenient when the inlet and outlet temperatures of the two fluids are known or can be determined by a heat balance. Then the surface area can be determined if U is known. However, when the temperatures of the fluids leaving the exchanger are not known and a given exchanger is to be used, a tedious trial-and-error procedure is necessary. To solve these cases, a method called the heat-exchanger effectiveness ε is used which does not involve any of the outlet temperatures.

The heat-exchanger effectiveness is defined as the ratio of the actual rate of heat transfer in a given exchanger to the maximum possible amount of heat transfer if an infinite heat-transfer area were available. The temperature profile for a counterflow heat exchanger is shown in Fig. 4.9-6.

2. Derivation of effectiveness equation

The heat balance for the cold (C) and hot (H) fluids is

Equation 4.9-7

Calling (mc_p)_H = C_H and (mc_p)_c = C_C, then in Fig. 4.9-6, C_H > C_C, and the cold fluid undergoes a greater temperature change than the hot fluid. Hence, we designate C_C as C_min or minimum heat capacity. Then, if there is an infinite area available for heat transfer, T_Co = T_Hi. Then the effectiveness ε is

Equation 4.9-8

If the hot fluid is the minimum fluid, T_Ho = T_Ci, and

Equation 4.9-9

In both equations the denominators are the same and the numerator gives the actual heat transfer:

Equation 4.9-10

Note that Eq. (4.9-10) uses only inlet temperatures, which is an advantage when inlet temperatures are known and it is desired to predict the outlet temperatures for a given existing exchanger.

For the case of a single-pass, counterflow exchanger, combining Eqs. (4.9-8) and (4.9-9),

Equation 4.9-11

We consider first the case when the cold fluid is the minimum fluid. Rewriting Eq. (4.5-25) using the present nomenclature,

Equation 4.9-12

Combining Eq. (4.9-7) with the left side of Eq. (4.9-11) and solving for T_Hi,

Equation 4.9-13

Subtracting T_Co from both sides,

Equation 4.9-14

From Eq. (4.9-7) for C_min = C_C and C_max = C_H,

Equation 4.9-15

This can be rearranged to give the following:

Equation 4.9-16

Substituting Eq. (4.9-13) into (4.9-16),

Equation 4.9-17

Finally, substituting Eqs. (4.9-14) and (4.9-17) into (4.9-12), rearranging, taking the antilog of both sides, and solving for ε,

Equation 4.9-18

We define NTU as the number of transfer units as follows:

Equation 4.9-19

The same result would have been obtained if C_H = C_min. For parallel flow we obtain

Equation 4.9-20

In Fig. 4.9-7, Eqs. (4.9-18) and (4.9-20) have been plotted in convenient graphical form. Additional charts are available for different shell-and-tube and cross-flow arrangements (K1).

EXAMPLE 4.9-2. Effectiveness of Heat Exchanger Water flowing at a rate of 0.667 kg/s enters a countercurrent heat exchanger at 308 K and is heated by an oil stream entering at 383 K at a rate of 2.85 kg/s (cp = 1.89 kJ/kg · K). The overall U = 300 W/m2 · K and the area A = 15.0 m2. Calculate the heat-transfer rate and the exit water temperature. Solution: Assuming that the exit water temperature is about 370 K, the cp for water at an average temperature of (308 + 370)/2 = 339 K is 4.192 kJ/kg · K (Appendix A.2). Then, (mcp)H = CH = 2.85(1.89 × 103) = 5387 W/K and (mcp)C = CC = 0.667(4.192 × 103) = 2796 W/K = Cmin. Since CC is the minimum, Cmin/Cmax = 2796/5387 = 0.519. Using Eq. (4.9-19), NTU = UA/Cmin = 300(15.0)/2796 = 1.607. Using Fig. (4.9-7a) for a counterflow exchanger, ε = 0.71. Substituting into Eq. (4.9-10), Using Eq. (4.9-7), Solving, TCo = 361.3 K.

4.9D. Fouling Factors and Typical Overall U Values

In actual practice, heat-transfer surfaces do not remain clean. Dirt, soot, scale, and other deposits form on one or both sides of the tubes of an exchanger and on other heat-transfer surfaces. These deposits offer additional resistance to the flow of heat and reduce the overall heat-transfer coefficient U. In petroleum processes coke and other substances can deposit. Silting and deposits of mud and other materials can occur. Corrosion products which could constitute a serious resistance to heat transfer may form on the surfaces. Biological growth such as algae can occur with cooling water and in the biological industries.

To avoid or lessen these fouling problems, chemical inhibitors are often added to minimize corrosion, salt deposition, and algae growth. Water velocities above 1 m/s are generally used to help reduce fouling. Large temperature differences may cause excessive deposition of solids on surfaces and should be avoided if possible.

The effect of such deposits and fouling is usually taken care of in design by adding a term for the resistance of the fouling on the inside and outside of the tube in Eq. (4.3-17) as follows:

Equation 4.9-21

where h_di is the fouling coefficient for the inside and h_do the fouling coefficient for the outside of the tube in W/m² · K. A similar expression can be written for U₀ using Eq. (4.3-18).

Fouling coefficients recommended for use in designing heat-transfer equipment are available in many references (P3, N1). A short tabulation of some typical fouling coefficients is given in Table 4.9-1.

Table 4.9-1. Typical Fouling Coefficients (P3, N1)

	hd (W/m2 · K)	hd (btu/h · ft2 · °F)
Distilled and seawater	11 350	2000
City water	5680	1000
Muddy water	1990–2840	350–500
Gases	2840	500
Vaporizing liquids	2840	500
Vegetable and gas oils	1990	350

In order to perform preliminary estimates of sizes of shell-and-tube heat exchangers, typical values of overall heat-transfer coefficients are given in Table 4.9-2. These values should be useful as a check on the results of the design methods described in this chapter.

Table 4.9-2. Typical Values of Overall Heat-Transfer Coefficients in Shell-and-Tube Exchangers (H1, P3, W1)

	U	U
	(W/m2· K)	(btu/h · ft2 · °F)
Water to water	1140–1700	200–300
Water to brine	570–1140	100–200
Water to organic liquids	570–1140	100–200
Water to condensing steam	1420–2270	250–400
Water to gasoline	340–570	60–100
Water to gas oil	140–340	25–60
Water to vegetable oil	110–285	20–50
Gas oil to gas oil	110–285	20–50
Steam to boiling water	1420–2270	250–400
Water to air (finned tube)	110–230	20–40
Light organics to light organics	230–425	40–75
Heavy organics to heavy organics	55–230	10–40