Chapter 2. Process Heat Transfer

In Figure 2.1, subscripts 1 and 2 indicate inlet and outlet conditions, respectively. The temperatures of the two streams are identified by uppercase and lowercase letters (T and t). For the simple case shown in Figure 2.1, there are no phase changes in either stream. In addition, the specific heat capacities of both streams are assumed to be constant, which gives rise to the linear temperature profiles shown in Figure 2.1. The energy balances for both streams are

where capital letters are used for the hot stream and lowercase letters are used for the cold stream. The term Q is the total rate of heat transferred between streams (energy/time), and are the mass flowrates, and C_p and c_p are the heat capacities of the streams. The streams are physically separated, usually by a metal wall, and heat is exchanged through this metal wall. The driving force for the heat exchange is, of course, the temperature difference (T_z − t_z) between the two streams at the given location, z, in the heat exchanger. In general, this driving force changes throughout the exchanger, and some appropriate average should be used. In order to obtain a relationship between Q and the inlet and exit temperatures of each stream a simple differential balance can be written on an element of the heat exchanger, as illustrated in the right-hand sketch of Figure 2.1, to give

where (T_z − t_z) = ΔT_z. For many applications, the overall heat transfer coefficient U does not vary substantially with the location in the exchanger and thus U_z = U. Since the heat capacities of both streams are assumed to be constant, the temperature difference between the streams (T_z − t_z) or ΔT_z is also linear with respect to the amount of heat transferred, Q, as illustrated in Figure 2.2.

From Figure 2.2, it can be seen that

Substituting Equation (2.4) into Equation (2.3) and dropping the subscript z gives

Integrating Equation (2.5) gives

or

where ΔT_lm is known as the log-mean temperature difference, or LMTD, and is given as

For heat-exchanger design and performance calculations, the LMTD is the correct average temperature to use when the streams flow countercurrently, and Equation (2.8) is the design equation for heat exchangers.

Note: For the special case when the or when there are constant-pressure phase changes for both streams, then the ΔT between the hot and cold streams is constant throughout the heat exchanger; that is, the T-Q lines for both fluids are parallel. For such cases, the expression for LMTD gives 0/ln(1), which is indeterminate, but by using L’Hôpital’s rule, it can be shown that LMTD = ΔT. Since the LMTD is simply an average temperature difference between the streams, it should make sense that the average between identical temperature differences is just that temperature difference.

A similar analysis to that given for the countercurrent exchanger could be carried out for the cocurrent configuration and would result in the same Equation (2.5). However, now ΔT₁ = (T₁− t₁) and ΔT₂ = T₂− t₂. The cocurrent configuration is less efficient in terms of heat exchange for two reasons. First, the average driving force is smaller than for countercurrent operation; therefore, the heat-exchange area for a given Q will be higher for cocurrent flow than for countercurrent flow. Second, the limiting temperature that either stream can reach is the exit temperature of the other stream, whereas for countercurrent operation, the limiting exit temperature is the inlet temperature of the other stream. Examples 2.1 and 2.2 illustrate these differences.

For all these cases, the LMTD should be used as the appropriate temperature driving force in heat-exchanger calculations. More complicated examples exist when both temperature and phase changes for one or both fluids occur within the same heat exchanger. This results in sensible heat changes () and phase changes () occurring within the same heat exchanger. For example, Figure 2.5 illustrates the case when Stream 2 is a subcooled liquid that is heated to saturation, boiled, and then the vapor is superheated using a superheated vapor (Stream 1) that cools, condenses, and subcools.

From Figure 2.5, it should be clear that using a single LMTD (shown by the dotted lines drawn between the end-point temperatures) based on the conditions of the streams entering and leaving the heat exchanger does not provide a realistic or accurate estimate of the correct average driving force within the heat exchanger. In cases such as these, it is necessary to divide the heat exchanger into sections in which the T versus Q line is a single straight line segment for each fluid in each section. For the case illustrated in Figure 2.5, this would result in five separate sections, known as zones, labeled A through E in the figure. The design (and performance) of the heat exchanger would then follow a “zoned analysis” that would consider each section or zone (A–E in Figure 2.5) separately.

Comparing Figure 2.6 and the analysis carried out in Section 2.1.1, it should be clear that since the T-Q lines for at least one of the streams is not straight, the integration of Equation (2.3) will no longer yield the LMTD and Equation (2.7). For these cases, the usual practice is to integrate Equation (2.3) numerically to give the appropriate form of Equation (2.8).

An alternative method is to approximate the curved T-Q line as straight line segments, and the LMTD for each segment can then be found, similar to a zoned analysis. Figure 2.7 illustrates this method.

where the term is the logarithmic mean of the product of U and ΔT. Example 2.3 illustrates the use of this equation.

• Shell and shell cover

• Tubes and tubesheet

• Channel and channel cover

• Baffles

• Inlet and outlet nozzles

A good review of each of these components is given by Mukherjee (1998), and a summary of the results of this work are presented in the next section. The TEMA designations for S-T heat exchangers are given in Figure 2.8, the notation for several examples of S-T exchangers are given in Figure 2.9, and some standard designs are given in Figure 2.10.

The arrangement or layout of tubes in an S-T exchanger is determined by the rotation or orientation of the tubes relative to the shell-side fluid, the spacing between the tube centers (the tube pitch), and the layout pattern (triangular or square). Some variations of these three factors are illustrated in Figure 2.12. In general, a given shell diameter accommodates more tubes if a triangular arrangement is used (Figure 2.12[c] and 2.12[d]), and this leads to more turbulence on the shell side and a correspondingly higher shell-side heat transfer coefficient. On the down side, when the typical tube pitch of 1.25 times the tube outer diameter is used, a triangular arrangement will not permit the mechanical cleaning of the tubes, because there are no access lanes between the tubes to allow for a brush or similar device to clean the accumulated scale from the outside surfaces of the tubes. Mechanical cleaning on the shell side using a triangular arrangement of tubes is possible if a larger tube pitch is used, but this requires a larger shell diameter and greater expense. When mechanical cleaning is desired on the shell side, a square or rotated square pitch is preferred.

• If one fluid is highly corrosive (compared to the other), this fluid should be placed in the tubes. This heuristic is due to the corrosive fluid needing to be in contact with a corrosion-resistant alloy, which is usually expensive. It makes the best economic sense to have the tubes (and connecting piping) made from this alloy. On the other hand, the shell (and baffles and tie rods, etc.) can be made out of a less expensive material (like carbon steel) that is only in contact with the shell-side (less corrosive) fluid.

• If the temperature and pressure conditions for one of the fluids require the use of specialty alloys, that fluid is placed in the tubes for the same reasons given for highly corrosive fluids.

• When one fluid is at much higher pressure than the other fluid, that fluid is placed in the tubes. Since higher pressure requires thicker walls, it makes sense to place the high-pressure fluid inside the tubes, while the shell can be designed for the lower-pressure fluid.

• If one fluid causes severe fouling or scaling, that fluid is placed in the tubes, but a U-tube (Type U) design is not used. Since straight tubes can be mechanically cleaned relatively easily, it makes sense to put the fouling fluid in the tubes.

• Fixed tubesheet exchangers have been used for temperature differences up to 110°C (200°F) with rolled expansion joints for moderate pressures in the shell (<10 bar).

• U-tube bundles provide a comprehensive solution to the thermal stress problem on the tube side, because they contract and expand independently of the shell, but tubesheet stresses still exist.

• If one fluid is very viscous, this fluid should be placed on the shell side, provided that by doing so the fluid is in the turbulent regime (Re_s > 100). If turbulent flow conditions cannot be achieved in the shell, the viscous fluid should be placed in the tubes.

• If one fluid requires a low-pressure drop through the exchanger, this fluid is placed on the shell side of the exchanger. In general, the shell-side pressure drop is lower than the tube-side pressure drop. In addition, as discussed in Section 2.2.1.1, many different shell configurations are available that lower the shell-side pressure drop.

To account for flow patterns that are not purely countercurrent, Equation (2.8) can be rewritten to include an LMTD correction factor, F_y, where the subscript refers to a type of flow such as cocurrent or cross-current or to the type of heat exchanger being used. Thus, the generalized working design equation for exchangers becomes:

By convention, the LMTD for pure countercurrent flow is taken as the reference configuration, and the correction factor, F_y, is always the correction relative to pure countercurrent flow. In this way, F_y is always less than or equal to 1.0. For pure cocurrent flow, the LMTD correction factor, F_cocurrent, is given by

where P is the ratio of the temperature change of the tube-side stream to the maximum temperature difference in the exchanger and R is the ratio of heat capacities of the tube-side to shell-side streams,

In the process industries, the workhorse of industrial heat exchangers is the S-T design. Many of the important design issues for these heat exchangers were covered in Section 2.2. However, in the next section, the effect of different, nonideal flow patterns on the LMTD correction factor (F_y) is covered.

Figure 2.18 is a schematic diagram illustrating the flow pattern for a 1–2, S-T heat exchanger. Many of the technical details of S-T heat exchangers are covered in Section 2.2, but a further, brief description of the fluid flow paths is given here. Referring to Figure 2.18, the tube-side fluid enters from the top of the diagram and passes into a head-channel. The fluid then flows horizontally (from left to right) through a series of parallel tubes into a second head-channel on the right of the diagram. The tube-side stream then reverses direction and passes from right to left through the bottom set of parallel tubes into the left-hand head-channel where it finally leaves the heat exchanger. The shell-side fluid enters the shell, which is separated from the two head-channels by a tubesheet at either end of the exchanger. The shell-side fluid passes from left to right in the shell and is guided by a set of baffles that direct the flow. This fluid finally leaves the shell at the bottom right in the diagram. An insert is shown at the top of Figure 2.18 that shows in more detail the flow patterns of the different streams. The insert identifies three regions (labeled i, ii, and iii). In Region i, the shell-side and tube-side fluids flow cocurrently with each other. In Region ii, the shell-side fluid flows perpendicular or cross-currently with the fluid in the tubes. Finally, in Region iii, the shell- and tube-side fluids flow counter-current to one another. From this description, it should be clear that the flow pattern in this type of heat exchanger does not fit either the countercurrent or cocurrent models described previously.

Despite the apparently complicated flow pattern shown in Figure 2.18, it has been shown by Bowman, Mueller, and Nagle (1940) that the flow pattern is closely approximated by half the exchanger with cocurrent flow (top half of tubes in Figure 2.18) and the other half with counter-current flow. With this assumption, an analytical expression for F_1–2 can be found, and is given as Equation (2.13) and Figure 2.19.

Figure 2.19 illustrates the correction factor, F_1–2, as a function of P and R for a 1–2, S-T exchanger. It should be noted that the correction factor for a 1–2, S-T exchanger is essentially the same (within 1% to 2%) for a single shell pass with any number of even tube passes, and the results from Equation (2.13) and Figure 2.19 may be used. Therefore, F_1–2 = F_1–4 = F_1–6 = F_1–8, and so on.

From Figure 2.19, it should be clear that for any given value of R, there exists a maximum value of P, P_max. The value of P_max is given by Equation (2.14):

For values of P > P_max no solution to Equation (2.13) exists, or to put it another way, a 1–2, S-T exchanger cannot always be used to make the temperature of the streams change as desired. The uses of the LMTD correction factor, Figure 2.19, and the limitations of 1–2, S-T exchangers are illustrated in Example 2.5.

Two important conclusions can be drawn from Example 2.5. First, by comparing the results for Parts (b) and (d), it can be seen that it does not matter what stream is chosen for the shell-side or tube-side fluid. Equation (2.13) and Figure 2.19 are essentially symmetrical in R and P, so that the choice of which fluid to put in the shell and which to put in the tubes does not affect the result and value of F_1–2. Other factors do affect which fluid goes where, but these were covered in Section 2.2.1.5. Secondly, it was seen that, as the exit temperature of the process fluid decreased, F_1–2 started to decrease. Specifically, when the outlet temperature of the hot stream drops below the outlet temperature of the cold stream (T₂ < t₂), the efficiency drops very rapidly, and for Part (c) no solution existed. These results can be explained by the fact that as the temperature between the hot exit and cold exit streams approaches zero, the efficiency of the 1–2, S-T design is reduced significantly. This effect is illustrated in Figure 2.20. For pure cocurrent flow, the highest temperature to which the cold stream can be heated is equal to the hot stream exit temperature. However, in a 1–2, S-T exchanger only half of the flow is cocurrent, so not surprisingly, the efficiency (F_1–2) of the heat exchanger starts to decrease rapidly when this condition is reached. Heat exchangers with multiple shell passes more closely mimic pure countercurrent flow, and these are covered next.

The line representing Equation (2.15) can be approximated by the condition F_1–2 = 0.8 on Figure 2.19. The usual design criterion for exchanger design is to limit the LMTD correction factor to a value ≥ 0.8. If the condition of F = 0.8 is taken as the reasonable practical limit of operation of a 1–2, S-T exchanger, the number of shells needed for any design can be calculated graphically using a McCabe-Thiele–type construction, illustrated in Figure 2.21. The analytical expression for the number of shells using this criterion if the specific heats of hot and cold streams are constant (lines on the T-Q diagram are straight) is given by

Analytical expressions for the LMTD correction factor, F, for higher shell and tube passes exist, and many of these are given by Bowman et al. (1940). For a 2–4, S-T exchanger, Equation (2.17) applies (this expression can also be used for a 2–4N, S-T exchanger, where N = 1, 2, 3,...):

For the case when R = 1, Equation (2.17) reduces to Equation (2.18):

For higher shell passes, the analytical expressions become more complicated. However, Bowman (1936) has shown that for any given values of F and R, the value of P for an exchanger with N shell-side and 2N tube-side passes can be related to P for a 1–2, S-T exchanger through Equation (2.19):

Equation (2.19) was used to generate the LMTD correction factors for 3–6, 4–8, 5–10, and 6–12 S-T exchangers, and the results are presented in Appendix 2.A. LMTD correction factors for many types of heat exchanger can also be found at http://checalc.com/solved/LMTD_Chart.html. It is observed that as the number of shell passes increases, the curve for the same value of R moves up, resulting in a higher F value. Example 2.6 illustrates the use of multiple shell heat exchangers.

For steady-state operations, the amount of heat transferred through each layer or resistance must the same. By writing the basic heat flux equation across each resistance and eliminating the intermediate temperatures, an overall expression of the heat transferred in terms of the bulk temperature driving force (which is used in the exchanger design equation) is obtained. This process is given for the planar geometry as follows:

Adding all the left-hand sides together gives

where

For circular (tubular) geometry with a single wall and scale formation on the inner and outer surfaces, a similar analysis gives

where the subscripts o and i refer to the outside and inside of the tube, respectively, and the resistances due to fouling are given in terms of an effective fouling heat transfer resistance, R_fo = 1/h_of and R_fi =1/h_if, respectively. The form of Equation (2.21) differs from that derived for planar geometry, because for circular geometry, the cross-sectional area through which the heat flows changes from the outside to the inside of the tube. By eliminating the intermediate temperatures and rearranging, the following working equations for circular geometry can be derived:

where

and

where D_i and D_o are the inside and outside diameters of the tube (or pipe), respectively.

From Equations (2.23) and (2.24), it is clear that for circular geometry, the overall heat transfer coefficient must be defined with respect to a specific surface but that the product UA is the same no matter what surface is chosen; that is, U_iA_i = U_oA_o. It should also be noted that Equation (2.22) was derived for a specific location where ΔT_Bulk is known. If the temperature difference between fluids varies with location, as it would in most heat exchangers, then an appropriate average temperature difference should be used. The analysis given in Section 2.1 and all the limitations discussed in Section 2.1 still apply. Thus the general working equation for heat transfer between fluids exchanging heat through tube walls becomes

In summary, for heat transfer between fluid streams separated by a wall, the overall heat transfer resistance is a function of the two film heat transfer coefficients, two fouling resistances, and a resistance due to the wall. In the next section, typical correlations and values required to determine all these resistances are presented.

Fouling resistances for water and chemical process streams as suggested by TEMA (2013) are shown in Tables 2.1 and 2.2.

For S-T heat exchanges, the sizes of standard tubes are given in Table 2.4. The third column heading is BWG, which stands for Birmingham Wire Gauge, which is a standard method, albeit a very old standard, of specifying tube wall thickness. Unlike schedule pipe sizes discussed in Chapter 1, where the nominal pipe size does not correspond to any actual dimension of the pipe, the stated diameter of BWG tubing is the outside diameter. Standard tube sizes are virtually always used in heat-exchanger design, because the cost of customized tubes and the associated fittings and tooling would be prohibitively expensive. Standard lengths for tubes used in heat-exchanger design are from 8 ft to 20 ft (2.438–6.096 m) in 2 ft increments. The number of tubes that can fit in standard shell diameters is given in the tube-count tables shown later in Section 2.7.

where Nu is the Nusselt number, Re is the Reynolds number, and Pr is the Prandtl number. The Prandtl number is the ratio of the rate of viscous diffusion to the rate of thermal diffusion. The Seider-Tate correlation, Equation (2.26), is valid for all fluids (except molten metals) for Re ≥ 6000. All the fluid properties (ρ, μ, c_p, k) should be evaluated at the average bulk temperature with the exception of μ_w, which is evaluated at the average wall temperature. The first term in parentheses on the right-hand side takes account of the entrance effects for short tubes. For the case when L/D > 72, the error in neglecting this term is <5%. The last term in parentheses on the right-hand side takes account of the viscosity change near the tube wall that affects the thickness of the thermal boundary layer. Usually this term is close to unity, partly due to the small exponent. However, the term may become important for highly viscous fluids, where large changes in viscosity may take place because of the temperature variation between the bulk fluid and the wall. The heat transfer coefficient determined from Equation (2.26) changes along the length of the heat exchanger as the viscosity at the wall changes with temperature and position; therefore, use of the Seider-Tate equation with the viscosity correction requires an iterative procedure.

Another common correlation is the Dittus-Boelter (1930) equation that is given by

where the value of the exponent n is 0.3 when the fluid is being cooled and 0.4 when the fluid is being heated. This equation does not require an iterative solution but is generally less accurate than Equation (2.26), especially for fluids undergoing large temperature changes.

For an annulus, the following equation can be used:

where D_i−o is the inside diameter of the outer pipe and D_o−i is the outside diameter of the inside pipe, and D_H is hydraulic diameter of the annular opening = (D_i−o − D_o−i), where D_H is defined in Equation (2.29).

For noncircular ducts (square, rectangle, triangular, and ellipsoidal), Equations (2.26) and (2.27) can still be used except that the hydraulic diameter, D_H, should be substituted for D_i in the equation (and in the definition of Re), where

The use of Equations (2.26) and (2.27) is illustrated in Example 2.7.

and Seider and Tate (1936),

are recommended.

It should be noted that Gz is a function of 1/L, so the heat transfer coefficient is strongly affected by the tube length. As the tube length becomes very long and Gz tends to zero, the Nusselt number tends to a limiting value of Nu_∞. When viscosity corrections are negligible, it can be seen that Nu_∞ = 3.66 assuming constant wall temperature.

Example 2.8 demonstrates the use of the equations for laminar flow.

Clearly, the heat transfer coefficients for laminar flow are much lower than for turbulent flow. In general, laminar flow should be avoided except for highly viscous liquids where pumping costs needed to give turbulent flow conditions may become prohibitively large.

For laminar flow in annuli, the heat transfer correlation of Chen, Hawkins, and Solberg (1946) is recommended:

where Re and Gz are based on the hydraulic diameter, D_H, μ_w is the viscosity at the inner wall of the annulus (at D_o−i), and D_i−o and D_o−i are the inside diameter of the outside tube and the outside diameter of the inside tube, respectively.

For flow in rectangular channels, the limiting value of Nu for a constant wall temperature and very long channels (Nu_∞) is a function of the ratio of the shorter to longer dimensions of the channel (0 < a/b < 1). The equivalent limiting value for tubes is 3.66, which is the first term on the right-hand side of Equation (2.31). The relationship between Nu_∞ and (a/b) for rectangular channels attributed to Clark and Kays (1953) is shown in Figure 2.25 and Equation (2.34).

It is suggested that Equation (2.31) be modified to use with rectangular channels as

where Nu_∞ is taken from Equation (2.34) and the hydraulic diameter is used in determining Re in Gz.

where l and v refer to saturated liquid and vapor conditions, s = 1.0 for water and 1.7 for other materials, and C_f is a constant that depends on the material of the heated surface and varies between 0.006 and 0.013.

This gives the equivalent heat transfer coefficient as

The application of Equations (2.47) and (2.48) is illustrated in Example 2.11.

For the flow in a vertical tube, from Figure 2.29(a), it can be seen that boiling starts at the bottom of the tube at some point above the inlet and results in a bubbly flow. As bubbles start to coalesce and form bigger spherical-cap-shaped bubbles, the flow pattern becomes more chaotic, and large slugs start to form and move rapidly upward. As more of the liquid becomes vaporized, there is a transition region where the continuous phase switches from liquid to vapor. The liquid tends to move up the tube at the walls, and the vapor forms a core at the center of the tube, in which nonvaporized liquid drops travel. Above the core-annular region, all the liquid at the walls has been vaporized, and the entrained drops of liquid in the vapor rapidly evaporate until only vapor exists. If the tube extends further vertically, then superheating of the vapor will occur. The situation for forced boiling in horizontal tubes is illustrated in Figure 2.29(b). The flow pattern in the horizontal flow mimics the vertical flow pattern in many ways. The big difference between the two orientations is that for horizontal flow, gravity tends to force the liquid to the bottom of the tube and the flow pattern is not radially symmetrical, as shown by the inserts above the figure. It should also be apparent that the local heat transfer coefficient varies widely over the range of conditions in the tube, which is also shown in Figure 2.29(b).

From the description of the flows given for these two orientations, it is clear that the heat transfer coefficient probably changes significantly over the length of the tube. Moreover, the flow patterns are very complex and depend on many parameters. Indeed, the accurate prediction of pressure drop for these two-phase flows is difficult. There have been many attempts to correlate heat transfer coefficients with the flow regimes and also in combination with parameters that describe the two-phase pressure drop. The approach used here is to give the results from a study by Gungor and Winterton (1986), in which correlations were developed for a large set of data taken from a wide variety of boiling liquids in horizontal and vertical tubes. The authors claim that the correlation is able to predict the data within ±25%. The general form of their correlation is

where h_cb is the overall convective boiling coefficient, h_l is the convective coefficient for liquid flow in the tube, and h_pb is the pool boiling coefficient. The factor f, which is greater than 1, accounts for the much higher velocities for two-phase flow compared with single-phase flow. This factor was correlated against the two-phase Martinelli parameter, X_tp, and is given by

where X_tp is the Martinelli two-phase flow parameter given by

where x is the stream quality (mass fraction of vapor in the stream), and density and viscosity are for the saturated liquid and vapor conditions. B_o is a boiling number and is given by

where Q/A is the heat flux, λ is the latent heat of vaporizations, and G is the superficial mass velocity flowing (axially) past the heating surface. In using Equations (2.51) and (2.52), it should be noted that as x → 0 or 1, the equations are no longer valid. These limits, however, are represented by the pure convective heat transfer coefficients for all liquid flow and all vapor flow, respectively.

The parameter s in Equation (2.49) is a “suppression” factor that is less than 1 and accounts for the lower superheat that is available in convective boiling compared to pool boiling alone. This parameter is given by

where Re_l is the Reynolds number for the flow inside the tube assuming that the flow is all liquid.

The convective coefficient in Equation (2.49), h_l, is taken to be from the Seider-Tate or Dittus-Boelter correlations (Equation [2.26] or [2.27]) for the liquid. The recommended expression for pool boiling is taken from the correlation attributed to Cooper (1984) and is given by

or

where P_r is the reduced pressure, M is the molecular weight (g/mol) and (Q/A) is the heat flux (W/m²), and the units of h_pb are W/m²/K. The pool boiling coefficient from Equation (2.48) could alternatively be used instead of Equation (2.54).

The sequence of Equations (2.49) to (2.54) must be solved to determine the convective boiling heat transfer coefficient and in general will require an iterative method because the heat flux (Q/A) is imbedded in the boiling number term B_o (and Equation [2.52]), which will not be known a priori. Moreover, the value of h_cb changes along the length of the tube as the vapor quality changes. An illustration of the use of equations for forced convection boiling is given in Example 2.12.

where the bar over the symbol represents an average heat transfer coefficient over the length of the tube. Often, the convective heat transfer coefficient in Equation (2.56) is augmented by a radiation component that becomes increasingly important as the absolute temperature difference between the surface and the boiling point of the liquid increases. When this is the case, the overall heat transfer coefficient, , should be calculated from

where is calculated from

where ɛ is the emissivity of the tube surface, σ is the Stefan-Boltzmann constant (5.6704 × 10⁻⁸ W/m⁻²/K⁻⁴), and the temperatures used in Equation (2.58) are absolute (Kelvin). It should be clear that when radiation is important, the calculation of the boiling film heat transfer coefficient using Equations (2.56), (2.57), and (2.58) is an iterative process.

When a vapor close to its saturation temperature, T_sat, is exposed to a cold surface at a temperature below T_sat, then condensation of the vapor occurs on the surface.

In heat exchangers, this process is often accomplished by exposing a saturated vapor on the shell side of an S-T exchanger to horizontal (or vertical) tubes through which a cooling liquid is passed. At steady state, the liquid forms a film around the circumference of the tube and falls by gravity onto tubes situated below it. This phenomenon is illustrated in Figure 2.30(a).

Shown in Figure 2.30(b) is the condensation of liquid on a flat, vertical plate. As the liquid falls, the liquid layer increases in thickness, and the resistance to heat transfer increases because the path for energy flow through the liquid film becomes longer. In Figure 2.30(c) the condensation on a stack of vertically aligned horizontal tubes is illustrated.

• Laminar flow of fluid with constant physical properties

• Gas is a pure vapor at T_sat. Condensation occurs only at the vapor-liquid interface, that is, no heat transfer resistance in gas

• Shear stress at the liquid-vapor boundary is zero, that is, no vapor velocity or vapor thermal boundary layer effects

• Heat transfer across the liquid film is only by conduction

Nusselt’s analysis leads to the following expression for the average heat transfer coefficient over the length, L, of the plate:

Alternatively, the average heat transfer coefficient can be written as

where, according to McAdams (1954), all the physical properties of the liquid should be evaluated at the average film temperature, T_f = T_sat − 0.75(T_sat − T_w).

The total rate at which liquid condenses can then be determined via an energy balance as

The condition for which the expression in Equation (2.60) is valid is determined on the basis of the value of an effective Reynolds number for the falling film, Re_δ, given by

where δ is the thickness of the falling film (Figure 2.30[b]) and is the mass flow rate of condensate falling down a plate of width W. Equation (2.60) is valid for Re_δ ≤ 30. For the region 30 ≤ Re_δ ≤ 1800, the flow down the plate is described as wavy laminar and there is a slight enhancement of the heat transfer coefficient such that from Equation (2.60) should be multiplied by E, where E is given by

where the latent heat is modified to include subcooling in the film as

For the case when the flow in the film is turbulent (and Re_δ,max > 1800), McAdams [1954] suggests the use of the following equation:

For film condensation on the outside of horizontally oriented tubes, the following expression was obtained by Nusselt:

Because the path length for the film formation (~πD_o/2) on tubes used for commercial heat exchangers is relatively small, turbulent flow rarely occurs in horizontal-tube condensers. An example to illustrate use of the correlations for condensing heat transfer is given in Example 2.14.

From Figure 2.31, it can be seen that a wide variety of fin geometries is possible, and the equations describing the temperature profile in the fin can be quite complicated. In subsequent sections, the equations for both simple and more complicated geometries are presented without derivation.

For the rectangular fin, L is the length of the fin and δ is the thickness of the fin; the fin effectiveness is given by Equation (2.68) and is plotted in Figure 2.32. The term m is a dimensionless parameter that is defined in Equation (2.69).

The value of ɛ_fin indicates the efficiency by which the additional fin surface area is being utilized. For example, if the thermal conductivity of the fin material is very high or the fin is short, then, all else being equal, the temperature everywhere along the length of the fin is close to the wall temperature, and temperature driving force between the fin and the surroundings will be close to T₀ − T_fluid. Therefore, the fin surface area is used effectively. If, on the other hand, the fin length, L, is very long compared with its thickness, δ, then the temperature of the fin over a major portion of its length is close to T_fluid, and the efficiency of the fin is low, since only a small portion of the fin (close to the wall) has a significant temperature driving force to exchange heat with the surrounding fluid. In other words, the good news is that the fin provides more area for heat transfer, but the bad news is that the driving force may decrease down the length of the fin, so all of the benefit of the increased area is not seen. The effectiveness factor quantifies this situation.

In the previous analysis, it was assumed that the tip of the fin did not lose any heat or that an adiabatic boundary condition at the end of the fin was used. This assumption is strictly correct only for fins with a very large value of L/δ. However, the results given in Equation (2.68) and Figure 2.32 can be used with great accuracy if a corrected fin length, L_c, is used, where

where and I_i and K_i are modified Bessel functions of the first and second kind

of order i, respectively. Equation (2.71) is plotted in Figure 2.33.

The efficiency for these fins, Figure 2.31(d), is given by

where , and Equation (2.72) is plotted in Figure 2.34.

If A_base is the area of the surface not taken up by fins (equal to Wb for the section of wall considered) and A_bare is the total surface area without fins (equal to W × (δ + b) for this example) and A_fin is the area of the fins (equal to 2WL for the single fin considered in this example), then the results from Equation (2.73) can be generalized as follows:

With fins

Without fins

The enhancement in heat transfer is found by comparing Equations (2.74) and (2.75):

This enhancement of heat transfer depends on the geometry of the fins and the fin efficiency and is illustrated in Example 2.15.

From Example 2.15, it is clear that for all the fins considered, there is a significant enhancement of heat transfer due to the presence of the fins. The enhancement factors for the cases with fins were 10.4, 9.3, 1.64, and 3.184 for cases b, c, e, and f, respectively. The enhancement factor can be considered a measure of the increase in effective film heat transfer coefficient that the fins provide. Thus, the required heat transfer area, based on the base area, is reduced significantly and varies in the range of 40% to 90% reduction for the cases considered in Example 2.15. The choice to use fins is an economic one, since the overall bare heat transfer area will be reduced, but the cost of the heat transfer area increases significantly.

where N_tp is the number of tube passes, u_i is the velocity inside the tubes, and D_i is the inside tube diameter. In Equation (2.77), the first term on the right-hand side accounts for the frictional pressure loss in the tubes, and the second term accounts for losses occurring at the ends of the exchanger where the fluid changes direction.

Once a given arrangement of tubes, shells, baffles, and so on, has been made, more sophisticated CFD (computational fluid dynamics) models can be constructed to determine more accurate estimates of the expected pressure drops through the exchanger.

1. Establish the energy balance around the heat exchanger:

2. Using the heuristics in Section 2.2.1.7, determine which fluid should go on the shell side and which on the tube side.

3. For the tube side, determine the mass flowrate, inlet temperature, outlet temperature, and enthalpy change.

4. For the shell side, determine the mass flowrate, inlet temperature, outlet temperature, and enthalpy change.

5. Determine all the fluid properties, μ, ρ, k, c_p, at the appropriate average temperatures.

6. Determine the maximum allowable pressure drops for the tube side and shell side.

7. Determine the number of shell passes, N_shells, using Equation (2.16) and the number of tube passes, N_tp = 2(or 4 or 6...)N_shells, and determine the LMTD correction factor, F_N−2N and LMTD.

8. Choose the shell and tube arrangement. Determine the following:

a. Tube length, L_tube: Standard lengths are 8, 12, 16, and 20 ft

b. Baffle cut, BC: 15% to 45% of D_s with 20% to 35% being “standard”

c. Baffle spacing: L_b ranges from a minimum of 0.2D_s or 2 in to a maximum of D_s

d. Tube diameter, D_o: Standard sizes are ¾ and 1 in

e. Tube thickness: Ranges from 12 to 18 BWG (see Table 2.4) and actual value will depend on the tube-side fluid pressure and temperature

f. Tube arrangement (square or triangular) and pitch, p: Typical values are p = 1.25D_o for triangular and p = D_o + 0.25 in for square

9. Choose a tube-side velocity between 1 and 3 m/s (this should not exceed 3 m/s), and from this value estimate the number of tubes per pass, the total number of tubes in the exchanger, N_tubes, and the inside heat transfer coefficient, h_i.

10. Using the tube count information from Tables 2.6 and 2.7, determine the shell diameter.

11. With the shell diameter from Step 10 and the information in Step 8, calculate the shell-side heat transfer coefficient using Equation (2.43).

12. Combine the heat transfer coefficients and fouling and tube metal resistance to give the overall outside heat transfer coefficient, U_o, Equation (2.23).

13. Determine the total heat transfer area required, A_o,new = Q/(U_oΔT_lmF_N−2N) and compare with the total area assumed from A_o = πD_oL_tubeN_tubes.

14. Determine the tube-side and shell-side pressure drops. If either of the pressure drops exceeds the maximum allowable values, adjust the shell/tube configuration to accommodate. Recommended actions include the following:

Tube side: Reduce the tube-side velocity by increasing the number of tubes.

Note that ΔP_tube ∝(1/N_tubes)²

Shell side: Increase the baffle spacing or tube pitch.

Note that ΔP_shell ∝ (1/L_b)³

If pressure drop is too low, consider increasing the number of tube passes (for increasing ΔP_tube) or decreasing the baffle spacing (for increasing ΔP_shell). These actions increase the overall heat transfer coefficient, which reduces heat transfer area and the cost of the heat exchanger.

15. Adjust the number of tubes based on the new estimated area and tube-side pressure drop and, if needed, adjust the baffle spacing based on shell-side pressure drop. Iterate from Step 8 until the solution converges—namely, A_o,new in Step 13 is between A_o and 1.2 A_o.

An example of off-design conditions will occur when there is high market demand for the chemical(s) produced from a process. Under these conditions, the process may have to be run to maximize the products (and profits) from the plant. Thus, the process might be pushed to 110% or 120% of its design capacity. Obviously, every heat exchanger in the plant will be operating quite far from its design conditions. The opposite might occur when there is weak market demand. The question addressed in this section is, for a given existing heat exchanger, how is the performance of that heat exchanger predicted?

It should be noted that these conditions are not the design conditions because the exchanger was built with approximately 10% additional area required to perform the design. Thus, more heat is transferred between the streams, and the crude leaves hotter and the kerosene leaves cooler than the design specification, which is to be expected because of the overdesign.

Now consider the case when the plant is to be scaled up. It is desired that the flows of crude and kerosene be increased by 20%. If the temperatures into the exchanger remain the same as in the design case, the exit temperatures of both streams must be determined. This problem is illustrated in Figure 2.36. For performance problems (where the equipment has been designed, installed, and is operating), the easiest way to determine the new operating conditions is via a ratio analysis, shown in the next section. The terms rating and performance are synonymous when considering how existing equipment performs. However, the term performance problem is preferred in this text.

If there are changes in phase, the enthalpy balances will have heats of vaporization instead of sensible heats, but the approach is similar. The assumption is then made that, for relatively small changes in flows, the physical properties of the fluids remain constant and also that the LMTD correction factors remain constant. The validity of these assumptions can be checked later. For the problem considered here, the three equations can be rewritten:

In Equations (2.81), (2.82), and (2.83), the ratios of the crude oil and kerosene flowrates for cases 1 and 2 are given as m and M, respectively. For the case of 20% scale-up, these ratios are 1.2 each. The ratio of the heat transferred is designated as Q, but it should be noted that Q is not 1.2 and must be determined.

In Equation (2.83), the ratio of the overall heat transfer coefficients must be determined. For the base case, from the last iteration in Table E2.16, h_o = 1009 W/m²/K, h_i 1121 W/m²/K, and

In Equation (2.84), the inside (tube) and outside (shell) heat transfer coefficients are dependent on the flows of the tube- and shell-side fluids. From Equations (2.27) and (2.36) for turbulent flow in the tubes and the shell, the heat transfer coefficients are related to the mass flow of fluids through the Reynolds number as follows:

It should be noted that in Kern’s method, the exponent on the Reynolds number is a little below 0.6, but this difference is small, and 0.6 can be used for shell-side flow. Using these relationships and noting that for constant physical properties Re is proportional to the mass flowrate, Equation (2.84) may be rewritten for the new case as

Substituting Equations (2.85) and (2.86) in (2.83) gives

Equations (2.81), (2.82), and (2.87) may now be solved simultaneously to give

Thus, the crude temperature decreases from 77.8°C to 76.4°C, and the kerosene temperature increases from 88°C to 91.3°C. Clearly, the average fluid properties would not change significantly for these temperature changes; similarly, the value of the LMTD correction factor hardly changes from 0.894 to 0.885, a change of ~1%. It should also be noted that the amount of heat transferred between the streams increases by only 16.5%, because the heat transfer coefficients are not linearly dependent on the mass flow rates.

Table 2.8 gives the dependence of individual heat transfer coefficients on the mass flow rate (effectively the Reynolds number, Re). Also shown in Table 2.8 are the relationships to be used when phase changes are considered. For phase changes, the mass flow rates do not affect the heat transfer coefficients, but the ΔT between the wall and fluid may be important.

It is important to note the implications of this result on the control and flexibility of operation for this process. If scale-down below 10% is required, then the only feasible way to achieve this is for the process temperature to drop below 55°C. This may not be a problem, but if, for example, the viscosity of the process fluid became too large at temperatures less than 55°C, then this design would need to be changed.

It should be noted that in Example 2.20, an additional relationship was used, namely, a flux balance across the wall given by Equations (E2.20a) and (E2.20b), in order to determine the unknown wall temperature T_w. The final problem considers a situation where the heat transfer coefficients and exchanger area are unknown, but one heat transfer coefficient is known to be limiting.

Bennett, C. O., and J. E. Myers. 1983. Momentum, Heat, and Mass Transfer, 3rd ed. New York: McGraw-Hill.

Bowman, R. A. 1936. “Mean Temperature Difference Correction in Multipass Exchangers.” Ind Eng Chem 28: 541–544.

Bowman, R. A., A. C. Mueller, and W. M. Nagle. 1940. “Mean Temperature Difference in Design.” Trans ASME 62: 283–294.

Bromley, L. A. 1950. “Heat Transfer in Stable Film Boiling.” Chem Eng Progr 46: 221–227.

Chen, C. Y., G. A. Hawkins, and H. L. Solberg. 1946. “Heat Transfer in Annuli.” Trans ASME 68: 99–106.

Churchill, S. W., and M. Bernstein. 1977. “A Correlating Equation for Forced Convection from Gases and Liquids to a Circular Cylinder in Crossflow.” J Heat Transfer, Trans ASME 99: 300–306.

Cichelli, M. T., and C. F. Bonilla. 1945. “Heat Transfer to Liquids Boiling Under Pressure.” Trans Am Inst Chem Engrs 41: 755–787.

Clark, S. H., and W. M. Kays. 1953. “Laminar Flow Forced Convection in Rectangular Ducts.” Trans ASME 75: 859–866.

Cooper, M. G. 1984. “Saturated Nucleate Pool Boiling—A Simple Correlation.” First UK National Heat Transfer Conf., IChemE Symp Ser, No. 86, 2, 785–793.

Couper, J. R., W. R. Penney, J. R. Fair, and S. M. Walas. 2012. Chemical Process Equipment—Selection and Design, 3rd ed. Oxford, UK: Butterworth-Heinmann.

Dittus, F. W., and L. M. K. Boelter. 1985. “Heat Transfer in Automobile Radiators of Tubular Type.” Int Comm Heat Mass Transfer 12: 3–22; modified from Univ. Calif. Publ. in Engineering 2: 443–461 (1930).

Geankoplis, C. J. 2003. Transport Processes and Separation Process Principles, 4th ed. Upper Saddle River, NJ: Prentice Hall.

Green, D. W., and R. H. Perry. 2008. Perry’s Chemical Engineers’ Handbook, 8th ed. New York: McGraw-Hill.

Gungor, K. E., and R. H. S. Winterton. 1986. “A General Correlation for Flow Boiling in Tubes and Annuli.” Int J Heat Mass Transfer 29: 351–358.

Hausen, Z. 1943. Ver Deut Beth Verfahrenstech 4: 91.

Kern, D. Q. 1950. Process Heat Transfer. New York: McGraw-Hill.

McAdams, W. H. 1954. Heat Transmission, 3rd ed. New York: McGraw-Hill.

Mukherjee, R. 1998. “Effectively Design Shell-and-Tube Heat Exchangers.” Chem Eng Prog 2: 21–37.

Rohsenow, W. M. (Ed.). 1964. Developments of Heat Transfer. Cambridge, MA: MIT Press.

Shelton, S. M. “Thermal Conductivity of Some Irons and Steels over the Temperature Range 100 to 500°C.” J Res Nat Stand 12 (April 1934).

Sieder, E. N., and G. E. Tate. 1936. “Heat Transfer and Pressure Drop of Liquids in Tubes.” Ind Eng Chem 28: 1429–1435.

Tinker, T. 1958. “Shellside Characteristics of Shell-Tube Heat Exchangers—A Simplified Rating System for Commercial Heat Exchangers.” Trans ASME 80: 36–52.

Tubular Exchanger Manufacturers Association. 2013. Standards of the Tubular Exchanger Manufacturers Association, 9th ed. Tarrytown, NY: Tubular Exchanger Manufacturers Association.

Žukauskas, A. 1972. “Heat Transfer from Tubes in Crossflow.” Advances in Heat Transfer 8: 93–160.

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For LMTD correction factors for other configurations, see CheCalc’s LMTD Correction Factor Charts at http://checalc.com/solved/LMTD_Chart.html.

The heat transfer process occurring in the fin consists of conduction through the fin and convectional loss from the side of the fin. The geometry and notation used in this derivation are shown in Figure B.1. A differential flux balance on a slice through the fin yields

where W is the width of the fin, δ is the fin thickness, h is the heat transfer coefficient between the fin and the surrounding fluid, and T_fluid is the temperature of the surrounding fluid. Equation (B.2.1) means that the heat flow into a differential section of the fin by conduction equals the heat flow out of the differential section of the fin by conduction down the length of the fin plus the heat exchanged with the surroundings in that differential section of the fin by convection. It is important to remember that heat is exchanged with the surroundings on both the top and bottom faces of the fin. Dividing through by δWΔx and taking the limit as Δx → 0 yields the following differential equation:

Substituting Fourier’s law of heat conduction in Equation (B.2.2) gives

Introducing the new variable T′ = T_x − T_fluid gives

The boundary condition at x = 0 is T = T₀ or T′ = T₀ − T_fluid. Several possible boundary conditions exist for x = L, the simplest being that dT′/dx = 0, which means that there is no heat lost from the tip of the fin (adiabatic fin tip). This is a good approximation for the case where the tip of the fin has a very small surface area compared to the faces of the fin. The second boundary condition is reasonable and accurate for long, thin fins, but for short thick fins, the amount of heat lost from the fin tip may be significant. Nevertheless, the adiabatic fin tip assumption may be used for all types of fin with an adjustment that is discussed in Section 2.6. With these boundary conditions, Equation (B.2.4) can be integrated as follows:

The total heat transferred into the fin is (δW)q_x=0 or

At this point, the efficiency of the fin is introduced and is defined as

Substituting the result from Equation (B.2.6) into Equation (B.2.7) gives

This result is plotted in Figure 2.32.

1. Why is it necessary to include an LMTD correction factor (F) into the design equation for most practical heat exchangers?

2. Explain why you agree or disagree with this statement: Because the heat transfer coefficients for gases are always much lower than for liquids, the limiting resistance for condensers (where a gas is turned into a liquid) is always on the condensing side of the heat exchanger.

3. For shell-and-tube exchangers in which the flow is turbulent and there is no phase change taking place for either stream, how does the heat transfer coefficient change with a 50% increase in the mass flowrate for the

a. Shell-side fluid?

b. Tube-side fluid?

4. Repeat Problem 2.3 assuming laminar flow for both fluids.

5. Would you expect the heat transfer coefficient for a given fluid to be higher if the fluid were in laminar flow or turbulent flow? Explain your answer.

6. Give three reasons for placing a specific fluid on the tube side of an S-T heat exchanger.

7. Give three reasons for placing a specific fluid on the shell side of an S-T heat exchanger.

8. Why are fins (extended surfaces) used in some heat exchangers?

9. Give explanations for the following terms for S-T heat exchangers:

a. Baffle cut

b. Number of tube passes

c. Tube pitch

d. Tube arrangement

10. State the basic design equation for an S-T heat exchanger in which there are no phase changes, and draw a sketch of the T-Q diagram with the relevant terms shown.

11. Draw the T-Q diagram for the following cases:

a. Condensing (pure) vapor using cooling water

b. Distillation reboiler using condensing steam as the heating media

c. Process liquid stream cooled by a another process stream

Problems to Solve

12. A process fluid (Stream 1) (C_p = 2100 J/kg/K) enters a heat exchanger at a rate of 3.4 kg/s and at a temperature of 135°C. This stream is to be cooled with another process stream (Stream 2) (c_p = 2450 J/kg/K) flowing at a rate of 2.65 kg/s and entering the heat exchanger at a temperature of 55°C. Determine the following (you may assume that the heat capacities of both streams are constant):

a. The exit temperature of Stream 1 if pure countercurrent flow occurs in the exchanger and the minimum approach temperature between the streams anywhere in the heat exchanger is 10°C

b. The exit temperature of Stream 1 if pure cocurrent flow occurs in the exchanger and the minimum approach temperature between the streams anywhere in the heat exchanger is 10°C

13. Repeat Problem 2.12 for the case when the specific heat capacities of both streams vary linearly with temperature with the following values:

a. Stream 1: C_p = 2000 + 3(T − 100) J/kg/K

b. Stream 2: c_p = 2425 + 5(T − 50) J/kg/K

14. The T-Q diagrams for three S-T heat-exchanger designs are given in Figure P2.14(a–c). Determine the number of S-T passes required to accomplish these designs.

15. At a given point in a thin-walled heat exchanger, the shell-side fluid is at 100°C and the tube side fluid is at 20°C. Ignoring any fouling resistances and the resistance of the wall, determine the wall temperature for the following cases:

a. The inside and outside heat transfer coefficients are equal.

b. The inside coefficient has a value 3 times that of the outside coefficient.

c. The inside coefficient has a value ⅓ that of the outside coefficient.

d. The inside coefficient is limiting.

16. In a heat exchanger, water (c_p = 4200 J/kg/K) flows at a rate of 1.5 kg/s, and toluene (c_p = 1953 J/kg/K) flows at a rate of 2.2 kg/s. The water enters at 50°C and leaves at 90°C, and the toluene enters at 190°C. For this situation, do the following:

a. Sketch the T–Q diagram if the flows are countercurrent.

b. Sketch the T–Q diagram if the flows are concurrent.

17. At a given location in a double-pipe heat exchanger, the bulk temperature of the fluid in the annulus is 100°C, and the bulk temperature of the fluid in the inner pipe is 20°C. The tube wall is very thin, so the resistance due to the metal wall may be ignored. Likewise, fluid fouling resistances may also be ignored. The individual heat transfer coefficients at this point in the heat exchanger are

a. What is the temperature of the wall at this location?

b. What is the heat flux across the wall at this location?

c. If there was a fouling resistance of 0.001 m²K/W on the inside surface of the inner pipe, what would the temperature of the wall be at this location?

18. A single phase fluid ( and c_p = 1800 J/kg/K) is to be cooled from 175°C to 75°C by exchanging heat with another single phase fluid ( and c_p = 2250 J/kg/K), which is to be heated from 50°C to 150°C. You have the choice of one of the following five heat exchangers:

Which exchanger do you recommend using for this service? Carefully explain your answer.

(For the following problems and where information on organic materials is not given in the problem, the physical properties may be obtained from a process simulator using the SRK thermodynamics package or from using suitable handbooks.)

19. What is the critical nucleate boiling flux for water at 20 atm?

20. Water flows inside a long ¾-in, 16 BWG tube at an average temperature of 110°F. Determine the inside heat transfer coefficient for the following cases:

a. Velocity = 0.1 ft/s

b. Velocity = 3 ft/s

21. Use Equations (2.47) and (2.48) to estimate the heat flux and heat transfer coefficient for boiling acetone at 1 atm pressure for a temperature driving force of 10°C. You may assume a C_f value of 0.01 for this problem.

22. Use the Sieder-Tate equation to determine the inside heat transfer coefficient for a fluid flowing inside a 16 ft long, 1-in tube (14 BWG) at a velocity of 2.5 ft/s that is being cooled and has an average temperature of 176°F and an average wall temperature of 104°F. Consider the following fluids:

a. Acetone (liquid)

b. Isopropanol (liquid)

c. Methane at 1 atm pressure—use a velocity of 50 ft/s

d. Compare the results from Parts (a), (b), and (c) using the Dittus-Boelter equation

23. Water at 5 atm pressure and 30°C flows at an inlet velocity of 1 m/s into a square duct 2.5 m long that has a cross section of 25 mm by 25 mm. The wall of the duct is maintained at a temperature of 120°C by condensing steam. What will the temperature of the water be when it exits the duct?

24. An S-T condenser contains six rows of five copper tubes per row on a square pitch. The tubes are ¾-in, 14 BWG, and 3 m long. Cooling water flows through the tubes such that h_i = 2000 W/m²/K. The water flow is high so that the temperature on the tube side may be assumed to be constant at 35°C. Pure, saturated steam at 2 bar is condensing on the shell side. Determine the capacity of the condenser (Q in kW) if the condenser tubes are oriented (a) vertically, and (b) horizontally.

25. Liquid dimethyl ether (DME) flows across the outside of a bank of tubes. It enters at 100°C and leaves at 50°C. The DME enters at a flowrate of 20 kg/s, the shell diameter is 15 in, the baffle spacing is 6 in, the baffle cut (BC) is 15%, and 1-in OD tubes on a 1.25-in pitch are used. The fluid (cooling water) inside the tubes may be assumed to be at a constant temperature of 35°C and the inside coefficient is expected to be much higher than the shell-side coefficient, and thus the wall temperature may be taken as 35°C.

Use Kern’s method to determine the average heat transfer coefficient for the shell side for the following arrangements:

a. Square pitch

b. Equilateral triangular pitch

26. A double-pipe heat exchanger consists of a length of 1-in, schedule-40 pipe inside an equal length of 3 in, schedule-40 pipe. Water flows at a velocity of 1.393 m/s in the annular region between the pipes and enters the heat exchanger at 30°C. Oil flows in the inner pipe at an average velocity of 1 m/s and enters at 100°C. The water and oil flow countercurrently. Following are the properties of the water and oil:

Determine how long the pipe lengths must be in order for the oil to leave the exchanger at 50°C. You may assume that both fluids are clean and that there is no fouling or tube wall resistances.

27. A 3 m long, 1.25-in, BWG 14 copper tube is used to condense ethanol at 3 bar pressure. Cooling water at 30°C flows through the inside of the tube at a high rate such that the wall temperature may be assumed to be 30°C and the inside heat transfer coefficient may be assumed to be much greater than the outside coefficient. Determine how much vapor will condense (kg/h) assuming no fouling resistance for the following cases:

a. The tube is oriented vertically.

b. The tube is oriented horizontally.

28. Air (1 atm and 30°C) flows crosswise over a bare copper tube (1-in, BWG 16). The approach velocity of the air is 20 m/s. Water enters the tube at 140°C and leaves the tube at an average temperature of 80°C. The average velocity of the water in the tube is 1 m/s. Determine the length of tube required to cool the water to the desired 80°C.

29. Oil flows inside a thin-walled copper tube of diameter D_i = 30 mm. Steam condenses on the outside of the tube, and the tube wall temperature may be assumed to be constant at the temperature of the steam (150°C). The oil enters the tube at 30°C and a flow rate of 1.6 kg/s. The properties of the oil are as follows:

a. Calculate the inside heat transfer coefficient, h_i, using the appropriate correlation. You should assume that the bulk oil temperature changes from 30°C to 50°C along the tube.

b. Using the result from Part (a), calculate the length of tube required to heat the oil from 30°C to 50°C.

30. Use the approach given in Example 2.12 to determine the length of tubes necessary to vaporize an organic liquid (acetic acid at 1 bar) flowing inside a set of vertical ¾-in, BWG 16 tubes using condensing steam on the outside of the tubes to provide the energy for vaporization. The major resistance to heat transfer is expected to be on the inside of the tubes, and the wall temperature, as a first approximation, may be assumed to be at the temperature of the condensing steam, which for this case is 125°C. It may be assumed that the value of the vapor quality, x, varies from 0.05 to 0.95 in the tube. The physical parameters for acetic acid are

ρ_v = 1.893 kg/m³, ρ_l = 939.7 kg/m³, μ_v = 11.32 × 10⁻⁶ kg/m/s, μ_l = 390.1 × 10⁻⁶ kg/m/s, T_sat = 117.6°C, P_c = 57.9 bar, M = 60, k_l = 0.1423 W/m/K, λ = 405 kJ/kg, c_p,l = 2.434 kJ/kg/K, c_p,v = 1.319 kJ/kg/K, λ = 0.04 kg/s/tube

31. Repeat Problem 2.28 to find the length of a tube fitted with 2-in diameter, 1/16-in thick annular fins spaced a distance of 3/16-in apart.

32. An air heater consists of a shell-and-tube heat exchanger with 24 longitudinal fins on the outside of the tubes. The tubes are 1.5-in, 14 BWG, and the fins are 0.75 mm thick and are 15 mm long. A total of 8 fins are spaced uniformly around the circumference of each tube. The tubes are made from carbon steel, and their length is 3 m. Air at 0.8 kg/s is being heated from 30°C to 200°C in the shell, and the heat transfer coefficient may be taken as h_o = 25 W/m²/K (without fins). Steam is condensing at 254°C in the tubes, and at that temperature λ = 1700 kJ/kg. The tube-side heat transfer coefficient may be taken as h_i = 6000 W/m²/K, and no fouling occurs for this service.

a. Calculate the overall heat transfer coefficient with and without fins.

b. Calculate the heat transfer area and the number of tubes needed with and without fins.

33. Your assignment is to design a replacement condenser for an existing distillation column. Space constraints dictate a vertical-tube condenser with a maximum height (equals tube length) of 3 m. An organic is condensed at a rate of 12,000 kg/h at a temperature of 75°C, and cooling water is used, entering at 30°C and exiting at 40°C. The person you are replacing has done some preliminary calculations suggesting that a 1–4 exchanger (water in tubes) using 1-in 16 BWG copper tubes on 1.25-in equilateral triangular pitch with a 37-in shell diameter would be suitable. However, there are only partial calculations to support this claim, and the person who performed the original design is unavailable for consultation. Complete the detailed heat transfer calculations to evaluate the suitability of this heat-exchanger design.

Data for condensing organic:

ρ_f = 800 kg/m³

ρ_g = 5.3 kg/m³

λ = 800 kJ/kg

c_pl = 2600 J/kg/K

k_f = 0.15 W/m/K

μ_f = 0.4 × 10⁻³ kg/m/s

Heat transfer coefficient for tube side: 6000 W/m²/K

Typical fouling coefficient for plant cooling water: 1200 W/m²/K (=1/R_fi)

Assume no fouling on condensing side

34. A 1–2 shell-and-tube heat exchanger has the following dimensions:

Tube length: 20 ft

Tube diameter: 1-in, BWG 14 carbon steel (k_cs = 45 W/m/K)

Number of tubes in shell: 608

Shell diameter: 35 in

Tube arrangement: Triangular pitch, center-to-center = 1.25 in

Number of baffles: 19

Baffle spacing: 1 ft

Baffle: Horizontal baffle with baffle cut = 25%

The fluids in the shell and tube sides of the exchanger have the following properties:

It may be assumed that neither fluid changes phase, and the viscosity correction factor at the wall may be ignored for both fluids. Determine the following:

a. The inside heat transfer coefficient

b. The shell-side heat transfer coefficient

c. The overall heat transfer coefficient (assuming that fouling may be ignored)

d. The exit temperatures of both fluids

35. A 1–2 S-T heat exchanger is used to cool oil in the tubes from 91°C to 51°C using cooling water at 30°C. In the design case, the water exits at 40°C. The resistances on the water and oil sides are equal. What are the new cooling water outlet temperature and the required cooling water flowrate if the oil throughput must be increased by 25% but the outlet temperature must be maintained at 51°C?

36. Repeat Problem 2.35 if the oil side provides 80% of the total resistance to heat transfer.

37. Repeat Problem 2.35 for the case when the oil throughput must be increased by 25% but the cooling water flow rate remains unchanged from the base case. Determine the new outlet temperatures for both the process and cooling water streams?

38. For the situation in Problem 2.35, suppose that the flowrate of the process stream must be reduced while keeping the process exit temperature at 51°C. Therefore, it will be necessary to reduce the flow of the cooling water stream. Determine the maximum scale-down of the process fluid that can occur without the exit cooling water temperature exceeding the limit of 45°C (when excessive fouling is known to occur). Plot the results as the ratio of the process stream flowrate to the base case value (x-axis) versus the cooling water exit temperature.

39. A reboiler is a heat exchanger used to add heat to a distillation column. In a typical reboiler, an almost pure material is vaporized at constant temperature, with the energy supplied by condensing steam at constant temperature. Suppose that steam is condensing at 254°C to vaporize an organic at 234°C. It is desired to scale up the throughput of the distillation column by 25%, meaning that 25% more organic must be vaporized. What will be the new operating conditions in the reboiler (numerical value for temperature, qualitative answer for pressure)? Suggest at least two possible answers.

40. In an S-T heat exchanger, initially h_o = 500 W/m²/K and h_i = 1500 W/m²/K. If the mass flowrate of the tube-side stream is increased by 30%, what change in the mass flowrate of the shell-side stream is required to keep the overall heat transfer coefficient constant?

41. A reaction occurs in an S-T reactor. One type of S-T reactor is essentially a heat exchanger with catalyst packed in the tubes. For an exothermic reaction, heat is removed by circulating a heat transfer fluid through the shell. In this situation, the reaction occurs isothermally at 510°C. The Dowtherm™ always enters the shell at 350°C. In the design (base) case, it exits at 400°C. In the base case, the heat transfer resistance on the reactor side is equal to that on the Dowtherm side. If it is required to increase throughput in the reactor by 25%, what is the required Dowtherm flowrate and the new Dowtherm exit temperature? You should assume that the reaction temperature remains at 510°C.

42. In Problem 2.41, the reactor is now a fluidized bed where Dowtherm circulates through tubes in the reactor with the reaction in the shell. The Dowtherm then flows to a heat exchanger in which boiler feed water is vaporized on the shell side to high-pressure steam at 254°C. This removes the heat absorbed by the Dowtherm stream in the reactor so the Dowtherm can be recirculated to the reactor. So, the Dowtherm is in a closed loop. The resistance in the steam boiler is all on the Dowtherm side. In the base case for the reactor, the resistance on the reactor side is four times that on the Dowtherm side. The desired increase in production can be accomplished by adding 25% more catalyst to the bed and operating at the same temperature, which is what is assumed to happen in this problem.

a. Write the six equations needed to model the performance of this system.

b. How many unknowns are there? Solve for all the unknowns.

c. If the temperature of the reactor is to be maintained at 510°C, determine the amount of process scale-up and all other unknowns for the following cases:

i. 10% increase in Dowtherm flowrate

ii. 25% increase in Dowtherm flowrate

iii. 50% increase in Dowtherm flowrate

d. Plot the results from Parts (b) and (c) in the form of a performance curve in which the amount of process scale-up (Q₂/Q₁ on the y-axis) is plotted as a function of the increase in Dowtherm flowrate (M_DT,2/M_DT,1 on the x-axis).

43. It is necessary to decrease the capacity of an existing distillation column by 30%. As a consequence, the amount of liquid condensed in the shell of the overhead condenser must decrease by the same amount (30%). In this condenser, cooling water (in tubes) is available at 30°C, and, under present operating conditions, exits the condenser at 40°C. The maximum allowable return temperature without a financial penalty assessed to your process is 45°C. Condensation takes place at 85°C.

a. If the limiting resistance is on the cooling water side, what is the maximum scale-down possible based on the condenser conditions without incurring a financial penalty? What is the new outlet temperature of cooling water? By what factor must the cooling water flow change?

b. Repeat Part (a) if the resistances are such that the cooling water heat transfer coefficient is three times the condensing heat transfer coefficient. You may assume that the value of the condensing heat transfer coefficient does not change appreciably from the design case. Does your solution exceed the maximum cooling water return temperature of 45°C? If so, can you suggest a way to decrease the condenser duty by 30% that would not violate the cooling water return temperature constraint?

44. A reaction occurs in a well-mixed fluidized bed reactor maintained at 450°C. Heat is removed by Dowtherm A™ circulating through a coil in the fluidized bed. In the design or base case, the Dowtherm enters and exits the reactor at 320°C and 390°C, respectively. In the base case, the heat transfer resistance on the reactor side is two times that on the Dowtherm side. It may be assumed that for the fluidized bed, the heat transfer coefficient on the fluidized side is essentially constant and independent of the throughput.

The Dowtherm is cooled in an external exchanger that produces steam at 900 psig (T_sat = 279°C). The Dowtherm pump limits the maximum increase in Dowtherm flowrate, so the pump limits the heat removal rate based on its pump and system curves. For the current situation, the maximum increase in Dowtherm flowrate through the reactor and boiler is estimated to be 34%. By how much can the process be scaled-up? You may assume that the limiting heat transfer coefficient in the steam boiler is Dowtherm that flows through the tube side of the exchanger.