HeatExchangers

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Chapter 12 Heat exchangers (Material presented in this chapter are based on those in Chapter 11, �Fundamentals of Heat and Mass Transfer�, Fifth Edition by Incropera and DeWitt) Heat exchanger is a device for heat exchange between two fluids, say hot and cold fluids. It is the workhorse of chemical, petrochemical, biochemical and power industries. Efficient energy transfer is an important aspect of power management in any industry. Therefore design of heat exchangers which facilitate energy transfer is very important. In this chapter, the functioning and characterization of heat exchangers and methods to assess their performance.

12.1

Classifications

Heat exchangers (HEs) are typically of two types based on the contact method, viz. direct contact HEs and two-phase HEs. Direct contact heat exchangers typically consists of wall between tubes, plates etc. This type of heat exchanger that has a wall between the two fluids is called transsmural heat oexchange. Two-phase heat exchangers are those in which two-phases are involved. Heat exchangers are classified based on the (geometrical) construction or flow arrangement. For instance heat exchangers can be concentric pipe heat exchangers, cross-flow heat exchangers, shell-tube heat exchangers. In all these cases, the hot and the cold may flow co-currently, that is, parallel-flow or counter-currently. Many different forms of shell-and-tube HEs may be possible, for example HE exchanger may involve multiple shell passes and 165

166

CHAPTER 12. HEAT EXCHANGERS

multiple tube passes. In the case of single shell-pass and multiple, say two tube pass case, the fluid stream that enters a HE, will be circulated two times the length of the heat exchanger. Such a configuration permits the exchange of heat from the tube-side fluid two times with the shell-side fluid for the same length of the heat exchanger. This presents an unique advantage to design heat exchangers of certain length that will permit heat exchange over lengths that are several fold longer than the length of the heat exchanger. A similar principle can be applied to the shell side fluid as well to expose the shell-side fluid to a higher heat transfer surface area. Additionally, in the cross-flow heat exchangers and shell-tube heat exchangers, fins may be attached in order to induce some mixing in the streams. The extent of heat exchange depends on the properties of the hot and cold fluids and the amount of heat that has to be transfered. Therefore, it is often an important question as to which configuration heat exchanger has to be used for heat exchange between two fluids. Another important type of heat exchanger is the compact heat exchanger which offers very high surface area per unit volume. If gas is one of the fluids for heat exchange, then a very high surface area per unit volume is required.

12.2

The overall heat transfer coefficient

Often it is useful for design calculations to obtain an overall heat transfer coefficient for a HE (see �Overall heat transfer coefficient� in section (9.2.4)). Overall heat transfer coefficient typically includes all the possible resistances for heat transfer in a given heat exchanger. The common resistances offered for heat exchange are those between hot fluid and the wall, between cold fluid and the wall, and wall resistance. Additionally, during the operation of HE, scales and films may form on the walls of the HE tubes. Formation of these is called fouling. Fouling of the HEs can offer certain resistance which is termed fouling resistances. Therefore, overall heat transfer coefficient U accounting for all these re-

12.3. LOG-MEAN TEMPERATURE DIFFERENCE

167

sistances is given by 1 1 1 = = = Rc,conv + Rc,f oul + Rwall + Rh,f oul + Rh,conv UA Uc Ac Uh Ah ” ” Rf,c Rf,h 1 1 = + + Rwall + (12.1) (ηhA)c (ηA)c (ηA)h (ηhA)h where, Rwall is the resistance offered by the wall, η is the surface efficiency, Rf” is the resistance offered by the fouling, subscripts h and c represent hot and cold fluid. The expression for heat transfer coefficient in Eq. (12.1) assumes that the heat transfer area in the hot fluid side and the cold fluids sides are different. Therefore the overall heat transfer coefficient depends on whether the coefficient is based on the hot fluid side or cold fluid side heat transfer surface area. According to Newton’s law of cooling, the overall heat exchanged by the two fluids (using the overall heat transfer coefficient) is given by q = U A∆Tlmtd

(12.2)

where the log-mean temperature difference ∆Tlmtd (Eq. 9.45) depends on the flow configuration.

12.3

Log-mean temperature difference

Consider the case of heat exchange between two fluids. Assume the hot fluid (with specific heat Cp,h to be flowing into the heat exchanger at temperature Th,i at a mass flow rate of m ˙ h and the cold fluid (with specific heat Cp,c to be flowing into the heat exchanger at temperature Tc,i at a mass flow rate of m ˙ c . If the cold and hot fluid out stream temperatures are Tc,o and Th,o , then the total heat lost by the hot stream is given by qh = m ˙ h Cp,h (Th,i − Th,o )

(12.3)

and the total heat gained by the cold stream is given by qc = m ˙ c Cp,c (Tc,o − Tc,i )

(12.4)

As the heat lost by the hot stream will be equal to the heat gained by the cold stream under steady state conditions, q = qh = qc = U A∆Tlmtd

(12.5)

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CHAPTER 12. HEAT EXCHANGERS

12.3.1

Parallel-flow heat exchanger

Consider the case of a parallel-flow heat exchanger in which the hot and cold fluids are flowing in two chambers separated by a wall (Fig. 12.1). Assume no axial conduction and constant properties. Consider hot and cold fluids flowing through the two parallel chambers at mass flow rates and temperatures shown in Fig. (12.1).

Figure 12.1: Parallel-flow heat exchanger Consider a differential element with differential area of heat transfer from the hot fluid stream to cold fluid stream to be dA. The heat exchanged is given by dq = −m ˙ h Cp,h dTh = m ˙ c Cp,c dTc = U ∆T dA (12.6) If the local temperatures on either side of the wall is Tc and Th , then the local temperature difference is given by ∆T = Th − Tc and therefore 1 1 d(∆T ) = dTh − dTc = −dq + (12.7) m ˙ h Cp,h m ˙ c Cp,c Using Eq. (12.2), Eq. (12.7) can be integrated from the inlet to the outlet of the HE to obtain ∆T Z out

∆Tin

ZA 1 1 = −U + dA m ˙ h Cp,h m ˙ c Cp,c 0 ∆Tin 1 1 ⇒ ln = −U A + (12.8) ∆Tout m ˙ h Cp,h m ˙ c Cp,c

12.3. LOG-MEAN TEMPERATURE DIFFERENCE

169

Using Eqs (12.4) and (12.3), Eq. (12.8) can be re-written as ln

∆Tin ∆Tout

= −U A [∆Tin − ∆Tout ]

(12.9)

where ∆Tin = Th,i − Tc,i and ∆Tout = Th,o − Tc,o . Equation (12.9) can be modified to obtain the expression for total heat transferred given by q = UA

[∆Tin − ∆Tout ] = U A∆TLM T D ∆Tin ln ∆T out

where log-mean temperature difference ∆TLM T D =

(12.10)

[∆T in −∆Tout ] ∆T ln ∆T in out

=

[∆T 1 −∆T 2 ] ∆T ln ∆T1 2

and subscripts 1 and 2 refer to the two ends of the HE. Figure (12.2) shows the typical temperature profile during parallel flow operation. By virtue of this design, irrespective of the length of the HE, the temperature of the outlet stream of the hot fluid cannot exceed the outlet cold fluid temperature. Therefore, there is an upper limit on the amount of heat that can be exchanged by the two fluids when using a parallel-flow configuration. This can be circumvented by using a counter-flow configuration.

Figure 12.2: Counterflow temperature profile

170

12.3.2

CHAPTER 12. HEAT EXCHANGERS

Counterflow heat exchanger

Cartoon and the temperatures and mass flow rate of a counterflow heat exchanger is presented in Fig. (12.3). The log-mean temperature difference for this case can be derived using the method as for the case Parallel-flow heat exchanger case in section (12.3.1). The log-mean temperature difference for a counterflow HE is given by ∆Tlmtd =

∆T1 − ∆T2 1 ln ∆T ∆T2

(12.11)

where ∆T1 = Th,i − Tc,o and ∆T2 = Th,o − Tc,i are the temperature differences at the two ends of the HE. Note that Tc,o can be greater than Th,o and

Figure 12.3: Counter flow heat exchanger therefore for the same temperatures of the hot and cold fluid streams and heat transfer surface area , the counterflow configuration will result in a higher heat exchange when compared with the parallel-flow configuration. This is because, ∆Tlmtd,CF > ∆Tlmtd,P F

(12.12)

where CF and P F stands for counterflow and parallel-flow. So, in order to achieve a certain heat exchange, for the same inlet and outlet temperatures of the hot and cold fluids, lowest heat transfer area is required in the case of CF as compared with the PF. Figure (12.4) shows the typical temperature profile in both fluid streams in a counterflow heat exchanger.

12.3. LOG-MEAN TEMPERATURE DIFFERENCE

171

Figure 12.4: Temperature distribution in a counterflow heat exchanger

12.3.3

Special cases

There are three special cases under which the HE may operate. • When m ˙ h Cp,h m ˙ c Cp,c or or condensation occurs (m ˙ h Cp,h → ∞), then the hot fluid temperature will remain constant. • When m ˙ c Cp,c m ˙ h Cp,h or evaporation occurs (m ˙ c Cp,c → ∞), then the cold fluid temperature will remain constant. • When m ˙ c Cp,c ≈ m ˙ h Cp,h , then the temperature difference between the hot and the cold fluid in all regions of the HE remains constant. Therefore, ∆T1 = ∆T2 = ∆Tlmtd .

12.3.4

Multipass and Cross-flow HEs

Log mean temperature difference for a multipass and cross-flow HEs is given by ∆Tlmtd = F ∆Tlmtd,CF (12.13) where ∆Tlmtd,CF is evaluated as though the flow is a counterflow. Expressions for the efficiency factor F have been derived for several differenct cases and are readily available in Incropera and DeWitt. Typically the F curves are presented as a function of temperature efficiency of the heat exchanger given

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CHAPTER 12. HEAT EXCHANGERS

by P =

Tt,o − Tt,i ∆Ttube = Ts,i − Tt,i ∆Tmax

(12.14)

where s and t represent the shell and tube sides and relative thermal capacim ˙ t Ct ties (R = m ) of the tube side and shell side fluids. When P or R → 0, then ˙ s Cs F → 1, that is if the temperature change of one fluid is negligible, multipass HEs behave line counterflow HE.

12.4

-NTU method

Suppose the inlet and outlet temperatures of both fluids are known then the ∆Tlmtd can be determined and LMTD method can be used to estimate the HE design parameters. However if only the inlet temperatures are known, then ∆Tlmtd cannot be determined and hence a tedious, iterative LMTD method will have to be used for obtaining the HE design parameters. An alternative method to estimate design parameters under these conditions is the −NTU method. The effectiveness of a HE is the ratio of the actual heat exchange between the two fluids that is permitted by the HE and the fraction of the maximum possible heat exchange between the two fluids. In order to find the effectiveness, maximum possible heat transfer rate has to be estimated. Parallel flow HE, even with infinite length cannot permit exchange of maximum heat transfer. However, counterflow HE can permit maximum heat exchange when the length is infinite. The question remains as to how to find this maximum heat exchange rate. Case I: Consider the case when m ˙ c Cp,c < m ˙ h Cp,h then, due to Eq. (12.6), |dTc | > |dTh |. As the length of the HE → ∞, the cold fluid outlet temperature must attain the hot fluid inlet temperature, that is, Tc,o = Th,i , and the maximal heat transfer is qmax = m ˙ c Cp,c (Tc,o − Tc,i = m ˙ c Cp,c (Th,i − Tc,i

(12.15)

Case II: Next, consider the case when m ˙ c Cp,c > m ˙ h Cp,h then, due to Eq. (12.6), |dTc | < |dTh |. As the length of the HE → ∞, the hot fluid outlet temperature must attain the cold fluid inlet temperature, that is, Th,o = Tc,i , and the maximal heat transfer is qmax = m ˙ h Cp,h (Th,i − Th,o = m ˙ c Cp,c (Th,i − Tc,i

(12.16)

12.4. -NTU METHOD

173

In general, from case I and case II, qmax = mC ˙ p )min (Th,i − Tc,i

(12.17)

Based on Eq. (12.17), the effectiveness is given by =

q qmax

=

m ˙ h Cp,h (Th,i − Th,o ) (mC ˙ p )min (Th,i − Tc,i )

(12.18)

and the heat rate is given by q = (mC ˙ p )min (Th,i − Tc,i )

12.4.1

(12.19)

Parallel flow heat exchanger

In the case of parallel flow, log-mean temperature difference can be written as Th,o − Tc,o UA (mC ˙ p )min ln =− 1+ (12.20) Th,i − Tc,i mC ˙ p )min (mC ˙ p )max Supposing, m ˙ h Cp,h is the minimum, then the heat transfer rate can be expressed as Th,i − Th,o (12.21) q= Th,i − Tc,i Defining the number of transfer units, NTU, as NT U =

UA (mC Ë™ p )min

(12.22)

which provides an estimate of the extent of heat that a heat exchanger can transfer given a maximum heat gain or loss, whichever applicable permitted by the fluid properties, Eq. (12.20) can be written as Th,o − Tc,o (mC ˙ p )min = exp −N T U 1 + (12.23) Th,i − Tc,i (mC ˙ p )max As (mC ˙ p )min = m ˙ h Cp,h , m ˙ h Cp,h (mC ˙ p )min Tc,o − Tc,i = = m ˙ c Cp,c (mC ˙ p )max Th,i − Th,o

(12.24)

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CHAPTER 12. HEAT EXCHANGERS

Using Eqs (12.24) and (12.21), in Eq. (12.23) can be expressed as h h ii mC ˙ p )min 1 − exp −N T U 1 + ((mC ˙ p )max = (12.25) (mC ˙ p )min 1 + (mC ˙ p )max If Cr =

(mC Ë™ p )min (mC Ë™ p )max

then =

1 − exp [−N T U (1 + Cr )] 1 + Cr

(12.26)

For several design calculations, it is also useful to have NTU expressed in terms of the effectiveness . The NTU for parallel flow in terms of is given by ln(1 − (1 + Cr )) NT U = − (12.27) 1 + Cr

12.5

Counterflow heat exchanger

The effectiveness as a function of NTU is given by =

1 − exp(−N T U (1 + Cr )) Cr < 1 1 − Cr exp(−N T U (1 − Cr )) NT U = Cr = 1 1 + NT U

(12.28)

and NTU in terms of is given by

−1 Cr < 1 Cr − 1 NT U = Cr = 1 1−

1 NT U = ln Cr − 1

12.5.1

(12.29)

Shell and tube heat exchanger

One shell pass, many tube passes Effectiveness, is given by h i −1   2 12   1 + exp −N T U (1 + C ) r 1  h i 1 = 2 1 + Cr + (1 + Cr2 ) 2  1  1 − exp −N T U (1 + Cr2 ) 2 

(12.30)

12.6. HEAT TRANSFER ANALYSIS CALCULATION METHODOLOGY175 and NTU is given by N T U = −(1 + where E =

2 −(1+Cr ) 1 (1+Cr2 )1 2

E−1 ln E+1

1 Cr2 )− 2

(12.31)

.

N shell passes, many tube passes Effectiveness is given by =

1− 1 Cr 1− 1

1− 1 Cr 1− 1

n

n

−1 (12.32)

− Cr

where 1 is the effectiveness for one shell pass and NTU is given by Eq. (12.31) with 1 F −1 Cr − 1 n 1 = ;F = (12.33) F − Cr −1

12.6

Heat transfer analysis calculation methodology

If the inlet and outlet temperatures for both the hot and cold fluid streams are known, then use log-mean temperature difference and find the area. While if the heat exchanger type and the inlet temperatures are known and the areas are known, then −NTU can be used to find the heat transfer rate and the temperatures of the outlet streams.

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