*2.3. Utility Cost and Capital Cost Calculation*

Besides the temperature and area ranges, the capital cost is another factor that is considered in designing and choosing a heat exchanger. The capital costs of heat exchangers are listed in Table 3.

**Heat Exchanger Type Capital Cost (\$)** Shell and tube *<sup>C</sup>st* = [0.9803 <sup>+</sup> 0.018( *<sup>P</sup>* <sup>×</sup> <sup>145</sup> <sup>100</sup> ) + 0.0017( *<sup>P</sup>* <sup>×</sup> <sup>145</sup> <sup>100</sup> ) 2 ] <sup>×</sup>[*<sup>a</sup>* + ( 10.76 <sup>×</sup> *<sup>A</sup>* <sup>100</sup> ) *b* ] <sup>×</sup> *<sup>e</sup>*11.667 <sup>−</sup> 0.8709 ln(10.76 <sup>×</sup> *<sup>A</sup>*) + 0.09005[ln(10.76 <sup>×</sup> *<sup>A</sup>*)]<sup>2</sup> (1) Double–pipe *Cdt* = <sup>2</sup> <sup>×</sup> [0.8510 + 0.1292( *<sup>P</sup>* <sup>×</sup> <sup>145</sup> <sup>100</sup> ) + 0.0198( *<sup>P</sup>* <sup>×</sup> <sup>145</sup> <sup>100</sup> ) 2 ] <sup>×</sup> *<sup>e</sup>*7.1460 <sup>+</sup> 0.16 ln(10.76 <sup>×</sup> *<sup>A</sup>*) (2) Spiral plate *Csp* = 6, 200 <sup>×</sup> (10.76 <sup>×</sup> *<sup>A</sup>*) 0.42 (3) Spiral tube *Cst* = *<sup>e</sup>*8.0757 <sup>+</sup> 0.4343 ln(10.76 <sup>×</sup> *<sup>A</sup>*) + 0.03812[ln(10.76 <sup>×</sup> *<sup>A</sup>*)]<sup>2</sup> (4) Plate and frame *Cp f* = 8, 880 <sup>×</sup> (10.76 <sup>×</sup> *<sup>A</sup>*) 0.42 (5)

**Table 3.** The capital cost of heat exchanger types [29] Reproduced from [29], John Wiley & Sons: 2010.

*Cst*, *Cdt*, *Csp*, *Cst*, *Cpf* are the capital costs of shell and tube, double-pipe, spiral plate, spiral tube, and plate and frame heat exchangers; *A* is the heat transfer area in m2; *P* is the shell-side pressure in MPa, and parameters *a* and *b* are materials of construction factors when the shell is made of carbon steel and the tube is made of Cr–Mo steel, the values of *a* and *b* are 1.55 and 0.05. For the shell and tube heat exchanger with floating head, carbon steel for shell, and Cr–Mo steel for tube, the capital cost calculation can be formulated as Equation (1). For the double-pipe heat exchanger, the capital cost equation for an outer pipe of carbon steel and an inner pipe of stainless steel is formulated as Equation (2). The capital cost of the spiral plate can be calculated by Equation (3). The capital cos of the spiral tube heat exchanger can be calculated by Equation (4). Finally, the plate and frame heat exchanger capital cost can be calculated by Equation (5).

Other fundamental equations used to determine the heat load, the overall heat transfer coefficient, *LMTD*, and the heat transfer area are listed as follows.

$$Q = A \times \mathcal{U} \times \text{LMTD} \tag{6}$$

where *Q* is the heat load, kW; *A* represents the heat transfer area, m2; *U* is the overall heat transfer coefficient, kW/(m2· ◦C); *LMTD* represents the logarithmic mean temperature difference, ◦C. The overall heat transfer coefficient can be calculated by Equation (7) if the inner and outer surfaces of the tube are almost identical. When the wall thickness of the tube is small and the thermal conductivity of the tube material is high, the overall heat transfer coefficient simplifies to Equation (8).

$$\frac{1}{\ell I} = \frac{1}{h\_{\rm li}} + \frac{d\chi\_w}{k} + \frac{1}{h\_c} \tag{7}$$

$$\frac{1}{M} = \frac{1}{h\_h} + \frac{1}{h\_c} \tag{8}$$

where *hh* and *hc* are the individual heat transfer coefficients of the connected hot and cold process streams, kW/(m2· ◦C); *dxw* represents the wall thickness, m; *k* represents the thermal conductivity of the material, kW/(m· ◦C).

The equations for calculating *LMTD* and the heat transfer area (*A*) are listed in Equations (9) and (10).

$$LMTD = \frac{(T\_{\rm in}^{H} - T\_{\rm out}^{C}) - (T\_{\rm out}^{H} - T\_{\rm in}^{C})}{\ln \frac{(T\_{\rm in}^{H} - T\_{\rm out}^{C})}{(T\_{\rm out}^{H} - T\_{\rm in}^{C})}} \tag{9}$$

$$A = \frac{Q}{U \times LMTD} \tag{10}$$

where *T<sup>H</sup> in* and *<sup>T</sup><sup>H</sup> out* are the inlet and outlet temperatures of hot streams, ◦C; *<sup>T</sup><sup>C</sup> in* and *<sup>T</sup><sup>C</sup> out* are the inlet and outlet temperatures of cold streams, ◦C.

For this illustrative example, as can be observed from Figure 3 to Figure 6, the utility cost of all four of these retrofit plans is the same. They can recover an additional 480 kW of heat compared to the existing HEN. The difference among these plans is the selection of heat exchanger types and their capital costs. The comparison of the capital cost for all retrofit plans is shown in Table 4. For plan 1, three types of heat exchangers (i.e., double-pipe, plate and frame, and shell and tube) are pre-selected in Section 2.2. The double-pipe can be excluded as the heat transfer area (38.3 m2) is higher than the upper limit of the area ranges of the double-pipe heat exchanger. The shell and tube heat exchanger is selected for plan 1 because the cost for the plate and frame exchanger is higher than the shell and tube heat exchanger. The total capital cost for the third retrofit plan is the cheapest. Two new double-pipe heat exchangers are selected. Another feasible plan is plan 1. One shell and tube heat exchanger should be implemented. For plan 2, the capital cost for the spiral tube heat exchanger is too high. Although it has a higher maximum bearing pressure, in this example, there is no need to use this type of heat exchanger. The plate and frame heat exchanger used in plan 4 has a relatively high cost for a small heat transfer area, and it is not recommended in this retrofit application.



Note: S&T refers to the shell and tube heat exchanger, D-P refers to the double-pipe heat exchanger, S-T refers to the spiral tube heat exchanger, and P-F refers to the plate and frame heat exchanger.
