**5. Optimal Design of WPHE for Ammonia Synthesis Column**

The mathematical model presented Section 3, validated in industrial conditions, can be used for the calculation of the optimal parameters of WPHE construction, as well as for WPHE optimal design based on plates of specific geometry. Here, the problem of WPHE optimisation consisting of finding the best construction parameters satisfying specified process conditions on heat transfer and pressure drop performance is considered. An approach discussed in [36] is used. The objective function is the heat transfer surface area of WPHE. There are a number of constraints that must be satisfied:


*Energies* **2020**, *13*, 2847

The estimation of the heat transfer performance of the WPHE is done through the determination of heat transfer effectiveness ε*T*. The total effectiveness of the heat exchanger with symmetric passes and countercurrent flow is determined by Equation (13) with the ratio of heat capacities flowrates *R* calculated by Equation (11). The value of effectiveness in one-pass ε*x*, required to satisfy the effectiveness of the whole WPHE, can be derived from Equation (13) as follows:

$$\varepsilon\_{X} = \left[ \left( \frac{1 - \varepsilon\_{T} \cdot \mathbf{R}}{1 - \varepsilon\_{T}} \right)^{\frac{1}{n}} - 1 \right] \cdot \left[ \left( \frac{1 - \varepsilon\_{T} \cdot \mathbf{R}}{1 - \varepsilon\_{T}} \right)^{\frac{1}{n}} - \mathbf{R} \right]^{-1} \tag{19}$$

The number of transfer units in one block of plates corresponding to one pass of crossflow WPHE from Equation (15):

$$\text{NTU}\_x^0 = -\frac{\ln(1 + 0.97 \cdot \text{R} \cdot \ln(1 - \varepsilon\_x))}{\text{R}} \tag{20}$$

It is required for fulfilment by WPHE of specified thermal process conditions. The number NTU*x* that can be obtained in one block of WPHE plates corresponding to one pass is according to the heat transfer ability of the block:

$$\text{NTU}\_{\text{x}} = \frac{\text{F}\_{\text{ax}} \cdot \text{U}}{\text{G}\_{2} \cdot \text{c}\_{p2}} \tag{21}$$

where F*ax* is the heat transfer area of the block of plates, m2.

Expressing F*ax* through the number of plates in one block and G2 through the number of channels and flow velocity in channel *w2* and channel cross-section area it is obtained:

$$\text{NTU}\_{\text{x}} = \frac{\text{2} \cdot \text{F}\_{\text{pl}} \cdot \text{U}}{c\_{p2} \cdot w\_2 \cdot \rho\_2 \cdot f\_{\text{cl}}} \tag{22}$$

where F*pl* is the heat transfer area of one plate, m2; <sup>ρ</sup><sup>2</sup> is the density of the cold fluid, kg/m3; *<sup>f</sup>* ch <sup>=</sup> *<sup>W</sup>*·*<sup>b</sup>* is the channel cross-section area, m2; *W* is the width of the channel, m.

The plate heat transfer surface area:

$$\mathbf{F}\_{pl} = \mathbf{L}\_{pl} \cdot \mathbf{W} \cdot \mathbf{F}\_{\mathbf{X}} \tag{23}$$

For the WPHE of ammonia synthesis column, the length of the plate L*pl* is equal to its width *W* and, from Equation (22), when the thermal constraint (iii) is completely satisfied (NTU*<sup>x</sup>* = NTU0*x*), Equation (24) follows:

$$\frac{\mathbf{I}\_{pl}}{b} = \frac{\mathbf{NTU}\_{\mathbf{x}}^{0} \cdot \mathbf{c}\_{p2} \cdot \mathbf{w}\_{2} \cdot \rho\_{2}}{\mathbf{2} \cdot \mathbf{U} \cdot \mathbf{F}\_{\mathbf{X}}} \tag{24}$$

Following the approach of [36], let us determine the plate length for the condition that pressure drop of the hot stream is strictly satisfied. From Equation (12), when the pressure drop ΔP1 is equal to the allowable pressure drop ΔP1 0,

$$\frac{\mathcal{L}\_{pl}}{b} = \frac{2}{\zeta\_1(w\_1)} \cdot \left(\frac{\Delta \mathcal{P}\_1^o \cdot 2}{\rho\_1 \cdot w\_1^2 \cdot n} - \zeta\_{\mathcal{D}Z1} \cdot \left(\frac{\mathcal{W}\_{\rm ch}}{\mathcal{W}\_{\rm emx}}\right)^2\right) \tag{25}$$

The flows velocities at the cold side (*w*2) and hot side (*w*1) are linked by Equation (26):

$$w\_2 = w\_1 \cdot \frac{\mathbf{G}\_2 \cdot \rho\_1}{\mathbf{G}\_1 \cdot \rho\_2} \tag{26}$$

When the hydraulic conditions by Equation (25) and thermal conditions by Equation (24) are satisfied simultaneously, the right sides of Equations (25) and (24) are equal. Accounting for Equation (26) for velocity in the hot channel, Equation (27) follows:

$$w\_1 = \sqrt{\frac{\Delta \mathbf{P}\_1^0}{n \cdot \rho\_1} \cdot \frac{1}{\frac{\zeta\_{\text{D} \text{T}\_1}{2} \cdot \left(\frac{\mathbf{W}\_{\text{ch}}}{\mathbf{W}\_{\text{ann}}}\right)^2 + \zeta\_1 (w\_1) \cdot \frac{\text{NTU}\_x^0 \mathbf{c}\_{p2} \cdot w\_1 \cdot \mathbf{C}\_2 \cdot \rho\_1}{8 \cdot \mathbf{U} (w\_2 w\_1) \cdot \mathbf{F}\_\mathbf{x} \cdot \mathbf{C}\_1}}}}\tag{27}$$

The variables ς and U at the right side of Equation (27) are the nonlinear functions of velocity *w*<sup>1</sup> expressed by Equations (4)–(7). The solution of Equation (27) gives the value of *w*<sup>1</sup> at which constraints (iii) and (iv) are satisfied as equalities and a corresponding value of overall heat transfer coefficient U is estimated. As a result, the value of WPHE heat transfer area for WPHE with *n* passes:

$$\mathbf{F}\_{\mathbf{d}} = \frac{\mathbf{NTU}\_{\mathbf{x}}^{0} \cdot \mathbf{G}\_{2} \cdot \mathbf{c}\_{p2}}{\mathbf{U}} \cdot n \tag{28}$$

The relations determining this objective function include nonlinear Equations and integer variable number of passes *n*. Besides, the numbers of plates and channels are also integers. To find the optimum of such function is the problem of Mixed-Integer Nonlinear Programming (MINLP). The numbers of plates and channels have rather big values and, on the first stage of finding the solution, can be treated as continues variables. The Equation (27) is a recurrent relation relative to velocity *w*1. Its solution by iterations gives the value of the velocity of a hot stream that, for a plate with specified geometrical parameters, strictly satisfies the thermal conditions and allowable pressure drop at the hot stream in a one-pass block of plates. When this velocity is obtained, the required plate length *Lpl* can be calculated by Equations (24) or (25). In case of a correct solution for *w*1, both are giving the same results. To satisfy the constraint (i), this plate length must be equal to the value specified for a given diameter of the column. The problem is solved by finding local optimums of plate spacing *bopt.i* at which constraint (i) for plate length is satisfied as equality. It is made for the series of increasing pass numbers *n*, starting from 2. The calculations are finished when *bopt.i* begin to increase. The value of *bopt.i* corresponding to a minimum of objective function F*<sup>a</sup>* and corresponding it passes number *n* are regarded as an optimal solution. After that, the number of plates in one block is rounded up, and rating design of WPHE performed. The method is implemented as a software for PC. The time of calculations for different conditions have not exceeded 200 s. The process of finding a solution is illustrated in the following case study.
