**3. Mathematical Modelling**

The reliable and accurate mathematical model of a heat exchanger is the base for the development of its design and optimisation of construction elements. The task of modelling WPHE thermal performance for its rating consists in finding its total effectiveness of heat transfer in WPHE ε*<sup>T</sup>* for the specified NTU, a value which is determined by a given WPHE construction and stream flowrates by Equation (9) in the assumption that thermo-physical properties in the calculation of heat transfer coefficients are taken for streams' average temperatures. For any pass's arrangement, the general method in the matrix form proposed for Compabloc WPHE in [18] can be used. But for the construction features of a considered WPHE for ammonia synthesis column, the number of pass combinations is limited. The bigger number of passes for one streamside has to be an integer multiple of the smaller passes number for another streamside or *X*<sup>1</sup> = *X*<sup>2</sup> × k, where k is an integer. An example of flow arrangements for a smaller number of passes, *n* = 4, is presented in Figure 6a for passes 4 × 4 and 8 × 4. For such conditions, a simplified mathematical model can be used based on Equation (13) proposed for total effectiveness ε*<sup>T</sup>* at equal passes arrangement with an overall countercurrent flow [35]. It is made considering the stream, with smaller pass numbers, *n* blocks of plates, with a flow arrangement of 1 × k and a heat transfer effectiveness, ε*x*, of individual blocks for R -1:

$$\varepsilon r = \frac{\left(\frac{1 - \varepsilon\_x \cdot \mathbf{R}}{1 - \varepsilon\_x}\right)^n - 1}{\left(\frac{1 - \varepsilon\_x \cdot \mathbf{R}}{1 - \varepsilon\_x}\right)^n - \mathbf{R}} \tag{13}$$

**Figure 6.** The schematic flow arrangements in 4 × 4 WPHE (**a**) and in 8 × 4 WPHE (**b**).

For the special case, when R = 1, to avoid uncertainty, it is calculated as:

$$\varepsilon\_T = \frac{n \cdot \varepsilon\_x}{1 + (n - 1) \cdot \varepsilon\_x} \tag{14}$$

For the symmetric arrangement of passes, when the passes numbers are equal *X*<sup>1</sup> = *X*<sup>2</sup> = *n* for both streams in WPHE, such a block of plates is a symmetric one-pass heat exchanger with heat transfer area Fa/*n*. Its heat transfer effectiveness, ε*x*, is equal to the effectiveness of one-pass heat exchanger, ε0, calculated in the following Equation (10) as:

$$\varepsilon\_x = \varepsilon\_{0s} = 1 - \mathbf{e}^{\frac{-1+\mathbf{e}^{-\mathsf{R}\cdot\mathsf{NTU}\_{0s}}{\mathsf{R}\cdot\mathsf{NTU}}}} \tag{15}$$

where NTU0s number in the symmetric block of plates is equal to NTU/*n*.

With unsymmetrical passes arrangements (see Figure 6b), one block consists of two sub-blocks of plates. For the cold side, there is one pass, and for the hot side, two sub-passes—as for k = 2—is shown in Figure 6b. For the last pass on the cold stream (first from the left in Figure 6b), the temperature drops of hot stream in the first sub-block is δt11 and, in the second, δt12:

$$
\delta\mathfrak{k}\_{11} = \Delta \cdot \varepsilon\_{02} \cdot \mathbb{R}\_{02} \tag{16}
$$

$$
\delta\mathfrak{t}\_{12} = (\Delta - \delta\mathfrak{t}\_{11}) \cdot \varepsilon\_{02} \cdot \mathbb{R}\_{02} \tag{17}
$$

where Δ = t11 − t24 is the difference of temperatures at streams incoming into considered sub-blocks; R02 = R/2 is the ratio of heat capacities flowrates in sub-blocks of plates; ε<sup>02</sup> is heat transfer effectiveness of the sub-block of plates which consist from 1/8 of the plates total number. The individual effectiveness of all sub-blocks is assumed the same, as all have the same heat transfer area and streams flowrates. The effectiveness of one block of plates consisting of two sub-blocks:

$$
\varepsilon\_{\chi} = \frac{\mathbf{t}\_{25} - \mathbf{t}\_{24}}{\Delta} = \frac{\delta \mathbf{t}\_{11} + \delta \mathbf{t}\_{12}}{\mathbf{R} \cdot \Delta} = \varepsilon\_{02} - \varepsilon\_{02}^{2} \cdot \frac{\mathbf{R}}{4} \tag{18}
$$

The heat transfer effectiveness of one sub-block, ε02, is determined by Equation (10) with NTU in one sub-block NTU02 = NTU/8, assuming its equal distribution in all WPHE and the ratio of heat capacities flowrates is R02. By substitution of *X*<sup>1</sup> = *2X*<sup>2</sup> received from Equation (18) into Equation (13), the total heat transfer effectiveness of WPHE is obtained for the case of unsymmetrical passes arrangement *X*<sup>1</sup> = 2·*X*2. The other values of k in relation *X*<sup>1</sup> = k·*X*<sup>2</sup> are not presenting practical interest and not considered in this paper. When the heat transfer effectiveness of WPHE ε<sup>T</sup> is found, the outlet temperatures can be calculated, as well as heat load and all other parameters required for WPHE thermal modelling.

The calculation of pressure drop in the hydraulic design of WPHE is made by Equation (12) for each of the streams.
