**2. Laboratory Experiment**

For the accurate modelling and design of WPHE, the reliable correlations for a calculation of heat transfer and pressure drop in channels of specific geometry are required. Such correlations for the models of criss-cross flow channels of PHEs with countercurrent flow are presented in [31], and the approach for their application for commercial plates in [13] (pp. 228–231). To check the validity of these correlations and to adopt them for specific construction of developed WPHE with crossflow of streams, the experimental data are needed.

The experimental study of heat transfer and pressure losses is performed with the model of WPHE having 15 plates welded together for modelling one pass of WPHE. There are 14 channels formed between plates; seven channels are for the hot stream, and the other seven channels are for the cold stream. The geometries of the channels at the hot and cold sides are different. At the cold stream, the side channel is formed by two plates, assembled in WPHE, which have different corrugation angles to the main flow direction. On one plate, corrugations are mainly (at 2/3 of plates area) situated parallel to the main flow. The corrugations on the other plate are inclined with the angle of 60◦ to the direction of the main flow. The average corrugation angle in this part of the channel is 30◦. In the other third of the channel area, the angle of corrugations on both plates forming the channel is 60◦. The averaged channel area angle of corrugations is β<sup>2</sup> = 40◦. The averaged on another channel area angle of corrugations to the main flow direction in this channel is β<sup>1</sup> = 50◦. These forms of corrugations at channel walls are made for facilitating the removal of possible dust in a stream, leaving a catalyser that can start to appear with catalyser aging. The geometrical parameters of the WPHE model, its plates and channels, are given in Table 1.

The thermal and hydraulic parameters of the WPHE model are determined from measurements at the test rig with distilled water flowing as hot and cold fluids. The structure of the test rig is shown in Figure 2. It has a closed water circuit. The water is pumped from its reservoir to the shell-and-tube steam heater to maintain specified temperature t11 of the hot stream at the model inlet. Inside the WPHE model it is being cooled to a temperature t12 at the outlet, and afterwards, it is directed to the additional PHE for cooling to temperature t21 by cold water from the centralised circuit with the cooling tower. With this temperature cold stream entering the WPHE model, it is heated to the cold stream outlet temperature t22 and is coming back into the water reservoir. The circulating water flowrate is measured with orifice meter. The temperatures at the entrances and exits of heat exchanging streams in the WPHE model are measured with an accuracy of ±0.05 ◦C by thermocouples copper–constantan. Two differential pressure gauges are used to measure pressure drops of hot- and cold-water streams. The tests are performed in a stable regime. The water temperature after the steam heater was varied from 55 to 85 ◦C. The temperature of cold water in the reservoir was in the range of 30 to 45 ◦C. The flowrate of circulating water was in the range of 0.8 and 4.5 kg/s.


**Table 1.** The parameters of WPHE experimental model.

**Figure 2.** Structure of the test rig.

The heat load Q of the model is determined as an average of the calculated for cold stream Q2 and for hot stream Q1, W:

$$\mathbf{Q}\_2 = \mathbf{G} \cdot \mathbf{c}\_{\mathbf{p}2} \cdot (\mathbf{t}\_{22} - \mathbf{t}\_{21}); \ \mathbf{Q}\_1 = \mathbf{G} \cdot \mathbf{c}\_{\mathbf{p}1} \cdot (\mathbf{t}\_{11} - \mathbf{t}\_{12}); \ \mathbf{Q} = (\mathbf{Q}\_1 + \mathbf{Q}\_2) / 2,\tag{1}$$

where G is mass flowrate of water, kg/s; cp1 is the specific heat capacity of a hot stream, J/(kg·K); cp2 is the specific heat capacity of a cold stream, J/(kg·K).

In a closed water circuit, the mass flowrates of hot and cold streams are the same, and the heat capacity difference is less 0.3%, so heat capacity flowrates are equal, and the mean temperature difference is

$$\Delta \mathbf{t\_m} = \frac{(\mathbf{t\_{11}} + \mathbf{t\_{12}}) - (\mathbf{t\_{21}} + \mathbf{t\_{22}})}{2} \tag{2}$$

The experimental value of overall heat transfer coefficient, W/(m2·K), is as follows:

$$\mathbf{U\_{ex}} = \frac{\mathbf{Q}}{\Delta \mathbf{t\_m} \cdot \mathbf{F\_{am}}} \tag{3}$$

The experimental results are compared with calculated by Equation (4):

$$\mathbf{U}\_{\rm cl} = \left(\frac{1}{\mathbf{h}\_1} + \frac{1}{\mathbf{h}\_2} + \frac{\delta\_{\rm w}}{\lambda\_{\rm w}} + \mathbb{R}\_{\rm f}\right)^{-1} \tag{4}$$

where *hi* are the film heat transfer coefficients for cold (index *<sup>i</sup>* <sup>=</sup> 2) and hot (index *<sup>i</sup>* <sup>=</sup> 1) sides, W/(m2·K); λ<sup>w</sup> is a thermal conductivity of plate metal, W/(m·K); δ<sup>w</sup> thickness of the plate metal, m; Rf is the fouling thermal resistance in WPHE, m2·K/W. In laboratory experiments, it is taken that Rf <sup>=</sup> 0.

The film heat transfer coefficients are calculated based on the modified analogy of heat and momentum transfer [31]:

$$\text{Nu}\_{i} = 0.065 \cdot \text{Re}\_{i}^{\frac{\phi}{\prime}} \cdot \left(\frac{\psi\_{i} \cdot \zeta\_{i}}{F\_{X}}\right)^{\frac{3}{7}} \cdot \text{Pr}\_{i}^{c} \cdot \left(\frac{\mu\_{i}}{\mu\_{\text{wi}}}\right)^{0.14} \tag{5}$$

where μ*<sup>i</sup>* is fluid dynamic viscosity at the main flow temperature, Pa·s; μwi is fluid dynamic viscosity at the wall temperature, Pa·s; Nu = *hi*·*d*e/λ*<sup>i</sup>* is Nusselt number; λ*<sup>i</sup>* is fluid thermal conductivity, W/(m·K); Pr*<sup>i</sup>* is Prandtl number; ζ*<sup>i</sup>* is the friction factor; ψ*<sup>i</sup>* is the ratio of friction pressure loss to a total loss of pressure; *F*<sup>X</sup> is the factor of area enlargement because of corrugations.

The value of ψ*<sup>i</sup>* is calculated by Equation proposed in [32]:

$$\psi\_i = \left\{ \frac{\text{Re}\_i}{380} \cdot \left[ \tan(\beta\_i) \right]^{1.75} \right\}^{-0.15 \cdot \sin(\beta\_i)} \ge 1 \tag{6}$$

For calculation of friction factor, Equation (7) is used [31]:

$$\begin{aligned} \zeta\_i &= 8 \cdot \left[ \left( \frac{12 + p\_{2i}}{\text{Re}\_i} \right)^{12} + \frac{1}{\left( A\_i + B\_i \right)^{\frac{3}{2}}} \right]^{\frac{1}{12}}, \\\ A\_i &= \left[ p\_{4i} \cdot \ln \left( \frac{p\_{5i}}{\left( \frac{p\_{5i}}{\text{Re}\_i} \right)^{0.9} + 0.27 \cdot 10^{-5}} \right) \right]^{16}, \ B\_i = \left( \frac{37530 \cdot p\_{1i}}{\text{Re}\_i} \right)^{16} \end{aligned} \tag{7}$$

where *p*1*i, p*2*i, p*3*i, p*4*i, p*<sup>5</sup>*<sup>i</sup>* are the parameters depending on the corrugation geometry:

$$\begin{aligned} p\_{1i} &= \text{e}^{-0.15705 \cdot \beta\_i}; p\_{2i} = \frac{\pi \cdot \text{\beta} \cdot \gamma\_i^2}{3}; p\_{3i} = \exp\left(-\pi \cdot \frac{\beta\_i}{180} \cdot \frac{1}{\gamma\_i^2}\right) \\ p\_{4i} &= \left[0.061 + (0.69 + \tan \beta\_i)^{-2.63}\right] \cdot \left[1 + (1 - \gamma\_i) \cdot 0.9 \cdot \beta\_i^{0.01}\right]; p\_{5i} = 1 + \frac{\beta\_i}{10}; p\_{5i} \end{aligned}$$

γ*<sup>i</sup>* = *2*·*b*/*S* is double height to pitch ratio of corrugation; β*<sup>i</sup>* is the angle of corrugation to the main flow direction, degrees; Re*<sup>i</sup>* = *wi*·*d*e·ρ*i*/μ*<sup>i</sup>* is Reynolds number; *wi* is the flow velocity, m/s; *de* = *2*·*b* is the channel equivalent diameter, m; <sup>μ</sup>*<sup>i</sup>* is fluid dynamic viscosity, Pa·s; <sup>ρ</sup>*<sup>i</sup>* is the fluid density, kg/m3.

The experimental and calculated values of overall heat transfer coefficients at different cold-water velocities in channels are compared. The calculated results are higher than experimental ones, with deviations up to +8%. The explanation is in a reduction of real average temperature difference compared to that calculated by Equation (2) because of the crossflow. The heat transfer effectiveness of the WPHE model is as follows:

$$\varepsilon = \frac{\mathbf{t}\_{22} - \mathbf{t}\_{21}}{\mathbf{t}\_{11} - \mathbf{t}\_{21}} \tag{8}$$

Effectiveness dependence from the number of heat transfer units (NTU) is presented in Figure 3. NTU is determined using calculated value Ucl of overall heat transfer coefficient by Equation (9):

$$\text{NTU} = \frac{\text{F}\_{\text{a}} \cdot \text{U}\_{\text{cl}}}{\text{G}\_{2} \cdot \text{c}\_{\text{F}^{2}}} \tag{9}$$

**Figure 3.** The relation ε-NTU for WPHE model: 1—experiment; 2—counter- current; 3—calculated by Equation (10) for crossflow in WPHE for the ammonia synthesis column.

The heat transfer effectiveness, ε, experimental values are smaller than calculated by Equation for countercurrent flow (curve 2 in Figure 3). It is better approximated by the Equation presented in [20] for cross flow and stream, with a higher corrugation angle β<sup>1</sup> = 50◦, mixed, and another with lower corrugation angle β<sup>1</sup> = 40◦, unmixed. For a better approximation, the corrected factor 0.97 is added:

$$\kappa = 1 - \mathbf{e}^{\frac{-1 + q^{-R\mathcal{N}\Gamma U}}{R\mathcal{R}\mathcal{W}}} \tag{10}$$

where R is the ratio of heat capacities flowrates:

$$\mathbf{R} = \frac{\mathbf{G}\_1 \cdot \mathbf{c}\_{\mathrm{pl}}}{\mathbf{G}\_2 \cdot \mathbf{c}\_{\mathrm{Pl}^2}} \tag{11}$$

Equation (10) is suitable for the calculation of one pass effectiveness when calculating multi-pass WPHE by the ε-NTU method described by Arsenyeva et al. [33].

Besides thermal efficiency, the hydraulic performance of the WPHE is of primary importance for its correct design. The experimental results on Euler number Eu*<sup>i</sup>* = ΔP*i*/(ρi·wi 2) for streams are presented in Figure 4.

For the channel with a bigger corrugation angle β<sup>1</sup> = 50◦, the pressure losses are higher than for another channel of β<sup>2</sup> = 40◦. Following the approach proposed in [34] for the estimation of pressure drop in plate-and-frame PHE at the main corrugation field and channel distribution zones at the entrance and exit separately, the total pressure drop at the stream in multi-pass WPHE can be summarised as follows:

$$
\Delta \mathbf{P}\_i = \left( \zeta\_i \cdot \frac{\mathbf{L}\_{pl}}{d\_\mathcal{E}} \cdot \frac{\rho\_i \cdot w\_i^2}{2} + \zeta\_{\text{DZ}i} \cdot \frac{\rho\_i \cdot w\_{\text{irr.},i}}{2} \right) \cdot X\_i \tag{12}
$$

where *winx.i* is the velocity at channel entrance/exit, m/s; ζ*Dzi* is the coefficient of local hydraulic resistance in entrance/exit zones; *Xi* is a number of passes.

**Figure 4.** The experimental data for pressure losses: 1—hot stream; 2—cold stream.

By comparison of the calculation by Equation (12) with the ζ*<sup>i</sup>* determined by Equation (7), the coefficients of local hydraulic resistance in the channels' distribution zones were obtained: ζ*DZ1* = 11, ζ*DZ2* = 17. The error of estimation by Equation (12)—the experimentally measured pressure drop in channels—does not exceed ±8%, as illustrated in Figure 5.

**Figure 5.** The comparison of experimental pressure drop calculated by Equation (12): 1—hot stream; 2—cold stream.
