Experimental-Numerical Method for Determining Heat Transfer Correlations in the Plate-and-Frame Heat Exchanger

Abstract

Plate heat exchangers are used in heat substations for domestic hot water preparation and building heating in municipal central heating systems. Water from the municipal water supply is heated by hot water from a district heating network. This paper presents a numerical method for simultaneously determining heat transfer correlations on the cold and hot water sides based on flow-thermal measurements of the plate heat exchanger. The unknown parameters in the functions approximating the Nusselt numbers, which depend on the Reynolds and Prandtl numbers, are determined using the least-squares method, so the sum of the squares of the differences in the calculated and measured temperatures at the heat exchanger outlet reaches a minimum. One or two correlations were sought for a plate heat exchanger, and the total number of parameters sought is between three and six. The limits of the 95% confidence intervals for all estimated parameters were also determined. Correlations for Nusselt numbers determined experimentally for a clean plate heat exchanger can be used in the online monitoring of the degree of fouling of plate heat exchangers installed in the substations of a large urban district heating network.

1. Introduction

Plate and frame (gasketed) heat exchangers are widely used in heating, air-conditioning, the food industry and many other applications [1]. Their main advantage is their high heat output with a small heat exchanger size.

Both in design and performance calculations, it is necessary to know the heat transfer correlation for calculating the Nusselt numbers (Nu), which allow for calculating the mean heat transfer coefficients (HTCs) on the side of each fluid [2].

The Wilson method [3] is a popular method used for experimentally determining correlations for Nusselt numbers on both the hot and cold fluid sides. Sekulić and Shah [4] present various methods for experimentally determining the HTC and correlations for Nusselt numbers on the side of one or both working fluids. They also present the practical application of Wilson’s plot method for determining correlations for Nusselt numbers on the gas and liquid sides in plate-fin and tube heat exchangers (PFTHE).

The Wilson method is also discussed by Taler [5] in the book on heat exchangers’ modelling and testing [4]. The Wilson method is labour-intensive because the determination of correlations for Nusselt numbers is iterative. Correlations are determined sequentially. First, the correlation for the Nusselt number on the side of one fluid, for example the hot fluid, is determined, while on the side of the other fluid, i.e., the cold fluid, the mass flow rate is constant so that the HTC is constant. In the second iteration, the HTC on the cold fluid side is determined, while the mass flow rate on the hot fluid side is kept constant. The iterative process is then repeated until the coefficients in the Nusselt number correlations do not change. When one or both working fluids are liquids, it is difficult to maintain a constant HTC on one side of the partition when the flow rate and fluid temperature on the other side change. Despite the constancy of the mass flow rate, the HTC changes because the Prandtl number changes as a result of changes in fluid temperature, and therefore, the HTC changes despite the constant mass flow rate of the fluid. Thus, the basic condition in the iterative process of determining the HTC, proposed by Wilson, is not maintained. This disadvantage of Wilson’s method is illustrated in [5] by determining the correlation for Nusselt numbers in a PFTHE. Another weakness of Wilson’s method is linearising the nonlinear correlations of the Nusselt number on the Reynolds (Re) and Prandtl (Pr) numbers. By taking the logarithm of this nonlinear power-type heat transfer correlation $Nu=f(\mathrm{Re},\mathrm{Pr})$ and introducing new variables, the determination of unknown coefficients is reduced to a linear least squares problem, usually linear regression. As a result of this operation, the unknown parameters in the correlation for the Nusselt number have different values than those obtained using the non-linear least squares method.

A commonly used technique for determining correlations on one side of a heat exchanger involves first estimating the overall HTC from experimental tests conducted on the entire heat exchanger. The HTC on the inner surface of the tubes is then calculated using well-established correlations for the Nusselt number in straight tubes, such as the Dittus–Boelter [6] or Gnielinski [7] equations. If transitional or turbulent flow occurs inside the exchanger tubes during experimental testing, Taler’s correlation [8] can be used to calculate the HTC on the inner surface of the tubes.

Knowing the HTC, the internal HTC, and the thermal resistance of the tube wall, the HTC on the outer surface of the tubes can be determined. Subsequently, an appropriate function is identified to approximate the outer surface Nusselt numbers as a function of the Reynolds and Prandtl numbers, or other relevant parameters influencing this coefficient.

Nasrfard et al. [9] provide an example of such an approach, presenting experimental results on the condensation heat transfer of R141b in an intermittent flow regime on a smooth horizontal tube. Their study proposed a power-law correlation for calculating the Nusselt number as a function of the equivalent Reynolds and Prandtl numbers. A similar methodology to determine the Nusselt number correlation on one side of the exchanger can be found in many other publications. Wilson’s method is used to derive correlations for the Nusselt number on the outer tube surfaces. Fu et al. [10] developed a tubular cross-flow serpentine tube heat exchanger for aero-engine cooling. A correlation was established for the Nusselt number on the cooling air side based on thermal flow tests of three different heat exchangers with different tube outer diameters: 2.2, 1.8, and 1.4 mm. The HTC from the flowing RP-3 kerosene to the inner surface of small-diameter tubes was calculated using Gnielinski’s equation. The Wilson method was used to determine the unknown parameters in the power correlation for the air side Nusselt number. Shah et al. [11] carried out an experimental test to find power-type heat transfer correlations on the inner surface of the tubes through which the water/glycol solution flowed. Ammonia was expanding in the space between the tubes of the shell-and-tube heat exchanger. The tubes with a 3-D structured outside surface enhanced condensation heat transfer. The heat transfer correlation for the tube side HTC was determined by the modified Wilson plot method. Nguyen et al. [12] experimentally studied the condensation of R-513a refrigerant in a vertical plate heat exchanger in which the plate surfaces were electrochemically etched. The correlation for the Nusselt number on the cooling water side was found using the modified Wilson plot method. The inverse HTC on the condensing side of R-513a was calculated by subtracting the thermal resistance of the wall and the inverse HTC on the water side from the inverse overall HTC. Next, a power correlation was determined to estimate the Nusselt number on the R-513a side. An experimental study by Li et al. [13] investigated the condensation of R134a in a horizontal shell-and-tube heat exchanger. Micro-fins on the outer surface of the tubes were used to increase the HTC on the condensing R134a side. The HTC on the side of the hot water flowing inside the tubes was calculated using Gnielinski’s equation [7], and the correlation for the outer HTC was established using the modified Wilson plot method. Shokouhmand et al. [14] investigated experimentally shell and coiled tube heat exchangers. The Wilson plot method was used to estimate the HTC on the outer surface of the coiled tube. The direct method was applied by Fronk and Garimella [15] to measure the HTC on the inner tube surface during condensation of ammonia and ammonia/water mixtures in minichannels. The HTC was calculated using its definition based on the measurement of the heat flux, wall and fluid temperatures.

Currently, instead of experimental research, flow and heat transfer modelling in heat exchangers is increasingly being carried out using various commercial Computational Fluid Dynamics (CFD) software. Based on computer simulations conducted at different mass flow rates and temperatures of both fluids, correlations are determined for the Nusselt number, usually on the side of one fluid. Taler D. et al. [16] established correlations for the air side Nusselt number on individual tube rows in plate-fin and tube heat exchangers (PFTHE). The HTC inside the tubes was calculated using the correlation available in the literature. The correlation proposed in [8] was used in [16] to estimate the HTC for water on the inner surface of the tubes. A similar procedure to that in [16] was used by Yu et al. [17] to determine the correlation for the Nusselt number on the flue gas side in a finned-tube heat exchanger in a gas-fired water boiler. The HTC on the water side was determined using Gnielinski’s equation [7], and the correlation for the Nusselt number on the gas side was determined based on CFD modelling. The Nusselt number determined based on CFD simulation closely approximates the Nusselt number determined on the experimental stand. Calculations of the HTC on the internal surfaces of tubes using experimental correlations available in the literature can be avoided in PFTHEs by assuming a constant external tube surface temperature or a constant tube and fin surface temperature [18]. A constant temperature of the finned tube surface can be assumed because, in the case of air or other gases, the HTC depends mainly on the gas flow velocity, and the influence of the tube and fin temperatures is negligible. CFD modelling allows for a simple and inexpensive way to determine correlations for the Nusselt number on the gas side of PFTHEs, as well as in plate heat exchangers. Zhu [19] investigated the influence of the chevron angle in plate heat exchangers on the thermal performance of plate heat exchangers using CFD simulation. New correlations for calculating the Nusselt number in plate heat exchangers, considering the chevron angle value, were proposed [19]. However, it is important to note the large discrepancies in Nusselt numbers determined using CFD simulations with different turbulence models. This is due to the lack of suitable turbulence models, despite the fact that there are several dozens of them. However, the results obtained using CFD modelling require experimental verification.

Taler D. [20] proposed a method for simultaneously determining correlations for Nusselt numbers on the air and water sides in PFTHE. Unknown parameters in Nusselt number correlations were selected so that the sum of squares of differences between calculated and measured water and air temperatures at the heat exchanger outlet was minimum. The minimised sum of squares took into account the sums of squares of the differences between the measured and calculated temperatures at the heat exchanger outlet, taking into account the measured water and air temperatures but with different weights. The weighting factors in the sum of squares are smaller for air, due to the lower accuracy of measurements on the air side compared to the accuracy of measurements on the water side. An analytical model was used to calculate the water and air temperatures at the outlet of the tested PFTHE.

An overview of methods for determining HTC on the inner surface of tubes in condensation flow is provided in the paper by Marchetto et al. [21]. The discussed works are divided into three groups devoted to the direct method, the Wilson plot method and the boundary layer thickness method.

Artificial neural networks or other data-driven methods can also be used to calculate HTCs in heat exchangers. However, methods based on artificial intelligence require a huge database of HTCs determined experimentally using other methods to predict HTCs with sufficient accuracy.

A literature review shows that in most experimental studies on heat exchangers, the correlation for the Nusselt number is determined only on one fluid’s side. The correlations available in the literature are usually used to determine the HTC on the side of the other fluid, for example, for the medium flowing in the tubes. Considering that the overall HTC is determined experimentally, the uncertainty of determining the average HTC using the formulas available in the literature may cause large errors in the correlations determined for the second medium using the Wilson method. It should be emphasised that the HTCs available in the literature were derived for straight tubes, while heat exchangers usually have multiple passes, which means that the direction of fluid flow in the tubes or manifolds is reversed. The fluid flow conditions inside the heat exchanger tubes differ from those in straight tubes.

The least squares method is used in the paper to determine unknown coefficients in correlations for Nusselt numbers on the hot and cold liquid sides. The Nusselt numbers are determined as nonlinear functions of the Reynolds and Prandtl numbers without transforming heat transfer correlations to obtain a linear least squares problem, as, for example, in Wilson’s method. This ensures high accuracy in determining unknown coefficients in both correlations for hot and cold fluids. In the method presented in this paper, heat transfer correlations can even be complex functions of Reynolds and Prandtl numbers. Correlations for the Nusselt numbers do not have to be power functions as in Wilson’s method.

2. Numerical-Experimental Method for Simultaneous Determination of the Heat Transfer Correlations on the Hot and Cold Water Side

The proposed method for determining heat transfer correlations is general and can be used to determine correlations for the Nusselt number on the hot and cold sides in various types of heat exchangers. For the determined correlations to be reliable, the measurement data must meet the following conditions:

• measurements of mass flow rates and temperatures of both working fluids should be carried out during steady-state operation of the heat exchanger,
• the range of inlet temperature changes of the hot and cold fluids should be large so that the range of Prandtl number changes for each fluid is large,
• the range of changes in the mass flow rates of the hot and cold fluids should be large so that the range of changes in the Reynolds number for each fluid is large,
• the uncertainties in the measurement of the temperature and mass flow rates of both working fluids should be small, i.e., the measuring instruments should be of high accuracy,
• the heat flow rates transferred from the hot liquid to the cold liquid, calculated based on measurement data, should not differ too much,
• the mathematical model of the heat exchanger should be highly accurate and closely approximate the measurement data,
• the number of measurement series should be large, in the order of several dozen measurement series with significantly different temperatures of the working fluids at the inlet to the heat exchanger and different mass flows of the working fluids,
• unknown parameters in correlations for the Nusselt number for hot and cold fluids should be selected so that the sum of the squares of the differences between the measured and calculated temperatures for all measurement series is as small as possible.

The plate heat exchanger under study is a counter-current heat exchanger. In the proposed method for determining the correlation for Nusselt number on the hot and cold water side, it is necessary to calculate the heat exchanger’s cold and hot water outlet temperatures multiple times for each measurement data set. This section presents a procedure for calculating the cold and hot water outlet temperatures of a heat exchanger based on the P-NTU (Effectiveness- Number of Transfer Units) method.

2.1. Determination of Hot and Cold Water Temperatures at the Outlet of the Heat Exchanger

The P-NTU method [22] was used to determine the cold and hot water temperatures at the outlet of the heat exchanger. Effectiveness $P_h$ is defined as follows (Figure 1)

$$P_h={\displaystyle \frac{{\dot{Q}}_h}{{\dot{Q}}_{h,\max }}}={\displaystyle \frac{{\dot{m}}_h{c}_h(T_{h1}-T_{h2})}{{\dot{m}}_h{c}_h(T_{h1}-T_{c2})}}={\displaystyle \frac{(T_{h1}-T_{h2})}{(T_{h1}-T_{c2})}}$$

(1)

where $P_h$—effectiveness of the heat exchanger calculated for hot water; ${\dot{Q}}_h$—rate of heat flow in the heat exchanger, W; ${\dot{Q}}_{h,\max }$—maximum rate of heat flow in the heat exchanger, W; ${\dot{m}}_h$—mass flow rate of hot water, kg/s; $c_h$—specific heat of hot water, J/(kgK); $T_{h1}$, $T_{h2}$—temperature of hot fluid at the inlet and outlet of the heat exchanger, °C; $T_{c2}$, $T_{c1}$—temperature of cold fluid at inlet and outlet of the heat exchanger, °C.

The equation for the effectiveness of a counter-flow heat exchanger has the following form [5]

$$P_h={\displaystyle \frac{1-\exp \left[-U_{A}A(\frac{1}{{\dot{m}}_h{c}_h}-\frac{1}{{\dot{m}}_c{c}_c})\right]}{1-\frac{{\dot{m}}_h{c}_h}{{\dot{m}}_c{c}_c}\exp \left[-U_{A}A(\frac{1}{{\dot{m}}_h{c}_h}-\frac{1}{{\dot{m}}_c{c}_c})\right]}}={\displaystyle \frac{1-\exp \left[-\left(NT{U}_h-NT{U}_c\right)\right]}{1-\frac{NT{U}_c}{NT{U}_h}\exp \left[-\left(NT{U}_h-NT{U}_c\right)\right]}}$$

(2)

where $U_A$—overall HTC related to the area $A$, m²; $NT{U}_c$, $NT{U}_h$—number of heat transfer units for cold and hot water; ${\dot{m}}_c$—mass flow rate of cold water, kg/s; $c_c$—specific heat of cold water, J/(kgK).

The numbers of heat transfer units for cold and hot water are given by the following formulas

$$NT{U}_c={\displaystyle \frac{U_{A}A}{{\dot{m}}_c{c}_c}}\hspace{1em}NT{U}_h={\displaystyle \frac{U_{A}A}{{\dot{m}}_h{c}_h}}$$

(3)

After calculating the effectiveness $P_h$ for hot water using Equation (2), the hot water temperature $T_{h2}$ at the outlet of the heat exchanger is determined from Equation (1)

$$T_{h2}=T_{h1}-P_h(T_{h1}-T_{c2})$$

(4)

By substituting Equation (2) into Formula (3), one obtains

$$\begin{array}{l}{T}_{h2}=T_{h1}-{\displaystyle \frac{(T_{h1}-T_{c2})\left\{1-\exp \left[-U_{A}A(\frac{1}{{\dot{m}}_h{c}_h}-\frac{1}{{\dot{m}}_c{c}_c})\right]\right\}}{1-\frac{{\dot{m}}_h{c}_h}{{\dot{m}}_c{c}_c}\exp \left[-U_{A}A(\frac{1}{{\dot{m}}_h{c}_h}-\frac{1}{{\dot{m}}_c{c}_c})\right]}}\\ \hspace{1em}=T_{h1}-{\displaystyle \frac{(T_{h1}-T_{c2})\left\{1-\exp \left[-\left(NT{U}_h-NT{U}_c\right)\right]\right\}}{1-\frac{NT{U}_c}{NT{U}_h}\exp \left[-\left(NT{U}_h-NT{U}_c\right)\right]}}\end{array}$$

(5)

The heat flow rate transferred from the hot water ${\dot{Q}}_h$ is equal to the heat flow rate absorbed by the cold water ${\dot{Q}}_c$

$${\dot{m}}_h{c}_h\left(T_{h1}-T_{h2}\right)={\dot{m}}_c{c}_c\left(T_{c1}-T_{c2}\right)$$

(6)

From Equation (4) we obtain

$$T_{h1}-T_{h2}=P_h(T_{h1}-T_{c2})$$

(7)

$$T_{c1}-T_{c2}=P_c(T_{h1}-T_{c2})$$

(8)

where $P_c$—effectiveness of the heat exchanger for cold water.

Substituting expressions (7) and (8) into Equation (6) gives

$${\dot{m}}_h{c}_h{P}_h\left(T_{h1}-T_{c2}\right)={\dot{m}}_c{c}_c{P}_c\left(T_{h1}-T_{c2}\right)$$

(9)

$${\displaystyle \frac{{\dot{m}}_h{c}_h}{{\dot{m}}_c{c}_c}}={\displaystyle \frac{P_c}{P_h}}$$

(10)

After simple transformations, Equation (10) gives

$$P_c={\displaystyle \frac{{\dot{m}}_h{c}_h}{{\dot{m}}_c{c}_c}}{P}_h={\displaystyle \frac{NT{U}_c}{NT{U}_h}}{P}_h$$

(11)

The cold water temperature $T_{c1}$ at the outlet of the heat exchanger, determined from Equation (6) is given by the formula

$$T_{c1}=T_{c2}+{\displaystyle \frac{{\dot{m}}_h{c}_h}{{\dot{m}}_c{c}_c}}\left(T_{h1}-T_{h2}\right)=T_{c2}+{\displaystyle \frac{NT{U}_c}{NT{U}_h}}\left(T_{h1}-T_{h2}\right)$$

(12)

Equation (8) can also be used to determine the temperature $T_{c1}$

$$T_{c1}=T_{c2}+P_c(T_{h1}-T_{c2})$$

(13)

Using the P-NTU [22] or ε-NTU (Effectiveness-Number of transfer units) [23] method to determine hot and cold medium outlet temperatures is advantageous, especially for heat exchangers with complex flow systems. Formulas for calculating effectiveness P can be found in [22] and for ε in [23] for many heat exchangers of different designs. The respective arithmetic averages of the inlet and outlet temperatures were used to calculate all physical properties of the hot and cold water.

2.2. Method for Determining Unknown Coefficients in Correlations for Nusselt Numbers for Hot and Cold Water

Knowledge of the HTCs on both sides of a wall, for example, a plate, plain or finned tube, is essential for design or performance calculations of heat exchangers as well as for online monitoring systems for heat transfer surface fouling.

The overall HTC for a clean plate and frame heat exchanger is determined using the following equation:

$${\displaystyle \frac{1}{U}}={\displaystyle \frac{1}{h_c}}+{\displaystyle \frac{\delta }{k_p}}+{\displaystyle \frac{1}{h_h}}$$

(14)

where $h_c$ and $h_h$—HTC for cold and hot water, respectively, $\mathrm{W}/{(\mathrm{m}}^2\mathrm{K})$; $k_p$—thermal conductivity of the plate, $\mathrm{W}/\left(\mathrm{m}\mathrm{K}\right)$; $\delta$—plate thickness, m.

Knowing the surface area A of the heat exchanger, the overall HTC $U$ and the inlet temperatures of the hot $T_{h1}$ and cold $T_{c2}$ fluid, the temperature of the hot $T_{h2}$ and cold fluid at the outlet of the heat exchanger can be calculated.

The HTCs $h_c$ and $h_h$ are calculated using the correlations for the Nusselt number on the cold and hot water side, which are determined based on experimental testing. The Nusselt number $N{u}_c$ on the cold water side is given by the following equation:

$$N{u}_c={\displaystyle \frac{h_c{D}_h}{k_c}}$$

(15)

The hydraulic diameter $D_h$ of the channel is defined as follows:

$$D_h={\displaystyle \frac{4A_{ch}}{P_w}}$$

(16)

where $A_{ch}$—channel flow area, $P_w$—wetted perimeter.

The Nusselt number on the hot water side is defined similarly to the cold water side

$$N{u}_h={\displaystyle \frac{h_h{D}_h}{k_h}}$$

(17)

The symbols $k_c$ and $k_h$ in Equations (15) and (17) represent the thermal conductivity of the cold and hot water.

The correlation for the Nusselt number on the hot water side was applied as follows

$$N{u}_h=x_{1,h}{\mathrm{Re}}_h^{x_{2,h}}{\mathrm{Pr}}_h^{x_{3,h}}$$

(18)

where the Reynolds number ${\mathrm{Re}}_h$ and Prandtl number ${\mathrm{Pr}}_h$ are defined as follows

$${\mathrm{Re}}_h={\displaystyle \frac{w_h{D}_h}{{\nu }_h}}$$

(19)

The following correlation was assumed for the Nusselt number on the cold water side:

$$N{u}_c=x_{1,c}{\mathrm{Re}}_h^{x_{2,c}}{\mathrm{Pr}}_h^{x_{3,c}}$$

(20)

where the Reynolds number ${\mathrm{Re}}_c$ and Prandtl number ${\mathrm{Pr}}_c$ are defined as follows

$${\mathrm{Re}}_c={\displaystyle \frac{w_c{D}_c}{{\nu }_c}}\hspace{1em}{\mathrm{Pr}}_c={\displaystyle \frac{c_c{\mu }_c}{k_c}}$$

(21)

In Equations (19)–(21), the following designations are adopted: $w_c,w_h$ average velocity of cold and hot water in the channel cross-section; ${\nu }_c,{\nu }_h$ kinematic viscosity of cold and hot water.

Unknown coefficients $x_{1,h}$ and power exponents $x_{2,h}$, $x_{3,h}$ in correlation (18) for hot water and unknown coefficients $x_{1,c}$ power exponents $x_{2,c}$, $x_{3,c}$ in correlation (20) for cold water were determined based on experimental studies.

First, the number n of coefficients required for the correlations per Nusselt number for hot and cold water is determined.

Three cases were considered:

(a)

the total number of unknown parameters in the correlation for cold water and for hot water is $n=n_h+n_c$, where the symbol $n_h$ indicates a number of unknown parameters in correlation for the Nusselt number for hot water, $n_c$ number of unknown parameters for cold water;

(b)

the total number of unknown parameters in the correlation for cold water and for hot water is $n=4$. There are two unknown parameters in each of the correlations for hot and cold water: $n_h=2$ and $n_c=2$;

(c)

in both correlations for the Nusselt numbers, on the hot and cold sides, there is the same number of unknown parameters, i.e., $n_h=n_c=n$. The total number n of unknown coefficients and power exponents on the cold and hot water side is $n=3$. It was assumed that: $x_{1,h}=x_{1,c}$, $x_{2,h}=x_{2,c}$ and $x_{3,h}=x_{3,c}$.

In the present study, $n_h=3$ and $n_c=3$. When different parameters are assumed in the correlations for the Nusselt number for hot and cold water, the total number of unknown parameters is $n=4$ or $n=6$.

The unknown coefficients (parameters) were determined from the condition of the minimum of the sum S of the differences of squares between the calculated and measured hot and cold water temperatures at the outlet of the heat exchanger

$$S={\displaystyle \sum _{i=1}^m\left[{\left(T_{h2,i}^{cal}-T_{h2,i}^{meas}\right)}^2+{\left(T_{c1,i}^{cal}-T_{c1,i}^{meas}\right)}^2\right]}\to \min$$

(22)

In Equation (22), the following designations are used, $m$—number of measuring series; $T_{h2,i}^{cal}, T_{c1,i}^{cal}$— calculated hot and cold water outlet temperatures of the exchanger for the i-th measuring series, respectively; $T_{h2,i}^{meas},T_{c1,i}^{meas}$—the measured temperature of the hot and cold water at the exchanger outlet for the i-th measuring series, respectively.

The values of the unknown parameters $x_1,\dots ,x_n$ were determined using the Levenberg–Marquardt method [24,25]. Temperatures $T_{h2,i}^{cal},T_{c1,i}^{cal}$ are functions of the parameters looked for $x_1,\dots ,x_n$.

The 95% confidence intervals of the determined parameters $x_1,\dots ,x_n$ were calculated using Gauss’s law of transfer of variance formulated for indirectly determined quantities [8,26]. The mean square deviation was determined on the basis of the minimum sum of squares (22).

A detailed derivation of the equations used to determine the 95% confidence intervals for the determined parameters can be found in [8].

2.3. Uncertainty Analysis

The uncertainties for the estimated parameters were determined using the Gauss variance propagation rule. Confidence intervals of the determined parameters in the correlations for the heat transfer coefficients at the sides of the air and water. The real values of the determined parameters ${\tilde{x}}_1$,…,${\tilde{x}}_n$ are found with the probability of $P=(1-\alpha )\cdot 100\%$ in the following intervals:

$$x_i-t_{2m-n}^{\alpha /2} s_t \sqrt{c_{ii}}\le {\tilde{x}}_i\le x_i+t_{2m-n}^{\alpha /2} s_t \sqrt{c_{ii}}$$

(23)

where $x_i$—parameter determined using the least squares method; $t_{2m-n}^{\alpha /2}$—quantile of the t-Student distribution for the confidence level $100(1-\alpha )\%$ and 2m−n degrees of freedom.

The least squares sum is characterized by the variance of the fit $s_t^2$, which is an estimate of the variance of the data ${\sigma }^2$ and is calculated according to

$$s_t=\sqrt{{\displaystyle \frac{S_{\min }}{2m-n}}}$$

(24)

where 2m denotes the number of measurement points and n stands for the number of searched parameters.

If the Levenberg–Marquardt iterative method [24,25,26] is used to solve the nonlinear least-squares problem, then the estimated variance-covariance matrix from the final iteration is

$$D_x^{\left(s\right)}=s_t{C}_x^{\left(s\right)}=s_t{\left[ {\left( J^{\left(s\right)} \right)}^T J^{\left(s\right)} \right]}^{-1}$$

(25)

where the matrix $C_x^{\left(s\right)}$ is

$$C_x^{\left(s\right)}={\left[ {\left( J^{\left(s\right)} \right)}^T{J}^{\left(s\right)} \right]}^{-1}$$

(26)

The superscript (s) denotes the number of the last iteration while J is the Jacobian matrix.

The Jacobian matrix J is given by

$$J={\displaystyle \frac{𝜕T^{cal}\left(x\right)}{𝜕x^T}}={\left[\left({\displaystyle \frac{𝜕T_i^{cal}\left(x\right)}{𝜕x_j}}\right)\right]}_{2m\times n},\hspace{1em}i=1 ,\dots , 2m,\hspace{1em}j=1 ,\dots , n$$

(27)

For the issue analyzed in this paper, the Jacobian matrix has the following form:

$$J={\left[\begin{array}{cccc}{\displaystyle \frac{𝜕T_{h2,1}^{cal}}{𝜕x_{1,h}}}& {\displaystyle \frac{𝜕T_{h2,1}^{cal}}{𝜕x_{2,h}}}& \dots & {\displaystyle \frac{𝜕T_{h2,1}^{cal}}{𝜕x_{n,h}}}\\ ⋮& ⋮& ⋮& ⋮\\ {\displaystyle \frac{𝜕T_{h2,m}^{cal}}{𝜕x_{1,h}}}& {\displaystyle \frac{𝜕T_{h2,m}^{cal}}{𝜕x_{2,h}}}& \dots & {\displaystyle \frac{𝜕T_{h2,m}^{cal}}{𝜕x_{n,h}}}\\ {\displaystyle \frac{𝜕T_{c1,1}^{cal}}{𝜕x_{1,c}}}& {\displaystyle \frac{𝜕T_{c1,1}^{cal}}{𝜕x_{2,c}}}& \dots & {\displaystyle \frac{𝜕T_{c1,1}^{cal}}{𝜕x_{n,c}}}\\ ⋮& ⋮& ⋮& ⋮\\ {\displaystyle \frac{𝜕T_{c1,m}^{cal}}{𝜕x_{1,c}}}& {\displaystyle \frac{𝜕T_{c1,m}^{cal}}{𝜕x_{2,c}}}& \dots & {\displaystyle \frac{𝜕T_{c1,m}^{cal}}{𝜕x_{n,c}}}\end{array}\right]}_{2m\times n}$$

(28)

The partial derivatives in the Jacobian matrix are determined for the calculated for m hot $T_{h2,i}^{cal}$ and for m cold $T_{c1,i}^{cal}$ water outlet temperatures of the heat exchanger were calculated using central differential quotients. The symbol $c_{ii}$ in Equation (23) denotes the diagonal element $c_{ii}$ of the matrix $C_x^{\left(s\right)}$.

Considering that the amount of measurement data is large and equals 2m = 68 then the quantiles $t_{2m-n}^{\alpha /2}$ are equal to two. Having solved the non-linear least squares problem, the temperature differences of the calculated and measured outlet temperatures are known. Next, the minimum sum S_min of the squared temperature differences given by Equation (22) and the 95% confidence intervals can be calculated from Equation (23).

3. Thermal-Fluid Measurements of a Plate and Frame Heat Exchanger

Thirty-four measurement series were carried out. The following parameters were measured: hot and cold water volume flow rate at the inlet to the heat exchanger, and hot and cold water temperatures at the inlet and outlet of the heat exchanger. The station is equipped with two tanks, each with a capacity of 800 litres, whose task is to provide a constant temperature of hot and cold water at the inlet to the exchanger (Figure 2). Maintaining a constant temperature of hot and cold water at the exchanger inlet allows for testing of the plate heat exchanger under steady-state conditions.

A thermal flow study of the XB12M-1-16 plate heat exchanger (Figure 3), consisting of 16 chevron plates manufactured by Danfoss, Nordborg, Denmark, was carried out. Volume flow rates and hot and cold water temperatures at the inlet and outlet of the exchanger were measured in m = 34 measurement series. Hot water flows counter-current to cold water. During the tests, the hot water mass flow rates varied from 7.86 × 10⁻² kg/s to 8.15 × 10⁻¹ kg/s, and the cold water from 8.16 × 10⁻² kg/s to 6.61 × 10⁻¹ kg/s. The volume flow rate of hot and cold media was measured using axial turbine flow meters with a measuring range of 4 to 160 L/min, with a measurement accuracy of ±2% of the measured value. A new plate heat exchanger with clean plate surfaces was installed at the station. Thermal flow investigations of the exchanger were carried out over two days, so it was unnecessary to clean the exchanger plates during the tests. The cold and hot water flow rates and the temperature of the hot water at the inlet to the exchanger were varied. The cold water temperature at the inlet to the heat exchanger varied from 11.0 to 13.0 °C. The hot water temperature at the inlet to the heat exchanger varied over a wider range than for cold water, from 53.9 to 75.6 °C.

The temperature of cold and hot water was measured using K-type (NiCr-Ni) sheathed thermocouples with an outer diameter of 1.5 mm. The thermocouples were pre-calibrated. The uncertainty of water temperature measurement was ±0.5 K.

The hot water temperature $T_{h1}$ at the exchanger inlet varied from 53.9 °C to 75.6 °C, and the cold water temperature from 11 °C to 13 °C. The measurements were carried out under steady-state conditions. After changing the operating parameters of the exchanger, for example, the flow rate of hot or cold water, it was necessary to wait until the operating parameters of the heat exchanger became independent of time.

The measurement data from the 34 measurement series are presented in

Table 1. Measurement data for the studied plate-and-frame heat exchanger.

$\mathit{i}$	${\dot{\mathit{V}}}_{\mathit{h}}$, L/min	${\mathit{T}}_{\mathit{h}1}$, °C	${\mathit{T}}_{\mathit{h}2}$, °C	${\dot{\mathit{V}}}_{\mathit{c}}$, L/min	${\mathit{T}}_{\mathit{c}1}$, °C	${\mathit{T}}_{\mathit{c}1}$, °C
1	20.1	65.1	40.3	10.4	12.5	60.3
2	20.2	64.6	34.1	15.0	12.5	53.4
3	20.1	63.8	29.3	20.0	12.3	46.7
4	20.1	63.6	26.6	24.9	12.7	42.4
5	20.0	62.7	23.4	29.9	12.3	38.4
6	20.1	62.5	22.9	34.8	12.3	35.1
7	20.1	61.6	21.1	39.7	12.4	32.8
8	40.1	63.7	57.4	4.9	12.6	63.6
9	40.0	63.6	51.1	10.1	12.7	62.1
10	39.6	61.4	43.8	15.1	12.0	57.3
11	39.8	60.1	40.2	19.2	12.1	53.3
12	39.8	57.6	35.0	24.9	11.6	47.3
13	39.9	56.0	31.8	30.0	11.5	43.6
14	39.7	54.8	29.6	34.8	11.4	40.0
15	39.8	53.9	28.0	39.7	11.7	37.6
16	4.8	61.6	13.0	20.0	12.3	23.9
17	10.1	62.6	18.7	20.0	12.3	34.4
18	15.0	62.6	24.3	20.0	12.3	40.8
19	20.0	61.6	28.3	20.0	12.3	45.5
20	25.0	61.1	33.4	19.9	13.0	47.7
21	30.0	60.3	35.2	20.2	12.2	49.3
22	34.9	59.6	37.3	20.0	12.2	50.9
23	39.9	58.5	38.7	20.0	12.2	51.2
24	45.0	57.6	39.7	20.2	12.2	51.1
25	5.0	61.8	11.8	25.0	11.2	21.2
26	9.9	62.2	15.2	25.1	11.2	29.5
27	15.0	61.8	20.7	25.1	11.1	35.7
28	20.1	71.2	28.1	25.0	12.5	46.8
29	25.0	72.6	31.4	25.0	12.5	53.3
30	30.0	59.3	31.2	25.0	11.0	44.4
31	35.1	75.6	41.4	25.0	12.9	60.7
32	39.9	70.3	36.5	29.6	12.1	57.4
33	44.6	75.2	44.5	25.1	12.1	64.4
34	50.2	75.0	46.7	26.3	12.1	62.5

.

3.1. Correlations for Calculating Nusselt Numbers on the Hot and Cold Water Sides of a Plate Heat Exchanger

Heat transfer correlations were derived based on the measurement data presented in Table 1.

First, heat transfer correlations were determined with six unknown parameters: $x_{1h}$, $x_{2h}$, $x_{3h}$ on the hot water side and $x_{1c}$, $x_{2c}$, $x_{3c}$ on the cold water side.

3.1.1. Six Unknown Parameters (Three on the Hot Water Side and Three on the Cold Water Side)

The Nusselt numbers were approximated by the following power-type correlations:

$$N{u}_h=x_{1,h}{\mathrm{Re}}_h^{x_{2,h}}{\mathrm{Pr}}_h^{x_{3,h}}\hspace{1em}55.44\le {\mathrm{Re}}_{\mathrm{h}}\le 852.96\hspace{1em}2.95\le {\mathrm{Pr}}_{\mathrm{h}}\le 4.6$$

(29)

$$N{u}_c=x_{1,c}{\mathrm{Re}}_c^{x_{2,c}}{\mathrm{Pr}}_c^{x_{3,c}}\hspace{1em}62.38\le {\mathrm{Re}}_c\le 378.89\hspace{1em}4.46\le {\mathrm{Pr}}_c\le 7.82$$

(30)

From the minimum sum-of-squares condition (24), the following parameter values were obtained:

• Hot water

$$x_{1,,h}=0.1902\hspace{1em}{x}_{2,h}=0.6353\hspace{1em}{x}_{3,h}=0.2990$$

• Cold water

$$x_{1,c}=0.0817\hspace{1em}{x}_{2,c}=0.8732\hspace{1em}{x}_{3,c}=0.3300$$

The minimum sum of squares is $S_{\min }=24.61{\mathrm{K}}^2$, and the root mean square error (RMSE)

$$s_t=\sqrt{S_{\min }/(m-n)}=\sqrt{24.61/(68-6)}=0.67\mathrm{K}$$

The 95% confidence intervals for the determined parameters calculated according to the Gaussian principle of variance transfer [8] are as follows

$$0.1808\le x_{1,h}\le 0.1996\hspace{1em}0.5790\le x_{2,h}\le 0.6917\hspace{1em}0.2246\le x_{3,h}\le 0.3734$$

$$0.0386\le x_{1,c}\le 0.1248\hspace{1em}0.8353\le x_{2,c}\le 0.9111\hspace{1em}0.2451\le x_{3,c}\le 0.4150$$

The determined correlations for the Nusselt numbers on the hot and cold water sides have good accuracy.

Table 2. Calculated and measured cold and hot water temperatures at the outlet of the exchanger and the relative differences between the calculated and measured temperatures at the outlet of the exchanger.

$\mathit{i}$	${\mathit{T}}_{\mathit{c}1}^{\mathit{c}\mathit{a}\mathit{l}}$, °C	${\mathit{T}}_{\mathit{c}1}^{\mathit{m}\mathit{e}\mathit{a}\mathit{s}}$, °C	${\mathit{T}}_{\mathit{h}2}^{\mathit{c}\mathit{a}\mathit{l}}$, °C	${\mathit{T}}_{\mathit{h}2}^{\mathit{m}\mathit{e}\mathit{a}\mathit{s}}$, °C	${\mathit{\epsilon }}_{{\mathit{T}}_{\mathit{c}}}$, %	${\mathit{\epsilon }}_{{\mathit{T}}_{\mathit{h}}}$, %
1	59.16	60.30	40.51	40.30	1.89	−0.52
2	53.19	53.40	33.83	34.10	0.40	0.80
3	46.85	46.70	28.79	29.30	−0.32	1.75
4	42.43	42.40	26.30	26.60	−0.07	1.92
5	38.01	38.40	23.56	23.40	1.02	−0.69
6	35.17	35.10	22.17	22.90	−0.20	2.16
7	32.54	32.80	19.87	20.45	0.78	0.02
8	63.32	63.60	57.40	57.4	0.44	0.00
9	61.67	62.10	51.02	51.10	0.70	0.15
10	56.83	57.30	44.02	43.80	0.83	−0.51
11	53.16	53.30	39.97	40.20	0.26	0.57
12	47.39	47.30	34.87	35.00	−0.18	0.38
13	43.30	43.60	31.74	31.80	0.69	0.19
14	39.97	40.00	29.39	29.60	0.07	0.70
15	37.45	37.60	27.86	28.00	0.41	0.50
16	23.60	23.90	13.65	13.00	1.27	−4.98
17	34.16	34.40	18.52	18.70	0.71	0.96
18	41.00	40.80	23.64	24.30	−0.50	2.16
19	45.21	45.50	27.90	28.30	0.63	0.65
20	48.50	47.70	32.37	33.40	−1.67	2.16
21	49.72	49.30	34.62	35.20	−0.86	2.16
22	50.97	50.90	37.03	37.30	−0.14	0.73
23	51.26	51.20	38.62	38.70	−0.12	0.21
24	51.30	51.10	39.79	39.70	−0.39	−0.22
25	20.90	21.20	17.00	18.20	1.42	−4.87
26	29.09	29.50	16.00	15.20	1.40	−5.26
27	35.46	35.70	20.30	20.70	0.68	1.93
28	46.94	46.80	27.39	27.90	−0.31	1.92
29	52.21	53.30	31.97	31.40	2.04	−1.81
30	44.97	44.40	30.53	31.20	−1.28	2.16
31	60.87	60.70	40.61	41.40	−0.27	1.92
32	54.90	57.40	37.86	36.50	4.35	−3.73
33	63.87	64.40	45.36	44.50	0.82	−1.94
34	64.33	62.50	46.99	46.70	−2.92	−0.62

summarises the calculated and measured hot and cold water temperatures at the heat exchanger outlet. The relative differences between the measured and calculated hot and cold water temperatures are also given.

The relative differences ${\epsilon }_{T_c}$ and ${\epsilon }_{T_h}$ between the measured and calculated temperatures shown in Table 2 were calculated from the following equations:

$${\epsilon }_{T_c}=100{\displaystyle \frac{T_{c1,i}^{meas}-T_{c1,i}^{cal}}{T_{c1,i}^{meas}}}\hspace{1em}{\epsilon }_{T_h}=100{\displaystyle \frac{T_{h2,i}^{meas}-T_{h2,i}^{cal}}{T_{h2,i}^{meas}}}$$

(31)

From the analysis of the results shown in Table 2, the agreement between the calculated and measured temperatures at the heat exchanger outlet is very good. The maximum absolute value of the difference ${\epsilon }_{T_c}$ is 2.92% for cold water, and the difference ${\epsilon }_{T_h}$ for hot water is 5.26%. The Root Mean Square Error (RMSE) for the cold water is 2.09% and for hot water, 1.24%.

Table 3. Measured and calculated heat flow rates transferred from hot to cold water, and the relative differences between the average measured and calculated heat flow rates.

$\mathit{i}$	${\dot{\mathit{Q}}}_{\mathit{c}}^{\mathit{m}\mathit{e}\mathit{a}\mathit{s}}$, W	${\dot{\mathit{Q}}}_{\mathit{h}}^{\mathit{m}\mathit{e}\mathit{a}\mathit{s}}$, W	${\dot{\mathit{Q}}}_{\mathit{m}}^{\mathit{m}\mathit{e}\mathit{a}\mathit{s}}$, W	${\dot{\mathit{Q}}}_{\mathit{m}}^{\mathit{c}\mathit{a}\mathit{l}}$, W	${\mathit{\epsilon }}_{\mathit{c}}^{\mathit{m}\mathit{e}\mathit{a}\mathit{s}}$, %	${\mathit{\epsilon }}_{\mathit{m},\mathit{i}}^{\dot{\mathit{Q}}}$, %
1	34,601.10	34,064.14	34,332.62	33,778.23	0.78	1.61
2	42,704.33	42,103.43	42,403.88	42,481.25	0.71	−0.18
3	47,898.54	47,402.77	47,650.66	48,108.81	0.52	−0.96
4	51,489.25	50,839.94	51,164.60	51,542.44	0.64	−0.74
5	54,346.77	53,753.79	54,050.28	53,534.17	0.55	0.95
6	55,264.24	54,440.22	54,852.23	55,437.32	0.75	−1.07
7	56,415.15	55,702.34	56,058.75	55,706.57	0.64	0.63
8	17,393.47	17,290.14	17,341.80	17,298.04	0.30	0.25
9	34,726.56	34,210.44	34,468.50	34,423.41	0.75	0.13
10	47,615.24	47,722.65	47,668.95	47,118.21	−0.11	1.16
11	55,066.15	54,258.92	54,662.54	54,880.80	0.74	−0.40
12	61,893.69	61,685.74	61,789.71	62,044.95	0.17	−0.41
13	67,059.88	66,264.81	66,662.34	66,428.02	0.60	0.35
14	69,318.64	68,693.64	69,006.14	69,253.81	0.45	−0.36
15	71,617.23	70,808.16	71,212.7	71,187.84	0.57	0.03
16	16,170.89	15,961.85	16,066.37	15,749.70	0.65	1.97
17	30,787.06	30,322.9	30,554.98	30,447.45	0.76	0.35
18	39,691.14	39,292.11	39,491.63	39,974.13	0.51	−1.22
19	46,229.25	45,576.95	45,903.10	45,829.58	0.71	0.16
20	48,067.50	47,408.06	47,737.78	49,169.80	0.69	−3.00
21	52,172.09	51,572.84	51,872.46	52,769.28	0.58	−1.73
22	53,881.41	53,325.05	53,603.23	53,978.85	0.52	−0.70
23	54,298.77	54,161.11	54,229.94	54,382.84	0.13	−0.28
24	54,701.24	55,247.63	54,974.44	54,982.70	−0.50	−0.02
25	17,433.93	17,104.11	17,269.02	16,907.62	0.96	2.09
26	32,010.68	31,827.39	31,919.04	31,286.44	0.29	1.98
27	43,015.63	42,180.19	42,597.91	42,590.11	0.98	0.02
28	59,696.88	58,979.68	59,338.28	59,946.27	0.60	−1.02
29	70,999.97	70,076.44	70,538.20	69,111.17	0.66	2.02
30	58,149.81	57,759.76	57,954.79	59,142.13	0.34	−2.05
31	83,170.48	81,567.99	82,369.23	83,458.74	0.97	−1.32
32	93,336.96	91,890.50	92,613.73	88,184.59	0.78	4.78
33	91,375.08	93,074.73	92,224.91	90,455.05	−0.92	1.92
34	92,265.31	96,594.41	94,429.86	95,609.80	−2.29	−1.25

compares the heat flow rate calculated from the hot water ${\dot{Q}}_h^{meas}$ and cold water sides ${\dot{Q}}_c^{meas}$, the arithmetic mean of both fluxes ${\dot{Q}}_m^{meas}$, and the relative difference ${\epsilon }_{m,i}^{\dot{Q}}$ between the measured mean flux and the calculated flux.

The values of the heat flow rates transferred from hot to cold water for the m = 34 measurement series shown in Table 3 were calculated from the following expressions

$$\begin{gathered} {\dot{Q}}_{h,i}^{meas}={\dot{m}}_{h,i}{c}_{h,i}(T_{h1,i}^{meas}-T_{h2,i}^{meas}) \\ {\dot{Q}}_{c,i}^{meas}={\dot{m}}_{c,i}{c}_{c,i}(T_{c1,i}^{meas}-T_{c2,i}^{meas}) \\ {\dot{Q}}_{m,i}^{meas}=({\dot{Q}}_{c,i}^{meas}+{\dot{Q}}_{h,i}^{meas})/2\hspace{1em}i=1,\dots ,m \end{gathered}$$

(32)

The relative differences between the heat flow rates calculated from the hot water ${\dot{Q}}_{h,i}^{meas}$ or cold water side ${\dot{Q}}_{c,i}^{meas}$ and the average heat flow rates ${\dot{Q}}_{m,i}^{meas}$ were calculated from the following formulas

$$\begin{gathered} {\epsilon }_{c,i}^{\dot{Q}}=100({\dot{Q}}_{c,i}^{meas}-{\dot{Q}}_{m,i}^{meas})/{\dot{Q}}_{m,i}^{meas} \\ {\epsilon }_{h,i}^{\dot{Q}}=100({\dot{Q}}_{h,i}^{meas}-{\dot{Q}}_{m,i}^{meas})/{\dot{Q}}_{m,i}^{meas}\hspace{1em}i=1,\dots ,m \end{gathered}$$

(33)

Between the differences, there is a relationship ${\epsilon }_{c,i}^{\dot{Q}}=-{\epsilon }_{h,i}^{\dot{Q}}$.

The calculated heat flow rate from hot and cold water are equal to each other

$${\dot{Q}}_i^{cal}={\dot{m}}_{h,i}{c}_{h,i}(T_{h1,i}^{meas}-T_{h2,i}^{cal})={\dot{m}}_{c,i}{c}_{c,i}(T_{c1,i}^{meas}-T_{c2,i}^{cal})$$

(34)

The differences between the average heat flow rate determined experimentally ${\dot{Q}}_{m,i}^{meas}$ and the calculated heat flow rates ${\dot{Q}}_i^{cal}$ were calculated from the equation

$${\epsilon }_{m,i}^{\dot{Q}}=100({\dot{Q}}_{m,i}^{meas}-{\dot{Q}}_i^{cal})/{\dot{Q}}_{m,i}^{meas}\hspace{1em}i=1,\dots ,m$$

(35)

In Equations (32)–(35) the following designations are used: ${\dot{Q}}_{c,i}^{meas}$, ${\dot{Q}}_{h,i}^{meas}$, ${\dot{Q}}_{m,i}^{meas}$—cold, hot and average heat flow rates, respectively, calculated for the i-th measurement series; ${\dot{Q}}_i^{cal}$—heat flow rate for the i-th measuring series calculated from the cold or hot water side; ${\epsilon }_{c,i}^{\dot{Q}}$, ${\epsilon }_{h,i}^{\dot{Q}}$—the difference between the experimentally determined heat flow rate; ${\dot{Q}}_{c,i}^{meas}$, ${\dot{Q}}_{h,i}^{meas}$ and the average value ${\dot{Q}}_{m,i}^{meas}$ for the i-th measurement series; ${\epsilon }_{m,i}^{\dot{Q}}$—the difference between the average heat flow rates measured ${\dot{Q}}_{m,i}^{meas}$ and calculated ${\dot{Q}}_i^{cal}$.

An analysis of the values presented in Table 3 shows that the heat flow rate calculated from the temperature measurements on the cold and hot water sides show very good agreement. The differences between the average flow rate and the flow rate on the cold or hot water side for most points do not exceed $\pm 1\%$. Only for measurement number 34 is the difference 2.29%. The calculated heat flow rate from hot to cold water also differs little from the average heat flow rate determined experimentally. The maximum difference is 4.78%.

Based on thirty-four measurement series in which the above-mentioned variables were measured, i.e., hot and cold water mass flow rates and inlet and outlet temperatures from the heat exchanger, unknown parameters were determined, i.e., coefficients and power exponents in correlations for the Nusselt number for hot and cold water. The values of parameters determined in correlations for Nusselt numbers clearly depend on the values of measured quantities, which means that the inverse problem is well-posed. Thus, the sensitivity coefficients determined for the Nusselt number in relation to the measured values should not be too small. A change in a given measured value should cause changes in the sought parameters appearing in the correlation for the Nusselt number and a change in the value of the Nusselt number. The sensitivity coefficients for the Nusselt number for the seventh measurement series (Table 1) are calculated below.

$$\begin{array}{l}{\displaystyle \frac{𝜕Nu}{𝜕{\dot{V}}_h}}={\displaystyle \frac{Nu({\dot{V}}_h+∆{\dot{V}}_h)-Nu({\dot{V}}_h-∆{\dot{V}}_h)}{2∆{\dot{V}}_h}}={\displaystyle \frac{Nu(20.1+2.01)-Nu(20.1-2.01)}{2\cdot 2.01}}=\\ ={\displaystyle \frac{9.252048-8.818098}{2\cdot 2.01}}=0.107948{\displaystyle \frac{\min }{\mathrm{l}}}\end{array}$$

$$\begin{array}{l}{\displaystyle \frac{𝜕Nu}{𝜕T_{h1}}}={\displaystyle \frac{Nu(T_{h1}+∆T_{h1})-Nu(T_{h1}-∆T_{h1})}{2∆T_{h1}}}={\displaystyle \frac{Nu(61.1+1.0)-Nu(61.1-1.0)}{2\cdot 1.0}}=\\ ={\displaystyle \frac{9.619821-9.633703}{2\cdot 1.0}}=-6.941\cdot {10}^{-3}{\displaystyle \frac{1}{\mathrm{K}}}\end{array}$$

$$\begin{array}{l}{\displaystyle \frac{𝜕Nu}{𝜕T_{h2}}}={\displaystyle \frac{Nu(T_{h2}+∆T_{h2})-Nu(T_{h2}-∆T_{h2})}{2∆T_{h2}}}={\displaystyle \frac{Nu(21.1+1.0)-Nu(21.1-1.0)}{2\cdot 1.0}}\\ ={\displaystyle \frac{9.262340-10.129780}{2\cdot 1.0}}=-0.43372{\displaystyle \frac{1}{\mathrm{K}}}\end{array}$$

$$\begin{array}{l}{\displaystyle \frac{𝜕Nu}{𝜕{\dot{V}}_c}}={\displaystyle \frac{Nu({\dot{V}}_c+∆{\dot{V}}_c)-Nu({\dot{V}}_c-∆{\dot{V}}_c)}{2∆{\dot{V}}_c}}={\displaystyle \frac{Nu(39.7+2.0)-Nu(39.7-2.0)}{2\cdot 2.0}}\\ ={\displaystyle \frac{27.938780-25.471890}{2\cdot 2.0}}=0.6167225{\displaystyle \frac{\min }{\mathrm{l}}}\end{array}$$

$$\begin{array}{l}{\displaystyle \frac{𝜕Nu}{𝜕T_{c1}}}={\displaystyle \frac{Nu(T_{c1}+∆T_{c1})-Nu(T_{c1}-∆T_{c1})}{2∆T_{c1}}}={\displaystyle \frac{Nu(12.4+1.0)-Nu(12.4-1.0)}{2\cdot 1.0}}\\ ={\displaystyle \frac{25.406170-27.587310}{2\cdot 1.0}}=-1.09057{\displaystyle \frac{1}{\mathrm{K}}}\end{array}$$

$$\begin{array}{l}{\displaystyle \frac{𝜕Nu}{𝜕T_{c2}}}={\displaystyle \frac{Nu(T_{c2}+∆T_{c2})-Nu(T_{c2}-∆T_{c2})}{2∆T_{c2}}}={\displaystyle \frac{Nu(32.8+1.0)-Nu(32.8-1.0)}{2\cdot 1.0}}\\ ={\displaystyle \frac{25.896690-27.438820}{2\cdot 1.0}}=-0.771065{\displaystyle \frac{1}{\mathrm{K}}}\end{array}$$

The determined sensitivity coefficient values are high, which is beneficial for the accuracy of the determined correlations for Nusselt numbers for cold and hot water.

Next, the determination of correlations for Nusselt numbers on the hot and cold water sides for two unknown parameters on each side of the plate will be presented.

3.1.2. Four Unknown Parameters (Two on the Hot Water Side and Two on the Cold Water Side)

Heat transfer correlations were also determined, assuming two unknown parameters in the correlation for hot water ($n_h$ = 2) and two unknown parameters on the cold water side ($n_c$ = 2).

The Nusselt numbers were approximated by the following power-type correlations

$$N{u}_h=x_{1,h}{\mathrm{Re}}_h^{x_{2,h}}{\mathrm{Pr}}_h^{0.29}\hspace{1em}55.44\le {\mathrm{Re}}_h\le 852.96\hspace{1em}2.95\le {\mathrm{Pr}}_h\le 4.6$$

(36)

$$N{u}_c=x_{1,c}{\mathrm{Re}}_c^{x_{2,c}}{\mathrm{Pr}}_c^{0.41}\hspace{1em}62.38\le {\mathrm{Re}}_c\le 378.89\hspace{1em}4.46\le {\mathrm{Pr}}_c\le 7.82$$

(37)

The minimum of the sum of squares defined by Equation (24)

• Hot water

$$x_{1,h}=0.2016\hspace{1em}{x}_{2,h}=0.6202\hspace{1em}$$

• Cold water

$$x_{1,c}=0.0622\hspace{1em}{x}_{2,c}=0.9098$$

The minimum sum of squares is $S_{\min }=25.03{\mathrm{K}}^2$, and the mean square error is

$$s_t=\sqrt{S_{\min }/(m-n)}=\sqrt{25.03/(68-4)}=0.63\mathrm{K}$$

The 95% confidence intervals for the determined parameters calculated according to the variance transfer principle [8] proposed by Gauss are as follows:

$$0.1266\le x_{1,h}\le 0.2765\hspace{1em}0.5213\le x_{2,h}\le 0.7191$$

$$0.0107\le x_{1,c}\le 0.1137\hspace{1em}0.8782\le x_{2,c}\le 0.9413\hspace{1em}$$

Heat transfer correlations were then determined assuming 3 unknown parameters $x_1,x_2,x_3$, the same on the hot and cold water sides.

3.1.3. Three Unknown Parameters (n = 3)

$$N{u}_h=x_1{\mathrm{Re}}_h^{x_2}{\mathrm{Pr}}_h^{x_3}\hspace{1em}55.44\le {\mathrm{Re}}_h\le 852.96\hspace{1em}2.95\le {\mathrm{Pr}}_h\le 4.6$$

(38)

$$N{u}_c=x_1{\mathrm{Re}}_c^{x_2}{\mathrm{Pr}}_c^{x_3}\hspace{1em}62.38\le {\mathrm{Re}}_c\le 378.89\hspace{1em}4.46\le {\mathrm{Pr}}_c\le 7.82$$

(39)

From the minimum sum-of-squares condition (24), the following parameter values were obtained

$$x_1=0.115374\hspace{1em}{x}_2=0.757967\hspace{1em}{x}_3=0.3334$$

The minimum sum of squares is $S_{\min }=26.81{\mathrm{K}}^2$, and the mean square error is

$$s_t=\sqrt{S_{\min }/(m-n)}=\sqrt{26.81/(68-3)}=0.64\mathrm{K}$$

The 95% confidence intervals for the determined parameters, calculated according to the variance transfer principle [8] proposed by Gauss are as follows:

$$0.1127\le x_1\le 0.1180\hspace{1em}0.7532\le x_2\le 0.7628\hspace{1em}0.3220\le x_3\le 0.3448\hspace{1em}$$

It can be seen that the 95% confidence intervals for all three parameters are narrow. This demonstrates the good accuracy of the mathematical model of the heat exchanger as well as being a good indication of the quality of the measurement data.

4. Discussion

This paper presents a method for simultaneously determining the correlation between the hot and cold water sides of the Nusselt numbers as a function of the Reynolds number and Prandtl number. Two approaches were used to determine the correlations on the hot and cold water sides. Firstly, separate correlations per Nusselt number for hot and cold water were proposed. Three unknown parameters were determined for the correlation on the hot water side, and three parameters on the cold water side.

The determined correlations differ significantly in the values of the power exponents at the Reynolds numbers if we assume two different correlations for hot and cold water. In the case of the six parameters determined, the exponents at the Reynolds number differ significantly and are $x_{2,h}=0.6353$ and $x_{2,c}=0.8732$. Slightly smaller differences are found in the exponents at Prandtl numbers $x_{3,h}=0.2990$ and $x_{3,c}=0.3300$.

The number of unknown parameters in correlations for the Nusselt number was reduced to four to check whether correct results could be obtained with fewer than six parameters. As in the case of the six determined parameters, the exponents for the Reynolds numbers differ significantly for the four determined parameters when the correlations on the hot and cold water sides differ. The values of the exponents for the Reynolds numbers are $x_{2,h}=0.6202$ and $x_{2,c}=0.9098$ for hot and cold water, respectively.

The exponents for the Prandtl numbers were selected by a trial and error method, so the sum of the squares of the calculated and measured temperature differences was the smallest. The exponent for the Prandtl number for hot water, equal to 0.29, and the exponent for cold water, equal to 0.41, are very close to the corresponding values of 0.3 and 0.4 adopted by Dittus and Boelter [2,3,4,5,6,7,8].

The third approach proposed a common correlation for cold and hot water with three unknown parameters. The following parameter values were obtained $x_1=0.115374$, $x_2=0.757967$ and $x_3=0.3334$. The value of the parameter $x_3$ is close to 1/3 of the values found in correlations for the Nusselt number based on Chilton-Colburn analogies [2,3,4,5,6,7,8].

All proposed correlations are valid in the following ranges of Reynolds and Prandtl number changes, namely $55.44\le {\mathrm{Re}}_h\le 852.96$, $2.95\le {\mathrm{Pr}}_h\le 4.6$ for hot water and $62.38\le {\mathrm{Re}}_c\le 378.89$, $4.46\le {\mathrm{Pr}}_c\le 7.82$ for cold water.

Table 4. Selected Nusselt number values for hot water determined using correlations (29), (36) and (38) for Prandtl number Pr = 3.0; correlation (29) was determined assuming six unknown parameters, correlation (36) assuming four unknown parameters, and correlation (38) assuming three parameters identical for hot and cold water; symbol $N{u}_m$ denotes the average Nusselt number from the values obtained using Equations (29), (36) and (38).

	Equation (29)	Equation (36)	Equation (38)
${\mathbf{Re}}_{\mathbf{h}}$	$\mathbf{Nu}=0.1902{\mathbf{Re}}^{0.6353}{\mathbf{Pr}}^{0.2990}$	$\mathbf{Nu}=0.2016{\mathbf{Re}}^{0.6202}{\mathbf{Pr}}^{0.2900}$	$\mathbf{Nu}=0.1153{\mathbf{Re}}^{0.7579}{\mathbf{Pr}}^{0.3334}$	$\mathit{N}{\mathit{u}}_{\mathit{m}}$
50	3.1712	3.1373	3.2281	3.1789
150	6.3730	6.2011	7.4231	6.6657
250	8.8162	8.5125	10.9330	9.4206
350	10.9173	10.4879	14.1091	11.8381
450	12.8072	12.2568	17.0698	14.0446
550	14.5486	13.8813	19.8740	16.1013
650	16.1776	15.3966	22.5567	18.0436
750	17.7172	16.8255	25.1410	19.8946
850	19.1836	18.1837	27.6429	21.6700

,

Table 5. Selected Nusselt number values for hot water determined using correlations (29), (36) and (38) for Prandtl number Pr = 3.9; correlation (29) was determined assuming six unknown parameters, correlation (36) assuming four unknown parameters, and correlation (38) assuming three parameters identical for hot and cold water; symbol $N{u}_m$ denotes the average Nusselt number from the values obtained using Equations (29), (36) and (38).

	Equation (29)	Equation (36)	Equation (38)
${\mathbf{Re}}_{\mathbf{h}}$	$\mathbf{Nu}=0.1902{\mathbf{Re}}^{0.6353}{\mathbf{Pr}}^{0.2990}$	$\mathbf{Nu}=0.2016{\mathbf{Re}}^{0.6202}{\mathbf{Pr}}^{0.2900}$	$\mathbf{Nu}=0.1153{\mathbf{Re}}^{0.7579}{\mathbf{Pr}}^{0.3334}$	${\mathbf{Nu}}_{\mathbf{m}}$
50	3.4300	3.3853	3.5232	3.4462
150	6.8930	6.6913	8.1017	7.2287
250	9.5356	9.1855	11.9324	10.2178
350	11.8082	11.3170	15.3989	12.8414
450	13.8524	13.2258	18.6301	15.2361
550	15.7359	14.9786	21.6907	17.4684
650	17.4978	16.6138	24.6187	19.5767
750	19.1631	18.1557	27.4391	21.5860
850	20.7491	19.6212	30.1698	23.5133

and

Table 6. Selected Nusselt number values for hot water determined using correlations (29), (36) and (38) for Prandtl number Pr = 4.8; correlation (29) was determined assuming six unknown parameters, correlation (36) assuming four unknown parameters, and correlation (38) assuming three parameters identical for hot and cold water; symbol $N{u}_m$ denotes the average Nusselt number from the values obtained using Equations (29), (36) and (38).

	Equation (29)	Equation (36)	Equation (38)
${\mathbf{Re}}_{\mathbf{h}}$	$\mathbf{Nu}=0.1902{\mathbf{Re}}^{0.6353}{\mathbf{Pr}}^{0.2990}$	$\mathbf{Nu}=0.2016{\mathbf{Re}}^{0.6202}{\mathbf{Pr}}^{0.2900}$	$\mathbf{Nu}=0.1153{\mathbf{Re}}^{0.7579}{\mathbf{Pr}}^{0.3334}$	${\mathbf{Nu}}_{\mathbf{m}}$
50	3.6497	3.5954	3.7757	3.6736
150	7.3345	7.1066	8.6824	7.7078
250	10.1464	9.7556	12.7877	10.8966
350	12.5646	12.0194	16.5026	13.6955
450	14.7397	14.0467	19.9656	16.2506
550	16.7438	15.9083	23.2455	18.6325
650	18.6185	17.6449	26.3833	20.8823
750	20.3905	19.2825	29.4060	23.0263
850	22.0781	20.8390	32.3323	25.0831

show the Nusselt numbers as a function of the Reynolds number for hot water for three selected Prandtl numbers, Pr = 3.0, Pr = 3.9 and Pr = 4.5. The Nusselt numbers were calculated using Equations (29), (36) and (38) derived for six, four and three unknown parameters, respectively, in correlations for the Nusselt number.

A comparison of the Nusselt numbers presented in Table 4, Table 5 and Table 6 shows that correlations (29) and (36) give similar results, while correlation (38), which contains the same three coefficients on the hot and cold water sides, significantly exceeds the Nusselt numbers obtained assuming six and four unknown parameters.

Similar comparisons of Nusselt number values to those presented in Table 4, Table 5 and Table 6 for hot water are illustrated in

Table 7. Selected Nusselt number values for cold water determined using correlations (30), (37) and (39) for Prandtl number Pr = 4.5; correlation (30) was determined assuming six unknown parameters, correlation (37) assuming four unknown parameters, and correlation (39) assuming three parameters identical for hot and cold water; symbol $N{u}_m$ denotes the average Nusselt number from the values obtained using Equations (30), (37) and (39).

	Equation (30)	Equation (37)	Equation (39)
${\mathbf{Re}}_{\mathbf{c}}$	$\mathbf{Nu}=0{.0817\mathbf{Re}}^{0.8732}{\mathbf{Pr}}^{0.3300}$	$\mathbf{Nu}=0{.0622\mathbf{Re}}^{0.9098}{\mathbf{Pr}}^{0.4100}$	$\mathbf{Nu}=0{.1153\mathbf{Re}}^{0.7579}{\mathbf{Pr}}^{0.3334}$	${\mathbf{Nu}}_{\mathbf{m}}$
60	4.7914	4.7794	4.2430	4.6046
100	7.4848	7.6069	6.2492	7.1136
140	10.0411	10.3313	8.0646	9.4790
180	12.5051	12.9854	9.7569	11.7491
220	14.9000	15.5863	11.3597	13.9487
260	17.2400	18.1447	12.8932	16.0926
300	19.5346	20.6677	14.3703	18.1909
340	21.7906	23.1605	15.8003	20.2505
380	24.0132	25.6268	17.1902	22.2767

,

Table 8. Selected Nusselt number values for cold water determined using correlations (30), (37) and (39) for Prandtl number Pr = 6.0; correlation (30) was determined assuming six unknown parameters, correlation (37) assuming four unknown parameters, and correlation (39) assuming three parameters identical for hot and cold water; symbol $N{u}_m$ denotes the average Nusselt number from the values obtained using Equations (30), (37) and (39).

	Equation (30)	Equation (37)	Equation (39)
${\mathbf{Re}}_{\mathbf{c}}$	$\mathbf{Nu}=0.0817{\mathbf{Re}}^{0.8732}{\mathbf{Pr}}^{0.3300}$	$\mathbf{Nu}=0.0622{\mathbf{Re}}^{0.9098}{\mathbf{Pr}}^{0.4100}$	$\mathbf{Nu}=0.1153{\mathbf{Re}}^{0.7579}{\mathbf{Pr}}^{0.3334}$	${\mathbf{Nu}}_{\mathbf{m}}$
60	5.2686	5.3777	4.6701	5.1054
100	8.2302	8.5592	6.8782	7.8892
140	11.0411	11.6246	8.8764	10.5140
180	13.7504	14.6110	10.7390	13.0335
220	16.3838	17.5375	12.5032	15.4749
260	18.9569	20.4162	14.1910	17.8547
300	21.4800	23.2551	15.8168	20.1840
340	23.9607	26.0599	17.3909	22.4705
380	26.4046	28.8350	18.9206	24.7200

and

Table 9. Selected Nusselt number values for cold water determined using correlations (30), (37) and (39) for Prandtl number Pr = 7.5; correlation (30) was determined assuming six unknown parameters, correlation (37) assuming four unknown parameters, and correlation (39) assuming three parameters identical for hot and cold water; symbol $N{u}_m$ denotes the average Nusselt number from the values obtained using Equations (30), (37) and (39).

	Equation (30)	Equation (37)	Equation (39)
${\mathbf{Re}}_{\mathbf{c}}$	$\mathbf{Nu}=0.0817{\mathbf{Re}}^{0.8732}{\mathbf{Pr}}^{0.3300}$	$\mathbf{Nu}=0.0622{\mathbf{Re}}^{0.9098}{\mathbf{Pr}}^{0.4100}$	$\mathbf{Nu}=0.1153{\mathbf{Re}}^{0.7579}{\mathbf{Pr}}^{0.3334}$	${\mathbf{Nu}}_{\mathbf{m}}$
60	5.6712	5.8929	5.0308	5.5316
100	8.8592	9.3792	7.4095	8.5493
140	11.8848	12.7383	9.5620	11.3950
180	14.8012	16.0108	11.5685	14.1268
220	17.6358	19.2177	13.4689	16.7742
260	20.4055	22.3722	15.2870	19.3549
300	23.1214	25.4830	17.0384	21.8810
340	25.7917	28.5565	18.7340	24.3607
380	28.4223	31.5975	20.3819	26.8006

for cold water.

In the case of cold water, correlation (39) with the same three coefficients for cold and hot water gives significantly lower Nusselt number values. It can be expected that the overall HTCs for the same Reynolds numbers will be similar regardless of the number of unknown parameters in the Nusselt number correlations. However, it should be emphasised that differences in Nusselt number correlations on the hot and cold water sides lead to different plate temperatures. In the case of high-temperature heat exchangers, the same correlation for cold and hot water may lead to an inaccurate estimate of the wall temperature and incorrect selection of plate material.

The average Nusselt number ${\mathrm{Nu}}_m$ in Figure 4a, Figure 5a, Figure 6a, Figure 7a, Figure 8a and Figure 9a was calculated using the equation

$${\mathrm{Nu}}_m\left(\mathrm{Re}\right)={\displaystyle \frac{{\mathrm{Nu}}^{6\mathrm{p}}\left(\mathrm{Re}\right)+{\mathrm{Nu}}^{4\mathrm{p}}\left(\mathrm{Re}\right)+{\mathrm{Nu}}^{3\mathrm{p}}\left(\mathrm{Re}\right)}{3}}$$

(40)

where the symbols ${\mathrm{Nu}}^{6\mathrm{p}}$, ${\mathrm{Nu}}^{4\mathrm{p}}$ and ${\mathrm{Nu}}^{3\mathrm{p}}$ denote the Reynolds numbers calculated using correlations containing six, four and three parameters, respectively.

The relative differences shown in Figure 4b, Figure 5b, Figure 6b, Figure 7b, Figure 8b and Figure 9b were calculated using the following equation

$$e\left(\mathrm{Re}\right)={\displaystyle \frac{\mathrm{Nu}\left(\mathrm{Re}\right)-{\mathrm{Nu}}_m\left(\mathrm{Re}\right)}{{\mathrm{Nu}}_m\left(\mathrm{Re}\right)}}\cdot 100\%$$

(41)

The average error value $e_m$ was calculated using the equation

$$e_m={\displaystyle \frac{{\displaystyle \sum _{i=1}^{n_f}\mathrm{Nu}\left({\mathrm{Re}}_i\right)}}{n_f}}$$

(42)

where $n_f$ = 86 is the number of points for hot water and $n_f$ = 33 is the number of points for cold water. The step for calculating the Reynolds number ${\mathrm{R}\mathrm{e}}_i$ was equal 10.

The average relative differences e_m between the values Nu (Re) and Nu_m, calculated using Equation (42), are as follows for hot water:

-

Pr = 3.0

−8.27%, −11.98%, and 20.25% in accordance with Equation (29) (six unknown parameters, separate correlations for hot and cold water), Equation (36) (4 unknown parameters, separate correlations for hot and cold water), and Equation (38) (3 unknown parameters, one common correlation for hot and cold water),

-

Pr = 3.9

−8.54%, −12.45%, and 20.98% in accordance with Equation (29) (six unknown parameters, separate correlations for hot and cold water), Equation (36) (4 unknown parameters, separate correlations for hot and cold water), and Equation (38) (3 unknown parameters, one common correlation for hot and cold water),

-

Pr = 4.8

−8.75%, −12.82%, and 21.56% in accordance with Equation (29) (six unknown parameters, separate correlations for hot and cold water), Equation (36) (4 unknown parameters, separate correlations for hot and cold water), and Equation (38) (3 unknown parameters, one common correlation for hot and cold water).

The average relative differences e_m for cold water are as follows:

-

Pr = 4.5

6.55%, 11.03%, and −17.58% in accordance with Equation (30) (six unknown parameters, separate correlations for hot and cold water), Equation (37) (4 unknown parameters, separate correlations for hot and cold water), and Equation (39) (3 unknown parameters, one common correlation for hot and cold water),

-

Pr = 6.0

5.1%, 12.61%, and −18.22% in accordance with Equation (30) (six unknown parameters, separate correlations for hot and cold water), Equation (37) (4 unknown parameters, separate correlations for hot and cold water), and Equation (39) (3 unknown parameters, one common correlation for hot and cold water),

-

Pr = 7.5

4.88%, 13.84%, and −18.72% in accordance with Equation (30) (six unknown parameters, separate correlations for hot and cold water), Equation (37) (4 unknown parameters, separate correlations for hot and cold water), and Equation (39) (3 unknown parameters, one common correlation for hot and cold water).

The analyses of the results presented in Table 4, Table 5, Table 6, Table 7 and Table 8 and Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 show that the HTCs on the hot and cold water sides should be calculated using different correlations. Separate correlations for hot and cold water better approximate the experimental data, which are characterised by different ranges of Reynolds and Prandtl number changes for cold and hot water.

It is recommended to determine separate correlations for the Nusselt number on the hot and cold fluid sides in order to better account for the range of changes in the Reynolds and Prandtl numbers on each side. The separate correlations for hot and cold water can be recommended for practical applications because of their better accuracy. They can also be used to calculate the heat transfer resistance from hot to cold water in the heat exchanger under test.

5. Conclusions

Knowledge of HTCs on the hot and cold sides is important not only in design and performance calculations of the heat exchanger, but is also essential for correctly determining the temperature of the plate and, in general, the temperature of the wall separating the hot and cold fluids.

The presented method of simultaneous determination of correlations for the Nusselt number on the hot and cold water sides is characterised by high accuracy. It is competitive with the Wilson method, which is more labour-intensive due to the need for iterative determination of correlations for Nusselt numbers on both sides of the partitions, in this case, plates. The advantage of the proposed method is the possibility of determining the uncertainty of the estimated parameters by determining 95% confidence intervals.

In the paper, three different heat transfer correlations for calculating the Nusselt number in a chevron plate heat exchanger are proposed. In the first correlation, six parameters were determined based on experimental data, four parameters were determined in the second correlation, and three were determined in the third correlation. The final correlation is identical for hot and cold water, while in the case of six and four parameters, separate correlations are used for hot and cold water. The correlations with six or four give similar Nusselt numbers for the same Reynolds and Prandtl numbers.

The accurate correlation with three or two parameters determined on each side of the plate results from different ranges of Reynolds and Prandtl number changes on the hot and cold water sides. The ranges of Reynolds and Prandtl numbers change on the hot water side are as follows: $55.44\le {\mathrm{Re}}_h\le 852.96$ and $2.95\le {\mathrm{Pr}}_h\le 4.6$, while on the cold water side, they are different: $62.38\le {\mathrm{Re}}_c\le 378.89$ and $4.46\le {\mathrm{Pr}}_c\le 7.82$.

However, in the third correlation, the three common parameters for hot and cold water are used. The analyses carried out in this study show that a single correlation for hot and cold water with three specified parameters does not give good results, as the experimental data for hot and cold water cover different ranges of Reynolds and Prandtl numbers. Since the determined parameters depend to a large extent on the range of changes in the Reynolds and Prandtl numbers, it is difficult to obtain satisfactory accuracy for hot and cold water using a single correlation for both.

It is recommended to determine separate correlations for the Nusselt number on the hot and cold fluid sides in order to better account for the range of changes in the Reynolds and Prandtl numbers on each side. The separate correlations for hot and cold water can be recommended for practical applications because of their accuracy. They can also be used to calculate the heat transfer resistance from hot to cold water in the heat exchanger under test.

6. Future Work

The general method presented in the article for simultaneously determining heat transfer correlations on the hot and cold sides is universal and can be used to determine correlations for Nusselt numbers in various types of heat exchangers. In the future, heat transfer correlations will be determined in coil, shell-and-tube heat exchangers and PFTHEs with round, elliptical and rectangular tubes and various types of plate fins.

The correlations derived in the paper for calculating HTCs for hot and cold water will be used to determine the heat transfer resistance from hot to cold water in plate heat exchangers with clean surfaces without scale deposits. Plate heat exchangers are used in heat substations for domestic hot water preparation and building heating in municipal central heating systems. Water from the municipal water supply is heated by hot water from a district heating network. Cold mains water is often characterised by a high content of mineral salts deposited on the plates’ surface in scale. For this reason, plate heat exchangers become fouled and must be cleaned periodically and replaced with new ones after several chemical cleanings. The build-up of limescale on the plate surface is accompanied by a decrease in the overall HTC, which decreases heat flow rate from hot to cold water. In addition, the active cross-sectional area between the plates decreases, which causes an increase in the pressure drop on the heated hot water side. Thermal resistance of scale deposits is the difference between the inverse of the HTC of a fouled heat exchanger and the HTC of a non-fouled heat exchanger with clean plate surfaces.

$$r_f={\displaystyle \frac{1}{U_f}}-{\displaystyle \frac{1}{U}}$$

(43)

where $r_f$—thermal resistance of scale deposits, (${\mathrm{m}}^2\mathrm{K}/\mathrm{W})$; $U_f$—overall HTC of fouled heat exchanger, $\mathrm{W}/{(\mathrm{m}}^2\mathrm{K})$; $U$—overall HTC of the clean heat exchanger, $\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n} \mathrm{b}\mathrm{y} \mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(14\right),\mathrm{W}/{(\mathrm{m}}^2\mathrm{K})$.

The overall HTC $U$ for the heat exchanger with clean plate surfaces is not constant and varies depending on the mass flow of hot and cold water and the temperature of hot and cold water at the inlet to the exchanger. When monitoring the thermal resistance $r_f$ of fouling, the overall HTC $U$ for a clean exchanger is calculated online using the Nusselt number correlation for cold and hot water proposed in the paper. The uncertainty in calculating the overall HTC $U$ for a clean heat exchanger is small, as this coefficient is calculated with high accuracy in the proposed method. Even if the $h_h$ and $h_c$ coefficients are calculated with less accuracy, this does not affect the determined $U$ value. If one of the heat transfer coefficients, e.g., $h_h$, is too high, then the $h_c$ coefficient must be lower so that the $U$ value remains the same.

The overall HTC of the fouled heat exchanger is determined online using the following equation:

$$U_f={\displaystyle \frac{\dot{Q}}{A∆T_{lm}}}$$

(44)

The symbol $\dot{Q}$ in Equation (44) represents the heat flow rate in the heat exchanger from hot to cold water; $∆T_{lm}$ is the logarithmic mean temperature difference (LMTD) between hot and cold water and $A$ designates the heat transfer area.

A numerical method for determining Nusselt numbers on the side of the cold and hot water in the clean plate-and-frame heat exchanger was developed based on flow-thermal measurements. The proposed method for determining the thermal resistance of scale deposits will be applied in a computer system to monitor heat substation fouling in large city’s district heating network. The online monitoring system for the fouling degree of heat exchangers will monitor several to several thousand heat exchangers in the central heating system. Installation of the system does not require any hardware modifications to the heating system. Unknown parameters in correlations for Nusselt numbers for a clean heat exchanger are determined based on flow and thermal tests of a clean exchanger in a laboratory or based on flow and thermal measurements of a clean heat exchanger installed in a municipal central heating system. Correlations for calculating HTCs are determined once. In the monitoring system, HTCs on the cold and hot water side are calculated using previously developed correlations, and then the overall HTC is calculated using Equation (14). The time required to calculate the overall heat transfer coefficient U and thermal resistance r_f is several to several dozen times shorter than the time step for collecting measurement data in the heat exchanger monitoring system.