Performance of Double Pipe Heat Exchanger—Partially Occupied by Metal Foam—Is Better Enhanced Using Robust Adaptive Barrier Function-Based Sliding Mode Control

Abstract

Numerous thermal practical applications utilize shell and tube heat exchanger appliances to transfer heat energy between hot and cold working fluids. Incorporating metal foam to the outer periphery of inner tube improves the heat transfer process from hot water in the tube side to cold water in the shell side and consequently improves heat exchanger performance. In this study, the integration of use of a porous material together with designing a robust adaptive controller could efficiently regulate the outlet cold water temperature to the desired value. This is achieved with respect to the time required for cold water to reach the desired temperature (settling time) and the amount of hot water volume flow during a certain time span. A barrier function-based adaptive sliding mode controller (BF-based adaptive SMC) is proposed, which requires only the information of temperature measurement of cold water. The stability of BF-based adaptive SMC is proved utilizing Lyapunov function analysis. The effectiveness of proposed controller is verified via numerical results, which showed that the proposed controller could achieve considerable accuracy of cold water temperature using suitable design parameters. In addition, the robustness of controller against variation in inlet temperature is also verified. Another improvement to performance of heat exchanger system is achieved by adding the metal foam of aluminum material on inner pipe perimeter with wide range of metal foam to outer inner pipe diameters ratio ($1\le s\le 1.8$). The results showed that the settling time is significantly reduced which enables outlet cold water to reach the required temperature faster. With respect of the case of non-adding metal foam on inner pipe outer circumference, when $s=1.2$, the settling time and hot water temperature are reduced by $1/2$ and $17.3\%$, respectively, while for $s=1.8$, they are decreased by $\le 1/20$ and $35.3\%$ correspondingly. Accordingly, the required volume flow for hot water is reduced considerably.

1. Introduction

Frequent applications include heat transfer operations; however, conversion, employment, and energy recovery in every manufacturing, commercial, and domestic application comprise a heat transmission process. In most of these demands, heat is transported through heat exchanger devices which are designed to effectively transmit heat between two fluids without pending into direct interaction. Nowadays, heat exchangers find extensive utilization across diverse industrial and engineering applications; power plants, air conditioning, thermal heating, refrigeration, manufacturing industries, etc. [1,2]. Consequently, improving thermal operation of heat exchange has an instant influence on the energy, material, and cost economy. Accordingly, augmenting heat transmission above that in traditional or normal practice can considerably enhance thermal efficiency in such implementations in addition to their design and operation [2]. There are widespread types of heat exchangers which can be simply factory-made, such as double-pipe or so-called shell and tube heat exchanger type. This type becomes the greatest noteworthy fragment in small and large engineering and technical applications. Also, it has attracted important attention from researchers, engineers, and industries due to numerous compensations, such as compact design, flexibility, versatility, modularity, easy cleaning and maintenance, and cost-efficacy [3,4].

Heat transmission in heat exchangers including the double-pipe heat exchanger can be enhanced by various techniques, such as heat sink appliances that can be combined with heat exchangers to improve heat transmission. These heat sink mechanisms comprise metal wires, disks, fins, rings, porous media, supercritical CO₂ with circular tetrahedral lattice structures, and other complex assemblies [5,6,7,8]. Metal foam which represents one of the porous media types can be utilized for supplementing heat transmission because of its small volume and low production cost. Additionally, metal foam has combinations of physical, structural, and mechanical properties that cannot be attained in other materials such as elevated mechanical strength, light weight, raised electrical and thermal conductivity, strong flow-mixing capability, and high specific surface area [9,10,11,12]. Augmentation of employing metal foam in flow and heat transfer appliances is due to its unique thermophysical properties such as high thermal conductivity, porosity, and fluid flow mixing in addition to the increase in the heat transfer surface area which raises the heat transmission rate as compared with components or surfaces without metal foam insertion [13,14,15,16,17,18,19,20].

Performance of double-pipe heat exchangers can be enhanced by inserting metal foam of different geometries on the inner and/or outer cylindrical periphery of outer tube (shell side) and/or inner tube (pipe side). Metal foam insertion can be accomplished fully or partially filled across required position according to the type of working fluids, shell- and tube-side dimensions, working fluids velocity, type of heat exchanger application, maintenance requirements, and many other factors that affect heat exchanger performance and efficiency. Recently, numerous studies have mainly concentrated on the heat transfer process in fully and partially foam-filled heat exchangers with different geometries such as metal foam fins, foam blocks, foam heat sinks, and foam wrapped tubes [12,21,22,23,24,25,26,27,28].

Increased efforts have been spent improving the quality of control in the chemical process industry, mainly for the temperature control systems in heat exchanger appliances. The designer is frequently faced with the problem of maintaining control system stability and performance (robustness) in the presence of significant plant model mismatch, nonlinearity, and external disturbances [29]. The Proportional–Integral–Derivative (PID) controller was utilized by many authors to design temperature control considering the linear model of heat exchanger [30,31,32]. Also, the fuzzy Proportional–Derivative (FPD) controller and PID controller were implemented by [33] and compared with the traditional PID controller. On the other hand, refs. [29,34] applied the PID controller for the feedback-linearized heat exchanger. Consequently, the PID controller is effective for linear model plant with constant perturbation, and so, for the case of a nonlinear model with non-constant perturbation, the controller efficiency degrades.

Nonlinear control design techniques were also applied to nonlinear models of heat exchanger and also for addressing the uncertainty in the models. The $H_{\infty }$ robust controller design theory is applied to the heat exchanger control system with defined uncertainties, like parametric uncertainty and performance, as specified in [35]. By utilizing the dynamic state feedback in [34], the controller was designed by linearizing the heat exchanger model and then linear control theory can be used to regulate temperature to the desired value. Moreover, the sliding mode control technique is a well-known robust control method which is used to control different types of systems. The SMC has several advantages, such as robustness against external disturbances and uncertainties in parameters. Thus, a first-order SMC with integral sliding variable was employed by [36], while a time-varying integral sliding variable was utilized by [37] to combine the benefit of control system robustness and convergence rate. Additionally, in [38], an adaptive fuzzy sliding mode controller was designed for strong robustness and high control precision. Afterwards, for first-order SMC, the convergence is asymptotic; however, for finite time stability, higher order SMCs have been utilized by many authors. Ref. [39] proposed three different terminal sliding mode controllers and was verified through numerical simulations and implemented using an experimental setup. Also, in [40], the super-twisting control (STC) was employed for the temperature control of the heat exchanger. They showed that the STC can be practically implemented for process control using mechanical positioning actuators in real-world applications. Furthermore, for a precise control for the outlet temperature of cold water, a Terminal Synergetic Control (TSC) was proposed in [41], where, in their work, it was shown that synergetic control-based techniques offer superior potential solutions for nonlinear feedback control problems. Finally, a Higher-Order Sliding Mode (HOSM) controller for a heat exchanger connected via a data communication network was proposed by [42]. Besides its ability for chattering attenuation, the HOSM controller compensates for the random communication network and process delays in the model uncertainties present in the heat exchanger.

Model predictive control (MPC) is a control method based on periodically solving a constrained optimal control problem in every time step to evaluate the optimal input signal. The MPC was modified by [43] for a constraint removal approach which achieves the robustness required for practical application to a laboratory-scaled heat exchanger. They analyzed the control performance of the heat exchanger from the industrial perspective considering computational time and energy consumption. Additionally, ref. [44] proposed modified single layer economic model predictive control which integrates the dynamic economic optimization with the objective function of conventional tracking model predictive control. Their results showed that the economic model predictive control provides good quality control at a lower operational cost. Moreover, two control strategies were performed by [45] to identify the best-performing strategy for a countercurrent flow-plate heat exchanger in a virtual environment. The first controller is based on an inverse model of the plant, with linear algebra techniques and numerical methods; the second controller is an MPC. They showed that the MPC controller has the shortest settling time and lowest steady-state error.

Sliding mode control is a robust design methodology based on a systematic scheme that ensures the attractiveness of sliding manifold [46]. During sliding motion, the system becomes invariant to system uncertainties and external disturbances. This is known as the invariance property, which represents the main advantage of sliding mode control [47]. Conversely, the main disadvantage is the chattering problem [48]. The chattering problem can be overcome by suitable selection of the control gain. The barrier function has been augmented with the SMC formula, and it represents an adaptive control gain [49] and also with ISMC [50,51]. With the aid of using the barrier function in SMC, there is no need for either the precise system model knowledge or the ultimate bounds on its uncertainty and disturbances [52]. Hence, an adaptive SMC is obtained and additionally the chattering is eliminated where the combination of the barrier function with the SMC makes the control law continuous [53].

The aim of the present work is to explore the effect and enhancement of adding metal foam to the outer perimeter of the inner pipe for a parallel flow double-pipe heat exchanger. The heat exchanger performance will be measured by fixing the cold outlet temperature via a robust controller. An adaptive SMC using the barrier function will be designed to regulate robustly the outlet cold water to the desired temperature. Then, performance of the controlled output heat exchanger will be measured via the time required to reach the required outlet cold water temperature and additionally the amount of hot water volume flow through the inner pipe during certain time span. The main contributions of the present work can be summarized in the following points:

• The double-pipe heat exchanger mathematical model has been derived from the first principles in the presence of metal foam insertion at the outer perimeter of the inner pipe (tube side). This also includes the derivation of overall thermal conductance;
• A robust barrier function-based sliding mode controller has been designed to enforce the outlet cold water temperature to follow a desired value;
• The insertion of metal foam on the outer periphery of double-pipe heat exchanger tube side could reduce both the settling time and hot water volume flow and hence improve the performance of the system;
• Improvement of the sustainability metric in terms of energy and consumed water, where both the volume and temperature of hot water have been decreased. By virtue of adding metal foam and fusing the proposed robust BF-based adaptive SMC, one can keep low consumption of hot water flow and, simultaneously, maintain lower temperature of the outlet hot water.

2. Double-Pipe Heat Exchanger Mathematical Model

A common double-pipe heat exchanger (shell and tube heat exchanger), shown in Figure 1, is composed of two concentric pipes with different diameters; the inner and outer pipes are so-called tube and shell sides, respectively. Two working fluids are flowing in the heat exchanger and heat transmission between them through the solid pipe wall is taking place. One of these fluids runs inside the tube side and the other fluid flows through the shell side (annular space between the two pipes). Two types of flow arrangement are probable for these working fluids, parallel flow and counter flow. In parallel flow, both hot and cold fluids enter the heat exchanger at the same end and move in a similar direction while working fluids in counter flow arrangement run in the opposite direction. In the present work, parallel flow arrangement is performed extensively.

To derive the mathematical model, the following assumptions are taken into consideration:

• The thermophysical properties of both cold and hot fluids under consideration are constant;
• The variation in fluid velocity and temperature radially is negligible, and thus the kinetic and potential energy changes are negligible;
• The outer surface of the heat exchanger is assumed to be perfectly insulated, so that there is no heat loss to the surrounding medium, and any heat transfer occurs between the two fluids only;
• Axial heat conduction along the tube is usually insignificant and can be considered negligible.

Referring to Figure 1, consider an incremental element $∆x$ and along $x$ to which the principle of energy conservation is applied. Energy balance is applied to differential element on tube side (either shell or inner tube). In the element at a particular distance $x-∆x$ and time t, the heat balance equation is given by [54,55] the following:

$$\left(\begin{array}{c}\mathrm{R}\mathrm{a}\mathrm{t}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e} \mathrm{o}\mathrm{f}\\ \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y} \mathrm{i}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{e}\\ \mathrm{e}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}∆\mathrm{x}\end{array}\right)=\left(\begin{array}{c}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{f}\mathrm{l}\mathrm{o}\mathrm{w} \mathrm{o}\mathrm{f}\\ \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y} \mathrm{i}\mathrm{n}\mathrm{t}\mathrm{o} \mathrm{t}\mathrm{h}\mathrm{e}\\ \mathrm{e}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}∆\mathrm{x}\end{array}\right)-\left(\begin{array}{c}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{f}\mathrm{l}\mathrm{o}\mathrm{w} \mathrm{o}\mathrm{f}\\ \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y} \mathrm{o}\mathrm{u}\mathrm{t} \mathrm{o}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e}\\ \mathrm{e}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}∆\mathrm{x}\end{array}\right)+\left(\begin{array}{c}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t} \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}\mathrm{e}\mathrm{r}\\ \mathrm{t}\mathrm{o} \mathrm{t}\mathrm{h}\mathrm{e}\\ \mathrm{e}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}∆\mathrm{x}\end{array}\right),$$

(1)

and it can be written as follows:

$$Q_{store,∆x}=Q_x-Q_{x+∆x}+Q_{radial, ∆x},$$

If the velocity of fluid averaged across the tube is constant, i.e., independent of $x$, then unsteady-state lumped energy balance is given by [34] the following:

$${\displaystyle \frac{d\left(\rho \forall CpT\right)}{dt}}=\rho Cp\dot{V}\left(T_{x-∆x}-T_x\right)+UA{∆T}_{LMTD},$$

(2)

where $\rho$ is fluid density; $\forall$ is volume of fluid; $Cp$ is fluid specific heat; $T$ is temperature; $t$ is time duration; $\dot{V}$ is fluid volume flow rate; $U$ is the overall heat transfer coefficient; $A$ is cross section area; and ${∆T}_{LMTD}$ is the logarithmic mean temperature difference.

Equation (2) can be utilized to derive the following equation for the shell side (cold fluid) as follows:

$${\displaystyle \frac{d{T}_c}{dt}}={\displaystyle \frac{{\dot{V}}_c}{{\forall }_c}}\left(T_{x-∆x}-T_x\right)+{\displaystyle \frac{UA}{{\rho }_c{\forall }_c{Cp}_c}}{∆T}_{LMTD},$$

(3)

And the tube side (hot fluid) is casted as follows:

$${\displaystyle \frac{d{T}_h}{dt}}={\displaystyle \frac{{\dot{V}}_h}{{\forall }_h}}\left(T_{x-∆x}-T_x\right)-{\displaystyle \frac{UA}{{\rho }_h{\forall }_h{Cp}_h}}{∆T}_{LMTD},$$

(4)

where subscripts $c$ and $h$ refer to cold and hot working fluid, respectively.

It must be noticed that the negative sign for the second term in the right side of Equation (4) is because heat transfer is transmitted from the element in tube side (hot fluid) to that in shell side (cold fluid).

Cold $T_c$ and hot $T_h$ fluid temperature in the left side of Equations (3) and (4), respectively, can be possessed as a bulk mean temperature of inlet and outlet temperature, $T_c={\displaystyle \frac{T_{ci}+T_{co}}{2}}$ and $T_h={\displaystyle \frac{T_{hi}+T_{ho}}{2}}$. Thus, Equations (3) and (4) become the following:

$${\displaystyle \frac{d\left(\frac{T_{ci}+T_{co}}{2}\right)}{dt}}={\displaystyle \frac{{\dot{V}}_c}{{\forall }_c}}\left(T_{ci}-T_{co}\right)+{\displaystyle \frac{UA}{{\rho }_c{\forall }_c{Cp}_c}}{∆T}_{LMTD},$$

(5)

$${\displaystyle \frac{d\left(\frac{T_{hi}+T_{ho}}{2}\right)}{dt}}={\displaystyle \frac{{\dot{V}}_h}{{\forall }_h}}\left(T_{hi}-T_{ho}\right)-{\displaystyle \frac{UA}{{\rho }_h{\forall }_h{Cp}_h}}{∆T}_{LMTD},$$

(6)

In the present work, $T_{ci}$ and $T_{hi}$ are considered constant. Thus, ${\displaystyle \frac{d{T}_{ci}}{dt}}=0$ and ${\displaystyle \frac{d{T}_{hi}}{dt}}=0$ and Equations (5) and (6) are presented as follows:

$${\displaystyle \frac{d{T}_{co}}{dt}}={\displaystyle \frac{2}{{\forall }_c}}\left[{\dot{V}}_c\left(T_{ci}-T_{co}\right)+{\displaystyle \frac{UA}{{\rho }_c{Cp}_c}}{∆T}_{LMTD}\right],$$

(7)

$${\displaystyle \frac{d{T}_{ho}}{dt}}={\displaystyle \frac{2}{{\forall }_h}}\left[{\dot{V}}_h\left(T_{hi}-T_{ho}\right)-{\displaystyle \frac{UA}{{\rho }_h{Cp}_h}}{∆T}_{LMTD}\right],$$

(8)

Meanwhile, if cold $T_c$ and hot $T_h$ fluid temperature in the left side of Equations (3) and (4) are taken equal to outlet temperature, $T_c=T_{co}$ and $T_h=T_{ho}$, then Equations (3) and (4) become the following:

$${\displaystyle \frac{d{T}_{co}}{dt}}={\displaystyle \frac{{\dot{V}}_c}{{\forall }_c}}\left(T_{ci}-T_{co}\right)+{\displaystyle \frac{UA}{{\rho }_c{\forall }_c{Cp}_c}}{∆T}_{LMTD},$$

(9)

$${\displaystyle \frac{d{T}_{ho}}{dt}}={\displaystyle \frac{{\dot{V}}_h}{{\forall }_h}}\left(T_{hi}-T_{ho}\right)-{\displaystyle \frac{UA}{{\rho }_h{\forall }_h{Cp}_h}}{∆T}_{LMTD},$$

(10)

Thus, Equations (9) and (10) are represented as follows:

$${\displaystyle \frac{d{T}_{co}}{dt}}={\displaystyle \frac{1}{{\forall }_c}}\left[{\dot{V}}_c\left(T_{ci}-T_{co}\right)+{\displaystyle \frac{UA}{{\rho }_c{Cp}_c}}{∆T}_{LMTD}\right],$$

(11)

$${\displaystyle \frac{d{T}_{ho}}{dt}}={\displaystyle \frac{1}{{\forall }_h}}\left[{\dot{V}}_h\left(T_{hi}-T_{ho}\right)-{\displaystyle \frac{UA}{{\rho }_h{Cp}_h}}{∆T}_{LMTD}\right],$$

(12)

In general form, Equations (7), (8), (11) and (12) can be casted as follows:

$${\displaystyle \frac{d{T}_{co}\left(t\right)}{dt}}={\displaystyle \frac{a}{{\forall }_c}}\left[{\dot{V}}_c\left(t\right)\left(T_{ci}-T_{co}\left(t\right)\right)+{\displaystyle \frac{UA}{{\rho }_c{Cp}_c}}{∆T}_{LMTD}\left(t\right)\right],$$

(13)

$${\displaystyle \frac{d{T}_{ho}\left(t\right)}{dt}}={\displaystyle \frac{a}{{\forall }_h}}\left[{\dot{V}}_h\left(t\right)\left(T_{hi}-T_{ho}\left(t\right)\right)-{\displaystyle \frac{UA}{{\rho }_h{Cp}_h}}{∆T}_{LMTD}\left(t\right)\right],$$

(14)

where $a$ may be either equal $1$ or $2$. In the present work, $a$ is possessed equal to $1$. Additionally, ${∆T}_{LMTD}\left(t\right)$ is determined using the following equation:

$${∆T}_{LMTD}\left(t\right)={\displaystyle \frac{\left(T_{hi}-T_{ci}\right)-\left(T_{ho}\left(t\right)-T_{co}\left(t\right)\right)}{\mathit{ln}\left[\left(T_{hi}-T_{ci}\right)/\left(T_{ho}\left(t\right)-T_{co}\left(t\right)\right)\right]}},$$

(15)

2.1. Overall Thermal Conductance

The overall heat transfer coefficient multiplied by surface area through which heat is transferred ($UA$) is known as overall thermal conductance. Generally, heat transfer rate in double-pipe heat exchanger can be assessed using the following equation:

$$Q={\displaystyle \frac{∆T}{R}}=U{A}_s{∆T}_{LMTD}=U_i{A}_i{∆T}_{LMTD}=U_o{A}_o{∆T}_{LMTD},$$

(16)

where $R$ is the overall thermal resistance; $U_i$ and $U_o$ are inner and outer overall heat transfer coefficient, respectively; $A_i$ and $A_o$ are inner and outer surface area correspondingly; and $A_s$ is the surface area.

By re-writing Equation (16), the total thermal resistance for inner tube with non-negligible wall thickness can be formulated as follows (shown in Figure 2):

$${\displaystyle \frac{1}{U{A}_s}}={\displaystyle \frac{1}{U_i{A}_i}}={\displaystyle \frac{1}{U_o{A}_o}}=R={\displaystyle \frac{1}{h_i{A}_i}}+{\displaystyle \frac{\mathit{ln}\left(D_o/D_i\right)}{2\pi kL}}+{\displaystyle \frac{1}{h_o{A}_o}},$$

(17)

where $k$ is the inner pipe thermal conductivity; $h_i$ and $h_o$ are heat transfer coefficient inside inner pipe and on outer periphery of inner pipe, respectively; $D_o$ and $D_i$ are inner and outer diameter of inner pipe correspondingly; and $L$ is the length of inner pipe.

As mentioned previously, metal foam can be added adjacent to the solid wall to enhance heat transmission. In the present work, a metal foam layer of different thicknesses $t_{foam}$ is supplemented on the outer perimeter of inner pipe (exposed in Figure 2) and its diameter $D_{foam}$ can be stated as follows:

$$D_{foam}=D_o+\left(2\times t_{foam}\right),$$

(18)

Consequently, Equation (17) can be presented as follows for the case of adding metal foam on the outer perimeter of inner tube by appending metal foam conduction thermal resistance ($R_{cond,foam}={\displaystyle \frac{\mathit{ln}\left(D_{foam}/D_o\right)}{2\pi k_{eff}L}}$) [54,55]:

$${\displaystyle \frac{1}{UA}}={\displaystyle \frac{1}{U_i{A}_i}}={\displaystyle \frac{1}{U_o{A}_o}}=R={\displaystyle \frac{1}{h_i{A}_i}}+{\displaystyle \frac{\mathit{ln}\left(D_o/D_i\right)}{2\pi kL}}+{\displaystyle \frac{\mathit{ln}\left(D_{foam}/D_o\right)}{2\pi k_{eff}L}}+{\displaystyle \frac{1}{h_o{A}_o}},$$

(19)

2.2. Calculation of Heat Transfer Coefficients

In order to estimate the overall heat transfer coefficient, inner heat transfer coefficient $h_i$ and outer heat transfer coefficient $h_o$ can be calculated for the two cases of inner pipe. These cases include without and with adding metal foam on the outer perimeter of inner pipe.

1.

The inner heat transfer coefficient $h_i$ for internal side of inner pipe (hot water) can be determined by employing the following equations [2,5,6,55,56]:

$$\mathrm{For} {Re}_i\le 2100: Nu=1.86{\left({\displaystyle \frac{{Re}_i {Pr}_i D_i}{L}}\right)}^{1/3}{\left({\displaystyle \frac{{\mu }_b}{{\mu }_s}}\right)}^{0.14},$$

(20)

$$\mathrm{For} 2100<{Re}_i<10000: Nu=1.86{\left({\displaystyle \frac{{Re}_i {Pr}_i D_i}{L}}\right)}^{1/3}{\left({\displaystyle \frac{{\mu }_b}{{\mu }_s}}\right)}^{0.14},$$

(21)

$$\mathrm{For} {Re}_i\ge 10000: Nu=0.023{Re}_i^{0.8}{Pr}_i^n,$$

(22)

where ${Re}_i={\displaystyle \frac{V_h{D}_i}{{\nu }_i}}$ is inner Reynolds number; $V_h={\displaystyle \frac{{\dot{m}}_h}{{\rho }_i{A}_c}}={\displaystyle \frac{{\dot{m}}_h}{{\rho }_i\left(\frac{1}{4}\pi D_i^2\right)}}$ is fluid velocity inside tube side; $Nu={\displaystyle \frac{h_i{D}_i}{k_i}}$ is Nusselt number; $k_i$ is tube-side fluid thermal conductivity; ${Pr}_i$ is tube-side Prandtl number; ${\mu }_b$ and ${\mu }_s$ are dynamic viscosity at bulk and surface temperature, respectively; ${\dot{m}}_i$ is tube-side fluid mass flow rate; ${\rho }_i$ is tube-side fluid density; $A_c$ is inner tube cross-section area; ${\nu }_i$ is tube-side fluid kinematic viscosity; and $n$ is a constant and it is taken as $n=0.4$ for heating and $0.3$ for cooling of fluid flowing through tube during heat exchanger operation.

2.

The outer heat transfer coefficient $h_o$ for the outer side of the inner pipe (cold water in annulus region) can be assessed by utilizing the following equations [5,56]:

$$V_c={\displaystyle \frac{{\dot{V}}_c}{{\rho }_o{A}_o}}={\displaystyle \frac{{\dot{V}}_c}{{\rho }_o\left[\frac{1}{4}\pi \left(D_{s,i}^2-D_{t,o}^2\right)\right]}},$$

(23)

Instead of using $D_{i,h}$ in the tube Equations (20)–(22), equivalent diameter equation $D_e$ can employed as given below:

$$D_e={\displaystyle \frac{4A_{s,c}}{P}}={\displaystyle \frac{4\pi \left(D_{s,i}^2-D_{t,o}^2\right)}{4\pi D_{t,o}}}={\displaystyle \frac{D_{s,i}^2-D_{t,o}^2}{D_{t,o}}},$$

(24)

where $V_c$ is cold fluid velocity inside shell side; $A_{s,c}$ is cross section area of annulus region (between outer diameter of pipe side and inner diameter of shell); $P$ is perimeter of annulus region; $D_{s,i}$ is inner diameter of outer heat exchanger pipe (shell side); and $D_{t,o}$ is outer diameter of inner pipe (tube side).

2.3. Thermally Steady Outlet Temperature

In this section, the thermally steady-state outlet temperature is derived by equating the steady-state heat transfer rate equations. These heat rate equations include the equation of heat transmission to cold water (shell side), given as follows:

$$Q_c={\dot{m}}_c{C}_{pc}(T_{c\_out}-T_{c\_in}),$$

(25)

The equation of heat transmission from hot water (tube side) is stated as follows:

$$Q_h={\dot{m}}_h{C}_{ph}(T_{h\_in}-T_{h\_out}),$$

(26)

The average heat transfer rate, which is represented as follows:

$$Q=UA {\displaystyle \frac{\left(T_{hi}-T_{ci}\right)-\left(T_{ho}-T_{co}\right)}{\mathit{ln}\left[\left(T_{hi}-T_{ci}\right)/\left(T_{ho}-T_{co}\right)\right]}},$$

(27)

As the steady-state condition is applied, the above three heat transfer rate Equations (25)–(27) can be equated because heat is transmitted from the hot fluid through inner pipe wall towards cold fluid. Thus,

$$Q=Q_c=Q_h,$$

(28)

By solving Equation (28) with respect to $T_{co}$ and $T_{ho}$, the outlet cold and hot temperature is obtained as function of $UA$, mass flow rate of cold and hot water, specific heat of water, and inlet cold and hot water temperature. An interesting point that can be deduced from heat balance Equation (28) for $T_{co}$ is that the maximum allowable outlet cold water can be estimated for a certain hot and cold input water temperature and also for specific cold and hot water mass flow rate. Accordingly, this is stated in the following lemma.

Lemma 1. 

Using the heat balance Equation (28) for certain cold and hot water inlet temperature and cold and hot water volumetric flow rates, the steady-state outlet hot water temperature is bounded by the following:

$$T_{co}<{\displaystyle \frac{\alpha T_{hi}+T_{ci}}{\left(1+\alpha \right)}},$$

(29)

where $\alpha ={\dot{V}}_h/{\dot{V}}_c$.

Proof. 

The two equations which are obtained from Equation (28) are given by the following:

$$0={\dot{V}}_c\left(T_{ci}-T_{co}\right)+{\displaystyle \frac{UA}{{\rho }_c{Cp}_c}}{∆T}_{LMTD},$$

(30)

$$0={\dot{V}}_h\left(T_{hi}-T_{ho}\right)-{\displaystyle \frac{UA}{{\rho }_h{Cp}_h}}{∆T}_{LMTD},$$

(31)

From which, the following equation can be attained:

$${\dot{V}}_c\left(T_{ci}-T_{co}\right)+{\dot{V}}_h\left(T_{hi}-T_{ho}\right)=0,$$

(32)

This leads to the following:

$$T_{co}={\displaystyle \frac{{\dot{V}}_h}{{\dot{V}}_c}}\left(T_{hi}-T_{ho}\right)+T_{ci},$$

(33)

Let $\alpha ={\dot{V}}_h/{\dot{V}}_c$ be the ratio of inlet hot water flow rate to inlet cold water flow rate, then,

$$T_{co}=-\alpha T_{ho}+\alpha T_{hi}+T_{ci},$$

(34)

with the assumption of no heat generation, the following equation is casted:

$$T_{co}<T_{ho},$$

(35)

Then, from Equation (34), the following is obtained:

$$T_{co}<-\alpha T_{co}+\alpha T_{hi}+T_{ci}$$

(36)

or it can be expressed as follows:

$$T_{co}<{\displaystyle \frac{\alpha T_{hi}+T_{ci}}{\left(1+\alpha \right)}},$$

(37)

□

From Lemma 1, it is clear that the bound on $T_{co}$ is function to $\alpha$ in addition to inlet cold and hot water temperature.

Remark 1. 

For a certain $\alpha ={\dot{V}}_h/{\dot{V}}_c$,$T_{hi}$, and $T_{ci}$, the outlet cold water temperature for which the heat exchanger is designed must be selected such that inequality Equation (37) does not violate.

Remark 2. 

For any$\alpha$, the time history of outlet cold temperature from the first instant to steady-state condition is included in the following set $\Sigma =\left\{T_{co}|T_{ci}<T_{co}<{\displaystyle \frac{\alpha T_{hi}+T_{ci}}{\left(1+\alpha \right)}}\right\}$.

Remark 3. 

An interesting estimation for the outlet cold water from inequality Equation (37) can be obtained for $\alpha =1$ which shows that $T_{co}$ will be less than the average temperature of inlet cold- and hot water temperature. Meanwhile, for $\alpha >>1$, $T_{co}$ approaches the inlet hot water temperature $T_{hi}$.

2.4. State Space Model

The double-pipe heat exchanger mathematical model given in Equations (13) and (14) with $a=1$ is re-written in the state space model by defining the state variables $x_1$ and $x_2$ as follows:

$$\left.\begin{array}{c}{x}_1=T_{co}\left(t\right)\\ x_2=T_{ho}\left(t\right)\end{array}\right\},$$

(38)

Hence, the double-pipe heat exchanger dynamics, in terms of state variables $x=\left(x_1,x_2\right)$, is expressed by the following:

$${\displaystyle \frac{d{x}_1}{dt}}=f_1\left(x\right),$$

(39)

$${\displaystyle \frac{d{x}_2}{dt}}=f_2\left(x\right)+g_2\left(x\right)u,$$

(40)

where $x=\left(x_1,x_2\right)$, $f_1\left(x\right)={\displaystyle \frac{1}{{\forall }_c}}\left[{\dot{V}}_c\left(T_{ci}-x_1\right)+{\displaystyle \frac{UA}{{\rho }_c{Cp}_c}}{∆T}_{LMTD}\right]$, $f_2\left(x\right)=-{\displaystyle \frac{1}{{\forall }_h}}{\displaystyle \frac{UA}{{\rho }_h{Cp}_h}}{∆T}_{LMTD}$, $g_2\left(x\right)={\displaystyle \frac{1}{{\forall }_h}}\left(T_{hi}-x_2\right)$, ${∆T}_{LMTD}={\displaystyle \frac{\left(T_{hi}-T_{ci}\right)-\left(x_2-x_1\right)}{\mathrm{l}\mathrm{n}\left(\frac{T_{hi}-T_{ci}}{x_2-x_1}\right)}}$, and finally $u={\dot{V}}_h$ is the volumetric flow rate of hot fluid which presents the control input.

The state space model can be put also in a general form as follows:

$${\displaystyle \frac{dx}{dt}}=f\left(x\right)+g\left(x\right)u,$$

(41)

where $f\left(x\right)={\left[f_1\left(x\right) f_2\left(x\right)\right]}^T$ and $g\left(x\right)={\left[0 { g}_2\left(x\right)\right]}^T$. The heat exchanger state model is state feedback linearizable system as stated in the following Lemma.

Lemma 2. 

The state space model of the double-pipe heat exchanger as given by Equations (39) and (40) is state feedback linearizable which can be transformed to a second-order canonical form via the state transformation $x\to y:$

$$\left.\begin{array}{c}{y}_1=x_1 \\ y_2={\dot{x}}_1 \end{array}\right\},$$

(42)

where $y_2$ is assumed to be measured or estimated.

Proof. 

By following the procedure of deriving the linearizing output $y=h\left(x\right)$ in [57], firstly the dimension of the matrix $\left\{g, [f,g]\right\}\left(x\right)$ must be two, as shown below:

$$im\left\{g\left(x\right), [f,g]\left(x\right)\right\}=dim\left[\begin{array}{cc}0& {\displaystyle \frac{\partial f_1}{\partial x_2}}{ g}_2\\ { g}_2& *\end{array}\right](x)=2,$$

(43)

where $[f,g]$ is the Lie bracket and ${\displaystyle \frac{\partial f_1}{\partial x_2}}{\left(g_2\right)}^2\ne 0 \forall x\ne 0$.

Secondly, since $span\left\{g\right\}$ is involutive, the double-pipe heat exchanger is a linearizable system. Finally, the linearizing output $h\left(x\right)$ must satisfy the following condition:

$${\displaystyle \frac{\partial h}{\partial x}}g=0; {\displaystyle \frac{\partial \left(L_{f}h\right)}{\partial x}}g\ne 0, and h\left(0\right)=0,$$

(44)

where $L_{f}h\left(x\right)$ is the Lee derivative of $h\left(x\right)$ with respect to $f$. So, ${\displaystyle \frac{\partial h}{\partial x}}g={\displaystyle \frac{\partial h}{\partial x_2}}{g}_2=0$, this leads to ${\displaystyle \frac{\partial h}{\partial x_2}}=0$, which means that $h=h\left(x_1\right).$ Eventually, let $h\left(x\right)=x_1$, which also satisfies the condition ${\displaystyle \frac{\partial \left(L_{f}h\right)}{\partial x}}g\ne 0$ because $L_{f}h\left(x\right)=f_1\left(x\right)$ then ${\displaystyle \frac{\partial \left(L_{f}h\right)}{\partial x}}={\displaystyle \frac{\partial f_1\left(x\right)}{\partial x_2}}{g}_2\ne 0 \forall x.$ Hence, the double-pipe heat exchanger state space model is linearizable with the linearizing output $h\left(x\right)=x_1$. Consequently, the change in variables is $y_1=h\left(x\right)$ and $y_2=L_{f}h\left(x\right)=f_1\left(x\right)$ which ends the proof. □

The linearized double-pipe heat exchanger state space model is specified by the following:

$$\left.\begin{array}{c}{\dot{y}}_1=y_2 \\ {\dot{y}}_2=F\left(x\right)+G\left(x\right)u \end{array}\right\},$$

(45)

where $F\left(x\right)={\displaystyle \frac{\partial f_1\left(x\right)}{\partial x}}\left[\begin{array}{c}{f}_1\left(x\right)\\ f_2\left(x\right)\end{array}\right]$, and $G\left(x\right)={\displaystyle \frac{\partial f_1\left(x\right)}{\partial x_2}}{g}_2\left(x\right)$. Also, it is assumed that $G\left(x\right)>0,\forall x$.

Note that it does not need to transform the control $u$ to a new control $v$ (using $u=-{\displaystyle \frac{1}{G\left(x\right)}} \left(-F\left(x\right)+v\right)$) in order to write Equation (42) as ${\dot{y}}_1=y_2$ and ${\dot{y}}_2=v$. This is because $F\left(x\right)$ and $G\left(x\right)$ are considered as uncertain functions and accordingly the controller needs to be robust and adaptive, as is stated in the next section.

3. Barrier Function-Based Sliding Mode Control Law

One challenging problem in control engineering is the design of controllers for uncertain systems. The model inaccuracies, due to un-modeled or neglected system dynamics, uncertainty in model parameters, and external disturbances, are the sources of uncertainties. One of the most efficient robust control methodologies that can deal with uncertain systems is the sliding mode control (SMC) [48].

In the present work, the proposed controller employs barrier function (BF) instead of the discontinuous control term. The barrier function is defined as follows:

Definition 1.

([49]) For some $ϵ > 0,$ which is known and fixed, the BF can be defined as an even continuous function $k_{bf}:z\in (-ϵ, ϵ) \to k_{bf}\left(z\right)\in [b,\infty ]$ strictly increasing on, $\left[0,\right.ϵ)$.

• $\underset{\left|\mathrm{x}\right|\to \mathsf{ϵ}}{\lim }{k}_{bf}\left(z\right)=+\infty$;
• $k_{bf}\left(z\right)$ has a unique minimum at zero and $k_{bf}\left(0\right)=b\ge 0$.

In the present study, a class of semi-definite BFs is considered as $k_{bf}\left(z\right)={\displaystyle \frac{\left|z\right|}{ϵ-\left|z\right|}}$. The SMC gain $k$ is used as a function of the sliding variable $\sigma$ via barrier function, i.e.,

$$k={\displaystyle \frac{\left|\sigma \right|}{ϵ-\left|\sigma \right|}},$$

(46)

By using the gain in Equation (46), the SMC law becomes continuous, adaptive, and with attenuating or eliminating chattering.

To present our proposed controller, we need first to define the sliding variable. Thus, the sliding variable $s$, with respect to Equation (45), is defined by the following:

$$\sigma =e_2+\lambda e_1, \lambda >0,$$

(47)

where $e_1$ and $e_2$ are the error functions which are defined as follows:

$$\left.\begin{array}{c}{e}_1=y_1-y_r=T_{co}\left(t\right)-T_{cor}\left(t\right)\\ e_2={\dot{e}}_1=y_2-{\dot{y}}_r={\dot{T}}_{co}\left(t\right)-{\dot{T}}_{cor}\left(t\right)\end{array}\right\},$$

(48)

Also, the time rate of change in $\sigma$ is given by the following:

$$\dot{\sigma }={\dot{y}}_2-{\ddot{y}}_r+\lambda \left(y_2-{\dot{y}}_r\right)=F\left(x,y\right)+G\left(x\right)u,$$

(49)

In the present work, we propose the use of a variable structure controller that consists of three controllers and the switching between them depends on the state position in state space. The controllers are presented as follows:

• The first controller uses the maximum hot water flow-rate as the control input when the state lies in the following region:

$${\mathsf{\Omega }}_I=\left\{x\left(t\right)|\sigma \left(t\right)\le -ϵ\right\},$$

(50)

where $ϵ$ is a small positive number. That is if $x\left(t\right)\in {\mathsf{\Omega }}_I$, then

$$u=u_{max},$$

(51)

2.

The second controller utilizes the barrier function as the adaptive gain law for the sliding mode controller. The control law for this controller is given by the following:

$$u=-k\left(\sigma \right)sign\left(\sigma \right),$$

(52)

where, as in Equation (48), $k\left(t,s\right)$ can be defined as follows:

$$k\left(\sigma \right)=k_o{\displaystyle \frac{\left|\sigma \right|}{ϵ-\left|\sigma \right|}}, k_o>0,$$

(53)

It is used when $x\left(t\right)\in {\mathsf{\Omega }}_{II}$ and $u\le u_{max}$ where

$${\mathsf{\Omega }}_{II}=\left\{x\left(t\right)|-ϵ<\sigma \left(t\right)<0\right\},$$

(54)

3.

The third controller is simply justifying the positive control input nature of the system. If $x\left(t\right)\in {\mathsf{\Omega }}_{III}$, where ${\mathsf{\Omega }}_{III}=\left\{x\left(t\right)|\sigma \left(t\right)\ge 0\right\}$, then the controller is given by the following:

$$u=0,$$

(55)

The three above regions ${\mathsf{\Omega }}_I$, ${\mathsf{\Omega }}_{II}$, and ${\mathsf{\Omega }}_{III}$ are shown in Figure 3 and the above variable structure controller is stated in the following theorem.

Theorem 1. 

For the double-pipe heat exchanger which is defined by Equations (39) and (40), the proposed variable structure controller in Equations (52)–(57) enforces output cold water to follow the desired temperature with ultimate error bound given by the following:

$$\left|e_1\right|\le {\displaystyle \frac{ϵ}{\lambda }},$$

(56)

Proof. 

Let us assume that the state is initiated in ${\mathsf{\Omega }}_{\mathrm{I}\mathrm{I}}$, then the second controllers, Equations (54) and (55), are used. Accordingly, we need to show that the error $e_1$ between the outlet cold water and desired outlet temperature is ultimately bounded by (58). To derive the inequality Equation (58), the following candidate Lyapunov function is employed:

$$V\left(\sigma \right)={\displaystyle \frac{1}{2}}{\sigma }^2,$$

(57)

Its time rate of change is given by the following:

$$\dot{V}\left(\sigma \right)=\sigma \dot{\sigma }=\sigma \left(e_2+\lambda {\dot{e}}_2\right),$$

(58)

Then for constant desired outlet cold temperature and the control law in Equations (52) and (53) we have the following:

$$\dot{V}\left(\sigma \right)=\sigma \left({\dot{y}}_2+\lambda y_2\right)=\sigma \left(F\left(x\right)+G\left(x\right)u+\lambda y_2\right),$$

$$=\sigma \left(F\left(x\right)-G\left(x\right)k\left(t,s\right)sign\left(s\right)+\lambda y_2\right),$$

$$\le -G\left(x\right)k\left(\sigma \right)\left|\sigma \right|+G\left(x\right)\left|H(x,y)\right|\left|\sigma \right|,$$

$$=-G\left(x\right)k_o{\displaystyle \frac{{\left|\sigma \right|}^2}{ϵ-\left|\sigma \right|}}\left|\sigma \right|+G\left(x\right)\left|H(x,y)\right|\left|\sigma \right|,$$

(59)

where $H\left(x,y\right)={\displaystyle \frac{F\left(x\right)+\lambda y_2}{G\left(x\right)}}$. Therefore,

$$\dot{V}\left(\sigma \right)<-\left(1-\theta \right)G\left(x\right)k_o{\displaystyle \frac{{\left|\sigma \right|}^2}{ϵ-\left|\sigma \right|}}, 0<\theta <1,$$

(60)

$\forall \left|\sigma \right|$ such that

$$-\theta k_o{\displaystyle \frac{{\left|\sigma \right|}^2}{ϵ-\left|\sigma \right|}}+\left|F\left(x,y\right)\right|\left|\sigma \right|<0,$$

(61)

The last inequality is solved for $\left|\sigma \right|$

$$\left|\sigma \right|>{\displaystyle \frac{\left|F\left(x,y\right)\right|}{\left(\theta k_o+\left|F\left(x,y\right)\right|\right)}}ϵ,$$

(62)

Hence, $\dot{V}\left(\sigma \right)<0$ for all state where $s$ satisfies this inequality. Accordingly, the ultimate bound is presented as follows:

$$\left|\sigma \right|\le b={\displaystyle \frac{\left|F\left(x,y\right)\right|}{\left(\theta k_o+\left|F\left(x,y\right)\right|\right)}}ϵ<ϵ,$$

(63)

Which proves that when the state in ${\mathsf{\Omega }}_{II}$, the sliding variable $s$ is bounded by $ϵ$. The ultimate bound with respect to the error $e_1=T_{co}\left(t\right)-T_{cor}\left(t\right)$ can be derived from Equation (9),

$$e_2+\lambda e_1=\sigma ,$$

(64)

by solving for $e_1$, we obtain

$$e_1\left(t\right)=e_1\left(0\right)\exp \left(-\lambda t\right)+\exp \left(-\mathsf{\lambda }\mathrm{t}\right){\int }_0^t\mathit{exp}\left(\lambda \tau \right)\sigma \left(\tau \right)d\tau ,$$

(65)

This leads to the following:

$$\left|e_1\left(t\right)\right|\le \left|e_1\left(0\right)\right|\exp \left(-\lambda t\right)+\exp \left(-\mathsf{\lambda }\mathrm{t}\right){\int }_0^t\mathit{exp}\left(\lambda \tau \right)\left|\sigma \left(\tau \right)\right|d\tau ,$$

$$\le \left|e_1\left(0\right)\right|\exp \left(-\lambda t\right)+ϵ\exp \left(-\mathsf{\lambda }\mathrm{t}\right){\int }_0^t\mathit{exp}\left(\lambda \tau \right)d\tau ,$$

$$=\left|e_1\left(0\right)\right|\exp \left(-\lambda t\right)+\left({\displaystyle \frac{ϵ}{\lambda }}\right)\left(1-\mathit{exp}\left(-\lambda t\right)\right),$$

(66)

As $\mathrm{t}\to \mathrm{\infty }$,

$$\underset{ t\to \infty }{\mathit{lim}}\left|e_1\left(t\right)\right|\le \left({\displaystyle \frac{ϵ}{\lambda }}\right),$$

(67)

This proves that the error function $e_1\left(t\right)$ is asymptotically approaching the ultimate bound $\left({\displaystyle \frac{ϵ}{\lambda }}\right)$.

To this end, it must be shown that ${\mathsf{\Omega }}_{II}$ is attractive if the state is initiated outside it. So, if the state is started in ${\mathsf{\Omega }}_I$ with $u=u_{max}$, $T_{co}\left(t\right)\in \Sigma$ according to Remark 2. That means from Lemma 1, if the desired outlet cold water $T_{cor}\left(t\right)$ lies in $\Sigma$ then with $u=u_{max}$, $T_{co}\left(t\right)$ will enter ${\mathsf{\Omega }}_{II}$ in finite time and this proves the attractiveness ${\mathsf{\Omega }}_{II}$ when the state is initiated in ${\mathsf{\Omega }}_I$.

Finally, it is required to show the attractiveness of ${\mathsf{\Omega }}_{II}$ when the state is initiated in the region where $s\left(t\right)\ge 0$. In this case, $u=0$, accordingly, Equation (34) with $\alpha =0$, the steady-state outlet cold water temperature becomes the following:

$$T_{co}=T_{ci},$$

(68)

Therefore, after a certain finite period of time, the state returns to enter ${\mathsf{\Omega }}_{II}$ since $T_{cor}\left(t\right)>T_{ci}$. This ends the proof where via switching between the tree control laws according to the specified region, the outlet cold water temperature is eventually forced to follow the desired value. □

Remark 4. 

In the proposed control law, $u_{max}$ is not necessary to be the maximum available hot water flow rate.The operator can use an appropriate flow rate value and it is assigned as $u_{max}$ but it is necessary to check that the condition $T_{cor}\left(t\right)\in \Sigma$ is satisfied.

Remark 5. 

The control law in ${\Omega }_{II}$ can be put in a more compact from as follows:

$$u={Sat}_{u_{max}}\left(k\left(\sigma \right)\right),$$

(69)

where

$${Sat}_{u_{max}}(k\left(\sigma \right))=\left\{\begin{array}{c}{u}_{max} if k\left(\sigma \right)\ge u_{max} \\ k\left(\sigma \right) if k\left(\sigma \right)<u_{max} \end{array}\right.,$$

(70)

Corollary 1. 

The proposed controller in theorem 1 is a continuous controller.

Proof. 

If the state is initiated in ${\mathsf{\Omega }}_{\mathrm{I}\mathrm{I}}$, then the control law is given by $\mathrm{u}=\mathrm{k}\left(\mathsf{\sigma }\right)={\mathrm{k}}_{\mathrm{o}}{\displaystyle \frac{\left|\mathsf{\sigma }\right|}{\mathsf{ϵ}-\left|\mathsf{\sigma }\right|}}$. The control formula in this region is a continuous function of $s$. But, if the state is initiated in ${\mathsf{\Omega }}_I$, then $u=u_{max}$ until the state enter ${\mathsf{\Omega }}_{II}$ where during a certain period of time after entering ${\mathsf{\Omega }}_{II}$, the computed control $u$ is greater than $u_{max}$ as stated in Remark 5. This is because it is initially very near to the boundary of barrier function and according to Equation (70), the control $u$ is fixed at $u_{max}$. After this period of time, the control $u$ becomes equal to $u_{max}$ and that is natural, since $s$ goes away from the barrier function boundary. Then, due to the continuity of control law in ${\mathsf{\Omega }}_{II}$, control effort starts decreasing until it reaches a certain value such that the thermal steady outlet temperature equation is satisfied based on the desired outlet cold water temperature. This proves the continuity of the proposed control law. □

Remark 6. 

The proposed control law in the above theorem is adaptive and works without the knowledge of system parameters or upper bound on their values. In fact, the structure of the mathematical model is the only thing that needs to be known, as given in Equations (11) and (12), in addition to the inlet and outlet cold and hot temperatures.

Figure 4 illustrates the mechanism of integrating the BF-based adaptive sliding mode controller to the heat exchanger.

4. Results and Discussions

This section is devoted to presenting the numerical simulations for the double-pipe heat exchanger utilizing the proposed controller using the barrier function based on adaptive SMC. In the first part of this section, the heat exchanger dynamic system is simulated for fixed hot water volume flowrate and for different metal foam diameter. Then, in the second part which is devoted to test control system performance, the heat exchanger model is simulated for different desired outlet cold water temperatures. The only information required is the measurements of the inlet and outlet cold and hot water temperature. Next, in the third part of the simulation experiments, double-pipe heat exchanger performance enhancement is examined through the results of numerical simulations after adding metal foam with different diameters.

The heat exchanger mathematical model parameters that are used in the simulation are listed in

Table 1. Heat exchanger model parameter values [2,21,55]. * Effective thermal conductivity for water-saturated metal foam of 10 PPI thermal [58].

Parameter	Value	Parameter	Value
$T_{hi}$	$85 °\mathrm{C}$	$D_{si}$	$0.108 \mathrm{m}$
$T_{ci}$	$20 °\mathrm{C}$	$\mu$	$855\times {10}^{-6} \mathrm{N}.\mathrm{s}/{\mathrm{m}}^2$
${\forall }_c$	$0.0029 {\mathrm{m}}^3$	$L$	$1.92 \mathrm{m}$
${\forall }_h$	$0.0043 {\mathrm{m}}^3$	$k$	$0.671 \mathrm{W}/{\mathrm{m}}^2.\mathrm{K}$
${\dot{V}}_c$	$2.5096\times {10}^{-6} {\mathrm{m}}^3/\mathrm{s}$	$k_{eff}$*	$7.6 \mathrm{W}/{\mathrm{m}}^2.\mathrm{K}$
${\rho }_c$	$966 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$	$D_{to}$	$0.0548 \mathrm{m}$
${\rho }_h$	$966 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$	$D_{ti}$	$0.0536 \mathrm{m}$
${Cp}_c$	$4204 \mathrm{J}/\mathrm{k}\mathrm{g}.\mathrm{K}$	$u_{max}$	$1.6667\times {10}^{-5} {\mathrm{m}}^3/\mathrm{s}$
${Cp}_h$	$4204 \mathrm{J}/\mathrm{k}\mathrm{g}.\mathrm{K}$

below, including the geometrical and the materials information required to calculate the overall thermal conductance $UA$ and the other state space model parameters in Equations (49) and (40).

4.1. Effect of Metal Foam Insertion on Outlet Cold Water Temperature for Fixed Cold and Hot Water Flow Rates

The effect of inserting metal foam layer on the outer periphery of inner pipe in a double-pipe heat exchanger is examined in the present part of the results. The results are based on simulating the heat exchanger with the parameters listed in Table 1.

The only variable parameter in the heat exchanger system is the diameter of inserted metal foam and for simplicity it was taken as a percent from the outlet diameter of the inner pipe (tube side) $D_{to}$, i.e., the metal foam diameter $D_{foam}$ is related to $D_{to}$ by the following relation:

$$D_{foam}=s.D_{to},$$

(71)

where $s=1$ refers to the heat exchanger without metal foam, while when $s=D_{si}/D_{to}$, the metal foam fills totally the gap between outer pipe $D_{to}$ and inner shell $D_{si}$ diameters.

For feasibility in engineering applications, it can be mentioned that if the annulus is filled completely with water without metal foam ($s=1$), the annulus mass is $13 \mathrm{kg}$ and if the annulus is fully filled with metal foam ($s=2$) the annulus mass becomes $15.25 \mathrm{kg}$. Therefore, the mass increase is about $2.25 \mathrm{kg}$ ($\approx 17\%$) and this amplification can be neglected with respect to the overall mass of heat exchanger and can be integrated with the design consideration of heat exchanger.

The effect of adding metal foam is examined for the range $1\le s\le 1.8<1.97=D_{si}/D_{to}$ with inlet hot water volume flow rate equal to $u_{max}$. Figure 5 reveals that the overall thermal conductance $UA$ is an increasing function with the increasing in the metal foam diameter $D_{foam}$. This behavior can be attributed to the reduction in thermal resistance as a result of increasing in effective thermal conductivity of metal foam. Note that in this paper, the results which are related to the metal foam diameter are referred to by their corresponding value of $s$.

With the same inlet hot water volume flow rate, the numerical simulation results for different metal foam diameters are plotted in Figure 6. This figure reveals that the effect of adding metal foam to the heat exchanger, where the outlet cold water temperature is at steady state, is increased with the increase in the metal foam diameter. As the thickness of inserted metal foam is increased, this tends to raise the heat transmissions from hot water in the tube side to cold water in the shell side. Consequently, this is a direct result of increasing $UA$ where a larger amount of heat energy is transformed from the hot water to the cold water.

4.2. Testing the SMC Performance

As mentioned above, the proposed barrier function-based SMC is first tested for the double-pipe heat exchanger using selected control design parameter values of $k_o=5u_{max}$, $\in =1$, and $\lambda =2$. The results in this part are obtained for the double heat exchanger with the addition of metal foam where $s=1.25$. Figure 7 shows the simulation results for three different, desired outlet cold water temperature where all the selected desired outlet cold temperatures are within the set $\Sigma$. It is observed that the control performance is well clarified where the outlet cold water converges to the desired temperature with a steady-state error less than ${\displaystyle \frac{ϵ}{\lambda }}=0.5 °\mathrm{C}$ (inequality Equation (69)). Also, it can be noticed from Figure 7 that the transient period for all required cold water temperature coincides. This is because the state is always initiated in ${\mathsf{\Omega }}_I$ and subsequently the control $u$ is $u_{max}$.

Also, in order to examine the control robustness simulation test, firstly it is demonstrated with different inlet cold and hot water temperatures, secondly through the different cold water inlet volume flow rate, and finally with different effective metal foam thermal conductivity. In Figure 8, the plot shows that the cold water converges to the desired temperature ($T_{cor}=45$) and satisfies inequality Equation (71) irrespective of the change in the inlet cold water temperature.

Meanwhile, in Figure 9, the outlet cold water temperature is plotted when the inlet cold water flow rate is varied ($\pm 10\%$ of the nominal cold water volume flow rate ${\dot{V}}_c=2.5096\ast {10}^{-6}{\mathrm{m}}^3/\mathrm{s}$) in order to accommodate the change in the cold-water volume flow rate due to metal foam insertion. Again, the proposed controller forces cold water to follow the desired temperature with a small variation in settling time.

For the case of different metal foam porosity (0.909, 0.937, and 0.971 [58]) and consequently different metal foam effective thermal conductivity, the proposed controller against this type of uncertainty is robust, as revealed in Figure 10.

For a stepwise desired outlet cold temperature, Figure 11 demonstrates the ability and performance of the proposed controller where the outlet cold water temperature follows the desired reference with the specified accuracy. Later, it is shown that by inserting metal foam, the settling time is reduced and consequently, the outlet cold water follows the required temperature quickly.

Finally, for the $45 °\mathrm{C}$ as a reference outlet cold water temperature, the hot water volume flow rate (the control input) is continuous, as can be seen from Figure 12 below where this result is proved in Corollary 1.

4.3. Testing the Enhancement of Adding Porous Material to the Heat Exchanger

In this part, the simulation results can answer the question about the evidence of adding porous material to the double-pipe heat exchanger with respect of the heat exchanger control system performance. Two factors are considered in evaluating the performance enhancement. The first factor is the settling time which is the time required to enter the positively invariant set $\mathsf{\Omega }$ that is a set where $\left|T_{co}\left(t\right)-T_{cor}\left(t\right)\right|\le {\displaystyle \frac{ϵ}{\lambda }}$. Meanwhile, the second factor is the hot water volume flow during a time span including the transient period and certain time interval throughout steady-state condition.

In Figure 13, the cold-water temperature state is plotted for different metal foam diameters. This figure clarifies that the settling time is reduced significantly with the increase in the metal foam diameter. The settling time is also shown in Figure 14 for different foam diameter where for $s=1$, the settling time is $10$,$800$ $\mathrm{s}$ while for $s=1.2$, the settling time is $4870 \mathrm{s}$ which means that the cold water with $s=1.2$ reaches the required temperature less than half the time spend for the case without metal foam. Additionally, when $s=1.8$, the settling time reduces greater than $20$ times the case of without adding metal foam ($s=1$), which proves the significance of adding the metal foam to the heat exchanger.

The reduction in the hot water volume (the second factor) is also shown in Figure 14. The addition of metal foam not only reduces significantly the settling time but also the required hot water volume, where it is measured here for a time span of 15,000 s. The settling time for different metal foam diameter ratio $s$ is replotted in Figure 15 where the significance of the addition of metal foam on the heat exchanger control system response is clarified. Finally, the performance enhancement via adding metal foam for a stepwise required temperature is clarified in Figure 16. The enhancement is clear when the result in this figure with $s=1.8$ is compared with those in Figure 11 where $s=1.25$.

Additionally, the effect of adding metal foam for various diameters on the sliding variable is shown in Figure 16, where the reaching time is decreased significantly. After that, the cold-water temperature asymptotically reaches the desired value during sliding motion according to the sliding variable Equation (49) with a time constant equal to $1/2$ as stated previously. As a result, the settling time which consists of reaching time and time spent during sliding phase is reduced and is illustrated in Figure 13.

An important contribution of adding metal foam, where large hot water volume flow rate reduction is obtained with increasing metal foam ratio, is proved in Figure 17. From this figure, it can be deduced that during the reaching phase, the hot water volume flow rate is constant and equal $u_{max}$ but after that it is reduced during sliding motion phase. The reaching time is decreased with respect of the metal foam diameter; therefore, the period of time during which $u=u_{max}$ is reduced and consequently less control effort is required. Moreover, this figure shows that the steady-state hot water volume flow rate is also reduced with increasing metal foam diameter. In context to the reduction in steady-state hot water volume flow rate, the steady-state outlet hot water temperature is reduced with rising metal foam diameter as depicted in Figure 18. For steady-state condition and from Figure 17 and Figure 18, both outlet hot water temperature and volume flow rate values are decreased with metal foam diameter increasing. Basically, according to this result, with both the outlet cold water temperature and volume flow rate values fixed, the thermal steady outlet temperature is satisfied and specifically Equation (32) is held. Accordingly, sustainable energy and consuming water requirements are attained as volume and temperature of hot water are decreased, which are a result of utilizing the current proposed control system and adding metal foam.

Finally, the performance enhancement via adding metal foam for a stepwise required temperature is clarified in Figure 19. The enhancement is clear when the result in this figure with $s=1.8$ is compared with those in Figure 11 where $s=1.25$.

5. Conclusions

The present work addresses the performance enhancement for the double-pipe heat exchanger results from adding the metal foam to the outer diameter. The thermal conductance $UA$ has been derived where Equation (19) showed that its value is increased with increasing metal foam diameter. The simulation results demonstrated the effect of the increase in in the thermal conductance $UA$, where it can be seen that the heat transfer from hot to cold water increased as in Figure 6. In order to study the effect of adding the metal foam, the adaptive and robust SMC has been designed to enforce the cold water to follow a certain desired value with the hot water volume flow rate play the role of the control input. The control stability and performance has been proved analytically through the use of Lyapunov function and also the simulation results reveal this fact. By using the designed controller, the simulation results reveal that a significant reduction occurred in the settling time and amount of hot water volume flow. Also, these results demonstrate that the enhancement in heat exchanger performance is achieved by inserting metal foam in addition to controlling the outlet cold water temperature. For $s=1.8$, it was found that the reduction in settling time reaches 1/20 times the case of non-adding metal foam ($s=1$). Finally, from these results, the insertion of the metal foam to the double-pipe heat exchanger control system is recommended.