# Rational Transfer Function Model for a Double-Pipe Parallel-Flow Heat Exchanger

## Abstract

**:**

## 1. Introduction

## 2. Distributed Parameter Model of the Heat Exchanger

#### 2.1. Governing PDEs

- exchanger is perfectly insulated from the environment;
- there are no internal thermal energy sources inside;
- the flows are sufficiently turbulent to cause effective heat transfer;
- only forced heat convection is considered (i.e., longitudinal heat conduction within the fluids and wall is neglected);
- pressure drops of fluids along the shell and the tube are negligible;
- the densities and heat capacities of the shell, tube and fluids are time and space invariant;
- the convective heat transfer coefficients are constant and uniform over each surface.

#### 2.2. Distributed Transfer Function Model

#### 2.3. Frequency-Domain Responses

#### 2.4. Steady-State Temperature Distribution

## 3. Approximate Model of the Heat Exchanger

#### 3.1. Mol Approximation

- Each nth section represents a third-order dynamical subsystem with the following vector of the state variables:$${\vartheta}_{n}\left(t\right)={\left[\begin{array}{ccc}{\vartheta}_{t,n}\left(t\right)& {\vartheta}_{w,n}\left(t\right)& {\vartheta}_{s,n}\left(t\right)\end{array}\right]}^{T},$$
- The output vector of each nth section is made of first and third components of its state vector in (34),$${\vartheta}_{o,n}\left(t\right)={\left[\begin{array}{cc}{\vartheta}_{to,n}\left(t\right)& {\vartheta}_{so,n}\left(t\right)\end{array}\right]}^{T}={\left[\begin{array}{cc}{\vartheta}_{t,n}\left(t\right)& {\vartheta}_{s,n}\left(t\right)\end{array}\right]}^{T}.$$
- Outputs of the last (Nth) section can be seen as approximations of the boundary outputs (7) of the heat exchanger system,$${\vartheta}_{o,N}\left(t\right)={\left[\begin{array}{cc}{\vartheta}_{t,N}\left(t\right)& {\vartheta}_{s,N}\left(t\right)\end{array}\right]}^{T}={\widehat{\vartheta}}_{o}\left(t\right)={\left[\begin{array}{cc}{\widehat{\vartheta}}_{to}\left(t\right)& {\widehat{\vartheta}}_{so}\left(t\right)\end{array}\right]}^{T},$$
- Inputs to the nth section, for $n=2,3,...,N$, are “connected” to the corresponding outputs of the section $n-1$,$${\vartheta}_{i,n}\left(t\right)={\left[\begin{array}{cc}{\vartheta}_{ti,n}\left(t\right)& {\vartheta}_{si,n}\left(t\right)\end{array}\right]}^{T}={\vartheta}_{o,n-1}\left(t\right)={\left[\begin{array}{cc}{\vartheta}_{to,n-1}\left(t\right)& {\vartheta}_{so,n-1}\left(t\right)\end{array}\right]}^{T}.$$
- Inputs to the first section ($n=1$) are given directly by the boundary inputs (6) to the heat exchanger system,$${\vartheta}_{i,1}\left(t\right)={\left[\begin{array}{cc}{\vartheta}_{ti,1}\left(t\right)& {\vartheta}_{si,1}\left(t\right)\end{array}\right]}^{T}={\vartheta}_{i}\left(t\right)={\left[\begin{array}{cc}{\vartheta}_{ti}\left(t\right)& {\vartheta}_{si}\left(t\right)\end{array}\right]}^{T},$$

#### 3.2. Rational Transfer Functions

#### 3.2.1. Single Section

#### 3.2.2. N-Section Approximation Model

#### 3.3. Approximate Frequency-Domain and Steady-State Responses

## 4. Results and Discussion

#### 4.1. Frequency Responses

#### 4.2. Steady-State Temperature Distribution

## 5. Conclusions and Future Work

## Funding

## Conflicts of Interest

## Symbols and Abbreviations

Roman Alphabets | |

A | state matrix |

a | characteristic polynomial coefficient of A |

B | input matrix |

b | transfer function numerator polynomial coefficient |

C | output matrix |

c | specific heat [J/(kg·K)] |

d | pipe diameter [m] |

i | imaginary unit |

g | transfer function, frequency response |

G | transfer function matrix, frequency response matrix |

h | heat transfer coefficient [W/(m^{2}·K)] |

k | constant parameter [1/s] |

L | heat exchanger length [m] |

${L}_{m}$ | logaritmic gain [dB] |

l | space variable [m] |

N | number of sections |

n | section number |

p | transfer function subexpression |

s | complex argument of Laplace transform |

t | time [s] |

v | fluid velocity [m/s] |

Greek Alphabets | |

$\alpha $ | first component of $\varphi $ |

$\beta $ | second component of $\varphi $ |

$\mathrm{\Delta}$ | difference |

$\vartheta $ | temperature [°C] |

$\rho $ | density [kg/m^{3}] |

$\tau $ | time delay [s] |

$\varphi $ | transfer function subexpression |

$\phi $ | phase shift [rad] |

$\omega $ | angular frequency [rad/s] |

Subscripts | |

i | inlet (temperature), inner (diameter) |

n | nth section |

o | outlet (temperature), outer (diameter) |

s | shell-side |

t | tube-side, time (Laplace transform) |

w | wall |

0 | initial (at t=0) |

Mathematical Signs | |

I | identity matrix |

$\mathrm{Im}\{.\}$ | imaginary part |

$\mathcal{L}\{.\}$ | Laplace transform |

$\mathbb{R}$ | set of real numbers |

$\mathrm{Re}\{.\}$ | real part |

$\overline{g}\left(l\right)$ | transfer function $g(l,s)$ at $s=0$ |

$\overline{\vartheta}\left(l\right)$ | steady state of $\vartheta (l,t)$ |

$\widehat{g}(l,s)$ | approximation of $g(l,s)$ |

$\widehat{\vartheta}(l,t)$ | approximation of $\vartheta (l,t)$ |

Acronyms | |

DPS | distributed parameter system |

LPS | lumped parameter system |

MOL | method of lines |

ODE | ordinary differential equation |

PDE | partial differential equation |

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**Figure 1.**Double-pipe heat exchanger laboratory setup at the Institute of Control Engineering, Opole University of Technology.

**Figure 2.**Schematic of a double-pipe heat exchanger. ${v}_{s},{v}_{t}$—shell-side and tube-side fluid velocities; ${\vartheta}_{s},{\vartheta}_{t}$—shell-side and tube-side fluid temperatures; ${\vartheta}_{w}$—wall temperature; ${\vartheta}_{si},{\vartheta}_{ti}$—shell-side and tube-side fluid inlet temperatures; ${\vartheta}_{so},{\vartheta}_{to}$—shell-side and tube-side fluid outlet temperatures; L—heat exchanger length; ${d}_{ti},{d}_{to}$—inner and outer diameters of the tube; ${d}_{si},{d}_{so}$—inner and outer diameters of the shell. Solid arrows show flow directions for the parallel-flow mode and dotted ones—for the counter-flow mode.

**Figure 3.**Block diagram of the distributed transfer function model for the double-pipe parallel-flow heat exchanger.

**Figure 4.**Block diagram of the approximation transfer function model for the double-pipe parallel-flow heat exchanger.

**Figure 5.**Frequency response ${g}_{ts}(L,i\omega )$ for the PDE-based model of the double-pipe parallel-flow heat exchanger vs. frequency responses ${\widehat{g}}_{ts}(L,i\omega )$ of its approximation models of different orders.

**Figure 6.**Frequency response ${g}_{tt}(L,i\omega )$ for the PDE-based model of the double-pipe parallel-flow heat exchanger vs. frequency responses ${\widehat{g}}_{tt}(L,i\omega )$ of its approximation models of different orders.

**Figure 7.**Steady-state temperature distribution for the PDE-based and approximation models of different orders for the parallel-flow heat exchanger with ${\overline{\vartheta}}_{ti}=100{\phantom{\rule{4pt}{0ex}}}^{\circ}\mathrm{C}$ and ${\overline{\vartheta}}_{si}=50{\phantom{\rule{4pt}{0ex}}}^{\circ}\mathrm{C}$.

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**MDPI and ACS Style**

Bartecki, K.
Rational Transfer Function Model for a Double-Pipe Parallel-Flow Heat Exchanger. *Symmetry* **2020**, *12*, 1212.
https://doi.org/10.3390/sym12081212

**AMA Style**

Bartecki K.
Rational Transfer Function Model for a Double-Pipe Parallel-Flow Heat Exchanger. *Symmetry*. 2020; 12(8):1212.
https://doi.org/10.3390/sym12081212

**Chicago/Turabian Style**

Bartecki, Krzysztof.
2020. "Rational Transfer Function Model for a Double-Pipe Parallel-Flow Heat Exchanger" *Symmetry* 12, no. 8: 1212.
https://doi.org/10.3390/sym12081212