**1. Introduction**

52 Heat Exchangers – Basics Design Applications

Yuan, P., and Kou, H.-S., 2001, Entropy Generation on a Three-Gas Crossflow Heat

Yuan, P., and Kou, H.-S., 2003, Entropy Generation on a Crossflow Heat Exchanger

Zhang, L., Yang, C., and Zhou, J., 2010, A Distributed Parameter Model and Its Application

Generation, International Journal of Thermal Sciences, 49 (8), pp. 1427-1436. Zhou, J.-h., Yang, C.-x., and Zhang, L.-n., 2009, Minimizing the Entropy Generation Rate of

Optimization, Applied Thermal Engineering, 29 (8-9), pp. 1872-1879. Zubair, S. M., Kadaba, P. V., and Evans, R. B., 1987, Second-Law-Based Thermoeconomic

Heat and Mass Transfer 28 (6), pp. 803-813.

(6), pp. 619-638.

Exchanger with Longitudinal Wall Conduction, International Communications in

Including Three Gas Streams with Different Arrangements and the Effect of Longitudinal Wall Conduction, Numerical Heat Transfer; Part A: Applications, 43

in Optimizing the Plate-Fin Heat Exchanger Based on the Minimum Entropy

the Plate-Finned Heat Sinks Using Computational Fluid Dynamics and Combined

Optimization of Two-Phase Heat Exchangers, ASME Journal of Heat Transfer, 109 (2), pp. 287-294. Also published as Zubair, S. M., Kadaba, P. V., and Evans, R. B., 1985, Design and Optimization of Two-Phase Heat Exchangers, Two-Phase Heat Exchanger Symposium, ed. Pearson, J. T. and Kitto, Jr., J. B., ASME, HTD, 44.

Two-fluid heat exchangers are widely used in almost every energy process such as those in power plants, gas turbines, air-conditioning systems, numerous chemical plants and home appliances. Every change of steady state or starting of a plant causes changes in the system which can considerably affect not only the observed process but also the safety of the plant's operations. In all above cases, it is important to know the dynamic behavior of a heat exchanger in order to choose the most suitable design, controls and operations. The traditional design based on stationary approach has become inadequate and nowadays, more attention is devoted to the analysis of the heat exchanger's dynamic behavior and its design is adjusted to such conditions of work. Although the process control technology has made considerable headway, its practical application requires the knowledge of the dynamic behavior of both the plant's components and the plant as a whole.

Ever since Profos (Profos, 1943) showed the first dynamic model of a simple heat exchanger and Takahashi (Takahashi, 1951) published the first transfer functions for ordinary heat exchangers, there have been numerous studies of the heat exchanger's dynamic behavior. The historic overview of dynamic modeling is given in (Kays & London, 1984) and (Roetzel & Xuan, 1999) thus, the attention of this paper will be directed exclusively towards the review of some significant works in this area and works which this paper has been influenced by.

The paper (Liapis & McAvoy, 1981) defines the conditions for obtaining analytical solutions of transient phenomena in the class of problems associated with heat and mass transfers in counter flow fluid streams. Their solutions take into account forced flow and the dependence of transient coefficient on the fluid's flow and do not involve the effect of wall finite heat capacity. The exact solution of dynamic behavior of a parallel heat exchanger in which wall heat capacity is negligible in relation to the fluid capacity was shown in (Li, 1986). These solutions are valid for both finite and nonfinite flow velocities. The paper (Romie, 1985) shows responses of outlet fluid temperatures for the equation of a step fluid inlet temperature change in a counter flow heat exchanger. The responses are determined by means of a finite difference method and involve the wall effect. The exact analytical solution for transient phenomena of a parallel flow heat exchanger for unit step change of inlet temperature of one of the fluids is given in (Romie, 1986). Although this solution includes the wall effect, it is limited to heat exchangers with equal fluid velocities or heat exchangers

Analytical Solution of Dynamic Response of Heat Exchangers 55

By respecting above assumptions, the energy balance for parallel, counter and cross flow

On the basis of simplified assumptions and by applying energy equations to both fluids and the wall, one can obtain three simultaneous partial differential equations in the coordinate system as shown in Fig. 1. It is obvious that both fluids flow in the same direction but on different sides of the heat exchanger's separating wall. Heat transfer areas and heat transfer

Differential equations describing fluid-temperature fields in the heat exchanger core are statements of "micro" energy balances for an arbitrary differential control volume of that

> 1 1 2 1 ( )( )( )( ) *<sup>w</sup> w w <sup>w</sup> <sup>w</sup> <sup>T</sup> <sup>M</sup> c hA T T hA T T*

1 1 1 1 1 1 1

2 2 2 2 2 2 2

<sup>1</sup> ( )( ) *p w T T mc L hA T T XU t* 

<sup>1</sup> ( )( ) *p w T T mc L hA T T XU t* 

Due to simplified standard assumptions underlying the theory, the mathematical model is

To define mathematical problem completely, inlet and initial conditions have to be prescribed:

*w r*

*T X T X T X T const*

0

(1)

(2)

coefficients from both sides are known. The length of the heat exchanger is L.

Fig. 1. Schematic Description of Parallel Flow Heat Exchanger.

particular core. The following set of partial differential equations:

*t*

represents the energy balance over the control volume shown in Fig. 1.

1 \*

1 2

(0, ) .

*T t T const*

*r*

*r*

*T for t T t*

(0, ) <sup>0</sup>

*T for t*

 

( ,0) ( ,0) ( ,0)

linear and tractable by available methods of calculus.

2

heat exchangers will be mathematically formulated.

**2.1 Parallel flow** 

in which both fluids are gases. The paper (Gvozdenac, 1987) shows analytical solution for transient response of parallel and counter flow heat exchangers. However, these solutions are limited to the case in which heat capacities of both fluids are negligibly small in relation to the heat exchanger's separating wall capacity. Moreover, it is important to mention that papers (Romie, 1983), (Gvozdenac, 1986), (Spiga & Spiga, 1987) and (Spiga & Spiga, 1988) deal with two-dimensional problems of transition for cross flow heat exchangers with both fluids unmixed throughout. The last paper is the most general one and provides opportunities for calculating transient temperatures of wall temperatures and of both fluids by an analytical method for finite flow velocities and finite wall capacity. The paper (Gvozdenac, 1990) shows analytical solution of transient response of the parallel heat exchanger with finite heat capacity of the wall. The procedure presented in the above paper is also used for resolving dynamic response of the cross flow heat exchanger with the finite wall capacity (Gvozdenac, 1991).

A very important book is that of Roetzel W and Xuan Y (Roetzel & Xuan, 1999) which provides detailed analysis of all important aspects of the heat exchanger's dynamic behavior in general. It also gives detailed overview and analysis of literature.

This paper shows solutions for energy functions which describe convective heat transfer between the wall of a heat exchanger and fluid streams of constant velocities. The analysis refers to parallel, counter and cross flow heat exchangers. Initial fluids and wall temperatures are equal but at the starting moment, there is unit step change of inlet temperature of one of the fluids. The presented model is valid for finite fluid velocities and finite heat capacity of the wall. The mathematical model is comprised of three linear partial differential equations which are resolved by manifold Laplace transforms. To a certain extent, this paper presents a synthesis of the author's pervious papers with some simplified and improved final solutions.

The availability of such analytical solutions enables engineers and designers a much better insight into the nature of heat transfers in parallel, counter and cross flow heat exchangers.

For the purpose of easier practical application of these solutions, the potential users are offered MS Excel program at the web address: www.peec.uns.ac.rs. This program is open and can be not only adjusted to special requirements but also modified.
