Thermal Hydraulics of Single Phase Flow in Multi Channel Systems

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A Note on Thermal-Hydraulics*** Single Phase – Heated Multi-Channel Problem ***


September 2010

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ABSTRACT

This paper describes a numerical method applicable for multi-channel thermal-hydraulic problems.

Only single phase fluid is considered, the channels are considered laterally non-communicating,

they are coupled each other only through common plena located at the bottom and on top of the

channels. All three balance equations are coupled each other, this results in a set of nonlinear

equation system to be solved my matrix operation.

The method described here can be used as a first approximation to analyze fuel assembly and/or

nuclear reactor core thermal-hydraulic characteristics under steady state conditions.

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CONTENT

ABSTRACT.........................................................................................................................................3

CONTENT...........................................................................................................................................4

LIST OF FIGURES..............................................................................................................................5

1. MULTI-CHANNEL THERMAL-HYDRAULIC SYSTEM.......................................................6

1.1. Governing Equations............................................................................................................6

1.2. Solution Method...................................................................................................................7

1.3. The Multi-Variable Newton-Raphson Method....................................................................9

2. ACKNOWLEDGMENT............................................................................................................11

3. REFERENCES...........................................................................................................................11

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LIST OF FIGURES

Figure 1. Multi-channel thermal hydraulic system..............................................................................6

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1. MULTI-CHANNEL THERMAL-HYDRAULIC SYSTEM

Consider a multi-channel thermal hydraulic system, in which fluid passes through several axially-

parallel, laterally non-interacting channels, connected only at lower and upper plena, as depicted on

the following figure:

UPPER PLENUM

LOWER PLENUM

1m2m 3m

1q 2q 3q

Tm

Tm

0inlet

p

r

0outlet

p

r

r

z

Figure 1. Multi-channel thermal hydraulic system

Note that in general, the diameter of each channel is not necessarily the same each other, an also,

the heat could be either added or extracted along each channel.

1.1. Governing Equations

Mass balance for channel :

Eq. 1

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Momentum balance for channel :

Eq. 2

Energy balance for channel :

Eq. 3

1.2. Solution Method

First let’s take a look at the momentum equation. Integrating the momentum equation from channel

inlet to channel outlet and rearranging the terms:

Eq. 4

Eq. 5

Where and are the axially-averaged value of friction factor and density, respectively. They

are defined as follow:

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Eq. 6

Eq. 7

For discretized domain, the integral becomes summation:

Eq. 8

Where is any discrete variable. A numerical method to calculate and at every axial mesh

is described on reference [1].

For most engineering problems, it has been a common practice to assume equal pressure drop for all

channels, by doing this, we can drop the subscript from the pressure variables:

Eq. 9

Based on Eq. 9, now we define a “momentum function” which depends on 2 variables as follow:

Eq. 10

Note that for N channel system, we will have N momentum functions.

Now let’s move to the mass balance equation. For steady state condition, the total mass flow rate

is constant, hence, the mass balance equation can be written as follow:

Eq. 11

Eq. 12

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Based on Eq. 12, now we define “mass function”, which depends on 2 variables as follow:

Eq. 13

Note that for any number of channels, we will only have a single mass function.

By examining the momentum function and the mass function, we can see that for N channel system,

our equation system has a total of N+1 equations and also N+1 unknowns, means that we have a

solvable, closed equation system.

Example for 3 channels system

Unknowns:

Eq. 10 and Eq. 13 form a multi-variable nonlinear equation system, we will use the Newton-

Raphson method to solve this equation system.

1.3. The Multi-Variable Newton-Raphson Method

For a single-variable nonlinear problem, a function can be solved by Newton-Raphson

iteration as follow:

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Eq. 14

Eq. 15

Eq. 16

Eq. 17

The iteration is carried out until the residuals become sufficiently small, for , and

for .

The multi-variable version of Newton-Raphson iteration is as follow:

Eq. 18

Example for 3 channels system:

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The matrix system shown in Eq. 18 can be solved by inverting the Jacobian matrix :

Eq. 19

The iteration is then carried out until all elements of matrix and become sufficiently

small. By performing this iteration, finally we will obtain a consistent flow distribution for all

channels, such that the pressure drop will be uniform. Note that to perform the actual calculation,

the method described on this paper needs to be combined with the method of solving single channel

problem described on reference [1].

A computer program implementing the method described here has been written in FORTRAN

language, it is freely available from http://wp.me/p61TQ-zt.

2. ACKNOWLEDGMENT

The author would like to thank Mr. Marco Pellegrini of Research Laboratory for Nuclear Reactors,

Tokyo Institute of Technology, for the fruitful discussions on this study.

3. REFERENCES

1. Thermal Hydraulics of Single Phase Flow in a Heated Channel. http://wp.me/p61TQ-w8

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Thermal Hydraulics of Single Phase Flow in Multi Channel Systems

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