Thermal-Hydraulics of Single Phase Flow in a Heated Channel

http://syeilendrapramuditya.wordpress.com

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


1 of 12


This document is freely downloadable in Microsoft Word 2003/XP format

The use of this document is ABSOLUTELY UNLIMITED

Some materials related to this document might be available here:

http://wp.me/p61TQ-w8

2 of 12



CONTENT

CONTENT...........................................................................................................................................3

LIST OF FIGURES..............................................................................................................................4

1. THE PROBLEM OF SINGLE HEATED CHANNEL................................................................5

1.1. Governing Equations............................................................................................................5

1.2. Solution Method...................................................................................................................6

1.3. Thermodynamic Equations of State for Liquid Sodium....................................................11

1.4. Computer Code...................................................................................................................12

3 of 12


LIST OF FIGURES

Figure 1. The problem of single heated channel..................................................................................5

Figure 2. Axial nodalization...............................................................................................................11

4 of 12


1. THE PROBLEM OF SINGLE HEATED CHANNEL

The system that we are going to analyzed is one the most basic thermal-hydraulics problems: the

case of single heated channel, in which a single phase fluid flows inside a heated pipe, as depicted

on the following figure:

Flu

id fl

ow

Hea

ted

Hea

ted

Figure 1. The problem of single heated channel

We are mainly interested in temperature and pressure of the fluid along the channel.

1.1. Governing Equations

To perform the analysis, we will use three balance equations in steady state form as described

below.

Mass balance:

Eq. 1

5 of 12


Momentum balance:

Eq. 2

Energy balance:

Eq. 3

The mass balance equation implies that the mass flux (and mass flow rate for constant area channel)

is constant under steady state condition. The energy equation will be solved for coolant temperature,

and the momentum equation will be solved for pressure.

1.2. Solution Method

Integrating Eq. 3:

Eq. 4

Applying trapezoid rule:

Eq. 5

6 of 12


Equation of state for is as follow:

Eq. 6

Where are some empirical constants.

Substituting Eq. 6 into Eq. 5:

Eq. 7

By rearranging Eq. 7, we get the following equation:

Eq. 8

Eq. 8 is a nonlinear equation that we need to solve to obtain coolant temperature. We will use the

Newton-Raphson technique to iteratively solve it.

Now let’s define as “Energy Function” for this case:

Eq. 9

Then the derivative of respect to is:

Eq. 10

7 of 12


The Newton-Raphson iteration is then performed as follow:

Eq. 11

We perform the iteration until the following condition is achieved:

Eq. 12

Where is a small number as a criterion of iteration convergence . We take the initial

guess for as simply the value of , which is already known.

Once we obtain coolant temperature at axial plane , we can obtain the following quantities:

Empirical correlation for dynamic viscosity:

Eq. 13

Reynolds number:

Eq. 14

Friction factor:

Eq. 15

Eq. 16

8 of 12


Note that for simplicity, we have ignored the transition region. We will need the above quantities to

solve the momentum equation as described below.

Now let’s move to the momentum equation. Integrating Eq. 2:

Eq. 17

Applying trapezoid rule:

Eq. 18

Equation of state for is as follow:

Eq. 19

Where are some empirical constants.

Then coolant densities for axial plane and are as follow, respectively:

Eq. 20

Eq. 21

By substituting Eq. 20 and Eq. 21 into Eq. 18, and rearranging the resulting equation, we will obtain

the following equation:

Eq. 22

9 of 12


Eq. 20 is a highly nonlinear equation, and we will use the Newton-Raphson method again to solve

it. Let’s define as “Momentum Function” for this case:

Eq. 23

Then the derivative of respect to is:

Eq. 24

The Newton-Raphson iteration is then performed as follow:

Eq. 25

We perform the iteration until the following condition is achieved:

Eq. 26

Where is a small number as a criterion of iteration convergence . We take the initial

guess for as simply the value of , which is already known.

10 of 12


The calculation is carried out sequentially from bottom to top of the pipe in a marching fashion. So

once the quantities at the inlet have been specified, their values at every axial position are

determined from the solution.

j = 1

j-1

j

j+1

j = n

Figure 2. Axial nodalization

To perform the calculation, first we need to specify the following quantities:

Coolant mass flux

Pipe diameter

Pressure and coolant temperature at inlet (cell )

Linear heat rate at every cell

Pipe length

Number of cell

Then the calculation is carried out start from to .

1.3. Thermodynamic Equations of State for Liquid Sodium

The following are some equations applicable to liquid sodium, one of the typical coolants for Fast

Breeder Reactors (FBRs).

Liquid density:

11 of 12


Eq. 27

Liquid specific heat capacity:

Eq. 28

Liquid dynamic viscosity:

Eq. 29

Saturation temperature:

Eq. 30

1.4. Computer Code

The method explained on this paper has been programmed in Fortran language, it is freely available

from the following URL: http://wp.me/p61TQ-w8

12 of 12


Thermal-Hydraulics of Single Phase Flow in a Heated Channel

Documents

Transcript of Thermal-Hydraulics of Single Phase Flow in a Heated Channel