Graphic-analytical calculation of Nusselt number

In the given calculation the general conditional formula is received.

The presented universal formula contains the power dependencies of the criteria Re, Pr, Gr. If one of the criteria in the calculation formula is absent (the influence of this criterion is not taken into account under the conditions under consideration), then its value is taken equal to one. Below is a table of exponent values that are available in various calculated dependencies to determine the number Nu and are used in the construction of a system of nomograms. For convenience, on the auxiliary nomogram, the exponent z is equal to the values of the exponents a and m, depending on which of the main graphs is being considered.

The system of nomograms for calculating the Nusselt number is built for the following ranges of criteria included in the formula:

Pr  - 0.05 ... 50

Re - 1 ... 2500

Gr - 0 ... 8000 

Nu - 0.4 ... 8,

I.e. the laminar mode of current is considered at low values of criteria Pr, Nu, Gr.

Systems of nomograms are broken on five variants on a complex C-ex B:

Design formulas for definition of Nusselt number for various conditions

1. Unrestricted motion of gas

 f  and  w - indexes, 

 f - index of a working bodyпараметр 

 w - index of environment

1.1 Heat irradiation in the not limited space

Horizontal pipes 10³ < (Gr_f * Pr_f) < 10⁸:

Nu_f = 0.50 * Gr_f^0.25 * Pr_f^0.5 / Pr_w^0.25,

Vertical surfaces (pipes, plates):

а)  10³ < (Gr_f * Pr_f) < 10⁹:

Nu_f = 0.76 * Gr_f^0.25 * Pr_f^0.5 / Pr_w^0.25,

б)  (Gr_f * Pr_f) > 10⁹ (turbulent flow):

Nu_f = 0.15 * Gr_f^0.33 * Pr_f^0.58 / Pr_w^0.25,

1.2 Heat dissipation in a confined space

In a limited space, the phenomena of heating and cooling of a liquid proceed close to each other and it is impossible to separate them; in this case, the whole process must be considered as a whole. Due to the limited space and the presence of ascending and descending flows, traffic conditions are greatly complicated here. They depend both on the shape and geometric dimensions of the space, and on the type of liquid or gas and the intensity of heat transfer.

2. Movement along a plate

2.1  Re_f > 4*10⁴ (turbulent flow):

- local heat irradiation:

Nu_f = 0.296 * Re_f^0.8 * Pr_f^0.68 / Pr_w^0.25,

- middle heat irradiation:

Nu_f = 0.037 * Re_f^0.8 * Pr_f^0.68 / Pr_w^0.25.

2.1  Re_f < 4*10⁴:

 Nu_f = 0.66 * Re_f^0.5 * Pr_f^0.68 / Pr_w^0.25.

3. Movement of gas in pipes

3.1 Laminar flow Re_f < 2·10³:

Nu_f = 0.15 * Re_f^0.33 * Pr_f^0.68 * Gr^0.1 / Pr_w^0.25,

w - the index determining criterion, concerning a wall of a pipe.

      Variant of the formula for air:

Nu_f = 0.13 * Re_f^0.33 * Gr^0.1 

3.2 Turbulent flow Re_f > 4*10⁴ и Pr = 0.6 ... 2500:

Nu_f = 0.021 * Re_f^0.8 * Pr_f^0.68  / Pr_w^0.25,

       Variant of the formula for air:

Nu_f = 0.018 * Re_f^0.8

4. Cross flow of pipes

4.1 Single pipes

- Re_f = 10 ... 10³:

Nu_f = 0.50 * Re_f^0.5 * Pr_f^0.58  / Pr_w^0.25,

variant of the formula for air:

Nu_f = 0.43 * Re_f^0.5.

-  Re_f = 10³ ... 2·10⁵:

Nu_f = 0.25 * Re_f^0.6 * Pr_f^0.58  / Pr_w^0.25,

variant of the formula for air:

Nu_f = 0.216 * Re_f^0.6.

Direction of current of a stream perpendicularly axes of a single pipe::

Nu_f = 0.43 + С * Re_fⁿ * Pr_f^m.

The values of the coefficient C and the exponent n depending on the Re_f criterion are given in the table

4.2 Tube bundle

Formulas are fair under condition -  Re_f =2·10² ... 2·10⁵_. 

Corridor system:

Nu_f = 0.23 * Re_f^0.65 * Pr_f^0.58  / Pr_w^0.25

variant of the formula for air:

Nu_f = 0.21 * Re_f^0.65.

Staggered tube bank:

Nu_f = 0.41 * Re_f^0.6 * Pr_f^0.58  / Pr_w^0.25,

variant of the formula for air:

Nu_f = 0.37 * Re_f^0.60.

5. Current of liquid metals

Free movement of heavy and alkaline metals, and also their alloys:

Nu_f = C * Gr_f^k * Pr_f^0.4.

For   Gr_f = 10² ... 10⁹ :   C = 0.52, k = 0.25.

For   Gr_f = 10⁹ ... 10¹³ :   C = 0.106, k = 0.33.

At the compelled turbulent movement:

Nu_f = 4,5 + 0.014 * Re_f^0.8 * Pr_f^0.8,

- the formula is fair at    Re_f = 10⁴ ... 10⁶;    Pr_f = 4 · 10^-3 ... 3.2 · 10^-2  и  L/d >30.

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