15 – Numerical evaluation of Nusselt´s number (Nu)

Nusselt’s number provide us a measure of heat transfer by convection on a surface, and it is defined as follows:
 

(15.1)

And :

Nu= Nusslet’s number;
h = convection coefficient;
L = length, and it has default value L = 1;
k= thermal conductivity coefficient of the fluid.

By the definition of Nu we see that we should evaluate it only on solid walls, therefore SinFlow considers as solid surfaces the faces of the domain, which have prescribed velocity equals to zero. This software also evaluates Nu on the surface of solid blocks.

Below we have a deduction to Nu local, i. e., evaluated on the surface of elemental volumes:
 
(15.2)
In which:
q”  =  flux per unit area at the interface;
hlocal  =  local convection coefficient;
Tsup  =  surface temperature given by the mean temperature of the neighbouring elemental volumes;
Tcarac =  characteristic temperature, which has as default value the mean temperature of the evaluated field, and it is calculated by the following expression:
 

(15.3)
It is important to remark that q” in (15.2) is the flux at the face of the elemental volume and it is evaluated multiplying q from (14.3) and (14.4) by the length of faces of the elemental volumes, i. e.:
 


(15.4)
 



(15.5)

Rearranging (15.2), it follows that:
 


(15.6)

And doing the properly substitutions in (15.1), we find:
 

(15.7)

Below we have a definition to average Nu (Num) over a determined surface:
 
(15.8)
In which:    
 

(15.9)
And    
 
(15.10)
 

(15.11)
Here,

hm = average convection coefficient at interface;
Tm = average temperature at interface;
q = total energy flux at interface;
S = total length of the interface;
DS = length of the face of the volume.
It is important to notice that prescribed fluxes on this section follow this signal convention:
  • Positive (+) - Entering in the physical domain or in a solid block;
  • Negative (-)  - Leaving the physical domain or a solid block.
 
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