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value oft and is given by solving following differentialequation
*3
*21 a St
St+ 0.083——(— ) = 6 (33)
Pr ax /'* •
X
x
(t=0atx=0 )
It is difficult to express the solution of Eq.(33) in a analytical form.
However it can be approximated by a simple analytical form by
*1.5
St ={ 1 — exp(-4.91 Pr x )}°•33(34)
In Fig.5, • a comparison is made between Eq.(34) and numerical solution of
Eq.(33) in St vs. Pro•33x* plot. As shown in this figure, Eq.(34) can
approximate the exact solution of Eq.(33) within 5 % errors.
On the other hand, for small time. the convection term in Eq.(32)
can be neglected and Eq.(32)can be rewritten by
St2+at(St2) = 6 (35)
(t ='0,at t = 0 )
The solution of this differential equation is easily obtained and given by
St =/{ 1 — exp(-t*)}(36)
In • view of Eqs.(34) and(36), the solution of the differential equation
(32) can be approximately given by
0.67
for -ln[1-{l-exp(-4.91P)}]< r xJ.1.5
t~
*1.5
St = { 1 — exp(-4.91 Pr x )}0;33(37)
for -ln[1-{1-exp(-4.91Prx" 5)}067] > t*
• St =/T1{ 1 - exp(-t*)}(38)
Then, from Egs.(20), (37) and (38), heat transfer coefficient is given by
122
, Eq.