Nu=φ(Re,Pr)

The forced convection heat transfer phenomenon can be influenced by the variables given in following table.

These seven variables are expressed in four primary dimensions (M,L,T,t) therefore, according to Buckingham pi theorem, the independent dimensionless group are:

= no. of variable affecting the phenomenon – No. of primary dimensions used.

=$7−4=3 (i.e. π_1,π_2,π_3)$

Writing these three group as,

$π_1=D^a ρ^b μ^c k_f^d u_∞$,

$π_2=D^e ρ^f μ^g k_f^h C_P$,

$π_3=D^i ρ^j μ^k k_f^l h$

Where D,,ρ,μ,$k_f$ from a core group(repeating variable) and $u_∞ C_p$ and h are as selected variable.

Since the groups $π_(1,) π_2,π_3$ are dimensionless hence certain exponents are applied on the repeating variable, which are to be determined,

Expressing the variable in their primary dimensions for $π_(1,)$

$π_1$=$L^a (ML^(-3) )^b (ML^(-1) t^(-1) )^c (MLt^(-3) T^(-1) )^d (Lt^(-1) )$

$M^0 L^0 T^0 t^0=M^(b+c+d) L^(a-3b-c+d+1) T^(-d) t^(-c-3d-1)$

Separating the exponents for dimensional homogeneity

M=b+c+d=0

L=a-3b-c+d+1=0

T=-d=0

t=-c-3d-1=0

Solving these simultaneous equations, we get

D=0,c=-1,b=1,a=1

Hence the dimensionless group is formed is

$π_1$=$\frac{(Dρu_∞)}{μ}=$$Re_D$ (Reynolds number) Expressing the primary dimension for variables of $π_2$, $π_2$=$L^e (ML^(-3) )^f (ML^(-1) t^(-1) )^g (MLt^(-3) T^(-1) )^h (L^2 t^(-2) T^(-1) )^1$

Separating the exponents for dimensional homogeneity.

M:0=f+g+h

L:0=e-3f-g+h+2

T:0=-h-1,

t:0=-g-3h-2

Solving these simultaneous equations, we get

H=-1,g=1,f=0,e=0

Hence the dimensionless group formed is,

$π_2$=$\frac{(μC_P)}{k_f}$ =Pr⁡(Prandtl Number)

Expressing the primary dimension for variables for $π_3$, $π_3$=$L^i (ML^(-3) )^j (ML^(-1) t^(-1) )^k (MLt^(-3) T^(-1) )^l (Mt^(-3) T^(-1) )$

M:0=j+k+l+1,

L:0=i-3j-k+l,

T:0=-l-1,

t: 0=-k-3l-3

Solving these simultaneous equations, we get

L=−1,k=0,j=0,i=1

Hence the dimensionless group formed is,

$π_3$=$\frac{hD}{k_f}$ =$Nu_D$ (Nusselt number)

Hence for forced convection,

$Nu_D$=φ($Re_D,Pr)$

hence proved.