In forced convective heat transfer, Stanton number (St), Nusselt number (Nu), Reynolds number (Re) and Prandtl number (Pr) are related as

The correct option is:

$$St=\frac{Nu}{RePr}$$

To understand why this relation holds, we can analyze the definitions of each of these dimensionless numbers used in forced convective heat transfer.

1. Nusselt Number (Nu):
The Nusselt number represents the ratio of convective heat transfer to conductive heat transfer across a boundary. It is defined as:
$$Nu=\frac{hL}{k}$$
where:
• $h$ is the convective heat transfer coefficient,
• $L$ is the characteristic length, and
• $k$ is the thermal conductivity of the fluid.

2. Reynolds Number (Re):
The Reynolds number represents the ratio of inertial forces to viscous forces in a fluid flow. It is defined as:
$$Re=\frac{\rho vL}{\mu }$$
where:
• $\rho$ is the fluid density,
• $v$ is the flow velocity, and
• $\mu$ is the dynamic viscosity of the fluid.

3. Prandtl Number (Pr):
The Prandtl number represents the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. It is defined as:
$$Pr=\frac{\mu C_p}{k}$$
where:
• $C_p$ is the specific heat capacity of the fluid at constant pressure.

4. Stanton Number (St):
The Stanton number measures the ratio of heat transferred into a fluid to the thermal capacity of the fluid. It is defined as:
$$St=\frac{h}{\rho v{C}_p}$$

Now, let us find the product of the Reynolds number and the Prandtl number ($Re·Pr$):
$$Re·Pr=\left(\frac{\rho vL}{\mu }\right)·\left(\frac{\mu C_p}{k}\right)$$
Simplifying by canceling the viscosity term ($\mu$) from the numerator and denominator:
$$Re·Pr=\frac{\rho vL{C}_p}{k}$$

Next, we divide the Nusselt number ($Nu$) by this product:
$$\frac{Nu}{Re·Pr}=\frac{\left({\displaystyle \frac{hL}{k}}\right)}{\left({\displaystyle \frac{\rho vL{C}_p}{k}}\right)}$$
By canceling out the common terms in the numerator and denominator ($L$ and $k$), we get:
$$\frac{Nu}{Re·Pr}=\frac{h}{\rho v{C}_p}$$

Since the right side is the exact definition of the Stanton number ($St$), we establish the relationship:
$$St=\frac{Nu}{RePr}$$