Question

The Sieder-Tate correlation for heat transfer in turbulent flow in a pipe gives \( \text{Nu} \propto \text{Re}^{0.8} \), where Nu is the Nusselt number and Re is the Reynolds number for the flow. Assuming that this relation is valid, the heat transfer coefficient varies with the pipe diameter (D) as:

• A. \( D^{-1.8} \)
• B. \( D^{-0.2} \)
• C. \( D^{0.2} \)
• D. \( D^{1.8} \)

Hint:

When evaluating proportionalities from dimensionless correlations, always verify the variable constraints: - At constant velocity (\( v \)): \( \text{Re} \propto D \rightarrow \text{Nu} \propto D^{0.8} \rightarrow \frac{h D}{k} \propto D^{0.8} \rightarrow h \propto D^{-0.2} \). - At constant mass flow rate (\( \dot{m} \)): \( \text{Re} \propto D^{-1} \rightarrow \text{Nu} \propto D^{-0.8} \rightarrow \frac{h D}{k} \propto D^{-0.8} \rightarrow h \propto D^{-1.8} \). Since Option (2) is highlighted as correct, the context implies constant velocity.

Answer 1

The Correct Option is B

Concept: The Sieder-Tate correlation is an empirical relationship used to determine the convective heat transfer coefficient (\( h \)) for turbulent fluid flows inside cylindrical channels. The standard definition framework uses dimensionless groups defined as:

• Nusselt number: \( \text{Nu} = \frac{h \cdot D}{k} \)

• Reynolds number: \( \text{Re} = \frac{\rho \cdot v \cdot D}{\mu} = \frac{4 \cdot \dot{m}}{\pi \cdot D \cdot \mu} \)
Where \( h \) is the convective coefficient, \( D \) is the inside pipe diameter, \( k \) is the thermal conductivity, \( \rho \) is the fluid density, \( v \) is the mean flow velocity, \( \mu \) is the dynamic fluid viscosity, and \( \dot{m} \) is the total mass flow rate.

Step 1: Rewriting the proportional dependencies using fundamental variables.
The problem statement provides the proportional relation from the Sieder-Tate correlation: \[ \text{Nu} \propto \text{Re}^{0.8} \] Let us substitute the explicit expressions for both the Nusselt number and the Reynolds number in terms of the pipe diameter \( D \). We evaluate this under the standard engineering constraint of a constant fluid mass flow rate (\( \dot{m} = \text{constant} \)) and uniform physical fluid properties (\( \rho, \mu, k \)): \[ \text{Nu} = \frac{h \cdot D}{k} \quad \Rightarrow \quad \text{Nu} \propto h \cdot D \] Now, let us examine the structural dependency of the Reynolds number on the internal diameter parameter \( D \) for a fixed mass flow rate condition: \[ \text{Re} = \frac{4 \cdot \dot{m}}{\pi \cdot D \cdot \mu} \quad \Rightarrow \quad \text{Re} \propto \frac{1}{D} = D^{-1} \]

Step 2: Combining the proportionalities to isolate the heat transfer coefficient.
We now substitute these proportional relations directly back into the core Sieder-Tate proportionality statement: \[ (h \cdot D) \propto \left( D^{-1} \right)^{0.8} \] Applying exponent rule properties to expand the right-hand expression: \[ h \cdot D \propto D^{-0.8} \] To isolate the local convective heat transfer coefficient parameter \( h \) on the left side of the relation, divide both sides by the pipe diameter variable \( D \) (which is equivalent to multiplying by \( D^{-1} \)): \[ h \propto \frac{D^{-0.8}}{D} \] \[ h \propto D^{-0.8} \cdot D^{-1} \] Combining the exponents according to standard algebraic rule systems: \[ h \propto D^{(-0.8 - 1)} \quad \Rightarrow \quad h \propto D^{-1.8} \] Wait, let us check if the assumption implies constant mass flow rate or constant velocity. Let us look at both engineering scenarios carefully to match the standard textbook question context. Alternative Assumption analysis: Constant flow velocity (\( v = \text{constant} \)).
If the baseline assumption instead implies a fixed velocity independent of diameter adjustments: \[ \text{Re} = \frac{\rho \cdot v \cdot D}{\mu} \quad \Rightarrow \quad \text{Re} \propto D^1 \] Substituting this alternative condition into the core relation: \[ h \cdot D \propto (D)^{0.8} \] Isolating the heat transfer coefficient variable: \[ h \propto \frac{D^{0.8}}{D^1} = D^{0.8 - 1} = D^{-0.2} \] Since option (2) matches \( D^{-0.2} \) perfectly and is marked as correct, the problem refers to the variation under conditions of constant linear flow velocity. Thus, we have: \[ h \propto D^{-0.2} \]