Question

An engineer is to design an experimental apparatus that consists of a \(25-\mathrm{mm}\)-diameter smooth tube, where different fluids at \(100^{\circ} \mathrm{C}\) are to flow through in fully developed laminar flow conditions. For hydrodynamically and thermally fully developed laminar flow of water, engine oil, and liquid mercury, determine \((a)\) the minimum tube length and \((b)\) the required pumping power to overcome the pressure loss in the tube at largest allowable flow rate.

Short answer

Answer: At a temperature of \(100^{\circ}\mathrm{C}\), the density and dynamic viscosity of water, engine oil, and liquid mercury are as follows: Water: - Density: \(\rho_{\text{Water}} = 958 \, \mathrm{kg/m^3}\) - Dynamic viscosity: \(\mu_{\text{Water}} = 2.82 \times 10^{-4} \, \mathrm{Pa \cdot s}\) Engine oil: - Density: \(\rho_{\text{Oil}} = 859 \, \mathrm{kg/m^3}\) - Dynamic viscosity: \(\mu_{\text{Oil}} = 7.17 \times 10^{-4} \, \mathrm{Pa \cdot s}\) Liquid mercury: - Density: \(\rho_{\text{Hg}} = 13534 \, \mathrm{kg/m^3}\) - Dynamic viscosity: \(\mu_{\text{Hg}} = 1.6 \times 10^{-3} \, \mathrm{Pa \cdot s}\)

Step-by-step solution

Determine fluid properties

For each fluid at a temperature of \(100^{\circ}\mathrm{C}\), we can find the density and dynamic viscosity values from the properties tables available in the literature. For water, engine oil, and liquid mercury, the values are as follows: Water: - Density: \(\rho_{\text{Water}} = 958 \, \mathrm{kg/m^3}\) - Dynamic viscosity: \(\mu_{\text{Water}} = 2.82 \times 10^{-4} \, \mathrm{Pa \cdot s}\) Engine oil: - Density: \(\rho_{\text{Oil}} = 859 \, \mathrm{kg/m^3}\) - Dynamic viscosity: \(\mu_{\text{Oil}} = 7.17 \times 10^{-4} \, \mathrm{Pa \cdot s}\) Liquid mercury: - Density: \(\rho_{\text{Hg}} = 13534 \, \mathrm{kg/m^3}\) - Dynamic viscosity: \(\mu_{\text{Hg}} = 1.6 \times 10^{-3} \, \mathrm{Pa \cdot s}\)

Confirm fully developed laminar flow

To confirm that the fluid flow is fully developed laminar, the Reynolds number should be less than 2000: $$\text{Re} = \frac{\rho vD}{\mu} < 2000$$ For each fluid, we will assume the largest allowable flow rate such that the above condition holds. Using the given tube diameter \(D=25\, \mathrm{mm}\), we can calculate the critical velocity for each fluid using the density and dynamic viscosity values we found in Step 1.

Determine the minimum tube length of each fluid

To determine the minimum tube length, first, we'll need to find the Nusselt number for each fluid. For laminar flow, it can be calculated using the following formula: $$\text{Nu} = 0.023 \; \text{Re}^{0.8}\, \text{Pr}^{n}$$ n = 0.4 for heating and 0.3 for cooling For this problem, let's assume the flow is cooled using a different analysis. Therefore, \(n\) will be 0.3. Next, we will calculate the Prandtl number, \(\text{Pr}\), for each fluid. Using the properties found in Step 1, we can calculate the specific heat at constant pressure, \(c_p\), for each fluid and thus finding the \(\text{Pr}\) using the formula: $$\text{Pr} = \frac{c_p\mu}{k}$$ Once we have the Prandtl number, we will determine the Nusselt number and find the ratios of entry lengths, \(L_{\text{entry}}\), for each fluid, and then find the minimum necessary value for each fluid.

Calculate required pumping power to overcome pressure loss

We will calculate the head loss for each fluid using the Hagen-Poiseuille equation in terms of pressure loss: $$\Delta P = \frac{128\mu QL}{\pi D^4}$$ Next, we will calculate the required pumping power, \(W\), and use the following formula for each fluid: $$W = \rho Q\Delta P$$ In conclusion, by following these steps and considering the properties of each fluid at the given temperature, we can determine the minimum tube length and required pumping power for fully developed laminar flow of water, engine oil, and liquid mercury.

Key concepts

Laminar Flow

In fluid dynamics, laminar flow describes a regime where fluid particles move in parallel paths, creating smooth and orderly layers. Unlike turbulent flow, where the fluid motion is chaotic, laminar flow is highly predictable and easy to model. It occurs at lower velocities and high fluid viscosity.

• Characteristics: Fluid layers slide past one another with little to no mixing and disturbance between them.
• Applications: Ideal for situations where precise, stable flow is needed, such as in chemical and biomedical engineering processes.

In the context of our problem, ensuring the flow remains laminar is crucial. It minimizes frictional losses in the tube, thereby reducing the effort needed to push the fluid through it. To ensure laminar flow, the velocity must be such that the Reynolds number remains below 2000.

Reynolds Number

The Reynolds number ( ext{Re}) is a dimensionless quantity used in fluid mechanics to determine whether fluid flow is laminar or turbulent. It is a ratio expressing the relative importance of inertial forces to viscous forces in the fluid's movement:
\[ ext{Re} = \frac{\rho vD}{\mu}\]

• \( \rho \) is the fluid density
• \( v \) is the velocity
• \( D \) is the characteristic length (e.g., diameter of the tube)
• \( \mu \) is the dynamic viscosity

For fully developed laminar flow in pipes, it is crucial that the Reynolds number is kept below 2000. This ensures that the flow maintains its orderly, laminar nature. In our problem, using given fluid properties, we can calculate the maximum velocity that keeps the flow laminar.

Hydrodynamic Entry Length

Hydrodynamic entry length refers to the distance a fluid must travel inside a pipe or tube before achieving a fully developed flow profile. Initially, as the fluid enters the pipe, it is influenced by the boundary and adjusts its velocity profile until it becomes steady.

• Importance: The entry length affects calculations for pressure drop and heat transfer. It's pivotal in designing pipe systems efficiently.
• Factors Influencing Length: Fluid properties and flow conditions such as Reynolds number.

In laminar flow, the hydrodynamic entry length can be estimated by:\[L_{entry} \approx 0.05 Re D\]By knowing this length, we can determine the minimum tube length required for maintaining consistent, fully developed flow.

Nusselt Number

The Nusselt number ( ext{Nu}) is a dimensionless quantity that indicates the enhancement of heat transfer through a fluid layer as a result of convection relative to conduction across the fluid layer. This ratio is crucial for evaluating heat transfer within fluids contained in tubes.
For our problem, the Nusselt number in laminar flows is determined using associated parameters through the equation:\[\text{Nu} = 0.023 Re^{0.8} Pr^n\]where \( Re \) is the Reynolds number, and \( Pr \) is the Prandtl number. The value of \( n \) differs based on cooling or heating conditions—here assumed to be 0.3 for cooling.

• Role: Helps in calculating the heat transfer coefficient, augmenting design of thermal systems.
• Calculation Consideration: Accounts for both velocity and thermal conductivities of the fluid and cylinder interaction.

Understanding the Nusselt number helps determine how effectively heat is transferred in the context of our exercise, providing insights into energy needed for fluid temperature control.