Problem 13 - Basic Heat And Mass Transfer

Question

Lithium enters a parallel-plate duct of spacing \(2 b\), with a uniform velocity \(U\) and a uniform temperature \(T_{0}\). The walls are maintained at a uniform temperature \(T_{s}\). If laminar plug flow is assumed, determine the temperature distribution downstream of the entrance. Also, if \(\operatorname{Re}_{D_{h}}=1500\) and \(\operatorname{Pr}=0.03\), determine the Nusselt number variation along the duct. (Hint: There is an analogous conduction problem in Chapter 3.)

Step-by-Step Solution

Answer

The temperature distribution follows the analogous solution to heat conduction: \(\theta(x)\), and \(Nu(x)\) can vary, stabilizing to specific values for laminar flow.

Step 1: Understand the Physical Setup

We have a parallel-plate duct with distance between the plates being \(2b\). Lithium enters with uniform velocity \(U\) and temperature \(T_0\). The duct walls are at uniform temperature \(T_s\). We need to determine the temperature distribution assuming laminar plug flow.

Step 2: Identify Governing Equations

For laminar plug flow, the governing equation is the energy equation for fluid flow. The relevant energy balance leads to a simplified heat conduction equation since the flow is assumed plug and doesn't depend on transverse advection. This can often be approximated using solutions to the analogous 2D unsteady heat conduction in a wall.

Step 3: Use Analogous Heat Conduction Problem

Refer to the Chapter 3 conduction problem that can be treated as similar to the present case. The solution to temperature within a semi-infinite solid under constant wall temperature may have a suitable analogous form: \(\theta = \frac{T-T_s}{T_0-T_s}\). The form usually involves error functions or exponential decay.

Step 4: Simplify Silicone Heat Equation

To find the temperature profile, use the analogy for transient heat conduction in a semi-infinite solid: \(\theta(x, t) = \text{function of } x \text{ and } t\), embodied by dimensionless temperature using \(\theta = \frac{T-T_s}{T_0-T_s}\). Parameters like \(t\) (time in conduction view) here can be replaced by \(x/U\) for convection.

Step 5: Calculate Nusselt Number for Plug Flow

The Nusselt number \(Nu\) describes the ratio of convective to conductive heat transfer. In laminar flow, and especially for low Prandtl numbers like \(\Pr = 0.03\), heat transfer is dominated by conduction near walls. Traditional correlations can be used with specifics for plug flow: \(Nu(x) = \text{constant or function of } x\).

Step 6: Apply Given Parameters

Given \(\operatorname{Re}_{D_h} = 1500\) and \(\Pr = 0.03\). Use a typical correlation for Nusselt number in laminar flow such as: \(Nu = 4.36\) for fully developed laminar flow. For thermally developing flow, transition to the final value.

Step 7: Solve for Nusselt Number

Calculate \(Nu(x)\) from known analogous temperature distribution and introduce specific boundary conditions if the complete correlation is needed. At the entrance, the value is higher and approaches a flat value described by the thermal profile.

Key Concepts

Laminar Flow

Laminar flow refers to the orderly and smooth flow of a fluid, characterized by parallel layers with no disruption between them. Imagine a gentle river flowing calmly with each layer of water sliding past adjacent ones without mixing turbulently. This describes laminar flow.
In the context of heat transfer, especially in narrow channels like parallel-plate ducts, laminar flow is crucial because it affects how heat moves from the fluid to the walls. The flow itself can be in the form of a plug flow where fluid particles move at the same velocity across any cross-section.
Laminar flow tends to occur at lower velocities and is described by a lower Reynolds number, typically less than 2000. In the given exercise, with a Reynolds number of 1500, we are well within the laminar regime.

Convection

Convection is a mode of heat transfer that involves the movement of fluid. There are two types, namely natural and forced convection. In our scenario, forced convection occurs as the lithium is pumped through the duct, gaining heat energy from the warmer duct walls.
Forced convection effectively demonstrates how energy is transferred from one part of the fluid to another, thanks to the motion of the fluid itself. For a laminar flow in a duct, the convection process conveys heat primarily through a combination of conduction near the walls and the bulk movement of fluid.
Convection in contexts like our exercise is further characterized by certain dimensionless numbers that provide insight into the behavior of heat transfer, such as the Nusselt number, which helps quantify the efficiency of this convection process.

Nusselt Number

The Nusselt number (4Nu4) is a dimensionless parameter used to describe the efficiency of convective heat transfer relative to conductive heat transfer within a fluid. It essentially gauges how effective a fluid is at transferring heat compared to if it simply conducted heat on its own.
For fully developed laminar flow, typical correlations give constant values like 4.36. However, for thermally developing laminar flow - where the flow and temperature fields are still changing - the Nusselt number can vary along the duct depending on the position and the relation between convective transport and conductive heat transfer.
Determining this variation helps predict how well the lithium inside the duct can absorb or discard heat, crucial information for systems requiring precise temperature control.

Energy Equation

The energy equation in fluid mechanics, specifically for laminar flow and heat transfer, describes how energy is conserved within a system. It links the transport of heat with the movement of fluid, balancing the input energy with the energy change within the system.
In the context of the exercise, the energy equation mirrors a heat conduction equation. This is because in the narrow duct flow, lateral mixing is limited, and axial conduction becomes a more prominent factor.
The analogy to a heat conduction problem allows us to simplify and solve for temperature distributions by comparing fluid flow to heat transfer in solid media. In our case, it involves semi-infinite solids, where solutions often use functions like error functions or exponential decay based on boundary conditions. This knowledge is instrumental in predicting temperature variations along the length of the duct.