Problem 11 - Basic Heat And Mass Transfer

Question

Consider laminar flow of high-viscosity oil in a circular tube with a uniform wall temperature. If viscous heating is significant, determine the temperature profile a long distance from the inlet. Assume constant properties.

Step-by-Step Solution

Answer

The temperature profile, considering viscous heating, forms as a quartic function and depends on radial position, peaking at the tube wall.

Step 1: Understand the problem

We are analyzing the temperature distribution in a circular tube with laminar flow of high-viscosity oil, where the wall temperature is uniform and viscous heating is considerable. We aim to find the temperature profile a long distance from the inlet, under the assumption that the oil properties (like viscosity and thermal conductivity) remain constant.

Step 2: Apply Energy Balance

In laminar flow through a tube, the energy equation, considering viscous heating, can be expressed as: \[ \rho C_p u \frac{dT}{dx} = k \frac{d^2T}{dr^2} + \frac{\mu}{2} \left( \frac{du}{dr} \right)^2 \]where \( \rho \) is density, \( C_p \) is specific heat, \( u \) is fluid velocity, \( T \) is temperature, \( k \) is thermal conductivity, \( \mu \) is dynamic viscosity, \( r \) is radial position, and \( x \) is axial position.

Step 3: Simplify with Assumptions

Since the flow is fully developed far from the inlet, the axial temperature gradient term \( \frac{dT}{dx} \) tends to zero due to uniform wall temperature and constant properties. Hence, we focus on the radial direction and the equation simplifies to:\[ k \frac{d^2T}{dr^2} = -\frac{\mu}{2} \left( \frac{du}{dr} \right)^2 \]

Step 4: Relate to Velocity Profile

For laminar flow, the velocity profile is parabolic, described by:\[ u(r) = u_{max} \left( 1 - \left( \frac{r}{R} \right)^2 \right) \]where \( u_{max} \) is the maximum velocity at the center and \( R \) is the radius of the tube. The velocity gradient is:\[ \frac{du}{dr} = -\frac{2 u_{max}}{R^2}r \]

Step 5: Substitute Velocity Gradient

Substituting the velocity gradient into the energy equation gives:\[ k \frac{d^2T}{dr^2} = -\frac{\mu}{2} \left( -\frac{2 u_{max}}{R^2}r \right)^2 \]Simplifying this:\[ k \frac{d^2T}{dr^2} = \frac{2 \mu u_{max}^2}{R^4} r^2 \]

Step 6: Solve Differential Equation

The solution to this differential equation is a function that varies with \( r \). Integrating twice with respect to \( r \), we obtain:\[ T(r) = \frac{\mu u_{max}^2}{R^4 k} r^4 + C_1 r + C_2 \]Where \( C_1 \) and \( C_2 \) are constants determined by the boundary conditions.

Step 7: Apply Boundary Conditions

Assume no heat flux at the center, \( \frac{dT}{dr} \bigg|_{r=0} = 0 \), and a constant wall temperature \( T(r=R) = T_w \). From \( \frac{dT}{dr} = 0 \) at \( r = 0 \), it follows that \( C_1 = 0 \). For \( T(R) = T_w \), substitute and solve for \( C_2 \).

Step 8: Final Temperature Profile

With constants, the temperature profile is:\[ T(r) = \frac{\mu u_{max}^2}{R^4 k} r^4 + T_w - \frac{\mu u_{max}^2}{R^4 k} R^4 \]Simplifying, the radial temperature profile adjusts according to the viscous heating effect, peaking at the wall.

Key Concepts

Viscous Heating

In the context of fluid flow inside a tube, viscous heating occurs when the fluid's viscosity generates heat as the fluid layers slide past each other due to shearing. This is particularly relevant in high-viscosity fluids, such as oils, where the internal friction causes significant energy dissipation as heat.
This heat generation can alter the temperature field of the fluid, impacting the flow characteristics and performance. Viscous heating is considerable when the Reynolds number is low, indicating laminar flow. In contrast, turbulence dominates in higher Reynolds numbers mitigating viscous heating effects.

• Viscous heating is more pronounced in laminar flows of thick fluids.
• The generated heat can modify the overall energy balance within the system.

In high-viscosity oil flowing through a tube, the viscous shear contributes a non-negligible heat source that needs full consideration when analyzing temperature distributions.

Temperature Distribution

The temperature distribution in a fluid flow through a tube depends heavily on various factors such as fluid properties, flow conditions, and heat generation.
In laminar flows, temperature profiles can usually be derived by solving the energy equation. Viscous heating adds an extra layer of complexity by serving as an additional heat source that influences this profile significantly.
Far from the tube inlet, where the flow becomes fully developed, the temperature gradient along the axial direction diminishes. Consequently, the main variation in temperature happens radially, from the center towards the tube wall.

• The temperature is highest near the walls due to viscous heating.
• It decreases moving towards the center of the tube.

This gives a distinctive radial temperature pattern because viscous dissipation of energy becomes significant relative to convective heat transfer.

Energy Equation

The energy equation in fluid dynamics is fundamental for describing how temperature varies within a flowing fluid. In the case of laminar flow with significant viscous heating, the energy equation accounts for conductive heat transfer and internal heat generation due to viscosity.
Expressed as:\[\rho C_p u \frac{dT}{dx} = k \frac{d^2T}{dr^2} + \frac{\mu}{2} \left( \frac{du}{dr} \right)^2\]This equation illustrates three primary energy aspects:

• Convective heat transfer along the flow direction (axial).
• Heat conduction in the radial direction.
• Heat generated by fluid friction (viscous heating).

In a fully developed flow, the axial temperature changes (\[\frac{dT}{dx} = 0\]) are negligible, enabling simplifications. Only conduction and viscous heat generation dictate the temperature change across the tube radius, pivotal for finding the temperature profile.

Boundary Conditions

Boundary conditions are essential to solve differential equations and rely on physical constraints or conditions at the boundaries of the flow domain.
For temperature profiles in laminar flow inside a tube with viscous heating, the common boundary conditions are:

• No heat flux at the center of the tube (\[\frac{dT}{dr}\bigg|_{r=0} = 0\]).
• Uniform wall temperature, often assumed as a constant since the walls are usually maintained at a specific temperature (e.g., \[T(r=R) = T_w\]).

Applying these boundary conditions aids in finding the integration constants of the energy equation. The boundary conditions capture the physical constraints of the system, like symmetries and externally applied thermal conditions which are critical for accurately predicting the temperature distribution in the fluid.