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When viscous dissipation is included, Equation \(8.48\) (multiplied by \(\rho c_{p}\) ) becomes $$ \rho c_{p} u \frac{\partial T}{\partial x}=\frac{k}{r} \frac{\partial}{\partial r}\left(r \frac{\partial T}{\partial r}\right)+\mu\left(\frac{d u}{d r}\right)^{2} $$ This problem explores the importance of viscous dissipation. The conditions under consideration are laminar, fully developed flow in a circular pipe, with \(u\) given by Equation 8.15. (a) By integrating the left-hand side over a section of a pipe of length \(L\) and radius \(r_{o}\), show that this term yields the right-hand side of Equation 8.34. (b) Integrate the viscous dissipation term over the same volume. (c) Find the temperature rise caused by viscous dissipation by equating the two terms calculated above. Use the same conditions as in Problem 8.9.

Step by step solution

01

Include laminar, fully developed flow velocity slope distribution

The velocity slope distribution in a circular pipe for laminar, fully developed flow is given by: \(u(r) = \frac{P}{4\mu L}(r_o^2 - r^2)\)

02

Multiply the given Equation with 蟻c_p

$$ \rho c_{p} u \frac{\partial T}{\partial x}=\frac{k}{r} \frac{\partial}{\partial r}\left(r \frac{\partial T}{\partial r}\right)+\mu\left(\frac{d u}{d r}\right)^{2} $$

03

Integrate the left-hand side over the section of a pipe

Integrating the left-hand side over the length L of the pipe and radius r_o: \(\int_{0}^{L} \int_{0}^{r_o} \rho c_{p} u \frac{\partial T}{\partial x} \, r dr dx\)

04

Apply substitution and evaluate the integral

Using the velocity distribution u(r) from step 1 and noting that \(\frac{\partial T}{\partial x} = const.\), the integral becomes: \(\rho c_{p} \frac{\partial T}{\partial x} \int_{0}^{L} \int_{0}^{r_o} r \left[\frac{P}{4\mu L}(r_o^2 - r^2)\right] dr dx\) Solve the integral to obtain: \(\rho c_{p} \frac{\partial T}{\partial x} = \frac{\rho c_{p} Pr_o^2}{8\mu}\), which matches the right-hand side of Equation 8.34. (b)

05

Integrate the viscous dissipation term

The viscous dissipation term is given by: \(\mu\left(\frac{d u}{d r}\right)^{2}\), Integrating this term over the same volume, we get: \(\int_{0}^{L} \int_{0}^{r_o} \mu\left(\frac{d u}{d r}\right)^{2} r dr dx\)

06

Substitute the velocity derivative

Differentiating the given velocity distribution with respect to r and substituting it into the integral for viscous dissipation: \(\int_{0}^{L} \int_{0}^{r_o} \mu \left[-\frac{P}{2\mu L} (r - r_{o})\right]^2 r dr dx\)

07

Evaluate the integral

Solve the integral to obtain: \(\frac{P^2 r_o^4}{32\mu L^2}\) (c)

08

Equate the two terms from parts (a) and (b)

To find the temperature rise caused by viscous dissipation, we equate the two terms from parts (a) and (b): \(\rho c_{p} \frac{\partial T}{\partial x} = \frac{P^2 r_o^4}{32\mu L^2}\)

09

Determine the temperature rise

Now, we use the conditions from Problem 8.9 to find the temperature rise: Water properties at 288K: \(\rho = 1000 kg/m^3\), \(c_p = 4190 J/kg^K\), \(\mu = 8.9 * 10^{-4} Pa.s\), and given values: \(P = 1000 Pa\), \(r_o = 0.01 m\), \(L = 1 m\) Substituting these values into Step 8's equation, we can solve for the temperature rise: \(\Delta T = \frac{P^2 r_o^4}{32\mu L^2 \rho c_p}\) \(\Delta T \approx 0.0033 K\) The temperature rise caused by viscous dissipation for the given conditions is approximately 0.0033 K.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Laminar Flow

Laminar flow refers to a type of fluid movement in which the fluid travels in smooth, parallel layers without mixing between layers. This kind of flow often occurs at lower velocities and is characterized by high predictability and orderliness. In laminar flow through a circular pipe, the velocity distribution is parabolic. Near the pipe walls, the fluid moves more slowly, while it speeds up towards the center. The law of conservation of momentum enables us to understand and predict these fluid pathways.

Some key characteristics of laminar flow include:

• Low Reynolds number, typically less than 2000.
• Streamlines that are parallel and undisturbed.
• Each layer of fluid slides past adjacent layers with minimal mixing.
• Predictable flow patterns useful for precise calculations like those in heat and mass transfer.

This is contrasted with turbulent flow, where mixing between layers creates chaotic fluid motion.

Circular Pipe

When discussing fluid flow, the geometry of the conduit is crucial. A circular pipe is one of the most common forms through which fluids are conveyed. With a cylindrical shape, the circular pipe offers certain mathematical conveniences, particularly when solving for flow velocity and heat transfer.

For laminar flow in such a pipe, the central axis tends to have the highest fluid velocity, decreasing towards the walls due to viscous resistance. This setup directly contributes to the parabolic velocity distribution formula specific to fully developed laminar flow:\(u(r) = \frac{P}{4\mu L}(r_o^2 - r^2)\).

Understanding flow in these pipes allows engineers and scientists to:

• Optimize systems for transportation of liquids or gases.
• Design efficient heating and cooling systems by harnessing the predictable nature of laminar flow.
• Calculate pressure drops and energy losses due to viscous effects more accurately.

Temperature Rise

Temperature rise in a flowing fluid is a phenomenon resulting from energy dissipation, particularly due to viscous effects. In a laminar flow setting, energy from mechanical input can convert to internal energy, leading to a slight increase in temperature. This process is captured by the term called viscous dissipation, a key focus of thermofluid dynamics.

The temperature increase is influenced by:

• The flow velocity and the specific heat capacity (\(c_p\)) of the fluid.
• The pipe material and its thermal conductivity.
• Energy input from external forces (such as pumps or pressure gradients).

Using the balance of energy equation, it鈥檚 possible to compute this rise mathematically. Under precise conditions, like those set in controlled environments, this increase might be small but crucial for evaluating thermal management systems.

Heat Transfer Equation

The heat transfer equation is central to calculating energy distribution in fluid flow. When viscous dissipation is considered, the heat transfer equation involves an additional term, representing heat generated internally within the fluid. This is particularly important in scenarios of laminar flow within circular pipes, where the flow conditions allow for clear evaluations.

The key components of this equation encompass:

• Convective heat transfer resulting from bulk fluid motion.
• Conductive heat transfer within the fluid and through the pipe wall.
• Viscous dissipation, described by \(\mu\left(\frac{du}{dr}\right)^{2}\).

Through integration over spatial parameters of the system (pipe length and radius), energy conservation principles help deduce and quantify temperature changes within the medium. This comprehensive approach is invaluable in thermal analysis and engineering applications like heat exchangers.