Question

A fluid is flowing in fully developed laminar conditions in a tube with diameter \(D\) and length \(L\) at a mass flow rate \(\dot{m}\). The tube is subjected to a surface heat flux that can be expressed as \(\dot{q}_{s}(x)=a \exp (-b x / 2)\), where \(a\) and \(b\) are constants. Determine an expression for the difference in mean temperature at the tube inlet and outlet.

Short answer

Question: Determine the expression for the difference in mean temperature at the tube inlet and outlet. Answer: \(\Delta T = -\frac{2 \pi a D}{b \dot{m} C_p} \left( \exp(-b L / 2) - 1 \right)\)

Step-by-step solution

Prepare the necessary equations for analysis

We know that energy conservation can be represented using the following equation: \(Q = \dot{m} \cdot C_p \cdot \Delta T\) where: - \(Q\) is the heat transfer into the fluid along the tube, - \(\dot{m}\) is the mass flow rate of the fluid, - \(C_p\) is the specific heat capacity of the fluid, and - \(\Delta T\) is the mean temperature difference between inlet and outlet. Since we know the surface heat flux, we can write a differential heat transfer equation, which is: \(dQ = \dot{q}_{s}(x) \cdot P \cdot dx\) where: - \(\dot{q}_{s}(x) = a \exp(-b x / 2)\) is the local surface heat flux in the tube, - \(P\) is the perimeter of the tube (\(P = \pi D\)), and - \(dx\) is an incremental length of the tube.

Integrate the differential heat transfer equation

Next, we'll integrate the differential heat transfer equation to find the total heat transfer (\(Q\)) as a function of tube length (\(x\)). \(Q = \int_{0}^{L} (\dot{q}_{s}(x) \cdot \pi D \cdot dx) = \int_{0}^{L} (a \exp(-b x / 2) \cdot \pi D \cdot dx)\) The integral can be solved as: \(Q = - \frac{2 \pi a D}{b} \left[ \exp(-b x / 2) \right]_0^L\) which simplifies to: \(Q = -\frac{2 \pi a D}{b} \left( \exp(-b L / 2) - 1 \right)\)

Find the expression for mean temperature difference

We can rearrange the energy conservation equation to find the expression for the mean temperature difference (\(\Delta T\)): \(\Delta T = \frac{Q}{\dot{m} \cdot C_p}\) Plugging the expression for \(Q\) from Step 2, we get: \(\Delta T = \frac{-\frac{2 \pi a D}{b} \left( \exp(-b L / 2) - 1 \right)}{\dot{m} \cdot C_p}\) Finally, we can simplify the expression for \(\Delta T\): \(\Delta T = -\frac{2 \pi a D}{b \dot{m} C_p} \left( \exp(-b L / 2) - 1 \right)\) This expression represents the mean temperature difference at the tube inlet and outlet as a function of the given parameters.

Key concepts

Energy Conservation Equation

Understanding the energy conservation equation is crucial when analyzing heat transfer in systems such as fluid flowing in a tube. It is essentially a statement of the first law of thermodynamics, which tells us that energy cannot be created or destroyed, but it can change forms. In the context of fluid flow and heat transfer, the energy conservation equation relates the thermal energy added or removed from a fluid to the change in the fluid's thermal state.

The basic form of the energy conservation equation is:

\(Q = \dot{m} \cdot C_p \cdot \Delta T\)

• \(Q\) represents the total heat added to or removed from the fluid,
• \(\dot{m}\) is the mass flow rate, the amount of mass passing through a given surface per time,
• \(C_p\) stands for the specific heat capacity, which indicates how much energy is required to raise the temperature of a unit mass of the substance by one degree,
• \(\Delta T\) is the mean temperature difference between two points, typically the inlet and outlet of the tube in our case.

This equation is fundamental for engineers and scientists because it allows them to calculate the amount of heat transfer based on measurable quantities like mass flow rate and temperature difference.

Surface Heat Flux

Surface heat flux refers to the rate at which heat is transferred per unit area of a surface. In the problem at hand, we're given a variable surface heat flux that changes along the length of the tube, which is mathematically expressed as \(\dot{q}_s(x) = a \exp(-bx/2)\), where \(a\) and \(b\) are constants.

Surface heat flux is an important concept because it helps quantify how much heat is being added to or removed from a surface at any given point. For instance, in our scenario, it provides a way to calculate the heat transfer rate from the tube's wall to the fluid inside it as a function of position along the tube's length. Here's how it ties into the differential form of the heat transfer equation:

\(dQ = \dot{q}_s(x) \cdot P \cdot dx\)

In this differential equation, \(P\) is the perimeter of the tube, which provides the contact area for heat flux, and \(dx\) is an increment along the tube's length. This microscopic view of heat transfer is integral to finding the total heat transfer over the entire length of the tube through integration.

Mean Temperature Difference

The mean temperature difference is a key term in the analysis of thermal systems and represents the average temperature variance between two points. For a fluid flowing through a tube, it's the difference between the fluid's temperature at the inlet and outlet. It's important to note, as highlighted in the exercise, that this is not a simple arithmetic average but a value that reflects the cumulative effect of the thermal energy transferred over the fluid's journey through the tube.

When dealing with variable heat flux, as in our example, determining the mean temperature difference becomes a bit more complex because the heat added to the fluid is not constant over the length of the tube. The heat transfer directly affects the temperature of the fluid and, consequently, its mean temperature. This change in temperature is captured by rearranging the energy conservation equation, which yields an expression for the mean temperature difference in terms of the heat transfer rate and the properties of the fluid.

Precisely, with our given data, the mean temperature difference helps quantify the thermal effect on the fluid by the tube walls.