Question

Consider a fluid with mean inlet temperature $T_i$ flowing through a tube of 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 ${\dot{q}}_s(x)=a+b\sin (x\pi /\mathrm{L})$, where $a$ and $b$ are constants. Determine an expression for the mean temperature of the fluid as a function of the $x$-coordinate.

Short answer

Answer: $T(x)=T_i+\frac{a}{\dot{m}{c}_p}x+\frac{bL}{\dot{m}{c}_p\pi }(1-\cos (\frac{x\pi }{L}))$

Step-by-step solution

Define the energy balance equation

: First, we need to consider an energy balance that takes into account the heat transfer from the tube to the fluid. The rate of heat transfer into the fluid can be expressed as the product of mass flow rate ($\dot{m}$), specific heat capacity of the fluid ($c_p$), and the temperature change of the fluid ($dT$). This equation is written as: $\dot{m}{c}_{p}dT={\dot{q}}_s(x)dx$

Substitute the given expression for surface heat flux

: Next, we substitute the given expression for surface heat flux (${\dot{q}}_s(x)=a+b\sin (\frac{x\pi }{L})$) into the energy balance equation: $\dot{m}{c}_{p}dT=(a+b\sin (\frac{x\pi }{L}))dx$

Integrate both sides of the equation

: Now, we will integrate both sides of the equation with respect to the respective variables ($dT$ and $x$). Since we want to find the mean temperature of the fluid as a function of the x-coordinate, we integrate the left side with respect to $dT$ (from $T_i$ to $T(x)$) and the right side with respect to $x$ (from $0$ to $x$): ${\int }_{T_i}^{T(x)}\dot{m}{c}_{p}dT={\int }_0^x(a+b\sin (\frac{x\pi }{L}))dx$

Perform the integration and solve for T(x)

: Perform the integration on both sides of the equation: $\dot{m}{c}_p(T(x)-T_i)=ax+\frac{bL}{\pi }(1-\cos (\frac{x\pi }{L}))$ Now we can isolate $T(x)$ on the left side of the equation: $T(x)=T_i+\frac{a}{\dot{m}{c}_p}x+\frac{bL}{\dot{m}{c}_p\pi }(1-\cos (\frac{x\pi }{L}))$

Provide the final expression for the mean temperature of the fluid

: The final expression for the mean temperature of the fluid as a function of the x-coordinate is as follows: $T(x)=T_i+\frac{a}{\dot{m}{c}_p}x+\frac{bL}{\dot{m}{c}_p\pi }(1-\cos (\frac{x\pi }{L}))$

Key concepts

Energy Balance Equation

Understanding the energy balance equation is essential in the study of heat transfer in fluid flow. In simple terms, this equation represents the principle of conservation of energy, where the heat added or removed from a system accounts for the change in internal energy of the fluid. For fluids flowing through a tube, as given in our problem, the equation translates to the rate of energy added to the fluid by heating being equal to the rate of increase in the fluid's enthalpy.

The general form of the energy balance for a flowing fluid can be written as $\dot{m}{c}_{p}dT={\dot{q}}_s(x)dx$, where $\dot{m}$ is the mass flow rate, $c_p$ is the specific heat capacity at constant pressure, and $dT$ represents an infinitesimal change in the fluid's temperature. Here, ${\dot{q}}_s(x)$ is the surface heat flux function, and $dx$ is an infinitesimal change in the position along the tube. This balance ensures that all heat transferred into the fluid is reflected as a change in the fluid's temperature.

Surface Heat Flux

Surface heat flux, denoted by ${\dot{q}}_s(x)$, is a key concept in studying heat transfer mechanisms involving fluid flow. It defines the amount of thermal energy passing through a surface per unit area per unit time. In our specific case, the fluid flowing through a tube is subjected to a surface heat flux that varies along the length of the tube.

The expression provided ${\dot{q}}_s(x)=a+b\sin (x\pi /\mathrm{L})$ indicates that the heat flux is not constant but rather varies sinusoidally with the position $x$ along the tube. This variable heat input can result in a complicated temperature profile within the fluid. When substituting this expression into the energy balance equation, we integrate it along the length of the tube to find out how it affects the temperature distribution in the fluid. The combination of constants $a$ and $b$ within the heat flux expression will shape this temperature profile.

Specific Heat Capacity

The specific heat capacity, $c_p$, has a fundamental role in thermal physics and heat transfer. It is the quantity of heat required to change the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin). In the context of our problem, the specific heat capacity of the fluid directly influences how much the fluid's temperature will rise for a given amount of heat added.

Furthermore, $c_p$ enables us to link the amount of heat energy transferred (due to surface heat flux) to the change in temperature of the fluid. A fluid with a high specific heat capacity, like water, will require more heat energy to achieve the same temperature change compared to a fluid with a lower specific heat capacity, like mercury. This property is significant in the integration process because it determines the rate of temperature change in the fluid as it absorbs heat from the tube's surface.

Integration in Heat Transfer

Integration is a mathematical tool that is fundamental in solving heat transfer problems, especially when dealing with varying conditions along a path, such as the length of a tube in our fluid flow scenario. The process of integration allows us to accumulate the effect of a variable heat flux along the tube's length, providing us with a total change in temperature from the inlet to any point $x$ within the tube.

In the given problem, we integrate the energy balance equation to transition from an infinitesimal change $dT$ to a finite change in temperature $T(x)-T_i$. The integral ${\int }_0^x(a+b\sin (\frac{x\pi }{L}))dx$ entails accumulating the effects of the sinusoidal heat flux over the segment from the tube's start (0) to any point $x$, providing a complete picture of the temperature distribution along the tube. This process is crucial for predicting the thermal behavior of the fluid, which has numerous practical applications in engineering and science.