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 / 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

Question: Determine the mean temperature of the fluid as a function of the x-coordinate for the given problem. Answer: The mean temperature of the fluid as a function of x can be expressed as: \(T(x) = T_{i} + \frac{a}{\dot{m}C_{p}}x + \frac{bL}{\dot{m}C_{p}\pi}\left(1-\cos(\frac{x\pi}{L})\right)\)

Step-by-step solution

Consider the Energy Balance Equation

We will use the energy balance equation to solve this problem, which can be expressed as: \(\dot{q}_{s}(x) = \dot{m}C_{p}(T - T_{i})\) where \(\dot{q}_{s}(x) = a + b\sin(\frac{x\pi}{L})\) is the heat flux as a function of x, \(\dot{m}\) is the mass flow rate, \(C_{p}\) is the specific heat capacity at constant pressure, \(T\) is the temperature of the fluid as a function of x, and \(T_{i}\) is the mean inlet temperature of the fluid.

Rearrange the Energy Balance Equation

To obtain the temperature \(T\) as a function of x, we need to rearrange the energy balance equation. This will result in: \((T - T_{i}) = \frac{\dot{q}_{s}(x)}{\dot{m}C_{p}}\)

Substitute the given heat flux function

Now, substitute the given surface heat flux, which is \(\dot{q}_{s}(x) = a + b\sin(\frac{x\pi}{L})\), into the energy balance equation: \((T - T_{i}) = \frac{a + b\sin(\frac{x\pi}{L})}{\dot{m}C_{p}}\)

Integrate the equation with respect to x

Now, to find the temperature \(T\) as a function of x, we are going to integrate the equation with respect to x: \((T - T_{i}) = \frac{a}{\dot{m}C_{p}}\int_{0}^{x}dx + \frac{b}{\dot{m}C_{p}}\int_{0}^{x}\sin(\frac{x'\pi}{L})dx'\) where \(x'\) is a dummy variable for integrating with respect to x.

Evaluate the integrals

Now, let's evaluate the integrals: \((T - T_{i}) = \frac{a}{\dot{m}C_{p}}x + \frac{bL}{\dot{m}C_{p}\pi}\left[\left.-\cos(\frac{x'\pi}{L})\right|_{0}^{x}\right]\)

Simplify the expression

By simplifying the expression, we obtain the mean temperature of the fluid as a function of x: \(T(x) = T_{i} + \frac{a}{\dot{m}C_{p}}x + \frac{bL}{\dot{m}C_{p}\pi}\left(1-\cos(\frac{x\pi}{L})\right)\) Thus, we have derived the mean temperature of the fluid as a function of the x-coordinate.