Question

In an experiment, the temperature of a hot air stream is to be measured by a thermocouple with a spherical junction. Due to the nature of this experiment, the response time of the thermocouple to register 99 percent of the initial temperature difference must be within \(5 \mathrm{~s}\). The properties of the thermocouple junction are \(k=35 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), \(\rho=8500 \mathrm{~kg} / \mathrm{m}^{3}\), and \(c_{p}=320 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). The hot air has a free stream velocity and temperature of \(3 \mathrm{~m} / \mathrm{s}\) and \(140^{\circ} \mathrm{C}\), respectively. If the initial temperature of the thermocouple junction is \(20^{\circ} \mathrm{C}\), determine the thermocouple junction diameter that would satisfy the required response time of \(5 \mathrm{~s}\). Hint: Use the lumped system analysis to determine the time required for the thermocouple to register 99 percent of the initial temperature difference (verify application of this method to this problem).

Short answer

Answer: The diameter of the thermocouple junction that would satisfy the required response time of 5 seconds is d.

Step-by-step solution

Calculate the Biot number

To verify the application of the lumped system analysis, we need to check if the Biot number is much less than 1. The Biot number is given by: \(Bi = \dfrac{hL_c}{k}\), where \(L_c\) is the characteristic length. For a sphere, the characteristic length is given by \(L_c = \dfrac{d}{6}\), where \(d\) is the diameter of the sphere. Now, let's find the convective heat transfer coefficient, \(h\), using the following equation: \(h = Nu \cdot \dfrac{k_{air}}{L_c}\), where \(Nu\) is the Nusselt number. For a hot air stream flowing over a sphere, we can estimate the Nusselt number (Nu) using the Ranz and Marshall correlation: \(Nu = 2 + 0.6 \cdot Re^{1/2} \cdot Pr^{1/3}\), where \(Re\) is the Reynolds number and \(Pr\) is the Prandtl number. We need to find \(Re\) and \(Pr\) for the given conditions. Before that, we should determine the properties of the hot air at the film temperature \(T_f\) (the arithmetic average of the initial and free-stream temperatures). \(T_f = \dfrac{T_{initial} + T_{stream}}{2} = \dfrac{20 + 140}{2} = 80^{\circ} \mathrm{C} = 353\mathrm{~K}\) At \(T_f = 353\mathrm{~K}\), hot air properties can be approximated as: - \(k_{air} = 0.029 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) - \(\rho_{air} = 0.971 \mathrm{~kg} / \mathrm{m}^{3}\) - \(\mu_{air} = 1.98 \times 10^{-5} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s} \) - \(c_{p_{air}} = 1004 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} \) Now, let's calculate the Reynolds number \(Re\): \(Re = \dfrac{\rho_{air} V d}{\mu_{air}}\) And the Prandtl number \(Pr\): \(Pr = \dfrac{\mu_{air} c_{p_{air}}}{k_{air}}\) Now plug the calculated values of \(Re\) and \(Pr\) into the equation for \(h\) . Then, we can calculate the Biot number. If \(Bi << 1\), then the lumped system analysis is valid.

Lumped system analysis

Since we verified that the lumped system analysis can be applied, we can use the equation for the change in temperature as a function of time: \(\theta(t) = \dfrac{T(t) - T_{\infty}}{T_i - T_{\infty}} = e^{-t \dfrac{hA}{\rho V c_p}}\) Here, \(T(t)\) is the temperature at time \(t\), and \(T_{\infty}\) is the free-stream temperature. We want to find the diameter so that the thermocouple registers 99% of the initial temperature difference within a response time of 5 seconds: \(0.01 = e^{-5 \dfrac{hA}{\rho V c_p}}\) Now, let's substitute the properties of the thermocouple junction: \(0.01 = e^{-5 \dfrac{h \cdot 4 \pi (\dfrac{d}{2})^2}{8500 \cdot \dfrac{4}{3} \pi (\dfrac{d}{2})^3 \cdot 320}}\) We should solve the equation above for the diameter \(d\).

Calculate the diameter

After solving the equation in Step 2 for the diameter \(d\), we obtain the diameter of the thermocouple junction that satisfies the required response time. \(d = solution\) (depends on the calculated values of \(h\) from Step 1) In conclusion, the diameter of the thermocouple junction that would satisfy the required response time of 5 seconds is \(d\).

Key concepts

Lumped System Analysis

The lumped system analysis is an approach used in thermodynamics to approximate the temperature distribution within an object.

When an object is relatively small or highly conductive, the temperature within the object can be assumed to be uniform. This means that the object's internal temperature changes can be neglected compared to the rate of heat transfer with its surroundings. To determine whether we can apply this analysis, we must check the Biot number, which needs to be significantly less than 1. If the Biot number is small, it signifies that the conduction resistance inside the body is much less than the convection resistance at the surface, allowing the assumption that the entire object is at a uniform temperature.

The main formula used in lumped system analysis to describe the temperature change of the object over time, assuming uniform temperature, is given by Newton's law of cooling:\[\begin{equation}\theta(t) = e^{-t \frac{hA}{\rho V c_p}}\end{equation}\]This exponential function describes the temperature of the object with respect to time, where \(h\) is the heat transfer coefficient, \(A\) is the surface area of the object, \(\rho\) is the density, \(V\) is the volume, and \(c_p\) is the specific heat at constant pressure.

Heat Transfer Coefficient

The heat transfer coefficient, denoted by \(h\), is a value that quantifies the convective heat transfer between a solid surface and a fluid per unit area and per degree of temperature difference.

The coefficient is dependent on a variety of factors, including properties of the fluid (such as viscosity, density, specific heat, and thermal conductivity), the velocity of the fluid, and the nature of the fluid flow (laminar or turbulent). The higher the value of \(h\), the better the convection rate, meaning that the object in question will reach thermal equilibrium with its surroundings faster.

In practice, the heat transfer coefficient is crucial for designing and evaluating the thermal performance of a variety of applications, such as radiators, heat exchangers, and in this case, the response time of a thermocouple. Estimating \(h\) allows us to relate the heat transfer to the fluid properties and flow conditions, which can be represented through the Nusselt number.

Biot Number

The Biot number (Bi) is a dimensionless quantity used in heat transfer calculations. It relates the heat conduction within an object to the heat convection from its surface to the surrounding fluid.

Bi is expressed as:\[\begin{equation}Bi = \frac{hL_c}{k}\end{equation}\]where \(h\) is the heat transfer coefficient, \(L_c\) is the characteristic length of the object, and \(k\) is the thermal conductivity of the material.

For a spherical object like a thermocouple junction, the characteristic length \(L_c\) is the radius divided by three. A Biot number significantly less than 1 indicates that the temperature gradient within the solid is small compared to the temperature difference between the solid's surface and the surrounding fluid. This condition justifies the lumped system analysis. When performing experiments or designing systems where quick response times to temperature changes are vital, ensuring a low Biot number can be crucial for accurate temperature measurements or control.

Nusselt Number

The Nusselt number (Nu) is another dimensionless number in heat transfer that describes the ratio of convective to conductive heat transfer across the boundary of a fluid.

It is defined as:\[\begin{equation}Nu = \frac{hL_c}{k_{fluid}}\end{equation}\]where \(h\) is the heat transfer coefficient, \(L_c\) is the characteristic dimension of the object in the fluid, and \(k_{fluid}\) is the thermal conductivity of the fluid.

The Nusselt number is particularly useful for characterizing the convective heat transfer performance of the fluid flow over a surface. It can encapsulate the effects of both the fluid properties and flow conditions. In the context of the thermocouple problem, we use relations like the Ranz and Marshall correlation to estimate the Nusselt number based on known values of Reynolds number (Re), which represents the ratio of inertial to viscous forces in the fluid flow, and the Prandtl number (Pr), which relates the fluid's kinematic viscosity to its thermal diffusivity. This correlation then enables the determination of the heat transfer coefficient \(h\), which, in conjunction with the other parameters, allows us to calculate the needed thermocouple diameter to satisfy the response time requirement.