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MAE 300: Engineering Instrumentation and Measurement Experiment No. 5 Experimental Investigation of Natural Convection from a Sphere, Using Lumped System Analysis Submitted By: Jack Martin Cristina Garcia Franco Allen Xavier Date Performed: 4-18-2022 Date Submitted: 5-2-2022

Abstract The objective of this experiment was to familiarize the students with the method of determining the convective heat transfer coefficient of a heated stainless steel sphere in an idle environment. The method used to find the convective heat transfer coefficient was the lumped system approach. Students recorded the temperature of the stainless steel sphere after it was heated to 124.8 °C and then students recorded the progressive decrease in temperature every 30 seconds over the span of 1500 seconds. A small torch was used to heat up the stainless steel sphere which was connected to a thermocouple which displayed the stainless steel sphere’s temperature on a display screen. The time was recorded by a phone stopwatch. After 1500 seconds, the temperature of the stainless steel sphere was 42.0 °C. After the experiment was conducted in the lab, students made all the calculations correlated to the lab. Students plotted a graph of ln((T-T∞)/(Ti-T∞)) vs time from the recorded data. Using the graph, a slope of -0.0011 was found. Using this slope, the overall effective convective heat transfer coefficient, h eff , was calculated for and was found to be 13.0432. Following this calculation, the total heat transfer out of the sphere, q, was calculated to be 0.9130. The average temperature of the sphere, T W , was found to be 83.4 ℃ . After these calculations, the lumped system assumption was verified by calculating for the Biot #. The biot number was calculated to be 0.0030 which is less than 0.01 therefore verifying the lumped system assumption. After this, the average theoretical convection heat transfer coefficient and its uncertainty, h_eff ± Δh, was calculated to be 13.04 ± 1.24.

Background Theory The method of finding the convective heat transfer of a lumped object is the lumped system approach. The lumped system approach for a sphere assumes that the temperature is uniform throughout the object and that it is a function of time only. The equation used to find the total heat transfer is: Another equation is used for the total heat transfer out of the sphere which is equal to the decrease of internal energy. The equation is as follows: After equating the two equations for heat transfer, rearranging them and deriving the with respect to temperature and time, you get the following equation: In this equation, T is the temperature measured as a function of time, T ∞ is the room temperature, T i is the initial temperature of the sphere at time t = 0, h eff is the heat transfer coefficient, A is the area of the sphere, ⍴ is the density of the sphere, V is the volume of the sphere, C p is the specific heat of the sphere, and t is the time respective to temperature T.

A graph can be plotted using ln((T-T∞)/(Ti-T∞)) vs time from which the slope of the line can be found to aid in finding the heat transfer coefficient. The equation used to find the heat transfer coefficient is: The lumped system approach can be verified by the Biot number: For a sphere, the equation for the Biot number is as follows: In order for the lumped system approach to be vald, the Biot # must be < 0.01. The constant K S for a brass sphere is 119 W/mK and 14 W/mK for a stainless steel sphere. The average theoretical heat transfer coefficient is found by calculating for the Rayleigh number and Nusselt number. The Rayleigh number is found by using the following equation: Where g is the value of gravitational acceleration, 𝛽 = 1/T f (where as T f = (T W +T ∞ )/2), D is the diameter of the sphere, T W is the sphere’s surface temperature, T ∞ is the room temperature, v is the kinematic viscosity, adn Pr is the

Prandtl number (which can be found using a table for the physical property of air at atmospheric pressure). The Nusselt number is found using the following equation: And finally the average theoretical convection heat transfer coefficient can be found using the following equation which is related to the Nusselt number, thermal conductivity of the air, and the diameter of the sphere: The uncertainty for the theoretical heat transfer coefficient can be found using the following equation: Where U h can be found using the following equation:

Aft6er finding both the average theoretical convection heat transfer coefficient and its uncertainty, they can written in the following form:

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