Heat and Fluid Flow
Purpose: When you complete this laboratory you will
· understand and be able to calculate the heat flow through an object due to a temperature difference.
· understand and be able to calculate the viscous flow of a fluid through a pipe due to a pressure difference (Poiseuille’s law).
Background: A scientific model is a construction we use to make sense of some data (which we obtain either using our senses or from some instruments) about a physical phenomenon. In physics a model is typically a verbal description plus a schematic diagram and finally a mathematical description. An interesting thing about models is that they may start out as very general concepts and frequently can be used to interpret a variety of different phenomena.
In this lab you will study the model of flow, which can be applied to two very different phenomena. The model is the following:
When an object is isolated and in a state of equilibrium, it can be characterized by some physical quantity. (For the case of heat flow, the isolated object is characterized by its temperature T.) A second isolated object may be in a different state (have a different temperature). But when the two objects are placed in contact through some sort of link, they are no longer in equilibrium (there is a temperature difference DT) and some physical quantity, which we’ll call Q, will flow through the link to bring the two systems into equilibrium. (For the example we are using, Q is the energy that flows.) The rate at which Q flows, DQ/Dt, where t is the time, is determined by a quantity R, the resistance, which is characteristic of the link.
Schematically, using the symbols for heat flow,
R
Q
T1 T2
Figure 1. Flow Model
And mathematically,
. (1)
The resistance is determined by three characteristics of the link: its length L, its cross-sectional area A and the material making up the link. In the simplest model of the link, we expect the resistance to be proportional to L, and inversely proportional to A:
, (2)
where a is a constant characteristic of the material making up the link, but independent of the dimensions of the link. [To see this, think of water flowing through a pipe. If a certain amount of water is flowing there will be a pressure drop along the pipe. If a second identical pipe is connected in series with the first, then the total pressure drop along both pipes will have to be twice as big to keep the flow constant. So the resistance of the two pipes together must be twice as big. On the other hand, if you connect a second pipe of the same length in parallel with the first, then twice as much water will flow, but the pressure drop will be unchanged, so the total resistance of the two pipes must only be half as big as for a single pipe.]
We now apply this model to two different phenomena:
Heat Flow: The temperature difference DT drives the flow of energy (loosely called heat) Q at a rate DQ/Dt through the link. The thermal resistance R is given by Eq. (2), where by convention a is replaced with k = 1/a. k is the thermal conductivity of the material in the link. Your textbook gives a table of values of k for various materials.
Viscous Fluid Flow: When you have a hollow tube connecting two containers filled with different levels of water, the pressure difference DP between the two ends of the tube causes water, mass m, to flow between the containers resulting in a water current Dm/Dt. The flow equation (Poiseuille’s law) is then:
. (6)
The resistance to the flow is caused by friction between the tube and the water and between the water molecules themselves. This friction is characterized by the coefficient of viscosity h (eta) of the fluid. Your textbook has a table of the coefficients of viscosity for several fluids. The units of h are N/m2.s = Pa.s. A commonly used unit, one-tenth as large, is the poise, P; 1 P = 0.1 Pa.s or the centipoise; 1 cP = 0.001 Pa.s. [The viscosity of water is about 1 cP.]
Non-viscous flow Viscous flow
Figure 2. Velocity Vectors for Fluid Flow in a Tube
Extrapolating from the first three cases, you might expect that R for viscous flow would be given by Eq. (2) with a = h. But this is not the case! The reason is that for the first three cases, how close to the wall the energy (or charge, or diffusing mass) travels does not affect the rate of flow. But for viscous flow, the interaction with the wall slows the fluid and, in fact, the fluid velocity at the wall is zero. Figure 2 shows how the velocity profile changes across the tube for viscous flow and for non-viscous flow such as occurs for electric current.
Poiseuille showed that because of this non-uniform “laminar” flow, the actual equation for the resistance for viscous flow is given by
, (7)
where r is the radius of the tube. [This is not an obvious result, but you can see that as the tube diameter increases less of the liquid is close to the wall, reducing the resistance. This gives the extra factor of 1/A.] . Think about the effect of the r-4 dependence on the flow rate. Suppose cholesterol deposits narrow the radius of an artery by one half. Then the resistance will decrease by a factor of 16!
Equipment: Heat flow apparatus, multi-meter, power supply, thermometer, resistance boards, water reservoir with various output tubes having various dimensions, beaker, electronic scale, computer, Logger Pro software.
Activity 1: Heat Flow. The apparatus you will use is shown in Fig. 3. It consists of an insulated metal rod with a heater and thermometer on one end. The other end fits into a metal block immersed in ice water. The thermometer will tell you DT, the temperature difference between the ends of the rod. The heater supplies energy that flows along the rod to the ice bath. You can measure its electrical resistance Rheater and the voltage V applied across it. [Be careful! This activity involves both electrical and thermal resistance. Don’t let that confuse you.] In order to calculate the energy flow you will need some physics that you will not study until later in the semester-- namely, that the electrical power (energy per unit time) converted into thermal energy by a resistor when a voltage is applied across it is given by V2/R. This energy flows through the metal rod to
temperature sensor
Heater
Thermal insulation
DT
Metal rod
Ice Water (0o C)
Support block
Figure 3. Heat Flow Apparatus
the ice bath and the rate of flow is equal to the heat flow DQ/Dt. Thus once the temperature of the heater reaches a constant (steady-state) value, the heat flow is:
. (8)
In the lab you’ll find several sets of apparatus that differ either in the material making up the metal rod, its length or its diameter. For your experiment, you might choose to measure DQ/Dt, DT, L and A, which will allow you to calculate the thermal conductivity of the rod. If you then repeat the experiment on a rod of the same material, but different dimensions or perhaps on the same rod but with different DQ/Dt, you should get the same thermal conductivity, verifying Eq. (1) and (2).
Use "temperature.xmbl" in the course folder for running the Logger Pro program.
Activity 2: Poiseuille’s law. The lab setup consists of a tank for water with an outlet at the bottom to which you can attach tubes of various lengths and diameters, Fig. 4. The pressure at the bottom of the tank is determined by the combined weight of the atmosphere and the column of water. [What is the pressure at the other end of the outlet tube?] The flow is determined by weighing the amount of water that flows into the beaker
Beaker or cup
Electronic scale
Figure 4. Poiseuille’s Flow Apparatus
in a given time. [The time interval should be short enough that the water level in the reservoir does not appreciably change. Why?] Your experiment might be to determine h for several different dimension tubes or for several different water heights and compare the values in order to see whether they are consistent.
Modeling Flow
Name: ______Section/TA: ______Date: _____
Preliminary Question to be completed before the start of class: Show your work.
1. The following table makes a comparison of heat flow and viscous flow. Fill in the empty cells following the pattern set for heat flow.
Type of flow / the quantity that flows Q; units / Current DQ/Dt; units / Driving “force”;units / Parameter characterizing the link;
units
heat
viscous
2. An aluminum rod 20 cm and 1 cm in diameter has a 2 Co temperature difference between its two ends. How much energy flows through the rod in 10 s? Use the table in your textbook for any constants that you may need. Show your work.
What should be the length of a copper rod for the same amount of energy to flow in 10 s if the diameter and the temperature difference are the same as for the aluminum rod?
Heat and Fluid Flow
Name: ______Section/TA: ______Date: ______
Partners (full names): ______
The purpose of these activities is to demonstrate that the particular phenomenon is adequately described by the flow model. In the following use complete sentences, good grammar and correct spelling. A diagram or sketch may be useful. Use tables where appropriate and attach any graphs that you made.
Activity 1. Heat Flow. Carefully describe the precise purpose of the experiment you have designed.
Describe the procedure you followed in carrying out the experiment.
Give your results.
What are your conclusions?
Activity 2: Poiseuille’s law. Carefully describe the precise purpose of the experiment you have designed.
Describe the procedure you followed in carrying out the experiment.
Give your results.
What are your conclusions?
Modeling Flow- 1