Heat Transfer and Energy

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1 What is Heat? Heat Transfer and Energy Heat is Energy in Transit. Recall the First law from Thermodynamics. U = Q - W What did we mean by all the terms? What is U? What is Q? What is W? What is Heat Transfer? Heat transfer is the transfer of Heat effected by a temperature difference. However, in contrast to what we may have done in Thermodynamics, we are concerned with the rate of heat transfer. For that, we need a different energy balance, one that accounts for the rate of transfer and production, a general energy balance. U = Q W We will derive such a balance in later lectures. Examples of Heat Transfer Pasteurizing milk Energy loss from a house by conduction through a window Melting of a polymer Cooling an automobile engine Cooling or heating a stream in a chemical process Condensing vapor leaving a distillation tower Solar heating of the earth Radiational cooling of a pond Air-conditioning Temperature control of the human body Ablational cooling of a space shuttle on reentry Spring

2 Conduction Modes of Heat Transfer Conduction Convection Radiation Conduction is the thermal equivalent of diffusion. In gases, energy is exchanged between molecules in collisions. You will recall from Thermodynamics that a measure of the molecular motion and the energy exchange is the temperature and that conduction is the transfer of energy from the more energetic molecules to the less energetic. In the presence of a temperature gradient, energy is then transferred from the high temperature ( more energetic) to the lower temperature(less energetic). For liquids and solids, the example is not as clear. Conduction in liquids the collisions are more frequent and not as energetic and in solids transfer is by vibrations, for example, in a crystal lattice. The constitutive equation describing the phenomenon is Fourier s law q = k T What we mean by the notation is the following: q = q i e i T = T x i e i Spring

3 q Conduction in one-dimension To give you some notion of what I mean, examine a simple onedimensional conduction problem. T T 1 T 2 x q x is the heat flux in watts/m 2 q x = k dt dx k is the thermal conductivity in units of watts/m- K If we make an energy balance over a differential element at steady state, q x x+ x q x x = 0 or q x = constant Then, we see that T = ax + b or dt dx = T 1 T 2 L It follows that at steady state L q x = k T 1 T 2 L Convection Convection in heat transfer is a process which involves the diffusion of heat and the advection of energy by flow. The process can be described by Newton s Law of Cooling q n = h T T b where h is the heat transfer coefficient as watts/m 2 - K Typical Values of Heat Transfer Coefficients h (watts/m 2 - K) Free Convection 5-25 Forced Convection gases liquids 50-20,000 Phase Change (boiling or condensation) ,000 Spring

4 What is the relation between h expressed in English units and SI units? In SI units, h = 10 watts/m 2 - K In English units, h = BTU/hr-ft 2 - F Quick estimates for conversion changes, divide watts/m 2 - K by 6 to get BTU/hr-ft 2 - F Radiation Thermal radiation is energy emitted by matter that is at a finite temperature. Emission occurs not only from solids, but from gases and liquids. The energy is carried by electromagnetic waves, originating at the expense of the internal energy of the matter. Conduction and convection depend on the presence of an intermediary. Radiation does not! The normal component of heat flux emitted by a surface is given by the Stefan-Boltzmann s Law q n = εσ T s 4 where ε is the emissivity σ is the Stefan-Boltzman constant The emissivity is the ratio of the energy emitted by the real surface compared to that emitted by an ideal surface (a black body). It is dimensionless. σ = 5.67 x 10-8 watts/m 2 -K 4 and T is in degrees Kelvin. The net rate of energy transfer between two surfaces q i j n j = F ij εσ T 4 4 i T j The view factor, F ij, depends on the distance R between and the orientation of the two surfaces. We will discuss Radiation in the last several weeks of the course. Spring

5 Problem Solving Your textbook outlines a "stock" procedure for addressing, formulating and solving problems in Heat Transfer. In my view, it is a sound technique and one that I would ask that you follow. It involves several steps, each with a prescribed formula. What folloiws is the outline. Problem Statement The statement of the problem and the results required Solution 1. Known Read the problem carefully, then state briefly and concisely what is know about the problem. This is not a simple restaement of the problem. 2. Find Briefly and concisely state the results that are required. 3. Schematic Draw a picture (schematic) of the system. Identify and label the heat transfer processes and the relevant boundaries of the system (control surfaces) 4. Assumptions List all the relevant assumptions you will use in formulating the problem. 5. Properties Compile the property valued for all the calculations to follow and identify the sources of the data. 6. Analysis Apply the appropriate conservation laws and appropriate constitutive equations ( rate laws). Develop the analysis as completely as you can without substituting numerical values. If it is feasible, make the equations dimensionless. Finally complete the calculation with numerical values. 7. Comments Discuss the results. This should include the key conclusions. Were the initial assumptions valid? What effect would there be if they were not? How sensitive is your solution to the parameters? Spring

6 Measurement of Thermal Conductivity A Design Problem Statement of the Problem Measure the conductivity of the metal in a metal rod connected to a constant temperature sink (an ice bath) Water flows through a well-mixed reservoir. In enters at T h1 and leaves at the same temperature as it is in the water reservoir, T h2. The rod, the two reservoirs are insulated from the surroundings. Derive a relationship for the thermal conductivity and recommend a flow rate for the water. Solution 1. Known The heat lost by the fluid in the lower reservoir is transferred to the ice batch through the rod. The temperature is known for the ce bath. The water flow is measured and known, and we measure the water temperatures in and out of the reservoir. 2. Find Derive a relationship for the thermal conductivity and recommend a flow rate for the water. Ice bath 3. Schematic F T h1 D TT h2 L = 6 in. D = 0.5 in. F T h2 Spring

7 4. Assumptions There are a number of assumptions in this energy balance. What are they? 5. Properties Some are given in the schematic, but what should we list? Data k (cal/s-cm- K) k (Btu/h-ft- F) Aluminum Copper Steel Conditions and Data T h1 = 80 F T c = 32 F k = 32BTU/fth- F C pw = 1BTU/lb- F = 62 lb./ft^3 L = 6in. D = 0.5in. Spring

8 6. Analysis Energy balance on the rod Heat exchanged from the water = heat transferred across the rod Q = ρ w F w C pw T h1 T h2 = k L πd2 4 T h2 We did not state the constitutive relations. What are they? T c There are a number of assumptions in this energy balance. What are they? We can solve for the thermal conductivity, k.. The result gives us an equation by which we might better design the experiment. k = 4L T πd 2ρ h1 wf w C pw T h2 T h2 T c The dimensions, L and D are fixed as are all the variables save T h1 and F w. What are the best choices for these variables? Dimensionless form for the solution. We can group parameters as θ, so that T h1 T h2 = T h2 T c πd 2 k 4Lρ w F w C pw = θ and the temperature rise can be expressed simply as T h1 T h2 = θ T h1 T c 1 + θ Spring

9 If we calculate the temperature rise as a function of flow rate we can obtain the following table. Flow rate T h1 T h2 T h1 T h2 T h1 T c cc/min In an experiment we can measure temperature no more precisely that 0.1 F so that the highest flow rate would be 5 ml/min. Some experimental questions How precise can you control and measure flow rates? How good is the well-stirred assumption? How precise are the temperature measurements? Given the precision of measurement, what is the precision of the estimation of the thermal conductivity? Spring

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