CBE 150A – Transport Spring Semester 2014 Heat Transfer Introduction

CBE 150A – Transport Spring Semester 2014 Goals : By the end of today’s lecture, you should be able to:  define the mechanisms for heat transfer  define Fourier's Law of heat conduction  describe the differences between series resistance and parallel resistance  calculate the total resistance for composite systems in series and parallel  distinguish between natural and forced convection Heat Transfer Introduction

CBE 150A – Transport Spring Semester 2014 Outline: I. Mechanisms of heat transfer II. Heat transfer by conduction III.Resistances in series and parallel IV.Example problem -- series resistances V.Convective heat transfer

CBE 150A – Transport Spring Semester 2014 I. Mechanisms of heat transfer Heat is transferred from an area of temperature to an area of temperature. What are the three mechanisms for heat transfer? High Low Conduction, Convection, Radiation

CBE 150A – Transport Spring Semester 2014 Name the key mode(s) of heat transfer for the following examples : The ceiling heating coil in a bathroom Heating a pan of water on the stove top Baking a pie The heating system in your car Steam radiator Heat exchanger

CBE 150A – Transport Spring Semester 2014 Heat Transfer is a ‘rate’ process The rate at which heat is transferred is critical in many processes. How can the rate of heat transfer be controlled?

CBE 150A – Transport Spring Semester 2014 II. Heat transfer by conduction For ease of argument, let's consider conduction through a solid. Before proceeding, we need a few definitions and some nomenclature. Heat fluxq = q has units of (Btu/ft 2 -hr) or (kcal/m 2 -hr) or (W/m 2 ). Is q a scalar or vector quantity? Vector

CBE 150A – Transport Spring Semester 2014 Heat transfer rate q has units of (Btu/ hr) or (kcal/hr) or (W). The heat transfer rate, q, measures the total amount of thermal energy flowing across a surface per unit time. On the other hand, Q is the net heat transfer into a control volume.

CBE 150A – Transport Spring Semester 2014 Fourier's Law (one-dimensional): What is the corresponding expression for heat flux (q)?

CBE 150A – Transport Spring Semester 2014 Thermal conductivity. The proportionality constant k is a physical property of the substance called the thermal conductivity. k has units of (Btu/hr-ft-°F) or (kcal/hr-m-°C) or (W/m-°K) What is the analogous physical property for momentum transfer? The thermal conductivity is a mild function of temperature, but is often assumed constant for most applications. Viscosity (  )

CBE 150A – Transport Spring Semester 2014 Typical values of thermal conductivity (Btu/hr-ft-°F) : air0.015 water0.35 glass0.45 steel25 stainless steel10 copper225

CBE 150A – Transport Spring Semester 2014

This implies what about the heat transfer rate, q? Substitute for q from Fourier's Law: Integration with respect to x, then using the boundary conditions that T = T 1 at x = 0 and T = T 2 at x =  x yields:

CBE 150A – Transport Spring Semester 2014 Thermal Resistance When doing heat transfer calculations, it proves convenient to define the thermal resistance, R: Comparing this definition with our previous result, yields the following relationship for R: [MSH Example 10.2]

CBE 150A – Transport Spring Semester 2014 For a cylindrical shell, the equivalent expression for the thermal resistance is: where A LM is the log mean area,

CBE 150A – Transport Spring Semester 2014

How does q 1 relate to q 2 ? We can express the rate of heat transfer as a function of (T 3 -T 1 ): q = We see that the total resistance (R T ) for the composite slab is simply the sum of the individual resistances. R T =  R i

CBE 150A – Transport Spring Semester 2014

Space Shuttle Thermal Protective System The shuttle's thermal protection system (TPS) is designed to keep the temperature of the shuttle's aluminum structure below 350 degrees Fahrenheit (177 C) while temperatures on some external surfaces exceed 2,300 Fahrenheit (1,260 C) degrees. This tile was removed from Columbia after the first shuttle mission, in April Example Problem For a 4 inch thickness, determine the required tile thermal conductivity for an energy flux of 300 BTU/hr ft 2. Compare this value to typical thermal conductivities given on Slide 11.

CBE 150A – Transport Spring Semester 2014 Example Problem -- Series resistances The wall of a downtown Berkeley bank is composed of a 5-inch layer of Texas granite (k = 1.5 Btu/hr-ft-°F) on the exterior, a 3/8-inch layer of wall board (gypsum, k = 0.25 Btu/hr-ft-°F), a 3-inch thick layer of air (k = Btu/hr-ft-°F), another 3/8- inch layer of wall board (gypsum, k = 0.25 Btu/hr-ft-°F), and finally, a 1/4- inch thick layer of oak paneling on the interior (k = 0.12 Btu/hr-ft-°F). A fire has been burning unnoticed on the inside of the bank for 20 min such that the interior oak paneling temperature has reached a steady state value of 700°F. Will anyone touching the exterior of the building experience pain or discomfort (note: scalding water ≈ 140°F)? Assume that the rate of heat loss is 50 Btu/hr-ft 2.

CBE 150A – Transport Spring Semester Minute Problem -- Series resistances A thick-walled tube of stainless steel having a k = W/m K with dimensions of m ID and m OD is covered with m thick layer of an insulation (k = W/m K). The inside-wall temperature of the pipe is 811 K and the outside surface of the insulation is K. For a m length of pipe, calculate the heat loss and the temperature at the interface between the metal and the insulation.