Newtons laws of cooling proposes that the rate of change of temperature is proportional to the temperature difference to the ambient (room) temperature. And can be modelled using the equation: dT/dt = -k (T-Ta) It can also be written as dT/T-Ta = -k dt Where: T = Temperature of material Ta = Ambient (room) temperature k = A cooling constant a) integrate both sides of the equation and show that the temperature difference is given by: (T-Ta) = CoE^-kt (Co is a constant for this problem) B) calculate Co if the initial temperature is 70 degrees C and Ta = 20 degrees C?

Newtons laws of cooling proposes that the rate of change of temperature is proportional to the temperature difference to the ambient (room) temperature. And can be modelled using the equation: dT/dt = -k (T-Ta) It can also be written as dT/T-Ta = -k dt Where: T = Temperature of material Ta = Ambient (room) temperature k = A cooling constant a) integrate both sides of the equation and show that the temperature difference is given by: (T-Ta) = CoE^-kt (Co is a constant for this problem) B) calculate Co if the initial temperature is 70 degrees C and Ta = 20 degrees C?

Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)

8th Edition

ISBN:9781305387102

Author:Kreith, Frank; Manglik, Raj M.

Publisher:Kreith, Frank; Manglik, Raj M.

Chapter1: Basic Modes Of Heat Transfer

Section: Chapter Questions

Problem 1.22P: 1.22 In order to prevent frostbite to skiers on chair lifts, the weather report at most ski areas...

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1.77 Explain each in your own words. (a) What is the mode of heat transfer through a large steel plate that has its surfaces at specified temperatures? (b) What are the modes when the temperature on one surface of the steel plate is not specified, but the surface is exposed to a fluid at a specified temperature?
Using the information in Problem 1.22, estimate the ambient air temperature that could cause frostbite on a calm day on the ski slopes. 1.22 In order to prevent frostbite to skiers on chair lifts, the weather report at most ski areas gives both an air temperature and the wind-chill temperature. The air temperature is measured with a thermometer that is not affected by the wind. However, the rate of heat loss from the skier increases with wind velocity, and the wind-chill temperature is the temperature that would result in the same rate of heat loss in still air as occurs at the measured air temperature with the existing wind. Suppose that the inner temperature of a 3-mm-thick layer of skin with a thermal conductivity of 0.35W/mKis35C and the air temperature is 20C. Under calm ambient conditions the heat transfer coefficient at the outer skin surface is about 20W/m2K (see Table 1.4), but in a 40-mph wind it increases to 75W/m2K. If frostbite occurs when the skin temperature drops to about 10C, do you advise the skier to wear a face mask? What is the skin temperature drop due to the wind?
To determine the thermal conductivity of a structural material, a large 15-cm-thick slab of the material is subjected to a uniform heat flux of 2500 W/m2 while thermocouples embedded in the wall at 2.5 cm. intervals are read over a period of time. After the system had reached equilibrium, an operator recorded the thermocouple readings shown below for two different environmental conditions: Distance from the Surface (cm) Temperature (C) Test 1 0 40 5 65 10 97 15 132 Test 2 0 95 5 130 10 168 15 208 From these data, determine an approximate expression for the thermal conductivity as a function of temperature between 40 and 208C.
For flow over a slightly curved isothermal surface, the temperature distribution inside the boundary layer t can be approximated by the polynomial T(y)=a+by+cy2+d3(yt), where y is the distance normal to the surface. (a) By applying appropriate boundary conditions, evaluate the constants a, b, c, and d. Fluid (b) Then obtain a dimensionless relation for the temperature distribution in the boundary layer.
3.9 The heat transfer coefficients for the flow of 26.6°C air over a sphere of 1.25 cm in diameter are measured by observing the temperature-time history of a copper ball the same dimension. The temperature of the copper ball was measured by two thermocouples, one located in the center and the other near the surface. The two thermocouples registered, within the accuracy of the recording instruments, the same temperature at any given instant. In one test run, the initial temperature of the ball was 66°C, and the temperature decreased by 7°C in 1.15 min. Calculate the heat transfer coefficient for this case.
The drag on an airplane wing in flight is known to be a function of the density of air (), the viscosity of air(), the free-stream velocity (U), a characteristic dimension of the wing (s), and the shear stress on the surface of the wing (s). Show that the dimensionless drag, sU2, can be expressed as a function of the Reynolds number, Us.
1.22 In order to prevent frostbite to skiers on chair lifts, the weather report at most ski areas gives both an air temperature and the wind-chill temperature. The air temperature is measured with a thermometer that is not affected by the wind. However, the rate of heat loss from the skier increases with wind velocity, and the wind-chill temperature is the temperature that would result in the same rate of heat loss in still air as occurs at the measured air temperature with the existing wind. Suppose that the inner temperature of a 3-mm-thick layer of skin with a thermal conductivity of 0.35 W/m K is and the air temperature is . Under calm ambient conditions the heat transfer coefficient at the outer skin surface is about (see Table 1.4), but in a 40-mph wind it increases to . (a) If frostbite occurs when the skin temperature drops to about , do you advise the skier to wear a face mask? (b) What is the skin temperature drop due to the wind?
The same engineer decides to look into rates of cooling for liquids toexperiment with different cooling solutions for servers. She finds thatthe rate of cooling for one liquid can be modelled by the equation:y = 48 × 0.99t(0 ≤ t ≤ 80)where y is the temperature of the liquid in degrees Celsius and t is thetime in minutes.(i) State whether the type of reduction for this model is linear orexponential. Describe how reduction rate differs between linear andexponential functions. (ii) Calculate the temperature when t = 20. [3](iii) Write down the scale factor and use this to find the percentagedecrease in the temperature per minute. (iv) Use the method shown in Subsection 5.2 of Unit 13 to find the timeat which the temperature is 30◦(v) Determine the halving time of the temperature.
An unmanned submarine is equipped with temperature and depth measuring instruments and hasradio equipment that can transmit the output readings of these instruments back to the surface.The submarine is initially floating on the surface of the sea with the instrument output readingsin steady state. The depth measuring instrument is approximately zero order and the temperaturetransducer first order with a time constant of 50 seconds. The water temperature on the seasurface, T0, is 20C and the temperature Tx at a depth of x metres is given by the relation:Tx = T0- 0.01x(a) If the submarine starts diving at time zero, and thereafter goes down at a velocity of 0.5metres/second, draw a table showing the temperature and depth measurements reported atintervals of 100 seconds over the first 500 seconds of travel. Show also in the table the error ineach temperature reading(b) What temperature does the submarine report at a depth of 1000 metres?
The temperature of an experimental heated enclosure is being ramped up from 80 to 450°F at the rate of 19°F/min. A thermocouple, embedded in a Thermowell for protection, is being used to monitor the oven temperature. The thermocouple has a time constant of 84 s. At t = 10 min, what is the difference between the actual temperature and the temperature indicated by the thermocouple? Final answer in whole number. Please show your complete solution and write readable.
Newton’s law of cooling states that the rate of change in the temperature T(t) of a body is proportional to the difference between the temperature of the medium M(t) and the temperature of the body. That is, dT/dt=K(M(t)−T(t)), where K is a constant. Let K= 0.04 (min)^−1 and the temperature of the medium be constant, M(t) = 293 kelvins. If the body isinitially at 368 kelvins, use Euler’s method with h= 3 min to approximatethe temperature of the body after 90 minutes. Please provide the details ofEuler’s method and the calculations.
According to the U.S. Geological survey, the volume of water in oceans, seas, and bays is 1.41 ✕ 1018 m3. The average concentration of naturally occurring uranium in seawater is 3.15 ✕ 10−6 kg/m3. If 0.670% of naturally occurring uranium is the fissionable isotope U-235 and each fission reaction yields 200 MeV, estimate the amount of time the uranium in the seawater on the planet could meet the planet's energy needs at an average usage of 1.94 ✕ 1013 J/s.