Fundamentals of Heat and Mass Transfer
7th Edition
ISBN: 9780470501979
Author: Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine
Publisher: Wiley, John & Sons, Incorporated
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Chapter 2, Problem 2.13P
A cylinder of radius
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You are asked to estimate the maximum human body temperature if the metabolic
heat produced in your body could escape only by tissue conduction and later on the surface by
convection. Simplify the human body as a cylinder of L=1.8 m in height and ro= 0.15 m in
radius. Further, simplify the heat transfer process inside the human body as a 1-D situation when
the temperature only depends on the radial coordinater from the centerline. The governing
dT
+q""=0
dr
equation is written as
1 d
k-
r dr
r = 0,
dT
dr
=0
dT
r=ro -k -=h(T-T)
dr
(k-0.5 W/m°C), ro is the radius of the cylinder (0.15 m), h is the convection coefficient at the
skin surface (15 W/m² °C), Tair is the air temperature (30°C). q" is the average volumetric heat
generation rate in the body (W/m³) and is defined as heat generated per unit volume per second.
The 1-D (radial) temperature distribution can be derived as:
T(r) =
q"¹'r² qr qr.
+
4k 2h
+
4k
+T
, where k is thermal conductivity of tissue
air
(A) q" can be calculated…
How long should it take to boil an egg? Model the egg as a sphere with radius of 2.3 cm that has properties similar to water with a density of = 1000 kg/m3 and thermal conductivity of k = 0.606 Watts/(mC) and specific heat of c = 4182 J/(kg C). Suppose that an egg is fully cooked when the temperature at the center reaches 70 C. Initially the egg is taken out of the fridge at 4 C and placed in the boiling water at 100 C. Since the egg shell is very thin assume that it quickly reaches a temperature of 100 C. The protein in the egg effectively immobilizes the water so the heat conduction is purely conduction (no convection). Plot the temperature of the egg over time and use the data tooltip in MATLAB to make your conclusion on the time it takes to cook the egg in minutes.
The temperature distribution across a wall 0.25 m thick at a certain instant of time is T(x) = a + bx + cx², where T is in degrees Celsius and x is
in meters, a = 200 C, b = -200 C/m, and c = 30 C/m². The wall has a thermal conductivity of 2.5 W/m.K. (a) Determine the heat flux into and
out of the wall (q"in and q'out). (b) If the cold surface is exposed to a fluid at 100 C, what is the convection coefficient h?
- Degree Celsius
200°C
q" In-
q'in
q'out=
h =
Choose...
Choose....
Choose...
L₂x
K = 2.5 W/m.k
T(x)-200-200 x +30x²
q" Out
142.7 C
11
L=0.25 m
Fluid
Too = 100 °C
h
Chapter 2 Solutions
Fundamentals of Heat and Mass Transfer
Ch. 2 - Assume steady-state, one-dimensional heat...Ch. 2 - Assume steady-state, one-dimensional conduction in...Ch. 2 - A hot water pipe with outside radius r, has a...Ch. 2 - A spherical shell with inner radius r1 and outer...Ch. 2 - Assume steady-state, one-dimensional heat...Ch. 2 - A composite rod consists of two different...Ch. 2 - A solid, truncated cone serves as a support for a...Ch. 2 - To determine the effect of the temperature...Ch. 2 - A young engineer is asked to design a thermal...Ch. 2 - A one-dimensional plane wall of thickness 2L=100mm...
Ch. 2 - Consider steady-state conditions for...Ch. 2 - Consider a plane wall 100 mm thick and of thermal...Ch. 2 - A cylinder of radius ro, length L, and thermal...Ch. 2 - In the two-dimensional body illustrated, the...Ch. 2 - Consider the geometry of Problem 2.14 for the case...Ch. 2 - Steady-state, one-dimensional conduction occurs in...Ch. 2 - An apparatus for measuring thermal conductivity...Ch. 2 - An engineer desires to measure the thermal...Ch. 2 - Consider a 300mm300mm window in an aircraft. For a...Ch. 2 - Consider a small but known volume of metal that...Ch. 2 - Use INT to perform the following tasks. Graph the...Ch. 2 - Calculate the thermal conductivity of air,...Ch. 2 - A method for determining the thermal conductivity...Ch. 2 - Compare and contrast the heat capacity cp of...Ch. 2 - A cylindrical rod of stainless steel is insulated...Ch. 2 - At a given instant of time, the temperature...Ch. 2 - A pan is used to boil water by placing it on a...Ch. 2 - Uniform internal heat generation at q=5107W/m3 is...Ch. 2 - Consider a one-dimensional plane wall with...Ch. 2 - The steady-state temperature distribution in a...Ch. 2 - The temperature distribution across a wall 0.3 m...Ch. 2 - Prob. 2.33PCh. 2 - One-dimensional, steady-state conduction with...Ch. 2 - Derive the heat diffusion equation, Equation 2.26,...Ch. 2 - Derive the heat diffusion equation, Equation 2.29....Ch. 2 - The steady-state temperature distribution in a...Ch. 2 - One-dimensional, steady-state conduction with no...Ch. 2 - One-dimensional, steady-state conduction with no...Ch. 2 - The steady-state temperature distribution in a...Ch. 2 - Prob. 2.41PCh. 2 - Prob. 2.42PCh. 2 - cylindrical system illustrated has negligible...Ch. 2 - Beginning with a differential control volume in...Ch. 2 - Prob. 2.45PCh. 2 - Prob. 2.46PCh. 2 - For a long circular tube of inner and outer radii...Ch. 2 - Passage of an electric current through a long...Ch. 2 - Two-dimensional. steady-state conduction occurs in...Ch. 2 - An electric cable of radius r1 and thermal...Ch. 2 - A spherical shell of inner and outer radii ri and...Ch. 2 - A chemically reacting mixture is stored in a...Ch. 2 - A thin electrical heater dissipating 4000W/m2 is...Ch. 2 - The one-dimensional system of mass M with constant...Ch. 2 - Consider a one-dimensional plane wall of thickness...Ch. 2 - A large plate of thickness 2L is at a uniform...Ch. 2 - The plane wall with constant properties and no...Ch. 2 - Consider the steady-state temperature...Ch. 2 - A plane wall has constant properties, no internal...Ch. 2 - A plane wall with constant properties is initially...Ch. 2 - Consider the conditions associated with Problem...Ch. 2 - Consider the steady-state temperature distribution...Ch. 2 - A spherical particle of radius r1 experiences...Ch. 2 - Prob. 2.64PCh. 2 - A plane wall of thickness L=0.1m experiences...Ch. 2 - Prob. 2.66PCh. 2 - A composite one-dimensional plane wall is of...Ch. 2 - Typically, air is heated in a hair dryer by...Ch. 2 - Prob. 2.69P
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- The initial temperature of a 50 cm long silver wire is 50 °C. The circumference of the wire in question is completely insulated, but both ends are kept at a temperature of 0 °C (zero degrees Celsius). Obtain the heat conduction along the wire as a function of time and position and, taking a single term in the solution, determine how many degrees Celsius the temperature in the middle of the rod will be after 7 minutes. (For silver wire, α=1.70 cm2/s.)arrow_forward= Consider a large plane wall of thickness L=0.3 m, thermal conductivity k = 2.5 W/m.K, and surface area A = 12 m². The left side of the wall at x=0 is subjected to a net heat flux of ɖo = 700 W/m² while the temperature at that surface is measured to be T₁ = 80°C. Assuming constant thermal conductivity and no heat generation in the wall, (a) express the differential equation and the boundary equations for steady one- dimensional heat conduction through the wall, (b) obtain a relation for the variation of the temperature in the wall by solving the differential equation, and (c) evaluate the temperature of the right surface of the wall at x=L. Ti до L Xarrow_forwardWhich formula is used to calculate the heat conduction in the AXIAL direction in a vertically located pipe segment whose inner and outer surfaces are perfectly insulated. Here r, is inner radius, r, outer radius, Tri pipe inner surface temperature, Tro pipe outer surface temperature, L is the length of the pipe, T the temperature on the lower surface, Ty the temperature on upper surface. Tu r; Tro rarrow_forward
- The time evolution of the temperature of an object follows the Newton's cooling laws dT dx = -k(T - Ts), where the term k = 2.2 (1/s) is the heat transfer constant, and Tg = 25.6° C is the ambient temperature. The initial temperature of the object at time t = = 0 is T(t = 0) = 200°C. °C Use the Euler's method, and a time step of h=0.2s, calculate: When t = = 0.2s, T = °C When t 1s, T =arrow_forwardunder steady-state conditions. If you are given T1 = 200 °C and T2 = 164 °C, determine: a) the conduction heat flux, q,.cond, in m2 W from x = 0 to x = L b) if the dimensions of the triangle ares 15 mm and h 13 mm, calculate the heat transfer due to convection, q,y, in W at x = L Finsulation T2 T T = 20°C h = 500 W/m2.K Triangular Prism x L x 0 L= 50 mm k = 100 W/m-Karrow_forwardHeat transferarrow_forward
- The temperature distribution across a wall 0.3 m thick at a certain instant of time is T(x) = a+ b+cx?, where T is in degrees Celsius and x is in meters, a = 200°C,b = -200°, and c = conductivity of 1 W /m · K. 30°C/m² . The wall has a thermal (a) On a unit surface area basis, determine the rate of heat transfer into and out of the wall and the rate of change of energy stored by the wall. (b) If the cold surface is exposed to a fluid at 100°C, what is the convection coefficient? k=1W/m•k T(x) =200-200x + 30x² 200°C- ĖST 142.7°C q"out | Fluid Too = 100°C,h 9"in |L-0.3marrow_forwardPLEASE HELP ANSWER THIS THERMODYNAMICS PRACTICE QUESTION THANK YOUarrow_forwardA certain material has a thickness of 30 cm and a thermal conductivity of 0.04 W/m- °C. At a particular instant in time, the temperature distribution with x, the distance from the left face, is T = 150x ^ 2 - 30x , where x is in meters. Calculate the heat-flow rates atx x = 0 and x = 30 cm . Is the solid heating up or cooling down?arrow_forward
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