Introduction to Heat Transfer
6th Edition
ISBN: 9780470501962
Author: Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine
Publisher: Wiley, John & Sons, Incorporated
expand_more
expand_more
format_list_bulleted
Concept explainers
Videos
Question
Chapter 4, Problem 4.45P
To determine
The finite difference equations for node
The finite difference equations for node
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
The heat flow per unit length of a thick cylindrical pipe is 772 W per meter. The pipe has radii ri = 12 cm, ro = 24 cm, outside surface temperature, To = 95 deg C and k = 0.05 + 0.0008T where T is in deg C and k is in W/(m K). Find the inside surface temperature of the pipe, assuming steady state conditions and accounting for the variation of thermal conductivity with temperature. Also determine the temperature of a point midway to the inside and outside radius.
Continuous temperature distribution in a semi-permeable material with laser radiation on it, thickness L and with a heat conduction coefficient k,
T(x)=-A/k.a^2.e^-ax+Bx+C
It is given by equality. Here A, a, B and C are known constants. For this case, the radiation absorption in the material manifests itself in a uniform heat generation term in the form q (x).
a) Obtain a relationship for the type that gives the conduction heat fluxes on the front and back surfaces.
b) get a correlation for q(x)
c) Obtain a relation that gives the radiation energy produced per unit surface area in the whole material.
A plane wall of thickness 2L = 30 mm and thermal conductivity k = 7 W/m-K experiences uniform volumetric heat generation at a
rate q, while convection heat transfer occurs at both of its surfaces (x = − L, + L), each of which is exposed to a fluid of
temperature T = 20°C. Under steady-state conditions, the temperature distribution in the wall is of the form
T(x) = a + bx + cx² where a = 82.0°C, b = -210°C/m, c = -2x 10°C/m², and x is in meters. The origin of the x-coordinate is at
the midplane of the wall.
(a) What is the volumetric rate à of heat generation in the wall?
(b) Determine the surface heat fluxes, q" (L)and q ( + L).
(c) What are the convection coefficients for the surfaces at x = - Land x = + L?
The volumetric rate of heat generation in the wall, in W/m³:
q = i
W/m³
The surface heat flux, in W/m²:
qx ( - L) = i
The surface heat flux, in W/m²:
q (+ L) = i
W/m²
W/m²
The convection coefficients for the surface at x = - L, in W/m²-K:
h(- L) = i
W/m².K
The convection…
Chapter 4 Solutions
Introduction to Heat Transfer
Ch. 4 - In the method of separation of variables (Section...Ch. 4 - A two-dimensional rectangular plate is subjected...Ch. 4 - Consider the two-dimensional rectangular plate...Ch. 4 - A two-dimensional rectangular plate is subjected...Ch. 4 - Prob. 4.5PCh. 4 - Prob. 4.6PCh. 4 - Free convection heat transfer is sometimes...Ch. 4 - Prob. 4.8PCh. 4 - Radioactive wastes are temporarily stored in a...Ch. 4 - Based on the dimensionless conduction heat rates...
Ch. 4 - Prob. 4.11PCh. 4 - A two-dimensional object is subjected to...Ch. 4 - Prob. 4.13PCh. 4 - Two parallel pipelines spaced 0.5 m apart are...Ch. 4 - A small water droplet of diameter D=100m and...Ch. 4 - Prob. 4.16PCh. 4 - Pressurized steam at 450 K flows through a long,...Ch. 4 - Prob. 4.19PCh. 4 - A furnace of cubical shape, with external...Ch. 4 - Prob. 4.21PCh. 4 - Prob. 4.22PCh. 4 - A pipeline, used for the transport of crude oil,...Ch. 4 - A long power transmission cable is buried at a...Ch. 4 - Prob. 4.25PCh. 4 - A cubical glass melting furnace has exterior...Ch. 4 - Prob. 4.27PCh. 4 - An aluminum heat sink k=240W/mK, used to coolan...Ch. 4 - Hot water is transported from a cogeneration power...Ch. 4 - Prob. 4.30PCh. 4 - Prob. 4.31PCh. 4 - Prob. 4.32PCh. 4 - An igloo is built in the shape of a hemisphere,...Ch. 4 - Consider the thin integrated circuit (chip) of...Ch. 4 - Prob. 4.35PCh. 4 - The elemental unit of an air heater consists of a...Ch. 4 - Prob. 4.37PCh. 4 - Prob. 4.38PCh. 4 - Prob. 4.39PCh. 4 - Prob. 4.40PCh. 4 - Prob. 4.41PCh. 4 - Determine expressions for...Ch. 4 - Prob. 4.43PCh. 4 - Prob. 4.44PCh. 4 - Prob. 4.45PCh. 4 - Derive the nodal finite-difference equations for...Ch. 4 - Prob. 4.47PCh. 4 - Prob. 4.48PCh. 4 - Consider a one-dimensional fin of uniform...Ch. 4 - Prob. 4.50PCh. 4 - Prob. 4.52PCh. 4 - Prob. 4.53PCh. 4 - Prob. 4.54PCh. 4 - Prob. 4.55PCh. 4 - Prob. 4.56PCh. 4 - Steady-state temperatures at selected nodal points...Ch. 4 - Prob. 4.58PCh. 4 - Prob. 4.60PCh. 4 - The steady-state temperatures C associated with...Ch. 4 - A steady-state, finite-difference analysis has...Ch. 4 - Prob. 4.64PCh. 4 - Consider a long bar of square cross section (0.8 m...Ch. 4 - Prob. 4.66PCh. 4 - Prob. 4.67PCh. 4 - Prob. 4.68PCh. 4 - Prob. 4.69PCh. 4 - Consider Problem 4.69. An engineer desires to...Ch. 4 - Consider using the experimental methodology of...Ch. 4 - Prob. 4.72PCh. 4 - Prob. 4.73PCh. 4 - Prob. 4.74PCh. 4 - Prob. 4.75PCh. 4 - Prob. 4.76PCh. 4 - Prob. 4.77PCh. 4 - Prob. 4.78PCh. 4 - Prob. 4.79PCh. 4 - Prob. 4.80PCh. 4 - Spheres A and B arc initially at 800 K, and they...Ch. 4 - Spheres of 40-mm diameter heated to a uniform...Ch. 4 - To determine which parts of a spiders brain are...Ch. 4 - Prob. 4.84P
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.Similar questions
- 2.15 Suppose that a pipe carrying a hot fluid with an external temperature of and outer radius is to be insulated with an insulation material of thermal conductivity k and outer radius . Show that if the convection heat transfer coefficient on the outside of the insulation is and the environmental temperature is , the addition of insulation actually increases the rate of heat loss if , and the maximum heat loss occurs when . This radius, is often called the critical radius.arrow_forwardA plane wall, 7.5 cm thick, generates heat internally at the rate of 105 W/m3. One side of the wall is insulated, and the other side is exposed to an environment at 90C. The convection heat transfer coefficient between the wall and the environment is 500 W/m2 K. If the thermal conductivity of the wall is 12 W/m K, calculate the maximum temperature in the wall.arrow_forward2.2 A small dam, which is idealized by a large slab 1.2 m thick, is to be completely poured in a short Period of time. The hydration of the concrete results in the equivalent of a distributed source of constant strength of 100 W/m3. If both dam surfaces are at 16°C, determine the maximum temperature to which the concrete will be subjected, assuming steady-state conditions. The thermal conductivity of the wet concrete can be taken as 0.84 W/m K.arrow_forward
- 2.29 In a cylindrical fuel rod of a nuclear reactor, heat is generated internally according to the equation where = local rate of heat generation per unit volume at r = outside radius = rate of heat generation per unit volume at the centerline Calculate the temperature drop from the centerline to the surface for a 2.5-cm-diameter rod having a thermal conductivity of if the rate of heat removal from its surface is 1.6 .arrow_forwardA square silicon chip 7mm7mm in size and 0.5-mm thick is mounted on a plastic substrate as shown in the sketch below. The top surface of the chip is cooled by a synthetic liquid flowing over it. Electronic circuits on the bottom of the chip generate heat at a rate of 5 W that must be transferred through the chip. Estimate the steady-state temperature difference between the front and back surfaces of the chip. The thermal conductivity of silicon is 150 W/m K. Problem 1.6arrow_forward2.30 An electrical heater capable of generating 10,000 W is to be designed. The heating element is to be a stainless steel wire having an electrical resistivity of ohm-centimeter. The operating temperature of the stainless steel is to be no more than 1260°C. The heat transfer coefficient at the outer surface is expected to be no less than in a medium whose maximum temperature is 93°C. A transformer capable of delivering current at 9 and 12 V is available. Determine a suitable size for the wire, the current required, and discuss what effect a reduction in the heat transfer coefficient would have. (Hint: Demonstrate first that the temperature drop between the center and the surface of the wire is independent of the wire diameter, and determine its value.)arrow_forward
- 1-D, steady-state conduction with uniform internal energy generation occurs in a plane wall with a thickness of 50 mm and a constant thermal conductivity of 5 W/m/K. The temperature distribution has the form T = a + bx + cx² °C. The surface at x=0 has a temperature of To = 120 °C and experiences convection with a fluid for which T.. surface at x= 50 mm is well insulated (no heat transfer). Find: (a) The volumetric energy generation rate q. (15) (b) Determine the coefficients a, b, and c. 20 °C and h 500 W/m² K. The To: = 120°C T = 20°C h = 500 W/m².K 111 Fluid T(x)- = q, k = 5 W/m.K L = 50 mmarrow_forwardProblem: Conduction related Uniform internal heat generation at q =6.0×10^7 W/m3 is occurring in a cylindrical nuclear reactor fuel rod of 60-mm diameter, and under steady-state conditions the temperature distribution is of the form T\left(r\right)=a+br^2T(r)=a+br2, where T is in degrees Celsius and r is in meters, while a = 900°C and b = -5.26 × 10^5 °C/m^2. The fuel rod properties are k = 30 W/m · K, ρ = 1,100 kg/m^3, and cp = 800 J/kg · K. (a) What is the rate of heat transfer per unit length of the rod at r = 0 (the centerline) and at r = 30 mm (the surface)? (b) If the reactor power level is suddenly increased to q2dot = 10^8 W/m^3, what is the initial time rate of temperature change at r = 0 and r = 30 mm?arrow_forwardA plane wall of thickness 2L = 2*33 mm and thermal conductivity k = 7 W/m-K experiences uniform volumetric heat generation at a rate q˙, while convection heat transfer occurs at both of its surfaces (x = −L, + L), each of which is exposed to a fluid of temperature T∞ = 31°C. Under steady-state conditions, the temperature distribution in the wall is of the form T(x) = a + bx + cx2 where a = 85°C, b = −-218°C/m, c = −-23,942°C/m2, and x is in meters. The origin of the x-coordinate is at the midplane of the wall. (a) Sketch the temperature distribution and identify significant physical features. (b) What is the volumetric rate of heat generation q˙ in the wall? (c) Obtain an expression for the heat flux distribution qx″(x). Is the heat flux zero at any location? Explain any significant features of the distribution. (d) Determine the surface heat fluxes, qx″(−L) and qx″(+L). How are these fluxes related to the heat generation rate? (e) What are the convection coefficients…arrow_forward
- Drive an expression for heat transfer and temperature distribution for steady state one dimensional heat conduction in a plan wall. The temperature is maintained at a temperature Ti at x=0, while the other face X-L is maintained at temperature T2, the thickness of the wall may be taken as L and the energy equation is given by: d²T/dx²=0.arrow_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(Q4) A 4m x 6m wall consists of 4 glass windows of 2m x 1.5m dimensions. The wall has thickness of 0.13m and a thermal conductivity of 0.5 W/m.K, while the glass windows are 6 mm thick with a thermal conductivity of 1.228 W/m.K. The values of intemal and external surface conductance for the wall (including glass) are 7.8 W/m? K and 34.4 W/m².K, respectively. The intemal and extemal temperatures are 22° C and 42°C, respectively. Calculate the total heat transfer rate through the wall. What percentage of this heat transfer is through the windows?arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Principles of Heat Transfer (Activate Learning wi...Mechanical EngineeringISBN:9781305387102Author:Kreith, Frank; Manglik, Raj M.Publisher:Cengage Learning
Principles of Heat Transfer (Activate Learning wi...
Mechanical Engineering
ISBN:9781305387102
Author:Kreith, Frank; Manglik, Raj M.
Publisher:Cengage Learning
Understanding Conduction and the Heat Equation; Author: The Efficient Engineer;https://www.youtube.com/watch?v=6jQsLAqrZGQ;License: Standard youtube license