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
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Question
Chapter 2, Problem 2.44P
To determine
The heat diffusion equation for one dimensional cylindrical, radial coordinate system with internal heat generation.
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b) Using the heat diffusion equation which you have derived in part (a).
Let consider a one-dimensional plane wall that separate of two fluids which
is illustrated in Figure lb with constant properties (e.g. thermal
conductivity, k) and uniform internal generation (e.g. no heat generation)
and steady state condition (e.g no change in the amount of energy storage)
i) Find the expression of temperature distribution, T(x)
ii) and the expression of heat flow, q
Ts.
Cold fluid
T2 h2
Hot fluid
T2
T1. h
Lox
x = L
Figure 1b
Consider a solid sphere of radius R with a fixed surface temperature, TR. Heat is generated within
the solid at a rate per unit volume given by q = ₁ + ₂r; where ₁ and ₂ are constants.
(a) Assuming constant thermal conductivity, use the conduction equation to derive an expression
for the steady-state temperature profile, T(r), in the sphere.
(b) Calculate the temperature at the center of the sphere for the following parameter values:
R=3 m 1₁-20 W/m³ TR-20 °C k-0.5 W/(m K) ₂-10 W/m³
.... Derive the general 3D heat conduction equation for a spherical coordinate.
Show step-by-step solution and schematic diagram.
Chapter 2 Solutions
Introduction to Heat 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 r1 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 - Prob. 2.9PCh. 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 - Prob. 2.13PCh. 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 - Prob. 2.17PCh. 2 - Prob. 2.18PCh. 2 - Consider a 300mm300mm window in an aircraft. For a...Ch. 2 - Prob. 2.20PCh. 2 - Use IHT 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 - Prob. 2.24PCh. 2 - Prob. 2.25PCh. 2 - At a given instant of time, the temperature...Ch. 2 - Prob. 2.27PCh. 2 - Uniform internal heat generation at q.=5107W/m3 is...Ch. 2 - Prob. 2.29PCh. 2 - The steady-state temperature distribution in a...Ch. 2 - The temperature distribution across a wall 0.3 m...Ch. 2 - Prob. 2.32PCh. 2 - Prob. 2.33PCh. 2 - Prob. 2.34PCh. 2 - Prob. 2.35PCh. 2 - Prob. 2.36PCh. 2 - Prob. 2.37PCh. 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 - One-dimensional, steady-state conduction with no...Ch. 2 - Prob. 2.42PCh. 2 - Prob. 2.43PCh. 2 - Prob. 2.44PCh. 2 - Beginning with a differential control volume in...Ch. 2 - A steam pipe is wrapped with insulation of inner...Ch. 2 - Prob. 2.47PCh. 2 - Prob. 2.48PCh. 2 - Two-dimensional, steady-state conduction occurs in...Ch. 2 - Prob. 2.50PCh. 2 - Prob. 2.51PCh. 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 - Prob. 2.57PCh. 2 - Prob. 2.58PCh. 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 - Prob. 2.62PCh. 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 - Prob. 2.68PCh. 2 - The steady-state temperature distribution in a...
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- Given a metallic block comprising of 2 unknown materials namely A and B (as shown in Figure 1 below). 1. You are tasked to determine the heat flux (W/cm2) for node at coordinate (2, 2) using finite-difference approximations using Elliptical Equation (Control Volume Approach) for the temperature gradients at this node. 2. Estimate the flux value in the horizontal direction in materials A and B, and determine if these two fluxes should be equal. 3. Calculate the vertical flux in materials A and B. Should these two fluxes be equal? The following values for the constants is as provided here: Δz = 0.5 cm, h =10 cm, ka = 0.3 W/cm · C, kb = 0.5 W/cm · C and nodal temperatures are T22 = 51.6oC, T21 = 74.2oC, T23 = 45.3oC, T32 = 38.6oC and T12 = 87.4oCarrow_forward8. The shown 2-D plate is in contact with a heat source at its upper edge, which supplies heat at a constant flux, qo, per unit length. -- a. Derive a finite-difference relationship to express the steady-state temperature at the shown boundary point TP, in terms of the temperatures at the surrounding points (TE, Tw, Ts) and the other quantities in the problem (e.g., k, qo, etc.). Follow the methodology outlined in the class notes (i.e., use energy balance). Assume Ax = Ay. y 90 b. Modify the relationship for the case when the upper edge is perfectly insulated (without heat addition).arrow_forward(a) Consider nodal configuration shown below. Derive the finite-difference equations under steady-state conditions for the following situations. (a) The boundary is insulated. (b) The boundary is subjected to a constant heat flux. m, n+1 Ay Im, n The side insulated m-1, n I I Ax- m, n-1arrow_forward
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