surface area per unit time is w(x, t). Derive the partial differential

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1 1.2 Conduction of Heat in One-Dimension Derive the diffusion equation for a chemical pollutant. (a) Consider the total amount of the chemical in a thin region between x and x + Ax. (b) Consider the total amount of the chemical between x = a and x = b Derive an equation for the concentration u(x, t) of a chemical pollutant if the chemical is produced due to chemical reaction at the rate of au(,3 - u) per unit volume Suppose that the specific heat is a function of position and temperature, c(x, u). (a) Show that the heat energy per unit mass necessary to raise the temperature of a thin slice of thickness Ax from 0 to u(x, t) is not c(x)u(x, t), but instead fo c(x, u) du. (b) Rederive the heat equation in this case. Show that (1.2.3) remains unchanged Consider conservation of thermal energy (1.2.4) for any segment of a onedimensional rod a < x < b. By using the fundamental theorem of calculus, a jb f (x) dx = f (b), ab derive the heat equation (1.2.9). * If u(x, t) is known, give an expression for the total thermal energy contained in a rod (0 < x < L) Consider a thin one-dimensional rod without sources of thermal energy whose lateral surface area is not insulated. (a) Assume that the heat energy flowing out of the lateral sides per unit surface area per unit time is w(x, t). Derive the partial differential equation for the temperature u(x, t). (b) Assume that w(x, t) is proportional to the temperature difference between the rod u(x, t) and a known outside temperature -y(x, t). Derive that cp at ax (Ko e / - A [u(x, t) - y(x, t))h(x), (1.2.15) where h(x) is a positive x-/dependent proportionality, P is the lateral perimeter, and A is the cross-sectional area. (c) Compare (1.2.15) to the equation for a one-dimensional rod whose lateral surfaces are insulated, but with heat sources. (d) Specialize (1.2.15) to a rod of circular cross section with constant thermal properties and 0 outside temperature.

2 18 Chapter 1. Heat Equation for any constant C2. Unlike the first example (with fixed temperatures at both ends), here there is not a unique equilibrium temperature. Any constant temperature is an equilibrium temperature distribution for insulated boundary conditions. Thus, for the time-dependent initial value problem, we expect slim u(x, t) = C2; 00 if we wait long enough, a rod with insulated ends should approach a constant temperature. This seems physically quite reasonable. However, it does not make sense that the solution should approach an arbitrary constant; we ought to know what constant it approaches. In this case, the lack of uniqueness was caused by the complete neglect of the initial condition. In general, the equilibrium solution will not satisfy the initial condition. However, the particular constant equilibrium solution is determined by considering the initial condition for the time-dependent problem (1.4.11). Since both ends are insulated, the total thermal energy is constant. This follows from the integral conservation of thermal energy of the entire rod [see (1.2.4)1: Since both ends are insulated, d /' L 8 au cpu dx = -Ko 87x (0, t) + Ko (L, t). ( ) dt f 8x 0 L cpu dx = constant. (1.4.20) 1 One implication of (1.4.20) is that the initial thermal energy must equal the final (limt.,,,) thermal energy. The initial thermal energy is ep f L f (x) dx since u(x, 0) = f (x), while the equilibrium thermal energy is cp LL C2 dx = cpc2l since the equilibrium temperature distribution is a constant u(x, t) = C2. The constant C2 is determined by equating these two expressions for the constant total thermal energy, cp fl f (x) dx = cpc2l. Solving for C2 shows that the desired unique steady-state solution should be t u(x) = C2 = L J f (x) dx, 0 (1.4.21) the average of the initial temperature distribution. It is as though the initial condition is not entirely forgotten. Later we will find a u(x, t) that satisfies ( ) and show that limt.,,. u(x, t) is given by (1.4.21). EXERCISES Determine the equilibrium temperature distribution for a one-dimensional rod with constant thermal properties with the following sources and boundary conditions:

3 1.4. Equilibrium Temperature Distribution 19 * (a) Q = 0, u(0) = 0, (b) Q = 0, u(0) = T, (c) Q = 0, (0) = 0, * (d) Q = 0, u(0) = T, (e) Ko = 1, u(0) = T1, u(l) = T2 * (f) Qc = x2, u(0) = T, 8 (L) = 0 (g) Q = 0, u(0) = T, ax (L) + u(l) = 0 *(h) Q=0, 8 (0)-[u(0)-TJ=0, ax(l)-a In these you may assume that u(x, 0) = f (x) Consider the equilibrium temperature distribution for a uniform one-dimensional rod with sources Q/Ko = x of thermal energy, subject to the boundary conditions u(0) = 0 and u(l) = 0. *(a) Determine the heat energy generated per unit time inside the entire rod. (b) Determine the heat energy flowing out of the rod per unit time at x = 0 and at x = L. (c) What relationships should exist between the answers in parts (a) and (b)? Determine the equilibrium temperature distribution for a one-dimensional rod composed of two different materials in perfect thermal contact at x = 1. For 0 < x < 1, there is one material (cp = 1, Ko = 1) with a constant source (Q = 1), whereas for the other 1 < x < 2 there are no sources (Q = 0, cp = 2, Ko = 2) (see Exercise 1.3.2) with u(o) = 0 and u(2) = If both ends of a rod are insulated, derive from the partial differential equation that the total thermal energy in the rod is constant Consider a one-dimensional rod 0 < x < L of known length and known constant thermal properties without sources. Suppose that the temperature is an unknoum constant T at x = L. Determine T if we know (in the steady state) both the temperature and the heat flow at x = The two ends of a uniform rod of length L are insulated. There is a constant source of thermal energy Qo 54 0, and the temperature is initially u(x, 0) _ f (x)-

4 20 Chapter 1. Heat Equation (a) Show mathematically that there does not exist any equilibrium temperature distribution. Briefly explain physically. (b) Calculate the total thermal energy in the entire rod For the following problems, determine an equilibrium temperature distribution (if one exists). For what values of,3 are there solutions? Explain physically. * (a) at z 8xz + 1, u x, 0 ) = A X ), ) =, 8x (L, a ( At 1 t) = Q (b) au 192U = 8xz, x, 0) = f ( x u ( ), au ax (0 t) = 1,, 8u (L, t) = Q ax ( c) & = 02U + 0) = P 8 (0 t = 0 8x ( L t z u ( x, X),, ),, ) Express the integral conservation law for the entire rod with constant thermal properties. Assume the heat flow is known to be different constants at both ends By integrating with respect to time, determine the total thermal energy in the rod. (Hint: use the initial condition.) (a) Assume there are no sources. (b) Assume the sources of thermal energy are constant Derive the integral conservation law for the entire rod with constant thermal properties by integrating the heat equation (1.2.10) (assuming no sources). Show the result is equivalent to (1.2.4) Suppose = e + 4, u(x, 0) = f (x), Ou (0, t) = 5, "u (L, t) = 6. Calculate the total thermal energy in the one-dimensional rod (as a function of time) Suppose = s + x, u(x, 0) = f (x), Ou (0, t) = Q, &u (L, t) = 7. (a) Calculate the total thermal energy in the one-dimensional rod (as a function of time). (b) From part (a), determine a value of Q for which an equilibrium exists. For this value of Q, determine lim u(x, t). t Suppose the concentration u(x, t) of a chemical satisfies Fick's law (1.2.13), and the initial concentration is given u(x, 0) = f (x). Consider a region 0 < x < L in which the flow is specified at both ends -kou (0, t) = a and -kou (L, t) _ 0. Assume a and # are constants. (a) Express the conservation law for the entire region. (b) Determine the total amount of chemical in the region as a function of time (using the initial condition). = 0

5 1.5. Heat Equation in Two or Three Dimensions 29 x=l Area magnified EXERCISES 1.5 Figure Spherical coordinates Let c(x, y, z, t) denote the concentration of a pollutant (the amount per unit volume). (a) What is an expression for the total amount of pollutant in the region R? (b) Suppose that the flow J of the pollutant is proportional to the gradient of the concentration. (Is this reasonable?) Express conservation of the pollutant. (c) Derive the partial differential equation governing the diffusion of the pollutant. * For conduction of thermal energy, the heat flux vector is 4 _ -KoVu. If in addition the molecules move at an average velocity V, a process called convection, then briefly explain why 0 _ -KoVu + cpuv. Derive the corresponding equation for heat flow, including both conduction and convection of thermal energy (assuming constant thermal properties with no sources) Consider the polar coordinates x=rcos9 y = r sin 9. (a) Since r2 = x2 + y2, show that O = cos 0, = sing, " = Tycos B and 8B sin 9 r ' 8x r (b) Show that r = cos Bi + sin 03 and B = - sin 0 + cos 63. (c) Using the chain rule, show that V = r" ar + 9,i- g and hence Vu = r"+ -r 8e 9. (d) If A = ArT + Ae6, show that r Tr_ (rar) + r 8 (AB), since 8r" /8B = 9 and 86/80 = -f follows from part (b).

6 30 Chapter 1. Heat Equation (e) Show that V2u = 1 r Or (r a;`) + 3 ae Using Exercise 1.5.3(a) and the chain rule for partial derivatives, derive the special case of Exercise 1.5.3(e) if u(r) only Assume that the temperature is circularly symmetric: u = u(r, t), where r2 = x2 + y2. We will derive the heat equation for this problem. Consider any circular annulus a < r < b. (a) Show that the total heat energy is 21r fq cpur dr. (b) Show that the flow of heat energy per unit time out of the annulus at r = h is --21rbKoau/ar 1,=b. A similar result holds at r = a. (c) Use parts (a) and (b) to derive the circularly symmetric heat equation without sources: au -k a r aul at r ar ' a, _ J Modify Exercise if the thermal properties depend on r Derive the heat equation in two dimensions by using Green's theorem, (1.5.16), the two-dimensional form of the divergence theorem If Laplace's equation is satisfied in three dimensions, show that Vu-ft ds = 0 Use for any closed surface. (Hint: the divergence theorem.) Give a physical interpretation of this result (in the context of heat flow) Determine the equilibrium temperature distribution inside a circular annulus (rl < r < r2): *(a) if the outer radius is at temperature T2 and the inner at T1 (b) if the outer radius is insulated and the inner radius is at temperature Ti Determine the equilibrium temperature distribution inside a circle (r < ro) if the boundary is fixed at temperature To. * Consider subject to at u ( r, 0) = f (r), _r 5T Crar) z a<r<b au au ar (a, t) = f3, and (b, t) = 1. 19r Using physical reasoning, for what value(s) of 0 does an equilibrium temperature distribution exist?

7 1.5. Heat Equation in Two or Three Dimensions Assume that the temperature is spherically symmetric, u = u(r, t), where r is the distance from a fixed point (r2 = x2 + y2 + z2). Consider the heat flow (without sources) between any two concentric spheres of radii a and b. (a) Show that the total heat energy is 47r fo cpur2 dr. (b) Show that the flow of heat energy per unit time out of the spherical shell at r = b is -4irb2Ko 8u/8r Ir=b. A similar result holds at r = a. (c) Use parts (a) and (b) to derive the spherically symmetric heat equation 8u k 8 T28u 8t r2 8r C? * Determine the steady-state temperature distribution between two concentric spheres with radii 1 and 4, respectively, if the temperature of the outer sphere is maintained at 80 and the inner sphere at 0 (see Exercise ) Isobars are lines of constant temperature. Show that isobars are perpendicular to any part of the boundary that is insulated Derive the heat equation in three dimensions assuming constant thermal properties and no sources Express the integral conservation law for any three-dimensional object. Assume there are no sources. Also assume the heat flow is specified, g(x, y, z), on the entire boundary and does not depend on time. By integrating with respect to time, determine the total thermal energy. (Hint: Use the initial condition.) Derive the integral conservation law for any three dimensional object (with constant thermal properties) by integrating the heat equation (1.5.11) (assuming no sources). Show that the result is equivalent to (1.5.1). Orthogonal curvilinear coordinates. A coordinate system (u, v, w) may be introduced and defined by x = x(u, v, w), y = y(u, v, w) and z = z(u, v, w). The radial vector r =_ At + yj + A. Partial derivatives of r with respect to a coordinate are in the direction of the coordinate. Thus, for example, a vector in the u-direction 8r/8u can be made a unit vector e in the u-direction by dividing by its length h = I8r/8ul called the scale factor: cu = - er/au Determine the scale factors for cylindrical coordinates Determine the scale factors for spherical coordinates The gradient of a scalar can be expressed in terms of the new coordinate system Vg = a 6)r/8u + b 8r/(7v + c Or/Ow, where you will determine the scalars a, b, c. Using dg = V9 dr, derive that the gradient in an orthogonal curvilinear coordinate system is given by Vg = 1 8g _ 1 8g 1 8g T" T. eu + h 8; e + hu, 8w ( )