Manometric Temperature Measurement - Part II Boundary Conditions for Instruction Only. End Temperatures = Zero

In this section, we will derive a solution to a homogeneous equation that is needed for a later solution, but which makes no apparent sense in the context of the problem. If you already know how to solve homogeneous boundary value problems with Fourier series solutions, then this section may be skipped, or lightly brushed over. Although slight changes have been made to make it applicable to this problem, the solution presented is largely copied from a textbook: William E. Boyce and Richard C. DiPrima, Elementary Differential Equations and Boundary Value Problems John Wiley & Sons, New York,1977. pp454-459.

In the first section on MTM derivation, we artificially fixed the problem so that the shelf quit adding heat to the ice instantly as the chamber was isolated by closing the valve. Because the shelf is already warm and there is no way to instantly remove that heat, the problem must be modified to account for the continued warming of the ice by the shelf throughout the time period of the pressure change measurement. Fortunately, "the conduction of heat in a solid body" is one of the classic partial differential equations of mathematical physics and for simple cases, it has been solved. Consider the ice as shown in the figure to have a distance axis, x, going from bottom to top. x = 0 and x= L are the two ends. We will also consider cross sections and for any cross section the temperature is uniform across the area of the section. So u(x,t ) is a function only of the distance along the axis and time. That is, at various times, t, the temperature for some given cross section will vary with time. Likewise, for different cross sections and some fixed time, the temperature along the length of the ice will be different.The differential equation that describes this problem is derived in a later section and is given as EQ#1.

			EQ#1

	EQ#2

Next, we must assume or measure an initial temperature distribution of the ice and express it as a function of distance, x, at time, t=0.

		EQ#3	Initial Conditions
where f(x) is the initial temperature distribution function At time zero, a suitable function for the temperature distribution might be given by the following.

In the language of Fourier Transforms, this approximates an 'Odd' function, but on close examination it is neither even nor odd.

f(x) is even if f(-x) = f(x) f(x) is odd if f(-x)=-f(x) cosines are even functions and sines are odd functions

Finally, we will start the solution by assuming that the ends of the ice are held at fixed temperatures. We know this to be incorrect. Indeed the entire solution is geared toward determining what the temperature at the top of the ice will be whenever the bottom of the ice is held at a constant temperature. Nonetheless, we will first solve for the case where Tb is the fixed temperature at the bottom of the ice and Ti is the fixed temperature at the top of the ice.

Indeed. To start, we will say that Tb = Ti = 0. In a later section, I will reduce this to a more general and realistic boundary condition.
In this notation, Ti = x(L)=the top of the ice. Tb = the bottom of the ice. Sometimes, we use T1 and T2, where T1 = bottom and T2 = top.

Boundary Conditions

EQ#4

Equations (1), (3), and (4) are both linear and homogeneous. Thus we can assume that u(x,t) is some function X(x)*T(t)
	Eq#5	Substituting EQ#5 into u(x,t) in EQ#1 yields,
	EQ#6	where primes refer to ordinary differentiation with respect to the independent variable, whether x or t. Equation 6 is equivalent to
	EQ#7	Separation of Variables
In order for EQ#7 to be valid for 0 < x < L, t < 0 , it is necessary that both sides of EQ#7 be equal to the same constant. Otherwise, by keeping one independent variable (say x) fixed and varying the other, one side (the left in this case) of EQ#7 would remain unchanged while the other varied, thus violating the equality. If we call this separation constant "S", then EQ#7 becomes
	EQ#8	Both sides equal the same constant
Hence we obtain the following two ordinary differential equations for X(x) and T(t):
	EQ#9
	EQ#10

The partial differential equation of EQ#1 has thus been replaced by two ordinary differential equations. Each of these equatiions can be readily solved for any value of the separation constant S. The product of two solutions of EQ#9 and EQ#10, respectively, for any value of S provides a solution of the partial differential equation, EQ#1. However, we are interested only in those solutions of EQ#1 also satisfying the boundary conditions, EQ#4. As we will now show, this severely restricts the possible values of S.

Substituting for u(x,t) from EQ#5 in the boundary condition at x=0, we obtain

EQ#11

If EQ#11 is satisfied by choosing T(t) to be zero for all t, then u(x,t) will be identically zero. That would be unacceptable, since it fails to satisfy the initial condition of EQ#3. Therefore, EQ#11 must be satisfied by requiring that

EQ#12

Similarly, the boundary condition at x = L requires that

EQ#13

We now want to consider EQ#9 subject to the boundary conditions of EQ#12 and EQ#13. Two cases must be distinguished, depending on whether S=0 or S <> 0
If S = 0, then the general solution to EQ#9 is

EQ#14

and, in order to satisfy the second boundary condition, we must have k₁ = 0. Hence X(x) is identically zero, and thus u(x,t) is also identically zero. As before, this is unacceptable, and we conclude that S must be nonzero.