Derive the expression for the temperature reached by a solid at a given time 10 't' in case of transient heat conduction
Consider a solid of thickness L and thermal conductivity k. Let the initial temperature of the solid be T1 and the temperature of the surrounding fluid be T2. At t=0, the entire solid is at temperature T1.
The heat conduction equation for the transient case is given by:
ρc ∂T/∂t = k ∂2T/∂x2
where ρ is the density of the solid, c is the specific heat capacity, T is the temperature distribution as a function of time 't' and position 'x'.
Assuming the heat flow to be one-dimensional, we can write the temperature distribution as:
T(x,t) = T2 + (T1 - T2) [1 - erf(x / 2√(k.t / ρ.c))]
Here, erf is the error function defined as:
erf(z) = 2/√π ∫0z e^(-t^2) dt
Substituting t=10t in the above expression, we get:
T(x,10t) = T2 + (T1 - T2) [1 - erf(x / 2√(k.10t / ρ.c))]
Thus, we have derived the expression for the temperature reached by a solid at a given time 10 't' in case of transient heat conduction.
To derive the expression for the temperature reached by a solid at a given time t in the case of transient heat conduction, we can use the 1D transient heat conduction equation, also known as the heat diffusion equation:
∂T/∂t = α (∂²T/∂x²)
Where:
- T is the temperature of the solid
- t is time
- x is the position along the solid
- α is the thermal diffusivity of the solid material
To solve this equation, we need to apply appropriate boundary conditions and initial conditions.
Let's assume that the solid has an initial temperature T(x,0) = T0, where T0 is the initial temperature of the solid. We'll also assume that the solid is insulated at both ends, meaning there is no heat transfer across the boundaries, so ∂T/∂x = 0 at x = 0 and x = L, where L is the length of the solid.
To solve the heat diffusion equation, we can use separation of variables method:
Assume T(x,t) = X(x)T(t)
Plugging this into the heat diffusion equation, we get:
(X(x)T'(t)) / T(t) = α (X''(x) / X(x))
Dividing both sides by T(x)X(x), we have:
[T'(t) / T(t)] = α [X''(x) / X(x)]
Since the left side of the equation only involves t and the right side only involves x, they must be equal to a constant, say -λ²:
[T'(t) / T(t)] = α [X''(x) / X(x)] = -λ²
Now we have two separate ordinary differential equations, one for T(t) and one for X(x):
T'(t) / T(t) = -λ²/α
X''(x) / X(x) = -λ²
Solving the equation for T(t), we have:
T'(t) = -λ²/α * T(t)
This is a first-order ordinary differential equation with the general solution:
T(t) = C1 * exp(-λ²/α * t)
Next, solving the equation for X(x), we have:
X''(x) + λ² * X(x) = 0
This is a second-order ordinary differential equation with the general solution:
X(x) = C2 * cos(λx) + C3 * sin(λx)
Now, applying the boundary conditions, we know that ∂T/∂x = 0 at x = 0 and x = L. This implies that X'(0) = X'(L) = 0.
Differentiating X(x) with respect to x, we get:
X'(x) = -C2 * λ * sin(λx) + C3 * λ * cos(λx)
Setting X'(0) = 0, we have:
-C2 * λ * sin(0) + C3 * λ * cos(0) = 0
C3 = 0
Setting X'(L) = 0, we have:
-C2 * λ * sin(λL) = 0
Since C2 cannot be zero, we must have sin(λL) = 0. This gives us the condition for λ:
λL = n * π, where n is an integer
Therefore, λ = n * π / L
Now, plugging the values of λ and α back into the equation for T(t), we have:
T(t) = C1 * exp(-(n * π / L)² * t / α)
The general solution for the temperature distribution is then:
T(x, t) = Σ[ C1n * exp(-(n * π / L)² * t / α) * cos(n * π * x / L) ]
where Σ denotes the sum for all possible values of n.
To find the values of C1n, we need to apply the initial condition T(x,0) = T0. This allows us to determine the Fourier cosine series coefficients C1n. The final solution will involve the summation of these terms.
Please note that the solution assumes homogeneous and isotropic material properties, and certain simplifying assumptions have been made. The solution can be more complex for specific cases or boundary conditions.
To derive the expression for the temperature reached by a solid at a given time (10 't') in the case of transient heat conduction, you need to use the heat conduction equation, which is a partial differential equation that describes how heat is transferred within a solid.
The general heat conduction equation for a solid is:
∂u/∂t = α (∂²u/∂x²)
Where:
- ∂u/∂t represents the rate of change of temperature with respect to time.
- α is the thermal diffusivity of the material (α = k / (ρ * Cp), with k being the thermal conductivity, ρ the density, and Cp the specific heat capacity of the material).
- ∂²u/∂x² represents the second derivative of the temperature with respect to the spatial coordinate x.
To solve this equation, we need an initial condition - the temperature distribution at t = 0, and boundary conditions - the temperature at the boundaries of the solid at any given time. Without specific values for these conditions, we cannot obtain a concrete expression for the temperature at 10t.
However, I can guide you on how to proceed with solving the transient heat conduction problem for a specific case:
1. Identify the boundary and initial conditions for the problem you are trying to solve.
2. Use appropriate mathematical techniques, such as separation of variables, Fourier series, or numerical methods (e.g., finite difference or finite element methods), to solve the heat conduction equation.
3. Apply the given boundary and initial conditions to determine the constants or coefficients in the solution.
4. Once you have obtained the general solution, substitute the specific values of time (10t) to find the temperature at that particular time.
Please note that the exact derivation and solution process vary depending on the specific problem and conditions involved. It is recommended to consult textbooks or academic resources on heat transfer to learn detailed steps and methods to solve transient heat conduction problems.