Heat conduction problem with variable temperature

[bancux]

Homework Statement

Find the the T(x). See attachment. The top, bottom as well as the left side are adiabatic. The fluid temperature on the side of convective heat transfer is a function of x. The dimension for the width is W.

Homework Equations

How to find T(x)

The Attempt at a Solution

To see my attempts, please open the pdf file in the attachment.

Thank you

 

[KataKoniK]

for posting this question. To find T(x), we can use the energy balance equation for a steady-state, one-dimensional heat transfer problem:

$$\frac{d}{dx}\left(k\frac{dT}{dx}\right)=-q''$$

where k is the thermal conductivity of the fluid, T is the temperature, and q'' is the heat flux at the convective boundary.

Since the top, bottom, and left sides are adiabatic, we can simplify the equation to:

$$\frac{d}{dx}\left(k\frac{dT}{dx}\right)=0$$

Integrating this equation twice and applying the boundary conditions, we get:

$$T(x)=\frac{q''}{k}x+C_1x+C_2$$

where C1 and C2 are constants of integration.

To solve for these constants, we can use the boundary conditions at the convective boundary:

$$T(x=W)=T_{convective}$$

$$\frac{dT}{dx}\bigg|_{x=W}=\frac{q''}{k}$$

Substituting these into the equation for T(x), we get:

$$T(x)=\frac{q''}{k}x+\left(T_{convective}-\frac{q''}{k}W\right)x+\left(T_{convective}-\frac{q''}{k}W\right)$$

Simplifying, we get the final equation for T(x):

$$T(x)=\left(T_{convective}-\frac{q''}{k}W\right)(x+1)+\frac{q''}{k}W$$

I hope this helps you solve for T(x). Let me know if you have any further questions or if you would like to discuss your attempts further. Good luck!