Conduction with Heat Generation and Transient Heat Conduction

Heat Generation

The process of conversion of electrical, nuclear, or chemical energy into heat (or thermal) energy is termed as 'Heat Generation'. It is a volumetric phenomenon and usually specified per unit volume whose unit is $W/m^3$.

General Heat Conduction Eqation

\[\boxed{\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} + \frac{q_g}{k} = \frac{ρ C_p}{k}\frac{\partial T}{\partial t}}\]

1. Steady state (Poisson Equation)             \(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} + \frac{q_g}{k} = 0 \)
2. No heat generation (Diffusion Equation)              \(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} = \frac{ρ C_p}{k}\frac{\partial T}{\partial t}\) 
3. No heat generation and steady state (Laplace Equation)             \(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2}  = 0\)

Thermal Diffusivity $\alpha$

It is a property of material and defined as the ratio of thermal conductivity(k) and thermal capacity. It is the ability of material to allow the heat energy to get diffuse through the medium. 

\[\alpha = \frac{k}{ρ C_p} \quad  \quad \quad m^2/s\]

Uniform Heat Generation in Plane Wall

\[\frac{\partial^2 T}{\partial x^2} + \frac{q_g}{k} = 0\]

Integration will give,

\[T = -\frac{q_g}{2k}x^2 + Ax + B\]

Boundary conditions:

T (x = -L) = Tw1

T (x = -L) = Tw2

\[T(x) = \frac{q_gL^2 }{2k}(1 - \frac{x^2}{L^2})+ \frac{T_{w1} - T_{w2}}{2}L + \frac{T_{w1} + T_{w2}}{2}\]

If Tw1 = Tw2 = Tw,

\[T(x) - T_w = \frac{q_gL^2 }{2k}\Big(1 - \frac{x^2}{L^2}\Big)\]

Note: 

• In case of same Tw at both side of wall, maximum temperature occurs at center of wall.

• Temperature equation for Cylinder (maintained surface temperature Ts) with uniform heat generation:

\[T(r) - T_s = \frac{q_gR^2 }{4k}\Big(1 - \frac{r^2}{R^2}\Big)\]

Transient heat conduction or Unsteady state heat conduction

Lumped system analysis

It is assumed that internal temperature gradient within the body are neglected i.e. at any instant of time the entire body has uniform temperature throughout such analysis is called lumped system analysis. 

Let's consider a body of mass m, volume V, density ρ, surface area A and specific heat Cp which is at initial temperature Ti is suddenly placed into a fluid of temperature $T_∞$. The body losses heat by convection to the fluid with a convective heat transfer coefficient h. Due to this heat loss, internal temperature of body decreases with the time.  

Ti = Temperature of body at time t = 0

T = Temperature of body at time 't'

Energy balance at time = t,

The rate of convection heat loss from the surface of body to fluid = Rate of decrease in internal energy of body

\[hA(T - T_∞) = -mCp(\frac{dT}{dt}) = -ρVCp(\frac{dT}{dt})\]

\[\int_{0}^{t}\frac{-hA}{ρVCp}dt = \int_{Ti}^{T} \frac{dT}{T - T_∞}\]

\[\boxed{\frac{Ti - T_∞}{T - T_∞} = exp(\frac{hA}{ρVCp}t)}\]

• Temperature of body exponentially decreases with the time.
• $\frac{hA}{ρVCp}$ is known as time constant. 

Criteria for Lumped system Analysis

\[\boxed{Bi < 0.1}\]

Biot Number (Bi)

\[Biot \quad Number (Bi) = \frac{Conduction \quad resistance \quad within \quad the \quad body}{Convection \quad resistance \quad at \quad the \quad surface\quad  of\quad  the \quad body}\]

\[Bi = \frac{\frac{Lc}{kA}}{\frac{1}{hA}} = \frac{hLc}{k}\]

\[\boxed{Bi = \frac{hLc}{k}}\]

where,  Lc = Characteristic length = volume of body / surface area of body

Therefore, small bodies with high thermal conductivity are good candidates for lumped system analysis especially when they are in a medium that is a poor conductor of heat (such as air or another gas) and motionless.

Characteristic length of Different bodies:

Sphere	\[Lc = \frac{\frac{4}{3} \pi R^3}{4 \pi R^2} = \frac{R}{3}\]
Long Cylinder  (L >> R)	\[Lc = \frac{ \pi R^2 L}{2 \pi R(R+L)} = \frac{R}{2}\]
Cube	\[Lc = \frac{ L^3}{6 L^2} = \frac{L}{6}\]
Rectangular plate of small thickness  't'	\[Lc = \frac{ lbt}{2lb} = \frac{t}{2}\]