Thank you for registering.

One of our academic counsellors will contact you within 1 working day.

Please check your email for login details.
MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping

Find the temperature difference between the two bodies at time t.

Find the temperature difference between the two bodies at time t.

Question Image
Grade:11

1 Answers

Arun
25763 Points
2 years ago
Let T₁, T₂ be the temperatures of object 1 and object 2 respectively and T₁₀ and the T₂₀ corresponding initial values. 

Let Object 2 transfer heat at flow rate H=dQ/dt, thus is rate of change of temperature is  
given by 
m₂∙s₂∙(dT₂/dt) = -H 
 
dT₂/dt = - H/(m₂∙s₂) 
Since connecting rod has not heat capacity is cannot store no energy. Therefore at same instant the same amunt of heat is absorbed by object 1. Thus, 
m₁∙s₁∙(dT₁/dt) = + H 
 
dT₁/dt = H/(m₁∙s₁) 

Because rod stores not heat we can assume quasi-stationary heat flow and apply Fourier's law in one-dimensional from. So heat flow between the objects is given by: 
H = (K∙A/L)∙(T₂ - T₁)  

Now introduce the temperature difference 
ΔT = T₂ - T₁ 

Hence; 
dΔT/dt = dT₂/dt - dT₁/dt = - H/(m₂∙s₂) - H/(m₁∙s₁) = - H∙(1/(m₂∙s₂) - 1/(m₁∙s₁)) 
 
dΔT/dt = - (K∙A/L)∙(T₂ - T₁)∙(1/(m₂∙s₂) - 1/(m₁∙s₁)) = - (K∙A/L)∙(1/(m₂∙s₂) - 1/(m₁∙s₁))∙ΔT 
 
dΔT/dt = - k∙ΔT 
where k = (K∙A/L)∙(1/(m₂∙s₂) - 1/(m₁∙s₁))∙ΔT 

To solve this differential equation separate variables and integrate 
(1/ΔT) dΔT = - k dt 
=> 
∫ (1/ΔT) dΔT = ∫ - k dt 
=> 
ln(ΔT) = -k∙t + c 
where c is the constant of integration 
 
ΔT = e^( -k∙t + c) = -k∙t + c = e^(c)∙e^(-k∙t) = C∙e^(-k∙t) 
where C = e^(c) 

at t=o the temperature difference is: 
ΔT₀ = T₂₀ - T₁₀  
that means 
ΔT₀ = C∙e^(-k∙0) = C 

So the temperature difference as function of time is given by 
ΔT = ΔT₀∙e^(-k∙t) 
= (T₂₀ - T₁₀)∙e^(- (K∙A/L)∙(1/(m₂∙s₂) - 1/(m₁∙s₁)) ∙ t)

Think You Can Provide A Better Answer ?

Provide a better Answer & Earn Cool Goodies See our forum point policy

ASK QUESTION

Get your questions answered by the expert for free