22 Dec '16 03:061 edit

While I was toping off the tub the other day for my daughters bath (The tub was partially filled at some cooler temp) I felt like figuring out how long it should take in general before the entire bath became the temperature of the replenishing stream. I got a little stuck on this one, but I belive I now have the proper function for the conditions. I'm just looking for some verification ( or denial) from anyone more confident than myself. Here goes.

The energy stored within a volume of water is given by:

E_st = m_cv * c_p * T ...Eq(1)

m_cv = mass of water

c_p = specific heat of water

T = temperature of water

Performing a power balance on the control volume ( assuming no heat loss by typical means)

d ( E_st )/dt = d(E_in)/dt ...Eq(2)

The water stored in the control volume is both changing mass and temperature, hence the stored energy is changing with both parameters. Differentiation of Equation (1) ( assuming constant specific heat) yields:

d(E_st)/dt = d( m_cv)/dt * c_p * T + m_cv * c_p * d(T)/dt ...Eq(2) LHS

Now for Eq(2) RHS

The power brought into the tub is assumed constant, and is given by:

d(m)/dt * c_p * T_in ...Eq(2) RHS

d(m_cv)/dt * c_p * T + m_cv * c_p * d(T)/dt = d(m)/dt * c_p * T_in ...Eq(2)

For simplification: d(m)/dt = constant = m'

Now applying a mass rate balance to the control volume:

d(m_cv)/dt = m' ...Eq(3)

(which states the rate of mass gained in the tub is the rate at which mass enters the tub by way of the faucet in this case)

Substitue (3) -> (2) and dividing out c_p

m' * T + m_cv * d(T)/dt = m' * T_in --->

m_cv * d(T)/dt = m' * T_in - m' * T --->

m_cv * d(T)/dt = m' * ( T_in - T ) ...Eq( 2' )

m_cv is a function of time "t". From Eq(3) with initial conditions t= 0, m_cv = m_o ( mass of water in tub @ t=0 )

m_cv = m' * t + m_o ...Eq(4)

Sub Eq(4) --> Eq( 2' )

( m' * t + m_o ) * d(T)/dt = m' * ( T_in - T ) ...Eq(5)

Separate:

d(T) / ( T_in - T ) = m' / ( m' * t + m_o )*dt ...Eq( 5' )

Change variables:

Φ = T_in - T

dΦ = -dT

U = m' * t + m_o

dU = m' * dt

Substitute:

- dΦ / Φ = dU / U ...Eq(6)

Integrate:

Φ_o / Φ = U / U_o ...Eq(7)

where,

Φ_o = T_in - T_o

U_o = m_o

Substitute into Eq(7) and solve for "T" (the instantaneous temperature of the tub)

T(t) = T_in - ( m_o / ( m' * t + m_o ) )*( T_in - T_o ) ...Desired Relationship

Sanity Check: T(0) = T_o & T( t--> ∞ ) = T_in

It seem to make sense, hoping I didn't flub something up too bad.

The energy stored within a volume of water is given by:

E_st = m_cv * c_p * T ...Eq(1)

m_cv = mass of water

c_p = specific heat of water

T = temperature of water

Performing a power balance on the control volume ( assuming no heat loss by typical means)

d ( E_st )/dt = d(E_in)/dt ...Eq(2)

The water stored in the control volume is both changing mass and temperature, hence the stored energy is changing with both parameters. Differentiation of Equation (1) ( assuming constant specific heat) yields:

d(E_st)/dt = d( m_cv)/dt * c_p * T + m_cv * c_p * d(T)/dt ...Eq(2) LHS

Now for Eq(2) RHS

The power brought into the tub is assumed constant, and is given by:

d(m)/dt * c_p * T_in ...Eq(2) RHS

d(m_cv)/dt * c_p * T + m_cv * c_p * d(T)/dt = d(m)/dt * c_p * T_in ...Eq(2)

For simplification: d(m)/dt = constant = m'

Now applying a mass rate balance to the control volume:

d(m_cv)/dt = m' ...Eq(3)

(which states the rate of mass gained in the tub is the rate at which mass enters the tub by way of the faucet in this case)

Substitue (3) -> (2) and dividing out c_p

m' * T + m_cv * d(T)/dt = m' * T_in --->

m_cv * d(T)/dt = m' * T_in - m' * T --->

m_cv * d(T)/dt = m' * ( T_in - T ) ...Eq( 2' )

m_cv is a function of time "t". From Eq(3) with initial conditions t= 0, m_cv = m_o ( mass of water in tub @ t=0 )

m_cv = m' * t + m_o ...Eq(4)

Sub Eq(4) --> Eq( 2' )

( m' * t + m_o ) * d(T)/dt = m' * ( T_in - T ) ...Eq(5)

Separate:

d(T) / ( T_in - T ) = m' / ( m' * t + m_o )*dt ...Eq( 5' )

Change variables:

Φ = T_in - T

dΦ = -dT

U = m' * t + m_o

dU = m' * dt

Substitute:

- dΦ / Φ = dU / U ...Eq(6)

Integrate:

Φ_o / Φ = U / U_o ...Eq(7)

where,

Φ_o = T_in - T_o

U_o = m_o

Substitute into Eq(7) and solve for "T" (the instantaneous temperature of the tub)

T(t) = T_in - ( m_o / ( m' * t + m_o ) )*( T_in - T_o ) ...Desired Relationship

Sanity Check: T(0) = T_o & T( t--> ∞ ) = T_in

It seem to make sense, hoping I didn't flub something up too bad.